INTRODUCTION TO BIOMEDICAL ENGINEERING

INTRODUCTION TO BIOMEDICAL ENGINEERING THIRD EDITION

This is a volume in the ACADEMIC PRESS SERIES IN BIOMEDICAL ENGINEERING JOSEPH BRONZINO, SERIES EDITOR Trinity College—Hartford, Connecticut

INTRODUCTION TO BIOMEDICAL ENGINEERING THIRD EDITION JOHN D. ENDERLE University of Connecticut Storrs, Connecticut

JOSEPH D. BRONZINO Trinity College Hartford, Connecticut

Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK # 2012 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. MATLABW and SimulinkW are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLABW and SimulinkW software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLABW and SimulinkW software. Library of Congress Cataloging-in-Publication Data Introduction to biomedical engineering / [edited by] John Enderle, Joseph Bronzino. – 3rd ed. p. ; cm. Includes bibliographical references and index. ISBN 978-0-12-374979-6 (alk. paper) 1. Biomedical engineering. I. Enderle, John D. (John Denis) II. Bronzino, Joseph D., 1937[DNLM: 1. Biomedical Engineering. QT 36] R856.I47 2012 610.28–dc22 2010046267 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Printed in the United State of America 11 12 13 14 9 8 7 6 5 4 3

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This book is dedicated to our families.

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Contents 2.13 Ethical Issues in Treatment Use 70 2.14 The Role of the Biomedical Engineer in the FDA Process 71 2.15 Exercises 72

Preface xi Contributors to the Third Edition xiii Contributors to the Second Edition xiv Contributors to the First Edition xv

3. Anatomy and Physiology

1. Biomedical Engineering: A Historical Perspective

SUSAN BLANCHARD AND JOSEPH D. BRONZINO

3.1 3.2 3.3 3.4 3.5 3.6

JOSEPH D. BRONZINO

1.1 The Evolution of the Modern Health Care System 2 1.2 The Modern Health Care System 9 1.3 What Is Biomedical Engineering? 16 1.4 Roles Played by the Biomedical Engineers 1.5 Recent Advances in Biomedical Engineering 23 1.6 Professional Status of Biomedical Engineering 29 1.7 Professional Societies 30 1.8 Exercises 32

Introduction 76 Cellular Organization 78 Tissues 93 Major Organ Systems 94 Homeostasis 126 Exercises 129

4. Biomechanics JOSEPH L. PALLADINO AND ROY B. DAVIS III

4.1 4.2 4.3 4.4 4.5

Introduction 134 Basic Mechanics 137 Mechanics of Materials 158 Viscoelastic Properties 166 Cartilage, Ligament, Tendon, and Muscle 170 4.6 Clinical Gait Analysis 175 4.7 Cardiovascular Dynamics 192 4.8 Exercises 215

2. Moral and Ethical Issues JOSEPH D. BRONZINO

2.1 Morality and Ethics: A Definition of Terms 36 2.2 Two Moral Norms: Beneficence and Nonmaleficence 44 2.3 Redefining Death 45 2.4 The Terminally Ill Patient and Euthanasia 49 2.5 Taking Control 52 2.6 Human Experimentation 53 2.7 Definition and Purpose of Experimentation 55 2.8 Informed Consent 57 2.9 Regulation of Medical Device Innovation 62 2.10 Marketing Medical Devices 64 2.11 Ethical Issues in Feasibility Studies 65 2.12 Ethical Issues in Emergency Use 67

5. Biomaterials LIISA T. KUHN

5.1 Materials in Medicine: From Prosthetics to Regeneration 220 5.2 Biomaterials: Types, Properties, and Their Applications 221 5.3 Lessons from Nature on Biomaterial Design and Selection 236 5.4 Tissue–Biomaterial Interactions 240 5.5 Biomaterials Processing Techniques for Guiding Tissue Repair and Regeneration 250

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CONTENTS

5.6 Safety Testing and Regulation of Biomaterials 258 5.7 Application-Specific Strategies for the Design and Selection of Biomaterials 263 5.8 Exercises 269

8.6 Enzyme Inhibition, Allosteric Modifiers, and Cooperative Reactions 497 8.7 Exercises 505

9. Bioinstrumentation JOHN D. ENDERLE

6. Tissue Engineering RANDALL E. MCCLELLAND, ROBERT DENNIS, LOLA M. REID, JAN P. STEGEMANN, BERNARD PALSSON, AND JEFFREY M. MACDONALD

6.1 6.2 6.3 6.4 6.5

What Is Tissue Engineering? 274 Biological Considerations 290 Physical Considerations 319 Scaling Up 339 Implementation of Tissue Engineered Products 343 6.6 Future Directions: Functional Tissue Engineering and the “-Omics” Sciences 347 6.7 Conclusions 349 6.8 Exercises 349

7. Compartmental Modeling JOHN D. ENDERLE

7.1 Introduction 360 7.2 Solutes, Compartments, and Volumes 360 7.3 Transfer of Substances between Two Compartments Separated by a Membrane 362 7.4 Compartmental Modeling Basics 379 7.5 One-Compartment Modeling 381 7.6 Two-Compartment Modeling 391 7.7 Three-Compartment Modeling 403 7.8 Multicompartment Modeling 418 7.9 Exercises 430

8. Biochemical Reactions and Enzyme Kinetics

9.1 Introduction 510 9.2 Basic Bioinstrumentation System 512 9.3 Charge, Current, Voltage, Power, and Energy 514 9.4 Resistance 520 9.5 Linear Network Analysis 531 9.6 Linearity and Superposition 537 9.7 The´venin’s Theorem 541 9.8 Inductors 544 9.9 Capacitors 548 9.10 A General Approach to Solving Circuits Involving Resistors, Capacitors, and Inductors 551 9.11 Operational Amplifiers 560 9.12 Time-Varying Signals 572 9.13 Active Analog Filters 578 9.14 Bioinstrumentation Design 588 9.15 Exercises 591

10. Biomedical Sensors YITZHAK MENDELSON

10.1 10.2 10.3 10.4 10.5 10.6 10.7

11. Biosignal Processing

JOHN D. ENDERLE

8.1 Chemical Reactions 448 8.2 Enzyme Kinetics 458 8.3 Additional Models Using the Quasi-Steady-State Approximation 467 8.4 Diffusion, Biochemical Reactions, and Enzyme Kinetics 473 8.5 Cellular Respiration: Glucose Metabolism and the Creation of ATP 485

Introduction 610 Biopotential Measurements 616 Physical Measurements 621 Blood Gas Sensors 639 Bioanalytical Sensors 647 Optical Sensors 651 Exercises 662

MONTY ESCABI

11.1 11.2 11.3 11.4 11.5

Introduction 668 Physiological Origins of Biosignals 668 Characteristics of Biosignals 671 Signal Acquisition 674 Frequency Domain Representation of Biological Signals 679 11.6 Linear Systems 700 11.7 Signal Averaging 721

CONTENTS

11.8 The Wavelet Transform and the Short-Time Fourier Transform 727 11.9 Artificial Intelligence Techniques 732 11.10 Exercises 741

14.3 Biomedical Heat Transport 975 14.4 Exercises 992

15. Radiation Imaging

12. Bioelectric Phenomena JOHN D. ENDERLE

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

Introduction 748 History 748 Neurons 756 Basic Biophysics Tools and Relationships 761 Equivalent Circuit Model for the Cell Membrane 773 The Hodgkin-Huxley Model of the Action Potential 783 Model of a Whole Neuron 797 Chemical Synapses 800 Exercises 808

13. Physiological Modeling JOHN D. ENDERLE

13.1 Introduction 818 13.2 An Overview of the Fast Eye Movement System 821 13.3 The Westheimer Saccadic Eye Movement Model 828 13.4 The Saccade Controller 835 13.5 Development of an Oculomotor Muscle Model 838 13.6 The 1984 Linear Reciprocal Innervation Saccadic Eye Movement Model 852 13.7 The 1995 Linear Homeomorphic Saccadic Eye Movement Model 864 13.8 The 2009 Linear Homeomorphic Saccadic Eye Movement Model 878 13.9 Saccade Neural Pathways 905 13.10 System Identification 910 13.11 Exercises 927

14. Biomedical Transport Processes GERALD E. MILLER

14.1 Biomedical Mass Transport 938 14.2 Biofluid Mechanics and Momentum Transport 957

JOSEPH D. BRONZINO

15.1 15.2 15.3 15.4 15.5

Introduction 995 Emission Imaging Systems 997 Instrumentation and Imaging Devices 1013 Radiographic Imaging Systems 1018 Exercises 1037

16. Medical Imaging THOMAS SZABO

16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9

Introduction 1040 Diagnostic Ultrasound Imaging 1042 Magnetic Resonance Imaging 1071 Magnetoencephalography 1099 Contrast Agents 1101 Comparison of Imaging Modes 1103 Image Fusion 1106 Summary 1107 Exercises 1108

17. Biomedical Optics and Lasers GERARD L. COTE´, LIHONG V. WANG, AND SOHI RASTEGAR

17.1 Introduction to Essential Optical Principles 1112 17.2 Fundamentals of Light Propagation in Biological Tissue 1118 17.3 Physical Interaction of Light and Physical Sensing 1130 17.4 Biochemical Measurement Techniques Using Light 1139 17.5 Fundamentals of the Photothermal Therapeutic Effects of Light Sources 1147 17.6 Fiber Optics and Waveguides in Medicine 1158 17.7 Biomedical Optical Imaging 1165 17.8 Exercises 1170

Appendix 1175 Index 1213

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Preface The purpose of the third edition remains the same as the first and second editions, that is, to serve as an introduction to and overview of the field of biomedical engineering. Many chapters have undergone major revision from the previous editions with new end-of-chapter problems added. Some chapters were eliminated completely, with several new chapters added to reflect changes in the field. Over the past fifty years, as the discipline of biomedical engineering has evolved, it has become clear that it is a diverse, seemingly all-encompassing field that includes such areas as bioelectric phenomena, bioinformatics, biomaterials, biomechanics, bioinstrumentation, biosensors, biosignal processing, biotechnology, computational biology and complexity, genomics, medical imaging, optics and lasers, radiation imaging, tissue engineering, and moral and ethical issues. Although it is not possible to cover all of the biomedical engineering domains in this textbook, we have made an effort to focus on most of the major fields of activity in which biomedical engineers are engaged. The text is written primarily for engineering students who have completed differential equations and a basic course in statics. Students in their sophomore year or junior year should be adequately prepared for this textbook. Students in the biological sciences, including those in the fields of medicine and nursing can also read and understand this material if they have the appropriate mathematical background.

Although we do attempt to be fairly rigorous with our discussions and proofs, our ultimate aim is to help students grasp the nature of biomedical engineering. Therefore, we have compromised when necessary and have occasionally used less rigorous mathematics in order to be more understandable. A liberal use of illustrative examples amplifies concepts and develops problem-solving skills. Throughout the text, MATLAB® (a matrix equation solver) and SIMULINK® (an extension to MATLAB® for simulating dynamic systems) are used as computer tools to assist with problem solving. The Appendix provides the necessary background to use MATLAB® and SIMULINK®. MATLAB® and SIMULINK® are available from: The Mathworks, Inc. 24 Prime Park Way Natick, Massachusetts 01760 Phone: (508) 647-7000 Email: [emailprotected] WWW: http://www.mathworks.com Chapters are written to provide some historical perspective of the major developments in a specific biomedical engineering domain as well as the fundamental principles that underlie biomedical engineering design, analysis, and modeling procedures in that domain. In addition, examples of some of the problems encountered, as well as the techniques used to solve them, are provided. Selected problems, ranging from simple to difficult, are presented at the end of each chapter in the same general order as covered in the text.

xii

PREFACE

The material in this textbook has been designed for a one-semester, two-semester, or three-quarter sequence depending on the needs and interests of the instructor. Chapter 1 provides necessary background to understand the history and appreciate the field of biomedical engineering. Chapter 2 presents the vitally important chapter on biomedically based morals and ethics. Basic anatomy and physiology are provided in Chapter 3. Chapters 4–11 provide the basic core biomedical engineering areas: biomechanics, biomaterials, tissue engineering, compartmental modeling, biochemical reactions, bioinstrumentation, biosensors, and biosignal processing. To assist instructors in planning the sequence of material they may wish to emphasize, it is suggested that the chapters on bioinstrumentation, biosensors and biosignal processing should be covered together as they are interdependent on each other. The remainder of the textbook presents material on biomedical systems and biomedical technology (Chapters 12–17). Readers of the text can visit http://www .elsevierdirect.com/9780123749796 to view extra material that may be posted there from time to time. Instructors can register at http://www .textbooks.elsevier.com for access to solutions and additional resources to accompany the text.

ACKNOWLEDGMENTS Many people have helped us in writing this textbook. Well deserved credit is due to the many contributors who provided chapters and worked under a very tight timeline. Special thanks go to our publisher, Elsevier, especially for the tireless work of the Publisher, Joseph Hayton and Associate Editor, Steve Merken. In addition, we appreciate the work of Lisa Lamenzo, the Project Manager. A great debt of gratitude is extended to Joel Claypool, the editor of the first edition of the book and Diane Grossman from Academic Press, and Christine Minihane, the editor of the second edition. Also, we wish to acknowledge the efforts of Jonathan Simpson, the first editor of this edition, who moved onto to other assignments before this project was complete. A final and most important note concerns our co-author of the first two editions of this book, Susan Blanchard. She decided that she wanted to devote more time to her family and not to continue as a co-author.

Contributors to the Third Edition Susan M. Blanchard Florida Gulf Coast University, Fort Meyers, Florida

Katharine Merritt Food and Drug Administration, Gaithersburg, Maryland

Joseph D. Bronzino Trinity College, Hartford, Connecticut

Gerald E. Miller Virginia Commonwealth University, Richmond, Virginia

Stanley A. Brown Food and Drug Administration, Gaithersburg, Maryland

Joseph Palladino Trinity College, Hartford, Connecticut

Gerard L. Cote´ Texas A&M University, College Station, Texas

Bernard Palsson University of California at San Diego, San Diego, California

Robert Dennis University of North Carolina, Chapel Hill, North Carolina

Sohi Rastegar National Science Foundation, Arlington, Virginia

John Enderle University of Connecticut, Storrs, Connecticut

Lola M. Reid University of North Carolina, Chapel Hill, North Carolina

Monty Escabı´ University of Connecticut, Storrs, Connecticut

Kirk K. Shung University of Southern California, Los Angeles, California

Liisa T. Kuhn University of Connecticut Health Center, Farmington, Connecticut

Jan P. Stegemann University of Michigan, Ann Arbor, Michigan

Jeffrey M. Macdonald University of North Carolina-Chapel Hill, Chapel Hill, North Carolina

Thomas Szabo Boston University, Boston, Massachusetts

Randall McClelland University of North Carolina, Chapel Hill, North Carolina

LiHong V. Wang Washington University in St. Louis, St. Louis, Missouri

Yitzhak Mendelson Worcester Polytechnic Institute, Worcester, Massachusetts

xiii

Contributors to the Second Edition Susan M. Blanchard Florida Gulf Coast University, Fort Meyers, Florida Joseph D. Bronzino Connecticut

Trinity College, Hartford,

Stanley A. Brown Food and Drug Administration, Gaithersburg, Maryland Gerard L. Cote´ Texas A&M University, College Station, Texas Charles Coward Drexel University, Philadelphia, Pennsylvania Roy B. Davis III Shriners Hospital for Children, Greenville, South Carolina Robert Dennis University of North Carolina, Chapel Hill, North Carolina John Enderle University of Connecticut, Storrs, Connecticut Monty Escabı´ University of Connecticut, Storrs, Connecticut Robert J. Fisher University of Massachusetts, Amherst, Massachusetts Liisa T. Kuhn University of Connecticut Health Center, Farmington, Connecticut Carol Lucas University of North CarolinaChapel Hill, Chapel Hill, North Carolina Jeffrey M. Macdonald University of North Carolina-Chapel Hill, Chapel Hill, North Carolina

Yitzhak Mendelson, PhD Worcester Polytechnic Institute, Worcester, Massachusetts Katharine Merritt Food and Drug Administration, Gaithersburg, Maryland Spencer Muse North Carolina State University, Raleigh, North Carolina H. Troy Nagle North Carolina State University, Raleigh, North Carolina Banu Onaral Drexel University, Philadelphia, Pennsylvania Joseph Palladino Trinity College, Hartford, Connecticut Bernard Palsson University of California at San Diego, San Diego, California Sohi Rastegar National Science Foundation, Arlington, Virginia Lola M. Reid University of North Carolina, Chapel Hill, North Carolina Kirk K. Shung University of Southern California, Los Angeles, California Anne-Marie Stomp North Carolina State University, Raleigh, North Carolina Thomas Szabo Boston University, Boston, Massachusetts Andrew Szeto San Diego State University, San Diego, California

Amanda Marley North Carolina State University, Raleigh, North Carolina

LiHong V. Wang Washington University in St. Louis, St. Louis, Missouri

Randall McClelland University of North Carolina, Chapel Hill, North Carolina

Melanie T. Young North Carolina State University, Raleigh, North Carolina

xiv

Contributors to the First Edition Susan M. Blanchard Florida Gulf Coast University, Fort Meyers, Florida

H. Troy Nagle North Carolina State University, Raleigh, North Carolina

Joseph D. Bronzino Trinity College, Hartford, Connecticut

Joseph Palladino Trinity College, Hartford, Connecticut

Stanley A. Brown Food and Drug Administration, Gaithersburg, Maryland

Bernard Palsson University of California at San Diego, San Diego, California

Gerard L. Cote´ Texas A&M University, College Station, Texas

Sohi Rastegar National Science Foundation, Arlington, Virginia

Roy B. Davis III Shriners Hospital for Children, Greenville, South Carolina

Daniel Schneck Virginia Polytechnic Institute & State University, Blacksburg, Virginia

John Enderle University of Connecticut, Storrs, Connecticut

Kirk K. Shung University of Southern California, Los Angeles, California

Robert J. Fisher University of Massachusetts, Amherst, Massachusetts

Anne-Marie Stomp North Carolina State University, Raleigh, North Carolina

Carol Lucas University of North CarolinaChapel Hill, Chapel Hill, North Carolina

Andrew Szeto San Diego State University, San Diego, California

Amanda Marley North Carolina State University, Raleigh, North Carolina

LiHong V. Wang Washington University in St. Louis, St. Louis, Missouri

Yitzhak Mendelson, PhD Worcester Polytechnic Institute, Worcester, Massachusetts

Steven Wright Texas A&M University, College Station, Texas Melanie T. Young North Carolina State University, Raleigh, North Carolina

Katharine Merritt Food and Drug Administration, Gaithersburg, Maryland

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C H A P T E R

1 Biomedical Engineering: A Historical Perspective Joseph D. Bronzino, PhD, PE O U T L I N E 1.1

The Evolution of the Modern Health Care System

1.2

The Modern Health Care System

1.3

What Is Biomedical Engineering?

1.4

Roles Played by the Biomedical Engineers

Recent Advances in Biomedical Engineering

1.5

1.6

Professional Status of Biomedical Engineering

1.7

Professional Societies

1.8

Exercises

Suggested Readings

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : biomedical engineers play in the health care delivery system.

• Identify the major role that advances in medical technology have played in the establishment of the modern health care system.

• Explain why biomedical engineers are professionals.

• Define what is meant by the term biomedical engineering and the roles

Introduction to Biomedical Engineering, Third Edition

1. BIOMEDICAL ENGINEERING

In the industrialized nations, technological innovation has progressed at such an accelerated pace that it has permeated almost every facet of our lives. This is especially true in the area of medicine and the delivery of health care services. Although the art of medicine has a long history, the evolution of a technologically based health care system capable of providing a wide range of effective diagnostic and therapeutic treatments is a relatively new phenomenon. Of particular importance in this evolutionary process has been the establishment of the modern hospital as the center of a technologically sophisticated health care system. Since technology has had such a dramatic impact on medical care, engineering professionals have become intimately involved in many medical ventures. As a result, the discipline of biomedical engineering has emerged as an integrating medium for two dynamic professions—medicine and engineering—and has assisted in the struggle against illness and disease by providing tools (such as biosensors, biomaterials, image processing, and artificial intelligence) that health care professionals can use for research, diagnosis, and treatment. Thus, biomedical engineers serve as relatively new members of the health care delivery team that seeks new solutions for the difficult problems confronting modern society. The purpose of this chapter is to provide a broad overview of technology’s role in shaping our modern health care system, highlight the basic roles biomedical engineers play, and present a view of the professional status of this dynamic field.

1.1 THE EVOLUTION OF THE MODERN HEALTH CARE SYSTEM Primitive humans considered diseases to be “visitations”—the whimsical acts of affronted gods or spirits. As a result, medical practice was the domain of the witch doctor and the medicine man and medicine woman. Yet even as magic became an integral part of the healing process, the cult and the art of these early practitioners were never entirely limited to the supernatural. Using their natural instincts and learning from experience, these individuals developed a primitive science based upon empirical laws. For example, through acquisition and coding of certain reliable practices, the arts of herb doctoring, bone setting, surgery, and midwifery were advanced. Just as primitive humans learned from observation that certain plants and grains were good to eat and could be cultivated, the healers and shamans observed the nature of certain illnesses and then passed on their experiences to other generations. Evidence indicates that the primitive healer took an active, rather than simply intuitive, interest in the curative arts, acting as a surgeon and a user of tools. For instance, skulls with holes made in them by trephiners have been collected in various parts of Europe, Asia, and South America. These holes were cut out of the bone with flint instruments to gain access to the brain. Although one can only speculate the purpose of these early surgical operations, magic and religious beliefs seem to be the most likely reasons. Perhaps this procedure liberated from the skull the malicious demons that were thought to be the cause of extreme pain (as in the case of migraines) or attacks of falling to the ground (as in epilepsy). That this procedure was carried out on living patients, some of whom actually survived, is

1.1 THE EVOLUTION OF THE MODERN HEALTH CARE SYSTEM

evident from the rounded edges on the bone surrounding the hole, which indicate that the bone had grown again after the operation. These survivors also achieved a special status of sanctity so that, after their death, pieces of their skull were used as amulets to ward off convulsive attacks. From these beginnings, the practice of medicine has become integral to all human societies and cultures. It is interesting to note the fate of some of the most successful of these early practitioners. The Egyptians, for example, have held Imhotep, the architect of the first pyramid (3000 BC), in great esteem through the centuries, not as a pyramid builder but as a doctor. Imhotep’s name signified “he who cometh in peace” because he visited the sick to give them “peaceful sleep.” This early physician practiced his art so well that he was deified in the Egyptian culture as the god of healing. Egyptian mythology, like primitive religion, emphasized the interrelationships between the supernatural and one’s health. For example, consider the mystic sign Rx, which still adorns all prescriptions today. It has a mythical origin: the legend of the Eye of Horus. It appears that as a child Horus lost his vision after being viciously attacked by Seth, the demon of evil. Then Isis, the mother of Horus, called for assistance to Thoth, the most important god of health, who promptly restored the eye and its powers. Because of this intervention, the Eye of Horus became the Egyptian symbol of godly protection and recovery, and its descendant, Rx, serves as the most visible link between ancient and modern medicine. The concepts and practices of Imhotep and the medical cult he fostered were duly recorded on papyri and stored in ancient tombs. One scroll (dated c. 1500 BC), which George Elbers acquired in 1873, contains hundreds of remedies for numerous afflictions ranging from crocodile bites to constipation. A second famous papyrus (dated c. 1700 BC), discovered by Edwin Smith in 1862, is considered to be the most important and complete treatise on surgery of all antiquity. These writings outline proper diagnoses, prognoses, and treatment in a series of surgical cases. These two papyri are certainly among the outstanding writings in medical history. As the influence of ancient Egypt spread, Imhotep was identified by the Greeks with their own god of healing: Aesculapius. According to legend, the god Apollo fathered Aesculapius during one of his many earthly visits. Apparently Apollo was a concerned parent, and, as is the case for many modern parents, he wanted his son to be a physician. He made Chiron, the centaur, tutor Aesculapius in the ways of healing (Figure 1.1). Chiron’s student became so proficient as a healer that he soon surpassed his tutor and kept people so healthy that he began to decrease the population of Hades. Pluto, the god of the underworld, complained so violently about this course of events that Zeus killed Aesculapius with a thunderbolt and in the process promoted Aesculapius to Olympus as a god. Inevitably, mythology has become entangled with historical facts, and it is not certain whether Aesculapius was in fact an earthly physician like Imhotep, the Egyptian. However, one thing is clear: by 1000 BC, medicine was already a highly respected profession. In Greece, the Aesculapia were temples of the healing cult and may be considered the first hospitals (Figure 1.1). In modern terms, these temples were essentially sanatoriums that had strong religious overtones. In them, patients were received and psychologically prepared, through prayer and sacrifice, to appreciate the past achievements of Aesculapius and his physician priests. After the appropriate rituals, they were allowed to enjoy “temple sleep.” During

1. BIOMEDICAL ENGINEERING

FIGURE 1.1 A sick child brought to the Temple of Aesculapius. Courtesy of http://www.nouveaunet.com/images/ art/84.jpg.

the night, “healers” visited their patients, administering medical advice to clients who were awake or interpreting dreams of those who had slept. In this way, patients became convinced that they would be cured by following the prescribed regimen of diet, drugs, or bloodletting. On the other hand, if they remained ill, it would be attributed to their lack of faith. With this approach, patients, not treatments, were at fault if they did not get well. This early use of the power of suggestion was effective then and is still important in medical treatment today. The notion of “healthy mind, healthy body” is still in vogue today. One of the most celebrated of these “healing” temples was on the island of Cos, the birthplace of Hippocrates, who as a youth became acquainted with the curative arts through his father, also a physician. Hippocrates was not so much an innovative physician as a collector of all the remedies and techniques that existed up to that time. Since he viewed the physician as a scientist instead of a priest, Hippocrates also injected an essential ingredient into medicine: its scientific spirit. For him, diagnostic observation and clinical treatment began to replace superstition. Instead of blaming disease on the gods, Hippocrates taught that disease was a natural process, one that developed in logical steps, and that symptoms were reactions of the body to disease. The body itself, he emphasized, possessed its own means of recovery, and the function of the physician was to aid these natural forces. Hippocrates treated each patient as an original case to be studied and documented. His shrewd

1.1 THE EVOLUTION OF THE MODERN HEALTH CARE SYSTEM

descriptions of diseases are models for physicians even today. Hippocrates and the school of Cos trained many individuals, who then migrated to the corners of the Mediterranean world to practice medicine and spread the philosophies of their preceptor. The work of Hippocrates and the school and tradition that stem from him constitute the first real break from magic and mysticism and the foundation of the rational art of medicine. However, as a practitioner, Hippocrates represented the spirit, not the science, of medicine, embodying the good physician: the friend of the patient and the humane expert. As the Roman Empire reached its zenith and its influence expanded across half the world, it became heir to the great cultures it absorbed, including their medical advances. Although the Romans themselves did little to advance clinical medicine (the treatment of the individual patient), they did make outstanding contributions to public health. For example, they had a well-organized army medical service, which not only accompanied the legions on their various campaigns to provide “first aid” on the battlefield but also established “base hospitals” for convalescents at strategic points throughout the empire. The construction of sewer systems and aqueducts were truly remarkable Roman accomplishments that provided their empire with the medical and social advantages of sanitary living. Insistence on clean drinking water and unadulterated foods affected the control and prevention of epidemics and, however primitive, made urban existence possible. Unfortunately, without adequate scientific knowledge about diseases, all the preoccupation of the Romans with public health could not avert the periodic medical disasters, particularly the plague, that mercilessly befell its citizens. Initially, the Roman masters looked upon Greek physicians and their art with disfavor. However, as the years passed, the favorable impression these disciples of Hippocrates made upon the people became widespread. As a reward for their service to the peoples of the Empire, Julius Caesar (46 BC) granted Roman citizenship to all Greek practitioners of medicine in his empire. Their new status became so secure that when Rome suffered from famine that same year, these Greek practitioners were the only foreigners not expelled from the city. On the contrary, they were even offered bonuses to stay! Ironically, Galen, who is considered the greatest physician in the history of Rome, was himself a Greek. Honored by the emperor for curing his “imperial fever,” Galen became the medical celebrity of Rome. He was arrogant and a braggart and, unlike Hippocrates, reported only successful cases. Nevertheless, he was a remarkable physician. For Galen, diagnosis became a fine art; in addition to taking care of his own patients, he responded to requests for medical advice from the far reaches of the empire. He was so industrious that he wrote more than 300 books of anatomical observations, which included selected case histories, the drugs he prescribed, and his boasts. His version of human anatomy, however, was misleading because he objected to human dissection and drew his human analogies solely from the studies of animals. However, because he so dominated the medical scene and was later endorsed by the Roman Catholic Church, Galen actually inhibited medical inquiry. His medical views and writings became both the “bible” and “the law” for the pontiffs and pundits of the ensuing Dark Ages. With the collapse of the Roman Empire, the Church became the repository of knowledge, particularly of all scholarship that had drifted through the centuries into the Mediterranean. This body of information, including medical knowledge, was literally scattered through the monasteries and dispersed among the many orders of the Church.

1. BIOMEDICAL ENGINEERING

The teachings of the early Roman Catholic Church and the belief in divine mercy made inquiry into the causes of death unnecessary and even undesirable. Members of the Church regarded curing patients by rational methods as sinful interference with the will of God. The employment of drugs signified a lack of faith by the doctor and patient, and scientific medicine fell into disrepute. Therefore, for almost a thousand years, medical research stagnated. It was not until the Renaissance in the 1500s that any significant progress in the science of medicine occurred. Hippocrates had once taught that illness was not a punishment sent by the gods but a phenomenon of nature. Now, under the Church and a new God, the older views of the supernatural origins of disease were renewed and promulgated. Since disease implied demonic possession, monks and priests would treat the sick through prayer, the laying on of hands, exorcism, penances, and exhibition of holy relics—practices officially sanctioned by the Church. Although deficient in medical knowledge, the Dark Ages were not entirely lacking in charity toward the sick poor. Christian physicians often treated the rich and poor alike, and the Church assumed responsibility for the sick. Furthermore, the evolution of the modern hospital actually began with the advent of Christianity and is considered one of the major contributions of monastic medicine. With the rise in 335 AD of Constantine I, the first of the Roman emperors to embrace Christianity, all pagan temples of healing were closed, and hospitals were established in every cathedral city. (The word hospital comes from the Latin hospes, meaning “host” or “guest.” The same root has provided hotel and hostel.) These first hospitals were simply houses where weary travelers and the sick could find food, lodging, and nursing care. The Church ran these hospitals, and the attending monks and nuns practiced the art of healing. As the Christian ethic of faith, humanitarianism, and charity spread throughout Europe and then to the Middle East during the Crusades, so did its “hospital system.” However, trained “physicians” still practiced their trade primarily in the homes of their patients, and only the weary travelers, the destitute, and those considered hopeless cases found their way to hospitals. Conditions in these early hospitals varied widely. Although a few were well financed and well managed and treated their patients humanely, most were essentially custodial institutions to keep troublesome and infectious people away from the general public. In these establishments, crowding, filth, and high mortality among both patients and attendants were commonplace. Thus, the hospital was viewed as an institution to be feared and shunned. The Renaissance and Reformation in the fifteenth and sixteenth centuries loosened the Church’s stronghold on both the hospital and the conduct of medical practice. During the Renaissance, “true learning,” the desire to pursue the true secrets of nature including medical knowledge, was again stimulated. The study of human anatomy was advanced, and the seeds for further studies were planted by the artists Michelangelo, Raphael Durer, and, of course, the genius Leonardo da Vinci. They viewed the human body as it really was, not simply as a text passage from Galen. The painters of the Renaissance depicted people in sickness and pain, sketched in great detail and, in the process, demonstrated amazing insight into the workings of the heart, lungs, brain, and muscle structure. They also attempted to portray the individual and to discover emotional as well as physical qualities. In this stimulating era, physicians began to approach their patients and the pursuit of medical knowledge in similar fashion. New medical schools, similar to the most famous of such institutions at

1.1 THE EVOLUTION OF THE MODERN HEALTH CARE SYSTEM

Salerno, Bologna, Montpelier, Padua, and Oxford, emerged. These medical training centers once again embraced the Hippocratic doctrine that the patient was human, disease was a natural process, and commonsense therapies were appropriate in assisting the body to conquer its disease. During the Renaissance, fundamentals received closer examination, and the age of measurement began. In 1592, when Galileo visited Padua, Italy, he lectured on mathematics to a large audience of medical students. His famous theories and inventions (the thermoscope and the pendulum, in addition to the telescopic lens) were expounded upon and demonstrated. Using these devices, one of his students, Sanctorius, made comparative studies of the human temperature and pulse. A future graduate of Padua, William Harvey, later applied Galileo’s laws of motion and mechanics to the problem of blood circulation. This ability to measure the amount of blood moving through the arteries helped to determine the function of the heart. Galileo encouraged the use of experimentation and exact measurement as scientific tools that could provide physicians with an effective check against reckless speculation. Quantification meant theories would be verified before being accepted. Individuals involved in medical research incorporated these new methods into their activities. Body temperature and pulse rate became measures that could be related to other symptoms to assist the physician in diagnosing specific illnesses or diseases. Concurrently, the development of the microscope amplified human vision, and an unknown world came into focus. Unfortunately, new scientific devices had little impact upon the average physician, who continued to blood-let and to disperse noxious ointments. Only in the universities did scientific groups band together to pool their instruments and their various talents. In England, the medical profession found in Henry VIII a forceful and sympathetic patron. He assisted the doctors in their fight against malpractice and supported the establishment of the College of Physicians, the oldest purely medical institution in Europe. When he suppressed the monastery system in the early sixteenth century, church hospitals were taken over by the cities in which they were located. Consequently, a network of private, nonprofit, voluntary hospitals came into being. Doctors and medical students replaced the nursing sisters and monk physicians. Consequently, the professional nursing class became almost nonexistent in these public institutions. Only among the religious orders did “nursing” remain intact, further compounding the poor lot of patients confined within the walls of the public hospitals. These conditions were to continue until Florence Nightingale appeared on the scene years later. Still another dramatic event was to occur. The demands made upon England’s hospitals, especially the urban hospitals, became overwhelming as the population of these urban centers continued to expand. It was impossible for the facilities to accommodate the needs of so many. Therefore, during the seventeenth century two of the major urban hospitals in London—St. Bartholomew’s and St. Thomas—initiated a policy of admitting and attending to only those patients who could possibly be cured. The incurables were left to meet their destiny in other institutions such as asylums, prisons, or almshouses. Humanitarian and democratic movements occupied center stage primarily in France and the American colonies during the eighteenth century. The notion of equal rights finally began, and as urbanization spread, American society concerned itself with the welfare of many of its members. Medical men broadened the scope of their services to include the

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“unfortunates” of society and helped to ease their suffering by advocating the power of reason and spearheading prison reform, child care, and the hospital movement. Ironically, as the hospital began to take up an active, curative role in medical care in the eighteenth century, the death rate among its patients did not decline but continued to be excessive. In 1788, for example, the death rate among the patients at the Hotel Dru in Paris, thought to be founded in the seventh century and the oldest hospital in existence today, was nearly 25 percent. These hospitals were lethal not only to patients but also to the attendants working in them, whose own death rate hovered between 6 and 12 percent per year. Essentially the hospital remained a place to avoid. Under these circumstances, it is not surprising that the first American colonists postponed or delayed building hospitals. For example, the first hospital in America, the Pennsylvania Hospital, was not built until 1751, and the city of Boston took over two hundred years to erect its first hospital, the Massachusetts General, which opened its doors to the public in 1821. A major advancement in the history of modern medicine came in the mid-nineteenth century with the development of the now well-known Germ Theory. Germ Theory simply states that infectious disease is caused by microorganisms living within the body. A popular example of early Germ Theory demonstration is that of John Snow and the Broad Street pump handle. When Cholera reached epidemic levels in the overcrowded Industrial Era streets of London, local physician John Snow was able to stop the spread of the disease with a street map. Snow plotted the cases of Cholera in the city, and he discovered an epicenter at a local water pump. By removing the handle, and thus access to the infected water supply, Snow illustrated Germ Theory and saved thousands of lives at the same time. French chemist Louis Pasteur is credited with developing the foundations of Germ Theory throughout the mid-nineteenth century. Not until the nineteenth century could hospitals claim to benefit any significant number of patients. This era of progress was due primarily to the improved nursing practices fostered by Florence Nightingale (Figure 1.2) on her return to England from the Crimean War. She demonstrated that hospital deaths were caused more frequently by hospital conditions than by disease. During the latter part of the nineteenth century, she was at the height of her influence, and few new hospitals were built anywhere in the world without her advice. During the first half of the nineteenth century, Nightingale forced medical attention to focus once more on the care of the patient. Enthusiastically and philosophically, she expressed her views on nursing: “Nursing is putting us in the best possible condition for nature to restore and preserve health. . . . The art is that of nursing the sick. Please mark, not nursing sickness.” Although these efforts were significant, hospitals remained, until the twentieth century, institutions for the sick poor. In the 1870s, for example, when the plans for the projected Johns Hopkins Hospital were reviewed, it was considered quite appropriate to allocate 324 charity and 24 pay beds. Not only did the hospital population before the turn of the century represent a narrow portion of the socioeconomic spectrum, but it also represented only a limited number of the types of diseases prevalent in the overall population. In 1873, for example, roughly half of America’s hospitals did not admit contagious diseases, and many others would not admit incurables. Furthermore, in this period, surgery admissions in general hospitals constituted only 5 percent, with trauma (injuries incurred by traumatic experience) making up a good portion of these cases.

1.2 THE MODERN HEALTH CARE SYSTEM

FIGURE 1.2 A portrait of Florence Nightingale. Courtesy of http://ginnger.topcities.com/cards/computer/nurses/ 765x525nightengale.gif.

American hospitals a century ago were rather simple in that their organization required no special provisions for research or technology and demanded only cooking and washing facilities. In addition, since the attending and consulting physicians were normally unsalaried, and the nursing costs were quite modest, the great bulk of the hospital’s normal operating expenses were for food, drugs, and utilities. Not until the twentieth century did “modern medicine” come of age in the United States. As we shall see, technology played a significant role in its evolution.

1.2 THE MODERN HEALTH CARE SYSTEM Modern medical practice actually began at the turn of the twentieth century. Before 1900, medicine had little to offer the average citizen, since its resources were mainly physicians, their education, and their little black bags. At this time physicians were in short supply, but for different reasons than exist today. Costs were minimal, demand was small, and many of the services provided by the physician could also be obtained from experienced amateurs residing in the community. The individual’s dwelling was the major site for treatment and recuperation, and relatives and neighbors constituted an able and willing nursing staff.

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Midwives delivered babies, and those illnesses not cured by home remedies were left to run their fatal course. Only in the twentieth century did the tremendous explosion in scientific knowledge and technology lead to the development of the American health care system, with the hospital as its focal point and the specialist physician and nurse as its most visible operatives. In the twentieth century, the advances made in the basic sciences (chemistry, physiology, pharmacology, and so on) began to occur much more rapidly. Discoveries in the physical sciences enabled medical researchers to take giant strides forward. For example, in 1903, William Einthoven devised the first electrocardiograph and measured the electrical changes that occurred during the beating of the heart (Figure 1.3). In the process, Einthoven initiated a new age for both cardiovascular medicine and electrical measurement techniques. Of all the new discoveries that followed one another like intermediates in a chain reaction, the most significant for clinical medicine was the development of x-rays. When W. K. Roentgen described his “new kinds of rays,” the human body was opened to medical inspection. Initially these x-rays were used in the diagnosis of bone fractures and dislocations. In the United States, x-ray machines brought this “modern technology” to most urban hospitals. In the process, separate departments of radiology were established, and the influence of their activities spread with almost every department of medicine (surgery, gynecology, and so forth) advancing with the aid of this new tool. By the 1930s, x-ray visualization

FIGURE 1.3 (a) An early electrocardiograph machine and

Continued

1.2 THE MODERN HEALTH CARE SYSTEM

FIGURE 1.3, cont’d

(b) a modern ECG setup. Computer technology and electronics advances have greatly simplified and strengthened the ECG as a diagnosis tool.

of practically all the organ systems of the body was possible by the use of barium salts and a wide variety of radiopaque materials. The power this technological innovation gave physicians was enormous. The x-ray permitted them to diagnose a wide variety of diseases and injuries accurately. In addition, being within the hospital, it helped trigger the transformation of the hospital from a passive receptacle for the sick poor to an active curative institution for all the citizens of American society. The introduction of sulfanilamide in the mid-1930s and penicillin in the early 1940s significantly reduced the main danger of hospitalization: cross-infection among patients. With these new drugs in their arsenals, surgeons were able to perform their operations without prohibitive morbidity and mortality due to infection. Also, despite major earlytwentieth-century advancements in the field of hematology (including blood type differentiation and the use of sodium citrate to prevent clotting), blood banks were not fully developed until the 1930s, when technology provided adequate refrigeration. Until that time, “fresh” donors were bled, and the blood was transfused while it was still warm. As technology in the United States blossomed, so did the prestige of American medicine. From 1900 to 1929, Nobel Prize winners in physiology or medicine came primarily from Europe, with no American among them. In the period 1930 to 1944, just before the end of World War II, 19 Americans were honored as Nobel Prize Laureates. During the postwar period (1945–1975), 102 American life scientists earned similar honors, and from 1975 to 2009, the number was 191. Thus, since 1930 a total of 312 American scientists, including some born abroad, have performed research that was significant enough to warrant the

1. BIOMEDICAL ENGINEERING

distinction of a Nobel Prize. Most of these efforts were made possible by the technology that was available to these clinical scientists. The employment of the available technology assisted in advancing the development of complex surgical procedures. The Drinker respirator was introduced in 1927, and the first heart-lung bypass was performed in 1939. In the 1940s, cardiac catheterization and angiography (the use of a cannula threaded through an arm vein and into the heart with the injection of radiopaque dye for the x-ray visualization of lung and heart vessels and valves) were developed. Accurate diagnoses of congenital and acquired heart disease (mainly valve disorders due to rheumatic fever) also became possible, and a new era of cardiac and vascular surgery began. The development and implementation of robotic surgery in the first decade of the twenty-first century have even further advanced the capabilities of modern surgeons. Neurosurgery, both peripheral and central, and vascular surgery have seen significant improvements and capabilities with this new technology (Figure 1.4). Another child of this modern technology, the electron microscope, entered the medical scene in the 1950s and provided significant advances in visualizing relatively small cells. Body scanners using early PET (positron-emission tomography) technology to detect tumors arose from the same science that brought societies reluctantly into the atomic age. These “tumor detectives” used radioactive material and became commonplace in newly established departments of nuclear medicine in all hospitals.

FIGURE 1.4 Changes in the operating room: (a) the surgical scene at the turn of the century, (b) the surgical scene in the late 1920s and early 1930s, and (c) the surgical scene today From J. D. Bronzino, Technology for Patient Care, St. Louis: Mosby, 1977; The Biomedical Engineering Handbook, CRC Press, 1995; 2000; 2005.

1.2 THE MODERN HEALTH CARE SYSTEM

The impact of these discoveries and many others was profound. The health care system that consisted primarily of the “horse and buggy” physician was gone forever, replaced by the doctor backed by and centered around the hospital, as medicine began to change to accommodate the new technology. Following World War II, the evolution of comprehensive care greatly accelerated. The advanced technology that had been developed in the pursuit of military objectives now became available for peaceful applications, with the medical profession benefiting greatly from this rapid surge of technological “finds.” For instance, the realm of electronics came into prominence. The techniques for following enemy ships and planes, as well as providing aviators with information concerning altitude, air speed, and the like, were now used extensively in medicine to follow the subtle electrical behavior of the fundamental unit of the central nervous system—the neuron—or to monitor the beating heart of a patient. The Second World War also brought a spark of innovation in the rehabilitation engineering and prosthetics fields. With advances in medical care technologies, more veterans were returning home alive—and disabled. This increase in need, combined with a surge in new materials development in the late 1940s, assisted the growth of assistive technologies during the post-WWII era. Science and technology have leapfrogged past each other throughout recorded history. Anyone seeking a causal relation between the two was just as likely to find technology the cause and science the effect, with the converse also holding true. As gunnery led to ballistics and the steam engine transformed into thermodynamics, so did powered flight lead to aerodynamics. However, with the advent of electronics this causal relation has been reversed; scientific research is systematically exploited in the pursuit of technical advancement. Just as World War II sparked an advancement in comprehensive care, the 1960s enjoyed a dramatic electronics revolution, compliments of the first lunar landing. What was considered science fiction in the 1930s and 1940s became reality. Devices continually changed to incorporate the latest innovations, which in many cases became outmoded in a very short period of time. Telemetry devices used to monitor the activity of a patient’s heart freed both the physician and the patient from the wires that previously restricted them to the four walls of the hospital room. Computers, similar to those that controlled the flight plans of the Apollo capsules, now completely inundate our society. Since the 1970s, medical researchers have put these electronic brains to work performing complex calculations, keeping records (via artificial intelligence), and even controlling the very instrumentation that sustains life. The development of new medical imaging techniques such as computerized tomography (CT) and magnetic resonance imaging (MRI) totally depended on a continually advancing computer technology. New imaging developments include functional MRI (Figure 1.5), a tool capable of illustrating active neural areas by quantifying oxygen consumption and blood flow in the brain. The citations and technological discoveries are so myriad that it is impossible to mention them all. “Spare parts” surgery is now routine. With the first successful transplantation of a kidney in 1954, the concept of “artificial organs” gained acceptance and officially came into vogue in the medical arena (Figure 1.6). Technology to provide prosthetic devices, such as artificial heart valves and artificial blood vessels, developed. Even an artificial heart program to develop a replacement for a defective or diseased human heart began.

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FIGURE 1.5 (a) A modern fMRI medical imaging facility and (b) an fMRI scan image. http://neurophilosophy .wordpress.com.

Eye Tissue Lungs Skin Heart

Heart Valves

Liver Kidneys Pancreas

Bowel

Bone Tendons

Veins

Ligaments

FIGURE 1.6

Transplantations performed today. http://www.transplant.bc.ca/images/what_organs.gif.

1.2 THE MODERN HEALTH CARE SYSTEM

With the neural function, resilience, and incredible mechanical strength and endurance of the human heart, complete replacement prosthetics have been only marginally successful. Left ventricular assist devices (LVAD), however, have seen success as a replacement for the “workhorse” region of the heart and are a popular temporary option for those waiting on a full heart transplant. Future directions for heart failure solutions will most likely involve more tissue and cellular level treatments, as opposed to macromechanical systems. These technological innovations have vastly altered surgical organization and utilization, even further enhancing the radical evolution hospitals have undergone from the low-tech institutions of just 100 years ago to the modern advanced medical centers of tomorrow. In recent years, technology has struck medicine like a thunderbolt. The Human Genome Project was perhaps the most prominent scientific and technological effort of the 1990s. Some of the engineering products vital to the effort included automatic sequencers, robotic liquid handling devices, and software for databasing and sequence assembly (See Figure 1.7). As a result, a major transition occurred, moving biomedical engineering to focus on the cellular and molecular level rather than solely on the organ system level. With the success of the “genome project,” completed in 2003 after a 13-year venture, new vistas have been opened. Stem cell research highlights this chemical and molecular level focus and has been on the

Test Drugs on Human Cells in Culture

Understand how to prevent and treat birth defects

Study Cell Differentiation

Test drugs before conducting clinical trials

Toxicity Testing

? Ectoderm

Mesoderm

Endoderm

Blood Cells

Neuron

Liver Cells

Generate Tissues and/or Cells for Transplantation

FIGURE 1.7

promise.htm.

Stem cell research—potential applications made possible. http://stemcells.nih.gov/info/media/

1. BIOMEDICAL ENGINEERING

FIGURE 1.8 Robotic surgery—a new tool in the arsenal of the physician. http://library.thinkquest.org/03oct/00760/

steve.jpg.

forefront of controversial scientific research since its conception. While the multitudes of possibilities defy imagination, the moral issues accompanying stem cells have received equal attention in recent years. Furthermore, advances in nanotechnology, tissue engineering, and artificial organs are clear indications that science fiction will continue to become reality. However, the social and economic consequences of this vast outpouring of information and innovation must be fully understood if this technology is to be exploited effectively and efficiently. As one gazes into the crystal ball, technology offers great potential for affecting health care practices (Figure 1.8). It can provide health care for individuals in remote rural areas by means of closed-circuit television health clinics with complete communication links to a regional health center. Development of multiphasic screening systems can provide preventative medicine to the vast majority of our population and restrict hospital admissions to those requiring the diagnostic and treatment facilities housed there. With the creation of a central medical records system, anyone moving or becoming ill away from home can have records made available to the attending physician easily and rapidly. These are just a few of the possibilities that illustrate the potential of technology in creating the type of medical care system that will indeed be accessible, high quality, and reasonably priced for all. (For an extensive review of major events in the evolution of biomedical engineering, see Nebeker, 2002.)

1.3 WHAT IS BIOMEDICAL ENGINEERING? Many of the problems confronting health professionals today are of extreme importance to the engineer because they involve the fundamental aspects of device and systems analysis, design, and practical application—all of which lie at the heart of processes that are fundamental to engineering practice. These medically relevant design problems can range

1.3 WHAT IS BIOMEDICAL ENGINEERING?

from very complex large-scale constructs, such as hospital information systems, to the creation of relatively small and “simple” devices, such as recording electrodes and transducers used to monitor the activity of specific physiological processes. The American health care system, therefore, encompasses many problems that represent challenges to certain members of the engineering profession, called biomedical engineers. Since biomedical engineering involves applying the concepts, knowledge, and approaches of virtually all engineering disciplines (e.g., electrical, mechanical, and chemical engineering) to solve specific health care–related problems, the opportunities for interaction between engineers and health care professionals are many and varied. Although what is included in the field of biomedical engineering is considered by many to be quite clear, many conflicting opinions concerning the field can be traced to disagreements about its definition. For example, consider the terms biomedical engineering, bioengineering, biological engineering, and clinical (or medical) engineer, which are defined in the Bioengineering Education Directory. While Pacela defined bioengineering as the broad umbrella term used to describe this entire field, bioengineering is usually defined as a basic-researchoriented activity closely related to biotechnology and genetic engineering—that is, the modification of animal or plant cells or parts of cells to improve plants or animals or to develop new microorganisms for beneficial ends. In the food industry, for example, this has meant the improvement of strains of yeast for fermentation. In agriculture, bioengineers may be concerned with the improvement of crop yields by treatment plants with organisms to reduce frost damage. It is clear that bioengineers for the future will have tremendous impact on the quality of human life. The full potential of this specialty is difficult to image. Typical pursuits include the following: • • • • • • • •

The development of improved species of plants and animals for food production The invention of new medical diagnostic tests for diseases The production of synthetic vaccines from clone cells Bioenvironmental engineering to protect human, animal, and plant life from toxicants and pollutants The study of protein-surface interactions Modeling of the growth kinetics of yeast and hybridoma cells Research in immobilized enzyme technology The development of therapeutic proteins and monoclonal antibodies

The term biomedical engineering appears to have the most comprehensive meaning. Biomedical engineers apply electrical, chemical, optical, mechanical, and other engineering principles to understand, modify, or control biological (i.e., human and animal) systems. When a biomedical engineer works within a hospital or clinic, he or she is more properly called a clinical engineer. However, this theoretical distinction is not always observed in practice, since many professionals working within U.S. hospitals today continue to be called biomedical engineers. The breadth of activity of biomedical engineers is significant. The field has moved significantly from being concerned primarily with the development of medical devices in the 1950s and 1960s to include a more wide-ranging set of activities. As shown in Figure 1.9, the field of biomedical engineering now includes many new career areas:

1. BIOMEDICAL ENGINEERING

Biomechanics Medical & Biological Analysis

Prosthetic Devices & Artificial Organs

Biosensors

Medical Imaging

Clinical Engineering

Biomaterials Biotechnology

Medical & Bioinformatics

Tissue Engineering Neural Engineering

Rehabilitation Engineering Physiological Modeling

Biomedical Instrumentation

Bionanotechnology

FIGURE 1.9 The world of biomedical engineering.

• Application of engineering system analysis (physiologic modeling, simulation, and control to biological problems) • Detection, measurement, and monitoring of physiologic signals (i.e., biosensors and biomedical instrumentation) • Diagnostic interpretation via signal-processing techniques of bioelectric data • Therapeutic and rehabilitation procedures and devices (rehabilitation engineering) • Devices for replacement or augmentation of bodily functions (artificial organs) • Computer analysis of patient-related data and clinical decision making (i.e., medical informatics and artificial intelligence) • Medical imaging—that is, the graphical display of anatomic detail or physiologic function • The creation of new biologic products (i.e., biotechnology and tissue engineering) Typical pursuits of biomedical engineers include the following: • • • • • • • • • • • • •

Research in new materials for implanted artificial organs Development of new diagnostic instruments for blood analysis Writing software for analysis of medical research data Analysis of medical device hazards for safety and efficacy Development of new diagnostic imaging systems Design of telemetry systems for patient monitoring Design of biomedical sensors Development of expert systems for diagnosis and treatment of diseases Design of closed-loop control systems for drug administration Modeling of the physiologic systems of the human body Design of instrumentation for sports medicine Development of new dental materials Design of communication aids for individuals with disabilities

1.3 WHAT IS BIOMEDICAL ENGINEERING?

• Study of pulmonary fluid dynamics • Study of biomechanics of the human body • Development of material to be used as replacement for human skin The preceding list is not intended to be all-inclusive. Many other applications use the talents and skills of the biomedical engineer. In fact, the list of activities of biomedical engineers depends on the medical environment in which they work. This is especially true for the clinical engineers—biomedical engineers employed in hospitals or clinical settings. Clinical engineers are essentially responsible for all the high-technology instruments and systems used in hospitals today, the training of medical personnel in equipment safety, and the design, selection, and use of technology to deliver safe and effective health care. Engineers were first encouraged to enter the clinical scene during the late 1960s in response to concerns about electrical safety of hospital patients. This safety scare reached its peak when consumer activists, most notably Ralph Nader, claimed, “At the very least, 1,200 Americans are electrocuted annually during routine diagnostic and therapeutic procedures in hospitals.” This concern was based primarily on the supposition that catheterized patients with a low-resistance conducting pathway from outside the body into blood vessels near the heart could be electrocuted by voltage differences well below the normal level of sensation. Despite the lack of statistical evidence to substantiate these claims, this outcry served to raise the level of consciousness of health care professionals with respect to the safe use of medical devices. In response to this concern, a new industry—hospital electrical safety—arose almost overnight. Organizations such as the National Fire Protection Association (NFPA) wrote standards addressing electrical safety specifically for hospitals. Electrical safety analyzer manufacturers and equipment safety consultants became eager to serve the needs of various hospitals that wanted to provide a “safety fix” and of some companies, particularly those specializing in power distribution systems (most notably isolation transformers). To alleviate these fears, the Joint Commission on the Accreditation of Healthcare Organizations (then known as the Joint Commission on Accreditation of Hospitals) turned to NFPA codes as the standard for electrical safety and further specified that hospitals must inspect all equipment used on or near a patient for electrical safety at least every six months. To meet this new requirement, hospital administrators considered a number of options, including (1) paying medical device manufacturers to perform these electrical safety inspections, (2) contracting for the services of shared-services organizations, or (3) providing these services with in-house staff. When faced with this decision, most large hospitals opted for in-house service and created whole departments to provide the technological support necessary to address these electrical safety concerns. As a result, a new engineering discipline—clinical engineering—was born. Many hospitals established centralized clinical engineering departments. Once these departments were in place, however, it soon became obvious that electrical safety failures represented only a small part of the overall problem posed by the presence of medical equipment in the clinical environment. At the time, this equipment was neither totally understood nor properly maintained. Simple visual inspections often revealed broken knobs, frayed wires, and even evidence of liquid spills. Many devices did not perform in accordance with manufacturers’

1. BIOMEDICAL ENGINEERING

specifications and were not maintained in accordance with manufacturers’ recommendations. In short, electrical safety problems were only the tip of the iceberg. By the mid-1970s, complete performance inspections before and after equipment use became the norm, and sensible inspection procedures were developed. In the process, these clinical engineering pioneers began to play a more substantial role within the hospital. As new members of the hospital team, they did the following: • Became actively involved in developing cost-effective approaches for using medical technology • Provided hospital administrators with advice regarding the purchase of medical equipment based on their ability to meet specific technical specifications • Started using modern scientific methods and working with standards-writing organizations • Became involved in the training of health care personnel regarding the safe and efficient use of medical equipment Then, during the 1970s and 1980s, a major expansion of clinical engineering occurred, primarily due to the following events: • The Veterans Administration (VA), convinced that clinical engineers were vital to the overall operation of the VA hospital system, divided the country into biomedical engineering districts, with a chief biomedical engineer overseeing all engineering activities in the hospitals in that district. • Throughout the United States, clinical engineering departments were established in most large medical centers and hospitals and in some smaller clinical facilities with at least three hundred beds. • Health care professionals—physicians and nurses—needed assistance in utilizing existing technology and incorporating new innovations. • Certification of clinical engineers became a reality to ensure the continued competence of practicing clinical engineers. During the 1990s, the evaluation of clinical engineering as a profession continued with the establishment of the American College of Clinical Engineering (ACCE) and the Clinical Engineering Division within the International Federation of Medical and Biological Engineering (IFMBE). Clinical engineers today provide extensive engineering services for the clinical staff and serve as a significant resource for the entire hospital (Figure 1.10). Possessing in-depth knowledge regarding available in-house technological capabilities as well as the technical resources available from outside firms, the modern clinical engineer enables the hospital to make effective and efficient use of most if not all of its technological resources. Biomedical engineering is thus an interdisciplinary branch of engineering heavily based in both engineering and the life sciences. It ranges from theoretical, nonexperimental undertakings to state-of-the-art applications. It can encompass research, development, implementation, and operation. Accordingly, like medical practice itself, it is unlikely that any single person can acquire expertise that encompasses the entire field. As a result, there has been an explosion of biomedical engineering specialties to cover this broad field. Yet, because of the interdisciplinary nature of this activity, there are considerable interplay and overlapping of interest and effort between them. For example, biomedical engineers engaged in the

1.4 ROLES PLAYED BY THE BIOMEDICAL ENGINEERS

Regulation Agencies ted ccep ty, A tice Safe Prac ical Med

Maintenance, Safety

Human

nical Requ irem and ents Relia bility

Doctors Engineering

Tech

Practice

Accepted Medical

CLINICAL ENGINEER

Power, Cabling

Leasing Agencies

Patients

Operation

Cost and Economics

Hospital Administration

Third-Party Payers

Clinical Research

Allied Health Professionals

Nurses Vendors

Hospital Environment

FIGURE 1.10 The range of interactions that a clinical engineer may be required to engage in a hospital setting.

development of biosensors may interact with those interested in prosthetic devices to develop a means to detect and use the same bioelectric signal to power a prosthetic device. Those engaged in automating the clinical chemistry laboratory may collaborate with those developing expert systems to assist clinicians in making clinical decisions based upon specific laboratory data. The possibilities are endless. Perhaps an even greater benefit of the utilization of biomedical engineers lies in the potential for implementing existing technologies to identify and solve problems within our present health care system. Consequently, the field of biomedical engineering offers hope in the continuing battle to provide high-quality health care at a reasonable cost. If properly directed toward solving problems related to preventative medical approaches, ambulatory care services, and the like, biomedical engineers can provide the tools and techniques to make our health care system more effective and efficient.

1.4 ROLES PLAYED BY THE BIOMEDICAL ENGINEERS In its broadest sense, biomedical engineering involves training essentially three types of individuals: the clinical engineer in health care, the biomedical design engineer for industry, and the research scientist. Presently, one might also distinguish among three specific roles these biomedical engineers can play. Each is different enough to merit a separate

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description. The first type, the most common, might be called the “problem solver.” This biomedical engineer (most likely the clinical engineer or biomedical design engineer) maintains the traditional service relationship with the life scientists who originate a problem that can be solved by applying the specific expertise of the engineer. For this problem-solving process to be efficient and successful, however, some knowledge of each other’s language and a ready interchange of information must exist. Biomedical engineers must understand the biological situation to apply their judgment and contribute their knowledge toward the solution of the given problem, as well as to defend their methods in terms that the life scientist can understand. If they are unable to do these things, they do not merit the “biomedical” appellation. The second type, which is less common, could be called the “technological entrepreneur” (most likely a biomedical design engineer in industry). This individual assumes that the gap between the technological education of the life scientist or physician and the present technological capability has become so great that the life scientist cannot pose a problem that will incorporate the application of existing technology. Therefore, technological entrepreneurs examine some portion of the biological or medical front and identify areas in which advanced technology might be advantageous. Thus, they pose their own problem and then proceed to provide the solution, at first conceptually and then in the form of hardware or software. Finally, these individuals must convince the medical community that they can provide a useful tool because, contrary to the situation in which problem solvers find themselves, the entrepreneur’s activity is speculative at best and has no ready-made customer for the results. If the venture is successful, however, whether scientifically or commercially, then an advance has been made much earlier than it would have been through the conventional arrangement. Because of the nature of their work, technological entrepreneurs should have a great deal of engineering and medical knowledge as well as experience in numerous medical systems. The third type of biomedical engineer—the “engineer-scientist” (most likely found in academic institutions and industrial research labs)—is primarily interested in applying engineering concepts and techniques to the investigation and exploration of biological processes. The most powerful tool at their disposal is the construction of an appropriate physical or mathematical model of the specific biological system under study. An example of this relationship can be found in the study of cardiac function. The engineer-scientist may be exploring the complexities of fluid flow through the incredible pump that is the human heart. Mathematical models may be created to model the kinematics of the heart during contraction and equations to define the behavior of fluid flow. Through simulation techniques and available computing machinery, they can use this model to understand features that are too complex for either analytical computation or intuitive recognition. In addition, this process of simulation facilitates the design of appropriate experiments that can be performed on the actual biological system. The results of these experiments can, in turn, be used to amend the model. Thus, increased understanding of a biological mechanism results from this iterative process. This mathematical model can also predict the effect of these changes on a biological system in cases where the actual experiments may be tedious, very difficult, or dangerous. The researchers are thus rewarded with a better understanding of the biological system,

1.5 RECENT ADVANCES IN BIOMEDICAL ENGINEERING

and the mathematical description forms a compact, precise language that is easily communicated to others. In the example of the cardiac researcher, the engineer must at all times consider the anatomical and physiological causes for the macro-model results—in this case, why the heart is pumping the way it is. The activities of the engineer-scientist inevitably involve instrument development because the exploitation of sophisticated measurement techniques is often necessary to perform the biological side of the experimental work. It is essential that engineer-scientists work in a biological environment, particularly when their work may ultimately have a clinical application. It is not enough to emphasize the niceties of mathematical analysis while losing the clinical relevance in the process. This biomedical engineer is a true partner of the biological scientist and has become an integral part of the research teams being formed in many institutes to develop techniques and experiments that will unfold the mysteries of the human organism. Each of these roles envisioned for the biomedical engineer requires a different attitude, as well as a specific degree of knowledge about the biological environment. However, each engineer must be a skilled professional with a significant expertise in engineering technology. Therefore, in preparing new professionals to enter this field at these various levels, biomedical engineering educational programs are continually being challenged to develop curricula that will provide an adequate exposure to and knowledge about the environment, without sacrificing essential engineering skills. As we continue to move into a period characterized by a rapidly growing aging population, rising social and economic expectations, and a need for the development of more adequate techniques for the prevention, diagnosis, and treatment of disease, development and employment of biomedical engineers have become a necessity. This is true not only because they may provide an opportunity to increase our knowledge of living systems but also because they constitute promising vehicles for expediting the conversion of knowledge to effective action. The ultimate role of the biomedical engineer, like that of the nurse and physician, is to serve society. This is a profession, not just a skilled technical service. To use this new breed effectively, health care practitioners and administrators should be aware of the needs for these new professionals and the roles for which they are being trained. The great potential, challenge, and promise in this endeavor offer not only significant technological benefits but humanitarian benefits as well.

1.5 RECENT ADVANCES IN BIOMEDICAL ENGINEERING Biomedical engineering is a vast field with a multitude of concentrations and research initiatives. While the technicians affiliated with clinical engineering and a number of other concentrations focus mainly on preexisting technologies, researchers enjoy the exhilaration of innovating the new. Biomedical engineering has grown exponentially since its acceptance as a field less than a century ago, to the extent that today there is not a branch of medicine untouched by the problem-solving skill set of the engineer. The objective of this section is not to make the reader aware of every cutting-edge technology in development today but rather to provide an introduction to a sample of these new adventures.

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1.5.1 Prosthetics Prosthetics are one of the oldest innovations of biomedical engineering. The assistive technology field, prosthetics especially, became a true engineering discipline in itself in the period following World War II, when an unprecedented number of veterans returned home alive, but disabled, due to advances in medicine. Prosthetics are defined as any “internal or external device(s) that replace lost parts or functions of the neuroskeletomotor system” and may be either orthopedic or externally controlled. Externally controlled devices may be powered by the body itself through myoelectricity or a separate power supply. Neural prosthetics represent the newest field in prosthetics and one of the fastest-developing topics in biomedical engineering today. Orthopedic Prosthetics In designing a “replacement” limb for the human body, an engineer is buried under an obscene amount of considerations and design constraints. The appendage must be functionally sufficient, a design unique to each individual, depending on the activities to be accomplished. It must be comfortable, aesthetically pleasing, convenient, and simple in attachment. Prosthetics and orthoses seeking to imitate the human body piece by piece tend to have a great amount of difficulty in development and implementation. Instead, the general application of the device should always be considered, with the user in mind. An example of this design strategy can be found in the flex foot, a prosthetic foot with no real resemblance to the natural appendage. Instead of struggling to recreate the biomechanics of the ankle, tarsals, metatarsals, and phalanges of the lower leg, designers created a prosthetic with a single contact piece, no joint, and consisting of only one material. The Cheetah Leg shown in Figure 1.11, is one type of such a prosthetic and has

FIGURE 1.11 Paralympic sprinter Oscar Pistorius with a prosthetic leg. Designing for overall function, as opposed to mirroring the human body, is often the more practical approach. Compliments of http://www.thefinalsprint.com/ images/2008/05/oscar-pistorius-double-amputee-sprinter.jpg.

1.5 RECENT ADVANCES IN BIOMEDICAL ENGINEERING

allowed paralympians like Oscar Pistorius to compete at a scale approaching that of ablebodied athletes. Actually, the Cheetah Leg allowed Pistorius, a double amputee, to compete at a level that became subject to controversy. In 2008, the South African sprinter battled courts for the opportunity to race with able-bodied athletes in the Beijing Olympic Games. While Pistorius ultimately did not qualify, his efforts fueled a debate as to whether his engineered prosthetics functioned better than a human leg, actually giving him an advantage over runners in the standard Olympic Games. Externally controlled prosthetics use external motors to power their operation. The C-Leg is an example of such a device. This prosthetic leg has a microprocessor-controlled knee; has force sensors throughout for angle, swing, and velocity; and lasts 25 to 30 hours without charging. Uneven terrain is tackled with the C-Leg, as are changes in walking pace and direction. In recent years, sensor and minimally sized motor developments have made devices such as the C-Leg possible. Neural Prosthetics Neural prosthetics present one of the newest and perhaps most exciting concentrations of biomedical engineering. These devices may be powered by the human body—that is, they operate from electrical signals sent via electrodes from an external source to the peripheral muscle neuron—or they may be powered externally. These systems that use functional electrical stimulation (FES) to “restore sensory or motor function”are the definition of neural prostheses. These NPs have the potential to assist victims of spinal cord or cervical column injury (SCI and CI), restoring function to the muscle and lower extremities. Stimulation via electrodes must reach a threshold frequency to achieve tetanization, or the smooth motion contraction of muscle. Stimulation below this frequency results in isolated twitches and muscle fatigue. Electrodes may be implanted transcutaneously (on the surface), percutaneously (stimulator outside the body connects to a stimulation point inside), or implanted. As opposed to the leg, where a series of fairly simple joints and large motor units provide sufficient function, the upper extremities prove a significant challenge in fine-tuned control requirements. The incredible strength and flexibility of complex hand function are difficult to reproduce. The newest in prosthetic design hopes to overcome some of these challenges. The Luke Arm (Figure 1.12) is the brainchild of Segway inventor Dean Kamen. The arm has just as many degrees of freedom as the human arm and is capable of lifting above the user’s head. The arm uses myoelectric signals originating from residual nerves in the upper body. Fine-tuned control is assisted by controls in the user’s shoe; by activating different “pedals,” the user can rotate the wrist or grasp or release an object. Sensory feedback, a constant issue with mechanical prosthetics, is provided via a pressure sensor on the fingertips, which feed back to a vibrating patch worn on the user’s back. Increased pressure is felt by the user by changes in vibration intensity. Clinical trials are underway. The design of prosthetics involves an intensive materials engineering background, as well as an in-depth understanding of kinematics modeling and physiology. The American Board for Certification in Orthotics, Prosthetics, and Pedorthics provides guidelines for certification as a licensed prosthetist. Those in the field are required to complete an accredited

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FIGURE 1.12

Dean Kamen’s Luke Arm, the most advanced neural prosthetic to date, which uses myoelectric signals. Clinical trials are presently underway. Courtesy of http://medgadget.com.

undergraduate program in prosthetics or a graduate program specializing in the field with an appropriate undergraduate degree. Neural prosthetic development involves a team of members from various backgrounds, including biomechanics, electronics, and mathematical modeling. While prosthetics provide a strong example of the evolution of assistive technology and present a number of interesting design innovations, in recent years the field of biomedical engineering has shifted from a focus on mechanical systems to biological and organic solutions. Whereas decades ago the primary objectives of the biomedical engineer consisted of device design, modern feats are more likely to involve biochemistry and gene therapy than screws, nuts, and bolts. Two prime examples of this shift in focus are tissue engineering and stem cell research.

1.5.2 Tissue Engineering Tissue engineering, a relatively new field in biomedical engineering, consists of the manufacture of biological tissue either ex vivo or in vitro (outside the body), or the incorporation of new advancements to aid in the repair and growth of existing tissues in vivo (inside the body). In ex vivo applications, bioartificial tissues (those composed of both synthetic and natural materials) are used as an alternative to organ transplant or developed to study tissue behavior in vitro. Some important issues within the field include cell isolation, control of cell organization and function, upscaling to full bioartificial tissues, and biomaterial fabrication. While the most well-known tissue engineering feats have been in epithelial tissues, clinical trials are also currently under way for reconstruction of cartilage, bone, neural, and liver tissues. Grafts are used for treatment of every type of skin damage, including burns, pressure sores, venous stasis ulcers, and diabetic ulcers. Polymeric tubes are implanted to assist in nerve regeneration due to central and peripheral nervous system damage or disorders.

1.5 RECENT ADVANCES IN BIOMEDICAL ENGINEERING

Tissue engineering also covers joint replacements, including connective tissue recreation and bone grafts. Artificial heart valves implement bovine and porcine tissues along with bioartificial substances. Organ failure is treated with innovations in the field as well, with treatment for everything from liver cancer to breast reconstruction. Blood transfusions and dental surgery advancements are just two more examples of the wide range of applications of tissue engineering technologies. Bone marrow transplantation works to regenerate the most prolific organ of the body. Marrow is responsible for the production of blood cells and is often damaged by myeloablative treatment regimens, such as chemotherapy and radiation. Modern methods involve harvesting patient samples of marrow prior to the therapy regimen and reinjecting them following treatment. The body regenerates its marrow supply, causing a temporary immunodeficiency. In the case of pancreatic and liver tissue development, a bioreactor model is used. Bioreactors are systems consisting of a large number of cells that take in an input of reactants and output a set of products. Bioreactors have also been implemented for blood cell production from hematopoietic tissue. The two types of bioreactor systems are hollow fiber and microcarrier-based systems. In the hollow fiber system, a large number of small-diameter, hollow tubes are bundled together by a larger shell tube. The small tubes are injected with organ-specific cells that are suspended in a collagen-based matrix. The matrix will contract, leaving space within the small tubes. The patient’s own blood or plasma is injected into the larger, encompassing tube and is allowed to nourish the hepatopoietic cells by flowing through the newly emptied space in the smaller tubes. In microcarrier-based systems, small beads (less than 500 mm) with surfaces specially treated for cell attachment are either positioned in a packed or fluidized bed or incorporated in hollow fiber cartridges. In the packed bed method, a column is filled with the beads and capped at each end with porous plates to allow perfusion. Success rates rely on fluid flow rate through the column, as well as the density of packed beads and dimension ratios of the column. Biomaterials play a significant role in tissue engineering. In each of the previous examples, biomaterials prove an integral component of tissue regeneration and reconstruction. From the obvious application of artificial valve design to the less apparent role of injection needle design in bone marrow transplantation, biomaterial development is a necessary step in the advancement of tissue engineering. Devices must provide mechanical support, prevent undesirable tissue interactions, and potentially allow for timely biodegradation. Biomaterial devices can be broken down into two types, each existing on a scale as small as a few hundred microns. Immunoprotective devices contain semipermeable membranes that prevent specific host immune system elements from entering the device. Open devices, in contrast, are designed for systems to be fully integrated with the host and have large pores (greater than 10 mm), allowing for free transport of cells and molecules. Pore sizes within a biomaterial directly correlate to the functions of the device. The structure of a pore is determined by the continuity of individual pores in the device, as well as the size and size distribution. The three classifications of porous materials are microporous, mesoporous, and macroporous. Microporous materials have pores with a diameter less than 2 nm and allow for transport of small molecules, including gases. Mesoporous materials allow for transport of small proteins and have pores with diameters ranging from 2 to 50 nm. Macroporous materials have pores with diameters greater than 50 nm and allow

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for large proteins, and possibly even cells, to pass through. Fibrovascular tissue will pass through any material with pores greater than 10 nm. Pore size and distribution are rarely regular, and abnormalities will both change general material properties as well as fluid flow rate across the device. A major focus in tissue engineering is controlling cell organization and regeneration. The more control the researcher has over cell development, the greater the capabilities and the wider the range of applications of the bioartificial tissue. Stem cells provide an opportunity for researchers to develop tissues essentially from scratch. Stem cells both build and maintain cells in vivo and possess the ability to be used for tissue generation ex vivo. Some background information on this new technology is provided following, and specific applications to tissue engineering are available in later chapters.

1.5.3 Stem Cell Research In recent years, stem cells have become the topic of both intense controversy and incredible excitement within the research community. The potential for stem cell technology is apparently limitless, with some known possibilities shown in Figure 1.7. Cells may be used to test drugs on different types of tissues, to understand how to prevent birth defects, and to potentially replace and regenerate damaged tissue in the body. The possibilities truly seem endless. In actuality there are two different types of stem cells. Embryonic stem cells come from embryos, which are mostly supplied by in vitro fertilization clinics four to five days following fertilization. At this point, stem cells will either self-regenerate or commit and differentiate. Self-renewal or regeneration means that the stem cell will reproduce with no developmental commitment. Essentially, the stem cell remains a stem cell. Differentiation is the expression of tissue or cell-specific genes. For the majority of tissues in the human body, cells will differentiate terminally. In some cases, however, dynamic operation is required, and, as such, a population of adult stem cells is maintained for regeneration purposes. The two most common types of adult stem cells are those of the hematopoietic system (blood renewal) and the intestinal epithelia. These cell types are similar in that they both occur in very large numbers and have short life spans. Stem cells are required to maintain this dynamic population. Researchers control stem cell development and differentiation within cultures by a number of means. For embryonic stem cells, the difference between self-regeneration and differentitation, surprisingly enough, is the concentration of a single essential protein, or growth factor. Leukemia inhibitory factor (LIF), in high enough concentrations, will cause embryonic stem cells to regenerate indefinitely in cultures. This is an interesting fact, because it proves that stem cell development is not an intrinsic predetermined state but rather is induced by extrinsic factors. With an executive order in 2009, President Barack Obama lifted an eight-and-a-half-year ban on government-funded stem cell research, earning praise from the science community for opening the door to potential cures for some of mankind’s most debilitating diseases. Both tissue engineering and stem cell research represent just a sampling of the breakthrough biologically focused ventures currently being explored by today’s biomedical engineers.

1.6 PROFESSIONAL STATUS OF BIOMEDICAL ENGINEERING

1.6 PROFESSIONAL STATUS OF BIOMEDICAL ENGINEERING Biomedical engineers are professionals, which are defined as an aggregate of people finding identity in sharing values and skills absorbed during a common course of intensive training. Whether individuals are professionals is determined by examining whether they have internalized certain given professional values. Furthermore, a professional is someone who has internalized professional values and is licensed on the basis of his or her technical competence. Professionals generally accept scientific standards in their work, restrict their work activities to areas in which they are technically competent, avoid emotional involvement, cultivate objectivity in their work, and put their clients’ interests before their own. The concept of a profession that is involved in the design, development, and management of medical technology encompasses three primary occupational models: science, business, and profession. Consider initially the contrast between science and profession. Science is seen as the pursuit of knowledge, its value hinging on providing evidence and communicating with colleagues. Profession, on the other hand, is viewed as providing a service to clients who have problems they cannot handle themselves. Scientists and professionals have in common the exercise of some knowledge, skill, or expertise. However, while scientists practice their skills and report their results to knowledgeable colleagues, professionals, such as lawyers, physicians, and engineers, serve lay clients. To protect both the professional and the client from the consequences of the layperson’s lack of knowledge, the practice of the profession is often regulated through such formal institutions as state licensing. Both professionals and scientists must persuade their clients to accept their findings. Professionals endorse and follow a specific code of ethics to serve society. On the other hand, scientists move their colleagues to accept their findings through persuasion. Consider, for example, the medical profession. Its members are trained in caring for the sick, with the primary goal of healing them. These professionals not only have a responsibility for the creation, development, and implementation of that tradition, but they are also expected to provide a service to the public, within limits, without regard to selfinterest. To ensure proper service, the profession closely monitors the licensing and certification process. Thus, medical professionals themselves may be regarded as a mechanism of social control. However, this does not mean that other facets of society are not involved in exercising oversight and control of physicians in their practice of medicine. A final attribute of professionals is that of integrity. Physicians tend to be both permissive and supportive in relationships with patients and yet are often confronted with moral dilemmas involving the desires of their patients and social interest. For example, how to honor the wishes of terminally ill patients while not facilitating the patients’ deaths is a moral question that health professionals are forced to confront. A detailed discussion of the moral issues posed by medical technology is presented in Chapter 2. One can determine the status of professionalization by noting the occurrence of six crucial events: the first training school, the first university school, the first local professional association, the first national professional association, the first state license law, and the first formal code of ethics. The early appearances of the training school and the university affiliation underscore the importance of the cultivation of a knowledge base. The strategic innovative role of the universities and early teachers lies in linking knowledge to practice and creating

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a rationale for exclusive jurisdiction. Those practitioners pushing for prescribed training then form a professional association. The association defines the tasks of the profession: raising the quality of recruits; redefining their function to permit the use of less technically skilled people to perform the more routine, less involved tasks; and managing internal and external conflicts. In the process, internal conflict may arise between those committed to previously established procedures and newcomers committed to change and innovation. At this stage, some form of professional regulation, such as licensing or certification, surfaces because of a belief that it will ensure minimum standards for the profession, enhance status, and protect the layperson in the process. The last area of professional development is the establishment of a formal code of ethics, which usually includes rules to exclude the unqualified and unscrupulous practitioners, rules to reduce internal competition, and rules to protect clients and emphasize the ideal service to society. A code of ethics usually comes at the end of the professionalization process. In biomedical engineering, all six critical steps have been clearly taken. The field of biomedical engineering, which originated as a professional group interested primarily in medical electronics in the late 1950s, has grown from a few scattered individuals to a very well-established organization. There are approximately 48 international societies throughout the world serving an increasingly expanding community of biomedical engineers. Today, the scope of biomedical engineering is enormously diverse. Over the years, many new disciplines such as tissue engineering, artificial intelligence, and so on, which were once considered alien to the field, are now an integral part of the profession. Professional societies play a major role in bringing together members of this diverse community to share their knowledge and experience in pursuit of new technological applications that will improve the health and quality of life. Intersocietal cooperation and collaborations, at both the national and international levels, are more actively fostered today through professional organizations such as the Biomedical Engineering Society (BMES), the American Institute for Medical and Biological Engineering (AIMBE), Engineering in Medicine and Biology Society (EMBS), and the Institute of Electrical and Electronic Engineers (IEEE).

1.7 PROFESSIONAL SOCIETIES 1.7.1 The American Institute for Medical and Biological Engineering The United States has the largest biomedical engineering community in the world. Major professional organizations that address various cross sections of the field and serve biomedical engineering professionals include the American College of Clinical Engineering, the American Institute of Chemical Engineers, the American Medical Informatics Association, the American Society of Agricultural Engineers, the American Society for Artificial Internal Organs, the American Society of Mechanical Engineers, the Association for the Advancement of Medical Instrumentation, the Biomedical Engineering Society, the IEEE Engineering in Medicine and Biology Society, an interdisciplinary Association for the Advancement of

1.7 PROFESSIONAL SOCIETIES

Rehabilitation and Assistive Technologies, and the Society for Biomaterials. In an effort to unify all the disparate components of the biomedical engineering community in the United States as represented by these various societies, the American Institute for Medical and Biological Engineering (AIMBE) was created in 1992. The primary goal of AIMBE is to serve as an umbrella organization in the United States for the purpose of unifying the bioengineering community, addressing public policy issues, and promoting the engineering approach in society’s effort to enhance health and the quality of life through the judicious use of technology. (For information, contact AIMBE, 1701 K Street, Suite 510, Washington, DC, 20036; http://www.aimbe.org/; e-mail: [emailprotected].)

1.7.2 IEEE Engineering in Medicine and Biology Society The Institute of Electrical and Electronic Engineers (IEEE) is the largest international professional organization in the world and accommodates 37 different societies and councils under its umbrella structure. Of these 37, the Engineering in Medicine and Biology Society (EMBS) represents the foremost international organization, serving the needs of over 8,000 biomedical engineering members around the world. The major interest of the EMBS encompasses the application of concepts and methods from the physical and engineering sciences to biology and medicine. Each year, the society sponsors a major international conference while cosponsoring a number of theme-oriented regional conferences throughout the world. Premier publications consist of a monthly journal (Transactions on Biomedical Engineering), three quarterly journals (Transactions on Neural Systems and Rehabilitation Engineering, Transactions on Information Technology in Biomedicine, and Transactions on Nanobioscience), as well as a bimonthly magazine (IEEE Engineering in Medicine and Biology Magazine). Secondary publications, authored in collaboration with other societies, include Transactions on Medical Imaging, Transactions on Neural Networks, and Transactions on Pattern Analysis and Machine Intelligence. (For more information, contact the IEEE EMBS Executive Office, IEEE, 445 Hoes Lane, Piscataway, NJ, 08855-1331; http:// www.embs.org/; e-mail: [emailprotected].)

1.7.3 The Biomedical Engineering Society Established in 1968, the Biomedical Engineering Society (BMES) was founded in order to address a need for a society that afforded equal status to representatives of both biomedical and engineering interests. With that in mind, the primary goal of the BMES, as stated in their Articles of Incorporation, is “to promote the increase of biomedical engineering knowledge and its utilization.” Regular meetings are scheduled biannually in both the spring and fall. Additionally, special interest meetings are interspersed throughout the year and are promoted in conjunction with other biomedical engineering societies such as AIMBE and EMBS. The primary publications associated with the BMES include Annals of Biomedical Engineering, a monthly journal presenting original research in several biomedical fields; BMES Bulletin, a quarterly newsletter presenting a wider array of subject matter relating both to biomedical engineering as well as BMES news and events; and the BMES Membership Directory, an annual publication listing the contact information of the society’s

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individual constituents. (For more information, contact the BMES directly: BMES, 8401 Corporate Drive, Suite 140, Landover, MD 20785-2224; http://www.bmes.org/default .asp; e-mail: [emailprotected].) The activities of these biomedical engineering societies are critical to the continued advancement of the professional status of biomedical engineers. Therefore, all biomedical engineers, including students in the profession, are encouraged to become members of these societies and engage in the activities of true professionals.

1.8 EXERCISES 1. Select a specific “medical technology” from the historical periods indicated. Describe the fundamental principles of operation and discuss their impact on health care delivery: (a) 1900–1939; (b) 1945–1970; (c) 1970–1980; (d) 1980–2003. 2. Provide a review of the effect that computer technology has had on health care delivery, citing the computer application and the time frame of its implementation. 3. The term genetic engineering implies an engineering function. Is there one? Should this activity be included in the field of biomedical engineering? 4. Discuss in some detail the role the genome project has had and is anticipated having on the development of new medical technology. 5. Using your crystal ball, what advances in engineering and/or life science do you think will have the greatest impact on clinical care or biomedical research? 6. The organizational structure of a hospital involves three major groups: the Board of Trustees, the administrators, and the medical staff. Specify the major responsibilities of each. In what group should a Department of Clinical Engineering reside? Explain your answer. 7. Based on its definition, what attributes should a clinical engineer have? 8. List at least seven (7) specific activities of clinical engineers. 9. Provide modern examples (i.e., names of individuals and their activities) of the three major roles played by biomedical engineers: (a) the problem solver; (b) the technological entrepreneur; and (c) the engineer scientist. 10. Do the following groups fit the definition of a profession? Discuss how they do or do not. (a) registered nurse; (b) biomedical technician; (c) respiratory therapist; (d) hospital administrator. 11. List the areas of knowledge necessary to practice biomedical engineering. Identify where in the normal educational process one can acquire knowledge. How best can administrative skills be acquired? 12. Prosthetic limbs are often created for specialized activities, such as mountain biking or driving. Create a design for an upper- or lower-extremity prosthetic for a particular specialty activity. 13. What steps must be taken to become a licensed prosthetician? 14. What are the two means of powering a neural prosthetic? 15. What is the difference between an adult stem cell and an embryonic stem cell? Where does each come from?

1.8 EXERCISES

16. While stem cell research has recently been granted federal funding, limitations on the types of research allowed are decided by the state. Research your home state’s policies on stem cell research, and provide a summary paragraph. 17. Provide a copy of the home page for a biomedical engineering professional society and a list of its major activities for the coming year. 18. Research a biomedical engineering professional society and provide three benefits of joining the society. 19. What is your view regarding the role biomedical engineers will play in the health care system of tomorrow? 20. Discuss the trade-offs in health care that occur as a result of limited financial resources. 21. Discuss whether medical technology is an economic cost factor, a benefit, or both.

Suggested Readings C. Aston, Biological Warfare Canaries, IEEE Spectrum (2001) 35–40. I.N. Bankman, Handbook of Medical Imaging, CRC Press, Boca Raton, FL, 2000. J.D. Bronzino, Biomedical Engineering Handbook, first and second ed., CRC Press, Boca Raton, FL, 1995; 2000; 2005. J.D. Bronzino, Management of Medical Technology: A Primer for Clinical Engineering, Butterworth, Boston, 1992. E. Carson, C. Cobelli, Modeling Methodology for Physiology and Medicine, Academic Press, San Diego, CA, 2001. D.J. DiLorenzo, J.D. Bronzino, Neuroengineering, CRC Press, Boca Raton, FL, 2008. C.T. Laurenchin, Repair and Restore with Tissue Engineering, EMBS Magazine 22 (5) (2003) 16–17. F. Nebekar, Golden Accomplishments in Biomedical Engineering, EMBS Magazine 21 (3) (2002) 17–48. A. Pacela, Bioengineering Education Directory, Quest Publishing Co., Brea, CA, 1990. B.O. Palsson, S.N. Bhatia, Tissue Engineering, Prentice Hall, Upper Saddle River, NJ, 2004. J.B. Park, J.D. Bronzino, Biomaterials: Principles and Applications, CRC Press, Boca Raton, FL, 2003. D. Serlin, Replaceable You, University of Chicago Press, Chicago, 2004. The IEEE/EMBS magazine published by the Institute of Electrical and Electronic Engineers, in: J. Enderle (Ed.), especially “Writing the Book on BME,” vol. 21, No. 3, 2002. M.L. Yarmush, M. Toner, Biotechnology for Biomedical Engineers, CRC Press, Boca Raton, FL, 2003.

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C H A P T E R

2 Moral and Ethical Issues Joseph D. Bronzino, PhD, PE O U T L I N E 2.1

Morality and Ethics: A Definition of Terms

Two Moral Norms: Beneficence and Nonmaleficence

2.3

Redefining Death

2.4

The Terminally Ill Patient and Euthanasia

2.5

Taking Control

2.6

Human Experimentation

2.7

Definition and Purpose of Experimentation

Informed Consent

2.2

2.8

2.9

Regulation of Medical Device Innovation

2.10

Marketing Medical Devices

2.11

Ethical Issues in Feasibility Studies

2.12

Ethical Issues in Emergency Use

2.13

Ethical Issues in Treatment Use

2.14

The Role of the Biomedical Engineer in the FDA Process

Exercises

2.15

Suggested Readings

A FT E R C O MP L E T IN G T H I S C H A PT E R , S T U D E N T S W I L L B E A BL E T O : • Present the codes of ethics for the medical profession, nursing profession, and biomedical engineering.

• Define and distinguish between the terms morals and ethics. • Present the rationale underlying two major philosophical schools of thought: utilitarianism and nonconsequentialism.

Introduction to Biomedical Engineering, Third Edition

• Identify the modern moral dilemmas, including redefining death, deciding how

2. MORAL AND ETHICAL ISSUES

to care for the terminally ill, and human experimentation, which arise from the two moral norms: beneficence (the provision of benefits) and nonmaleficence (the avoidance of harm).

• Discuss the moral judgments associated with present policies regarding the regulation of the development and use of new medical devices.

The tremendous infusion of technology into the practice of medicine has created a new medical era. Advances in material science have led to the production of artificial limbs, heart valves, and blood vessels, thereby permitting “spare parts” surgery. Numerous patient disorders are now routinely diagnosed using a wide range of highly sophisticated imaging devices, and the lives of many patients are being extended through significant improvements in resuscitative and supportive devices, such as respirators, pacemakers, and artificial kidneys. These technological advances, however, have not been benign. They have had significant moral consequences. Provided with the ability to develop cardiovascular assist devices, perform organ transplants, and maintain the breathing and heartbeat of terminally ill patients, society has been forced to reexamine the meaning of such terms as death, quality of life, heroic efforts, and acts of mercy, and to consider such moral issues as the right of patients to refuse treatment (living wills) and to participate in experiments (informed consent). As a result, these technological advances have made the moral dimensions of health care more complex and have posed new and troubling moral dilemmas for medical professionals, the biomedical engineer, and society at large. The purpose of this chapter is to examine some of the moral questions related to the use of new medical technologies. The objective, however, is not to provide solutions or recommendations for these questions. Rather, the intent is to demonstrate that each technological advance has consequences that affect the very core of human values. Technology and ethics are not foreigners; they are neighbors in the world of human accomplishment. Technology is a human achievement of extraordinary ingenuity and utility and is quite distant from the human accomplishment of ethical values. They face each other rather than interface. The personal face of ethics looks at the impersonal face of technology in order to comprehend technology’s potential and its limits. The face of technology looks to ethics to be directed to human purposes and benefits. In the process of making technology and ethics face each other, it is our hope that individuals engaged in the development of new medical devices, as well as those responsible for the care of patients, will be stimulated to examine and evaluate critically “accepted” views and to reach their own conclusions. This chapter, therefore, begins with some definitions related to morality and ethics, followed by a more detailed discussion of some of the moral issues of special importance to biomedical engineers.

2.1 MORALITY AND ETHICS: A DEFINITION OF TERMS From the very beginning, individuals have raised concerns about the nature of life and its significance. Many of these concerns have been incorporated into the four fundamental questions posed by the German philosopher Immanuel Kant (1724–1804): What can I know?

2.1 MORALITY AND ETHICS: A DEFINITION OF TERMS

What ought I to do? What can I hope? What is man? Evidence that early societies raised these questions can be found in the generation of rather complex codes of conduct embedded in the customs of the earliest human social organization: the tribe. By 600 BC, the Greeks were successful in reducing many primitive speculations, attitudes, and views on these questions to some type of order or system and integrating them into the general body of wisdom called philosophy. Being seafarers and colonizers, the Greeks had close contact with many different peoples and cultures. In the process, struck by the variety of customs, laws, and institutions that prevailed in the societies that surrounded them, they began to examine and compare all human conduct in these societies. This part of philosophy they called ethics. The term ethics comes from the Greek ethos, meaning “custom.” On the other hand, the Latin word for custom is mos, and its plural, mores, is the equivalent of the Greek ethos and the root of the words moral and morality. Although both terms (ethics and morality) are often used interchangeably, there is a distinction between them that should be made. Philosophers define ethics as a particular kind of study and use morality to refer to its subject matter. For example, customs that result from some abiding principal human interaction are called morals. Some examples of morals in our present society are telling the truth, paying one’s debts, honoring one’s parents, and respecting the rights and property of others. Most members of society usually consider such conduct not only customary but also correct or right. Thus, morality encompasses what people believe to be right and good and the reasons they give for it. Most of us follow these rules of conduct and adjust our lifestyles in accordance with the principles they represent. Many even sacrifice life itself rather than diverge from them, applying them not only to their own conduct but also to the behavior of others. Individuals who disregard these accepted codes of conduct are considered deviants and, in many cases, are punished for engaging in an activity that society as a whole considers unacceptable. For example, individuals committing “criminal acts” (defined by society) are often “outlawed” and, in many cases, severely punished. These judgments regarding codes of conduct, however, are not inflexible; they must continually be modified to fit changing conditions and thereby avoid the trauma of revolution as the vehicle for change. While morality represents the codes of conduct of a society, ethics is the study of right and wrong, of good and evil in human conduct. Ethics is not concerned with providing any judgments or specific rules for human behavior, but rather with providing an objective analysis about what individuals “ought to do.” Defined in this way, it represents the philosophical view of morals, and, therefore, is often referred to as moral philosophy. Consider the following three questions: “Should badly deformed infants be kept alive?”; “Should treatment be stopped to allow a terminally ill patient to die?”; “Should humans be used in experiments?” Are these questions of morality or ethics? In terms of the definitions just provided, all three of these inquiries are questions of moral judgment. Philosophers argue that all moral judgments are considered to be “normative judgments”— that is, they can be recognized simply by their characteristic evaluative terms such as good, bad, right, wrong, and so on. Typical normative judgments include the following: • Stealing is wrong. • Everyone ought to have access to an education. • Voluntary euthanasia should not be legalized.

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Each of these judgments expresses an evaluation; that is, it conveys a negative or positive attitude toward some state of affairs. Each, therefore, is intended to play an action-guiding function. Arriving at moral judgments, however, requires knowledge of valid moral standards in our society. Nevertheless, how is such knowledge obtained? The efforts to answer this question lie in two competing schools of thought that currently dominate normative ethical theory: utilitarianism, a form of consequentialism, and Kantianism, a form of nonconsequentialism. Consequentialism holds that the morally right action is always the one among the available options that has the best consequences. An important implication of consequentialism is that no specific actions or courses of conduct are automatically ruled out as immoral or ruled in as morally obligatory. The rightness or wrongness of an action is wholly contingent upon its effects. According to utilitarianism, there are two steps to determining what ought to be done in any situation. First, determine which courses of action are open. Second, determine the consequences of each alternative. When this has been accomplished, the morally right course of action is the one that maximizes pleasure, minimizes pain, or both—the one that does the “greatest good for the greatest number.” Because the central motivation driving the design, development, and use of medical devices is improvement of medicine’s capacity to protect and restore health, an obvious virtue of utilitarianism is that it assesses medical technology in terms of what many believe makes health valuable: the attainment of well-being and the avoidance of pain. Utilitarianism, therefore, advocates that the end justifies the means. As long as any form of treatment maximizes good consequences, it should be used. Many people, though, believe that the end does not always justify the means and that individuals have rights that are not to be violated no matter how good the consequences might be. In opposition to utilitarianism stands the school of normative ethical thought known as nonconsequentialism. Proponents of this school deny that moral evaluation is simply and wholly a matter of determining the consequences of human conduct. They agree that other considerations are relevant to moral assessment and so reject the view that morally right conduct is whatever has the best consequences. Based largely on the views of Immanuel Kant, this ethical school of thought insists that there is something uniquely precious about human beings from the moral point of view. According to Kant’s theory, humans have certain “rights” that do not apply to any other animal. For example, the moral judgments that we should not kill and eat one another for food or hunt one another for sport or experiment on one another for medical science are all based on this view of human rights. Humans are, in short, owed a special kind of respect simply because they are people. These two philosophies may be extended to apply to animal testing in scientific research as well. On the utilitarianism side of the argument for animal experimentation, the health care advancements for humans made possible through animal research far outweigh the majority of arguments against the practice. In contrast, nonconsequentialism would state that maltreatment of innocent and unprotected living beings is morally unjust and as such is an inappropriate means to the ends of better health care for people. Ultimately researchers must decide for themselves, based on their own beliefs and reasoning, which philosophy wins out. In terms of human experimentation, to better understand the Kantian perspective, it may be helpful to recognize that Kant’s views are an attempt to capture in secular form a basic

2.1 MORALITY AND ETHICS: A DEFINITION OF TERMS

tenet of Christian morality. What makes human beings morally special entities deserving a unique type of respect? Christianity answers in terms of the doctrine of ensoulment. This doctrine holds that only human beings are divinely endowed with an eternal soul. According to Christian ethics, the soul makes humans the only beings with intrinsic value. Kant’s secular version of the doctrine of ensoulment asserts that human beings are morally unique and deserve special respect because of their autonomy. Autonomy is taken by Kant to be the capacity to make choices based on rational deliberation. The central task of ethics then is to specify what human conduct is required to respect the unique dignity of human beings. For most Kantians, this means determining what limits human beings must observe in the way they treat one another, and this, in turn, is taken to be a matter of specifying each individual’s fundamental moral rights. These two ethical schools of thought, therefore, provide some rationale for moral judgments. However, when there is no clear moral judgment, one is faced with a dilemma. In medicine, moral dilemmas arise in those situations that raise fundamental questions about right and wrong in the treatment of sickness and the promotion of health in patients. In many of these situations, the health professional usually faces two alternative choices, neither of which seems to be a satisfactory solution to the problem. For example, is it more important to preserve life or to prevent pain? Is it right to withhold treatment when doing so may lead to a shortening of life? Does an individual have the right to refuse treatment when refusing it may lead to death? All these situations seem to have no clear-cut imperative based on our present set of convictions about right and wrong. That is the dilemma raised by Kant: What ought I do?

CASE STUDY: STEM CELL RESEARCH At the moment of conception—that is, when a sperm penetrates an egg—the process of fertilization occurs. The formation of an embryo is initiated. Once the sperm enters the egg, there is an immediate opening of ion channels, which depolarizes the plasma membrane of the cell and prevents other sperm from fusing with it. DNA replication then begins, and the first cell division occurs approximately 36 hours later. As the process continues, the cell begins to experience cleavage, where the cells repeatedly divide, cycling between the S (DNA synthesis) and M (mitosis) phases of cell division, essentially skipping the G1 and G2 phases, when most cell growth normally occurs. Thus, there is no net growth of the cells, merely subdivision into smaller cells, individually called blastomeres. Five days after fertilization, the number of cells composing the embryo is in the hundreds, and the cells form tight junctions characteristic of a compact epithelium, which is arranged around a central cavity. This is the embryonic stage known as the blastocyst. Within the cavity exists a mass of cells, which protrude inward. These cells are known as the inner cell mass and become the embryo. The exterior cells are the trophoblast and eventually form the placenta. It is the cells from the inner cell mass of the blastocyst, however, that, when isolated and grown in a culture, are identified as embryonic stem cells. It is important to note that if cell division continues, determination and differentiation happen. Differentiation occurs when a cell begins to exhibit the specific attributes of a predestined specialized cellular role. Determination is related to differentiation but is somewhat dissimilar.

2. MORAL AND ETHICAL ISSUES

Totipotent Cells

Blastocyst

Fetus

Primordial Germ Cells

Inner Cell Mass

Cultured Pluripotent Stem Cells

FIGURE 2.1 Using the inner cell mass to form pluripotent stem cells. Courtesy of http://www.nih.gov/news/ stemcell/primer.htm.

When a cell group that has been determined is transplanted, it will not assimilate with the other cells but will grow into cells that comprised the original organ it was destined to become. Since the process of obtaining embryonic stem cells (Figure 2.1) destroys the embryo, the following questions arise: 1. Is the embryo a living human being, entitled to all of the same rights that a human at any other age would be granted? Discuss the answer to this question from a Utilitarian and Kantian point of view. 2. Should any research that is potentially beneficial to the well-being of mankind be pursued? In 2009, President Obama passed groundbreaking legislation entitled “Executive Order 13505—Removing Barriers to Responsible Scientific Research Involving Human Stem Cells.” The order calls for a review of NIH (National Institute of Health) guidelines for stem cell research and, more importantly, removes the requirement of President Action to approve NIH-funded stem cell investigations. 3. Should the federal government support (i.e., use tax dollars to fund) such research? Or, in contrast, should the government be allowed to interfere?

In the practice of medicine, moral dilemmas are certainly not new. They have been present throughout medical history. As a result, over the years there have been efforts to provide a set of guidelines for those responsible for patient care. These efforts have resulted in the development of specific codes of professional conduct. Let us examine some of these codes or guidelines. For the medical profession, the World Medical Association adopted a version of the Hippocratic Oath entitled the Geneva Convention Code of Medical Ethics in 1949. This declaration contains the following statements:

2.1 MORALITY AND ETHICS: A DEFINITION OF TERMS

I solemnly pledge myself to consecrate my life to the services of humanity; I will give to my teachers the respect and gratitude which is their due; I will practice my profession with conscience and dignity; The health of my patient will be my first consideration; I will respect the secrets which are confided in me; I will maintain by all the means in my power, the honour and the noble traditions of the medical profession; My colleagues will be my brothers; I will not permit considerations of religion, nationality, race, party politics, or social standing to intervene between my duty and my patient; I will maintain the utmost respect for human life from the time of conception, even under threat; I will not use my medical knowledge contrary to the laws of humanity; I make these promises solemnly, freely, and upon my honour. In the United States, the American Medical Association (AMA) adopted a set of Principles of Medical Ethics in 1980 and revised them in June 2001. A comparison of the two sets of principles is provided following.

Revised Principles Version adopted by the AMA House of Delegates, June 17, 2001 The medical profession has long subscribed to a body of ethical statements developed primarily for the benefit of the patient. As a member of this profession, a physician must recognize responsibility to patients first and foremost, as well as to society, to other health professionals, and to self. The following Principles adopted by the American Medical Association are not laws but standards of conduct that define the essentials of honorable behavior for the physician. I. A physician shall be dedicated to providing competent medical care, with compassion and respect for human dignity and rights. II. A physician shall uphold the standards of professionalism, be honest in all professional interactions, and strive to report physicians deficient in character or

Previous Principles As adopted by the AMA’s House of Delegates, 1980 The medical profession has long subscribed to a body of ethical statements developed primarily for the benefit of the patient. As a member of this profession, a physician must recognize responsibility not only to patients, but also to society, to other health professionals, and to self. The following Principles adopted by the American Medical Association are not laws but standards of conduct that define the essentials of honorable behavior for the physician. I. A physician shall be dedicated to providing competent medical service with compassion and respect for human dignity. II. A physician shall deal honestly with patients and colleagues, and strive to expose those physicians deficient in character or competence, or who engage in fraud or deception.

2. MORAL AND ETHICAL ISSUES

III.

IV.

VI.

VII.

VIII.

IX.

competence, or engaging in fraud or deception, to appropriate entities. A physician shall respect the law and also recognize a responsibility to seek changes in those requirements which are contrary to the best interests of the patient. A physician shall respect the rights of patients, colleagues, and other health professionals, and shall safeguard patient confidences within the constraints of the law. A physician shall continue to study, apply, and advance scientific knowledge; make relevant information available to patients, colleagues, and the public; obtain consultation; and use the talents of other health professionals when indicated. A physician shall, in the provision of appropriate patient care, except in emergencies, be free to choose whom to serve, with whom to associate, and the environment in which to provide medical care. A physician shall recognize a responsibility to participate in activities contributing to the improvement of the community and the betterment of public health. A physician shall, while caring for a patient, regard responsibility to the patient as paramount. A physician shall support access to medical care for all people.

III. A physician shall respect the law and also recognize a responsibility to seek changes in those requirements which are contrary to the best interests of the patient. IV. A physician shall respect the rights of patients, of colleagues, and of other health professionals, and shall safeguard patient confidences within the constraints of the law. V. A physician shall continue to study, apply and advance scientific knowledge; make relevant information available to patients, colleagues, and the public; obtain consultation; and use the talents of other health professionals when indicated. VI. A physician shall, in the provision of appropriate patient care, except in emergencies, be free to choose whom to serve, with whom to associate, and the environment in which to provide medical services. VII. A physician shall recognize a responsibility to participate in activities contributing to an improved community.

For the nursing profession, the American Nurses Association formally adopted in 1976 the Code For Nurses, whose statements and interpretations provide guidance for conduct and relationships in carrying out nursing responsibilities: PREAMBLE: The Code for Nurses is based on belief about the nature of individuals, nursing, health, and society. Recipients and providers of nursing services are viewed as individuals and groups who possess basic rights and responsibilities, and whose values and circumstances

2.1 MORALITY AND ETHICS: A DEFINITION OF TERMS

command respect at all times. Nursing encompasses the promotion and restoration of health, the prevention of illness, and the alleviation of suffering. The statements of the Code and their interpretation provide guidance for conduct and relationships in carrying out nursing responsibilities consistent with the ethical obligations of the profession and quality in nursing care. The nurse provides services with respect for human dignity and the uniqueness of the client, unrestricted by considerations of social or economic status, personal attributes, or the nature of health problems. The nurse safeguards the client’s right to privacy by judiciously protecting information of a confidential nature. The nurse acts to safeguard the client and the public when health care and safety are affected by the incompetent, unethical, or illegal practice of any person. The nurse assumes responsibility and accountability for individual nursing judgments and actions. The nurse maintains competence in nursing. The nurse exercises informed judgment and uses individual competence and qualifications as criteria in seeking consultation, accepting responsibilities, and delegating nursing activities to others. The nurse participates in activities that contribute to the ongoing development of the profession’s body of knowledge. The nurse participates in the profession’s efforts to implement and improve standards of nursing. The nurse participates in the profession’s efforts to establish and maintain conditions of employment conducive to high-quality nursing care. The nurse participates in the profession’s effort to protect the public from misinformation and misrepresentation and to maintain the integrity of nursing. The nurse collaborates with members of the health professions and other citizens in promoting community and national efforts to meet the health needs of the public. These codes take as their guiding principle the concepts of service to humankind and respect for human life. When reading these codes of conduct, it is difficult to imagine that anyone could improve on them as summary statements of the primary goals of individuals responsible for the care of patients. However, some believe that such codes fail to provide answers to many of the difficult moral dilemmas confronting health professionals today. For example, in many situations, all the fundamental responsibilities of the nurse cannot be met at the same time. When a patient suffering from a massive insult to the brain is kept alive by artificial means and this equipment is needed elsewhere, it is not clear from these guidelines how “nursing competence is to be maintained to conserve life and promote health.” Although it may be argued that the decision to treat or not to treat is a medical and not a nursing decision, both professions are so intimately involved in the care of patients that they are both concerned with the ultimate implications of any such decision. For biomedical engineers, an increased awareness of the ethical significance of their professional activities has also resulted in the development of codes of professional ethics. Typically consisting of a short list of general rules, these codes express both the minimal standards to which all members of a profession are expected to conform and the ideals for which all members are expected to strive. Such codes provide a practical guide for the ethical conduct of the profession’s practitioners. Consider, for example, the code of ethics endorsed by the American College of Clinical Engineers:

2. MORAL AND ETHICAL ISSUES

As a member of the American College of Clinical Engineering, I subscribe to the established Code of Ethics in that I will: • Accurately represent my level of responsibility, authority, experience, knowledge, and education. • Strive to prevent a person from being placed at risk due to the use of technology. • Reveal conflicts of interest that may affect information provided or received. • Respect the confidentiality of information. • Work toward improving the delivery of health care. • Work toward the containment of costs by the better management and utilization of technology. • Promote the profession of clinical engineering. Although these codes can be useful in promoting ethical conduct, such rules obviously cannot provide ethical guidance in every situation. A profession that aims to maximize the ethical conduct of its members must not limit the ethical consciousness of its members to knowledge of their professional code alone. It must also provide them with resources that will enable them to determine what the code requires in a particular concrete situation and thereby enable them to arrive at ethically sound judgments in situations in which the directives of the code are ambiguous or simply do not apply.

2.2 TWO MORAL NORMS: BENEFICENCE AND NONMALEFICENCE Two moral norms have remained relatively constant across the various moral codes and oaths that have been formulated for health care providers since the beginnings of Western medicine in classical Greek civilization. They are beneficence, which is the provision of benefits, and nonmaleficence, which is the avoidance of doing harm. These norms are traced back to a body of writings from classical antiquity known as the Hippocratic Corpus. Although these writings are associated with the name of Hippocrates, the acknowledged founder of Western medicine, medical historians remain uncertain whether any of them, including the Hippocratic Oath, were actually his work. Although portions of the Corpus are believed to have been authored during the sixth century BC, other portions are believed to have been written as late as the beginning of the Christian era. Medical historians agree that many of the specific moral directives of the Corpus represent neither the actual practices nor the moral ideals of the majority of physicians of Ancient Greece and Rome. Nonetheless, the general injunction “As to disease, make a habit of two things: (1) to help or, (2) at least, to do no harm” was accepted as a fundamental medical ethical norm by at least some ancient physicians. With the decline of Hellenistic civilization and the rise of Christianity, beneficence and nonmaleficence became increasingly accepted as the fundamental principles of morally sound medical practice. Although beneficence and nonmaleficence were regarded merely as concomitant to the craft of medicine in classical Greece and Rome, the emphasis upon compassion and the brotherhood of humankind, central to Christianity, increasingly made these norms the only acceptable motives for medical practice. Even today, the provision of benefits and the avoidance of doing harm are stressed just as much in virtually all contemporary Western codes of conduct for health professionals as they were in the oaths and codes that guided the health-care providers of past centuries.

2.3 REDEFINING DEATH

Traditionally, the ethics of medical care has given greater prominence to nonmaleficence than to beneficence. This priority was grounded in the fact that, historically, medicine’s capacity to do harm far exceeded its capacity to protect and restore health. Providers of health care possessed many treatments that posed clear and genuine risks to patients and that offered little prospect of benefit. Truly effective therapies were all too rare. In this context, it is surely rational to give substantially higher priority to avoiding harm than to providing benefits. The advent of modern science changed matters dramatically. Knowledge acquired in laboratories, tested in clinics, and verified by statistical methods has increasingly dictated the practice of medicine. This ongoing alliance between medicine and science became a critical source of the plethora of technologies that now pervade medical care. The impressive increases in therapeutic, preventive, and rehabilitative capabilities that these technologies have provided have pushed beneficence to the forefront of medical morality. Some have even gone so far as to hold that the old medical ethic of “Above all, do no harm” should be superseded by the new ethic “The patient deserves the best.” However, the rapid advances in medical technology capabilities have also produced great uncertainty as to what is most beneficial or least harmful for the patient. In other words, along with increases in ability to be beneficent, medicine’s technology has generated much debate about what actually counts as beneficent or nonmaleficent treatment. Having reviewed some of the fundamental concepts of ethics and morality, let us now turn to several specific moral issues posed by the use of medical technology.

2.3 REDEFINING DEATH Although medicine has long been involved in the observation and certification of death, many of its practitioners have not always expressed philosophical concerns regarding the beginning of life and the onset of death. Since medicine is a clinical and empirical science, it would seem that health professionals had no medical need to consider the concept of death: the fact of death was sufficient. The distinction between life and death was viewed as the comparison of two extreme conditions separated by an infinite chasm. With the advent of technological advances in medicine to assist health professionals to prolong life, this view has changed. There is no doubt that the use of medical technology has in many instances warded off the coming of the grim reaper. One need only look at the trends in average life expectancy for confirmation. For example, in the United States today, the average life expectancy for males is 74.3 years and for females 76 years, whereas in 1900 the average life expectancy for both sexes was only 47 years. Infant mortality has been significantly reduced in developed nations where technology is an integral part of the culture. Premature births no longer constitute a threat to life because of the artificial environment that medical technology can provide. Today, technology has not only helped individuals avoid early death but has also been effective in delaying the inevitable. Pacemakers, artificial kidneys, and a variety of other medical devices have enabled individuals to add many more productive years to their lives. Technology has been so successful that health professionals responsible for the care of critically ill patients have been able to maintain their “vital signs of life” for extensive periods of time. In the process, however, serious philosophical questions concerning the quality of the life provided these patients have arisen.

2. MORAL AND ETHICAL ISSUES

Consider the case of the patient who sustains a serious head injury in an automobile accident. To the attendants in the ambulance who reached the scene of the accident, the patient was unconscious but still alive with a beating heart. After the victim was rushed to the hospital and into the emergency room, the resident in charge verified the stability of the vital signs of heartbeat and respiration during examination and ordered a computerized tomography (CT) scan to indicate the extent of the head injury. The results of this procedure clearly showed extensive brain damage. When the EEG was obtained from the scalp electrodes placed about the head, it was noted to be significantly abnormal. In this situation, then, the obvious questions arise: What is the status of the patient? Is the patient alive? Alternatively, consider the events encountered during one open-heart surgery. During this procedure, the patient was placed on the heart bypass machine while the surgeon attempted to correct a malfunctioning valve. As the complex and long operation continued, the EEG monitors that had indicated a normal pattern of electrical activity at the onset of the operation suddenly displayed a relatively straight line indicative of feeble electrical activity. However, since the heart-lung bypass was maintaining the patient’s so-called vital signs, what should the surgeon do? Should the medical staff continue on the basis that the patient is alive, or is the patient dead? The increasing occurrence of these situations has stimulated health professionals to reexamine the definition of “death.” In essence, advances in medical technology that delay death actually hastened its redefinition. This should not be so surprising because the definition of death has always been closely related to the extent of medical knowledge and available technology. For many centuries, death was defined solely as the absence of breathing. Since it was felt that the spirit of the human being resided in the spiritus (breath), its absence became indicative of death. With the continuing proliferation of scientific information regarding human physiology and the development of techniques to revive a nonbreathing person, attention turned to the pulsating heart as the focal point in determination of death. However, this view was to change through additional medical and technological advances in supportive therapy, resuscitation, cardiovascular assist devices, and organ transplantation. As understanding of the human organism increased, it became obvious that one of the primary constituents of the blood is oxygen and that any organ deprived of oxygen for a specified period of time will cease to function and die. The higher functions of the brain are particularly vulnerable to this type of insult, since the removal of oxygen from the blood supply even for a short period of time (three minutes) produces irreversible damage to the brain tissues. Consequently, the evidence of “death” began to shift from the pulsating heart to the vital, functioning brain. Once medicine was provided with the means to monitor the brain’s activity (i.e., the EEG), another factor was introduced in the definition of death. Advocates of the concept of brain death argued that the human brain is truly essential to life. When the brain is irreversibly damaged, so are the functions that are identified with self and our own humanness: memory, feeling, thinking, knowledge, and so on. As a result, it became widely accepted that the meaning of clinical death implies that the spontaneous activity of the lungs, heart, and brain is no longer present. The irreversible cessation of functioning of all three major organs—the heart, lungs, and brain—was required before anyone was pronounced dead. Although damage to any other organ system such as the liver or kidney may ultimately cause the death of the individual through a fatal effect on the essential functions of the heart, lungs, or brain, this aspect was not included in the definition of clinical death.

2.3 REDEFINING DEATH

With the development of modern respirators, however, the medical profession encountered an increasing number of situations in which a patient with irreversible brain damage could be maintained almost indefinitely. Once again, a new technological advance created the need to reexamine the definition of death. The movement toward redefining death received considerable impetus with the publication of a report sponsored by the Ad Hoc Committee of the Harvard Medical School in 1968, in which the committee offered an alternative definition of death based on the functioning of the brain. The report of this committee was considered a landmark attempt to deal with death in light of technology. In summary, the criteria for death established by this committee included the following: (1) the patient must be unreceptive and unresponsive—that is, in a state of irreversible coma; (2) the patient must have no movements of breathing when the mechanical respirator is turned off; (3) the patient must not demonstrate any reflexes; and (4) the patient must have a flat EEG for at least 24 hours, indicating no electrical brain activity. When these criteria are satisfied, then death may be declared. At the time, the committee also strongly recommended that the decision to declare the person dead and then to turn off the respirator should not be made by physicians involved in any later efforts to transplant organs or tissues from the deceased individual. In this way, a prospective donor’s death would not be hastened merely for the purpose of transplantation. Thus, complete separation of authority and responsibility for the care of the recipient from the physician or group of physicians who are responsible for the care of the prospective donor is essential. The shift to a brain-oriented concept involved deciding that much more than just biological life is necessary to be a human person. The brain death concept was essentially a statement that mere vegetative human life is not personal human life. In other words, an otherwise intact and alive but brain-dead person is not a human person. Many of us have taken for granted the assertion that being truly alive in this world requires an “intact functioning brain.” Yet, precisely this issue was at stake in the gradual movement from using heartbeat and respiration as indices of life to using brain-oriented indices instead. Indeed, total and irreparable loss of brain function, referred to as “brainstem death,” “whole brain death,” or, simply, “brain death,” has been widely accepted as the legal standard for death. By this standard, an individual in a state of brain death is legally indistinguishable from a corpse and may be legally treated as one even though respiratory and circulatory functions may be sustained through the intervention of technology. Many take this legal standard to be the morally appropriate one, noting that once destruction of the brainstem has occurred, the brain cannot function at all, and the body’s regulatory mechanisms will fail unless artificially sustained. Thus mechanical sustenance of an individual in a state of brain death is merely postponement of the inevitable and sustains nothing of the personality, character, or consciousness of the individual. It is simply the mechanical intervention that differentiates such an individual from a corpse, and a mechanically ventilated corpse is a corpse nonetheless. Even with a consensus that brainstem death is death, and thus that an individual in such a state is indeed a corpse, difficult cases remain. Consider the case of an individual in a persistent vegetative state, the condition known as “neocortical death.” Although severe brain injury has been suffered, enough brain function remains to make mechanical sustenance of respiration and circulation unnecessary. In a persistent vegetative state, an individual exhibits no purposeful response to external stimuli and no evidence of self-awareness. The eyes

2. MORAL AND ETHICAL ISSUES

may open periodically, and the individual may exhibit sleep-wake cycles. Some patients even yawn, make chewing motions, or swallow spontaneously. Unlike the complete unresponsiveness of individuals in a state of brainstem death, a variety of simple and complex responses can be elicited from an individual in a persistent vegetative state. Nonetheless, the chances that such an individual will regain consciousness are remote. Artificial feeding, kidney dialysis, and the like make it possible to sustain an individual in a state of neocortical death for decades. If brainstem death is death, is neocortical death also death? Again, the issue is not a straightforward factual matter. For it, too, is a matter of specifying which features of living individuals distinguish them from corpses and so make treatment of them as corpses morally impermissible. Irreparable cessation of respiration and circulation, the classical criterion for death, would entail that an individual in a persistent vegetative state is not a corpse and so, morally speaking, must not be treated as one. The brainstem death criterion for death would also entail that a person in a state of neocortical death is not yet a corpse. On this criterion, what is crucial is that brain damage be severe enough to cause failure of the regulatory mechanisms of the body. Is an individual in a state of neocortical death any less in possession of the characteristics that distinguish the living from cadavers than one whose respiration and circulation are mechanically maintained? It is a matter that society must decide. And until society decides, it is not clear what counts as beneficent or nonmaleficent treatment of an individual in a state of neocortical death.

CASE STUDY: TERRI SCHIAVO AND THE BRAIN DEATH DEBATE In February 1990, an otherwise healthy 27-year-old Terri Schiavo suffered heart failure in her home and fell into a coma. While Schiavo ultimately woke and initially proved responsive, after a year of multiple rehabilitation facilities and nursing homes, the by then 28-year-old was diagnosed as in an irreversible persistent vegetative state (PVS). In 1998, Schiavo’s husband, Michael Schiavo, made a petition to the Florida courts to remove his wife from life support, a petition fought vehemently by the woman’s parents. In 2001, after a doctor confirmed brain death with a report of significant brainstem damage and 80 percent loss of upper brain function, Schiavo’s feeding tube was removed, but was replaced days later, following a Court Appeal by her parents. Ultimately, the feeding tube was ordered to be removed on three separate occasions, each time her legal guardian and husband fighting to allow his wife to “die in peace,” while her parents insisted that their daughter maintained cognitive function and requested more tests. Finally in 2005, 15 years after her injury, and under constant national media coverage, Schiavo died from dehydration two weeks after her tube had been removed for the final time and while her case was still pending with the highest word in the nation: the Supreme Court. 1. Without a Living Will, who is responsible for deciding the would-be intentions of a victim of brain death? Who is responsible for mediation when loved ones disagree? 2. Who is responsible for the years of health care costs of a potentially brain dead individual? In 2006, Rom Houben, a man presumed brain dead for 23 years, was discovered to have full brain function after a series of advanced brain scan imaging tests. Houben was paralyzed in an accident

2.4 THE TERMINALLY ILL PATIENT AND EUTHANASIA

in 1983 at the age of 20. A formal martial arts enthusiast, Houben was assumed to be in a PVS for over two decades. After therapy he is now able to communicate via typing, and he reads books while lying down, using an assistive device. “I want to read,” says Houben, via keyboard, “talk with my friends via the computer, and enjoy my life now that people know I am not dead.” 3. With diagnosis technologies constantly in development, should PVS victims ever be “allowed to die”?

2.4 THE TERMINALLY ILL PATIENT AND EUTHANASIA Terminally ill patients today often find themselves in a strange world of institutions and technology devoted to assisting them in their fight against death. However, at the same time, this modern technologically oriented medical system may cause patients and their families considerable economic, psychological, and physical pain. In enabling medical science to prolong life, modern technology has in many cases made dying slower and more undignified. As a result of this situation, there is a moral dilemma in medicine. Is it right or wrong for medical professionals to stop treatment or administer a lethal dose to terminally ill patients? This problem has become a major issue for our society to consider. Although death is all around us in the form of accidents, drug overdoses, alcoholism, murders, and suicides, for most of us, the end lies in growing older and succumbing to some form of chronic illness. As the aged approach the end of life’s journey, they may eventually wish for the day when all troubles can be brought to an end. Such a desire, frequently shared by a compassionate family, is often shattered by therapies provided with only one concern: to prolong life regardless of the situation. As a result, many claim a dignified death is often not compatible with today’s standard medical view. Consider the following hypothetical version of the kind of case that often confronts contemporary patients, their families, health care workers, and society as a whole. Suppose a middle-aged man suffers a brain hemorrhage and loses consciousness as a result of a ruptured aneurysm. Suppose that he never regains consciousness and is hospitalized in a state of neocortical death, a chronic vegetative state. His life is maintained by a surgically implanted gastronomy tube that drips liquid nourishment from a plastic bag directly into his stomach. The care of this individual takes seven and one-half hours of nursing time daily and includes shaving, oral hygiene, grooming, attending to his bowels and bladder, and so forth. Suppose further that his wife undertakes legal action to force his caregivers to end all medical treatment, including nutrition and hydration so complete bodily death of her husband will occur. She presents a preponderance of evidence to the court to show that her husband would have wanted just this result in these circumstances. The central moral issue raised by this sort of case is whether the quality of the individual’s life is sufficiently compromised to make intentional termination of that life morally permissible. While alive, he made it clear to both family and friends that he would prefer to be allowed to die rather than be mechanically maintained in a condition of irretrievable loss of consciousness. Deciding whether his judgment in such a case should be allowed requires deciding which capacities and qualities make life worth living, which qualities are sufficient to endow it with value worth sustaining, and whether their absence justifies deliberate termination of a life, at least when this would be the wish of the individual in question. Without this decision, the traditional norms of medical ethics, beneficence and

2. MORAL AND ETHICAL ISSUES

nonmaleficence, provide no guidance. Without this decision, it cannot be determined whether termination of life support is a benefit or harm to the patient. For many individuals, the fight for life is a correct professional view. They believe that the forces of medicine should always be committed to using innovative ways of prolonging life for the individual. However, this cannot be the only approach to caring for the terminally ill. Certain moral questions regarding the extent to which physicians engaged in heroic efforts to prolong life must be addressed if the individual’s rights are to be preserved. The goal of those responsible for patient care should not solely be to prolong life as long as possible by the extensive use of drugs, operations, respirators, hemodialyzers, pacemakers, and the like, but rather to provide a reasonable “quality of life” for each patient. It is out of this new concern that euthanasia has once again become a controversial issue in the practice of medicine. The term euthanasia is derived from two Greek words meaning “good” and “death.” Euthanasia was practiced in many primitive societies in varying degrees. For example, on the island of Cos, the ancient Greeks assembled elderly and sick people at an annual banquet to consume a poisonous potion. Even Aristotle advocated euthanasia for gravely deformed children. Other cultures acted in a similar manner toward their aged by abandoning them when they felt these individuals no longer served any useful purpose. However, with the spread of Christianity in the Western world, a new attitude developed toward euthanasia. Because of the Judeo-Christian belief in the biblical statements “Thou shalt not kill” (Exodus 20: 13) and “He who kills a man should be put to death” (Leviticus 24: 17), the practice of euthanasia decreased. As a result of these moral judgments, killing was considered a sin, and the decision about whether someone should live or die was viewed solely as God’s responsibility, not humans’. In today’s society, euthanasia implies to many “death with dignity,” a practice to be followed when life is merely being prolonged by machines and no longer seems to have value. In many instances, it has come to mean a contract for the termination of life in order to avoid unnecessary suffering at the end of a fatal illness and, therefore, has the connotation of relief from pain. Discussions of the morality of euthanasia often distinguish active from passive euthanasia, a distinction that rests upon the difference between an act of commission and an act of omission. When failure to take steps that could effectively forestall death results in an individual’s demise, the resultant death is an act of omission and a case of letting a person die. When a death is the result of doing something to hasten the end of a person’s life (for example, giving a lethal injection), that death is caused by an act of commission and is a case of killing a person. The important difference between active and passive euthanasia is that in passive euthanasia, the physician does not do anything to bring about the patient’s death. The physician does nothing, and death results due to whatever illness already afflicts the patient. In active euthanasia, however, the physician does something to bring about the patient’s death. The physician who gives the patient with cancer a lethal injection has caused the patient’s death, whereas if the physician merely ceases treatment, the cancer is the cause of death. In active euthanasia, someone must do something to bring about the patient’s death, and in passive euthanasia, the patient’s death is caused by illness rather than by anyone’s conduct. Is this notion correct? Suppose a physician deliberately decides not to treat a patient who is terminally ill, and the patient dies. Suppose further that the physician were to attempt to exonerate himself by saying, “I did nothing. The patient’s death was the result

2.4 THE TERMINALLY ILL PATIENT AND EUTHANASIA

of illness. I was not the cause of death.” Under current legal and moral norms, such a response would have no credibility. The physician would be blameworthy for the patient’s death as surely as if he or she had actively killed the patient. Thus, the actions taken by a physician to continue treatment to the very end are understood. Euthanasia may also be classified as involuntary or voluntary. An act of euthanasia is involuntary if it hastens the individual’s death for his or her own good, but against their wishes. Involuntary euthanasia, therefore, is no different in any morally relevant way from unjustifiable homicide. However, what happens when the individual is incapable of agreeing or disagreeing? Suppose that a terminally ill person is unconscious and cannot make his or her wishes known. Would hastening their death be permissible? It would be if there was substantial evidence that the individual has given prior consent. The individual may have told friends and relatives that, under certain circumstances, efforts to prolong their life should not be undertaken or continued and might even have recorded their wishes in the form of a living will or an audio- or videotape. When this level of substantial evidence of prior consent exists, the decision to hasten death would be morally justified. A case of this sort would be a case of voluntary euthanasia. For a living will to be valid, the person signing it must be of sound mind at the time the will is made and shown not to have altered their opinion in the interim between its signing and the onset of the illness. In addition, the witnesses must not be able to benefit from the individual’s death. As the living will itself states, it is not a legally binding document. It is essentially a passive request and depends on moral persuasion. Proponents of the will, however, believe that it is valuable in relieving the burden of guilt often carried by health professionals and the family in making the decision to allow a patient to die. Those who favor euthanasia point out the importance of individual rights and freedom of choice and look on euthanasia as a kindness ending the misery endured by the patient. The thought of a dignified death is much more attractive than the process of continuous suffering and gradual decay into nothingness. Viewing each person as a rational being possessing a unique mind and personality, proponents argue that terminally ill patients should have the right to control the ending of their own life. On the other hand, those opposed to euthanasia demand to know who has the right to end the life of another. Some use religious arguments, emphasizing that euthanasia is in direct conflict with the belief that only God has the power to decide when a human life ends. Their view is that anyone who practices euthanasia is essentially acting in the place of God and that no human should ever be considered omnipotent. Others turn to the established codes, reminding those responsible for the care of patients that they must do whatever is in their power to save a life. Their argument is that health professionals cannot honor their pledge and still believe that euthanasia is justified. If terminally ill patients are kept alive, there is at least a chance of finding a cure that might be useful to them. Opponents of euthanasia feel that legalizing it would destroy the bonds of trust between doctor and patient. How would sick individuals feel if they could not be sure that their physician and nurse would try everything possible to cure them, knowing that if their condition worsened, they would just lose faith and decide that death would be better? Opponents of euthanasia also question whether it will be truly beneficial to the suffering person or will only be a means to relieve the agony of the family. They believe that destroying life (no matter how minimal) merely to ease the emotional suffering of others is indeed unjust.

2. MORAL AND ETHICAL ISSUES

Many fear that if euthanasia is legalized, it will be difficult to define and develop clearcut guidelines that will serve as the basis for euthanasia to be carried out. Furthermore, once any form of euthanasia is accepted by society, its detractors fear that many other problems will arise. Even the acceptance of passive euthanasia could, if carried to its logical conclusion, be applied in state hospitals and institutions for the mentally handicapped and the elderly. Such places currently house thousands of people who have neither hope nor any prospect of a life that even approaches normality. Legalization of passive euthanasia could prompt an increased number of suits by parents seeking to end the agony of incurably afflicted children or by children seeking to shorten the suffering of aged and terminally ill parents. In Nazi Germany, for example, mercy killing was initially practiced to end the suffering of the terminally ill. Eventually, however, the practice spread so even persons with the slightest deviation from the norm (e.g., the mentally ill, minority groups such as Jews and others) were terminated. Clearly, the situation is delicate and thought provoking.

2.5 TAKING CONTROL Medical care decisions can be tremendously difficult. They often involve unpleasant topics and arise when we are emotionally and physically most vulnerable. Almost always these choices involve new medical information that feels alien and can seem overwhelming. In an attempt to assist individuals to make these decisions, it is often helpful to follow these steps: 1. Obtain all the facts—that is, clarify the medical facts of the situation. 2. Understand all options and their consequences. 3. Place a value on each of the options based upon your own set of personal values.

A LIVING WILL TO MY FAMILY, MY PHYSICIAN, MY CLERGYMAN, MY LAWYER: If the time comes when I can no longer take part in decisions about my own future, let this statement stand as testament of my wishes: If there is no reasonable expectation of my recovery from physical or mental disability, I request that I be allowed to die and not be kept alive by artificial means or heroic measures. Death is as much a reality as birth, growth, maturity, and old age—it is the one certainty. I do not fear death as much as I fear the indignity of deterioration, dependence, and hopeless pain. I ask that drugs be mercifully administered to me for the terminal suffering even if they hasten the moment of death. This request is made after careful consideration. Although this document is not legally binding, you who care for me will, I hope, feel morally bound to follow its mandate. I recognize that it places a heavy burden of responsibility upon you, and it is with the intention of sharing that responsibility and of mitigating any feelings of guilt that this statement is made. Signed ______________________________________________________ Date ______________________________________________________ Witnessed by ________________________________________________

2.6 HUMAN EXPERIMENTATION

The three-step facts/options/values path concerns the “how” of decisions, but equally important is the “who.” Someone must make every single medical decision. Ideally it will be made by the person most intimately involved: the patient. Very often, however, it is made by someone else—spouse, family, physician—or a group of those people acting on behalf of the patient. It is, therefore, important to recognize the four concentric circles of consent: • The first, and primary, circle is the patient. • The second circle is the use of advance directives—that is, choosing in advance through the use of such documents as the living will. • The third circle is others deciding for the patient—that is, the move from personal control to surrogate control. • The fourth and final circle is the courts and bureaucrats. It is the arena of last resort where our society has decreed that we go when the patient may be incapacitated, where there is no clear advance directive, and where it is not clear who should make the decision. These three steps and four circles are simply attempts to impose some order on the chaos that is medical decision making. They can help individuals take control.

2.6 HUMAN EXPERIMENTATION Medical research has long held an exalted position in our modern society. It has been acclaimed for its significant achievements that range from the development of the Salk and Sabin vaccines for polio to the development of artificial organs. In order to determine their effectiveness and value, however, these new drugs and medical devices eventually are used on humans. The issue is, therefore, not only whether humans should be involved in clinical studies designed to benefit themselves or their fellow humans but also clarifying or defining more precisely the conditions under which such studies are to be permitted. For example, consider the case of a 50-year-old female patient suffering from severe coronary artery disease. What guidelines should be followed in the process of experimenting with new drugs or devices that may or may not help her? Should only those procedures viewed as potentially beneficial to her be tried? Should experimental diagnostic procedures or equipment be tested on this patient to evaluate their effectiveness when compared to more accepted techniques, even though they will not be directly beneficial to the patient? On the other hand, consider the situation of conducting research on the human fetus. This type of research is possible as a result of the legalization of abortion in the United States, as well as the technological advances that have made fetal studies more practical than in the past. Under what conditions should medical research be conducted on these subjects? Should potentially hazardous drugs be given to women planning to have abortions to determine the effect of these drugs on the fetus? Should the fetus, once aborted, be used in any experimental studies? Although these questions are difficult to answer, clinical researchers responsible for the well-being of their patients must face the moral issues involved in testing new equipment and procedures and at the same time safeguard the individual rights of their patients.

2. MORAL AND ETHICAL ISSUES

CASE STUDY: NEONATAL INTENSIVE CARE UNIT (NICU) Throughout time, low birth weight, oftentimes arising from premature birth, has been a major factor affecting infant survival. Underweight infants, who are typically classified as either low birth weight (LBW) (less than 1,500 g) or very low birth weight (VLBW) (less than 1,000 g), must be treated with the utmost caution and care in order to maximize their chances of survival. Advances in premature-infant medical care, such as improved thermoregulation and ventilation techniques, have greatly decreased the mortality rate among LBW and VLBW infants. Included in these advances was the creation of the NICU (Figure 2.2), where all the necessary equipment needed to sustain the life of the child could be kept conveniently in close proximity to one another. One of the most important devices used in the NICU is the incubator. This device, typically molded of see-through plastic, is used to stabilize the body temperature of the infant. In essence, the incubator allows the medical staff to keep the newborn warm without having to wrap it in

FIGURE 2.2 A Neonatal Intensive Care Unit. Courtesy of http://www.pediatrics.ucsd.edu/Divisions/Neonatology/ Pictures/Image%20Library/NICU%20Bed.jpg.

2.7 DEFINITION AND PURPOSE OF EXPERIMENTATION

blankets. The incubator also aids in preventing infection, as well as in stabilizing the humidity of the child’s environment. By keeping the temperature and humidity levels of the newborn’s environment static, the baby remains well hydrated and water loss is kept to a minimum. A complication that many preterm infants suffer from is the inability to breathe normally on their own. The child may be completely unable to breathe for himself, or he may suffer from a condition known as apnea, where the breathing pattern is either aperiodic or irregular. In these cases, children susceptible to an apneic event are closely monitored so if they stop breathing, nurses can rush to the bedside and wake them up. However, it is often minutes before the nurse can arrive at the scene. To facilitate the process of waking the infant experiencing an apneic event, biomedical engineers developed a tactile vibrator that when triggered by such an event vibrates against the infant’s foot and wakes her. In order to prove that the device is effective and safe, a human experiment must be initiated. In this case, the following questions need to be resolved: 1. 2. 3. 4.

Who is responsible for proposing the conduction of this study? What should the process of approval of such a study include? What should the policy be related to informed consent? Should changes that were made in the device during the course of the study, which would alter the nature of the initially proposed device, be allowed?

2.7 DEFINITION AND PURPOSE OF EXPERIMENTATION One may ask, what exactly constitutes a human experiment? Although experimental protocols may vary, it is generally accepted that human experimentation occurs whenever the clinical situation of the individual is consciously manipulated to gather information regarding the capability of drugs and devices. In the past, experiments involving human subjects have been classified as either therapeutic or nontherapeutic. A therapeutic experiment is one that may have direct benefit for the patient, while the goal of nontherapeutic research is to provide additional knowledge without direct benefit to the person. The central difference is a matter of intent or aim rather than results. Throughout medical history, there have been numerous examples of therapeutic research projects. The use of nonconventional radiotherapy to inhibit the progress of a malignant cancer, of pacemakers to provide the necessary electrical stimulation for proper heart function, or of artificial kidneys to mimic nature’s function and remove poisons from the blood were all, at one time, considered novel approaches that might have some value for the patient. In the process, they were tried and found not only to be beneficial for the individual patient but also for humankind. Nontherapeutic research has been another important vehicle for medical progress. Experiments designed to study the impact of infection from the hepatitis virus or the malarial parasite or the procedures involved in cardiac catheterization have had significant impacts on the advancement of medical science and the ultimate development of appropriate medical procedures for the benefit of all humans. In the mid-1970s, the National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research offered the terms practice and research to replace the

2. MORAL AND ETHICAL ISSUES

conventional therapeutic and nontherapeutic distinction just mentioned. Quoting the commission, Alexander Capron in 1986 wrote the following: The term practice refers to interventions that are designed solely to enhance the well-being of an individual patient or client and that have a reasonable expectation of success. In the medical sphere, practices usually involve diagnosis, preventive treatment, or therapy; in the social sphere, practices include governmental programs such as transfer payments, education, and the like. By contrast, the term research designates an activity designed to test a hypothesis, to permit conclusions to be drawn, and thereby to develop or contribute to generalizable knowledge (expressed, for example, in theories, principles, or statements of relationships). In the polar cases, then, practice uses a proven technique in an attempt to benefit one or more individuals, while research studies a technique in an attempt to increase knowledge.

Although the practice/research dichotomy has the advantage of not implying that therapeutic activities are the only clinical procedures intended to benefit patients, it is also based on intent rather than outcome. Interventions are “practices” when they are proven techniques intended to benefit the patient, while interventions aimed at increasing generalizable knowledge constitute research. What about those interventions that do not fit into either category?

CASE STUDY: THE ARTIFICIAL HEART In the early 1980s, a screening committee had been set up to pick the first candidate for the “Jarvik 7,” a new (at the time) artificial heart (Figure 2.3). It was decided that the first recipient had to be someone so sick that death was imminent. It was thought unethical to pick someone who might have another year to live when the artificial heart might well kill the patient immediately.

FIGURE 2.3 The Jarvik-7 artificial heart, 1985. Courtesy of http://www.smithsonianlegacies.si.edu/objectdescription

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2.8 INFORMED CONSENT

1. Is this an example of nonvalidated practice? 2. Is informed consent still required? A week after the operation, Barney Clark began having seizures from head to toe. Suffering a seizure, Clark’s unconscious body quivered for several hours. The seizures and spells of mental confusion continued throughout the next months. As a result, Clark expressed a desire to die. Although he did issue a positive statement during a videotaped interview, Clark was not a happy man, tethered to a huge machine, barely conscious, and in some pain. In March 1983, Barney Clark died of multiple organ collapse. 3. Discuss in detail the notions of “criteria for success” and quality of life in this case. 4. Barney Clark suffered a great deal. In response to this, who should be the responsible party in deciding what is right for the patient? When both sides hope for positive results, is it possible to make an unbiased decision based on what’s best for the patient?

One such intervention is “nonvalidated practice,” which may encompass prevention as well as diagnosed therapy. The primary purpose of the use of a nonvalidated practice is to benefit the patient while emphasizing that it has not been shown to be safe and efficacious. For humans to be subjected to nonvalidated practice, they must be properly informed and give their consent.

2.8 INFORMED CONSENT Informed consent has long been considered by many to be the most important moral issue in human experimentation. It is the principal condition that must be satisfied in order for human experimentation to be considered both lawful and ethical. All adults have the legal capacity to give medical consent (unless specifically denied through some legal process). As a result, issues concerning legal capability are usually limited to minors. Many states, if not all, have some exceptions that allow minors to give consent. Informed consent is an attempt to preserve the rights of individuals by giving them the opportunity for self-determination—that is, to determine for themselves whether they wish to participate in any experimental effort. In 1964, the World Medical Association (WMA) in Finland endorsed a code of ethics for human experimentation as an attempt to provide some guidelines in this area. In October 2000, the 52nd WMA General Assembly in Edinburgh, Scotland, revised these guidelines. Because it is often essential to use the results obtained in human experiments to further scientific knowledge, the World Medical Association prepared the following recommendations to serve as a guide to physicians all over the world. However, it is important to point out that these guidelines do not relieve physicians, scientists, and engineers from criminal, civil, and ethical responsibilities dictated by the laws of their own countries.

2.8.1 Basic Principles • Biomedical research involving human subjects must conform to generally accepted scientific principles and should be based on adequately performed laboratory and animal experimentation and on a thorough knowledge of the scientific literature.

2. MORAL AND ETHICAL ISSUES

• The design and performance of each experimental procedure involving human subjects should be clearly formulated in an experimental protocol, which should be transmitted to a specially appointed independent committee for consideration, comment, and guidance. • Biomedical research involving human subjects should be conducted only by scientifically qualified persons and under the supervision of a clinically competent medical person. The responsibility for the human subject must always rest with a medically qualified person and never rest on the subject of the research, even though the subject has given his or her consent. • Biomedical research involving human subjects cannot legitimately be carried out unless the importance of the objective is in proportion to the inherent risk to the subject. • Every biomedical research project involving human subjects should be preceded by careful assessment of predictable risks in comparison with foreseeable benefits to the subject or to others. Concern for the interests of the subject must always prevail over the interests of science and society. • The right of the research subject to safeguard his or her integrity must always be respected. Every precaution should be taken to respect the privacy of the subject and to minimize the impact of the study on the subject’s physical and mental integrity and on the personality of the subject. • Doctors should abstain from engaging in research projects involving human subjects unless they are satisfied that the hazards involved are believed to be predictable. Doctors should cease any investigation if the hazards are found to outweigh the potential benefits. • In publication of the results of his or her research, the doctor is obliged to preserve the accuracy of the results. Reports of experimentation not in accordance with the principles laid down in this Declaration should not be accepted for publication. • In any research on human beings, each potential subject must be adequately informed of the aims, methods, anticipated benefits and potential hazards of the study and the discomfort it may entail. He or she should be informed that he or she is at liberty to abstain from participation in the study and that he or she is free to withdraw his or her consent to participation at any time. The doctor should then obtain the subject’s freely-given informed consent, preferably in writing. • When obtaining informed consent for the research project, the doctor should be particularly cautious if the subject is in a dependent relationship to him or her or may consent under duress. In that case, the informed consent should be obtained by a doctor who is not engaged in the investigation and who is completely independent of this official relationship. • In the case of legal incompetence, informed consent should be obtained from the legal guardian in accordance with national legislation. Where physical or mental incapacity makes it impossible to obtain informed consent, or when the subject is a minor, permission from the responsible relative replaces that of the subject in accordance with national legislation. • The research protocol should always contain a statement of the ethical considerations involved and should indicate that the principles enunciated in the present Declaration are complied with.

2.8 INFORMED CONSENT

2.8.2 Medical Research Combined with Professional Care • In the treatment of the sick person, the doctor must be free to use a new diagnostic and therapeutic measure if in his or her judgment it offers hope of saving life, reestablishing health, or alleviating suffering. • The potential benefits, hazards, and discomfort of a new method should be weighed against the advantages of the best current diagnostic and therapeutic methods. • In any medical study, every patient—including those of a control group, if any— should be assured of the best-proven diagnostic and therapeutic method. • The refusal of the patient to participate in a study must never interfere with the doctorpatient relationship. • If the doctor considers it essential not to obtain informed consent, the specific reasons for this proposal should be stated in the experimental protocol for transmission to the independent committee. • The doctor can combine medical research with professional care, the objective being the acquisition of new medical knowledge, only to the extent that medical research is justified by its potential diagnostic or therapeutic value for the patient.

2.8.3 Nontherapeutic Biomedical Research Involving Human Subjects • In the purely scientific application of medical research carried out on a human being, it is the duty of the doctor to remain the protector of the life and health of that person on whom biomedical research is being carried out. • The subjects should be volunteers—that is, either healthy persons or patients for whom the experimental design is not related to the patient’s illness. • The investigator or the investigating team should discontinue the research if in his/her or their judgment it may, if continued, be harmful to the individual. • In research on humans, the interest of science and society should never take precedence over considerations related to the well-being of the subject. These guidelines generally converge on six basic requirements for ethically sound human experimentation. First, research on humans must be based upon prior laboratory research and research on animals, as well as upon established scientific fact, so the point under inquiry is well focused and has been advanced as far as possible by nonhuman means. Second, research on humans should use tests and means of observation that are reasonably believed to be able to provide the information being sought by the research. Methods that are not suited for providing the knowledge sought are pointless and rob the research of its scientific value. Third, research should be conducted only by persons with the relevant scientific expertise. Fourth, all foreseeable risks and reasonably probable benefits, to the subject of the investigation and to science, or more broadly to society, must be carefully assessed, and the comparison of those projected risks and benefits must indicate that the latter clearly outweighs the former. Moreover, the probable benefits must not be obtainable through other less risky means. Fifth, participation in research should be based on informed and voluntary consent. Sixth, participation by a subject in an experiment should be halted immediately if the subject finds continued participation undesirable or a prudent investigator has cause to believe that the experiment is likely to result in

2. MORAL AND ETHICAL ISSUES

injury, disability, or death to the subject. Conforming to conditions of this sort probably does limit the pace and extent of medical progress, but society’s insistence on these conditions is its way of saying that the only medical progress truly worth having must be consistent with a high level of respect for human dignity. Of these conditions, the requirement to obtain informed and voluntary consent from research subjects is widely regarded as one of the most important protections. A strict interpretation of the criteria mentioned above for subjects automatically rules out whole classes of individuals from participating in medical research projects. Children, the mentally retarded, and any patient whose capacity to think is affected by illness are excluded on the grounds of their inability to comprehend exactly what is involved in the experiment. In addition, those individuals having a dependent relationship to the clinical investigator, such as the investigator’s patients and students, would be eliminated based on this constraint. Since mental capacity also includes the ability of subjects to appreciate the seriousness of the consequences of the proposed procedure, this means that even though some minors have the right to give consent for certain types of treatments, they must be able to understand all the risks involved. Any research study must clearly define the risks involved. The patient must receive a total disclosure of all known information. In the past, the evaluation of risk and benefit in many situations belonged to the medical professional alone. Once made, it was assumed that this decision would be accepted at face value by the patient. Today, this assumption is not valid. Although the medical staff must still weigh the risks and benefits involved in any procedure they suggest, it is the patient who has the right to make the final determination. The patient cannot, of course, decide whether the procedure is medically correct, since that requires more medical expertise than the average individual possesses. However, once the procedure is recommended, the patient then must have enough information to decide whether the hoped-for benefits are sufficient to risk the hazards. Only when this is accomplished can a valid consent be given. Once informed and voluntary consent has been obtained and recorded, the following protections are in place: • It represents legal authorization to proceed. The subject cannot later claim assault and battery. • It usually gives legal authorization to use the data obtained for professional or research purposes. Invasion of privacy cannot later be claimed. • It eliminates any claims in the event that the subject fails to benefit from the procedure. • It is defense against any claim of an injury when the risk of the procedure is understood and consented to. • It protects the investigator against any claim of an injury resulting from the subject’s failure to follow safety instructions if the orders were well explained and reasonable.

CASE STUDY: CONFIDENTIALITY, PRIVACY, AND CONSENT Integral to the change currently taking place in the United States health care industry is the application of computer technology to the development of a health care information system. Most major hospitals in the United States have now updated their systems to entirely electronic databases. Patient medications are scheduled and followed by a nurse on a computer module present

2.8 INFORMED CONSENT

MRI scans are just one of the PPI elements available on electronic databases. Courtesy of http://images .medicinenet.com/images/SlideShow/dementia_s21_mri_doctor.jpg.

FIGURE 2.4

in every hospital room. MRI scans (Figure 2.4) are no longer printed in film but are uploaded to the patient’s file, where physicians with proper approval can access the images. Entire medical histories are stored on patient databases. 1. Discuss the benefits of the electronic system and the potential risks. 2. Discuss in detail where and how the issue of consent to access should be handled. While access to the patient information database is limited to accredited physicians and employees, the issue of illegal access from the inside is a prominent one. Hospital employees with access, be they physicians or researchers, may be capable of accessing family accounts or those of friends. With a paper system, protected patient information (PPI) had the potential to be leaked as well via lost files and irresponsible handling. With the electronic system, however, more intentional breach of privacy may be possible to those with access to the system. 3. How can a hospital employee with a medical record on the system be guaranteed privacy from colleagues? 4. Should patients be allowed to decide personally whether their information is stored electronically? Would an integrated system function efficiently?

Nevertheless, can the aims of research ever be reconciled with the traditional moral obligations of physicians? Is the researcher/physician in an untenable position? Informed and voluntary consent once again is the key only if subjects of an experiment agree to participate in the research. What happens to them during and because of the experiment is then a product of their own decision. It is not something that is imposed on them but rather, in a very real sense, something they elected to have done to themselves. Because their autonomy is thus respected, they are not made a mere resource for the benefit of others. Although they may suffer harm for the benefit of others, they do so of their own volition as a result of the exercise of their own autonomy, rather than as a result of having their autonomy limited or diminished. For consent to be genuine, it must be truly voluntary and not the product of coercion. Not all sources of coercion are as obvious and easy to recognize as physical violence.

2. MORAL AND ETHICAL ISSUES

A subject may be coerced by fear that there is no other recourse for treatment, by the fear that nonconsent will alienate the physician on whom the subject depends for treatment, or even by the fear of disapproval of others. This sort of coercion, if it truly ranks as such, is often difficult to detect and, in turn, to remedy. Finally, individuals must understand what they are consenting to do. Therefore, they must be given information sufficient to arrive at an intelligent decision concerning whether to participate in the research. Although a subject need not be given all the information a researcher has, it is important to determine how much should be provided and what can be omitted without compromising the validity of the subject’s consent. Another difficulty lies in knowing whether the subject is competent to understand the information given and to render an intelligent opinion based upon it. In any case, efforts must be made to ensure that sufficient relevant information is given and that the subject is sufficiently competent to process it. These are matters of judgment that probably cannot be made with absolute precision and certainty, but rigorous efforts must be made in good faith to prevent research on humans from involving gross violations of human dignity.

2.9 REGULATION OF MEDICAL DEVICE INNOVATION The Food and Drug Administration (FDA) is the sole federal agency charged by Congress with regulating medical devices to ensure their safety and effectiveness. Unlike food and drugs, which have been regulated by the FDA since 1906, medical devices first became subject to FDA regulation in 1938. At that time, the FDA’s major concern was to ensure that legitimate medical devices were in the marketplace and were truthfully labeled, not misbranded. Over time, the scope of FDA review of medical devices has evolved, as has the technology employed by medical devices). The first substantive legislative attempt to address the premarket review of all medical devices occurred with the Medical Device Amendment of 1976 (Pub. L. No. 94-295, 90 Stat. 539). This statute requires approval from the FDA before new devices are marketed and imposes requirements for the clinical investigation of new medical devices on human subjects. For details related to the FDA process, visit http://www.fda.gov/. The FDA is organized into five major program centers: the Center for Biologics Evaluation and Research, the Center for Drug Evaluation and Research, the Center for Food Safety and Applied Nutrition, the Center for Veterinary Medicine, and the Center for Devices and Radiological Health (CDRH). Each FDA program center has primary jurisdiction over a different subject area. According to the FDA, the CDRH is responsible for ensuring the safety and effectiveness of medical devices and eliminating unnecessary human exposure to man-made radiation from medical, occupational, and consumer products. The CDRH has six distinct offices: the Office of Systems and Management, the Office of Compliance, the Office of Science and Technology, the Office of Health and Industry Programs, the Office of Surveillance and Biometrics, and the Office of Device Evaluation (ODE). The ODE has several principal functions, including the following: • Advising the CDRH director on all premarket notification 510(k) submissions, premarket approvals (PMAs), device classifications, and investigational device exemptions (IDEs).

2.9 REGULATION OF MEDICAL DEVICE INNOVATION

• Planning, conducting, and coordinating CDRH actions regarding approval, denial, and withdrawals of 510(k)s, PMAs, and IDEs. • Ongoing review, surveillance, and medical evaluation of the labeling, clinical experience, and required reports submitted by sponsors of approval applications. • Developing and interpreting regulations and guidelines regarding the classification of devices, 510(k)’s, PMAs, and IDEs. • Participating in the development of national and international consensus standards. Everyone who develops or markets a medical device will likely have multiple interactions with ODE before, during, and after the development of a medical device. In principle, if a manufacturer makes medical claims about a product, it is considered a device, and may be subject to FDA pre- and postmarket regulatory controls (Figure 2.5). The device definition distinguishes a medical device from other FDA-regulated products, such as drugs. According to the FDA, a medical device is: An instrument, apparatus, machine, contrivance, implant, in vitro reagent, or other similar or related article intended for use in the diagnosis of disease or other conditions, or in the cure, mitigation, treatment, or prevention of disease in man or other animals OR intended to affect the structure or any function of the body of man or other animals, and which does not achieve any of its primary intended purposes through chemical action or is not dependent upon being metabolized.

Needs & Intended Uses

Design Input Process

Review Requirements

Initial Design Stage

Stage 1 Design Output

Verification

Validation FIGURE 2.5

Possible Interim Reviews

...

Nth Design Stage

Final Design Output

Production

Test Articles

The purpose of the regulatory process is to conduct product review to ensure (1) device safety and effectiveness, (2) quality of design, and (3) surveillance to monitor device quality. Therefore, the review process results in verification and validation of the medical device.

2. MORAL AND ETHICAL ISSUES

2.10 MARKETING MEDICAL DEVICES The four principal routes to marketing a medical device in the United States are as follows. Premarket Approval (PMA) A marketing approach for high-risk (Class III) medical devices must be accomplished through a PMA unless the device can be marketed through the 510(k) process (see following). The PMA hinges on the FDA determining that the medical device is safe and effective. The PMA process can be quite costly. The collection of the data required for a PMA may costs hundreds of thousands, if not several million, dollars. Moreover, the timeline for a PMA applicant to collect the requisite data could take several years. However, an approved PMA is akin to a private license granted to the applicant to market a particular medical device, because other firms seeking to market the same type of device for the same use must also have an approved PMA. Investigational Device Exemption (IDE) The IDE is an approved regulatory mechanism that permits manufacturers to receive an exemption for those devices solely intended for investigational use on human subjects (clinical evaluation). Because an IDE is specifically for clinical testing and not commercial distribution, the FDCA authorizes the FDA to exempt these devices from certain requirements that apply to devices in commercial distribution. The clinical evaluation of all devices may not be cleared for marketing, unless otherwise exempt by resolution, requires an IDE. An IDE may be obtained either by an institutional review board (IRB), or an IRB and the FDA. Product Development Protocol (PDP) An alternative to the IDE and PMA processes for Class III devices subject to premarket approval, the PDP is a mechanism allowing a sponsor to come to early agreement with the FDA as to what steps are necessary to demonstrate the safety and effectiveness of a new device. In the years immediately subsequent to the enactment of the Medical Device Amendment, the FDA did not focus its energies on the PDP but worked to effectively implement the major provisions of the Amendment, including device classification systems, and the 510(k) and PMA processes. 510(k) Notification Unless specifically exempted by federal regulation, all manufacturers are required to give the FDA 90 days’ notice before they intend to introduce a device to the U.S. market by submitting a 510(k). During that 90-day period, the FDA is charged with determining whether the device is or is not substantially equivalent to a pre-Amendment device. The premarket notification is referred to in the industry as a 510(k) because 510(k) is the relevant section number of the FDCA. The 510(k) is used to demonstrate that the medical device is or is not substantially equivalent to a legally marketed device. With respect to clinical research on humans, the FDA distinguishes devices into two categories: devices that pose significant risk and those that involve insignificant risk. Examples of the former included orthopedic implants, artificial hearts, and infusion pumps.

2.11 ETHICAL ISSUES IN FEASIBILITY STUDIES

Examples of the latter include various dental devices and contact lenses. Clinical research involving a significant risk device cannot begin until an institutional review board (IRB) has approved both the protocol and the informed consent form and the FDA itself has given permission. This requirement to submit an IDE application to the FDA is waived in the case of clinical research where the risk posed is insignificant. In this case, the FDA requires only that approval from an IRB be obtained certifying that the device in question poses only insignificant risk. In deciding whether to approve a proposed clinical investigation of a new device, the IRB and the FDA must determine the following: 1. That risk to subjects is minimized. 2. That risks to subjects are reasonable in relation to anticipated benefit and knowledge to be gained. 3. That subject selection is equitable. 4. That informed consent materials and procedures are adequate. 5. That provisions for monitoring the study and protecting patient information are acceptable. The FDA allows unapproved medical devices to be used without an IDE in three types of situations: feasibility studies, emergency use, and treatment use.

2.11 ETHICAL ISSUES IN FEASIBILITY STUDIES In a feasibility study, or “limited investigation,” human research involving the use of a new device would take place at a single institution and involve no more than ten human subjects. The sponsor of a limited investigation is required to submit to the FDA a “Notice of Limited Investigation,” which includes a description of the device, a summary of the purpose of the investigation, the protocol, a sample of the informed consent form, and a certification of approval by the responsible medical board. In certain circumstances, the FDA could require additional information or require the submission of a full IDE application or suspend the investigation. Investigations of this kind are limited to (1) investigations of new uses for existing devices, (2) investigations involving temporary or permanent implants during the early developmental stages, and (3) investigations involving modification of an existing device. To comprehend adequately the ethical issues posed by clinical use of unapproved medical devices outside the context of an IDE, it is necessary to use the distinctions among practice, nonvalidated practice, and research elaborated upon in the previous pages. How do those definitions apply to feasibility studies? Clearly, the goal of the feasibility study, which is a generalizable knowledge, makes it an instance of research rather than practice. Manufacturers seek to determine the performance of a device with respect to a particular patient population in an effort to gain information about its efficacy and safety. Such information is important in order to determine whether further studies (animal or human) need to be conducted, whether the device needs modification before further use, and the like. The main difference between using an unapproved device in a feasibility study and using it under the terms of an IDE is that the former would be subject to significantly less intensive FDA review than the latter. This, in turn, means

2. MORAL AND ETHICAL ISSUES

that the responsibility for ensuring that the use of the device is ethically sound would fall primarily to the IRB of the institution conducting the study. The ethical concerns posed here can be best comprehended only with a clear understanding of what justifies research in the first place. Ultimately, no matter how much basic research and animal experimentation has been conducted on a given device, the risks and benefits it poses for humans cannot be adequately determined until it is actually used on humans. The benefit of research on humans lies primarily in the generalizable information that is provided. This information is crucial to medical science’s ability to generate new modes of medical treatment that are both efficacious and safe. Therefore, one condition for experimentation to be ethically sound is that it must be scientifically sound. Although scientific soundness is a necessary condition of ethically sound research on humans, it is not of and by itself sufficient. The human subjects of such research are at risk of being mere research resources—that is, having value only for the ends of the research. Human beings are not valuable wholly or solely for the uses to which they can be put. They are valuable simply by being the kinds of entities they are. To treat them as such is to respect them as people. Treating individuals as people means respecting their autonomy. This requirement is met by ensuring that no competent person is subjected to any clinical intervention without first giving voluntary and informed consent. Furthermore, respect for people means that the physician will not subject a human to unnecessary risks and will minimize the risks to patients in required procedures. Much of the scrutiny that the FDA imposes upon use of unapproved medical devices in the context of an IDE addresses two conditions of ethically sound research: Is the experiment scientifically sound? and Does it respect the rights of the human subjects involved? Medical ethicists argue that decreased FDA scrutiny will increase the likelihood that either or both of these conditions will not be met. This possibility exists because many manufacturers of medical devices are, after all, commercial enterprises, companies that are motivated to generate profit and thus to get their devices to market as soon as possible with as little delay and cost as possible. These self-interest motives are likely, at times, to conflict with the requirements of ethically sound research and thus to induce manufacturers to fail to meet these requirements. Profit is not the only motive that might induce manufacturers to contravene the requirements of ethically sound research on humans. A manufacturer may sincerely believe that its product offers great benefit to many people and be prompted to take shortcuts that compromise the quality of the research. Whether the consequences being sought by the research are desired for reasons of self-interest, altruism, or both, the ethical issue is the same. Research subjects may be placed at risk of being treated as mere objects rather than as people. What about the circumstances under which feasibility studies would take place? Are these not sufficiently different from the “normal” circumstances of research to warrant reduced FDA scrutiny? As just noted, manufacturers seek to engage in feasibility studies in order to investigate new uses of existing devices, to investigate temporary or permanent implants during the early developmental stages, and to investigate modifications to an existing device. As also noted, a feasibility study would take place at only one institution and would involve no more than ten human subjects. Given these circumstances, is the sort of research that is likely to occur in a feasibility study less likely to be scientifically sound or to fail to respect people than normal research upon humans in “normal” circumstances?

2.12 ETHICAL ISSUES IN EMERGENCY USE

Research in feasibility studies would be done on a very small subject pool, and the harm of any ethical lapses would likely affect fewer people than if such lapses occurred under more usual research circumstances. Yet even if the harm done is limited to ten or fewer subjects in a single feasibility study, the harm is still ethically wrong. To wrong ten or fewer people is not as bad as to wrong in the same way more than ten people, but it is to engage in wrongdoing nonetheless. Are ethical lapses more likely to occur in feasibility studies than in studies that take place within the requirements of an IDE? Although nothing in the preceding discussion provides a definitive answer to this question, it is a question to which the FDA should give high priority. The answer to this question might be quite different when the device at issue is a temporary or permanent implant than when it is an already approved device being put to new uses or modified in some way. Whatever the contemplated use under the feasibility studies mechanism, the FDA would be ethically advised not to allow this kind of exception to IDE use of an unapproved device without a reasonably high level of certainty that research subjects would not be placed in greater jeopardy than in “normal” research circumstances.

2.12 ETHICAL ISSUES IN EMERGENCY USE What about the mechanism for avoiding the rigors of an IDE for emergency use? The FDA has authorized emergency use in instances where an unapproved device offers the only alternative for saving the life of a dying patient. However, what if an IDE has not yet been approved for the device or its use, or an IDE has been approved but the physician who wishes to use the device is not an investigator under the IDE? The purpose of emergency use of an unapproved device is to attempt to save a dying patient’s life under circumstances where no other alternative is available. This sort of use constitutes practice rather than research. Its aim is primary benefit to the patient rather than provision of new and generalizable information. Because this sort of use occurs before the completion of clinical investigation of the device, it constitutes a nonvalidated practice. What does this mean? First, it means that while the aim of the use is to save the life of the patient, the nature and likelihood of the potential benefits and risks engendered by use of the device are far more speculative than in the sort of clinical intervention that constitutes validated practice. In validated practice, thorough investigation of a device, including preclinical studies, animal studies, and studies on human subjects, has established its efficacy and safety. The clinician thus has a well-founded basis upon which to judge the benefits and risks such an intervention poses for the patient. It is precisely this basis that is lacking in the case of a nonvalidated practice. Does this mean that emergency use of an unapproved device should be regarded as immoral? This conclusion would follow only if there were no basis upon which to make an assessment of the risks and benefits of the use of the device. The FDA requires that a physician who engages in emergency use of an unapproved device must have substantial reason to believe that benefits will exist. This means that there should be a body of preclinical and animal tests allowing a prediction of the benefit to a human patient.

2. MORAL AND ETHICAL ISSUES

CASE STUDY: MEDICAL EXPERT SYSTEMS Expert systems have been developed in various disciplines, including clinical decision making. These systems have been designed to simulate the decision-making skills of physicians. Their adaptability, however, depends on the presence of an accepted body of knowledge regarding the “prescribed path” physicians would take given specific input data. These systems have been viewed as “advisory systems” providing the clinician with suggested/recommended courses of action. The ultimate decision remains with the physician. Consider one such system designed to monitor drug treatment in a psychiatric clinic. This system, designed and implemented by biomedical engineers working with clinicians, begins by the entry of a specific diagnosis and immediately recommends the appropriate drugs to be considered for the treatment of someone who has that mental disorder (Figure 2.6). The physician selects

Begin Treatment

Monitor Activities: Medical History/ Mental Status Exam Check Diagnosis DSM-III Diagnosis Check Lab Results Laboratory Pretreatment Verify Selection Protocol: Select Procedure (drug) Check Dosage Prescription Monitor Results

Follow Up Labs Rating Scales Clinical Examinations Check If Therapeutic

Yes

Therapeutic Procedure?

Yes

FIGURE 2.6 The drug treatment process followed by clinicians.

2.12 ETHICAL ISSUES IN EMERGENCY USE

one of the recommended drugs and conducts a dose regimen to determine the effectiveness of the drug for the particular patient. During the treatment, blood tests are conducted to ascertain the presence of drug toxicity, and other psychiatric measures obtained to determine if the drug is having the desired effect. As these data elements are entered, they are compared with standard expected outcomes, and if the outcomes are outside the expected limits, an alert is sent to the physician indicating further action needs to be taken. In this situation: 1. Who is liable for mistreatment—the clinician, the programmer, or the systems administrator? 2. What constitutes mistreatment? 3. What is the role of the designers of such a system; in other words, what constitutes a successful design? 4. How does the clinic evaluate the performance of a physician using the system, as well as the system itself?

Thus, although the benefits and risks posed by use of the device are highly speculative, they are not entirely speculative. Although the only way to validate a new technology is to engage in research on humans at some point, not all nonvalidated technologies are equal. Some will be largely uninvestigated, and assessment of their risks and benefits will be wholly or almost wholly speculative. Others will at least have the support of preclinical and animal tests. Although this is not sufficient support for incorporating use of a device into regular clinical practice, it may, however, represent sufficient support to justify use in the desperate circumstances at issue in emergency situations. Desperate circumstances can justify desperate actions, but desperate actions are not the same as reckless actions, hence the ethical soundness of the FDA’s requirement that emergency use be supported by solid results from preclinical and animal tests of the unapproved device. A second requirement that the FDA imposes on the emergency use of unapproved devices is the expectation that physicians “exercise reasonable foresight with respect to potential emergencies and . . . make appropriate arrangements under the IDE procedures.” Thus, a physician should not “create” an emergency in order to circumvent IRB review and avoid requesting the sponsor’s authorization of the unapproved use of a “device.” From a Kantian point of view, which is concerned with protecting the dignity of people, this is a particularly important requirement. To create an emergency in order to avoid FDA regulations is to treat the patient as a mere resource whose value is reducible to service to the clinician’s goals. Hence, the FDA is quite correct to insist that emergencies are circumstances that reasonable foresight would not anticipate. Also especially important here is the nature of the patient’s consent. Individuals facing death are especially vulnerable to exploitation and deserve greater measures for their protection than might otherwise be necessary. One such measure would be to ensure that the patient, or his legitimate proxy, knows the highly speculative nature of the intervention being offered—that is, to ensure that it is clearly understood that the clinician’s estimation of the intervention’s risks and benefits is far less solidly grounded than in the case of validated practices. The patient’s consent must be based on an awareness that the device whose use is contemplated has not undergone complete and rigorous testing on humans and that

2. MORAL AND ETHICAL ISSUES

estimations of its potential are based wholly on preclinical and animal studies. Above all, the patient must not be led to believe that the risks and benefits of the intervention are not better understood than they in fact are. Another important point is to ensure that the patient understands all of the options—not simply life or death, but also a life with severely impaired quality. Although desperate circumstances may legitimate desperate actions, the decision to take such actions must rest upon the informed and voluntary consent of the patient, certainly for an especially vulnerable patient. It is important here for a clinician involved in emergency use of an unapproved device to recognize that these activities constitute a form of practice, albeit nonvalidated, and not research. Hence, the primary obligation is to the well-being of the patient. The patient enters into the relationship with the clinician with the same trust that accompanies any normal clinical situation. Treating this sort of intervention as if it were an instance of research and, thus, justified by its benefits to science and society would be an abuse of this trust.

2.13 ETHICAL ISSUES IN TREATMENT USE The FDA has adopted regulations authorizing the use of investigational new drugs in certain circumstances where a patient has not responded to approved therapies. This “treatment use” of unapproved new drugs is not limited to life-threatening emergency situations but is also available to treat “serious” diseases or conditions. The FDA has not approved treatment use of unapproved medical devices, but it is possible that a manufacturer could obtain such approval by establishing a specific protocol for this kind of use within the context of an IDE. The criteria for treatment use of unapproved medical devices would be similar to criteria for treatment use of investigational drugs: (1) the device is intended to treat a serious or lifethreatening disease or condition; (2) there is no comparable or satisfactory alternative product available to treat that condition; (3) the device is under an IDE or has received an IDE exemption, or all clinical trials have been completed and the device is awaiting approval; and (4) the sponsor is actively pursuing marketing approval of the investigational device. The treatment use protocol would be submitted as part of the IDE and would describe the intended use of the device, the rationale for use of the device, the available alternatives and why the investigational product is preferable, the criteria for patient selection, the measures to monitor the use of the device and to minimize risk, and technical information that is relevant to the safety and effectiveness of the device for the intended treatment purpose. Were the FDA to approve treatment use of unapproved medical devices, what ethical issues would be posed? First, because such use is premised on the failure of validated interventions to improve the patient’s condition adequately, it is a form of practice rather than research. Second, since the device involved in an instance of treatment use is unapproved, such use would constitute nonvalidated practice. As such, like emergency use, it should be subject to the FDA’s requirement that prior preclinical tests and animal studies have been conducted that provide substantial reason to believe that patient benefit will result. As with emergency use, although this does not prevent assessment of the intervention’s benefits and risks from being highly speculative, it does prevent assessment from being totally speculative. Here, too, although desperate circumstances can justify desperate action, they do not

2.14 THE ROLE OF THE BIOMEDICAL ENGINEER IN THE FDA PROCESS

justify reckless action. Unlike emergency use, the circumstances of treatment use involve serious impairment of health rather than the threat of premature death. Hence, an issue that must be considered is how serious such impairment must be to justify resorting to an intervention whose risks and benefits have not been solidly established. In cases of emergency use, the FDA requires that physicians not create an exception to an IDE to avoid requirements that would otherwise be in place. As with emergency use of unapproved devices, the patients involved in treatment uses would be particularly vulnerable patients. Although they are not dying, they are facing serious medical conditions and are thereby likely to be less able to avoid exploitation than patients under less desperate circumstances. Consequently, here too it is especially important that patients be informed of the speculative nature of the intervention and of the possibility that treatment may result in little to no benefit to them.

2.14 THE ROLE OF THE BIOMEDICAL ENGINEER IN THE FDA PROCESS On November 28, 1991, the Safe Medical Devices Act of 1990 (Public Law 101-629) went into effect. This regulation requires a wide range of health care institutions, including hospitals, ambulatory-surgical facilities, nursing homes, and outpatient treatment facilities, to report information that “reasonably suggests” the likelihood that the death, serious injury, or serious illness of a patient at that facility was caused or contributed to by a medical device. When a death is device-related, a report must be made directly to the FDA and to the manufacturer of the device. When a serious illness or injury is device-related, a report must be made to the manufacturer or to the FDA in cases where the manufacturer is not known. In addition, summaries of previously submitted reports must be submitted to the FDA on a semiannual basis. Prior to this regulation, such reporting was wholly voluntary. This new regulation was designed to enhance the FDA’s ability to learn quickly about problems related to medical devices and supplements the medical device reporting (MDR) regulations promulgated in 1984. MDR regulations require that manufacturers and importers submit reports of device-related deaths and serious injuries to the FDA. The new law extends this requirement to users of medical devices along with manufacturers and importers. This act gives the FDA authority over device-user facilities. The FDA regulations are ethically significant because by attempting to increase the FDA’s awareness of medical device-related problems, it attempts to increase that agency’s ability to protect the welfare of patients. The main controversy over the FDA’s regulation policies is essentially utilitarian in nature. Skeptics of the law are dubious about its ability to provide the FDA with much useful information. They worry that much of the information generated by this new law will simply duplicate information already provided under MDR regulations. If this were the case, little or no benefit to patients would accrue from compliance with the regulation. Furthermore, these regulations, according to the skeptics, are likely to increase lawsuits filed against hospitals and manufacturers and will require device-user facilities to implement formal systems for reporting device-related problems and to provide personnel to operate those systems. This would, of course, add to the costs of health care and thereby exacerbate the problem of access to care, a situation that many

2. MORAL AND ETHICAL ISSUES

believe to be of crisis proportions already. In short, the controversy over FDA policy centers upon the worry that its benefits to patients will be marginal and significantly outweighed by its costs. Biomedical engineers need to be aware of FDA regulations and the process for FDA approval of the use of medical devices and systems. These regulatory policies are, in effect, society’s mechanism for controlling the improper use of these devices.

2.15 EXERCISES 1. Explain the distinction between the terms ethics and morality. Provide examples that illustrate this distinction in the medical arena. 2. Explain the distinction between the terms beneficence and nonmaleficence, and provide a realworld example of each. Which has been favored by medicine in the ethical sense? 3. Provide three examples of medical moral judgments. 4. What do advocates of the utilitarian school of thought believe? 5. What does Kantianism expect in terms of the patient’s rights and wishes? 6. Discuss how the code of ethics for clinical engineers provides guidance to practitioners in the field. 7. Discuss what is meant by brainstem death. How is this distinguished from neocortical death? 8. In response to the Schiavo and Houben case studies, what steps, if any, can be taken to guarantee brain death? Should there be set procedures for determination? 9. Distinguish between active and passive euthanasia, as well as voluntary and involuntary euthanasia. In your view, which, if any, are permissible? Provide your reasoning and any conditions that must be satisfied to meet your approval. 10. Should the federal government be able to require an individual to sign a living will in case of an accident? 11. If the family of a patient in the intensive care unit submits the individual’s “living will,” should it be honored immediately, or should there be a discussion between physicians and the family? Who should make the decision? Why? 12. What constitutes a human experiment? Under what conditions are they permitted? What safeguards should hospitals have in place? 13. Should animal experimentation be required prior to human experimentation? How does animal research play into the philosophies of nonconsequentialism and utilitarianism? 14. Discuss the relationship between cost (or risk) and benefit in the decision for a patient to participate in a human experiment. 15. In the event of unfavorable and potentially painful results in consented human experimentation, who should be held liable? Why? 16. A biomedical engineer has designed a new sleep apnea monitor. Discuss the steps that should be taken before it is used in a clinical setting. 17. Discuss the distinctions among practice, research, and nonvalidated practice. Provide examples of each in the medical arena. 18. What are the two major conditions for ethically sound research?

2.15 EXERCISES

19. Informed consent is one of the essential factors in permitting humans to participate in medical experiments. What ethical principles are satisfied by informed consent? What should be done to ensure it is truly voluntary? What information should be given to human subjects? 20. What are the distinctions between feasibility studies and emergency use? 21. In the practice of medicine, health care professionals use medical devices to diagnose and treat patients. Therefore, the clinical staff must not only become knowledgeable and skilled in their understanding of human physiology, but they must also be competent in using the medical tools at their disposal. This requirement often results in litigation when a device fails. The obvious question is, “Who is to blame?” Consider the case of a woman undergoing a surgical procedure that requires the use of a ground plate—an 8 11-inch pad that serves as a return path for any electrical current that comes from electrosurgical devices used during the procedure. As a result of the procedure, this woman received a major burn that seriously destroyed tissue at the site of the ground plate. (a) Discuss the possible individuals and/or organizations that may have been responsible for this injury. (b) Outside of seeking the appropriate responsible party, are there specific ethical issues here?

Suggested Readings N. Abrams, M.D. Buckner (Eds.), Medical Ethics, MIT Press, Cambridge, MA, 1983. J.D. Bronzino, V.H. Smith, M.L. Wade, Medical Technology and Society, MIT Press, Cambridge, MA, 1990. J.D. Bronzino, Management of Medical Technology, Butterworth, Boston, 1992. A.R. Chapman, Health Care and Information Ethics: Protecting Fundamental Human Rights, Sheed and Ward, Kansas City, KS, 1997. N. Dubler, D. Nimmons, Ethics on Call, Harmony Books, New York, 1992. A.R. Jonsen, The New Medicine and The Old Ethics, Harvard University Press, Cambridge, MA, 1990. J.C. Moskop, L. Kopelman (Eds.), Ethics and Critical Care Medicine, D. Reidel Publishing Co., Boston, 1985. G.E. Pence, Classic Cases in Medical Ethics, McGraw-Hill, New York, 1990. J. Rachels, Ethics at the End of Life: Euthanasia and Morality, Oxford University Press, Oxford, 1986. J. Reiss, Bringing Your Medical Device to Market, FDLI Publishers, Washington, DC, 2001. E.G. Seebauer, R.L. Barry, Fundamentals of Ethics for Scientists and Engineers, Oxford Press, NY, 2001.

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C H A P T E R

3 Anatomy and Physiology Susan Blanchard, PhD, and Joseph D. Bronzino, PhD, PE O U T L I N E 3.1

Introduction

3.5

Homeostasis

126

3.2

Cellular Organization

3.6

Exercises

129

3.3

Tissues

Suggested Readings

131

3.4

Major Organ Systems

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : • Define anatomy and physiology and explain why they are important to biomedical engineering.

• List and describe the functions of the major organelles found within mammalian cells.

• Define important anatomical terms.

• Describe the similarities, differences, and purposes of replication, transcription, and translation.

• Describe the cell theory. • List the major types of organic compounds and other elements found in cells.

• List and describe the major components and functions of five organ systems: cardiovascular, respiratory, nervous, skeletal, and muscular.

• Explain how the plasma membrane maintains the volume and internal concentrations of a cell.

• Define homeostasis and describe how feedback mechanisms help maintain it.

• Calculate the internal osmolarity and ionic concentrations of a model cell at equilibrium.

Introduction to Biomedical Engineering, Third Edition

3. ANATOMY AND PHYSIOLOGY

3.1 INTRODUCTION Since biomedical engineering is an interdisciplinary field based in both engineering and the life sciences, it is important for biomedical engineers to have knowledge about and be able to communicate in both areas. Biomedical engineers must understand the basic components of the body and how they function well enough to exchange ideas and information with physicians and life scientists. Two of the most basic terms and areas of study in the life sciences are anatomy and physiology. Anatomy refers to the internal and external structures of the body and their physical relationships, whereas physiology refers to the study of the functions of those structures. Figure 3.1a shows a male body in anatomical position. In this position, the body is erect and facing forward, with the arms hanging at the sides and the palms facing outward. This particular view shows the anterior (ventral) side of the body, whereas Figure 3.1c illustrates the posterior (dorsal) view of another male body that is also in anatomical position, and Figure 3.1b presents the lateral view of the female body. In clinical practice, directional terms are used to describe the relative positions of various parts of the body. Proximal parts are nearer to the trunk of the body or to the attached end of a limb than are distal parts (Figure 3.1a). Parts of the body that are located closer to the head than other parts when the body is in anatomical position are said to be superior (Figure 3.1b), whereas those located closer to the feet than other parts are termed inferior. Medial implies that a part is toward the midline of the body, whereas lateral means away from the midline (Figure 3.1c). Parts of the body that lie in the direction of the head are said to be in the cranial direction,

SUPERIOR INFERIOR PROXIMAL

DISTAL

MEDIAL LATERAL

(a)

FIGURE 3.1

(b)

(c)

(a) Anterior view of male body in anatomical position. (b) Lateral view of female body. (c) Posterior view of male body in anatomical position. Relative directions (proximal and distal, superior and inferior, and medial and lateral) are also shown.

3.1 INTRODUCTION

CRANIAL DIRECTION

FRONTAL PLANE SAGITTAL PLANE MIDSAGITTAL PLANE TRANSVERSE PLANE

CAUDAL DIRECTION

FIGURE 3.2 The body can be divided into sections by the frontal, sagittal, and transverse planes. The midsagittal plane goes through the midline of the body.

whereas those parts that lie in the direction of the feet are said to be in the caudal direction (Figure 3.2). Anatomical locations can also be described in terms of planes. The plane that divides the body into two symmetric halves along its midline is called the midsaggital plane (Figure 3.2). Planes that are parallel to the midsaggital plane but do not divide the body into symmetric halves are called sagittal planes. The frontal plane is perpendicular to the midsaggital plane and divides the body into asymmetric anterior and posterior portions. Planes that cut across the body and are perpendicular to the midsaggital and frontal planes are called transverse planes. Human bodies are divided into two main regions: axial and appendicular. The axial part consists of the head, neck, thorax (chest), abdomen, and pelvis, while the appendicular part consists of the upper and lower extremities. The upper extremities, or limbs, include the shoulders, upper arms, forearms, wrists, and hands, while the lower extremities include the hips, thighs, lower legs, ankles, and feet. The abdominal region can be further divided into nine regions or four quadrants. The cavities of the body hold the internal organs. The major cavities are the dorsal and ventral body cavities, while smaller ones include the nasal, oral, orbital (eye), tympanic (middle ear), and synovial (movable joint) cavities. The dorsal body cavity includes the

3. ANATOMY AND PHYSIOLOGY

cranial cavity that holds the brain and the spinal cavity that contains the spinal cord. The ventral body cavity contains the thoracic and abdominopelvic cavities that are separated by the diaphragm. The thoracic cavity contains the lungs and the mediastinum, which contains the heart and its attached blood vessels, the trachea, the esophagus, and all other organs in this region except for the lungs. The abdominopelvic cavity is divided by an imaginary line into the abdominal and pelvic cavities. The former is the largest cavity in the body and holds the stomach, small and large intestines, liver, spleen, pancreas, kidneys, and gallbladder. The latter contains the urinary bladder, the rectum, and the internal portions of the reproductive system. The anatomical terms described previously are used by physicians, life scientists, and biomedical engineers when discussing the whole human body or its major parts. Correct use of these terms is vital for biomedical engineers to communicate with health care professionals and to understand the medical problem of concern or interest. While it is important to be able to use the general terms that describe the human body, it is also important for biomedical engineers to have a basic understanding of some of the more detailed aspects of human anatomy and physiology.

3.2 CELLULAR ORGANIZATION Although there are many smaller units such as enzymes and organelles that perform physiological tasks or have definable structures, the smallest anatomical and physiological unit in the human body that can, under appropriate conditions, live and reproduce on its own is the cell. Cells were first discovered more than 300 years ago shortly after Antony van Leeuwenhoek, a Dutch optician, invented the microscope. With his microscope, van Leeuwenhoek was able to observe “many very small animalcules, the motions of which were very pleasing to behold” in tartar scrapings from his teeth. Following the efforts of van Leeuwenhoek, Robert Hooke, a Curator of Instruments for the Royal Society of England, in the late 1600s further described cells when he used one of the earliest microscopes to look at the plant cell walls that remain in cork. These observations and others led to the cell theory developed by Theodor Schwann and Matthias Jakob Schleiden and formalized by Rudolf Virchow in the mid-1800s. The cell theory states that (1) all organisms are composed of one or more cells, (2) the cell is the smallest unit of life, and (3) all cells come from previously existing cells. Thus, cells are the basic building blocks of life. Cells are composed mostly of organic compounds and water, with more than 60 percent of the weight in a human body coming from water. The organic compounds—carbohydrates, lipids, proteins, and nucleic acids—that cells synthesize are the molecules that are fundamental to sustaining life. These molecules function as energy packets, storehouses of energy and hereditary information, structural materials, and metabolic workers. The most common elements found in humans (in descending order based on percent of body weight) are oxygen, carbon, hydrogen, nitrogen, calcium, phosphorus, potassium, sodium, chlorine, magnesium, sulfur, iron, and iodine. Carbon, hydrogen, oxygen, and nitrogen contribute more than 99 percent of all the atoms in the body. Most of these elements are incorporated into organic compounds, but some exist in other forms, such as phosphate groups and ions. Carbohydrates are used by cells not only as structural materials but also to transport and store energy. The three classes of carbohydrates are monosaccharides (e.g., glucose),

3.2 CELLULAR ORGANIZATION

oligosaccharides (e.g., lactose, sucrose, maltose), and polysaccharides (e.g., glycogen). Lipids are greasy or oily compounds that will dissolve in each other but not in water. They form structural materials in cells and are the main reservoirs of stored energy. Proteins are the most diverse form of biological molecules. Specialized proteins, called enzymes, make metabolic reactions proceed at a faster rate than would occur if the enzymes were not available and enable cells to produce the organic compounds of life. Other proteins provide structural elements in the body, act as transport channels across plasma membranes, function as signals for changing activities, and provide chemical weapons against disease-carrying bacteria. These diverse proteins are built from a small number (20) of essential amino acids. Nucleotides and nucleic acids make up the last category of important biological molecules. Nucleotides are small organic compounds that contain a five-carbon sugar (ribose or deoxyribose), a phosphate group, and a nitrogen-containing base that has a single or double carbon ring structure. Adenosine triphosphate (ATP) is the energy currency of the cell and plays a central role in metabolism. Other nucleotides are subunits of coenzymes that are enzyme helpers. The two nucleic acids are deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). DNA (Figure 3.3) is a unique, helical molecule that contains chains of paired nucleotides that run in opposite directions. Each nucleotide contains either a pyrimidine base—thymine (T) or cytosine (C)—with a single ring structure or a purine base— adenine (A) or guanine (G)—with a double ring. In the double helix of DNA, thymine always pairs with adenine (T-A) and cytosine always pairs with guanine (C-G). RNA is similar to DNA except that it consists of a single helical strand, contains ribose instead of deoxyribose, and has uracil (U) instead of thymine. All cells are surrounded by a plasma membrane that separates, but does not isolate, the cell’s interior from its environment. Animal cells, such as those found in humans, are eukaryotic cells. A generalized animal cell is shown in Figure 3.4. In addition to the plasma membrane, eukaryotic cells contain membrane-bound organelles and a membrane-bound nucleus. Prokaryotic cells, such as bacteria, lack membrane-bound structures other than the plasma membrane. In addition to a plasma membrane, all cells have a region that contains DNA (which carries the hereditary instructions for the cell) and cytoplasm (which is a semifluid substance that includes everything inside the plasma membrane except for the DNA).

(a)

FIGURE 3.3

(b)

(a) DNA consists of two chains of paired nucleotides that run in opposite directions and form a helical structure. (b) Thymine pairs with adenine (T-A) and cytosine pairs with guanine (C-G) due to hydrogen bonding between the bases.

3. ANATOMY AND PHYSIOLOGY

NUCLEOLUS

ENDOPLASMIC RETICULUM NUCLEAR ENVELOPE

GOLGI APPARATUS

NUCLEOPLASM PLUS DNA

VESICLE

MITOCHONDRIA

CENTRIOLE

GOLGI APPARATUS

ENDOPLASMIC RETICULUM

FIGURE 3.4 Animal cells are surrounded by a plasma membrane. They contain a membrane-bound region, the nucleus, which contains DNA. The cytoplasm lies outside of the nucleus and contains several types of organelles that perform specialized functions.

3.2.1 Plasma Membrane The plasma membrane performs several functions for the cell. It gives mechanical strength, provides structure, helps with movement, and controls the cell’s volume and its activities by regulating the movement of chemicals in and out of the cell. The plasma membrane is composed of two layers of phospholipids interspersed with proteins and cholesterol (Figure 3.5). The proteins in the plasma membranes of mammalian cells provide

HYDROPHOBIC TAIL

PERIPHERAL PROTEIN

CHOLESTEROL

PHOSPHOLIPID BILAYER

HYDROPHILIC HEAD INTEGRAL PROTEIN

FIGURE 3.5

The plasma membrane surrounds all cells. It consists of a double layer of phospholipids interspersed with proteins and cholesterol.

3.2 CELLULAR ORGANIZATION

binding sites for hormones, recognition markers for identifying cells as one type or another, adhesive mechanisms for binding adjacent cells to each other, and channels for transporting materials across the plasma membrane. The phospholipids are arranged with their “water loving” (hydrophilic) heads pointing outward and their “water fearing” (hydrophobic) tails pointing inward. This double-layer arrangement of phospholipids interspersed with protein channels helps maintain the internal environment of a cell by controlling the substances that move across the membrane, whereas the cholesterol molecules act as stabilizers to prevent extensive lateral movement of the lipid molecules. Some molecules, such as oxygen, carbon dioxide, and water, can easily cross the plasma membrane, whereas other substances, such as large molecules and ions, must move through the protein channels. Osmosis is the process by which substances move across a selectively permeable membrane such as a cell’s plasma membrane, whereas diffusion refers to the movement of molecules from an area of relatively high concentration to an area of relatively low concentration. Substances that can easily cross the plasma membrane achieve diffusion equilibrium when there is no net movement of these substances across the membrane; that is, the concentration of the substance inside the cell equals the concentration of the substance outside of the cell. Active transport, which requires an input of energy usually in the form of ATP, can be used to move ions and molecules across the plasma membrane and is often used to move them from areas of low concentration to areas of high concentration. This mechanism helps maintain concentrations of ions and molecules inside a cell that are different from the concentrations outside the cell. A typical mammalian cell has internal sodium ion (Naþ) concentrations of 12 mM (12 moles of Naþ per 1,000 liters of solution) and extracellular Naþ concentrations of 120 mM, whereas intracellular and extracellular potassium ion (Kþ) concentrations are on the order of 125 mM and 5 mM, respectively. In addition to positively charged ions (cations), cells also contain negatively charged ions (anions). A typical mammalian cell has intracellular and extracellular chloride ion (Cl) concentrations of 5 mM and 125 mM and internal anion (e.g., proteins, charged amino acids, sulfate ions, and phosphate ions) concentrations of 108 mM. These transmembrane ion gradients are used to make ATP, to drive various transport processes, and to generate electrical signals.

EXAMPLE PROBLEM 3.1 How many molecules of sodium and potassium ions would a cell that has a volume of 2 nl contain?

Solution Assuming that the intracellular concentrations of Naþ and Kþ are 12 mM and 125 mM, respectively, the number of molecules for each can be determined by using the volume of the cell and Avogadro’s number. Naþ : 12

moles molecules 6:023 1023 2 109 liters ¼ 1:45 1013 molecules 1; 000 liters mole

Kþ : 125

moles molecules 6:023 1023 2 109 liters ¼ 1:51 1014 molecules 1; 000 liters mole

3. ANATOMY AND PHYSIOLOGY

The plasma membrane plays an important role in regulating cell volume by controlling the internal osmolarity of the cell. Osmolarity is defined in terms of concentration of dissolved substances. A 1 osmolar (1 Osm) solution contains 1 mole of dissolved particles per liter of solution, while a 1 milliosmolar (1 mOsm) solution has 1 mole of dissolved particles per 1,000 liters of solution. Thus, solutions with high osmolarity have low concentrations of water or other solvents. For biological purposes, solutions with 0.1 Osm glucose and 0.1 Osm urea have essentially the same concentrations of water. It is important to note that a 0.1 M solution of sodium chloride (NaCl) will form a 0.2 Osm solution, since NaCl dissociates into Naþ and Cl ions and thus has twice as many dissolved particles as a solution of a substance—for example, glucose—that does not dissociate into smaller units. Two solutions are isotonic if they have the same osmolarity. One solution is hypotonic to another if it has a lower osmolarity and hypertonic to another if it has a higher osmolarity. It is important to note that tonicity (isotonic, hypotonic, or hypertonic) is only determined by those molecules that cannot cross the plasma membrane, since molecules that can freely cross will eventually reach equilibrium with the same concentration inside and outside of the cell. Consider a simple model cell that consists of a plasma membrane and cytoplasm. The cytoplasm in this model cell contains proteins that cannot cross the plasma membrane and water that can. At equilibrium, the total osmolarity inside the cell must equal the total osmolarity outside the cell. If the osmolarity inside and the osmolarity outside of the cell are out of balance, there will be a net movement of water from the side of the plasma membrane where it is more highly concentrated to the other until equilibrium is achieved. For example, assume that a model cell (Figure 3.6) contains 0.2 M protein and is placed in a hypotonic solution that contains 0.1 M sucrose. The plasma membrane of this model cell is impermeable to proteins and sucrose but freely permeable to water. The volume of the PLASMA MEMBRANE

WATER

WATER 0.2 M PROTEINS

0.1 M SUCROSE

FIGURE 3.6 A simple model cell that consists of cytoplasm, containing 0.2 M proteins, and a plasma membrane is placed in a solution of 0.1 M sucrose. The plasma membrane is insoluble to proteins and sucrose but allows water to pass freely in either direction. The full extent of the extracellular volume is not shown and is much larger than the cell’s volume of 1 nl.

3.2 CELLULAR ORGANIZATION

cell, 1 nl, is very small relative to the volume of the solution. In other words, changes in the cell’s volume have no measurable effect on the volume of the external solution. What will happen to the volume of the cell as it achieves equilibrium? At equilibrium, the osmolarity inside the cell must equal the osmolarity outside the cell. The initial osmolarity inside the cell is 0.2 Osm, since the proteins do not dissociate into smaller units. The osmolarity outside the cell is 0.1 Osm due to the sucrose solution. A 0.2 Osm solution has 0.2 moles of dissolved particles per liter of solution, while a 0.1 Osm solution has half as many moles of dissolved particles per liter. The osmolarity inside the cell must decrease by a factor of 2 in order to achieve equilibrium. Since the plasma membrane will not allow any of the protein molecules to leave the cell, this can only be achieved by doubling the cell’s volume. Thus, there will be a net movement of water across the plasma membrane until the cell’s volume increases to 2 nl and the cell’s internal osmolarity is reduced to 0.1 Osm—the same as the osmolarity of the external solution. The water moves down its concentration gradient by diffusing from where it is more highly concentrated in the 0.1 M sucrose solution to where it is less concentrated in the 0.2 M protein solution in the cell.

EXAMPLE PROBLEM 3.2 What would happen to the model cell in Figure 3.6 if it were placed in pure water?

Solution Water can pass through the plasma membrane and would flow down its concentration gradient from where it is more concentrated (outside of the cell) to where it is less concentrated (inside of the cell). Eventually, enough water would move into the cell to rupture the plasma membrane, since the concentration of water outside the cell would be higher than the concentration of water inside the cell as long as there were proteins trapped within the cell.

EXAMPLE PROBLEM 3.3 Assume that the model cell in Figure 3.6 has an initial volume of 2 nl and contains 0.2 M protein. The cell is placed in a large volume of 0.2 M NaCl. In this model, neither Naþ nor Cl can cross the plasma membrane and enter the cell. Is the 0.2 M NaCl solution hypotonic, isotonic, or hypertonic relative to the osmolarity inside the cell? Describe what happens to the cell as it achieves equilibrium in this new environment. What will be the final osmolarity of the cell? What will be its final volume?

Solution The osmolarity inside the cell is 0.2 Osm. The osmolarity of the 0.2 M NaCl solution is 0.4 Osm (0.2 Osm Naþ þ 0.2 Osm Cl). Thus, the NaCl solution is hypertonic relative to the osmolarity inside the cell (osmolarityoutside > osmolarityinside). Since none of the particles (protein, Naþ, and Cl) can cross the membrane, water will move out of the cell until the osmolarity inside Continued

3. ANATOMY AND PHYSIOLOGY

the cell is 0.4 Osm. This will be achieved when the volume inside the cell has been reduced from 2 nl to 1 nl. C1 V1 ¼ C2 V2 0:2 Osm 2nl ¼ V2 0:4 Osm 1nl ¼ V2

Real cells are much more complex than the simple model just described. In addition to achieving osmotic balance at equilibrium, real cells must also achieve electrical balance with regard to the ions that are present in the cytoplasm. The principle of electrical neutrality requires that the overall concentration of cations in a biological compartment—for example, a cell—must equal the overall concentration of anions in that compartment. Consider another model cell (Figure 3.7) with internal and external cation and anion concentrations similar to those of a typical mammalian cell. Is the cell at equilibrium if the plasma membrane is freely permeable to Kþ and Cl but impermeable to Naþ and the internal anions? The total osmolarity inside the cell is 250 mOsm (12 mM Naþ, 125 mM Kþ, 5 mM Cl, 108 mM anions), while the total osmolarity outside the cell is also 250 mOsm (120 mM Naþ, 5 mM Kþ, 125 mM Cl), so the cell is in osmotic balance—that is, there will be no net movement of water across the plasma membrane. If the average charge per molecule of the anions inside the cell is considered to be –1.2, then the cell is also approximately in electrical equilibrium (12 þ 125 positive charges for Naþ and Kþ; 5 þ 1.2 * 108 negative charges for Cl and the other anions). Real cells, however, cannot maintain this equilibrium without expending energy, since real cells are slightly permeable to Naþ. In order to maintain equilibrium and keep Naþ from accumulating intracellularly, mammalian cells must actively pump Naþ out of the cell against its diffusion and electrical gradients. Since Naþ INTRACELLULAR FLUID

PLASMA MEMBRANE WATER K+

WATER K+ Cl− 108 mM ANIONS 12 mM Na− 125 mM K+ 5 mM Cl−

Cl−

120 mM Na+ 5 mM K+ 125 mM Cl−

EXTRACELLULAR FLUID

FIGURE 3.7

A model cell with internal and external concentrations similar to those of a typical mammalian cell. The full extent of the extracellular volume is not shown and is much larger than the cell’s volume.

3.2 CELLULAR ORGANIZATION

is pumped out through specialized protein channels at a rate equivalent to the rate at which it leaks in through other channels, it behaves osmotically as if it cannot cross the plasma membrane. Thus, mammalian cells exist in a steady state, rather than at equilibrium, since energy in the form of ATP must be used to prevent a net movement of ions across the plasma membrane.

EXAMPLE PROBLEM 3.4 Consider a simple model cell, such as the one in Figure 3.7, that has the following ion concentrations. Is the cell at equilibrium? Explain your answer. Ion

Intracellular Concentration (mM)

Extracellular Concentration (mM)

Kþ Naþ Cl A

158 20 52 104

4 163 167 —

Solution Yes. The cell is both electrically and osmotically at equilibrium because the charges within the inside and outside compartments are equal, and the osmolarity inside the cell equals the osmolarity outside of the cell. Inside Positive Negative Osmolarity

158 þ 20 ¼ 178 mM 52 þ 1.2 * 104 ¼ 177 mM 178 mMpos 177 mMneg 158 þ 20 þ 52 þ 104 ¼ 334 mM 334 mMinside ¼ 334

Outside 4 þ 163 ¼ 167 mM 167 mM 167 mMpos ¼ 167 mMneg 4 þ 163 þ 167 ¼ 334 mM mMoutside

One of the consequences of the distribution of charged particles in the intracellular and extracellular fluids is that an electrical potential exists across the plasma membrane. The value of this electrical potential depends on the intracellular and extracellular concentrations of ions that can cross the membrane and will be described more fully in Chapter 11. In addition to controlling the cell’s volume, the plasma membrane also provides a route for moving large molecules and other materials into and out of the cell. Substances can be moved into the cell by means of endocytosis (Figure 3.8a) and out of the cell by means of exocytosis (Figure 3.8b). In endocytosis, material—for example, a bacterium—outside of the cell is engulfed by a portion of the plasma membrane that encircles it to form a vesicle. The vesicle then pinches off from the plasma membrane and moves its contents to the inside of the cell. In exocytosis, material within the cell is surrounded by a membrane to form a vesicle. The vesicle then moves to the edge of the cell, where its membrane fuses with the plasma membrane and its contents are released to the exterior of the cell.

3. ANATOMY AND PHYSIOLOGY

PLASMA MEMBRANE

(a)

(b)

VESICLE

FIGURE 3.8 Substances that are too large to pass through the integral proteins in the plasma membrane can be moved into the cell by means of endocytosis (a) and out of the cell by means of exocytosis (b).

3.2.2 Cytoplasm and Organelles The cytoplasm contains fluid (cytosol) and organelles. Ions (such as Naþ, Kþ, and Cl) and molecules (such as glucose) are distributed through the cytosol via diffusion. Membranebound organelles include the nucleus, rough and smooth endoplasmic reticulum, the Golgi apparatus, lysosomes, and mitochondria. Nonmembranous organelles include nucleoli, ribosomes, centrioles, microvilli, cilia, flagella, and the microtubules, intermediate filaments, and microfilaments of the cytoskeleton. The nucleus (see Figure 3.4) consists of the nuclear envelope (a double membrane) and the nucleoplasm (a fluid that contains ions, enzymes, nucleotides, proteins, DNA, and small amounts of RNA). Within its DNA, the nucleus contains the instructions for life’s processes. Nuclear pores are protein channels that act as connections for ions and RNA, but not proteins or DNA, to leave the nucleus and enter the cytoplasm and for some proteins to enter the nucleoplasm. Most nuclei contain one or more nucleoli. Each nucleolus contains DNA, RNA, and proteins and synthesizes the components of the ribosomes that cells use to make proteins. The smooth and rough endoplasmic reticulum (ER), Golgi apparatus, and assorted vesicles (Figures 3.4, 3.9a, and 3.9b) make up the cytomembrane system, which delivers proteins and lipids for manufacturing membranes and accumulates and stores proteins and lipids for specific uses. The ER also acts as a storage site for calcium ions. The rough ER differs from the smooth ER in that it has ribosomes attached to its exterior surface. Ribosomes provide the platforms for synthesizing proteins. Those that are synthesized on the rough ER are passed into its interior, where nonproteinaceous side chains are attached to them. These modified proteins move to the smooth ER, where they are packaged in vesicles. The smooth ER also manufactures and packages lipids into vesicles and is responsible for releasing stored calcium ions. The vesicles leave the smooth ER and become attached to the Golgi apparatus, where their contents are released, modified, and repackaged into new vesicles. Some of these vesicles, called lysosomes, contain digestive enzymes that are used to break down materials that move into the cells via endocytosis. Other vesicles contain proteins, such as hormones and neurotransmitters, that are secreted from the cells by means of exocytosis.

3.2 CELLULAR ORGANIZATION

ROUGHER

SMOOTHER (a)

VESICLE

RIBOSOME

OUTER COMPARTMENT

VESICLE (b)

INNER COMPARTMENT (c)

FIGURE 3.9 Subcellular organelles. The endoplasmic reticulum (a), the Golgi apparatus (b), and vesicles (b) make up the cytomembrane system in the cell. The small circles on the endoplasmic reticulum (ER) represent ribosomes. The area containing ribosomes is called the rough ER, while the area that lacks ribosomes is called the smooth ER. The mitochondria (c) have a double membrane system that divides the interior into two compartments that contain different concentrations of enzymes, substrates, and hydrogen ions (Hþ). Electrical and chemical gradients between the inner and outer compartments provide the energy needed to generate ATP.

The mitochondria (Figures 3.9c and 3.10) contain two membranes: an outer membrane that surrounds the organelle and an inner membrane that divides the organelle’s interior into two compartments. Approximately 95 percent of the ATP required by the cell is produced in the mitochondria in a series of oxygen-requiring reactions that produce carbon dioxide as a by-product. Mitochondria are different from most other organelles in that they contain their own DNA. The majority of the mitochondria in sexually reproducing organisms, such

FIGURE 3.10 Scanning electron micrograph of a normal mouse liver at 8,000X magnification. The large round organelle on the left is the nucleus. The smaller round and oblong organelles are mitochondria that have been sliced at different angles. The narrow membranes in parallel rows are endoplasmic reticula. The small black dots on the ERs are ribosomes. Photo courtesy of Valerie Knowlton, Center for Electron Microscopy, North Carolina State University.

3. ANATOMY AND PHYSIOLOGY

MICROTUBULES

MICROVILLI (a)

(b)

(c)

FIGURE 3.11 Centrioles (a) contain microtubules and are located at right angles to each other in the cell’s centrosome. These organelles play an important part in cell division by anchoring the microtubules that are used to divide the cell’s genetic material. Microvilli (b), which are extensions of the plasma membrane, line the villi, tiny fingerlike protrusions in the mucosa of the small intestine, and help increase the area available for the absorption of nutrients. Cilia (c) line the respiratory tract. The beating of these organelles helps move bacteria and particles trapped in mucus out of the lungs.

as humans, come from the mother’s egg cell, since the father’s sperm contributes little more than the DNA in a haploid (half) set of chromosomes to the developing offspring. Microtubules, intermediate filaments, and microfilaments provide structural support and assist with movement. Microtubules are long, hollow, cylindrical structures that radiate from microtubule organizing centers and, during cell division, from centrosomes, a specialized region of the cytoplasm that is located near the nucleus and contains two centrioles (Figures 3.4 and 3.11a) oriented at right angles to each other. Microtubules consist of spiraling subunits of a protein called tubulin, whereas centrioles consist of nine triplet microtubules that radiate from their centers like the spokes of a wheel. Intermediate filaments are hollow and provide structure to the plasma membrane and nuclear envelope. They also aid in cell-to-cell junctions and in maintaining the spatial organization of organelles. Myofilaments are found in most cells and are composed of strings of protein molecules. Cell movement can occur when actin and myosin, protein subunits of myofilaments, interact. Microvilli (Figure 3.11b) are extensions of the plasma membrane that contain microfilaments. They increase the surface area of a cell to facilitate absorption of extracellular materials. Cilia (Figure 3.11c) and flagella are parts of the cytoskeleton that have shafts composed of nine pairs of outer microtubules and two single microtubules in the center. Both types of shafts are anchored by a basal body that has the same structure as a centriole. Flagella function as whiplike tails that propel cells such as sperm. Cilia are generally shorter and more profuse than flagella and can be found on specialized cells such as those that line the respiratory tract. The beating of the cilia helps move mucus-trapped bacteria and particles out of the lungs.

3.2.3 DNA and Gene Expression DNA (see Figure 3.3) is found in the nucleus and mitochondria of eukaryotic cells. In organisms that reproduce sexually, the DNA in the nucleus contains information from both parents, while that in the mitochondria comes from the organism’s mother. In the nucleus, the DNA is wrapped around protein spools, called nucleosomes, and is organized into pairs of chromosomes. Humans have 22 pairs of autosomal chromosomes and two sex

3.2 CELLULAR ORGANIZATION

X Y

FIGURE 3.12 This karyotype of a normal human male shows the 22 pairs of autosomal chromosomes in descending order based on size, as well as the X and Y sex chromosomes.

chromosomes, XX for females and XY for males (Figure 3.12). If the DNA from all 46 chromosomes in a human somatic cell—that is, any cell that does not become an egg or sperm cell—was stretched out end to end, it would be about 2 nm wide and 2 m long. Each chromosome contains thousands of individual genes that are the units of information about heritable traits. Each gene has a particular location in a specific chromosome and contains the code for producing one of the three forms of RNA (ribosomal RNA, messenger RNA, and transfer RNA). The Human Genome Project was begun in 1990 and had as its goal to first identify the location of at least 3,000 specific human genes and then to determine the sequence of nucleotides (about 3 billion!) in a complete set of haploid human chromosomes (one chromosome from each of the 23 pairs). See Chapter 13 for more information about the Human Genome Project. DNA replication occurs during cell division (Figure 3.13). During this semiconservative process, enzymes unzip the double helix, deliver complementary bases to the nucleotides, and bind the delivered nucleotides into the developing complementary strands. Following replication, each strand of DNA is duplicated so two double helices now exist, each consisting of one strand of the original DNA and one new strand. In this way, each daughter cell gets the same hereditary information that was contained in the original dividing cell. During replication, some enzymes check for accuracy, while others repair pairing mistakes so the error rate is reduced to approximately one per billion. Since DNA remains in the nucleus, where it is protected from the action of the cell’s enzymes, and proteins are made on ribosomes outside of the nucleus, a method (transcription) exists for transferring information from the DNA to the cytoplasm. During transcription (Figure 3.14), the sequence of nucleotides in a gene that codes for a protein is transferred to messenger RNA (mRNA) through complementary base pairing of the nucleotide sequence in the gene. For example, a DNA sequence of TACGCTCCGATA would become AUGCGAGGCUAU in the mRNA. The process is somewhat more complicated, since the transcript produced directly from the DNA contains sequences of nucleotides, called introns, that are removed before the final mRNA is produced. The mRNA also has a tail, called a poly-A tail, of about 100–200 adenine nucleotides attached to one end. A cap with a nucleotide that has a methyl group and phosphate groups bonded to it is attached at the other end of the mRNA. Transcription differs from replication in that (1) only a certain stretch of DNA acts as the template and not the whole strand, (2) different enzymes are used, and (3) only a single strand is produced.

3. ANATOMY AND PHYSIOLOGY

5' Sugar-phosphate backbone

3' C

A G

Sugar-phosphate backbone

(a)

5' (b)

3' 3'

5' A

A 3'

(c)

FIGURE 3.13

During replication, DNA helicase shown as a black wedge in (b) unzips the double helix (a). Another enzyme, DNA polymerase, then copies each side of the unzipped chain in the 5’ to 3’ direction. One side of the chain (5’ to 3’) can be copied continuously, while the opposite side (3’ to 5’) is copied in small chunks in the 5’ to 3’ direction that are bound together by another enzyme, DNA ligase. Two identical double strands of DNA are produced as a result of replication.

After being transcribed, the mRNA moves out into the cytoplasm through the nuclear pores and binds to specific sites on the surface of the two subunits that make up a ribosome (Figure 3.15). In addition to the ribosomes, the cytoplasm contains amino acids and another form of RNA: transfer RNA (tRNA). Each tRNA contains a triplet of bases, called an anticodon, and binds at an area away from the triplet to an amino acid that is specific for that particular anticodon. The mRNA that was produced from the gene in the nucleus also contains bases in sets of three. Each triplet in the mRNA is called a codon. The four

3.2 CELLULAR ORGANIZATION

FIGURE 3.14 During transcription, RNA is formed from genes in the cell’s DNA by complementary base pairing to one of the strands. RNA contains uracil (U) rather than thymine (T), so the Ts in the first two pairs of the DNA become Us in the single-stranded RNA.

possibilities for nucleotides (A, U, C, G) in each of the three places give rise to 64 (43) possible codons. These 64 codons make up the genetic code. Each codon codes for a specific amino acid, but some amino acids are specified by more than one codon (Table 3.1). For example, AUG is the only mRNA codon for methionine (the amino acid that always signals the starting place for translation—the process by which the information from a gene is used to produce a protein), while UUA, UUG, CUU, CUC, CUA, and CUG are all codons for leucine. The anticodon on the tRNA that delivers the methionine to the ribosome is UAC, whereas tRNAs with anticodons of AAU, AAC, GAA, GAG, GAU, and GAC deliver leucine. During translation, the mRNA binds to a ribosome and tRNA delivers amino acids to the growing polypeptide chain in accordance with the codons specified by the mRNA. Peptide bonds are formed between each newly delivered amino acid and the previously delivered one. When the amino acid is bound to the growing chain, it is released from the tRNA, and the tRNA moves off into the cytoplasm, where it joins with another amino acid that is specified by its anticodon. This process continues until a stop codon (UAA, UAG, or UGA) is reached on the mRNA. The protein is then released into the cytoplasm or into the rough ER for further modifications.

3. ANATOMY AND PHYSIOLOGY

Cytoplasm Nucleus C--G

T--A

A--T

G--C

DNA

mRNA Ribosome

mRNA

tRNA amino acid

Polypeptide

FIGURE 3.15 Following transcription from DNA and processing in the nucleus, mRNA moves from the nucleus to the cytoplasm. In the cytoplasm, the mRNA joins with a ribosome to begin the process of translation. During translation, tRNA delivers amino acids to the growing polypeptide chain. Which amino acid is delivered depends on the three-base codon specified by the mRNA. Each codon is complementary to the anticodon of a specific tRNA. Each tRNA binds to a particular amino acid at a site that is opposite the location of the anticodon. For example, the codon CUG in mRNA is complementary to the anticodon GAC in the tRNA that carries leucine and will result in adding the amino acid leucine to the polypeptide chain.

EXAMPLE PROBLEM 3.5 Consider a protein that contains the amino acids asparagine, phenylalanine, histidine, and serine in sequence. Which nucleotide sequences on DNA (assuming that there were no introns) would result in this series of amino acids? What would be the anticodons for the tRNAs that delivered these amino acids to the ribosomes during translation?

Solution The genetic code (see Table 3.1) provides the sequence for the mRNA codons that specify these amino acids. The mRNA codons can be used to determine the sequence in the original DNA and the anticodons of the tRNA, since the mRNA bases must pair with the bases in both DNA and tRNA. Note that DNA contains thymine (T) but no uracil (U) and that both mRNA and tRNA contain U and not T. See Figures 3.3 and 3.14 for examples of base pairing.

mRNA codon DNA tRNA anticodon

Asparagine (Asn)

Phenylalanine (Phe)

Histidine (His)

Serine (Ser)

AAU or AAC TTA or TTG UUA or UUG

UUU or UUC AAA or AAG AAA or AAG

CAU or CAC GTA or GTG GUA or GUG

UC(A, G, U, or C) AG(T, C, A, or G) AG(U, C, A, or G)

3.3 TISSUES

Second Base

Third Base

TABLE 3.1 The Genetic Code First Base

Lys

Ile

Arg

Thr

Lys

Met - Start

Arg

Thr

Asn

Ile

Ser

Thr

Asn

Ile

Ser

Thr

Stop

Leu

Stop

Ser

Stop

Leu

Trp

Ser

Tyr

Phe

Cys

Ser

Tyr

Phe

Cys

Ser

Glu

Val

Gly

Ala

Glu

Val

Gly

Ala

Asp

Val

Gly

Ala

Asp

Val

Gly

Ala

Gln

Leu

Arg

Pro

Gln

Leu

Arg

Pro

His

Leu

Arg

Pro

His

Leu

Arg

Pro

Amino acid 3-letter and 1-letter codes: Ala (A) ¼ Alanine; Arg (R) ¼ Arginine; Asn (N) ¼ Asparagine; Asp (D) ¼ Aspartic acid; Cys (C) ¼ Cysteine; Glu (E) ¼ Glutamic acid; Gln (Q) ¼ Glutamine; Gly (G) ¼ Glycine; His (H) ¼ Histidine; Ile (I) ¼ Isoleucine; Leu (L) ¼ Leucine; Lys (K) ¼ Lysine; Met (M) ¼ Methionine; Phe (F) ¼ Phenylalanine; Pro (P) ¼ Proline; Ser (S) ¼ Serine; Thr (T) ¼ Threonine; Trp (W) ¼ Tryptophan; Tyr (Y) ¼ Tyrosine; Val (V) ¼ Valine.

3.3 TISSUES Groups of cells and surrounding substances that function together to perform one or more specialized activities are called tissues (Figure 3.16). The four primary types of tissue in the human body are epithelial, connective, muscle, and nervous. Epithelial tissues are either composed of cells arranged in sheets that are one or more layers thick or are organized into glands that are adapted for secretion. They are also characterized by having a free surface— for example, the inside surface of the intestines or the outside of the skin—and a basilar membrane. Typical functions of epithelial tissue include absorption (lining of the small intestine), secretion (glands), transport (kidney tubules), excretion (sweat glands), protection (skin, Figure 3.16a), and sensory reception (taste buds). Connective tissues are the most abundant and widely distributed. Connective tissue proper can be loose (loosely woven fibers found around and between organs), irregularly dense (protective capsules around organs), and regularly dense (ligaments and tendons), whereas specialized connective tissue includes blood

3. ANATOMY AND PHYSIOLOGY

HAIR SHAFT EPIDERMIS

WHITE BLOOD CELLS

DERMIS

FAT (a)

RED BLOOD CELLS

HAIR FOLLICLE

SEBACEOUS GLAND SWEAT GLAND

ARRECTOR PILI MUSCLE

(b)

DENDRITES CARDIAC

AXON SKELETAL CELL BODY SMOOTH

NUCLEUS

NODE OF RANVIER PRESYNAPTIC TERMINALS

(c)

(d)

FIGURE 3.16

Four tissue types. Skin (a) is a type of epithelial tissue that helps protect the body. Blood (b) is a specialized connective tissue. The three types of muscle tissue (c) are cardiac, skeletal, and smooth. Motor neurons (d) are a type of nervous tissue that conducts electrical impulses from the central nervous system to effector organs such as muscles.

(Figure 3.16b), bone, cartilage, and adipose tissue. Muscle tissue provides movement for the body through its specialized cells that can shorten in response to stimulation and then return to their uncontracted state. Figure 3.16c shows the three types of muscle tissue: skeletal (attached to bones), smooth (found in the walls of blood vessels), and cardiac (found only in the heart). Nervous tissue consists of neurons (Figure 3.16d) that conduct electrical impulses and glial cells that protect, support, and nourish neurons.

3.4 MAJOR ORGAN SYSTEMS Combinations of tissues that perform complex tasks are called organs, and organs that function together form organ systems. The human body has 11 major organ systems: integumentary, endocrine, lymphatic, digestive, urinary, reproductive, circulatory, respiratory, nervous, skeletal, and muscular. The integumentary system (skin, hair, nails, and various

3.4 MAJOR ORGAN SYSTEMS

glands) provides protection for the body. The endocrine system (ductless glands such as the thyroid and adrenals) secretes hormones that regulate many chemical actions within cells. The lymphatic system (glands, lymph nodes, lymph, lymphatic vessels) returns excess fluid and protein to the blood and helps defend the body against infection and tissue damage. The digestive system (stomach, intestines, and other structures) ingests food and water, breaks food down into small molecules that can be absorbed and used by cells, and removes solid wastes. The urinary system (kidneys, ureters, urinary bladder, and urethra) maintains the fluid volume of the body, eliminates metabolic wastes, and helps regulate blood pressure and acid-base and water-salt balances. The reproductive system (ovaries, testes, reproductive cells, and accessory glands and ducts) produces eggs or sperm and provides a mechanism for the production and nourishment of offspring. The circulatory system (heart, blood, and blood vessels) serves as a distribution system for the body. The respiratory system (airways and lungs) delivers oxygen to the blood from the air and carries away carbon dioxide. The nervous system (brain, spinal cord, peripheral nerves, and sensory organs) regulates most of the body’s activities by detecting and responding to internal and external stimuli. The skeletal system (bones and cartilage) provides protection and support as well as sites for muscle attachments, the production of blood cells, and calcium and phosphorus storage. The muscular system (skeletal muscle) moves the body and its internal parts, maintains posture, and produces heat. Although biomedical engineers have made major contributions to understanding, maintaining, and/or replacing components in each of the 11 major organ systems, only the last 5 listed will be examined in greater detail.

3.4.1 Circulatory System The circulatory system (Figure 3.17) delivers nutrients and hormones throughout the body, removes waste products from tissues, and provides a mechanism for regulating temperature and removing the heat generated by the metabolic activities of the body’s internal organs. Every living cell in the body is no more than 10–100 mm from a capillary (small blood vessels with walls only one cell thick that are 8 mm in diameter, approximately the same size as a red blood cell). This close proximity allows oxygen, carbon dioxide, and most other small solutes to diffuse from the cells into the capillary or from the capillary into the cells, with the direction of diffusion determined by concentration and partial pressure gradients. Accounting for about 8 þ/– 1 percent of total body weight, averaging 5,200 ml, blood is a complex, heterogeneous suspension of formed elements—the blood cells, or hematocytes— suspended in a continuous, straw-colored fluid called plasma. Nominally, the composite fluid has a mass density of 1.057 þ/– 0.007 g/cm3, and it is six times as viscous as water. The hematocutes include three basic types of cells: red blood cells (erythrocytes, totaling nearly 95 percent of the formed elements), white blood cells (leukocytes, averaging less than .15 percent of all hematocytes), and platelets (thrombocytes, on the order of 5 percent of all blood cells). Hematocytes are all derived in the active (“red”) bone marrow (about 1,500 g) of adults from undifferentiated stem cells called hemocytoblasts, and all reach ultimate maturity via a process called hematocytopoiesis. The primary function of erythrocytes is to aid in the transport of blood gases—about 30 to 34 percent (by weight) of each cell consisting of the oxygen- and carbon dioxide–carrying protein hemoglobin (64,000 MW 68,000) and a small portion of the cell containing the

3. ANATOMY AND PHYSIOLOGY

COMMON CAROTID ARTERIES RIGHT SUBCLAVIAN ARTERY DESCENDING AORTA COMMON ILIAC ARTERIES

(a)

RIGHT SUBCLAVIAN VEIN

LEFT SUBCLAVIAN ARTERY SUPERIOR VENA CAVA ASCENDING AORTA HEART

INTERNAL JUGULAR VEINS

LEFT SUBCLAVIAN VEIN INFERIOR VENA CAVA COMMON ILIAC VEINS

(b)

FIGURE 3.17

(a) The distribution of the main arteries in the body that carry blood away from the heart. (b) The distribution of the main veins in the body that return the blood to the heart.

enzyme carbonic anhydrase, which catalyzes the reversible formation of carbonic acid from carbon dioxide and water. The primary function of lukocytes is to endow the human body with the ability to identify and dispose of foreign substances (such as infectious organisms) that do not belong there—agranulocytes (lymphocytes and monocytes) essentially doing the “identifying” and granulocytes (neutrophils, basophils, and esinophils) essentially doing the “disposing.” The primary function of platelets is to participate in the bloodclotting process. Removal of all hematocytes from blood centrifugation or other separating techniques leaves behind the aqueous (91 percent water by weight, 94.8 percent water by volume), saline (0.15N) suspending medium called plasma—which has an average mass density of 1.035 þ/– 0.005 g/cm3 and a viscosity 1½ to 2 times that of water. Some 6.5 to 8 percent by weight of plasma consists of the plasma proteins, of which there are three major types—albumin, the globulins, and fibrinogen—and several of lesser prominence. The primary functions of albumin are to help maintain the osmotic (oncotic) transmural pressure differential that ensures proper mass exchange between blood and interstitial fluid at the capillary level and to serve as a transport carrier molecule for several hormones and other small biochemical constituents (such as some metal ions). The primary function of the globulin class of proteins is to act as transport carrier molecules (mostly of the a and b) for large biochemical substances, such as fats (lipoproteins) and certain carbohydrates (muco- and glycoproteins) and heavy metals (mineraloproteins), and to work together with leukocytes in the body’s immune system. The latter function is primarily the responsibility

3.4 MAJOR ORGAN SYSTEMS

of the g class of immunoglobulins, which have antibody activity. The primary function of fibrinogen is to work with thrombocytes in the formation of a blood clot—a process also aided by one of the most abundant of the lesser proteins, prothrombin. Of the remaining 2 percent or so (by weight) of plasma, just under half consists of minerals (inorganic ash), trace elements, and electrolytes, mostly the cations sodium, potassium, calcium, and magnesium and the anions chlorine, bicarbonate, phosphate, and sulfate—the latter three helping as buffers to maintain the fluid at a slightly alkaline pH between 7.35 and 7.45 (average 7.4). What is left, about 1,087 mg materials per deciliter of plasma, includes (1) mainly three major types of fat—cholesterol (in a free and esterified form), phospholipids (a major ingredient of cell membranes), and triglyceride—with lesser amounts of the fat-soluble vitamins (A, D, E, and K), free fatty acids, and other lipids, and (2) “extractives” (0.25 percent by weight), of which about two-thirds include glucose and other forms of carbohydrate, the remainder consisting of the water-soluble vitamins (B-complex and C), certain enzymes, nonnitrogenous and nitrogenous waste products of metabolism (including urea, creatine, and creatinine), and many smaller amounts of other biochemical constituents—the list seeming virtuously endless. It is easy to understand why blood is often referred to as the “river of life.” This river is made to flow through the vascular piping network by two central pumping stations arranged in series: the left and right sides of the human heart. The heart (Figure 3.18), the pumping station that moves blood through the blood vessels, consists of two pumps: the right side and the left side. Each side has one chamber (the atrium) that receives blood and another chamber (the ventricle) that pumps the blood away from the heart. The right side moves deoxygenated blood that is loaded with carbon dioxide from the body to the lungs, and the left side receives oxygenated blood that has had most of its carbon dioxide removed from the lungs and pumps it to the body. The vessels that lead to and from the lungs make up the pulmonary circulation, and those that lead to and from the rest of the tissues in the body make up the systemic circulation (Figure 3.19). SUPERIOR VENA CAVA

RIGHT ATRIUM

AORTIC ARCH SUPERIOR VENA CAVA LEFT ATRIUM

PULMONARY ARTERY LEFT RIGHT ATRIUM ATRIUM

MITRAL VALVE

TRICUSPID VALVE

RIGHT VENTRICLE (a)

PULMONARY SEMILUNAR VALVE RIGHT VENTRICLE SEPTUM

LEFT VENTRICLE

(b)

FIGURE 3.18 (a) The outside of the heart as seen from its anterior side. (b) The same view after the exterior surface of the heart has been removed. The four interior chambers—right and left atria and right and left ventricles—as well as several valves are visible.

3. ANATOMY AND PHYSIOLOGY

VENULES SYSTEMIC CIRCULATION VEINS SUPERIOR VENA CAVA PULMONARY CIRCULATION RIGHT ATRIUM RIGHT VENTRICLE INFERIOR VENA CAVA SYSTEMIC CIRCULATION VEINS

ARTERIOLES SYSTEMIC CIRCULATION ARTERIES ASCENDING AORTA AORTIC ARCH PULMONARY CIRCULATION PULMONARY ARTERY LEFT ATRIUM LEFT VENTRICLE DESCENDING AORTA SYSTEMIC CIRCULATION ARTERIES

FIGURE 3.19

Oxygenated blood leaves the heart through the aorta. Some of the blood is sent to the head and upper extremities and torso, whereas the remainder goes to the lower torso and extremities. The blood leaves the aorta and moves into other arteries, then into smaller arterioles, and finally into capillary beds, where nutrients, hormones, gases, and waste products are exchanged between the nearby cells and the blood. The blood moves from the capillary beds into venules and then into veins. Blood from the upper part of the body returns to the right atrium of the heart through the superior vena cava, whereas blood from the lower part of the body returns through the inferior vena cava. The blood then moves from the right atrium to the right ventricle and into the pulmonary system through the pulmonary artery. After passing through capillaries in the lungs, the oxygenated blood returns to the left atrium of the heart through the pulmonary vein. It moves from the left atrium to the left ventricle and then out to the systemic circulation through the aorta to begin the same trip over again.

Blood vessels that carry blood away from the heart are called arteries, while those that carry blood toward the heart are called veins. The pulmonary artery is the only artery that carries deoxygenated blood, and the pulmonary vein is the only vein that carries oxygenated blood. The average adult has about 5 L of blood with 80 to 90 percent in the systemic circulation at any one time; 75 percent of the blood is in the systemic circulation in the veins, 20 percent is in the arteries, and 5 percent is in the capillaries. Because of the anatomic proximity of the heart to the lungs, the right side of the heart does not have to work very hard to drive blood through the pulmonary circulation, so it functions as a low-pressure (P 40 mmHg gauge) pump compared with the left side of the heart, which does most of its work at a high pressure (up to 140 mmHg gauge or more) to drive blood through the entire systemic circulation to the furthest extremes of the organism. In order of size, the somewhat spherically shaped left atrium is the smallest chamber— holding about 45 ml of blood (at rest). The pouch-shaped right atrium is next (63 ml of blood), followed by the conical/cylindrically shaped left ventricle (100 ml of blood) and the crescent-shaped right ventricle (about 130 ml of blood). Altogether, then, the heart chambers collectively have a capacity of some 325 to 350 ml, or about 6.5 percent of the total blood volume in a “typical” individual—but these values are nominal, since the organ alternately fills and expands, contracts, and then empties.

3.4 MAJOR ORGAN SYSTEMS

During the 480-ms or so filling phase—diastole—of the average 750-ms cardiac cycle, the inlet valves of the two ventricles (3.8-cm-diameter tricuspid valve from right atrium to right ventricle; 3.1-cm-diameter bicuspid or mitral valve from left atrium to left ventricle) are open, and the outlet valves (2.4-cm-diameter pulmonary valve and 2.25-cm-diameter aortic semilunar valve, respectively) are closed—the heart ultimately expanding to its enddiastolic-volume (EDV), which is on the order of 140 ml of blood for the left ventricle. During the 270-ms emptying phase—systole—electrically induced vigorous contraction of cardiac muscle drives the intraventricular pressure up, forcing the one-way inlet valves closed and the unidirectional outlet valves open as the heart contracts to its end-systolicvolume (ESV), which is typically on the order to 70 ml of blood for the left ventricle. Thus, the ventricles normally empty about half their contained volume with each heartbeat, the remainder being termed the cardiac reserve volume. More generally, the difference between the actual EDV and the actual ESV, called the stroke volume (SV), is the volume of blood expelled from the heart during each systolic interval, and the ratio of SV to EDV is called the cardiac ejection faction, or ejection ratio (0.5 to 0.75 is normal, 0.4 to 0.5 signifies mild cardiac damage, 0.25 to 0.40 implies moderate heart damage, and less than 0.25 warns of severe damage to the heart’s pumping ability). If the stroke volume is multiplied by the number of systolic intervals per minute, or heart rate (HR), one obtains the total cardiac output (CO): CO ¼ HR ðEDV ESVÞ where EDV-ESV to the stroke volume. Several investigations have suggested that the cardiac output (in milliliters per minute) is proportional to the weight W (in kilograms) of an individual according to the equation CO 224W3=4 and that “normal” heart rate obeys very closely the relation HR ¼ 229W1=4 For a “typical” 68.7-kg individual (blood volume ¼ 5,200 ml), these equations yield CO ¼ 5,345 ml/min, HR ¼ 80 beats/min (cardiac cycle period ¼ 754 ms) and SV ¼ CO/HR ¼ 229W1/4/CO-224W3/4 ¼ 0.978W ¼ 67.2 ml/beat, which are very reasonable values. Furthermore, assuming this individual lives to be about 75 years old, his or her heart will have cycled over 3.1536 billion times, pumping a total of 0.2107 billion liters of blood (55.665 million gallons, or 8,134 quarts per day) within their lifetime. In the normal heart, the cardiac cycle, which refers to the repeating pattern of contraction (systole) and relaxation (diastole) of the chambers of the heart, begins with a self-generating electrical pulse in the pacemaker cells of the sinoatrial node (Figure 3.20). This rapid electrical change in the cells is the result of the movement of ions across their plasma membranes. The permeability of the plasma membrane to Naþ changes dramatically and allows these ions to rush into the cell. This change in the electrical potential across the plasma membrane from one in which the interior of the cell is more negative than the extracellular fluid (approximately –90 mV) to one in which the interior of the cell is more positive than the extracellular fluid (approximately 20 mV) is called depolarization. After a very short period of time (6 mm) before they can reach the respiratory zone. Epithelial cells that line the trachea and bronchi have cilia that beat in a coordinated fashion to move mucus toward the pharynx, where it can be swallowed or expectorated. The respiratory zone, consisting of respiratory bronchioles with outpouchings of alveoli and terminal clusters of alveolar sacs, is where gas exchange between air and blood occurs (Figure 3.24b). The respiratory zone comprises most of the mass of the lungs. Conduction of air begins at the larynx, or voice box, at the entrance to the trachea, which is a fibromuscular tube 10 to 12 cm in length and 1.4 to 2.0 cm in diameter. At a location called the carina, the trachea terminates and divides in the left and right bronchi. Each bronchus has a discontinuous cartilaginous support in its wall. Muscle fibers capable of controlling airway diameter are incorporated into the walls of the bronchi, as well as in those of air passages closer to the alveoli. Smooth muscle is present throughout the respiratory bronchiolus and alveolar ducts but is absent in the last alveolar duct, which terminates in one to several alveoli. The alveolar walls are shared by other alveoli and are composed of highly pliable and collapsible squamous epithelium cells. The bronchi subdivide into subbronchi, which further subdivide into bronchioli, which further subdivide, and so on, until finally reaching the alveolar level. Movement of gases in the respiratory airways occurs mainly by bulk flow (convection) throughout the region from the mouth to the nose to the fifteenth generation. Beyond the

NASAL CAVITY

TERMINAL BRONCHIOLE

PHARYNX LARYNX

ORAL CAVITY

TRACHEA

ALVEOLUS

LEFT LUNG

ALVEOLAR DUCT

RIGHT LUNG

ALVEOLAR SAC BRONCHIOLES (a)

FIGURE 3.24

(b)

(a) The respiratory system consists of the passageways that are used to move air into and out of the body and the lungs. (b) The terminal bronchioles and alveolar sacs within the lungs have alveoli where gas exchange occurs between the lungs and the blood in the surrounding capillaries.

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fifteenth generation, gas diffusion is relatively more important. With the low gas velocities that occur in diffusion, dimensions of the space over which diffusion occurs (alveolar space) must be small for adequate oxygen delivery into the walls; smaller alveoli are more efficient in transfer of gas than are larger ones. Thus, animals with high levels of oxygen consumption are found to have smaller-diameter alveoli compared with animals with low levels of oxygen consumption. Alveoli are the structures through which gases diffuse to and from the body. To ensure that gas exchange occurs efficiently, alveolar walls are extremely thin. For example, the total tissue thickness between the inside of the alveolus to pulmonary capillary blood plasma is only about 0.4 106 m. Consequently, the principal barrier to diffusion occurs at the plasma and red blood cell level, not at the alveolar membrane. Molecular diffusion within the alveolar volume is responsible for mixing of the enclosed gas. Due to small alveolar dimensions, complete mixing probably occurs in less than 10 ms, fast enough that alveolar mixing time does not limit gaseous diffusion to or from the blood. Of particular importance to proper alveolar operation is a thin surface coating of surfactant. Without this material, large alveoli would tend to enlarge and small alveoli would collapse. It is the present view that surfactant acts like a detergent, changing the stressstrain relationship of the alveolar wall and thereby stabilizing the lung. Certain physical properties, such as compliance, elasticity, and surface tension, are characteristic of lungs. Compliance refers to the ease with which lungs can expand under pressure. A normal lung is about 100 times more distensible than a toy balloon. Elasticity refers to the ease with which the lungs and other thoracic structures return to their initial sizes after being distended. This aids in pushing air out of the lungs during expiration. Surface tension is exerted by the thin film of fluid in the alveoli and acts to resist distention. It creates a force that is directed inward and creates pressure in the alveolus, which is directly proportional to the surface tension and inversely proportional to the radius of the alveolus (Law of Laplace). Thus, the pressure inside an alveolus with a small radius would be higher than the pressure inside an adjacent alveolus with a larger radius and would result in air flowing from the smaller alveolus into the larger one. This could cause the smaller alveolus to collapse. This does not happen in normal lungs because the fluid inside the alveoli contains a phospholipid that acts as a surfactant. The surfactant lowers the surface tension in the alveoli and allows them to get smaller during expiration without collapsing. Premature babies often suffer from respiratory distress syndrome because their lungs lack sufficient surfactant to prevent their alveoli from collapsing. These babies can be kept alive with mechanical ventilators or surfactant sprays until their lungs mature enough to produce surfactant. Breathing, or ventilation, is the mechanical process by which air is moved into (inspiration) and out of (expiration) the lungs. A normal adult takes about 15 to 20 breaths per minute. During inspiration, the inspiratory muscles contract and enlarge the thoracic cavity, the portion of the body where the lungs are located. This causes the alveoli to enlarge and the alveolar gas to expand. As the alveolar gas expands, the partial pressure within the respiratory system drops below atmospheric pressure by about 3 mmHg so air easily flows in (Boyle’s Law). During expiration, the inspiratory muscles relax and return the thoracic cavity to its original volume. Since the volume of the gas inside the respiratory system

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has decreased, its pressure increases to a value that is about 3 mmHg above atmospheric pressure. Air now moves out of the lungs and into the atmosphere. The primary purpose of the respiratory system is gas exchange. In the gas-exchange process, gas must diffuse through the alveolar space, across tissue, and through plasma into the red blood cell, where it finally chemically joins to hemoglobin. A similar process occurs for carbon dioxide elimination. As long as intermolecular interactions are small, most gases of physiologic significance can be considered to obey the ideal gas law: pV ¼ nRT where p V n R T

¼ ¼ ¼ ¼ ¼

pressure, N/m3 volume of gas, m3 number of moles, mol gas constant, (N m)/(mol K) absolute temperature, K

The ideal gas law can be applied without error up to atmospheric pressure; it can be applied to a mixture of gases, such as air, or to its constituents, such as oxygen or nitrogen. All individual gases in a mixture are considered to fill the total volume and have the same temperature but reduced pressures. The pressure exerted by each individual gas is called the partial pressure of the gas. Dalton’s law states that the total pressure is the sum of the partial pressures of the constituents of a mixture: P¼

N X

i¼1

where pi ¼ partial pressure of the ith constituent, N/m3 N ¼ total number of constituents Dividing the ideal gas law for a constituent by that for the mixture gives Pi V ni Ri T ¼ PV nRT so that pi ni Ri ¼ p nR which states that the partial pressure of a gas may be found if the total pressure, mole fraction, and ratio of gas constants are known. For most respiratory calculations, p will be considered to be the pressure of 1 atmosphere, 101 kN/m2. Avogadro’s principle states that different gases at the same temperature and pressure contain equal numbers of molecules: V1 nR1 R1 ¼ ¼ V2 nR2 R2

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Thus, pi Vi ¼ p V where Vi/V is the volume fraction of a constituent in air and is therefore dimensionless. Table 3.2 provides individual gas constants, as well as volume fractions of constituent gases of air. Lung mechanics refers to the study of the mechanical properties of the lung and chest wall, whereas lung statics refers to the mechanical properties of a lung in which the volume is held constant over time. Understanding lung mechanics requires knowledge about the volumes within the lungs. Lung capacities contain two or more volumes. The tidal volume (TV) is the amount of air that moves in and out of the lungs during normal breathing (Figure 3.25). The total lung capacity (TLC) is the amount of gas contained within the lungs at the end of a maximum inspiration. The vital capacity (VC) is the maximum amount of air that can be exhaled from the lungs after inspiration to TLC. The residual volume (RV) is the amount of gas remaining in the lungs after maximum exhalation. The amount of gas that can be inhaled after inhaling during tidal breathing is called the inspiratory reserve volume (IRV). The amount of gas that can be expelled by a maximal exhalation after exhaling during tidal breathing is called the expiratory reserve volume (ERV). The inspiratory capacity (IC) is the maximum amount of gas that can be inspired after a normal exhalation during tidal breathing, and the functional residual capacity (FRC) is the amount of gas that remains in the lungs at this time (Table 3.3). All of the volumes and capacities except those that include the residual volume can be measured with a spirometer. The classic spirometer is an air-filled container that is constructed from two drums of different sizes. One drum contains water, and the other air-filled drum is inverted over an air-filled tube and floats in the water. The tube is connected to a mouthpiece used by the patient. When the patient inhales, the level of the

TABLE 3.2 Molecular Masses, Gas Constants, and Volume Fractions for Air and Constituents Constituent

Molecular Mass kg/mol

Gas Constant N m/(mol K)

Volume Fraction in Air m3/m3

Air

29.0

286.7

1.0000

Ammonia

17.0

489.1

0.0000

Argon

39.9

208.4

0.0093

Carbon Dioxide

44.0

189.0

0.0003

Carbon Monoxide

28.0

296.9

0.0000

Helium

4.0

2,078.6

0.0000

Hydrogen

2.0

4,157.2

0.0000

Nitrogen

28.0

296.9

0.7808

Oxygen

32.0

259.8

0.2095

Note: Universal gas constant is 8314.43 N m/kg K.

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2,000

1,000

EXPIRATORY RESERVE VOLUME

FUNCTIONAL RESIDUAL CAPACITY

TOTAL LUNG CAPACITY

3,000

TIDAL VOLUME

4,000

VITAL CAPACITY

VOLUME (ML)

5,000

MAXIMUM INSPIRATION

INSPIRATORY CAPACITY

6,000

INSPIRATORY RESERVE VOLUME

3. ANATOMY AND PHYSIOLOGY

RESIDUAL VOLUME 0 MAXIMUM EXPIRATION TIME

FIGURE 3.25

Lung volumes and capacities, except for residual volume, functional residual capacity, and total lung capacity, can be measured using spirometry.

floating drum drops. When the patient exhales, the level of the floating drum rises. These changes in floating drum position can be recorded and used to measure lung volumes.

EXAMPLE PROBLEM 3.9 The total lung capacity of a patient is 5.9 liters. If the patient’s inspiratory capacity was found to be 3.3 liters using spirometry, what would be the patient’s functional residual capacity? What would you need to measure in order to determine the patient’s residual volume?

Solution From Figure 3.25, total lung capacity (TLC) is equal to the sum of inspiratory capacity (IC) and functional residual capacity (FRC). TLC ¼ IC þ FRC 5:9 l ¼ 3:3 l þ FRC FRC ¼ 2:6 l TLC, which cannot be determined by means of spirometry, and vital capacity (VC), which can be measured using spirometry, must be known in order to determine residual volume (RV), since TLC VC ¼ RV

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TABLE 3.3 Typical Lung Volumes for a Normal, Healthy Male Lung Volume

Normal Values

Total lung capacity (TLC)

6.0 103m3

(6,000 cm3)

Residual volume (RV)

1.2 103m3

(1,200 cm3)

Vital capacity (VC)

4.8 103m3

(4,800 cm3)

Inspiratory reserve volume (IRV)

3.6 103m3

(3,600 cm3)

Expiratory reserve volume (ERV)

1.2 103m3

(1,200 cm3)

Functional residual capacity (FRC)

2.4 103m3

(2,400 cm3)

Anatomic dead volume (VD)

1.5 104m3

(150 cm3)

Upper airways volume

8.0 105m3

(80 cm3)

Lower airways volume

7.0 105m3

(70 cm3)

Physiologic dead volume (VD)

1.8 104m3

(180 cm3)

Minute volume (Ve) at rest

1.0 104m3/s

(6,000 cm3/m)

Respiratory period (T) at rest

Tidal volume (VT) at rest

4.0 104m3

(400 cm3)

Alveolar ventilation volume (VA) at rest

2.5 104m3

(250 cm3)

Minute volume during heavy exercise

1.7 103m3/s

(10,000 m3/m)

Respiratory period during heavy exercise

1.2s

Tidal volume during heavy exercise

2.0 103m3

(2,000 cm3)

Alveolar ventilation during heavy exercise

1.8 103m3

(1,820 cm3)

Since spirograms record changes in volume over time, flow rates can be determined for different maneuvers. For example, if a patient exhales as forcefully as possible to residual volume following inspiration to TLC, then the forced expiratory volume (FEV1.0) is the total volume exhaled at the end of 1 s. The FEV1.0 is normally about 80 percent of the vital capacity. Restrictive diseases, in which inspiration is limited by reduced compliance of the lung or chest wall or by weakness of the inspiratory muscles, result in reduced values for FEV1.0 and vital capacity, but their ratio remains about the same. In obstructive diseases, such as asthma, the FEV1.0 is reduced much more than the vital capacity. In these diseases, the TLC is abnormally large, but expiration ends prematurely. Another useful measurement is the forced expiratory flow rate (FEF25–75 percent), which is the average flow rate measured over the middle half of the expiration—that is, from 25 to 75 percent of the vital capacity. Flow-volume loops provide another method for analyzing lung function by relating the rate of inspiration and expiration to the volume of air that is moved during each process.

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The TLC can be measured using the gas dilution technique. In this method, patients inspire to TLC from a gas mixture containing a known amount of an inert tracer gas, such as helium, and hold their breath for 10 s. During this time, the inert gas becomes evenly distributed throughout the lungs and airways. Due to conservation of mass, the product of initial tracer gas concentration (which is known) times the amount inhaled (which is measured) equals the product of final tracer gas concentration (which is measured during expiration) times the TLC. Body plethysmography, which provides the most accurate method for measuring lung volumes, uses an airtight chamber in which the patient sits and breathes through a mouthpiece. This method uses Boyle’s Law, which states that the product of pressure and volume for gas in a chamber is constant under isothermal conditions. Changes in lung volume and pressure at the mouth when the patient pants against a closed shutter can be used to calculate the functional residual capacity. Since the expiratory reserve volume can be measured, the residual volume can be calculated by subtracting it from the functional residual capacity.

EXAMPLE PROBLEM 3.10 A patient is allowed to breathe a mixture from a 2-liter reservoir that contains 10 percent of an inert gas—that is, a gas that will not cross from the lungs into the circulatory system. At the end of a period that is sufficient for the contents of the reservoir and the lungs to equilibrate, the concentration of the inert gas is measured and is found to be 2.7 percent. What is the patient’s total lung capacity?

Solution The total amount of inert gas is the same at the beginning and end of the measurement, but its concentration has changed from 10 percent (C1) to 2.7 percent (C2). At the beginning, it is confined to a 2-liter reservoir (V1). At the end, it is in both the reservoir and the patient’s lungs (V2 ¼ V1 þ TLC). C1 V1 ¼ C2 V2 ð0:1Þð2 lÞ ¼ ð0:027Þð2 l þ TLCÞ 0:2 l 0:054 l ¼ 0:027 TLC 5:4 l ¼ TLC

External respiration occurs in the lungs when gases are exchanged between the blood and the alveoli (Figure 3.26). Each adult lung contains about 3.5 108 alveoli, which results in a large surface area (60–70 m2) for gas exchange to occur. Each alveolus is only one cell layer thick, so the air-blood barrier is only two cells thick (an alveolar cell and a capillary endothelial cell), which is about 2 mm. The partial pressure of oxygen in the alveoli is higher than the partial pressure of oxygen in the blood, so oxygen moves from the alveoli into the blood. The partial pressure of carbon dioxide in the alveoli is lower than the partial pressure of carbon dioxide in the blood, so carbon dioxide moves from the blood into the alveoli. During internal respiration, carbon dioxide and oxygen move between the blood and the extracellular fluid surrounding the body’s cells. The direction and rate of

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3.4 MAJOR ORGAN SYSTEMS

CARBON DIOXIDE

ALVEOLUS

CAPILLARY

RED BLOOD CELL

OXYGEN

FIGURE 3.26

During external respiration, oxygen moves from the alveoli to the blood, and carbon dioxide moves from the blood to the air within the alveoli.

movement of a gas depends on the partial pressures of the gas in the blood and the extracellular fluid, the surface area available for diffusion, the thickness of the membrane that the gas must pass through, and a diffusion constant that is related to the solubility and molecular weight of the gas (Fick’s Law). Mechanical ventilators can be used to deliver air or oxygen to a patient. They can be electrically or pneumatically powered and can be controlled by microprocessors. Negative pressure ventilators, such as iron lungs, surround the thoracic cavity and force air into the lungs by creating a negative pressure around the chest. This type of ventilator greatly limits access to the patient. Positive pressure ventilators apply high-pressure gas at the entrance to the patient’s lungs so air or oxygen flows down a pressure gradient and into the patient. These ventilators can be operated in control mode to breathe for the patient at all times or in assist mode to help with ventilation when the patient initiates the breathing cycle. This type of ventilation changes the pressure within the thoracic cavity to positive during inspiration, which affects venous return to the heart and cardiac output (the amount of blood the heart moves with each beat). High-frequency jet ventilators deliver very rapid (60–90 breaths per minute) low-volume bursts of air to the lungs. Oxygen and carbon dioxide are exchanged by molecular diffusion rather than by the mass movement of air. This method causes less interference with cardiac output than does positive pressure ventilation. Extracorporeal membrane oxygenation (ECMO) uses the technology that was developed for cardiopulmonary bypass machines. Blood is removed from the patient and passed through an artificial lung, where oxygen and carbon dioxide are exchanged. It is warmed to body temperature before being returned to the patient. This technique allows the patient’s lungs to rest and heal themselves and has been used successfully on some cold water drowning victims and on infants with reversible pulmonary disease.

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3.4.3 The Nervous System The most exciting and mysterious part of the human body is the magical 3½ pounds of tissue we carry around inside our skulls: the brain. For centuries, the brain has frustrated those daring enough to explore its secrets. Encased not only in its bony protective covering but also in a shroud of mysticism, it has been an extremely difficult structure to study. Even with the invention of the microscope and the discovery of electricity, generations of Western scientists refrained from investigating the activity of the human brain out of respect for it as the seat of a human’s immortal soul. In recent years, this convoluted mass, the source of all thought and emotion, has been the focal point of intense scientific investigation. There has been a great flurry of activity in assembling interdisciplinary teams consisting of physiologists, psychologists, biochemists, and engineers in order to gain a better understanding of brain function. To many of these individuals, the brain represents a symbolic Mt. Everest, an obstacle to be scaled and conquered before it will be better understood. And yet, in spite of all the efforts to date, we are still only in the foothills of such a climb. The mechanisms and processes that enable the brain to convert the variety of electrical and chemical activity occurring within it into thoughts, feeling, dreams, and memories—the fundamental awareness of self—are still beyond our understanding. However, in spite of the difficulties encountered and the frustrations experienced by explorers in this world of the mind, significant progress has been made in deciphering the cryptic flow of electrical energy coming from the brain. In reviewing these electrical signals, it has been possible to detect the presence of certain patterns or rhythms that occur in the brain that represent a “language” that can be recognized and understood by neural circuits in the brain itself. The fundamental building block of this neuronal communication network is the individual nerve cell: the neuron. Figure 3.16d is a schematic drawing of just such a cell. It consists of three major components: the cell body itself, or soma; the receptor zone, or dendrites; and a long fiber called the axon, which carries electrical signals from the main body of the cell to the muscles, glands, or other neurons. Numbering approximately 20 billion in each human being, these tiny cells come in a variety of sizes and shapes. However, nowhere is more variety displayed than in the length of the axonal terminating fiber. In the human body, it ranges from a few thousandths of an inch up to three feet or more, depending on the type of neuron involved. Consider, for example, the long pathways from the extremities to the brain. In these communication channels between the periphery and the “central data processor” that we call the brain, only a few neurons may be connected to one another. As a result, the axon of these nerve cells may be as long as 2 or 3 feet, even though the cell body is quite small. Some axons are surrounded by sheaths of myelin that are formed by specialized nonneural cells called Schwann cells. Gaps, called “Nodes of Ranvier,” in the myelin sheath allow the action potential generated by the neuron to travel more rapidly by essentially jumping from one node to the next. Neurons are anatomically distinct units with no physical continuity between them. The transmitting portion of a neuron, its axon, ends in a series of synapses, thereby making contact with other neurons (see Figure 3.28). Under the microscope this often stands out as a spherical enlargement at the end of the axon to which various names have been given, for example, boutons, end-feet, or synaptic terminals. This ending does not actually make contact with the soma or the dendrite but is separated by a narrow cleft (gap) that is, on

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113

average, 100 to 200 Angstroms (109 meters) wide. This is known as the synaptic cleft. Each of these synaptic endings contains a large number of submicroscopic spherical structures (synaptic vesicles) that can be detected only under the electron microscope. These synaptic vesicles, in turn, are essentially “chemical carriers” containing transmitter substance that is released into the synaptic clefts on excitation. With this information in hand, let us consider the sequence of events that enables one neuron to communicate with another. When an individual neuron is excited, an electrical signal is transmitted along its axon to many tiny branches, diverging fibers near its far end. These axonal terminals end or synapse close to the “input terminals” (the dendrites and cell body) of a large number of other neurons. When an electrical pulse arrives at the synapse, it triggers the release of a tiny amount of transmitter substance. This chemical carrier floats across the synaptic cleft between the axonal fiber and the cell body, thereby altering the status of the receiving neuron. For example, the chemical emissions may urge the receiving neuron into a state whereby this second cell is activated and conducts a similar electrical pulse to its axon. In this way, the initial electrical signal may be propagated to a still more remote part of the other hand. If the surfaces of muscle cells lie close enough to a number of such terminals to receive a substantial supply of these chemical carriers, the muscle will experience a resulting electrochemical reaction of its own that will cause it to contract and thereby perform some mechanical chore. In a similar manner, a gland can be stimulated to secrete the chemical characteristic to its activity. Neurons with the ability to cause a muscular or glandular reaction are known as effector neurons or motoneurons. For most of us, the most pleasurable sensations come from our perception of the world around us. This sense of awareness is made possible by still another group of specialized neurons known as receptor cells. Acting as input devices, these neurons accept and convert various sensory information into appropriate electrical impulses that can then be properly processed within the nervous system. These receptors measure such quantities as pressure, warmth, cold, and displacement, as well as the presence of specific chemicals. Considering that every minute of one’s life the brain is virtually bombarded by such a voluminous amount of incoming information, it is astounding that it can function at all. So as we have seen, nerve cells are responsible for the following variety of essential functions: 1. Accepting and converting sensory information into a form that can be processed with the nervous system by other neurons. 2. Processing and analyzing this information so an “integrated portrait” of the incoming data can be obtained. 3. Translating the final outcome or “decision” of this analysis process into an appropriate electrical or chemical form needed to stimulate glands or activate muscles. The nervous system, which is responsible for the integration and control of all the body’s functions, has been divided by neuroscientists into the central nervous system (CNS) and the peripheral nervous system (PNS) (Figure 3.27). The former consists of all nervous tissue enclosed by bone (e.g., the brain and spinal cord), and the latter consists of all nervous tissue not enclosed by bone, which enables the body to detect and respond to both internal and external stimuli. The peripheral nervous system consists of the 12 pairs of cranial and 31 pairs of spinal nerves with afferent (sensory) and efferent (motor) neurons.

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BRAIN SPINAL CORD SPINAL NERVES

SCIATIC NERVE

FIGURE 3.27 The central nervous system (CNS) consists of all nervous tissue that is enclosed by bone—that is, the brain and spinal cord—whereas the peripheral nervous system (PNS) consists of the nervous tissue that is not encased by bone.

MITOCHONDRION

AXON TERMINAL

VESICLES SYNAPSE

NEUROTRANSMITTER

FIGURE 3.28 Following stimulation, vesicles in the axon terminal move to the synapse by means of exocytosis and release neurotransmitters into the space between the axon and the next cell, which could be the dendrite of another neuron, a muscle fiber, or a gland. The neurotransmitters diffuse across the synapse and elicit a response from the adjacent cell.

3.4 MAJOR ORGAN SYSTEMS

115

Clusters of nerve cells located in the CNS are called nuclei, and clusters of nerve cells in the PNS are called ganglion. On the other hand, nucleons located in the PNS have been designated as nerves, while those in the CNS are called tracts. The nervous system has also been divided into the somatic and autonomic nervous systems. Each of these systems consists of components from both the central and peripheral nervous systems. For example, the somatic peripheral nervous system consists of the sensory neurons, which convey information from receptors for pain, temperature, and mechanical stimuli in the skin, muscles, and joints to the central nervous system, and the motor neurons, which return impulses from the central nervous system to these same areas of the body. The autonomic nervous system is concerned with the internal meter of the body, including involuntary regulation of smooth muscle, cardiac muscle, and glands and is further divided into the sympathetic and parasympathetic divisions. The sympathetic division causes blood vessels in the viscera and skin to constrict, vessels in the skeletal muscles to dilate, and the heart rate to increase, whereas the parasympathetic division has the opposite effect on the vessels in the viscera and skin, provides no innervation to the skeletal muscles, and causes the heart rate to decrease. Thus, the sympathetic division prepares the body for “fight or flight,” while the parasympathetic division returns the body to normal operating conditions. Brain function is dependent on neuronal circuits. Neurons interconnect in several different types of circuits. In a divergent circuit, each branch in the axon of the presynaptic neuron connects with the dendrite of a different postsynaptic neuron. In a convergent circuit, axons from several presynaptic neurons meet at the dendrite(s) of a single postsynaptic neuron. In a simple feedback circuit, the axon of a neuron connects with the dendrite of an interneuron that connects back with the dendrites of the first neuron. A two-neuron circuit is one in which a sensory neuron synapses directly with a motor neuron, whereas a three-neuron circuit consists of a sensory neuron, an interneuron in the spinal cord, and a motor neuron. Both of these circuits can be found in reflex arcs (Figure 3.29). The reflex arc is a special type of neural circuit that begins with a sensory neuron at a receptor (e.g., a pain receptor in the fingertip) and ends with a motor neuron at an effector (e.g., a skeletal muscle).

FIGURE 3.29

This reflex arc begins with a sensory neuron in the finger that senses pain when the fingertip is pricked by the pin. An action potential travels from the sensory neuron to an interneuron and then to a motor neuron that synapses with muscle fibers in the finger. The muscle fibers respond to the stimulus by contracting and removing the fingertip from the pin.

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Withdrawal reflexes are elicited primarily by stimuli for pain and heat great enough to be painful and are also known as protective or escape reflexes. They allow the body to respond quickly to dangerous situations without taking additional time to send signals to and from the brain and to process the information. The brain is a large, soft mass of nervous tissue and consists of the cerebrum, the diencephalon, the mesencephalon (midbrain), and the brain stem and cerebellum. The cerebrum (Figure 3.30a), which is divided into two hemispheres, is the largest and most obvious portion of the brain and consists of many convoluted ridges (gyri), narrow grooves (sulci), and deep fissures, which result in a total surface area of about 2.25 m2. The outer layer of the cerebrum, the cerebral cortex, is composed of gray matter (neurons with unmyelinated HYPOTHALAMUS CORPUS CALLOSUM CEREBRUM

THALAMUS PINEAL BODY MIDBRAIN

CEREBELLUM LATERAL SULCUS

PONS MEDULLA OBLONGATA (b)

(a)

White matter Gray matter Posterior median sulcus Central canal Anterior median fissure Pia mater

Posterior rootlets Posterior root Posterior root ganglion Spinal nerve Anterior root

Subarachnoid space Arachnoid

Anterior rootlets

Dura mater

(c)

FIGURE 3.30 the spinal cord.

Anterior view

(a) The exterior surface of the brain. (b) A midsagittal section through the brain. (c) Structure of

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axons) that is 2–4 mm thick and contains over 50 billion neurons and 250 billion glial cells called neuroglia. The thicker inner layer is the white matter that consists of interconnecting groups of myelinated axons that project from the cortex to other cortical areas or from the thalamus (part of the diencephalon) to the cortex. The connection between the two cerebral hemispheres takes place via the corpus callosum (Figure 3.30b). The left side of the cortex controls motor and sensory functions from the right side of the body, whereas the right side controls the left side of the body. Association areas that interpret incoming data or coordinate a motor response are connected to the sensory and motor regions of the cortex. Fissures divide each cerebral hemisphere into a series of lobes that include the frontal lobe, the parietal lobe, the temporal lobe, and the occipital lobe. Each of these lobes has different functions. The functions of the frontal lobe include initiating voluntary movement of the skeletal muscles, analyzing sensory experiences, providing responses relating to personality, and mediating responses related to memory, emotions, reasoning, judgment, planning, and speaking. The parietal lobe responds to stimuli from cutaneous (skin) and muscle receptors throughout the body. The temporal lobes interpret some sensory experiences, store memories of auditory and visual experiences, and contain auditory centers that receive sensory neurons from the cochlea of the ear. The occipital lobes integrate eye movements by directing and focusing the eye and are responsible for correlating visual images with previous visual experiences and other sensory stimuli. The insula is a deep portion of the cerebrum that lies under the parietal, frontal, and temporal lobes. Little is known about its function, but it seems to be associated with gastrointestinal and other visceral activities. The diencephalon is the deep part of the brain that connects the midbrain of the brain stem with the cerebral hemispheres. Its main parts include the thalamus, hypothalamus, and epithalamus (Figure 3.30b). The thalamus, the major switchboard of the brain, is involved with sensory and motor systems, general neural background activity, and the expression of emotion and uniquely human behaviors. Due to its two-way communication with areas of the cortex, it is linked with thought, creativity, interpretation and understanding of spoken and written words, and identification of objects sensed by touch. The hypothalamus is involved with integration within the autonomic nervous system, temperature regulation, water and electrolyte balance, sleep-wake patterns, food intake, behavioral responses associated with emotion, endocrine control, and sexual responses. The epithalamus contains the pineal body that is thought to have a neuroendocrine function. The brain stem connects the brain with the spinal cord and automatically controls vital functions such as breathing. Its principal regions include the midbrain, pons, and medulla oblongota (Figure 3.30b). The midbrain connects the pons and cerebellum with the cerebrum and is located at the upper end of the brain stem. It is involved with visual reflexes, the movement of eyes, focusing of the lenses, and the dilation of the pupils. The pons is a rounded bulge between the midbrain and medulla oblongata that functions with the medulla oblongata to control respiratory functions, acts as a relay station from the medulla oblongata to higher structures in the brain, and is the site of emergence of cranial nerve V. The medulla oblongata is the lowermost portion of the brain stem and connects the pons to the spinal cord. It contains vital centers that regulate heart rate, respiratory rate, constriction and dilation of blood vessels, blood pressure, swallowing, vomiting, sneezing, and coughing. The cerebellum is located behind the pons and is the second largest part of the brain. It processes sensory information that is used by the motor systems and is involved with

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coordinating skeletal muscle contractions and impulses for voluntary muscular movement that originate in the cerebral cortex. The cerebellum is a processing center that is involved with coordination of balance, body positions, and the precision and timing of movements. The gray matter of the spinal cord is divided into the dorsal and ventral horns. In a human, standing upright, the “dorsal” horn is posterior and the “ventral” horn is anterior. Dorsal horn neurons receive and process sensory information from the skin, while ventral horn neurons participate in the control of skeletal muscle contraction. The gray matter is surrounded by columns (funiculi) of white matter containing ascending and descending axons. The spinal cord communicates with the periphery via the dorsal and ventral root fibers that exit between the bony vertebra. Dorsal root fibers bring information to the spinal cord, and ventral root fibers carry information away from the spinal cord (Figure 3.30c).

3.4.4 The Skeletal System The average adult skeleton contains 206 bones, but the actual number varies from person to person and decreases with age as some bones become fused. Like the body, the skeletal system is divided into two parts: the axial skeleton and the appendicular skeleton (Figure 3.31). The axial skeleton contains 80 bones (skull, hyoid bone, vertebral column, and thoracic cage), whereas the appendicular skeleton contains 126 (pectoral and pelvic girdles and upper and lower extremities). The skeletal system protects and supports the

SKULL CLAVICLE MANDIBLE SCAPULA HUMERUS STERNUM ULNA RADIUS CARPALS

RIB VERTEBRAL COLUMN PELVIS SACRUM COCCYX

METACARPALS PHALANGES

FEMUR PATELLA TIBIA FIBULA TARSALS

METATARSALS

PHALANGES

FIGURE 3.31 The skull, hyoid bone (not shown), vertebral column, and thoracic cage (ribs, cartilage, and sternum) make up the axial skeleton, whereas the pectoral (scapula and clavicle) and pelvic girdles and upper and lower extremities make up the appendicular skeleton.

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body, helps with movement, produces blood cells, and stores important minerals. It is made up of strong, rigid bones that are composed of specialized connective tissue, bear weight, and form the major supporting elements of the body. Some support also comes from cartilage that is a smooth, firm, resilient, nonvascular type of connective tissue. Since the bones of the skeleton are hard, they protect the organs, such as the brain and abdominal organs, that they surround. There are 8 cranial bones that support, surround, and protect the brain. Fourteen facial bones form the face and serve as attachments for the facial muscles that primarily move skin rather than bone. The facial bones, except for the lower jaw (mandible), are joined with each other and with the cranial bones. There are 6 auditory ossicles, 3 in each ear, that transmit sound waves from the external environment to the inner ear. The hyoid bone, which is near the skull but is not part of it, is a small U-shaped bone that is located in the neck just below the lower jaw. It is attached to the skull and larynx (voice box) by muscles and ligaments and serves as the attachment for several important neck and tongue muscles. The vertebral column starts out with approximately 34 bones, but only 26 independent ones are left in the average human adult. There are 7 cervical bones including the axis, which acts as a pivot around which the head rotates, and the atlas, which sits on the axis and supports the “globe” of the head. These are followed by 5 cervical, 12 thoracic, and 5 lumbar vertebrae and then the sacrum and the coccyx. The last two consist of 5 fused vertebrae. The vertebral column supports the weight of and allows movement of the head and trunk, protects the spinal cord, and provides places for the spinal nerves to exit from the spinal cord. There are four major curves (cervical, thoracic, lumbar, and sacral/coccygeal) in the adult vertebral column that allow it to flex and absorb shock. While movement between any two adjacent vertebrae is generally quite limited, the total amount of movement provided by the vertebral column can be extensive. The thoracic cage consists of 12 thoracic vertebrae (which are counted as part of the vertebral column), 12 pairs of ribs and their associated cartilage, and the sternum (breastbone). It protects vital organs and prevents the collapse of the thorax during ventilation. Bones are classified as long, short, flat, or irregular, according to their shape. Long bones, such as the femur and humerus, are longer than they are wide. Short bones, such as those found in the ankle and wrist, are as broad as they are long. Flat bones, such as the sternum and the bones of the skull, have a relatively thin and flattened shape. Irregular bones do not fit into the other categories and include the bones of the vertebral column and the pelvis. Bones make up about 18 percent of the mass of the body and have a density of 1.9 g/cm3. The two types of bone are spongy and compact (cortical). Spongy bone forms the ends (epiphyses) of the long bones and the interior of other bones and is quite porous. Compact bone forms the shaft (diaphysis) and outer covering of bones and has a tensile strength of 120 N/mm2, compressive strength of 170 N/mm2, and Young’s modulus of 1.8 104 N/mm2. The medullary cavity, a hollow space inside the diaphysis, is filled with fatty yellow marrow or red marrow that contains blood-forming cells. Bone is a living organ that is constantly being remodeled. Old bone is removed by special cells, osteoclasts, and new bone is deposited by osteoblasts. Bone remodeling occurs during bone growth and in order to regulate calcium availability. The average skeleton is totally remodeled about three times during a person’s lifetime. Osteoporosis is a disorder in which old bone is broken down faster than new bone is produced so the resulting bones are weak and brittle.

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(a)

FIGURE 3.32

(b)

(c)

Bones of the skeletal system are attached to each other at (a) fibrous, (b) cartilaginous, or (c) syno-

vial joints.

The bones of the skeletal system are attached to one another at fibrous, cartilaginous, or synovial joints (Figure 3.32). The articulating bones of fibrous joints are bound tightly together by fibrous connective tissue. These joints can be rigid and relatively immovable to slightly movable. This type of joint includes the suture joints in the skull. Cartilage holds together the bones in cartilaginous joints. These joints allow limited motion in response to twisting or compression and include the joints of the vertebral system and the joints that attach the ribs to the vertebral column and to the sternum. Synovial joints, such as the knee, are the most complex and varied and have fluid-filled joint cavities, cartilage that covers the articulating bones, and ligaments that help hold the joints together. Synovial joints are classified into six types, depending on their structure and the type of motion they permit. Gliding joints (Figure 3.33) are the simplest type of synovial joint, allow

FIGURE 3.33

GLIDING

PIVOT

SADDLE

BALL-ANDSOCKET

CONDYLOID

HINGE

Synovial joints have fluid-filled cavities and are the most complex and varied types of joints. Each synovial joint is classified into one of six types, depending on its structure and type of motion.

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back-and-forth or side-to-side movement, and include the intercarpal articulations in the wrist. Hinge joints, such as the elbow, permit bending in only one plane and are the most common type of synovial joint. The atlas and axis provide an example of a pivot joint that permits rotation. In condyloid articulations, an oval, convex surface of one bone fits into a concave depression on another bone. Condyloid joints, which include the metacarpophalangeal joints (knuckles) of the fingers, permit flexion-extension and rotation and are considered to be biaxial because rotation is limited to two axes of movement. The saddle joint, represented by the joint at the base of the thumb, is a modified condyloid joint that permits movement in several directions (multiaxial). Ball-and-socket joints allow motion in many directions around a fixed center. In these joints, the ball-shaped head of one bone fits into a cuplike concavity of another bone. This multiaxial joint is the most freely movable of all and includes the shoulder and hip joints. Biomedical engineers have helped develop artificial joints that are routinely used as replacements in diseased or injured hips, shoulders, and knees (Figure 3.34).

3.4.5 Muscular System The muscular system (Figure 3.35) is composed of 600–700 skeletal muscles, depending on whether certain muscles are counted as separate or as pairs, and makes up 40 percent of the body’s mass. The axial musculature makes up about 60 percent of the skeletal muscles in the body and arises from the axial skeleton (see Figure 3.31). It positions the head and spinal column and moves the rib cage during breathing. The appendicular musculature moves or stabilizes components of the appendicular skeleton. The skeletal muscles in the muscular system maintain posture, generate heat to maintain the body’s temperature, and provide the driving force that is used to move the bones and joints of the body and the skin of the face. Muscles that play a major role in accomplishing a movement are called prime movers, or agonists. Muscles that act in opposition to a prime

(a)

(b)

FIGURE 3.34 Diseased or damaged hip (a) and knee (b) joints that are nonfunctional or extremely painful can be replaced by prostheses. Artificial joints can be held in place by a special cement (polymethylmethacrylate [PMMA]) and by bone ingrowth. Special problems occur at the interfaces due to the different elastic moduli of the materials (110 GPa for titanium, 2.2 GPa for PMMA, and 20 GPa for bone).

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TRAPEZIUS PECTORALIS MAJOR BICEPS BRACHII ANTERIOR FOREARM MUSCLES

QUADRICEPS FEMORIS VASTUS LATERALIS RECTUS FEMORIS VASTUS MEDIALIS

STERNOCLEIDOMASTOID

DELTOID SERRATUS ANTERIOR EXTERNAL ABDOMINAL OBLIQUE TENSOR FASCIAE LATTAE SARTORIUS PATELLAR LIGAMENT

GASTROCNEMIUS TIBIALIS ANTERIOR SOLEUS

FIGURE 3.35

Some of the major skeletal muscles on the anterior side of the body.

mover are called antagonists, whereas muscles that assist a prime mover in producing a movement are called synergists. The continual contraction of some skeletal muscles helps maintain the body’s posture. If all of these muscles relax, which happens when a person faints, the person collapses. A system of levers, which consists of rigid lever arms that pivot around fixed points, is used to move skeletal muscle (Figure 3.36). Two different forces act on every lever: the weight to be moved—that is, the resistance to be overcome—and the pull or effort applied—that is, the applied force. Bones act as lever arms, and joints provide a fulcrum. The resistance to be overcome is the weight of the body part that is moved, and the applied force is generated by the contraction of a muscle or muscles at the insertion, the point of attachment of a muscle to the bone it moves. An example of a first-class lever, one in which the fulcrum is between the force and the weight, is the movement of the facial portion of the head when the face is tilted upward. The fulcrum is formed by the joint between the atlas and the occipital bone of the skull, and the vertebral muscles inserted at the back of the head generate the applied force that moves the weight, the facial portion of the head. A second-class lever is one in which the weight is between the force and the fulcrum. This can be found in the body when a person stands on “tip toe.” The ball of the foot is the fulcrum, and the applied force is generated by the calf muscles on the back of the leg. The weight that is moved is that of the whole body. A third-class lever is one in which the force is between the weight and the fulcrum. When a person has a bent elbow and holds

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CLASS 2

CLASS 1

CLASS 3

FIGURE 3.36

Depending on the muscle in use, the location of the load, and the location of the fulcrum, the humerus can act as a class 1 lever, a class 2 lever, or a class 3 lever.

a ball in front of the body, the applied force is generated by the contraction of the biceps brachii muscle. The weight to be moved includes the ball and the weight of the forearm and hand, and the elbow acts as the fulcrum. The three types of muscle tissue—cardiac, skeletal, and smooth—share four important characteristics: contractility, the ability to shorten; excitability, the capacity to receive and respond to a stimulus; extensibility, the ability to be stretched; and elasticity, the ability to return to the original shape after being stretched or contracted. Cardiac muscle tissue is found only in the heart, whereas smooth muscle tissue is found within almost every other organ, where it forms sheets, bundles, or sheaths around other tissues. Skeletal muscles are composed of skeletal muscle tissue, connective tissue, blood vessels, and nervous tissue. Each skeletal muscle is surrounded by a layer of connective tissue (collagen fibers) that separates the muscle from surrounding tissues and organs. These fibers come together at the end of the muscle to form tendons, which connect the skeletal muscle to bone, to skin (face), or to the tendons of other muscles (hand). Other connective tissue fibers divide the skeletal muscles into compartments called fascicles that contain bundles of muscle fibers. Within each fascicle, additional connective tissue surrounds each skeletal muscle fiber and ties adjacent ones together. Each skeletal muscle fiber has hundreds of nuclei just beneath the cell membrane. Multiple nuclei provide multiple copies of the genes that direct the production of enzymes and structural proteins needed for normal contraction so contraction can occur faster.

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SARCOPLASMIC RETICULUM ACTIN AND MYOSIN FILAMENTS

SARCOLEMMA

TRANSVERSE TUBULE

SARCOPLASM

NUCLEUS MYOFIBRIL

MITOCHONDRIA

FIGURE 3.37

Skeletal muscles are composed of muscle fascicles that are composed of muscle fibers such as the one shown here. Muscle fibers have hundreds of nuclei just below the plasma membrane—the sarcolemma. Transverse tubules extend into the sarcoplasm, the cytoplasm of the muscle fiber, and are important in the contraction process because they deliver action potentials that result in the release of stored calcium ions. Calcium ions are needed to create active sites on actin filaments so cross-bridges can be formed between actin and myosin and the muscle can contract.

In muscle fibers, the plasma membrane is called the sarcolemma, and the cytoplasm is called the sarcoplasm (Figure 3.37). Transverse tubules (T tubules) begin at the sarcolemma and extend into the sarcoplasm at right angles to the surface of the sarcolemma. The T tubules, which play a role in coordinating contraction, are filled with extracellular fluid and form passageways through the muscle fiber. They make close contact with expanded chambers, cisternae, of the sarcoplasmic reticulum, a specialized form of the ER. The cisternae contain high concentrations of calcium ions that are needed for contraction to occur. The sarcoplasm contains cylinders 1 or 2 mm in diameter that are as long as the entire muscle fiber and are called myofibrils. The myofibrils are attached to the sarcolemma at each end of the cell and are responsible for muscle fiber contraction. Myofilaments—protein filaments consisting of thin filaments (primarily actin) and thick filaments (mostly myosin)— are bundled together to make up myofibrils. Repeating functional units of myofilaments are called sarcomeres (Figure 3.38). The sarcomere is the smallest functional unit of the muscle fiber and has a resting length of about 2.6 mm. The thin filaments are attached to dark bands, called Z lines, which form the ends of each sarcomere. Thick filaments containing double-headed myosin molecules lie between the thin ones. It is this overlap of thin and thick filaments that gives skeletal muscle its banded, striated appearance. The I band is the area in a relaxed muscle fiber that just contains actin filaments, and the H zone is the area that just contains myosin filaments. The H zone and the area in which the actin and myosin overlap form the A band. When a muscle contracts, myosin molecules in the thick filaments form cross-bridges at active sites in the actin of the thin filaments and pull the thin filaments toward the center of

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SARCOMERE SARCOMERE

A BAND

MYOSIN

H ZONE I BAND

Z LINE ACTIN

FIGURE 3.38 The sarcomere is the basic functional unit of skeletal muscles and extends from one Z line to the next. Actin filaments are attached to the Z lines and extend into the A band, where they overlap with the thicker myosin filaments. The H zone is the portion of the A band that contains no overlapping actin filaments. When the muscle changes from its extended, relaxed position (left panel) to its contracted state (right panel), the myosin filaments use cross-bridges to slide past the actin filaments and bring the Z lines closer together. This results in shorter sarcomeres and a contracted muscle.

the sarcomere. The cross-bridges are then released and reformed at a different active site further along the thin filament. This results in a motion that is similar to the hand-overhand motion that is used to pull in a rope. This action, the sliding filament mechanism, is driven by ATP energy and results in shortening of the muscle. Shortening of the muscle components (contraction) results in bringing the muscle’s attachments (e.g., bones) closer together (Figure 3.38). Muscle fibers have connections with nerves. Sensory nerve endings are sensitive to length, tension, and pain in the muscle and send impulses to the brain via the spinal cord, whereas motor nerve endings receive impulses from the brain and spinal cord that lead to excitation and contraction of the muscle. Each motor axon branches and supplies several muscle fibers. Each of these axon branches loses its myelin sheath and splits up into a number of terminals that make contact with the surface of the muscle. When the nerve is stimulated, vesicles in the axon terminals release a neurotransmitter, acetylcholine, into the synapse between the neuron and the muscle. Acetylcholine diffuses across the synapse and binds to receptors in a special area, the motor end plate, of the sarcolemma. This causes the sodium channels in the sarcolemma to open up, and an action potential is produced in the muscle fiber. The resulting action potential spreads over the entire sarcolemmal surface and travels down all of the T tubules, where it triggers a sudden massive release of calcium by the cisternae. Calcium triggers the production of active sites on the thin filaments so cross-bridges with myosin can form and contraction occurs. Acetylcholinesterase breaks down the acetylcholine while the contraction process is under way so the original relatively low permeability of the sarcolemma to sodium is restored.

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A motor unit consists of a motor neuron and the muscle fibers that it innervates. All the muscle fibers in a single motor unit contract at the same time, whereas muscle fibers in the same muscle but belonging to different motor units may contract at different times. When a contracted muscle relaxes, it returns to its original (resting) length if another contracting muscle moves it or if it is acted upon by gravity. During relaxation, ATP is expended to move calcium back to the cisternae. The active sites that were needed for cross-bridge formation become covered so actin and myosin can no longer interact. When the cross-bridges disappear, the muscle returns to its resting length—that is, it relaxes. The human body contains two different types of skeletal muscle fibers: fast and slow. Fast fibers can contract in 10 ms or less following stimulation and make up most of the skeletal muscle fibers in the body. They are large in diameter and contain densely packed myofibrils, large glycogen reserves (used to produce ATP), and relatively few mitochondria. These fibers produce powerful contractions that use up massive amounts of ATP and fatigue (can no longer contract in spite of continued neural stimulation) rapidly. Slow fibers take about three times as long to contract as fast fibers. They can continue to contract for extended periods of time because they contain (1) a more extensive network of capillaries, so they can receive more oxygen; (2) a special oxygen-binding molecule called myoglobin; and (3) more mitochondria, which can produce more ATP than fast fibers. Muscles contain different amounts of slow and fast fibers. Those that are dominated by fast fibers (e.g., chicken breast muscles) appear white, while those that are dominated by slow fibers (e.g., chicken legs) appear red. Most human muscles appear pink because they contain a mixture of both. Genes determine the percentage of fast and slow fibers in each muscle, but the ability of fast muscle fibers to resist fatigue can be increased through athletic training.

3.5 HOMEOSTASIS Organ systems work together to maintain a constant internal environment within the body. Homeostasis is the process by which physical and chemical conditions within the internal environment of the body are maintained within tolerable ranges even when the external environment changes. Body temperature, blood pressure, and breathing and heart rates are some of the functions that are controlled by homeostatic mechanisms that involve several organ systems working together. Extracellular fluid—the fluid that surrounds and bathes the body’s cells—plays an important role in maintaining homeostasis. It circulates throughout the body and carries materials to and from the cells. It also provides a mechanism for maintaining optimal temperature and pressure levels, the proper balance between acids and bases, and concentrations of oxygen, carbon dioxide, water, nutrients, and many of the chemicals that are found in the blood. Three components—sensory receptors, integrators, and effectors—interact to maintain homeostasis (Figure 3.39). Sensory receptors, which may be cells or cell parts, detect stimuli—that is, changes to their environment—and send information about the stimuli to integrators. Integrators are control points that pull together information from one or more sensory receptors. Integrators then elicit a response from effectors. The brain is an

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3.5 HOMEOSTASIS

RESPONSE

RECEPTOR

INTEGRATOR

EFFECTOR

STIMULUS

FIGURE 3.39 Feedback mechanisms are used to help maintain homeostatis. A stimulus is received by a receptor that sends a signal (messenger) to an effector or to an integrator that sends a signal to an effector. The effector responds to the signal. The response feeds back to the receptor and modifies the effect of the stimulus. In negative feedback, the response subtracts from the effect of the stimulus on the receptor. In positive feedback, the response adds to the effect of the stimulus on the receptor.

integrator that can send messages to muscles or glands or both. The messages result in some type of response from the effectors. The brain receives information about how parts of the body are operating and can compare this to information about how parts of the body should be operating. Positive feedback mechanisms are those in which the initial stimulus is reinforced by the response. There are very few examples of this in the human body, since it disrupts homeostasis. Childbirth provides one example. Pressure from the baby’s head in the birth canal stimulates receptors in the cervix, which send signals to the hypothalamus. The hypothalamus responds to the stimulus by releasing oxytocin, which enhances uterine contractions. Uterine contractions increase in intensity and force the baby further into the birth canal, which causes additional stretching of the receptors in the cervix. The process continues until the baby is born, the pressure on the cervical stretch receptors ends, and the hypothalamus is no longer stimulated to release oxytocin. Negative feedback mechanisms result in a response that is opposite in direction to the initiating stimulus. For example, receptors in the skin and elsewhere in the body detect the body’s temperature. Temperature information is forwarded to the hypothalamus in the brain, which compares the body’s current temperature to what the temperature should be (approximately 37 C). If the body’s temperature is too low, messages are sent to contract the smooth muscles in blood vessels near the skin (reducing the diameter of the blood vessels and the heat transferred through the skin), to skeletal muscles to start contracting rapidly (shivering), and to the arrector pili muscles (see Figure 3.16a) to erect the hairs and form “goose bumps.” The metabolic activity of the muscle contractions generates heat and warms the body. If the body’s temperature is too high, messages are sent to relax the smooth muscles in the blood vessels near the skin (increasing the diameter of the blood vessels and the amount of heat transferred through the skin) and to sweat glands to release moisture and thus increase evaporative cooling of the skin. When the temperature of circulating blood changes to such an extent in the appropriate direction that it reaches the set point of the system, the hypothalamus stops sending signals to the effector muscles and glands. Another example of a negative feedback mechanism in the body involves the regulation of glucose in the bloodstream by clusters of cells, the pancreatic islets (Figure 3.40). There are between 2 105 and 2 106 pancreatic islets scattered throughout the adult

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HIGH BLOOD GLUCOSE

LOW BLOOD GLUCOSE

PANCREAS

GLUCAGON RELEASED BY ALPHA CELLS OF PANCREAS

LIVER RELEASES GLUCOSE INTO BLOOD

INSULIN RELEASED BY BETA CELLS OF PANCREAS

FAT CELLS TAKE IN GLUCOSE FROM BLOOD

ACHIEVE NORMAL BLOOD GLUCOSE LEVELS

FIGURE 3.40 Two negative feedback mechanisms help control the level of glucose in the blood. When blood glucose levels are higher than the body’s set point (stimulus), beta cells (receptors) in the pancreatic islets produce insulin (messenger), which facilitates glucose transport across plasma membranes and enhances the conversion of glucose into glycogen for storage in the liver (effector). This causes the level of glucose in the blood to drop. When the level equals the body’s set point, the beta cells stop producing insulin. When blood glucose levels are lower than the body’s set point (stimulus), alpha cells (receptors) in the pancreatic islets produce glucagon (messenger), which stimulates the liver (effector) to convert glycogen into glucose. This causes the level of glucose in the blood to increase. When the level equals the body’s set point, the alpha cells stop producing glucagon.

pancreas. When glucose levels are high, beta cells in the islets produce insulin, which facilitates glucose transport across plasma membranes and into cells and enhances the conversion of glucose into glycogen that is stored in the liver. During periods of fasting, or whenever the concentration of blood glucose drops below normal (70–110 mg/dl), alpha cells produce glucagon, which stimulates the liver to convert glycogen into glucose and the formation of glucose from noncarbohydrate sources such as amino acids and lactic acid. When glucose levels return to normal, the effector cells in the pancreatic islets stop producing their respective hormone—that is, insulin or glucagon. Some biomedical

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engineers are working on controlled drug delivery systems that can sense blood glucose levels and emulate the responses of the pancreatic islet cells, whereas other biomedical engineers are trying to develop an artificial pancreas that would effectively maintain appropriate blood glucose levels.

3.6 EXERCISES 1. Using as many appropriate anatomical terms as apply, write sentences that describe the positional relationship between your mouth and (a) your left ear, (b) your nose, and (c) the big toe on your right foot. 2. Using as many appropriate anatomical terms as apply, describe the position of the stomach in the body and its position relative to the heart. 3. Search the Internet to find a transverse section of the body that was imaged using computerized tomography (CT) or magnetic resonance imaging (MRI). Print the image and indicate its web address. 4. Search the Internet to find a frontal section of the body that was imaged using CT or MRI. Print the image and indicate its web address. 5. Name and give examples of the four classes of biologically important organic compounds. What are the major functions of each of these groups? 6. What are the molarity and osmolarity of a 1-liter solution that contains half a mole of calcium chloride? How many molecules of chloride would the solution contain? 7. Consider a simple model cell, such as the one in Figure 3.6, that consists of cytoplasm and a plasma membrane. The cell’s initial volume is 2 nl and contains 0.2 M protein. The cell is placed in a large volume of 0.05 M CaCl2. Neither Caþþ nor Cl can cross the plasma membrane and enter the cell. Is the 0.05 M CaCl2 solution hypotonic, isotonic, or hypertonic relative to the osmolarity inside the cell? Describe what happens to the cell as it achieves equilibrium in this new environment. What will be the final osmolarity of the cell? What will be its final volume? 8. What does the principle of electrical neutrality mean in terms of the concentration of ions within a cell? 9. Consider the same model cell that was used in problem 7, but instead of being placed in 0.05 M CaCl2, the cell is placed in 0.2 M urea. Unlike Caþþ and Cl, urea can cross the plasma membrane and enter the cell. Describe what happens to the cell as it achieves equilibrium in this environment. What will be the final osmolarity of the cell? What will be its final volume? 10. Briefly describe the path that a protein, such as a hormone, that is manufactured on the rough ER would take in order to leave the cell. 11. What major role do mitochondria have in the cell? Why might it be important to have this process contained within an organelle? 12. List and briefly describe three organelles that provide structural support and assist with cell movement. 13. Find a location on the Internet that describes the Human Genome Project. Print its home page and indicate its web address. Find and print an ideogram of a chromosome that shows a gene that causes cystic fibrosis. Continued

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14. Briefly describe the major differences between replication and transcription. 15. Describe how the hereditary information contained in genes within the cell’s DNA is expressed as proteins that direct the cell’s activities. 16. Six different codons code for leucine, while only one codes for methionine. Why might this be important for regulating translation and producing proteins? 17. Insulin (Figure 3.41) was the first protein to be sequenced biochemically. Assuming that there were no introns involved in the process, what are the possible DNA sequences that produced the last four amino acids in the molecule?

Phe

Gly

Val

Ile

Asn

Val

Gln

Glu

His

Gln

Leu

Cys -S-S- Cys

Leu Ser

Cys -S-S- Cys Ala

Gly

Tyr Gln

Val

Leu

Ser

Glu Asn

Ser

Tyr Leu

His

Val Leu

Cys -S-S- Cys

Leu Ala

Gly

Val Glu Phe

Tyr

Asn

Glu Gly Arg

Phe Tyr Thr Pro Lys Ala

FIGURE 3.41 Bovine insulin consists of two polypeptide chains that are joined by two disulfide bonds (-S-S-). Hydrogen bonds also exist between the chains and between segments of the same chain. The threeletter names stand for different amino acids (see Table 3.1).

3.6 EXERCISES

131

18. Copy the title page and abstract of five peer-reviewed journal articles that discuss engineering applications for five different organ systems in the body (one article per organ system). Review articles, conference proceeding papers, copies of keynote addresses and other speeches, book chapters, articles from the popular press and newspapers, and editorials are not acceptable. Good places to look are the Annals of Biomedical Engineering, the IEEE Transactions on Biomedical Engineering, the IEEE Engineering in Medicine and Biology Magazine, and Medical and Biological Engineering and Computing. What information in the article indicates that it was peer-reviewed? 19. Trace the path of a single red blood cell from a capillary bed in your right hand to the capillary beds of your right lung and back. What gases are exchanged? Where are they exchanged during this process? 20. Draw and label a block diagram of pulmonary and systemic blood flow that includes the chambers of the heart, valves, major veins and arteries that enter and leave the heart, the lungs, and the capillary bed of the body. Use arrows to indicate the direction of flow through each component. 21. Find an example of an ECG representing normal sinus rhythm on the Internet and use it to demonstrate how heart rate is determined. 22. Why are R waves (Figure 3.22) used to determine heart rate rather than T waves? 23. How could the stroke volume be determined if a thermal dilution technique is used to determine cardiac output? 24. What would be the pulse pressure and mean arterial pressure for a hypertensive person with a systolic pressure of 145 mmHg and a diastolic pressure of 98 mmHg? 25. The total lung capacity of a patient is 5.5 liters. Find the patient’s inspiratory reserve volume if the patient’s vital capacity was 4.2 liters, the tidal volume was 500 ml, and the expiratory reserve volume was 1.2 liters. 26. What would you need to know or measure in order to determine the residual volume of the patient described in Example Problem 3.10? 27. Briefly describe the functions and major components of the central, peripheral, somatic, automatic, sympathetic, and parasympathetic nervous systems. Which ones are subsets of others? 28. Explain how sarcomeres shorten and how that results in muscle contraction. 29. How do the muscular and skeletal systems interact to produce movement? 30. Draw a block diagram to show the negative feedback mechanisms that help regulate glucose levels in the blood. Label the inputs, sensors, integrators, effectors, and outputs.

Suggested Readings B.H. Brown, R.H. Smallwood, D.C. Barber, P.V. Lawford, D.R. Hose, Medical Physics and Biomedical Engineering, Institute of Physics Publishing, Bristol and Philadelphia, 1999. G.M. Cooper, The Cell—A Molecular Approach, second ed., ASM Press, Washington, D.C., 2000. S. Deutsch, A. Deutsch, Understanding the Nervous System: An Engineering Perspective, IEEE Press, New York, 1993. S.I. Fox, Human Physiology, eighth ed., McGraw-Hill, Boston, 2004. W.J. Germann, C.L. Stanfield, Principles of Human Physiology, second ed., Pearson Benjamin Cummings, San Francisco, 2005.

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A.C. Guyton, Basic Neuroscience. Anatomy & Physiology, W. B. Saunders Company, Philadelphia, 1991. A.C. Guyton, J.E. Hall, Textbook of Medical Physiology, tenth ed., W. B. Saunders Company, Philadelphia, 2000. F.M. Harold, The Way of the Cell—Molecules, Organisms and the Order of Life, Oxford University Press, Inc., New York, 2001. G. Karp, Cell and Molecular Biology—Concepts and Experiments, third ed., John Wiley & Sons, Inc., New York, 2002. A.M. Katz, Physiology of the Heart, Raven Press, New York, 1986. R.D. Keynes, D.J. Aidley, Nerve & Muscle, second ed., Cambridge University Press, Cambridge, 1991. A.R. Leff, P.T. Schumacker, Respiratory Physiology. Basics and Applications, W. B. Saunders Company, Philadelphia, 1993. H. Lodish, A. Berk, S.L. Zipursky, P. Matsudaira, D. Baltimore, J. Darnell, Molecular Cell Biology, fourth ed., W. H. Freeman and Company, New York, 2000. F.H. Martini, Fundamentals of Anatomy & Physiology, fifth ed., Prentice Hall, Upper Saddle River, NJ, 2001. G.G. Matthews, Cellular Physiology of Nerve and Muscle, Blackwell Scientific Publications, Boston, 1991. G.H. Pollack, Cells, Gels and the Engines of Life—A New Unifying Approach to Cell Function, Ebner & Sons, Seattle, WA, 2001. R. Rhoades, R. Pflanzer, Human Physiology, fourth ed., Thomson Learning, Inc., Pacific Grove, CA, 2003. D.U. Silverthorn, Human Physiology—An Integrated Approach, third ed., Pearson Benjamin Cummings, San Francisco, 2004. A. To¨zeren, S.W. Byers, New Biology for Engineers and Computer Scientists, Pearson Education, Inc., Upper Saddle River, NJ, 2004. K.M. Van De Graaff, S.I. Fox, K.M. LaFleur, Synopsis of Human Anatomy & Physiology, Wm. C. Brown Publishers, Dubuque, IA, 1997. J.B. West, Respiratory Physiology—The Essentials, fourth ed., Williams & Wilkins, Baltimore, 1990. E.P. Widmaier, H. Raff, K.T. Strang, Vander, Sherman, & Luciano’s Human Physiology—The Mechanisms of Body Function, McGraw-Hill, Boston, 2004.

C H A P T E R

4 Biomechanics Joseph L. Palladino, PhD, and Roy B. Davis III, PhD O U T L I N E 4.1

Introduction

134

4.6

Clinical Gait Analysis

175

4.2

Basic Mechanics

137

4.7

Cardiovascular Dynamics

192

4.3

Mechanics of Materials

158

4.8

Exercises

215

4.4

Viscoelastic Properties

166

References

217

4.5

Cartilage, Ligament, Tendon, and Muscle

Suggested Readings

218

170

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : • Understand the application of engineering kinematic relations to biomechanical problems.

• Understand how kinematic equations of motion are used in clinical analysis of human gait.

• Understand the application of engineering kinetic relations to biomechanical problems.

• Understand how kinetic equations of motion are used in clinical analysis of human gait.

• Understand the application of engineering mechanics of materials to biological structures.

• Explain how biomechanics applied to human gait is used to quantify pathological conditions, to suggest surgical and clinical treatments, and to quantify their effectiveness.

• Use MATLAB to write and solve biomechanical static and dynamic equations.

• Understand basic rheology of biological fluids.

• Use Simulink to study viscoelastic properties of biological tissues.

• Understand the development of models that describe blood vessel mechanics.

Introduction to Biomedical Engineering, Third Edition

133

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• Understand basic heart mechanics. • Explain how biomechanics applied to the cardiovascular system is used to quantify the

effectiveness of the heart as a pump, to study heart-vessel interaction, and to develop clinical applications.

4.1 INTRODUCTION Biomechanics combines engineering and the life sciences by applying principles from classical mechanics to the study of living systems. This relatively new field covers a broad range of topics, including strength of biological materials, biofluid mechanics in the cardiovascular and respiratory systems, material properties and interactions of medical implants and the body, heat and mass transfer into biological tissues, biocontrol systems regulating metabolism or voluntary motion, and kinematics and kinetics applied to study human gait. The great breadth of the field of biomechanics arises from the complexities and variety of biological organisms and systems. The goals of this chapter are twofold: to apply basic engineering principles to biological structures and to develop clinical applications. Section 4.2 provides a review of concepts from introductory statics and dynamics. Section 4.3 presents concepts from mechanics of material that are fundamental for engineers and accessible to those with only a statics/ dynamics background. Section 4.4 introduces viscoelastic complexities characteristic of biological materials, with the concepts further applied in Section 4.5. The last two sections bring all of this information together in two “real-world” biomechanics applications: human gait analysis and cardiovascular dynamics. The human body is a complex machine, with the skeletal system and ligaments forming the framework and the muscles and tendons serving as the motors and cables. Human gait biomechanics may be viewed as a structure (skeleton) composed of levers (bones) with pivots (joints) that move as the result of net forces produced by pairs of agonist and antagonist muscles, a concept with origins as early as 1680, as depicted in Figure 4.1 from Borelli’s De Motu Animalium (On the Motion of Animals). Consequently, the strength of the structure and the action of muscles will be of fundamental importance. Using a similar functional model, the cardiovascular system may be viewed as a complex pump (heart) pumping a complex fluid (blood) into a complex set of pipes (blood vessels). An extensive suggested reading list for both gait and cardiovascular dynamics permits the reader to go beyond the very introductory nature of this textbook. The discipline of mechanics has a long history. For lack of more ancient records, the history of mechanics starts with the ancient Greeks and Aristotle (384–322 BC). Hellenic mechanics devised a correct concept of statics, but those of dynamics, fundamental in living systems, did not begin until the end of the Middle Ages and the beginning of the modern era. Starting in the sixteenth century, the field of dynamics advanced rapidly with work by Kepler, Galileo, Descartes, Huygens, and Newton. Dynamic laws were subsequently codified by Euler, LaGrange, and LaPlace (see A History of Mechanics by Dugas).

4.1 INTRODUCTION

135

FIGURE 4.1 Plate reproduced from Borelli’s De Motu Animalium, showing animal (human) motion resulting from the action of muscle pairs on bones, serving as levers, allowed to move at joints. From Images from the History of Medicine (IHM), National Library of Medicine, National Institutes of Health, http://www.nlm.nih.gov/hmd/ihm/.

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In Galileo’s Two New Sciences (1638), the subtitle Attenenti all Mecanica & i Movimenti Locali (Pertaining to Mechanics and Local Motions) refers to force, motion, and strength of materials. Since then, “mechanics” has been extended to describe the forces and motions of any system, ranging from quanta, atoms, molecules, gases, liquids, solids, structures, stars, and galaxies. The biological world is consequently a natural object for the study of mechanics. The relatively new field of biomechanics applies mechanical principles to the study of living systems. The eminent professor of biomechanics Dr. Y. C. Fung describes the role of biomechanics in biology, physiology, and medicine as follows: Physiology can no more be understood without biomechanics than an airplane can without aerodynamics. For an airplane, mechanics enables us to design its structure and predict its performance. For an organ, biomechanics helps us to understand its normal function, predict changes due to alteration, and propose methods of artificial intervention. Thus, diagnosis, surgery, and prosthesis are closely associated with biomechanics.1

Clearly, biomechanics is essential to assessing and improving human health. The following is a brief list of biomechanical milestones, especially those related to the topics in this chapter: • Galen of Pergamon (129–199) Published extensively in medicine, including De Motu Muscularum (On the Movements of Muscles). He realized that motion requires muscle contraction. • Leonardo da Vinci (1452–1519) Made the first accurate descriptions of ball-and-socket joints, such as the shoulder and hip, calling the latter the “polo dell’omo” (pole of man). His drawings depicted mechanical force acting along the line of muscle filaments. • Andreas Vesalius (1514–1564) Published De Humani Corporis Fabrica (The Fabric of the Human Body). Based on human cadaver dissections, his work led to a more accurate anatomical description of human musculature than Galen’s and demonstrated that motion results from the contraction of muscles that shorten and thicken. • Galileo Galilei (1564–1642) Studied medicine and physics, integrated measurement and observation in science, and concluded that mathematics is an essential tool of science. His analyses included the biomechanics of jumping and the gait analysis of horses and insects, as well as dimensional analysis of animal bones. • Santorio Santorio (1561–1636) Used Galileo’s method of measurement and analysis and found that the human body changes weight with time. This observation led to the study of metabolism and, thereby, ushered in the scientific study of medicine. • William Harvey (1578–1657) Developed an experimental basis for the modern circulation concept of a closed path between arteries and veins. The structural basis, the capillary, was discovered by Malpighi in 1661. • Giovanni Borelli (1608–1679) A mathematician who studied body dynamics, muscle contraction, animal movement, and motion of the heart and intestines. He published De Motu Animalium (On the Motion of Animals) in 1680.

Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Y. C. Fung, 1993.

4.2 BASIC MECHANICS

137

• Jan Swammerdam (1637–1680) Introduced the nerve-muscle preparation, stimulating muscle contraction by pinching the attached nerve in the frog leg. He also showed that muscles contract with little change in volume, refuting the previous belief that muscles contract when “animal spirits” fill them, causing bulging. • Robert Hooke (1635–1703) Devised Hooke’s Law, relating the stress and elongation of elastic materials, and used the term “cell” in biology. • Isaac Newton (1642–1727) Not known for biomechanics work, but he developed calculus, the classical laws of motion, and the constitutive equation for viscous fluid, all of which are fundamental to biomechanics. • Nicholas Andre´ (1658–1742) Coined the term “orthopaedics” at the age of 80 and believed that muscular imbalances cause skeletal deformities. • Stephen Hales (1677–1761) Was likely the first to measure blood pressure, as described in his book Statistical Essays: Containing Haemostaticks, or an Account of Some Hydraulick and Hydrostatical Experiments Made on the Blood and Blood-Vessels of Animals; etc., in 1733. • Leonard Euler (1707–1783) Generalized Newton’s laws of motion to continuum representations that are used extensively to describe rigid body motion, and studied pulse waves in arteries. • Thomas Young (1773–1829) Studied vibrations and voice, wave theory of light and vision, and devised Young’s modulus of elasticity. • Ernst Weber (1795–1878) and Eduard Weber (1806–1871) Published Die Mechanik der meschlichen Gerwerkzeuge (On the Mechanics of the Human Gait Tools) in 1836, pioneering the scientific study of human gait. • Hermann von Helmholtz (1821–1894) Studied an immense array of topics, including optics, acoustics, thermodynamics, electrodynamics, physiology, and medicine, including ophthalmoscopy, fluid mechanics, nerve conduction speed, and the heat of muscle contraction. • Etienne Marey (1830–1904) Analyzed the motion of horses, birds, insects, fish, and humans. His inventions included force plates to measure ground reaction forces and the “Chronophotographe a pellicule,” or motion picture camera. • Wilhelm Braune and Otto Fischer (research conducted from 1895–1904) Published Der Gang des Menschen (The Human Gait), containing the mathematical analysis of human gait and introducing methods still in use. They invented “cyclography” (now called interrupted-light photography with active markers), pioneered the use of multiple cameras to reconstruct 3-D motion data, and applied Newtonian mechanics to estimate joint forces and limb accelerations.

4.2 BASIC MECHANICS This section reviews some of the main points from any standard introductory mechanics (statics and dynamics) course. Good references abound, such as Engineering Mechanics by Merriam and Kraige (2008). A review of vector mathematics is followed by matrix coordinate transformations, a topic new to some students. Euler’s equations of motion (see Section 4.2.5) may also be new material. For both topics, Principles of Dynamics by Greenwood provides a comprehensive reference.

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y F Fy θ Fx

FIGURE 4.2

Two-dimensional representation of vector F.

4.2.1 Vector Mathematics Forces may be written in terms of scalar components and unit vectors, of magnitude equal to one, or in polar form with magnitude and direction. Figure 4.2 shows that the two-dimensional vector F is composed of the i component, Fx, in the x-direction, and the j component, Fy, in the y-direction, or F ¼ Fx i þ Fy j

ð4:1Þ

as in 20i þ 40j lb. In this chapter, vectors are set in bold type. This same vector may be written in polar form in terms of the vector’s magnitude jFj, also called the norm, and the vector’s angle of orientation, y: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:2Þ jFj ¼ F2x þ F2y y ¼ arctan

Fy Fx

ð4:3Þ

yielding jFj ¼ 44:7 lb and y ¼ 63:4 . Vectors are similarly represented in three dimensions in terms of their i, j, and k components: F ¼ Fx i þ Fy j þ Fz k

ð4:4Þ

with k in the z-direction. Often, a vector’s magnitude and two points along its line of action are known. Consider the three-dimensional vector in Figure 4.3. F has magnitude of 10 lb, and its line of action passes from the origin (0,0,0) to the point (2,6,4). F is written as the product of the magnitude jFj and a unit vector eF that points along its line of action: F ¼ jFjeF 0

B 2i þ 6j þ 4k C ¼ 10 lb@pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 22 þ 62 þ 42 F ¼ 2:67i þ 8:02j þ 5:34k lb

139

4.2 BASIC MECHANICS

F θz θy θx

4ft y 2ft

6ft

FIGURE 4.3 Three-dimensional vector defined by its magnitude and line of action.

The quantity in parentheses is the unit vector of F, or ! 2i þ 6j þ 4k eF ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:267i þ 0:802j þ 0:534k 22 þ 62 þ 42 and the magnitude of F is jFj ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:672 þ 8:022 þ 5:342

¼ 10 lb The vector F in Figure 4.3 may also be defined in 3-D space in terms of the angles between its line of action and each coordinate axis. Consider the angles yx, yy, and yz that are measured from the positive x, y, and z axes, respectively, to F. Then Fx jFj Fy cos yy ¼ jFj Fz cos yz ¼ jFj

ð4:5Þ

cos yx ¼

ð4:6Þ ð4:7Þ

These ratios are termed the direction cosines of F. The unit vector eF is equivalent to eF ¼ cos yx i þ cos yy j þ cos yz k or, in general

ð4:8Þ

BFx i þ Fy j þ Fz kC eF ¼ @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A F2x þ F2y þ F2z

ð4:9Þ

140

4. BIOMECHANICS

The angles yx, yy, and yz for this example are consequently ! 2:67 yx ¼ arccos ¼ 74:5 10 ! 8:02 yy ¼ arccos ¼ 36:7 10 ! 5:34 ¼ 57:7 yz ¼ arccos 10 Vectors are added by summing their components: A ¼ Ax i þ Ay j þ Az k B ¼ B x i þ B y j þ Bz k C ¼ A þ B ¼ ðAx þ Bx Þi þ ðAy þ By Þj þ ðAz þ Bz Þk In general, a set of forces may be combined into an equivalent force denoted the resultant R, where X X X R¼ Fx i þ Fy j þ Fz k ð4:10Þ as will be illustrated in subsequent sections. Vectors are subtracted similarly by subtracting vector components. Vector multiplication consists of two distinct operations: the dot and cross products. The dot, or scalar, product of vectors A and B produces a scalar via A B ¼ AB cos y

ð4:11Þ

where y is the angle between the vectors. For an orthogonal coordinate system, where all axes are 90 apart, all like terms alone remain, since ii¼jj¼kk¼1 ij¼jk¼ki¼¼0

ð4:12Þ

For example: A ¼ 3i þ 2j þ k ft B ¼ 2i þ 3j þ 10k lb A B ¼ 3ð2Þ þ 2ð3Þ þ 1ð10Þ ¼ 10 ft lb Note that the dot product is commutative—that is, A B B A. The physical interpretation of the dot product A B is the projection of A onto B, or, equivalently, the projection of B onto A. For example, work is defined as the force that acts in the same direction as the motion of a body. Figure 4.4 (left) shows a force vector F dotted with a direction of motion vector d. The work W done by F is given by F d Fd cos y. Dotting F with d yields the component of F acting in the same direction as d.

141

4.2 BASIC MECHANICS

y F

F O

r θ d

FIGURE 4.4

(Left) The dot, or scalar, product of vectors F and d is equivalent to the projection of F onto d. (Right) The cross, or vector, product of vectors r and F is a vector that points along the axis of rotation, the z-axis coming out of the page.

The moment of a force about a point or axis is a measure of its tendency to cause rotation. The cross, or vector, product of two vectors yields a new vector that points along the axis of rotation. For example, Figure 4.4 (right) shows a vector F acting in the x-y plane at a distance from the body’s coordinate center O. The vector r points from O to the line of action of F. The cross product r F is a vector that points in the z direction along the body’s axis of rotation. If F and r are three-dimensional, thereby including k components, their cross product will have additional components of rotation about the x and y axes. The moment M resulting from crossing r into F is written M ¼ Mx i þ My j þ Mz k

ð4:13Þ

where Mx, My, and Mz cause rotation of the body about the x, y, and z axes, respectively. Cross products may be taken by crossing each vector component term by term—for example: A B ¼ 3ð2Þi i þ 3ð3Þi j þ 3ð10Þi k þ 2ð2Þj i þ 2ð3Þj j þ 2ð10Þj k þ 1ð2Þk i þ 1ð3Þk j þ 1ð10Þk k The magnitude jA Bj ¼ AB sin y, where y is the angle between A and B. Consequently, for an orthogonal coordinate system, the cross products of all like terms equal zero, and i j ¼ k, j k ¼ i, k i ¼ j, i k ¼ j, and so on. The previous example yields A B ¼ 9k 30j þ 4k þ 20i 2j 3i ¼ 17i 32j þ 13k lb ft Note that the cross product is not commutative—in other words, A B 6¼ B A. Cross products of vectors are commonly computed using matrices. The previous example AB is given by the matrix

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i j k A B ¼ Ax Ay Az Bx By Bz i j k ¼ 3 2 1 2 3 10 ¼ i½ð2Þð10Þ ð1Þð3Þ j½ð3Þð10Þ ð1Þð2Þ þ k½ð3Þð3Þ ð2Þð2Þ ¼ ið20 3Þ jð30 þ 2Þ þ kð9 þ 4Þ

ð4:14Þ

¼ 17i 32j þ 13k lb ft

EXAMPLE PROBLEM 4.1 The vector F in Figure 4.5 has a magnitude of 10 kN and points along the dashed line as shown. (a) Write F as a vector. (b) What is the component of F in the x-z plane? (c) What moment does F generate about the origin (0,0,0)? y

15m

12m z

FIGURE 4.5

Force vector F has magnitude of 10 kN.

Solution This example problem is solved using MATLAB. The >> prompt denotes input, and the percent sign, %, precedes comments ignored by MATLAB. Lines that begin without the >> prompt are MATLAB output. Some spaces in the following output were omitted to conserve space. >> %(a) First write the direction vector d that points along F >> % as a 1D array: >> d ¼ [12 -15 9] d ¼ 12 -15 9 >> % Now write the unit vector of F, giving its direction: >> unit_vector ¼ d/norm(d) unit_vector ¼ 0.5657 -0.7071 0.4243

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4.2 BASIC MECHANICS

>> % F consists of the magnitude 10 kN times this unit vector >> F ¼ 10*unit_vector F ¼ 5.6569 -7.0711 4.2426 >> % Or, more directly >> F ¼ 10*(d/norm(d)) F ¼ 5.6569 -7.0711 4.2426 >> % (b) First write the vector r_xz that points from the origin >> % to the intersection of F and the xz plane: >> r_xz ¼ [12 0 9] r_xz ¼ 12 0 9 >> >> >> >> >>

% The dot product is given by the sum of all the term by term % multiplications of elements of vectors F and r_xz % F_dot_r_xz ¼ sum(F.*r_xz) % or simply, dot(F,r_xz) F_dot_r_xz ¼ dot(F,r_xz)

F_dot_r_xz ¼ 106.0660 >> >> >> >>

% (c) Cross F with a vector that points from the origin to % any point along the line of action of vector F. % The cross product is given by the cross function r_xz_cross_F ¼ cross(r_xz,F)

r_xz_cross_F ¼ 63.6396 0 -84.8528 >> % Note that the cross product is not commutative >> % resulting in different þ- signs. >> cross(F,r_xz) ans ¼ -63.6396 0 84.8528

EXAMPLE PROBLEM 4.2 Pointers are sometimes used in biomechanics labs to measure the location of a point in space. The pointer in Figure 4.6 consists of a rod equipped with two reflective markers, A and B. The locations of the two reflective markers are provided by a camera-based motion capture system. Given marker locations A ¼ (629, 35, 190) mm and B ¼ (669, 191, 120) mm, determine the location of the pointer tip, T, if marker B is a fixed distance, D, of 127 mm from the pointer tip.

Solution Given marker locations A ¼ (629, –35, 190) mm B ¼ (669, 191, 120) mm Continued

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4. BIOMECHANICS

A B z

D y x

FIGURE 4.6

Pointer with reflective markers A and B and with tip T located a fixed distance D ¼ 127 mm

from marker B.

then the vector from marker A to marker B is rB=A ¼ rB rA ¼ ð669i þ 191j þ 120kÞ ð629i 35j þ 190kÞ mm ¼ 40i þ 226j 70k mm with an associated unit vector, eB=A ¼

rB=A jrB=A j

40i þ 226j 70k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð40Þ2 þ ð226Þ2 þ ð70Þ2 ¼ 0:167i þ 0:942j 0:292k Pointer tip T (Figure 4.7) is located by rT ¼ rB þ rT=B ¼ rB þ D eT=B rB þ D eB=A ¼ ð669i þ 191j þ 120kÞ þ 127ð0:167i þ 0:942j 0:292kÞ mm ¼ 690i þ 311j þ 83k mm A

eB/A rB/A B rA

rT/B D

FIGURE 4.7 B to T.

Pointer tip T can be located using vector rB, the unit vector eB/A and the distance D from

145

4.2 BASIC MECHANICS

4.2.2 Coordinate Transformations 3-D Direction Cosines When studying the kinematics of human motion, it is often necessary to transform body or body segment coordinates from one coordinate system to another. For example, coordinates corresponding to a coordinate system determined by markers on the body, a moving coordinate system, must be translated to coordinates with respect to the fixed laboratory, an inertial coordinate system. These three-dimensional transformations use direction cosines that are computed as follows. Consider the vector A measured in terms of the uppercase coordinate system XYZ, shown in Figure 4.8 in terms of the unit vectors I, J, K: A ¼ Ax I þ Ay J þ Az K

ð4:15Þ

The unit vectors I, J, K can be written in terms of i, j, k in the xyz system I ¼ cos yxX i þ cos yyX j þ cos yzX k J ¼ cos yxY i þ cos yyY j þ cos yzY k K ¼ cos yxZ i þ cos yyZ j þ cos yzZ k where yxX is the angle between i and I, and similarly for the other angles. Substituting Eqs. (4.16)–(4.18) into Eq. (4.15) gives A ¼ Ax cos yxX i þ cos yyX j þ cos yzX k þAy cos yxY i þ cos yyY j þ cos yzY k þAz cos yxZ i þ cos yyZ j þ cos yzZ k or A ¼ Ax cos yxX þ Ay cos yxY þ Az cos yxZ i þ Ax cos yyX þ Ay cos yyY þ Az cos yyZ j þ Ax cos yzX þ Ay cos yzY þ Az cos yzZ k

ð4:16Þ ð4:17Þ ð4:18Þ

ð4:19Þ

ð4:20Þ

Consequently, A may be represented in terms of I, J, K or i, j, k. z

A x X

Y y

FIGURE 4.8 Vector A, measured with respect to coordinate system XYZ is related to coordinate system xyz via the nine direction cosines of Eq. (4.20).

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Euler Angles The coordinates of a body in one orthogonal coordinate system may be related to another orthogonal coordinate system via Euler angle transformation matrices. For example, one coordinate system might correspond to markers placed on the patient’s pelvis, and the other coordinate system might correspond to the patient’s thigh. The two coordinate systems are related by a series of rotations about each original axis in turn. Figure 4.9 shows z

z' θy

x θy x' z' z'' θx

y'' θx y'

y''' θz y''

θz x''' x'' The unprimed coordinate system xyz undergoes three rotations: about the y-axis (top), about the x-axis (middle), and about the z-axis (bottom), yielding the new triple-primed coordinate system x 000 y 000 z000 for a y-x-z rotation sequence.

FIGURE 4.9

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4.2 BASIC MECHANICS

the xyz coordinate axes with a y-x-z rotation sequence. First, xyz is rotated about the y-axis (top), transforming the ijk unit vectors into the i0 j0 k0 unit vectors, via the equations i0 ¼ cos yy i sin yy k

ð4:21Þ

j ¼j k0 ¼ sin yy i þ cos yy k

ð4:22Þ ð4:23Þ

This new primed coordinate system is then rotated about the x-axis (Figure 4.9, middle), giving the double-primed system i00 ¼ i0 j ¼ cos yx j0 þ sin yx k0 k00 ¼ sin yx j0 þ cos yx k0

ð4:24Þ ð4:25Þ ð4:26Þ

Finally, the double-primed system is rotated about the z-axis, giving the triple-primed system i000 ¼ cos yz i00 þ sin yz j00 j000 ¼ sin yz i00 þ cos yz j00 k000 ¼ k00

ð4:27Þ ð4:28Þ ð4:29Þ

The three rotations may be written in matrix form to directly 2 000 3 2 32 32 cos yy cos yz 1 0 0 sin yz 0 i 6 000 7 6 76 0 76 sin yx 54 4 j 5¼4 sin yz cos yz 0 54 0 cos yx 000 sin yy 0 sin yx cos yx 0 0 1 k 2

cos yz

sin yz cos yx cos yz cos yx

sin yx

6 ¼4 sin yz

32 cos yy sin yz sin yx 76 0 cos yz sin yx 54 sin yy cos yx

translate ijk into i000 j000 k000 : 32 3 0 sin yy i 7 6 1 0 54 j 7 5 ð4:30Þ 0 cos yy k

sin yy

1 0

0 cos yy

32 3 i 76 7 54 j 5 k

3 2 32 3 cos yz cos yy þ sin yz sin yx sin yy sin yz cos yx cos yz sin yy þ sin yz sin yx cos yy i i000 6 000 7 6 sin y cos y þ cos y sin y sin y cos y cos y 7 6 sin yz sin yy þ cos yz sin yx cos yy 54 j 7 4 j 5¼4 5 z y z x y z x 000 cos y sin y sin y cos y cos y k k x y x x y ð4:31Þ If the angles of coordinate system rotation (yx , yy , yz ) are known, coordinates in the xyz system can be transformed into the x000 y000 z000 system. Alternatively, if both the unprimed and triple-primed coordinates are known, the angles may be computed as follows: k000 j ¼ sin yx

yx ¼ arcsinðk000 jÞ

ð4:32Þ

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4. BIOMECHANICS

k000 i ¼ cos yx2sin yy 3 k000 i 5 yy ¼ arcsin4 cos yx

ð4:33Þ

i000 j ¼ sin yz2cos yx 3 i000 j 5 yz ¼ arcsin4 cos yx

ð4:34Þ

Walking produces tri-planar hip, knee, and ankle motion: flexion/extension (FE), abduction/ adduction (AA), and internal/external transverse rotation (TR). Euler angles offer an opportunity to quantify these coordinated motions. The order of the Euler angle rotation sequence FE-AA-TR, corresponding to a y-x-z axis rotation sequence, or tilt-obliquity-rotation, was chosen to correlate to the largest to smallest joint excursions during walking [6]. More recently, research suggests that for pelvic motion, an Euler angle z-x-y rotation sequence corresponding to rotation-obliquity-tilt is more consistent with clinical observations [1].

EXAMPLE PROBLEM 4.3 Write the Euler angle transformation matrices for the y-x-z rotation sequence using the MATLAB symbolic math toolbox.

Solution The following MATLAB script, or m-file, is a collection of MATLAB commands that can be run by invoking the m-file name “eulerangles” in the command line. % % % % % % %

eulerangles.m Euler angles for y-x-z rotation sequence using MATLAB symbolic math toolbox x, y and z are thetax, thetay and thetaz, respectively First define them as symbolic variables

syms x y z % Writing equations 4.21-23 as a matrix A A ¼ [ cos(y), 0, -sin(y); 0, 1, 0; sin(y), 0, cos(y)] % equations 4.24-26 as matrix B B¼[1 0, 0; 0, cos(x), sin(x); 0, -sin(x), cos(x)]

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4.2 BASIC MECHANICS

% and equations 4.27-29 as matrix C C ¼ [ cos(z), sin(z), 0; -sin(z), cos(z), 0; 0, 0, 1] % The matrix equation 4.30 is created by multiplying matrices C, B % and A D¼C*B*A

The resulting transformation matrix from the preceding m-file is D¼ [cos(z)*cos(y)þsin(z)*sin(x)*sin(y), sin(z)*cos(x), -cos(z) *sin(y)þsin(z)*sin(x)*cos(y)] [-sin(z)*cos(y)þcos(z)*sin(x)*sin(y), cos(z)*cos(x), sin(z) *sin(y)þcos(z)*sin(x)*cos(y)] [cos(x)*sin(y), -sin(x), cos(x)*cos(y)]

which is the same as Eq. (4.31). The Euler transformation matrices are used differently depending on the available data. For example, if the body coordinates in both the fixed (unprimed) and body (triple primed) systems are known, the body angles yx, yy, and yz can be computed, for example, using Eqs. (4.32)–(4.34) for a y-x-z rotation sequence. Alternatively, the body’s initial position and the angles yx, yy, and yz may be used to compute the body’s final position.

EXAMPLE PROBLEM 4.4 An aircraft undergoes 30 degrees of pitch (yx), then 20 degrees of roll (yy), and finally 10 degrees of yaw (yz). Write a MATLAB function that computes the Euler angle transformation matrix for this series of angular rotations.

Solution Since computers use radians for trigonometric calculations, first write two simple functions to compute cosines and sines in degrees: function y ¼ cosd(x) %COSD(X) cosines of the elements of X measured in degrees. y ¼ cos(pi*x/180); function y ¼ sind(x) %SIND(X) sines of the elements of X measured in degrees. y ¼ sin(pi*x/180);

Next write the x-y-z rotation sequence transformation matrix: function D ¼ eulangle(thetax, thetay, thetaz) %EULANGLE matrix of rotations by Euler’s angles. % EULANGLE(thetax, thetay, thetaz) yields the matrix of

Continued

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4. BIOMECHANICS

% rotation of a system of coordinates by Euler’s % angles thetax, thetay and thetaz, measured in degrees. % Now the first rotation is about the x-axis, so we use eqs. 4.24-26 A¼[1 0 0

0 cosd(thetax) -sind(thetax)

0 sind(thetax) cosd(thetax) ];

% Next is the y-axis rotation (eqs. 4.21-23) B ¼ [ cosd(thetay) 0 sind(thetay)

0 1 0

-sind(thetay) 0 cosd(thetay) ];

% Finally, the z-axis rotation (eqs. 4.27-29) C ¼ [ cosd(thetaz) sind(thetaz) 0 -sind(thetaz) cosd(thetaz) 0 0 0 1 ]; % Multiplying rotation matrices C, B and A as in Eq. 4.30 gives the solution: D¼C*B*A;

Now use this function to compute the numerical transformation matrix: >> eulangle(30,20,10) ans ¼ 0.9254 0.3188 -0.2049 -0.1632 0.8232 0.5438 0.3420 -0.4698 0.8138

This matrix can be used to convert any point in the initial coordinate system (premaneuver) to its position after the roll, pitch, and yaw maneuvers have been executed.

4.2.3 Static Equilibrium Newton’s equations of motion applied to a structure in static equilibrium reduce to the following vector equations: X F¼0 ð4:35Þ X M¼0 ð4:36Þ These equations are applied to biological systems in the same manner as standard mechanical structures. Analysis begins with a drawing of the free-body diagram of the body segments of interest with all externally applied loads and reaction forces at the supports. Orthopedic joints can be modeled with appropriate ideal joints, such as hinge, ball-andsocket, and so forth, as discussed in Chapter 3 (see Figure 3.33).

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4.2 BASIC MECHANICS

EXAMPLE PROBLEM 4.5 Figure 4.10 (top) shows a Russell’s traction rig used to apply an axial tensile force to a fractured femur for immobilization. (a) What magnitude weight w must be suspended from the free end of the cable to maintain the leg in static equilibrium? (b) Compute the average tensile force applied to the thigh under these conditions.

Solution The free-body diagram for this system is shown in the lower panel of Figure 4.10. If the pulleys are assumed frictionless and of small radius, the cable tension T is constant throughout. Using Eq. (4.35), F1 þ F2 þ F3 þ Ffemur mgj ¼ 0 Writing each force in vector form, F1 ¼ F1 i ¼ Ti F2

¼ ðF2 cos 30 Þi þ ðF2 sin 30 Þj ¼ ðT cos 30 Þi þ ðT sin 30 Þj ¼ ðF3 cos 40 Þi þ ðF3 sin 40 Þj ¼ ðT cos 40 Þi þ ðT sin 40 Þj

Ffemur ¼ ðFfemur cos 20 Þi ¼ ðFfemur sin 20 Þj

30⬚ A

40⬚

20⬚ Ffemur F3

F2 B

30⬚ F1 y

40⬚

A 20⬚ x

Ffemur mg = 0.061w = 9.2 Ib

FIGURE 4.10

(Top) Russell’s traction mechanism for clinically loading lower-extremity limbs. (Bottom) Free-body diagram of the leg in traction. Adapted from [5].

Continued

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4. BIOMECHANICS

Using Table 4.1, and neglecting the weight of the thigh, the weight of the foot and leg is 0.061 multiplied by total body weight, yielding mgj ¼ ð0:061Þð150j lbÞ ¼ 9:2j lb Summing the x components gives T T cos 30 þ T cos 40 þ Ffemur cos 20 ¼ 0 Summing the y components gives T sin 30 þ T sin 40 Ffemur sin 20 mg ¼ 0 The last two expressions may be solved simultaneously, giving both T, which is equal to the required externally applied weight, and the axial tensile force, Ffemur T ¼ 12:4 lb Ffemur ¼ 14:5 lb

EXAMPLE PROBLEM 4.6 The force plate depicted in Figure 4.11 has four sensors, one at each corner, that read the vertical forces F1 , F2 , F3 , and F4 . If the plate is square with side of length l and forces F1 F4 are known, write two expressions that will give the x and y locations of the resultant force R.

Solution The resultant magnitude R can be computed from the sum of forces in the z-direction: P Fz ¼ 0 F1 þ F2 þ F3 þ F4 R ¼ 0 R ¼ F1 þ F2 þ F3 þ F4 z

y F4 x

F3 F2

A square force plate with sides of length l is loaded with resultant force R and detects the vertical forces at each corner, F1 F4 .

FIGURE 4.11

The force plate remains horizontal, so the sum of the moments about the x and y axes must each be zero. Taking moments about the x-axis, X Mx ¼ 0 F2 l þ F3 l Ry ¼ 0 y¼

ðF2 þ F3 Þl R

4.2 BASIC MECHANICS

153

Similarly, summing moments about the y-axis, X My ¼ 0 F1 l þ F2 l Rx ¼ 0 x¼

ðF1 þ F2 Þl R

The coordinates x and y locate the resultant R.

4.2.4 Anthropomorphic Mass Moments of Inertia A body’s mass resists linear motion; its mass moment of inertia resists rotation. The resistance of a body, or a body segment such as a thigh in gait analysis, to rotation is quantified by the body or body segment’s moment of inertia I: Z ð4:37Þ I ¼ r2 dm m

where m is the body mass and r is the moment arm to the axis of rotation. The elemental mass dm can be written rdV. For a body with constant density r, the moment of inertia can be found by integrating over the body’s volume V: Z ð4:38Þ I ¼ r r2 dV V

This general expression can be written in terms of rotation about the x, y, and z axes: R Ixx ¼ V ðy2 þ z2 ÞrdV R Iyy ¼ V ðx2 þ z2 ÞrdV ð4:39Þ R 2 Izz ¼ V ðx þ y2 ÞrdV The radius of gyration k is the moment arm between the axis of rotation and a single point where all of the body’s mass is concentrated. Consequently, a body segment may be treated as a point mass with moment of inertia I ¼ mk2

ð4:40Þ

where m is the body segment mass. The moment of inertia with respect to a parallel axis I is related to the moment of inertia with respect to the body’s center of mass Icm via the parallel axis theorem: I ¼ Icm þ md2

ð4:41Þ

where d is the perpendicular distance between the two parallel axes. Anthropomorphic data for various body segments are listed in Table 4.1.

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4. BIOMECHANICS

TABLE 4.1 Anthropomorphic Data Center Mass/ Segment Length

Radius Gyration/ Segment Length

Segment Weight/ Body Weight

Proximal

Distal

Proximal

Segment

Definition

Hand

Wrist axis/knuckle II middle finger

0.006

0.506

0.494

0.587

0.577

Forearm

Elbow axis/ulnar styloid

0.016

0.430

0.570

0.526

0.647

Upper arm

Glenohumeral axis/elbow axis

0.028

0.436

0.564

0.542

0.645

Forearm and hand

Elbow axis/ulnar styloid

0.022

0.682

0.318

0.827

0.565

Total arm

Glenohumeral joint/ulnar styloid

0.050

0.530

0.470

0.645

0.596

Foot

Lateral malleolus/head metatarsal II

0.0145

0.50

0.690

Leg

Femoral condyles/medial malleolus

0.0465

0.433

0.567

0.528

0.643

Thigh

Greater trochanter/femoral condyles

0.100

0.433

0.567

0.540

0.653

Foot and leg

Femoral condyles/medial malleolus

0.061

0.606

0.394

0.735

0.572

Total leg

Greater trochanter/medial malleolus

0.161

0.447

0.553

0.560

0.650

Head and neck

C7-T1 and 1st rib/ear canal

0.081

1.000

Shoulder mass

Sternoclavicular joint/ glenohumeral axis

Thorax

C7-T1/T12-L1 and diaphragm

Abdomen

Distal

1.116

0.712

0.288

0.216

0.82

0.18

T12-L1/L4-L5

0.139

0.44

0.56

Pelvis

L4-L5/greater trochanter

0.142

0.105

0.895

Thorax and abdomen

C7-T1/L4-L5

0.355

0.63

0.37

Abdomen and pelvis

T12-L1/greater trochanter

0.281

0.27

0.73

Trunk

Greater trochanter/glenohumeral joint

0.497

0.50

Trunk, head, neck

Greater trochanter/glenohumeral joint

0.578

0.66

0.34

0.830

0.607

Head, arm, trunk

Greater trochanter/glenohumeral joint

0.678

0.626

0.374

0.798

0.621

Adapted from Winter, 2009.

4.2 BASIC MECHANICS

155

EXAMPLE PROBLEM 4.7 A person weighing 150 pounds has a thigh length of 17 inches. Find the moment of inertia of this body segment with respect to its center of mass in SI units.

Solution Thigh length in SI units is lthigh ¼ 17 in ¼ 0:432 m Table 4.1 lists ratios of segment weight to body weight for different body segments. Starting with body mass, mbody ¼ ð150 lbÞð0:454 kg=lbÞ ¼ 68:1 kg the thigh segment mass is mthigh ¼ ð0:100Þð68:1 kgÞ ¼ 6:81 kg Table 4.1 also lists body segment center of mass and radius of gyration as ratios with respect to segment length for each body segment. Table 4.1 gives both proximal and distal segment length ratios. Note that “proximal” for the thigh refers toward the hip and “distal” refers toward the knee. Consequently, the proximal thigh segment length is the distance between the thigh center of mass and the hip, and the distal thigh segment length is the distance between the thigh center of mass and the knee. The moment of inertia of the thigh with respect to the hip is therefore Ithigh=hip ¼ mk2 ¼ ð6:81 kgÞ½ð0:540Þð0:432 mÞ2 ¼ 0:371 kg m2 The thigh’s moment of inertia with respect to the hip is related to the thigh’s moment of inertia with respect to its center of mass via the parallel axis theorem (Eq. (4.41)). Consequently, Table 4.1 data can be used to compute segment moments of inertia with respect to their centers of mass: Ithigh=hip ¼ Ithigh=cm þ md2 so Ithigh=cm ¼ Ithigh=hip md2 In this case, distance d is given by the proximal segment length data: d ¼ ð0:432 mÞð0:433Þ ¼ 0:187 m and the final result is Ithigh=cm ¼ 0:371 kg m2 ð6:81 kgÞð0:187 mÞ2 ¼ 0:133 kg m2

EXAMPLE PROBLEM 4.8 A person weighing 160 pounds is holding a 10-lb weight in his palm, with the elbow fixed at 90 flexion (Figure 4.12 (top)). (a) What force must the biceps generate to hold the forearm in static equilibrium? (b) What force(s) does the forearm exert on the humerus? Continued

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4. BIOMECHANICS

humerus FB 75°

10-lb. ball radius

ulna

2 in.

10.5 in. 14 in.

10 lb.

75° P

O 2 in.

mg = 0.022w = 3.5 lb. 0.682L = 7.2 in. 14 in.

FIGURE 4.12 (Top) The forearm held statically fixed in 90 flexion while holding a 10 lb weight at the hand. (Bottom) Free-body diagram of the forearm system. Adapted from [5].

Solution Figure 4.12 (bottom) shows the free-body diagram of this system. Due to the increased number of unknowns compared to the previous example, both Eqs. (4.35) and (4.36) will be used. From the anthropometric relationships in Table 4.1, the segment weight (forearm and hand) is approximated as 2.2 percent of total body weight, with the segment mass located 68.2 percent of the segment length away from the elbow axis. Note that the segment length for the “forearm and hand” segment in Table 4.1 is defined as the distance between the elbow axis and the ulnar styloid. P Summing moments about the elbow at point O, the equilibrium equation M ¼ 0 can be written as rOE FA þ rOB ð10 lbÞj þ rOP ð3:5 lbÞj ¼ 0 ð2 inÞi ðFA Þj þ ð12 inÞi ð10 lbÞj þ ð5:2 inÞi ð3:5 lbÞj ¼ 0 ð2 inÞFA k ð120 lb inÞk ð18:2 lb inÞk ¼ 0

4.2 BASIC MECHANICS

157

Solving this last expression for the one unknown, FA, the vertical force at the elbow: FA ¼ 69:1 lb To find the unknown horizontal force at the elbow, FC, and the unknown force the biceps must P generate, FB, the other equation of equilibrium F ¼ 0 is used: FC i FA j þ ðFB cos 75 i þ FB sin 75 jÞ 10 lbj 3:5 lbj ¼ 0 Summing the x and y components gives FC FB cos ð75 Þ ¼ 0 FA þ FB sin ð75 Þ 10 lb 3:5 lb ¼ 0

Solving these last two equations simultaneously and using FA ¼ 69:1 lb gives the force of the biceps muscle, FB, and the horizontal elbow force, FC: FB ¼ 85:5 lb FC ¼ 22:1 lb

4.2.5 Equations of Motion Vector equations of motion are used to describe the translational and rotational kinetics of bodies. Newton’s Equations of Motion Newton’s second law relates the net force F and the resulting translational motion as F ¼ ma

ð4:42Þ

where a is the linear acceleration of the body’s center of mass for translation. For rotation M ¼ Ia

ð4:43Þ

where Ia is the body’s angular momentum. Hence, the rate of change of a body’s angular momentum is equal to the net moment M acting on the body. These two vector equations of motion are typically written as a set of six x, y, and z component equations. Euler’s Equations of Motion Newton’s equations of motion describe the motion of the center of mass of a body. More generally, Euler’s equations of motion describe the motion of a rigid body with respect to its center of mass. For the special case where the xyz coordinate axes are chosen to coincide with the body’s principal axes, that is, a cartesian coordinate system whose origin is located at the body’s center of mass, Euler’s equations are X Mx ¼ Ixx ax þ ðIzz Iyy Þoy oz ð4:44Þ X ð4:45Þ My ¼ Iyy ay þ ðIxx Izz Þoz ox

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4. BIOMECHANICS

X Mz ¼ Izz az þ ðIyy Ixx Þox oy

ð4:46Þ

Mi is the net moment, Iii is the body’s moment of inertia with respect to the principal axes, and ai and oi are the body’s angular acceleration and angular velocity, respectively. Euler’s equations require angular measurements in radians. Their derivation is outside the scope of this chapter but may be found in an intermediate dynamics book—for example, [12]. Equations (4.44)–(4.46) will be used in Section 4.6 to compute intersegmental or joint moments.

4.3 MECHANICS OF MATERIALS Just as kinematic and kinetic relations may be applied to biological bodies to describe their motion and its associated forces, concepts from mechanics of materials may be used to quantify tissue deformation, to study distributed orthopedic forces, and to predict the performance of orthopedic implants and prostheses and of surgical corrections. Since this topic is very broad, some representative concepts will be illustrated with the following examples. An orthopedic bone plate is a flat segment of stainless steel used to screw two failed sections of bone together. The bone plate in Figure 4.13 has a rectangular cross section, A, measuring 4.17 mm by 12 mm and made of 316L stainless steel. An applied axial load, F, of 500 N produces axial stress, s, (force/area): s¼ ¼

F A 500 N ¼ 10 MPa 3 ð4:17 10 mÞð12 103 mÞ

ð4:47Þ

The maximum shear stress, tmax , occurs at a 45 angle to the applied load t max ¼

F45 A45

ð500 NÞ cos 45 3 ¼ 5 MPa ¼2 mÞ5 4ð0:00417 mÞð0:012 cos 45

ð4:48Þ

which is 0:5s, as expected from mechanics of materials principles. Prior to loading, two points were punched 15 mm apart on the long axis of the plate, as shown. After the 500 N load is applied, those marks are an additional 0.00075 mm apart. The plate’s strain, e, relates the change in length, Dl to the original length, l:

4.3 MECHANICS OF MATERIALS

159

4.17

12.00

15.00

FIGURE 4.13 Bone plate used to fix bone fractures, with applied axial load. Dimensions are in mm. Adapted from [2].

Dl l 0:00075 mm ¼ 50 106 ¼ 15 mm

e¼

ð4:49Þ

often reported as 50 m, where m denotes microstrain (106). The elastic modulus, E, relates stress and strain and is a measure of a material’s resistance to distortion by a tensile or compressive load. For linearly elastic (Hookean) materials, E is a constant, and a plot of s as a function of e is a straight line with slope E: E¼

s e

ð4:50Þ

For the bone plate, E¼

10 106 Pa ¼ 200 GPa 50 106

Materials such as metals and plastics display linearly elastic properties only in limited ranges of applied loads. Biomaterials have even more complex elastic properties. Figure 4.14 shows tensile stress-strain curves measured from longitudinal and transverse sections of

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4. BIOMECHANICS

120 110

Longitudinal

100

Stress [MPa]

90 80 70 60

Transverse

50 40 30 20 10 0 0

5000

10000 Strain [μ]

15000

20000

FIGURE 4.14 Tensile stress-strain curves for longitudinal and transverse sections of bone. Adapted from [2].

TABLE 4.2 Tensile Yield and Ultimate Stresses, and Elastic Moduli E for Some Common Orthopedic Materials Material

syield [MPa]

sultimate [MPa]

E [GPa]

Stainless steel

700

850

180

Cobalt alloy

490

700

200

Titanium alloy

1,100

1,250

110

Bone

120

PMMA (fixative) UHMWPE (bearing) Patellar ligament

Data from [2].

bone. Taking the longitudinal curve first, from 0 to 7,000m, bone behaves as a purely elastic solid, with E 12 GPa. At a tensile stress of approximately 90 MPa, the stress-strain curve becomes nonlinear, yielding into the plastic region of deformation. This sample ultimately fails around 120 MPa. Table 4.2 shows elastic moduli, yield stresses, and ultimate stresses for some common orthopedic materials, both natural and implant. Figure 4.14 also shows that the elastic properties of bone differ depending on whether the sample is cut in the longitudinal or transverse direction—that is, bone is anisotropic. Bone is much weaker and less stiff in the transverse compared to the longitudinal direction, as is illustrated by the large differences in the yield and ultimate stresses and the slopes of the stress-strain curves for the two samples.

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4.3 MECHANICS OF MATERIALS

Stress [MPa]

B 130 120 110 100 90 80 70 60 50 40 30 20 10 0

5000

FIGURE 4.15

10000 Strain [μ]

15000

Bone shows hysteresis and shifting of stress-strain curves with repeated loading and unloading.

Adapted from [2].

Figure 4.15 shows that the elastic properties of bone also vary depending on whether the load is being applied or removed, displaying hysteresis. From a thermodynamic view, the energy stored in the bone during loading is not equal to the energy released during unloading. This energy difference becomes greater as the maximum load increases (curves A to B to C). The “missing” energy is dissipated as heat due to internal friction and damage to the material at high loads. The anisotropic nature of bone is sufficient in that its ultimate stress in compression is 200 MPa, while in tension it is only 140 MPa and in torsion 75 MPa. For torsional loading, the shear modulus or modulus of rigidity, denoted G, relates the shear stress to the shear strain. The modulus of rigidity is related to the elastic modulus via Poisson’s ratio, n, where etransverse ð4:51Þ n¼ elongitudinal Typically, n 0:3, meaning that longitudinal deformation is three times greater than transverse deformation. For linearly elastic materials, E, G, and n are related by G¼

E 2ð1 þ nÞ

ð4:52Þ

One additional complexity of predicting biomaterial failure is the complexity of physiological loading. For example, consider “boot-top” fractures in skiing. If the forward motion of a skier is abruptly slowed or stopped, such as by suddenly running into wet or soft snow, his forward momentum causes a moment over the ski boot top, producing three-point bending of the tibia. In this bending mode the anterior tibia undergoes compression, while the posterior is in tension and potentially in failure, since bone is much stronger in compression than in tension. Contraction of the triceps surae muscle produces high compressive stress at the posterior side, reducing the amount of bone tension and helping to prevent injury. Example Problem 4.9 shows how topics from statics and mechanics of materials may be applied to biomechanical problems.

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4. BIOMECHANICS

EXAMPLE PROBLEM 4.9 Figure 4.16 (left) shows an orthopedic nail-plate used to fix an intertrochanteric fracture. The hip applies an external force of 400 N during static standing, as shown. The nail-plate is rectangular stainless steel with cross-sectional dimensions of 10 mm (width) by 5 mm (height), and is well-fixed with screws along its vertical axis and friction fit into the trochanteric head (along the x-axis). What forces, moments, stresses, and strains will develop in this orthopedic device?

Solution As for any statics problem, the first task is constructing a free-body diagram, including all applied forces and moments and all reaction forces and moments that develop at the supports. Because of the instability at the fracture site, the nail-plate may be required to carry the entire 400 N load. Consequently, one reasonable model of the nail-plate is a cantilever beam of length 0.06 m with a combined loading, as depicted in Figure 4.16 (right, top). The applied 400 N load consists of both axial and transverse components: Fx ¼ 400 N cos 20 ¼ 376 N Fy ¼ 400 N sin 20 ¼ 137 N y 400 N

20⬚

y x

Ax Ma

0.06m

135⬚

cm y

M 376N 8.22 Nm

N x

137N

FIGURE 4.16 (Left) An intertrochanteric nail plate used in bone repair. To the right is the free-body diagram of the upper section of this device, and below it is the free-body diagram of a section of this beam cut at a distance x from the left-hand support. Adapted from [2].

The axial load produces compressive normal stress; from Eq. (4.47), sx ¼ ¼

Fx A 376 N ¼ 7:52 MPa ð0:005 mÞð0:01 mÞ

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4.3 MECHANICS OF MATERIALS

in compression, which is only about 1 percent of the yield stress for stainless steel (see Table 4.2). The maximum shear stress due to the axial load is sx tmax ¼ ¼ 3:76 MPa 2 and occurs at 45 from the long axis. The axial strain can be computed using the elastic modulus for stainless steel, E¼

s F=A ¼ e Dl=l

giving an expression for strain: e¼ ¼

F EA 376 N ¼ 41:8 106 180 10 Pað0:005 mÞð0:01 mÞ 9

From this strain the axial deformation can be computed Dlaxial ¼ el ¼ 2:51 106 m which is negligible. The transverse load causes the cantilever section to bend. The equations describing beam bending can be found in any mechanics of materials text (e.g., [26]). Consider the beam in the left panel of Figure 4.17. If this beam is fixed at the left-hand side and subjected to a downward load on the right, it will bend with the top of the beam elongating and the bottom shortening. Consequently, the top of the beam is in tension, and the bottom in compression. The point of transition, where there is no bending force, is denoted the neutral axis, located at distance c. For a symmetric rectangular beam of height h, c is located at the midline h/2. The beam resists bending via its area moment of inertia I. For a rectangular 1 bh3 , depicted in the right panel of Figure 4.17. cross section of width b and height h, I ¼ 12 Beam tip deflection dy is equal to dy ¼

Fx2 ð3L xÞ 6EI

ð4:53Þ

y I = 1/12 bh3 c

Tension x

h c

Compression b

FIGURE 4.17 (Left) A beam fixed on the left and subjected to a downward load on the right undergoes bending, with the top of the beam in tension and the bottom in compression. The position where tension changes to compression is denoted the neutral axis, located at c. (Right) A beam of rectangular cross section with width b 1 bh3 . and height h resists bending via the area moment of inertia I ¼ 12

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where x is the axial distance along the beam, L is the total beam length, and I is the beam’s cross-sectional area moment of inertia. For this example, I¼

1 ð10 103 mÞð5 103 mÞ3 ¼ 1:042 1010 m4 12

Maximum deflection will occur at x ¼ L, dymax ¼ ¼

FL3 3EI 137 Nð0:06 mÞ3 3ð180 109 N=m2 Þð1:042 1010 m4 Þ

ð4:54Þ

¼ 5:26 104 m ¼ 0:526 mm which is also negligible. Computation of maximum shear and bending stresses requires maximum shear force V and bending moment M. Starting by static analysis of the entire free-body P Fx : Ax 376 N ¼ 0 P Fy : Ay 137 N ¼ 0 P MA : Ma 137 Nð0:06 mÞ ¼ 0 Solving these equations gives Ax ¼ 376 N, Ay ¼ 137 N, and Ma ¼ 8:22 N m. Taking a cut at any point x to the right of A and isolating the left-hand section gives the free-body in Figure 4.16 (right, bottom). Applying the equations of static equilibrium to this isolated section yields P 376 N N ¼ 0 Fx : NðxÞ P Fy

376 N

137 N V ¼ 0

VðxÞ P MA

137 N

8:22 N m ð137 NÞðx mÞ þ M ¼ 0

MðxÞ

ð137 N mÞx 8:22 N m

These last equations can be plotted easily using MATLAB, giving the axial force, shear force, and bending moment diagrams shown in Figure 4.18. % Use MATLAB to plot axial force, shear force and bending moment % diagrams for Example Problem 9 x ¼ [0:0.01:0.06]; N ¼ x.*0 þ 376;

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4.3 MECHANICS OF MATERIALS

V ¼ x.*0 þ 137; M ¼ 137.*x - 8.22; figure subplot(3,1,1), plot(x,N,x,N,’x’) xlabel(’x [m]’) ylabel(’N [N]’) title(’Axial Force N’) subplot(3,1,2), plot(x,V,x,V,’x’) xlabel(’x [m]’) ylabel(’V [N]’) title(’Shear Force V’) subplot(3,1,3), plot(x,M,x,M,’x’) xlabel(’x [m]’) ylabel(’M [N-m]’) title(’Bending Moment M’) Axial Force N 377 N [N]

376.5 376 375.5 375

0.01

0.02

0.03 x [m]

0.04

0.05

0.06

0.04

0.05

0.06

0.04

0.05

0.06

Shear Force V

V [N]

138

137

136

0.01

0.02

0.03 x [m] Bending Moment M

M [N-m]

−5 −10

0.01

0.02

0.03 x [m]

FIGURE 4.18 Axial force N (top), shear force V (middle), and bending moment M (bottom) computed for the nail plate in Figure 4.16 as functions of the distance x along the plate.

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The maximum bending and shear stresses follow as Mmax c I where c, the distance to the beam’s neutral axis, is h/2 for this beam: sbmax ¼

sbmax ¼

8:22 Nm½0:5ð5 103 mÞ ¼ 197 MPa 1:042 1010 m4

tbmax ¼

Vmax h2 8I

ð4:55Þ

ð4:56Þ

137 Nð5 103 mÞ2 ¼ 4:11 MPa 8ð1:042 1010 m4 Þ

All of these stresses are well below syield ¼ 700 MPa for stainless steel.

4.4 VISCOELASTIC PROPERTIES The Hookean elastic solid is a valid description of materials only within a narrow loading range. For example, an ideal spring that relates force and elongation by a spring constant k is invalid in nonlinear low-load and high-load regions. Further, if this spring is coupled to a mass and set into motion, the resulting perfect harmonic oscillator will vibrate forever, which experience shows does not occur. Missing is a description of the system’s viscous or damping properties. In this case, energy is dissipated as heat in the spring and air friction on the moving system. Similarly, biomaterials all display viscoelastic properties. Different models of viscoelasticity have been developed to characterize materials with simple constitutive equations. For example, Figure 4.19 shows three such models that consist of a series ideal spring and dashpot (Maxwell), a parallel spring and dashpot (Voight), and a series spring and dashpot with a parallel spring (Kelvin). Each body contains a dashpot, which generates force in proportion to the derivative of its elongation. Consequently, the resulting models exhibit stress and strain properties that vary in time.

(a) Maxwell

FIGURE 4.19

(b)

Voight

(c)

Kelvin

Three simple viscoelastic models: (a) the Maxwell model, (b) the Voight model, and (c) the Kelvin body or standard linear solid model.

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4.4 VISCOELASTIC PROPERTIES

The dynamic response of each model can be quantified by applying a step change in force F and noting the model’s resulting change in length, or position x, denoted the creep response. The converse experiment applies a step change in x and measures the resulting change in F, denoted stress relaxation. Creep and stress relaxation tests for each dynamic model can be carried out easily using the Simulink program. Figure 4.20 shows a purely elastic material subjected to a step change in applied force F. The material’s subsequent position x follows the change in force directly. This material exhibits no creep. Figure 4.21 shows the purely elastic material subjected to a step change in position x. Again, the material responds immediately with a step change in F (i.e., no stress relaxation is observed). James Clerk Maxwell (1831–1879) used a series combination of ideal spring and dashpot to describe the viscoelastic properties of air. Figure 4.22 shows the Maxwell viscoelastic model subjected to a step change in applied force, and Figure 4.23 shows the Maxwell model’s stress relaxation response. The latter exhibits an initial high stress followed by stress relaxation back to the initial stress level. The creep response, however, shows that this model is not bounded in displacement, since an ideal dashpot may be extended forever. Woldemar Voight (1850–1919) used the parallel combination of an ideal spring and dashpot in his work with crystallography. Figure 4.24 shows the creep test of the Voight viscoelastic model. Figure 4.25 shows that this model is unbounded in force. That is, when a step change in length is applied, force goes to infinity, since the dashpot cannot immediately respond to the length change. Elastic Model − simple spring 1

Creep F

1/K

6 5

Displacement x

4 3 2 1 0 −1 0

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

FIGURE 4.20 Simulink model of the creep test for a purely elastic material (an ideal spring). This model solves the equation x¼F/K, where x is displacement, F is applied force, and K is the spring constant. Below is the elastic creep response to a step increase in applied force F with K¼1 and force changed from 0 to 5 (arbitrary units). The displacement x linearly follows the applied force.

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4. BIOMECHANICS

Elastic Model − simple spring 1

Stress Relaxation x

6 5

Force F

4 3 2 1 0 −1

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

FIGURE 4.21 Simulink model (top) and stress relaxation test of the purely elastic model, which solves F¼Kx. Applied step displacement x¼5 and spring constant K¼1. Force linearly follows displacement.

Maxwell Model of Viscoelasticity − series spring and dashpot Creep F/B = dx1/dt 1 1/B

x1 + F/K = x

x1 1 s Integrator

Sum

F/K x = F/K + integral(F/B)

1 1/K 30

Displacement x

25 20 15 10 5 0 0

FIGURE 4.22

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

Creep of the Maxwell viscoelastic model, a series combination of ideal spring and dashpot (Figure 4.19a). The spring constant K¼1, and dashpot damping coefficient B¼1 (arbitrary R units). This system is subjected to a step change in force, and displacement x arises by solving x ¼ F=K þ F=B. The spring instantly responds, followed by creep of the ideal dashpot, which may extend as long as force is applied.

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4.4 VISCOELASTIC PROPERTIES

Maxwell Model of Viscoelasticity − series spring and dashpot Stress Relaxation x−x1

K(x−x1) = F 1

Sum x1

1 s Integrator

F/B = dx1/dt

1 1/B

7.5

Force F

2.5

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

5 R

FIGURE 4.23 Stress relaxation of the Maxwell viscoelastic model. This model solves F ¼ K½x F=B, again

with K¼B¼1 and arbitrary units. The ideal spring instantly responds followed by stress relaxation via the dashpot to the steady-state force level.

William Thompson (Lord Kelvin, 1824–1907) used the three-element viscoelastic model (Figure 4.19c) to describe the mechanical properties of different solids in the form of a torsional pendulum. Figure 4.26 shows the three-element Kelvin model’s creep response. This model has an initial rapid jump in position with subsequent slow creep. Figure 4.27 shows the Kelvin model stress relaxation test. Initially, the material is very stiff, with subsequent stress decay to a nonzero steady-state level that is due to the extension of the dashpot. The three-element Kelvin model is the simplest lumped viscoelastic model that is bounded both in extension and force. The three-element viscoelastic model describes the basic features of stress relaxation and creep. Biological materials often exhibit more complex viscoelastic properties. For example, plotting hysteresis as a function of frequency of applied strain gives discrete curves for the lumped viscoelastic models. Biological tissues demonstrate broad, distributed hysteresis properties. One solution is to describe biomaterials with a distributed network of three-element models. A second method is to use the generalized viscoelastic model of Westerhof and Noordergraaf (1970) to describe the viscoelastic wall properties of blood vessels. Making the elastic modulus mathematically complex yields a model that includes the frequency dependent elastic modulus, stress relaxation, creep, and hysteresis exhibited by arteries. Further, the Voight and Maxwell models emerge as special (limited) cases of this general approach.

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4. BIOMECHANICS

Voight Model of Viscoelasticity − parallel spring and dashpot Creep F/B 1 F

F/B − K/B x = dx/dt

1/B

Sum

1 x s dx/dt

1 K/B 6 5

Displacement x

4 3 2 1 0 −1

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

FIGURE 4.24

Creep of the Voight viscoelastic model, a parallel combination of ideal spring and dashpot (Figure 4.19b). This model solves the differential equation dx=dt ¼ 1=B½F Kx for x. K¼B¼1, and the step applied force is 5 arbitrary units. Displacement slowly creeps toward its steady-state value.

4.5 CARTILAGE, LIGAMENT, TENDON, AND MUSCLE The articulating surfaces of bones are covered with articular cartilage, a biomaterial composed mainly of collagen. Collagen is the main structural material of hard and soft tissues in animals. Isolated collagen fibers have high tensile strength that is comparable to nylon (50–100 MPa) and an elastic modulus of approximately 1 GPa. Elastin is a protein found in vertebrates and is particularly important in blood vessels and the lungs. Elastin is the most linearly elastic biosolid known, with an elastic modulus of approximately 0.6 MPa. It gives skin and connective tissue their elasticity.

4.5.1 Cartilage Cartilage serves as the bearing surfaces of joints. It is porous, and its complex mechanical properties arise from the motion of fluid in and out of the tissue when subjected to joint loading. Consequently, articular cartilage is strongly viscoelastic, with stress relaxation times

171

4.5 CARTILAGE, LIGAMENT, TENDON, AND MUSCLE

Voight Model of Viscoelasticity − parallel spring and dashpot Stress Relaxation x du/dt x

x dot

B x dot 1 B

Derivative

Sum1

F = K x + B x dot

1 K 30 25

Force F

20 15 10 5 0 −5

0.5

1.5

2 2.5 3 Time t [s]

3.5

4.5

FIGURE 4.25 Stress relaxation of the Voight viscoelastic model. This model solves the equation F ¼ Kx þ Bdx=dt. Since the dashpot is in parallel with the spring, and since it cannot respond immediately to a step change in length, the model force goes to infinity.

in compression on the order of 1 to 5 seconds. Cartilage is anisotropic and displays hysteresis during cyclical loading. The ultimate compressive stress of cartilage is on the order of 5 MPa.

4.5.2 Ligaments and Tendons Ligaments join bones together and consequently serve as part of the skeletal framework. Tendons join muscles to bones and transmit forces generated by contracting muscles to cause movement of the jointed limbs. Tendons and ligaments primarily transmit tension, so they are composed mainly of parallel bundles of collagen fibers and have similar mechanical properties. Human tendon has an ultimate stress of 50–100 MPa and exhibits very nonlinear stress-strain curves. The middle stress-strain range is linear, with an elastic modulus of approximately 1–2 GPa. Both tendons and ligaments exhibit hysteresis, viscoelastic creep, and stress relaxation. These materials may also be “preconditioned,” whereby initial tensile loading can affect subsequent load-deformation curves. The material properties shift due to changes in the internal tissue structure with repeated loading.

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4. BIOMECHANICS

6 5

Displacement x

4 3 2 1 0 −1

0.5

1.5

2.5

3.5

4.5

Time t [s]

FIGURE 4.26

Creep of the Kelvin three-element viscoelastic model. This model’s equations of motion are left to the reader to derive. After a step change in force, this model has an initial immediate increase in displacement, with a subsequent slow creep to a steady-state level.

15 12.5 10

Force F

7.5 5 2.5 0 −2.5 −5

0.5

1.5

2.5

3.5

4.5

Time t [s]

FIGURE 4.27 Stress relaxation of the Kelvin viscoelastic model. This model has an initial immediate increase in force followed by slower stress relaxation to a steady-state force level.

4.5.3 Muscle Mechanics Chapter 3 introduced muscle as an active, excitable tissue that generates force by forming cross-bridge bonds between the interdigitating actin and myosin myofilaments. The quantitative description of muscle contraction has evolved into two separate foci: lumped descriptions based on A. V. Hill’s contractile element and cross-bridge models based on A. F. Huxley’s description of a single sarcomere [22]. The earliest quantitative descriptions of muscle are lumped whole muscle models, with the simplest mechanical description being a

4.5 CARTILAGE, LIGAMENT, TENDON, AND MUSCLE

173

purely elastic spring. Potential energy is stored when the spring is stretched, and shortening occurs when it is released. The idea of muscle elastance can be traced back to Ernst Weber [29], who considered muscle as an elastic material that changes state during activation via conversion of chemical energy. Subsequently, investigators retained the elastic description but ignored metabolic alteration of muscle stiffness. A purely elastic model of muscle can be refuted on thermodynamic grounds, since the potential energy stored during stretching is less than the sum of the energy released during shortening as work and heat. Still, efforts to describe muscle by a combination of traditional springs and dashpots continued. In 1922, Hill coupled the spring with a viscous medium, thereby reintroducing viscoelastic muscle descriptions that can be traced back to the 1840s. Quick stretch and release experiments show that muscle’s viscoelastic properties are strongly time dependent. In general, the faster a change in muscle length occurs, the more severely the contractile force is disturbed. Muscle contraction clearly arises from a more sophisticated mechanism than a damped elastic spring. In 1935, Fenn and Marsh added a series elastic element to Hill’s damped elastic model and concluded that “muscle cannot properly be treated as a simple mechanical system.” Subsequently, Hill embodied the empirical hyperbolic relation between load and initial velocity of shortening for skeletal muscle as a model building block, denoted the contractile element. Hill’s previous viscoelastic model considered muscle to possess a fixed amount of potential energy whose rate of release is controlled by viscosity. Energy is now thought to be controlled by some undefined internal mechanism rather than by friction. This new feature of muscle dynamics varying with load was a step in the right direction; however, subsequent models, including heart studies, built models based essentially on the hyperbolic curve that was measured for tetanized skeletal muscle. This approach can be criticized on two grounds: (1) embodiment of the contractile element by a single force-velocity relation sets a single, fixed relation between muscle energetics and force; and (2) it yields no information on the contractile mechanism behind this relation. Failure of the contractile element to describe a particular loading condition led investigators to add passive springs and dashpots liberally, with the number of elements reaching at least nine by the late 1960s. Distributed models of muscle contraction, generally, have been conservative in design and have depended fundamentally on the Hill contractile element. Recent models are limited to tetanized, isometric contractions or to isometric twitch contractions. A second, independent focus of muscle contraction research works at the ultrastructural level, with the sliding filament theory serving as the most widely accepted contraction mechanism. Muscle force generation is viewed as the result of crossbridge bonds formed between thick and thin filaments at the expense of biochemical energy. The details of bond formation and detachment are under considerable debate, with the mechanism for relaxation particularly uncertain. Prior to actual observation of crossbridges, A. F. Huxley [13] devised the crossbridge model based on structural and energetic assumptions. Bonds between myofilaments are controlled via rate constants f and g that dictate attachment and detachment, respectively. One major shortcoming of this idea was the inability to describe transients resulting from rapid changes in muscle length or load, similar to the creep and stress relaxation tests previously discussed. Subsequent models adopt increasingly complex bond attachment and detachment rate functions and are often limited in scope to description of a single pair of myofilaments. Each tends

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4. BIOMECHANICS

to focus on description of a single type of experiment. No model has been shown to broadly describe all types of contractile loading conditions. Crossbridge models have tended to rely on increasingly complex bond attachment and detachment rate functions. This trend has reversed the issue of describing complex muscle dynamics from the underlying, simpler crossbridges to adopting complex crossbridge dynamics to describe a particular experiment. Alternatively, Palladino and Noordergraaf [22] proposed a large-scale, distributed muscle model that manifests both contraction and relaxation as the result of fundamental mechanical properties of crossbridge bonds. As such, muscle’s complex contractile properties emerge from its underlying ultrastructure dynamics—that is, function follows from structure. Bonds between myofilaments, which are biomaterials, are described as viscoelastic material. The initial stimulus for contraction is electrical. Electrical propagation through cardiac muscle occurs at finite speed, implying spatial asynchrony of stimulation. Furthermore, Caþþ release from the sarcoplasmic reticulum depends on diffusion for availability at the myosin heads. These effects, as well as nonuniformity of structure, strongly suggest that contraction is asynchronous throughout the muscle. Recognition of muscle’s distributed properties by abandoning the assumption of perfect synchrony in contraction and consideration of myofilament mass allow for small movements of thick with respect to thin filaments. Such movements lead to bond detachment and heat production. Gross movement such as muscle shortening exacerbates this process. Quick transients in muscle length or applied load have particularly strong effects and have been observed experimentally. Muscle relaxation is thereby viewed as a consequence of muscle’s distributed properties. The distributed muscle model is built from the following main features: sarcomeres consist of overlapping thick and thin filaments connected by crossbridge bonds that form during activation and detach during relaxation. Figure 4.28 shows a schematic of a muscle fiber composed of a string of series sarcomeres. Crossbridge bonds are each described as three-element viscoelastic solids, and myofilaments as masses. Force is generated due to viscoelastic crossbridge bonds that form and are stretched between the interdigitating matrix of myofilaments. sarcomere 1

sarcomere 2

sarcomere N ...

bond 1,1

bond 1,2

bond 1,3

bond 1,4

bond 1,2N-1

bond 1,2N

bond 2,1

bond 2,2

bond 2,3

bond 2,4

bond 2,2N-1

bond 2,2N

bond M,4

...

bond M,3

...

bond M,2

...

bond M,1

...

bond M,2N-1

bond M,2N

FIGURE 4.28 Schematic diagram of a muscle fiber built from a distributed network of N sarcomeres. Each sarcomere has M parallel pairs of crossbridge bonds. Adapted from [22].

4.6 CLINICAL GAIT ANALYSIS

175

The number of bonds formed depends on the degree of overlap between thick and thin filaments and is dictated spatially and temporally due to finite electrical and chemical activation rates. Asynchrony in bond formation and unequal numbers of bonds formed in each half sarcomere, as well as mechanical disturbances such as muscle shortening and imposed length transients, cause small movements of the myofilaments. Since myofilament masses are taken into account, these movements take the form of damped vibrations with a spectrum of frequencies due to the distributed system properties. When the stress in a bond goes to zero, the bond detaches. Consequently, myofilament motion and bond stress relaxation lead to bond detachment and produce relaxation without assumption of bond detachment rate functions. In essence, relaxation results from inherent system instability. Although the model is built from linear, time-invariant components (springs, dashpots, and masses), the highly dynamic structure of the model causes its mechanical properties to be highly nonlinear and time-varying, as is found in muscle fibers and strips. Sensitivity of the model to mechanical disturbances is consistent with experimental evidence from muscle force traces, aequorin measurements of free calcium ion, and highspeed x-ray diffraction studies, which all suggest enhanced bond detachment. The model is also consistent with sarcomere length feedback studies in which reduced internal motion delays relaxation, and it predicted muscle fiber (cell) dynamics prior to their experimental measurement. This model proposes a structural mechanism for the origin of muscle’s complex mechanical properties and predicts new features of the contractile mechanism—for example, a mechanism for muscle relaxation and prediction of muscle heat generation. This approach computes muscle’s complex mechanical properties from physical description of muscle anatomical structure, thereby linking subcellular structure to organ-level function. This chapter describes some of the high points of biological tissues’ mechanical properties. More comprehensive references include Fung’s Biomechanics: Mechanical Properties of Living Tissues, Nigg and Herzog’s Biomechanics of the Musculo-Skeletal System, and Mow and Hayes’s Basic Orthopaedic Biomechanics. Muscle contraction research has a long history, as chronicled in the book Machina Carnis by Needham. For a more comprehensive history of medicine, see Singer and Underwood’s book. The next two sections apply biomechanics concepts introduced in Sections 4.2–4.5 to human gait analysis and to the quantitative study of the cardiovascular system.

4.6 CLINICAL GAIT ANALYSIS An example of applied dynamics in human movement analysis is clinical gait analysis. Clinical gait analysis involves the measurement of the parameters that characterize a patient’s gait pattern, the interpretation of the collected and processed data, and the recommendation of treatment alternatives. It is a highly collaborative process that requires the cooperation of the patient and the expertise of a multidisciplinary team that typically includes a physician, a physical therapist or kinesiologist, and an engineer or technician. The engineer is presented with a number of challenges. The fundamental objective in data collection is to monitor the patient’s movements accurately and with sufficient precision for clinical use without altering the patient’s typical performance. While measurement devices for clinical gait analysis are established to some degree and are commercially available, the

176

4. BIOMECHANICS

protocols for the use of the equipment continue to develop. The validity of these protocols and associated models and the care with which they are applied ultimately dictate the meaning and quality of the resulting data provided for interpretation. This is one area in which engineers in collaboration with their clinical partners can have a significant impact on the clinical gait analysis process. Generally, data collection for clinical gait analysis involves the placement of highly reflective markers on the surface of the patient’s skin. These external markers reflect light to an array of video-based motion cameras that surround the measurement volume. The instantaneous location of each of these markers can then be determined stereometrically based on the images obtained simultaneously from two or more cameras. Other aspects of gait can be monitored as well, including ground reactions via force platforms embedded in the walkway and muscle activity via electromyography with either surface or intramuscular fine wire electrodes, depending on the location of the particular muscle. In keeping with the other material presented in this chapter, the focus of this section will pertain to the biomechanical aspects of clinical gait analysis and includes an outline of the computation of segmental and joint kinematics and joint kinetics, and a brief illustration of how the data are interpreted.

4.6.1 The Clinical Gait Model The gait model is the algorithm that transforms the data collected during walking trials into the information required for clinical interpretation. For example, the gait model uses the data associated with the three-dimensional displacement of markers on the patient to compute the angles that describe how the patient’s body segment and lower-extremity joints are moving. The design of the gait model is predicated on a clear understanding of the needs of the clinical interpretation team—for example, the specific aspects of gait dynamics of interest. To meet these clinical specifications, gait model development is constrained both by the technical limitations of the measurement system and by the broad goal of developing protocols that may be appropriate for a wide range of patient populations that vary in age, gait abnormality, walking ability, and so on. An acceptable model must be sufficiently general to be used for many different types of patients (e.g., adults and children with varying physical and cognitive involvement), be sufficiently sophisticated to allow detailed biomechanical questions to be addressed, and be based on repeatable protocols that are feasible in a clinical setting.

4.6.2 Kinematic Data Analysis Reflective markers placed on the surface of the patient’s skin are monitored or tracked in space and time by a system of video-based cameras. These marker trajectories are used to compute coordinate systems that are anatomically aligned and embedded in each body segment under analysis. These anatomical coordinate systems provide the basis for computing the absolute spatial orientation, or attitude, of the body segment or the angular displacement of one segment relative to another, such as joint angles. For this analysis, at least three noncollinear markers or points of reference must be placed on or identified for each body segment included in the analysis. These markers form a plane from which a segmentally fixed coordinate system may be derived. Any three markers will allow the segment motion to be monitored, but unless these markers are referenced to the subject’s anatomy, such

177

4.6 CLINICAL GAIT ANALYSIS

kinematic quantification is of limited clinical value. Markers must either be placed directly over palpable bony landmarks on the segment or at convenient (i.e., visible to the measurement cameras) locations on the segment that are referenced to the underlying bones. An examination of the pelvic and thigh segments illustrates these two alternatives. Pelvic Anatomical Coordinate System For the pelvis, markers placed over the right and left anterior-superior-iliac-spine (ASIS) and either the right or left posterior-superior-iliac-spine (PSIS) will allow for the computation of an anatomically aligned coordinate system, as described in Example Problem 4.10.

EXAMPLE PROBLEM 4.10 Given the following three-dimensional locations in meters for a set of pelvic markers expressed relative to an inertially fixed laboratory coordinate system (Figure 4.29), Right ASIS : RASIS ¼ 0:850i 0:802j þ 0:652k Left ASIS : LASIS ¼ 0:831i 0:651j þ 0:652k PSIS ¼ 1:015i 0:704j þ 0:686k compute an anatomical coordinate system for the pelvis. r3 PSIS

LASIS RASIS

epaz

PSIS

epay

r2 H

RASIS

LASIS

epax H

TW MK LK

K r7 r4

TW r8

ettz etaz r5 MK e tay K etty etax r6 ettx r9

FIGURE 4.29

Kinematic markers used to define pelvis and thigh coordinate systems. For the pelvis, PSIS denotes posterior-superior-iliac-spine, H is hip center, and RASIS and LASIS denote right and left anteriorsuperior-iliac-spine markers, respectively. For the thigh, TW is thigh wand, K is knee center, and MK and LK are medial and lateral knee (femoral condyle) markers, respectively.

Continued

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4. BIOMECHANICS

Solution These three anatomical markers form a plane. The line between the right ASIS and left ASIS represents one coordinate system axis. Another coordinate axis is perpendicular to the pelvic plane. The third coordinate axis is computed to be orthogonal to the first two: 1. Subtract vector RASIS from vector LASIS, LASIS RASIS ¼ ð0:831 ð0:850ÞÞi þ ð0:651 ð0:802ÞÞj þ ð0:652 0:652Þk to find r1 ¼ 0:0190i þ 0:1510j þ 0:0000k and its associated unit vector: 0:019i þ 0:151j þ 0:000k er1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0192 þ 0:1512 þ 0:0002 er1 ¼ 0:125i þ 0:992j þ 0:000k Unit vector er1 represents the medial-lateral direction or y-axis for the pelvic anatomical coordinate system epay (Figure 4.29). 2. A second vector in the pelvic plane is required to compute the coordinate axis that is perpendicular to the plane. Consequently, subtract vector RASIS from vector PSIS to find r2 ¼ 0:165i þ 0:098j þ 0:034k 3. Take the vector cross product epay r2 to yield i j k r3 ¼ 0:125 0:992 0:000 0:165 0:098 0:034 ¼ ½ð0:992Þð0:034Þ ð0:000Þð0:098Þi þ ½ð0:000Þð0:165Þ ð0:125Þð0:034Þj þ ½ð0:125Þð0:098Þ ð0:992Þð0:165Þk ¼ 0:034i 0:004j þ 0:176k and its associated unit vector: er3 ¼ epaz ¼ 0:188i 0:024j þ 0:982k Unit vector er3 represents the anterior-superior direction or z-axis of the pelvic anatomical coordinate system epaz (Figure 4.29). 4. The third coordinate axis is computed to be orthogonal to the first two. Take the vector cross product epay epaz to compute the fore-aft direction, or x-axis, of the pelvic anatomical coordinate system: epax ¼ 0:974i 0:123j 0:190k

4.6 CLINICAL GAIT ANALYSIS

179

For this example, the anatomical coordinate system for the pelvis can be expressed as follows: 32 3 2 3 2 epax i 0:974 0:123 0:190 0:992 0:000 54 j 5 fepa g ¼ 4 epay 5 ¼ 4 0:125 epaz k 0:188 0:024 0:982 Note that the coefficients associated with these three axes represent the direction cosines that define the orientation of the pelvic coordinate system relative to the laboratory coordinate system. In summary, by monitoring the motion of the three pelvic markers, the instantaneous orientation of an anatomical coordinate system for the pelvis, fepa g, comprising axes epax, epay, and epaz, can be determined. The absolute angular displacement of this coordinate system can then be computed via Euler angles as pelvic tilt, obliquity, and rotation using Eqs. (4.32)–(4.34). An example of these angle computations is presented later in this section.

Thigh Anatomical Coordinate System The thigh presents a more significant challenge than the pelvis because three bony anatomical landmarks are not readily available as reference points during gait. A model based on markers placed over the medial and lateral femoral condyles and the greater trochanter is appealing but ill-advised. A marker placed over the medial femoral condyle is not always feasible during gait—for example, with patients whose knees make contact while walking. A marker placed over the greater trochanter is often described in the literature but should not be used as a reference because of its significant movement relative to the underlying greater trochanter during gait—that is, skin motion artifact [3]. In general, the approach used to quantify thigh motion, and the shank and foot, is to place additional anatomical markers on the segments during a static subject calibration process. Then the relationship between these static anatomical markers, which are removed before gait data collection, and the motion markers that remain on the patient during gait data collection may be calculated. It is assumed that this mathematical relationship remains constant during gait—that is, the instrumented body segments are assumed to be rigid. This process is illustrated in the Example Problem 4.11.

EXAMPLE PROBLEM 4.11 Given the following marker coordinate data that have been acquired while the patient stands quietly, also in meters, lateral femoral condyle marker LK ¼ 0:881i 0:858j þ 0:325k medial femoral condyle marker MK ¼ 0:855i 0:767j þ 0:318k compute an anatomical coordinate system for the thigh.

Solution A thigh plane is formed based on three anatomical markers or points: the hip center, the lateral femoral condyle marker LK, and the medial femoral condyle marker MK. The knee center location can then be estimated as the midpoint between LK and MK. With these points, the vector Continued

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4. BIOMECHANICS

from the knee center to the hip center represents the longitudinal axis of the coordinate system. A second coordinate axis is perpendicular to the thigh plane. The third coordinate axis is computed to be orthogonal to the first two. The location of the knee center of rotation may be approximated as the midpoint between the medial and lateral femoral condyle markers, LK þ MK ð0:881Þ þ ð0:855Þ ð0:858Þ þ ð0:767Þ ð0:325Þ þ ð0:318Þ ¼ iþ jþ k 2 2 2 2 yielding knee center location K ¼ 0:868i 0:812j þ 0:321k The location of the center of the head of the femur, referred to as the hip center, is approximated based either on patient anthropometry and a statistical model of pelvic geometry [6] or from data collected while the subject executes a specific hip movement [8]. In this example, the hip center can be located at approximately hip center location H ¼ 0:906i 0:763j þ 0:593k Now the anatomical coordinate system for the thigh may be computed as follows: 1. Subtract the vector K from H, giving r4 ¼ 0:038i þ 0:049j þ 0:272k and its associated unit vector er4 ¼ etaz ¼ 0:137i þ 0:175j þ 0:975k Unit vector er4 represents the longitudinal direction, or z-axis, of the thigh anatomical coordinate system etaz. 2. As with the pelvis, a second vector in the thigh plane is required to compute the coordinate axis that is perpendicular to the plane. Consequently, subtract vector LK from MK: r5 ¼ 0:026i þ 0:091j 0:007k 3. Form the vector cross product r5 etaz to yield r6 ¼ 0:090i 0:024j þ 0:017k and its associated unit vector er6 ¼ etax ¼ 0:949i 0:258j þ 0:180k Unit vector er6 represents the fore-aft direction, or x-axis, of the thigh anatomical coordinate system etax. 4. Again, the third coordinate axis is computed to be orthogonal to the first two. Determine the medial-lateral or y-axis of the thigh anatomical coordinate system, etay, from the cross product etaz etax : etay ¼ 0:284i þ 0:950j 0:131k

4.6 CLINICAL GAIT ANALYSIS

181

For this example, the anatomical coordinate system for the thigh can be expressed as 32 3 2 3 2 i 0:949 0:258 0:180 etax 0:950 0:131 54 j 5 feta g ¼ 4 etay 5 ¼ 4 0:284 k 0:137 0:175 0:975 etaz This defines an anatomical coordinate system fixed to the thigh, feta g, comprising axes etax, etay, and etaz. Its basis, however, includes an external marker (medial femoral condyle MK) that must be removed before the walking trials. Consequently, the location of the knee center cannot be computed as described in the preceding example. This dilemma is resolved by placing another marker on the surface of the thigh such that it also forms a plane with the hip center and lateral knee marker. These three reference points can then be used to compute a “technical” coordinate system for the thigh to which the knee center location may be mathematically referenced.

EXAMPLE PROBLEM 4.12 Continuing Example Problem 4.11, and given the coordinates of another marker placed on the thigh but not anatomically aligned, thigh wand marker TW ¼ 0:890i 0:937j þ 0:478k compute a technical coordinate system for the thigh.

Solution A technical coordinate system for the thigh can be computed as follows: 1. Compute the longitudinal direction, or z-axis, of the technical thigh coordinate system ett. Start by subtracting vector LK from the hip center H to form r7 ¼ 0:025i þ 0:094j þ 0:268k and its associated unit vector er7 ¼ ettz ¼ 0:088i þ 0:330j þ 0:940k Unit vector er7 represents the z-axis of the thigh technical coordinate system, ettz. 2. To compute the axis that is perpendicular to the plane formed by LK, H, and TW, subtract vector LK from TW to compute r8 ¼ 0:009i 0:079j þ 0:153k 3. Calculate the vector cross product r7 r8 to yield r9 ¼ 0:036i þ 0:001j þ 0:003k with its associated unit vector er9 ¼ ettx ¼ 0:996i þ 0:040j þ 0:079k Unit vector er9 represents the fore-aft direction, or x-axis, of the thigh technical coordinate system ettx. Continued

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4. The third coordinate axis is computed to be orthogonal to the first two axes. Compute the vector cross product ettz ettx to determine the medial-lateral direction, or y-axis, of the thigh technical coordinate system: etty ¼ ettz ettx ¼ 0:012i þ 0:943j þ 0:332k For this example, the technical coordinate system for the thigh can be expressed as 32 3 2 3 2 i 0:996 0:040 0:079 ettx 0:943 0:332 54 j 5 fett g ¼ 4 etty 5 ¼ 4 0:012 ettz k 0:088 0:330 0:940 Note that this thigh technical coordinate system fett g computed during the standing subject calibration can also be computed from each camera frame of walking data. That is, its computation is based on markers (the lateral femoral condyle and thigh wand markers) and an anatomical landmark (the hip center) that are available for both the standing and walking trials. Consequently, the technical coordinate system fett g becomes the embedded reference coordinate system to which other entities can be related. The thigh anatomical coordinate system feta g can be related to the thigh technical coordinate system fett g by using either direction cosines or Euler angles, as described in Section 4.2.2. Also, the location of markers that must be removed after the standing subject calibration (e.g., the medial femoral condyle marker MK), or computed anatomical locations (e.g., the knee center) can be transformed into the technical coordinate system fett g and later retrieved for use in walking trial data reduction.

Segment and Joint Angles Tracking the anatomical coordinate system for each segment allows for the determination of either the absolute angular orientation, or attitude, of each segment in space or the angular position of one segment relative to another. In the preceding example, the three pelvic angles that define the position of the pelvic anatomical coordinate system fepa g relative to the laboratory (inertially fixed) coordinate system can be computed from the Euler angles, as described in Section 4.2.2 with Eqs. (4.32)–(4.34). Note that in these equations the laboratory coordinate system represents the proximal (unprimed) coordinate system, and the pelvic anatomical coordinate system fepa g represents the distal (triple primed) coordinate system. Consequently, Eq. (4.32) yx ¼ arcsinðk000 jÞ becomes yx ¼ arcsinðepaz jÞ ¼ arcsinðð0:188i 0:024j þ 0:982kÞ jÞ ¼ arcsinð0:024Þ ¼ 1 of pelvic obliquity

4.6 CLINICAL GAIT ANALYSIS

Similarly, Eq. (4.33)

becomes

183

000 ðk iÞ yy ¼ arcsin cos yx 0 1 ðepaz iÞA yy ¼ arcsin@ cos yx 0 1 ð0:188i 0:024j þ 0:982kÞ iA ¼ arcsin@ cos 1 0 1 0:188 A ¼ arcsin@ cos 1 ¼ 11 of anterior pelvic tilt

and Eq. (4.34)

yz ¼ arcsin

becomes

ði000 jÞ cos yx

0 1 ðepax jÞA yz ¼ arcsin@ cos yx 0 1 ð0:974i 0:123j 0:190kÞ j A ¼ arcsin@ cos 1 0 1 0:123 A ¼ arcsin@ cos 1 ¼ 7 of pelvic rotation

This Euler angle computation may be repeated to solve for the three hip angles that define the position of the thigh anatomical coordinate system feta g relative to the pelvic anatomical coordinate system fepa g. For the hip angles, the proximal (unprimed) coordinate system is the pelvis and the distal (triple-primed) coordinate system is the thigh. Substituting the values of fepa g and feta g from Example Problems 4.10 and 4.11 into Eq. (4.32) yields: yz ¼ arcsinðetaz epay Þ ¼ arcsinðð0:137i þ 0:175j þ 0:975kÞ ð0:125i þ 0:992j þ 0:000kÞÞ ¼ arcsinð0:156Þ ¼ 9 of hip abduction-adduction

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The negative sign is associated with hip adduction of the left thigh or hip abduction of the right thigh. Further substitution of values of fepa g and feta g into Eqs. (4.33) and (4.34) yields hip flexion-extension yy ¼ 20 hip internal-external rotation yz ¼ 8 For hip internal-external rotation, the negative sign is associated with hip internal rotation of the left thigh or hip external rotation of the right thigh. A negative hip flexion-extension angle corresponds to hip extension, independent of side. This process may be repeated for other body segments such as the shank (lower leg), foot, trunk, arms, and head with the availability of properly defined anatomical coordinate systems.

4.6.3 Kinetic Data Analysis The marker displacement or motion data provide an opportunity to appreciate segment and joint kinematics. Kinematic data can be combined with ground reaction data—that is, forces and torques and their points of application, which are referred to as the centers of pressure. Combined with estimates of segment mass and mass moments of inertia, the net joint reaction forces and moments may then be computed. To illustrate the details of this computational process, consider the following determination of the reactions at the ankle (Figure 4.30) for an individual with mass of 25.2 kg. Data for one instant in the gait cycle are shown in the following table.

Ankle center location Toe marker location Center of pressure location Ground reaction force vector Ground reaction torque vector Foot anatomical coordinate system

Foot linear acceleration vector Foot angular velocity vector Foot angular acceleration vector Ankle angular velocity vector

Symbol

Units

xlab

A T CP Fg Tg efax efay efaz afoot vfoot afoot vankle

[m] [m] [m] [N] [N-m]

0.357 0.421 0.422 3.94 0.000 0.977 0.0815 0.195 2.09 0.0420 0.937 0.000759

[m/s2] [rad/s] [rad/sec2] [rad/s]

ylab 0.823 0.819 0.816 15.21 0.000 –0.0624 0.993 0.102 0.357 2.22 8.85 1.47

zlab 0.056 0.051 0.000 242.36 0.995 0.202 0.0877 0.975 0.266 0.585 5.16 0.0106

Anthropomorphic relationships presented in Table 4.1 are used to estimate the mass and mass moments of inertia of the foot, as well as the location of its center of gravity. The mass of the foot, mfoot, may be estimated to be 1.45 percent of the body mass, or 0.365 kg, and the location of the center of gravity is approximated as 50 percent of the foot length. The length

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4.6 CLINICAL GAIT ANALYSIS

FA z' A CG x'

r1 Z

y' r2 mfootg

k i

X Fg

FIGURE 4.30

Ankle A and toe T marker data are combined with ground reaction force data Fg and segment mass and mass moment of inertia estimates to compute the net joint forces and moments.

of the foot lfoot may be approximated as the distance between the ankle center and the toe marker, determined as follows: T A ¼ ð0:421 0:357Þi þ ð0:819 0:823Þj þ ð0:051 0:056Þk lfoot

¼ 0:064i 0:004j 0:005k ¼ jT Aj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð0:064Þ2 þ ð0:004Þ2 þ ð0:005Þ2 ¼ 0:064 m

Then the location of the center of gravity can be determined relative to the ankle center as

lfoot ðT AÞ 0:064 0:064i 0:004j 0:005k Aþ ¼ ð0:357i þ 0:823j þ 0:056kÞ þ 2 jT Aj 2 0:064 giving the location of the center of gravity: CG ¼ 0:389i þ 0:821j þ 0:054k which allows computation of position vectors r1 and r2 (see Figure 4.30). With a foot length of 0.064 m, a foot mass of 0.365 kg, and a proximal radius of gyration per segment length of 0.690, the mass moment of inertia relative to the ankle center may be estimated with Eq. (4.40) as Ifoot=ankle ¼ ð0:365 kgÞ½ð0:690Þð0:064 mÞ2 ¼ 7:12 104 kg m2 The centroidal mass moment of inertia, located at the foot’s center of mass, may then be estimated using the parallel axis theorem (Eq. (4.41)): Ifoot=cm ¼ Ifoot=ankle mfoot d2 Note that the center of mass is equivalent to the center of gravity in a uniform gravitational field. In this case, d is the distance between the foot’s center of mass and the ankle.

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Table 4.1 shows the ratio of the foot center of mass location relative to its proximal end to be 0.5, so d ¼ 0:5ðlfoot Þ ¼ 0:032 m. Therefore, Ifoot=cm ¼ ð7:12 104 kg m2 Þ ð0:365 kgÞð0:032 mÞ2 ¼ 3:38 104 kg m2 Ifoot=cm represents the centroidal mass moment of inertia about the transverse principal axes of the foot (y 0 and z 0 in Figure 4.30). Consequently, Iy 0 y 0 ¼ 3:38 104 kg m2 Iz 0 z 0 ¼ 3:38 104 kg m2 The foot is approximated as a cylinder with a length to radius ratio of 6. The ratio of transverse to longitudinal (x 0 ) mass moments of inertia can be shown to be approximately 6.5. Then the longitudinal mass moment of inertia (about x 0 in Figure 4.30) may be estimated as Ix 0 x 0 ¼ 5:20 105 kg m2 Having estimated the anthropomorphic values for the foot, the kinetic analysis may now begin. The unknown ankle reaction force, FA, is found by using Newton’s Second Law, or P F ¼ ma: Fg þ FA mfoot gk ¼ mfoot afoot FA ¼ mfoot afoot Fg þ mfoot gk ¼ ð0:365 kgÞ½2:09i 0:357j 0:266k m=s ð3:94i 15:21j þ 242:4kÞ N þ ð0:365 kgÞð9:81 m=s2 Þk ¼ 3:18i þ 15:08j 238:9k N Euler’s equations of motion (Eqs. (4.44)–(4.46)) are then applied to determine the unknown ankle moment reaction MA. Euler’s equations are defined relative to the principal axes fixed to the segment—that is, x 0 , y 0 , and z 0 fixed to the foot. It is noted, however, that the data required for the solution presented previously—for example, vfoot and afoot —are expressed relative to the laboratory coordinate system (x, y, z). Consequently, vectors required for the solution of Euler’s equations must first be transformed into the foot coordinate system. In the preceding data set, the foot anatomical coordinate system was given as efax ¼ 0:977i 0:0624j 0:202k efay ¼ 0:0815i þ 0:993j þ 0:0877k efaz ¼ 0:195i 0:102j þ 0:975k where efax, efay, and efaz correspond to x 0 , y 0 , and z 0 , or i0 , j0 , and k0 . Recall from the discussion in Section 4.2.2 that coefficients in the expression for efax represent the cosines of the angles between x 0 and x, x 0 and y, and x 0 and z, respectively. Similarly, the coefficients in the expression for efay represent the cosines of the angles between y 0 and x, y 0 and y, and y 0 and z, and

4.6 CLINICAL GAIT ANALYSIS

187

the coefficients in the expression for efaz represent the cosines of the angles between z 0 and x, z 0 and y, and z 0 and z. Consequently, these relationships can be transposed as i ¼ 0:977i0 þ 0:0815j0 þ 0:195k0 j ¼ 0:0624i0 þ 0:993j0 0:102k0 k ¼ 0:202i0 þ 0:0877j0 þ 0:975k0 In this form, these relationships can be used to transform vectors expressed in terms of lab coordinates: A ¼ Ax i þ Ay j þ Az k into foot coordinates: A ¼ Ax i0 þ Ay j0 þ Az k0 To demonstrate this process, consider the foot angular velocity vector vfoot ¼ 0:042i þ 2:22j 0:585k rad=s Substituting the relationships for the lab coordinate system in terms of the foot coordinate system, vfoot becomes vfoot ¼ 0:042ð0:977i0 þ 0:0815j0 þ 0:195k0 Þ þ 2:22ð0:0624i0 þ 0:993j0 0:102k0 Þ 0:585ð0:202i0 þ 0:0877j0 þ 0:975k0 Þ ¼ 0:0210i0 þ 2:16j0 0:789k0 rad=s In a similar manner, the other vectors required for the computation are transformed into the foot coordinate system: r1 ¼ 0:032i þ 0:002j þ 0:002k ¼ 0:032i0 0:004k0 m r2 ¼ 0:033i 0:005j 0:054k ¼ 0:0435i0 0:007j0 0:0457k0 m Fg ¼ 3:94i 15:21j þ 242:36k ¼ 44:16i0 þ 6:47j0 þ 238:62k0 N Tg ¼ 0:995k ¼ 0:201i0 þ 0:0873j0 þ 0:970k0 N m FA ¼ 3:18i þ 15:1j 239k vfoot

¼ 44:2i0 6:23j0 235k0 N ¼ 0:0420i þ 2:22j 0:585k

afoot

¼ 0:021i0 þ 2:16j0 0:789k0 rad=s ¼ 0:937i þ 8:85j 5:16k ¼ 0:425i0 þ 8:26j0 6:116k0 rad=s2

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Expanding Euler’s equations of motion (Eqs. (4.44)–(4.46)), MAx0 þ ðr1 FA Þx 0 þ ðr2 Fg Þx 0 þ Tgx0 ¼ Ix 0 x 0 ax0 þ ðIz 0 z 0 Iy 0 y 0 Þoy0 oz0 MAy0 þ ðr1 FA Þy 0 þ ðr2 Fg Þy 0 þ Tgy0 ¼ Iy 0 y 0 ay0 þ ðIx 0 x 0 Iz 0 z 0 Þoz0 ox0 MAz0 þ ðr1 FA Þz 0 þ ðr2 Fg Þz 0 þ Tgz0 ¼ Iz 0 z 0 az0 þ ðIy 0 y 0 Ix 0 x 0 Þox0 oy 0 where ðr1 FA Þx0 represents the x0 component of r1 FA , ðr2 Fg Þx 0 represents the x0 component of r2 Fg , and so forth. Substitution of the required values and arithmetic reduction yields MA0 ¼ 1:50i0 þ 15:9j0 1:16k0 N m which can be transformed back into fixed lab coordinates, MA ¼ 2:54i þ 15:9j 0:037k N m By combining the ankle moment with the ankle angular velocity, the instantaneous ankle power may be computed as MA vankle ¼ ð2:54i þ 15:9j 0:037k N mÞ ð0:000759i þ 1:47j þ 0:0106k rad=sÞ ¼ 23:3 Watts or MA0 vankle0 ¼ ð1:50i0 þ 15:9j0 1:16k0 N mÞ ð0:0946i0 þ 1:46j0 0:140k0 rad=sÞ ¼ 23:3 Watts which is thought to represent a quantitative measure of the ankle’s contribution to propulsion.

4.6.4 Clinical Gait Interpretation The information and data provided for treatment decision making in clinical gait analysis include not only the quantitative variables described previously—3-D kinematics such as angular displacement of the torso, pelvis, hip, knee and ankle/foot, and 3-D kinetics, such as moments and power of the hip, knee, and ankle—but the following as well: • • • •

Clinical examination measures Biplanar video recordings of the patient walking Stride and temporal gait data such as step length and walking speed Electromyographic (EMG) recordings of selected lower extremity muscles

Generally, the interpretation of gait data involves the identification of abnormalities, the determination of the causes of the apparent deviations, and the recommendation of treatment alternatives. As each additional piece of data is incorporated, a coherent picture of the patient’s walking ability is developed by correlating corroborating data sets and resolving apparent contradictions in the information. Experience allows the team to distinguish a gait anomaly that presents the difficulty for the patient from a gait compensatory mechanism that aids the patient in circumventing the gait impediment to some degree. To illustrate aspects of this process, consider the data presented in Figures 4.31–4.33, which were measured from a 9-year-old girl with cerebral palsy spastic diplegia. Cerebral

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4.6 CLINICAL GAIT ANALYSIS

Pelvic Tilt

30 Ant. 20 10 Post. 0

Hip Flexion-Extension 45 Flex. 25 5 Ext. −15 Knee Flexion-Extension 70 Flex. 40 10 Ext. −20 Ankle Plantar-Dorsiflexion

30 Dors. 10 −10 Plnt. −30

50 75 % Gait Cycle

100

FIGURE 4.31 Sagittal plane kinematic data for the left side of a 9-year-old patient with cerebral palsy spastic diplegia (solid curves). Shaded bands indicate þ/ one standard deviation about the performance of children with normal ambulation. Stance phase is 0–60 percent of the gait cycle, and swing phase is 60–100 percent, as indicated by the vertical solid lines.

palsy is a nonprogressive neuromuscular disorder that is caused by an injury to the brain during or shortly after birth. The neural motor cortex is most often affected. In the ambulatory patient, this results in reduced control of the muscles required for balance and locomotion, causing overactivity, inappropriately timed activity, and muscle spasticity. Treatment options include physical therapy, bracing (orthoses), spasmolytic medications such as botulinum toxin and Baclofen, and orthopedic surgery and neurosurgery. The sagittal plane kinematics for the left side of this patient (Figure 4.31) indicate significant involvement of the hip and knee. Her knee is effectively “locked” in an excessively

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4. BIOMECHANICS

L Rectus Femoris

L Vastus Lateralis

L Vastus Medialis

L Hamstrings

L Anterior Tibialis

L Gastroc/Soleus 0

50 % Gait Cycle

100

FIGURE 4.32

Electromyogram (EMG) data for the same cerebral palsy patient as in Figure 4.31. Plotted are EMG activity signals for each of six left lower extremity muscles, each plotted as functions of percent of gait cycle. Gray bars represent mean normal muscle activation timing.

flexed position throughout stance phase (0–60 percent of the gait cycle) when her foot is contacting the floor. Knee motion in swing phase (60–100 percent) is also limited, with the magnitude and timing of peak knee flexion in swing reduced and delayed. The range of motion of her hip during gait is less than normal, failing to reach full extension at the end of stance phase. The motion of her pelvis is significantly greater than normal, tilting anteriorly in early stance coincident with extension of the hip, and tilting posteriorly in swing coincident with flexion of the hip. The deviations noted in these data illustrate neuromuscular problems commonly seen in this patient population. Inappropriate hamstring tightness, observed during the clinical examination, and inappropriate muscle activity during stance, seen in Figure 4.32, prevent the knee from properly extending. This flexed knee position also impedes normal extension of the hip in stance due to hip extensor weakness, also observed during the clinical examination. Hip extension is required in stance to allow the thigh to rotate under the advancing pelvis and upper body. To compensate for her reduced ability to extend the hip, she rotates her pelvis anteriorly in early stance to help move the thigh through its arc of motion. The biphasic pattern of the pelvic curve indicates that this is a bilateral issue to some degree. The limited knee flexion in swing combines with the plantar flexed ankle position to result in foot clearance problems during swing phase. The inappropriate activity of the rectus femoris muscle (Figure 4.32) in midswing suggests that spasticity of that muscle, a

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4.6 CLINICAL GAIT ANALYSIS

Hip Flexion 45 Joint Rotation 25 (degrees) 5 Extension −15 Extensor

Joint Moment (N-m/kg)

Flexor

−1

Knee

Ankle

−10

−15

−30

Generation 3 2 Joint Power (Watts/kg)

1 0 −1

Absorption −2 0

25 50 75 % Gait Cycle

100 0

25 50 75 % Gait Cycle

100 0

25 50 75 % Gait Cycle

100

FIGURE 4.33 Sagittal joint kinetic data for the same cerebral palsy patient of Figure 4.31. Joint rotation, joint moment, and joint power are plotted as functions of percent of gait cycle for the hip, knee, and ankle. Dark and light solid curves denote right and left sides, respectively. Bands indicate þ/ one standard deviation of the normal population.

knee extensor (also observed during clinical examination), impedes knee flexion. Moreover, the inappropriate activity of the ankle plantar flexor, primarily the gastrocnemius muscle, in late swing suggests that it is overpowering the pretibial muscles, primarily the anterior tibialis muscle, resulting in plantar flexion of the ankle or “foot drop.” The sagittal joint kinetics for this patient (Figure 4.33) demonstrate asymmetrical involvement of the right and left sides. Of special note is that her right knee and hip are compensating for some of the dysfunction observed on the left side. Specifically, the progressively increasing right knee flexion beginning at midstance (first row, center) and continuing into swing aids her contralateral limb in forward advancement during swing—that is, her pelvis can rock posteriorly along with a flexing hip to advance the thigh. One potentially adverse consequence of this adaptation is the elevated knee extensor moment in late stance that increases patella-femoral loading with indeterminate effects over time. The asymmetrical power production at the hip also illustrates clearly that the right lower extremity, in particular the muscles that cross the hip, provides the propulsion for gait with significant power generation early in stance to pull the body forward and elevate its center of gravity. Moreover, the impressive hip power generation, both with respect to magnitude and timing, at toe-off accelerates the stance limb into swing and facilitates knee flexion in spite of the elevated knee extensor moment magnitude. This is important to appreciate given the bilateral spastic response of the plantar flexor muscles, as evidenced by the premature ankle power

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4. BIOMECHANICS

generation and the presentation of a spastic stretch reflex in the clinical examination. This girl uses her hip musculature, right more than left, to a much greater degree than her ankle plantar flexors to propel herself forward during gait. This cursory case examination illustrates the process whereby differences from normal gait are recognized and the associated biomechanical etiology is explored. Some of the effects on gait of neuromuscular pathology in the sagittal plane have been considered in this discussion. Clinical gait analysis can also document and elucidate gait abnormalities associated with static bony rotational deformities. It also is useful in areas of clinical research by documenting treatment efficacy associated with bracing, surgery, and so forth. It should be noted, however, that although engineers and applied physicists have been involved in this work for well over a hundred years, there remains significant opportunity for improvement in the biomechanical protocols and analytical tools used in clinical gait analysis; in other words, there remains much to learn.

4.7 CARDIOVASCULAR DYNAMICS One major organ system benefiting from the application of mechanics principles is cardiovascular system dynamics, or hemodynamics, the study of the motion of blood. From a functional point of view, the cardiovascular system is driven by a complex pump, the heart, that generates pressure resulting in the flow of a complex fluid, blood, through a complex network of complex pipes, the blood vessels. Cardiovascular dynamics focuses on the measurement and analysis of blood pressure, volume, and flow within the cardiovascular system. The complexity of this elegant system is such that mechanical models, typically formulated as mathematical equations, are relied on to understand and integrate experimental data, to isolate and identify physiological mechanisms, and to lead ultimately to new clinical measures of heart performance and health and guide clinical therapies. As described in Chapter 3, the heart is a four-chambered pump connected to two main collections of blood vessels: the systemic and pulmonary circulations. This pump is electrically triggered and under neural and hormonal control. One-way valves control blood flow. Total human blood volume is approximately 5.2 liters. The left ventricle, the strongest chamber, pumps 5 liters per minute at rest, almost the body’s entire blood volume. With each heartbeat, the left ventricle pumps 70 ml, with an average of 72 beats per minute. During exercise, left ventricular output may increase sixfold, and heart rate more than doubles. The total combined length of the circulatory system vessels is estimated at 100,000 km, a distance two and one half times around the earth. The left ventricle generates approximately 1.7 watts of mechanical power at rest, increasing threefold during heavy exercise. One curious constant is the total number of heartbeats in a lifetime, around one billion in mammals [31]. Larger animals have slower heart rates and live longer lives, and vice versa for small animals.

4.7.1 Blood Rheology Blood is composed of fluid, called plasma, and suspended cells, including erythrocytes (red blood cells), leukocytes (white cells), and platelets. From a mechanical point of view, a fluid is distinguished from a solid as follows. Figure 4.34 shows a two-dimensional block of solid material (left panel) subjected to two opposite, parallel, transverse external forces, depicted by the solid arrows at the top and bottom surfaces. This applied shear force is resisted by

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4.7 CARDIOVASCULAR DYNAMICS

Solid

Fluid

FIGURE 4.34 (Left) A solid material resists applied external shear stress (solid vectors) via internally generated reaction shear stress (dashed vectors). (Right) A fluid subjected to applied shear stress is unable to resist and instead flows (dashed lines).

the solid via internally generated reaction forces, depicted by the dashed arrows. When applied to a fluid (right panel), the fluid cannot resist the applied shear but rather flows. The applied shear forces lead to shear stresses, force per area, and the measure of flow can be quantified by the resulting shear strain rate. In essence, the harder one pushes on a fluid (higher shear stress), the faster the fluid flows (higher shear strain rate). The relationship between shear stress, t, and shear strain rate, g_ , is the fluid’s viscosity, m. Viscosity is sometimes written as in biomedical applications. As shown in Figure 4.35, many fluids, including water, are characterized by a constant, linear viscosity and are called Newtonian. Others possess nonlinear shear stress-strain rate relations, and are non-Newtonian fluids. For example, fluids that behave more viscously as shear strain rate increases are called dilatant, or shear thickening. One example of dilatant behavior is Dow Corning 3179 dilatant compound, a silicone polymer commonly known as “Silly Putty.” When pulled slowly, this fluid stretches (plastic deformation); when pulled quickly, it behaves as a solid and fractures. Fluids that appear less viscous with higher shear strain rates are called pseudoplastic, or shear thinning. For example, no-drip latex paint flows when applied with a brush or roller that provides shear stress, but it does not flow after application. Bingham Plastic Casson Equation

Shear Stress τ

Dilatant Newtonian Pseudoplastic

τ0

•

Shear Strain Rate γ

Newtonian fluids exhibit a constant viscositym ¼ t=_g, arising from the linear relation between shear stress and shear strain rate. Non-Newtonian fluids are nonlinear. Blood is often characterized with a Casson equation but under many conditions may be described as Newtonian.

FIGURE 4.35

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4. BIOMECHANICS

Biological fluids are typically non-Newtonian. Blood plasma is Newtonian and is very similar in physical properties to water. Whole blood behaves as a Bingham plastic, whereby a nonzero shear stress, denoted yield stress t0 , is required before this fluid begins to flow. Blood is often characterized by a power law function, of the form t ¼ k_gn

ð4:57Þ

where k and n are constants derived from a straight-line fit of ln t plotted as a function of ln g_ , since ln t ¼ ln k þ n ln g_ Another common description of blood’s viscosity is the Casson equation: 1

t2 ¼ t20 þ k_g2

ð4:58Þ

From a Casson plot, the yield stress t0 can be measured. Rheology, the study of deformation and flow of fluids, focuses on these often complex viscous properties of fluids. Textbooks with rheological data for biofluids include Basic Transport Phenomena in Biomedical Engineering by Fournier [9] and Biofluid Mechanics by Chandran and colleagues [4].

EXAMPLE PROBLEM 4.13 The following rheological data were measured on a blood sample: Shear Strain Rate [s1] 1.5 2.0 3.2 6.5 11.5 16.0 25.0 50.0 100

Shear Stress [dyne/cm2] 12.5 16.0 25.2 40.0 62.0 80.5 120 240 475

Fit the data to a power law function using a MATLAB m-file.

Solution % Power Law Fit of Blood Data % % Store shear strain rate and stress data in arrays alpha ¼ [1.5,2,3.2,6.5,11.5,16,25,50,100]; T ¼ [12.5,16,25.2,40,62,80.5,120,240,475]; % Take natural logs of both x ¼ log(alpha); y ¼ log(T);

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4.7 CARDIOVASCULAR DYNAMICS

% Use MATLAB’s polyfit function to do linear curve fit coeff ¼ polyfit(x,y,1) % Write curve fit coefficients as a new x-y function for plotting x1¼[0;0.01;5] y1¼polyval(coeff,x1) % Plot the original data as ’o’ points plot(x,y,’o’) hold on % Overlay a plot of the curve-fit line plot(x1,y1) grid on title(’Power Law Function’) xlabel(’ln Strain Rate [ln(1/s)]’) ylabel(’ln Shear Stress [ln dyne/cm2]’) %

The resulting plot appears in Figure 4.36. Power Law Function

6.5

ln Shear Stress [ln dyne/cm2]

6 5.5 5 4.5 4 3.5 3 2.5 2 0

0.5

1.5 2 2.5 3 3.5 ln Strain Rate [ln(1/s)]

4.5

FIGURE 4.36 The power law curve-fit using MATLAB of the rheological blood data in Example Problem 4.13.

When subjected to very low shear rates, blood’s apparent viscosity is higher than expected. This is due to the aggregation of red blood cells, called rouleaux. Such low shear rates are lower than those typically occurring in major blood vessels or in medical devices. In small tubes of less than 1 mm diameter, blood’s apparent viscosity at high shear rates is smaller than in larger tubes. This Fahraeus-Lindquist effect arises from plasma–red blood cell dynamics. Beyond these two special cases, blood behaves as a Newtonian fluid and is widely accepted as such. We shall see that the assumption of Newtonian fluid greatly simplifies mechanical description of the circulation.

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4.7.2 Arterial Vessels Mechanical description of blood vessels has a long and somewhat complicated history. Much of the advanced mathematics and applied mechanics associated with this work is beyond the scope of this textbook. This section will therefore give an overview of some of the main developments and will present a simplified, reduced arterial system model for use in the following subsection. The reader is referred to the following textbooks for more in-depth coverage: Circulatory System Dynamics [21] by Noordergraaf, Hemodynamics [15] by Milnor, and Biofluid Mechanics [4] by Chandran and colleagues, and for basic fluid mechanics, Fluid Mechanics [30] by White. Study of the mechanical properties of the heart as a pump requires the computation of pressures and flows arising from forces and motion of the underlying heart muscle. Consequently, general equations of motion in the cardiovascular system typically arise from the conservation of linear momentum. The Reynold’s transport theorem, a conservation equation from fluid mechanics, applied to linear momentum yields the following general equation of motion for any fluid: dV ð4:59Þ rg —p þ — tij ¼ r dt where r is fluid density (mass/volume), p is pressure, tij are viscous forces, and V is velocity. — is the differential operator @ @ @ —¼i þj þk @x @y @z The general velocity V is a vector function of position and time and is written Vðx, y, z, tÞ ¼ uðx, y, z, tÞi þ vðx, y, z, tÞj þ wðx, y, z, tÞk where u, v, and w are the local velocities in the x, y, and z directions, respectively. Equation (4.59) comprises four terms: gravitational, pressure, and viscous forces, plus a time-varying term. Note that this is a vector equation and so can be expanded in x, y, and z components as the set of three equations: @p @txx @tyx @tzx @u @u @u @u rgx þ þ ¼r þ þu þv þw ð4:60Þ @x @y @z @x @t @x @y @z @p @txy @tyy @tzy @v @v @v @v þ þ ¼r rgy þ þu þv þw ð4:61Þ @x @y @z @y @t @x @y @z @p @txz @tyz @tzz @w @w @w @w þ þ ¼r þ þu þv þw ð4:62Þ rgz @x @y @z @z @t @x @y @z This set of nonlinear, partial differential equations is general but not solvable; solution requires making simplifying assumptions. For example, if the fluid’s viscous forces are neglected, Eq. (4.59) reduces to Euler’s equation for inviscid flow. The latter, when integrated along a streamline, yields the famous Bernoulli equation relating pressure and flow. In application, Bernoulli’s inviscid, and consequently frictionless, origin is sometimes forgotten. If flow is steady, the right-hand term of Eq. (4.59) goes to zero. For incompressible fluids, including liquids, density r is constant, which greatly simplifies integration of the

4.7 CARDIOVASCULAR DYNAMICS

197

gravitational and time-varying terms that contain r. Similarly, for Newtonian fluids, viscosity m is constant. In summary, although we can write perfectly general equations of motion, the difficulty of solving these equations requires making reasonable simplifying assumptions. Two reasonable assumptions for blood flow in major vessels are those of Newtonian and incompressible behavior. These assumptions reduce Eq. (4.59) to the Navier-Stokes equations: 2 @p @ u @2 u @2u du rgx þ þ þm ð4:63Þ ¼r @x @x2 @y2 @z2 dt 2 @p @ v @2v @2v dv þm ð4:64Þ ¼r þ þ rgy 2 2 2 @y @x @y @z dt 2 @p @ w @2w @2w dw rgz þm ð4:65Þ þ þ 2 ¼r @z @x2 @y2 @z dt Blood vessels are more easily described using a cylindrical coordinate system rather than a rectangular one. Hence, the coordinates x, y, and z may be transformed to radius r, angle y, and longitudinal distance x. If we assume irrotational flow, y ¼ 0 and two Navier-Stokes equations suffice: 2 dP dw dw dw d w 1 dw d2 w þ ð4:66Þ ¼r þu þw þ m dx dt dr dx dr2 r dr dx2 2 dP du du du d u 1 du d2 u u þ ¼r þu þw m þ ð4:67Þ dr dt dr dx dr2 r dr dx2 r2 where w is longitudinal velocity dx/dt, and u is radial velocity dr/dt. Most arterial models also use the continuity equation, arising from the conservation of mass: du u dw þ þ ¼0 dr r dx

ð4:68Þ

In essence, the net rate of mass storage in a system is equal to the net rate of mass influx minus the net rate of mass efflux. Noordergraaf and his colleagues [20] rewrote the Navier-Stokes Eq. (4.66) as

dP dQ ¼ RQ þ L dx dt

ð4:69Þ

where P is pressure, Q is the volume rate of flow, R is an equivalent hydraulic resistance, and L is fluid inertance. The Navier-Stokes equations describe fluid mechanics within the blood vessels. Since arterial walls are elastic, equations of motion for the arterial wall are also required. The latter have evolved from linear elastic and linear viscoelastic, to complex viscoelastic (see [21]). The most general mechanical description of linear anisotropic arterial wall material requires 21 parameters (see [10]), most of which have never been measured. Noordergraaf and colleagues divided the arterial system into short segments and combined the fluid mechanical equation (Eq. (4.69)) with the continuity equation for each vessel segment. The arterial wall elasticity leads to a time-varying amount of blood stored in the

198

4. BIOMECHANICS

vessel as it bulges with each heartbeat. For a segment of artery, the continuity equation becomes dQ dP ¼ GP þ C ð4:70Þ dx dt where G is leakage through the blood vessel wall. This pair of hydraulic equations— Eqs. (4.69) and (4.70)—was used to describe each of 125 segments of the arterial system and was the first model sufficiently detailed to explain arterial pressure and flow wave reflection. Arterial branching leads to reflected pressure and flow waves that interact in this pulsatile system. Physical R-L-C circuits were constructed and built into large transmission line networks with measured voltages and currents corresponding to hydraulic pressures and flows, respectively. If distributed arterial properties such as pulse wave reflection are not of interest, the arterial system load seen by the heart can be much reduced, as an electrical network may be reduced to an equivalent circuit. The most widely used arterial load is the three-element model shown in Figure 4.37. The model appears as an electrical circuit due to its origin prior to the advent of the digital computer. Z0 is the characteristic impedance of the aorta, in essence the aorta’s flow resistance. Cs is transverse arterial compliance, the inverse of elastance, and describes stretch of the arterial system in the radial direction. Rs is the peripheral resistance, describing the systemic arteries’ flow resistance downstream of the aorta. This simple network may be used to represent the systemic arterial load seen by the left ventricle. The following ordinary differential equation relates pressure at the left-hand side, p(t), to flow, Q(t): dp 1 Z0 dQ ð4:71Þ þ Z0 Cs Cs þ pðtÞ ¼ QðtÞ 1 þ Rs dt Rs dt

Q Cs

FIGURE 4.37

Equivalent systemic arterial load. Circuit elements are described in the text.

EXAMPLE PROBLEM 4.14 Using basic circuit theory, derive the differential equation (Eq. (4.71)) from Figure 4.37.

Solution Define node 1 as shown in Figure 4.38. By Kirchoff’s current law, the flow Q going into node 1 is equal to the sum of the flows Q1 and Q2 coming out of the node: Q ¼ Q1 þ Q2

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4.7 CARDIOVASCULAR DYNAMICS

We can write Q1 and Q2 as Q 1 ¼ Cs Q

dp1 dt Q2

p Q1 Rs

FIGURE 4.38 Nodal analysis of three-element arterial load. Q2 ¼

p1 Rs

so Q ¼ Cs

dp1 p1 þ dt Rs

From Ohm’s law, p p1 ¼ QZ0 Solving the last expression for p1 and substituting back into the flow expression: Q ¼ Cs

d 1 ½p QZ0 þ ½p QZ0 dt Rs

¼ Cs

dp dQ 1 Z0 Z0 Cs þ p Q dt dt Rs Rs

Grouping terms for Q on the left and p on the right gives Eq. (4.71).

4.7.3 Heart Mechanics Mechanical performance of the heart, more specifically the left ventricle, is typically characterized by estimates of ventricular elastance. The heart is an elastic bag that stiffens and relaxes with each heartbeat. Elastance is a measure of stiffness, classically defined as the differential relation between pressure and volume: Ev ¼

dpv dVv

ð4:72Þ

Here, pv and Vv denote ventricular pressure and volume, respectively. For any instant in time, ventricular elastance Ev is the differential change in pressure with respect to volume. Mathematically, this relation is clear. Measurement of Ev is much less clear.

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In the 1970s Kennish and colleagues tried to estimate the differential relation of Eq. (4.72) using the ratio of finite changes in ventricular pressure and volume: Ev ¼

Dpv DVv

ð4:73Þ

This approach leads to physically impossible results. For example, before the aortic valve opens, the left ventricle is generating increasing pressure while there is not yet any change in volume. The ratio in Eq. (4.73) gives an infinite elastance when the denominator is zero. Suga and Sagawa [28] used the ratio of pressure to volume itself, rather than differential or discrete changes, to estimate elastance: Ev ðtÞ ¼

pv ðtÞ Vv ðtÞ Vd

ð4:74Þ

In this equation, Vd is a dead volume that remains constant. All the other terms are allowed to be varying with time. Ventricular elastance measured in this way leads to elastance curves as depicted in Figure 4.39. These curves show wide variation, as suggested by the large error bars. The distinctive asymmetric shape leads to a major contradiction. A simple experiment involves clamping the aorta, thereby preventing the left ventricle from ejecting blood, denoted an isovolumic beat. Equation (4.74) shows that under isovolumic conditions (Vv is constant) ventricular pressure pv must have the same shape as elastance Ev(t). However, experiments show that isovolumic pressure curves are symmetric, unlike Figure 4.39. A further complication is the requirement of ejecting beats for measuring Ev(t), which requires not only the heart (a ventricle) but also a circulation (blood vessels). Hence, time-varying elastance curves such as Figure 4.39 are measures of both a particular heart, the source, combined with a particular circulation, its load. Experiments show that elastance curves measured in this way are subject to vascular changes, as well as the desired ventricular properties. As such, this approach cannot uniquely separate out ventricular from vascular properties. Consequently, a new measure of the heart’s mechanical properties is required. The problems just described—inconsistent isovolumic and ejecting behavior and combined heart-blood vessel properties—led to the development of a new mechanical description of the

Normalized E

1.0

0.5

1.0

1.5

Normalized Time

FIGURE 4.39 Time-varying ventricular elastance curves measured using the definition in Eq. (4.74). Measured elastance curves exhibit distinctive asymmetry. Adapted from [28].

4.7 CARDIOVASCULAR DYNAMICS

201

FIGURE 4.40 Isolated canine left ventricle used to develop a new biomechanical model of the heart. Photo courtesy of Dr. Jan Mulier, Leuven, Belgium.

left ventricle [17, 24, 23]. This model should be simple and versatile and should have direct physiological significance, in contrast with simulations, which merely mimic physiological behavior. The model was developed using isolated canine heart experiments, as depicted in Figure 4.40. The left ventricle was filled with an initial volume of blood, subjected to different loading conditions, stimulated, and allowed to beat. Ventricular pressure, and in some experiments ventricular outflow, was then measured and recorded. Experiments began with measurement of isovolumic ventricular pressure. For each experiment the isolated left ventricle was filled with an initial end-diastolic volume and the aorta was clamped to prevent outflow of blood. The ventricle was stimulated, and generated ventricular pressure was measured and recorded. The ventricle was then filled to a new end-diastolic volume and the experiment was repeated. As in the famous experiments of Otto Frank (c. 1895), isovolumic pressure is directly related to filling. Figure 4.41 shows a set of isovolumic pressure curves measured on a normal canine left ventricle. These isovolumic pressure curves were then described by the following equation. Ventricular pressure pv is a function of time t and ventricular volume Vv according to " ttb a # t a ð1 eðtc Þ Þeð tr Þ 2 ð4:75Þ pv ¼ aðVv bÞ þ ðcVv dÞ tp a tp tb a ð1 eðtc Þ Þeð tr Þ or written more compactly, pv ðt, Vv Þ ¼ aðVv bÞ2 þ ðcVv dÞf ðtÞ

ð4:76Þ

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4. BIOMECHANICS

mmHg

Left Ventricular Pressure

150

100

sec

0 0.0

0.2

0.4

0.6

Time

FIGURE 4.41 Isovolumic ventricular pressure curves. For each curve, the left ventricle is filled with a fixed initial volume, the heart is stimulated, and generated ventricular pressure is measured with respect to time. Other curves arise from different fixed initial volumes.

where f(t) is the activation function in square brackets in Eq. (4.75). The constants a, b, c, d, tp , tc , tr , and a were derived from the isolated canine ventricle experiments. Physiologically, Eq. (4.76) says that the ventricle is a time- and volume-dependent pressure generator. The term to the left of the plus sign, including constants a and b, describes the ventricle’s passive elastic properties. The term to the right, including c and d, describes its active elastic properties, arising from the active generation of force in the underlying heart muscle. Representative model quantities measured from canine experiments are given in Table 4.3. This model was adapted to describe the human left ventricle using quantities in the right-hand column [24, 25]. TABLE 4.3 Ventricle Model Quantities Measured from Animal Experiments and Adapted for the Human Analytical Model Quantity

Dog (Measured)

Human (Adapted) 2

0.003 [mmHg/ml ]

1.0 [ml]

3.0 [mmHg/ml]

20.0 [mmHg]

0.164 [s]

0.264

0.271 [s]

0.371

0.199 [s]

0.299

0.233 [s]

0.258

2.88

0.0007 20.0 2.5 80.0

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4.7 CARDIOVASCULAR DYNAMICS

EXAMPLE PROBLEM 4.15 Solve Eq. (4.75) and plot ventricular pressure pv(t) for one human heartbeat. Use initial ventricular volume of 150 ml and the parameter values in Table 4.3.

Solution The following MATLAB m-file will perform the required computation and plot the results, shown in Figure 4.42. Isovolumic Ventricular Pressure

350

Ventricular Pressure Pv [mmHg]

300 250 200 150 100 50 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

FIGURE 4.42

Isovolumic ventricular pressure computed for a human heartbeat.

% ventricle.m % % MATLAB m-file to compute isovolumic pressure using ventricle model % % Initial conditions: % delt ¼ 0.001; % The iteration time step delta t a ¼ 7e-4; b ¼ 20.; c ¼ 2.5; d ¼ 80.; tc ¼ 0.264; tp ¼ 0.371; tr ¼ 0.299; tb ¼ 0.258; alpha ¼ 2.88;

Continued

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Vv0 ¼ 150; % Initial (end-diastolic) ventricular volume % % Compute intermediate term denom % to simplify computations: % denom ¼ ((1.-exp(-(tp/tc)âlpha))*exp(-((tp-tb)/tr)âlpha)); % % Compute for initial time t¼0 (MATLAB does not allow 0 index) % t(1) ¼ 0.; Vv(1) ¼ Vv0; edp ¼ a*((Vv0 -b))^2; pdp ¼ c*Vv0 - d; pp ¼ pdp/denom; t1 ¼ 0.; % Time step for first exponential t2 ¼ 0.; % Time step for second exponential e1 ¼ exp(-(t1/tc)âlpha); e2 ¼ exp(-(t2/tr)âlpha); pv0 ¼ edp þ pp*((1.-e1)*e2); % % Main computation loop: % for j¼2:1000 t(j) ¼ t(j-1) þ delt; Vv(j) ¼ Vv(j-1); % edp ¼ a*((Vv(j) -b))^2; pdp ¼ c*Vv(j) - d; pp ¼ pdp/denom; t1 ¼ t(j); % Second exponential begins at t > tb t2 ¼ t(j) - tb; if (t2 < 0.) ; t2 ¼ 0.; end e1 ¼ exp(-(t1/tc)âlpha); e2 ¼ exp(-(t2/tr)âlpha); pv(j) ¼ edp þ pp*((1.-e1)*e2); end % plot(t,pv) grid on title(’Isovolumic Ventricular Pressure’) xlabel(’Time [s]’) ylabel(’Ventricular Pressure Pv [mmHg]’)

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4.7.4 Cardiovascular Mechanics This concise generalized pressure model of the left ventricle (Eq. (4.75)) was coupled to the reduced arterial load model of Figure 4.37 and allowed to eject blood. Model parameter values for a normal arterial load are given in Table 4.4. Figure 4.43 shows results for a normal canine left ventricle ejecting into a normal arterial system. The solid curves (left ordinate) describe ventricular pressure pv and root aortic pressure as functions of time. Clinically, arterial pressure is reported as two numbers—for example, 110/60. This corresponds to the maximum and minimum root arterial pulse pressures—in this case, about 120/65 mmHg. The dashed curve (right ordinate) shows ventricular outflow. The ventricle was filled with an end-diastolic volume of 45 ml, and it ejected 30 ml (stroke volume), giving an ejection fraction of 66 percent, which is about normal for this size animal. The same ventricle may be coupled to a pathological arterial system—for example, one with doubled peripheral resistance Rs. This change is equivalent to narrowed blood vessels. TABLE 4.4 Representative Systemic Arterial Model Element Values Symbol

Control Value

Characteristic aorta impedance

0.1 mmHg-s/ml

Systemic arterial compliance

1.5 ml/mmHg

Peripheral arterial resistance

1.0 mmHg-s/ml

FIGURE 4.43

200

300

180 250

160 Ved = 45ml SV = 30ml EF = 66%

140 120

200 150

100 80

100

60 40

Ventricular Outflow [ml/s]

Ventricular and Root Aortic Pressures [mmHg]

Element

20 0

0 0

0.25

0.5 Time [s]

0.75

Ventricular and root aortic pressures (solid curves, left ordinate) and ventricular outflow (dashed curve, right ordinate) computed using the model of Eq. (4.75) for a normal canine left ventricle pumping into a normal arterial circulation. The topmost solid curve corresponds to a clamped aorta (isovolumic). The ventricle has initial volume of 45 ml and pumps out 30 ml, for an ejection fraction of 66 percent, which is about normal.

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300

180 250

160 Ved = 45ml SV = 23ml EF = 50%

140 120

200

150

100 80

100 60 40

Ventricular Outflow QLV [ml/s]

Ventricular and Root Aortic Pressures [mmHg]

200

20 0

0 0

0.25

0.5 Time [s]

0.75

FIGURE 4.44

The same normal canine ventricle of Figure 4.43 now pumping into an arterial system with doubled peripheral (flow) resistance. As expected, increased resistance, corresponding to narrowed vessels, leads to increased arterial pulse pressure. Stroke volume is reduced from 66 to to 50 percent.

As expected, increased peripheral resistance raises arterial blood pressure to 140/95 mmHg and impedes the ventricle’s ability to eject blood (Figure 4.44). The ejection fraction decreases to 50 percent in this experiment. Other experiments, such as altered arterial stiffness, may be performed. The model’s flexibility allows description of heart pathology as well as changes in blood vessels. This one ventricular equation with one set of measured parameters is able to describe the wide range of hemodynamics observed experimentally [24, 23]. The previous expressions for ventricular elastance defined in Eqs. (4.73) and (4.74) have the same units as elastance defined classically as Eq. (4.72), but are mathematically not the same. Since ventricular pressure is defined as an analytical function (Eq. (4.75)), ventricular elastance, Ev, defined in the classical sense, may now be calculated as @pv =@Vv : " ttb a # t a ð1 eðtc Þ Þeð tr Þ Ev ðt, Vv Þ ¼ 2aðVv bÞ þ c ð4:77Þ tp a tp tb a ð1 eðtc Þ Þeð tr Þ or Ev ðt, Vv Þ ¼ 2aðVv bÞ þ cf ðtÞ

ð4:78Þ

Figure 4.45 shows ventricular elastance curves computed using this new analytical definition of elastance (Eq. (4.77)). Elastance was computed for a wide range of ventricular and arterial states, including normal and pathological ventricles, normal and pathological arterial systems, and isovolumic and ejecting beats. These elastance curves are relatively

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Ventricular Elastance Ev [mmHg/ml]

3.5

Control Ventricle

3 2.5 2

Decreased Contractile State

1.5 1 0.5

0.25

0.5

0.75

Time [s]

FIGURE 4.45 Ventricular elastance curves computed using the new analytical function of Eq. (4.77). Elastance curves computed in this way are representative of the ventricle’s contractile state—that is, its ability to pump blood.

invariant and cluster in two groups: either normal or weakened ventricle contractile state. Consequently, this new measure of elastance may now effectively assess the health of the heart alone, separate from blood vessel pathology. Chapter 3 gives a brief overview of the circulatory system, a mass and heat transfer system that circulates blood throughout the body. Figure 3.18 shows the four chambers of the heart, the major blood vessels, and valves between the two. From a mechanical point of view, the contracting heart chambers generate pressures that propel blood into the downstream blood vessels. This process is depicted in detail for the most important chamber, the left ventricle, in Figure 4.46. Plotted are representative waveforms for an isolated canine left ventricle, with ventricular pressure and root aortic pressure (top), ventricular volume (middle) and ventricular outflow (bottom), all as functions of time. The numbers 1–4 at the top of the figure correspond to four major phases of the contraction cycle, marked by dashed lines. At time 1, filling is complete, the mitral valve closes, and the ventricle begins to contract isovolumically (no change in volume). At time 2, ventricular pressure exceeds root aortic pressure, the aortic valve opens, and blood ejection begins. Heart valves are passive, and the outflow of blood results simply from the pressure difference across the valve. When ventricular pressure falls below aortic pressure, the aortic valve closes (time 3) and outflow ends. The initial volume (time 1) is denoted end-diastolic volume, EDV, and the volume at the end of ventricular ejection (time 3) is end-systolic volume, ESV. At time 4, ventricular pressure falls below left atrial pressure (not shown), the mitral valve opens, and filling begins in preparation for the next heartbeat. The difference between

208

Ventricular Outflow [ml/s]

Ventricular Volume [ml]

Ventricular, Aortric Pressures [mmHg]

4. BIOMECHANICS

100

0.2

0.4 0.6 Time t [s]

0.8

0.2

0.4 0.6 Time t [s]

0.8

0.2

0.4 0.6 Time t [s]

0.8

90 80 70 60 50 40 30

500 400 300 200 100 0 –100

FIGURE 4.46

Representative left ventricular and root aortic pressures (top), ventricular volume (middle), and ventricular outflow (bottom) for an isolated canine heart. The numbers 1–4 at top correspond to distinct phases of the cardiac cycle, described in the text.

end-diastolic and end-systolic volumes, EDV-ESV, is denoted stroke volume, SV, and is the amount of blood pumped in one heartbeat. The ratio SV/EDV is the ejection fraction, EF, and is approximately 50 percent in this example. Clinical blood pressure corresponds to the maximum and minimum arterial pressures—in this example, around 125/75 mmHg. Clinicians often study heart performance via pressure-volume work loops, plotting ventricular pressure as a function of ventricular volume, as in Figure 4.47. The four phases of the heart cycle just described are plotted for the same heart as in Figure 4.46, with isovolumic contraction between points 1 and 2, ejection from 2 to 3, isovolumic relaxation between 3 and 4, and

209

4.7 CARDIOVASCULAR DYNAMICS

Left Ventricular Pressure [mmHg]

140 120 100

3 2

80 SV 60 40 20 0 30

ESV

50 60 70 Left Ventricular Volume [ml]

EDV

FIGURE 4.47 Pressure-volume work loop corresponding to the left ventricular data in Figure 4.46.

refilling from 4 to 1. Also shown are volumes EDV, ESV, and SV. The area bounded by the work loop corresponds to mechanical work performed by the left ventricle. Figure 4.48 shows work loops calculated from the generalized pressure model of the left ventricle (Eq. (4.75)) pumping into the three-element arterial load model (Figure 4.37) using human parameter values for both the ventricle and arterial load. The left ventricle was filled from a constant pressure reservoir. The control work loop (center) may be compared to those corresponding to a 10 percent increase and 10 percent decrease in filling pressure.

Ventricular Pressure pLV [mmHg]

140 +10% filling pressure

120 –10%

100 80 60 40 20 0 60

100 120 Ventricular Volume VLV [ml]

140

160

FIGURE 4.48 Pressure-volume work loops computed for the human left ventricle (Eq. (4.75)) and arterial load (Figure 4.37) for control (middle curve) and varied preload (filling) conditions.

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4. BIOMECHANICS

These changes directly affect the amount of filling of the ventricle, denoted preload. Stroke volume increases with increased preload, and vice versa. As the ventricle is filled more, the end-diastolic volume shifts to the right and the work loop area increases due to the FrankStarling mechanism. In essence, increased filling stretches the constituent heart muscle, which allows the muscle to generate more force via the force-length relation, the direct relation between muscle length and maximum force of contraction (up to a point). There is also a small shift of end-systolic volume to the right with increased preload. As the heart ejects more blood (SV), the load against which the heart must work is increased. Figure 4.49 shows work loops for the same ventricle and arterial load for direct variations in afterload, the load against which the ventricle pumps. Afterload changes were achieved by varying the value of arterial peripheral resistance RS. Stroke volume decreases as afterload is increased due to the underlying force-velocity relation of muscle. Higher peripheral resistance forces muscle to operate at a lower velocity of shortening, so less time is available for the ventricle to eject blood. Figure 4.49 shows a small shift of end-diastolic volume to the right with increased afterload. Since stroke volume is reduced, there is remaining blood available for the next heartbeat. This effect is more pronounced in the natural system, which can be examined by extending the model as follows. The left ventricle model of Eq. (4.75) was used to describe each of the four chambers of the human heart, depicted in Figure 4.50 [25, 23]. This complete model of the circulatory system displays a remarkable range of cardiovascular physiology with a small set of equations and parameters. Changes in blood vessel properties may be studied alone or in combination with altered heart properties. Other system parameters such as atrial performance, as well as other experiments, may be examined. The modular form of this model allows its expansion for more detailed studies of particular sites in the circulatory system. Figure 4.51 shows work loops computed for the left ventricle in the complete circulatory system model for three different preloads, achieved by increasing and decreasing total

Ventricular Pressure pLV [mmHg]

160 140 120

control

100 80

–50%

60 40 20 0 50

FIGURE 4.49

+50% afterload

75 100 125 Ventricular Volume VLV [ml]

150

Pressure-volume work loops computed for the human left ventricle (Eq. (4.75)) and arterial load (Figure 4.37) for control and varied afterload, achieved by varying peripheral resistance RS.

211

4.7 CARDIOVASCULAR DYNAMICS

RLA

ZSO

RSA

RSV

CSA

CSV

RRA

Systemic Circulation

ZPO

RPA

CPA

CPV

RPV

Pulmonary Circulation

LA = Left Atrium RLA = Left Atrial Resistance MV = Mitral Valve LV = Left Ventricle AV = Aortic Valve ZSO = Systemic Characteristic Impedance CSA = Systemic Arterial Compliance RSA = Systemic Peripheral Resistance CSV = Systemic Venous Compliance RSV = Systemic Venous Resistance

RA = Right Atrium RRA = Right Atrial Resistance TV = Tricuspid Valve RV = Right Ventricle PV = Pulmonic Valve ZPO = Pulmonic Characteristic Impedance CPA = Pulmonic Arterial Compliance RPA = Pulmonic Peripheral Resistance CPV = Pulmonic Venous Compliance RPV = Pulmonic Venous Resistance

FIGURE 4.50

Application of the canine left ventricle model to hemodynamic description of the complete human cardiovascular system. Adapted from [25].

150 Ventricular Pressure pLV [mmHg]

+10% volume

100

–10%

0 50

100 110 70 80 90 Ventricular Volume VLV [ml]

120

130

FIGURE 4.51 Left ventricular pressure-volume work loops computed for the complete human circulation model (Figure 4.50) for control (middle curve) and varied preload, achieved by varying total blood volume (5 liters) 10 percent.

blood volume, 5 liters, by 10 percent. As for the isolated left ventricle model, stroke volume increases with increased preload via the Frank-Starling mechanism, and this increase is moderated by increased end-systolic volume due to increased afterload. Preload may be increased in the natural system by increased central venous pressure, resulting from decreased venous compliance caused by sympathetic venoconstriction, or by augmented

212

4. BIOMECHANICS

160 Ventricular Pressure pLV [mmHg]

+50% afterload

140 120

control

100 –50%

80 60 40 20 0 40

80 100 Ventricular Volume VLV [ml]

120

FIGURE 4.52

Left ventricular pressure-volume work loops computed for the complete human circulation model (Figure 4.50) for control and varied afterload, achieved by changing both systemic (RSA) and pulmonic (RPA) peripheral resistances 50 percent.

venous return due, for example, to gravity via head-down tilt. Preload may also change with total blood volume, which may, for example, decrease as the direct consequence of hemorrhage, or change with renal regulation. Figure 4.52 shows work loops computed for the left ventricle in the complete circulation model for three different values of afterload, achieved by varying both systemic peripheral resistance, RSA, and pulmonic peripheral resistance, RPA. Stroke volume is inversely related to peripheral resistance, but the Frank-Starling mechanism partially compensates. For increased afterload, smaller SV results in increased filling for the subsequent beat, and this increased EDV moderates the reduction in SV. This compensatory mechanism is more pronounced in the full circulation model than for the isolated left ventricle (Figure 4.49). Afterload commonly increases in the natural system via increased aortic pressure with increased systemic vascular resistance. The latter occurs, for example, when arterial vessel diameter is reduced associated with chronic hypertension. Afterload also increases with aortic valve stenosis, the narrowing of the valve orifice. The interdependence of preload and afterload is manifested in treatment of heart failure with vasodilator drugs. These drugs decrease afterload, allowing the ventricle to eject blood more rapidly via muscle’s force-velocity relation, which increases stroke volume. As SV increases, less blood remains to fill the ventricle for the next beat, but this decrease in EDV is less than the reduction of ESV, resulting in a net increase in stroke volume. The heart’s contractile state may be changed by varying the parameter c in the model (Eq. (4.75)). Figure 4.53 shows left ventricular work loops for such variations in inotropy, executed by changing the contractile parameter c for each of the four heart chambers. Increased inotropy causes an increase in stroke volume, with a decrease in end-systolic volume due to the more strongly contracting heart. End-diastolic volume decreases a small amount

213

4.7 CARDIOVASCULAR DYNAMICS

Ventricular Pressure pLV [mmHg]

150 +25% control –25% inotropy

100

0 40

80 100 Ventricular Volume VLV [ml]

120

FIGURE 4.53 Left ventricular pressure-volume work loops computed for the complete human circulation model (Figure 4.50) for control and varied inotropy, achieved by changing the contractile parameter c in Eq. (4.75) for each heart chamber 25 percent.

since the more forcefully ejecting heart leaves less volume available for the subsequent heartbeat. Inotropy is regulated by the autonomic nerves, augmented by sympathetic adrenergic effects of circulating catecholamines, such as epinephrine, the so-called “fight or flight” response to stress. Inotropic drugs used clinically to stimulate the heart in acute and chronic heart failure include digoxin and the beta-adrenoceptor agonists dopamine, dobutamine, epinephrine, and isoproterenol. The complete circulatory system model may be used to study interactions between left (systemic) and right (pulmonary) circulations. Figure 4.54 shows left and right ventricle work loops for the normal heart ejecting into the normal (control) circulatory system, depicted by solid curves. The right ventricle work loop is smaller, as expected, than the left. Figure 4.54 also shows the same two work loops for a weakened left ventricle (dashed curves). As expected, this left ventricle work loop is diminished in size and shifts to the right on the volume axis. Since the weaker ventricle ejects less blood, more remains to fill the heart more for the subsequent beat (EDV of 192 instead of 119 ml). This increased filling partially compensates for the weakened ventricle via Starling’s law (increased pressure for increased filling). Table 4.5 shows examples of congestive heart failure, resulting from decreases in c for the left ventricle and for the right ventricle. Decreasing left ventricular contractile state to one-third of the control value lowers the left ventricular ejection fraction from 55 to 27 percent, and root aortic pulse pressure decreases from 131/58 to 102/47 mmHg. Left ventricular stroke volume decreases less, from 66 to 53 ml, since it is compensated for by the increased left enddiastolic volume (192 ml) via Starling’s law. Decreasing left ventricular contractile state is equivalent to left congestive heart failure. Consequently, pulmonary venous volume increases from 1,540 ml to 2,057 ml (not shown), indicating pulmonary congestion for this case.

214

4. BIOMECHANICS

Ventricular Pressure pV [mmHg]

140 LV

120

100 80 RV

60 40

20 0 50

100 150 Ventricular Volume VV [ml]

200

FIGURE 4.54

Computed work loops for the left and right ventricles under control conditions (solid curves) and for the case of a weakened left ventricle (dashed curves).

Similar changes are noted when the right ventricle’s contractile state is halved. The right ventricular ejection fraction drops from 55 to 37 percent, root pulmonary artery pulse pressure decreases from 54/18 to 40/15 mmHg, and right stroke volume decreases from 66 to 56 ml, with an increased end-diastolic volume of 153 ml, from 119 ml. Conversely, c can be increased in any heart chamber to depict administration of an inotropic drug. Although not plotted, pressures, flows, and volumes are available at any circuit site, all as functions of time. In summary, the left ventricle may be described as a dynamic pressure generator. A small number of experimentally derived parameters are sufficient to describe the wide range of observed cardiovascular dynamics. This approach links experiment and theory, leading to new ideas and experiments. It also links underlying muscle dynamics to heart performance. Work is under way to devise a new measure of cardiovascular health using this model. TABLE 4.5 Cardiovascular Performance for a Normal Heart and for Weakened Left and Right Ventricles SV [ml]

EDV [ml]

EF [%]

pAO

pPU [mmHg]

Control

119

131/58

54/18

Weak LV

192

109

102/47

54/24

Weak RV

106

153

117/54

40/15

Note: SV denotes stroke volume, EDV denotes end-diastolic volume, EF is ejection fraction, and pAO and pPU are root aorta and root pulmonary artery pressures, respectively, for the left (LV) and right (RV) ventricles. Note that SV left and right are equal under all conditions.

4.8 EXERCISES

215

The field of biomechanics applies physical principles to living systems using the language of mathematics. Hemodynamics studies the human cardiovascular system, which comprises a complex pump moving complex fluid around an extensive network of complex pipes. In developing hemodynamic principles, experiments and analysis go hand-in-hand, ensuring the validity of principles with experiments and with analysis clarifying, modifying, and often preceding experiments. In this fashion, interpretations of cardiovascular health are further defined.

4.8 EXERCISES 1. Write and evaluate all the vector expressions of Eqs. (4.1) through (4.14) using MATLAB. 2. The measurement error associated with the pointer marker data presented in Example Problem 4.2 is 0:5 mm in all three coordinate directions. Given the geometry of the pointer in the example problem, what is the measurement error associated with the pointer tip, point T? If you wanted to minimize the measurement error at point T, how would you design the pointer with respect to the distances between markers A and B, and between marker B and the pointer tip T? 3. Repeat Example Problem 4.3 using a z-x-y rotation sequence. 4. Write the free-body diagrams for each of the three orientations of the humerus in Figure 3.36. For a particular load and fixed position, write and solve the equations of static equilibrium. 5. The force plate in Figure 4.11 is 70 cm wide in the x-direction and 80 cm long in the y-direction. At a particular instant of the gait cycle each transducer reads F1 ¼ 150 N, F2 ¼ 180 N, F3 ¼ 220 N, and F4 ¼ 210 N. Compute the resultant force and its location. 6. Solve Example Problem 4.8 for forearm orientations angled y from the horizontal position. Let y vary from 0 to 70 down from the horizontal in 5 increments. Using MATLAB, plot the required biceps muscle force FB for static equilibrium as a function of y. By how much does this force vary over this range? 7. Repeat Problem 6, this time plotting forces FA, FB, and FC over the same range of angles y. 8. Considering the previous problem, explain why Nautilus weight machines at the gym use asymmetric pulleys. 9. Solve Example Problem 4.7 using the moment of inertia of the thigh with respect to the knee. 10. For your own body, compute the mass moment of inertia of the body segments: Forearm, Total Arm, Thigh, Foot, and Trunk in Table 4.1 with respect to their centers of mass. 11. Repeat Example Problem 4.9 using a cobalt alloy rod with circular cross-sectional diameter of 10 mm. 12. Write the Simulink models of the three-element Kelvin viscoelastic description and perform the creep and stress relaxation tests, the results of which appear in Figures 4.26 and 4.27. 13. Use the three-element Kelvin model to describe the stress relaxation of a biomaterial of your choice. Using a stress response curve from the literature, find the model spring constants K1 and K2, and the viscous damping coefficient b. 14. Write and solve the kinematic equations defining an anatomically referenced coordinate system for the pelvis, fepa g, using MATLAB. 15. Using the kinetic data of Section 4.6.3, compute the instantaneous ankle power of the 25.2 kg patient using MATLAB. Continued

216

4. BIOMECHANICS

16. Pelvic obliquity is commonly associated with an angular displacement of the pelvis as seen from the front of the patient and is estimated as the angle formed between the vector from right to left anterior superior iliac spine (ASIS) and the horizon. Given the pelvic position data below, calculate the clinician’s estimate of pelvic obliquity. Note: The clinician would be looking down the x-axis of the global coordinate system at the pelvic data in the y-z plane. Right ASIS Left ASIS PSIS

17. 18.

19.

20.

¼ ¼ ¼

0:400i þ 0:400j þ 0:580k m 0:390i þ 0:435j þ 0:600k m 0:100i þ 0:418j þ 0:820k m

Which Euler angle algorithm, one based on a y-x-z rotation sequence or one based on a z-x-y rotation sequence, would provide a value of pelvic tilt that would best match the clinician’s estimate in this case? Given the pelvis and thigh anatomical coordinate systems defined in Example Problems 4.10 and 4.11, compute the pelvis tilt, obliquity, and rotation angles using a z-x-y rotation sequence. Repeat Problem 17 (using a z-x-y rotation sequence) solving for hip flexion/extension, hip abduction/adduction, and internal/external hip rotation. Hint: feta g is the triple-primed coordinate system, and fepa g is the unprimed system in this case. Using MATLAB, determine the effect that a 10 mm perturbation in each coordinate direction would have on ankle power amplitude (Section 4.6.3). Hint: Increase the foot anatomical coordinate system fefa g by 0.010 m in each direction. Reflective markers placed over the anterior superior iliac spine (ASIS) and the posterior superior iliac spine (PSIS) define the pelvic anatomical coordinate system. An easy mistake to make in the placement of these markers on an overweight or obese subject is to place the ASIS marker(s) at the horizontal level of the ASIS, but not in the pelvic plane, as shown in the figure below. The same type of error can be made with the placement of the PSIS. If the actual pelvic tilt for a patient is 34 and reflective markers are placed in the following pelvic locations relative to the fixed laboratory coordinate system, what is the magnitude of the error in pelvic tilt, if any?

PSIS

ASIS

REFERENCES

217

21. Fit the blood rheological data of Example Problem 4.13 to a Casson model and find the yield stress t0 for blood. 22. Solve Eq. (4.71) for pressure p(t) when the aortic valve is closed. Using the parameter values in Table 4.4, plot p as a function of time for one heartbeat (t ¼ 0 – 1 sec). 23. Compute isovolumic ventricular pressure pv(t) for the canine heart with initial volumes Vv ¼ 30, 40, 50, 60, 70 ml. Overlay these plots as in Figure 4.41. 24. Write a MATLAB m-file to compute ventricular elastance using Eq. (4.77). Compute and plot Ev(t) for the parameter values in Example Problem 4.15. 25. For the three pressure-volume work loops in Figure 4.52, measure end-diastolic volume EDV, end-systolic volume ESV, stroke volume SV, and ejection fraction EF. Estimate the total mechanical power done by the ventricle in units of watts.

References [1] R. Baker, Pelvic angles: a mathematically rigorous definition which is consistent with a conventional clinical understanding of the terms, Gait Posture 13 (2001) 1–6. [2] A.H. Burstein, T.M. Wright, Fundamentals of Orthopaedic Biomechanics, Williams & Wilkins, Baltimore, MD, 1994. [3] A. Cappozzo, Gait analysis methodology, Hum. Mov. Sci. 3 (1984) 27–50. [4] K.B. Chandran, S.E. Rittgers, A.P. Yoganathan, Biofluid Mechanics: The Human Circulation, CRC Taylor & Francis Group, Boca Raton, FL, 2007. [5] R.B.D. Davis III, Musculoskeletal biomechanics: Fundamental measurements and analysis, Chpt. 6 in: J.D. Bronzino (Ed.), Biomedical Engineering and Instrumentation, PWS Engineering, Boston, MA, 1986. ˜ unpuu, D.J. Tyburski, J.R. Gage, A gait analysis data collection and reduction technique, Hum. [6] R.B. Davis, S. O Mov. Sci. 10 (1991) 575–587. [7] R. Dugas, A History of Mechanics, Dover Publications, New York, NY, 1988, reprinted from a 1955 text. [8] R.M. Ehrig, W.R. Taylor, G.N. Duda, M.O. Heller, A survey of formal methods for determining functional joint axes, J. Biomech. 40 (2007) 2150–2157. [9] R.L. Fournier, Basic Transport Phenomena in Biomedical Engineering, Taylor & Francis, Philadelphia, PA, 1999. [10] Y.C. Fung, A First Course in Continuum Mechanics, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1977. [11] Y.C. Fung, Biomechanics: Mechanical properties of living tissues, second ed., Springer-Verlag, New York, NY, 1993. [12] D.T. Greenwood, Principles of Dynamics, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1988. [13] A.F. Huxley, Muscle structure and theories of contraction, Prog. Biophys. 7 (1957) 255–318. [14] A. Kennish, E. Yellin, R.W. Frater, Dynamic stiffness profiles in the left ventricle, J. Appl. Physiol. 39 (1975) 665. [15] W.R. Milnor, Hemodynamics, second ed., Williams and Wilkins, Baltimore, MD, 1989. [16] V.C. Mow, W.C. Hayes, Basic Orthopaedic Biomechanics, second ed., Lippencott-Rave, Philadelphia, PA, 1997. [17] J.P. Mulier, Ventricular pressure as a function of volume and flow, Ph.D. dissertation, Univ. of Leuven, Belgium, 1994. [18] D.M. Needham, Machina Carnis: The Biochemistry of Muscular Contraction in its Historical Development, Cambridge University Press, Cambridge, U.K., 1971. [19] B.M. Nigg, W. Herzog, Biomechanics of the Musculo-Skeletal System, third ed., John-Wiley, New York, NY, 2007. [20] A. Noordergraaf, Hemodynamics, Chpt. 5 in: H.P. Schwan (Ed.), Biological Engineering, McGraw-Hill, New York, NY, 1969. [21] A. Noordergraaf, Circulatory System Dynamics, Academic Press, New York, NY, 1978.

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4. BIOMECHANICS

[22] J.L. Palladino, A. Noordergraaf, Muscle contraction mechanics from ultrastructural dynamics, Chpt. 3 in: G.M. Drzewiecki, J.K.J. Li (Eds.), Analysis and Assessment of Cardiovascular Function, Springer-Verlag, New York, NY, 1998. [23] J.L. Palladino, A. Noordergraaf, Functional requirements of a mathematical model of the heart, in: Proc. IEEE Eng. Med. Biol. Conf., Minneapolis, MN, 2009, pp. 4491–4494. [24] J.L. Palladino, J.P. Mulier, A. Noordergraaf, Closed-loop circulation model based on the Frank mechanism, Surv. Math. Ind 7 (1997) 177–186. [25] J.L. Palladino, L.C. Ribeiro, A. Noordergraaf, Human circulatory system model based on Frank’s mechanism, in: J.T. Ottesen, M. Danielsen (Eds.), Mathematical Modelling in Medicine, IOS Press, Amsterdam, Netherlands, 2000, pp. 29–39. [26] R.J. Roark, Formulas for Stress and Strain, sixth ed., McGraw-Hill, New York, NY, 1989. [27] C. Singer, E.A. Underwood, A Short History of Medicine, second ed., Oxford Univ. Press, New York, NY, 1962. [28] H. Suga, K. Sagawa, Instantaneous pressure-volume relationship under various enddiastolic volume, Circ. Res. 35 (1974) 117–126. [29] E. Weber, B.R. Wagner (Eds.), Handwo¨rterbuch der Physiologie, vol. 3, Vieweg, Braunschweig, 1846. [30] F.M. White, Fluid Mechanics, sixth ed., McGraw-Hill, New York, NY, 2008. [31] S. Vogel, Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems, Oxford Univ. Press, New York, NY, 1992.

Suggested Readings P. Allard, I.A.F. Stokes, J.P. Blanchi (Eds.), Three-Dimensional Analysis of Human Movement, Human Kinetics, Champagne, IL, 1995. R. Davis, P. DeLuca, Clinical Gait Analysis: Current Methods and Future Directions, in: G. Harris, P. Smith (Eds.), Human Motion Analysis: Current Applications and Future Directions, IEEE Press, Piscataway, NJ, 1996, pp. 17–42. J.R. Gage, M.H. Schwartz, S.E. Koop, T.F. Novacheck, The Identification and Treatment of Gait Problems in Cerebral Palsy, MacKeith Press, London, U.K., 2009. W.F. Ganong, Review of Medical Physiology, twenty-second ed., McGraw-Hill, New York, NY, 2005. J.L. Meriam, L.G. Kraige, Engineering Mechanics, sixth ed., John Wiley, New York, NY, 2008. W.R. Milnor, Cardiovascular Physiology, Oxford Univ. Press, New York, NY, 1990. A.D. McCulloch, Cardiac Biomechanics, Chpt. 8 in: D.R. Peterson, J.D. Bronzino (Eds.), Biomechanics Principles and Applications, CRC Taylor & Francis Group, Boca Raton, FL, 2008. W.W. Nichols, M.F. O’Rourke, McDonald’s Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles, third ed., Edward Arnold, London, U.K., 1990. J. Rose, J.G. Gamble, Human Walking, Lippincott Williams & Wilkins, Philadelphia, PA, 2006. N. Westerhof, A. Noordergraaf, Arterial Viscoelasticity: A Generalized Model, J. Biomech. 3 (1970) 357–379. D.A. Winter, Biomechanics and Motor Control of Human Movement, fourth ed., John Wiley, New York, NY, 2009.

C H A P T E R

5 Biomaterials Liisa T. Kuhn, PhD With contributions from Katharine Merritt, PhD, and Stanley Brown, EngD O U T L I N E 5.1 5.2 5.3

Materials in Medicine: From Prosthetics to Regeneration

220

Biomaterials: Types, Properties, and Their Applications

221

5.6 5.7

Safety Testing and Regulation of Biomaterials

258

Application-Specific Strategies for the Design and Selection of Biomaterials

263

Lessons from Nature on Biomaterial Design and Selection

236

5.8

Exercises

269

5.4

Tissue–Biomaterial Interactions

240

Suggested Readings

270

5.5

Biomaterials Processing Techniques for Guiding Tissue Repair and Regeneration 250

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : • Understand the complexity of natural tissue structure that biomaterials scientists seek to replace with biomaterials. • Describe several different types of biological responses to implanted materials. • Understand the benefits and differences among the various classes of biomaterials used in medicine. • Design bio-inspired medical device features to enhance or modify cellular interactions.

Introduction to Biomedical Engineering, Third Edition

• Explain a variety of methods to fabricate scaffolds for tissue engineering. • Understand the rationale for selecting particular chemistries and structures for several different medical product applications. • Know where to find the appropriate testing protocols to demonstrate medical product safety.

219

220

5. BIOMATERIALS

5.1 MATERIALS IN MEDICINE: FROM PROSTHETICS TO REGENERATION Throughout the ages, materials used in medicine (biomaterials) have made an enormous impact on the treatment of injury and disease of the human body. Biomaterials use increased rapidly in the late 1800s, particularly after the advent of aseptic surgical technique by Dr. Joseph Lister in the 1860s. The first metal devices to fix bone fractures were used as early as the late eighteenth to nineteenth centuries, the first total hip replacement prosthesis was implanted in 1938, and in the 1950s and 1960s, polymers were introduced for cornea replacements and as blood vessel replacements. Today, biomaterials are used throughout the body (Figure 5.1). Estimates of the numbers of biomedical devices incorporating biomaterials used in the United States in 2006 include the following: Total hip joint replacements: 579,271 Knee joint replacements: 1,349,641 Shoulder joint replacements: 49,000 Dental implants: 1,040,172 Coronary stents: 1,489,980 Coronary catheters: 1,648,235 Millions of lives have been saved due to biomaterials, and the quality of life for millions more is improved every year due to biomaterials. The field remains a rich area for research and invention because no one material is suitable for all biomaterial applications, and new applications are continually being developed as medicine advances. In addition, there are still many unanswered questions regarding the biological response to biomaterials and

Impact of Biomaterials Artificial ear Cochlear implant Nasal implants Dental materials Mandibular mesh Artificial skin Pacemaker Pectus implant Birth control implant Vascular grafts Artificial liver Spinal fixation Cartilage replacement Artificial leg Ankle implant

Hydrocephalus shunt Ocular lens, contact lens Orbital floor Artificial chin Blood substitutes Shoulder prosthesis Artificial heart, Heart valves Breast prosthesis Artificial kidney Glucose biosensor Dialysis shunts, catheters Adsorbable pins Temporary tendons Hip implant Finger joint

Testicular prosthesis

FIGURE 5.1

Biomaterials have made an enormous impact on the treatment of injury and disease and are used throughout the body.

5.2 BIOMATERIALS: TYPES, PROPERTIES, AND THEIR APPLICATIONS

221

the optimal role of biomaterials in tissue regeneration that continue to motivate biomaterials research and new product development. Over most of history, minimal understanding of the biological mechanisms of tissues meant that the biomedical engineering approach was to completely replace the damaged body part with a prosthetic—a simple, nonbiologically active piece of hardware. As our understanding of developmental biology, disease, and healthy tissue structure and function improved, the concept of attempting to repair damaged tissues emerged. More recently, with the advent of stem cell research, the medical field believes it will be possible to regenerate damaged or diseased tissues by cell-based tissue engineering approaches (see Chapter 6). The notion of a biomaterial has evolved over time in step with changing medical concepts. Williams in 1987 defined a biomaterial as “a nonviable material used in a medical device, intended to interact with biological systems.” This definition still holds true today and encompasses the earliest use of biomaterials for replacing form (e.g., wooden leg, glass eye), as well as the current use of biomaterials in regenerative medical devices such as a biodegradable scaffold used to deliver cells for tissue engineering. While the definition has remained the same, there have been dramatic changes in understanding of the level of interaction of biomaterials with the biological system (in this case, the human body). The expectations for biomaterial function have advanced from remaining relatively inert in the body, to being “bioactive” and not blocking regeneration, to providing biological cues that initiate and guide regeneration. Now there are biomaterials that can initiate a biological response after implantation such as cell adhesion, proliferation, or more excitingly, the differentiation of a stem cell that may one day lead to regeneration of a whole organ. Due to the complexity of cell and tissue reactions to biomaterials, it has proven advantageous to look to nature for guidance on biomaterials design, selection, synthesis, and fabrication. This approach is known as biomimetics. Within the discipline of biomaterials, biomimetics involves imitating aspects of natural materials or living tissues such as their chemistry, microstructure, or fabrication method. This does not always lead to the desired outcome, since many of the functionalities of natural tissues are as yet unknown. Furthermore, the desirable or optimal properties of a biomaterial vary enormously, depending on where they will be used in the body. Therefore, in addition to presenting general strategies for guiding tissue repair by varying the chemistry, structure, and properties of biomaterials, this chapter includes application-specific biomaterials solutions for several of the major organ systems in the body and for drug delivery applications. This chapter also includes a section on the regulatory approval process and the testing required that play an essential role in establishing and ensuring the safety and efficacy of medical products.

5.2 BIOMATERIALS: TYPES, PROPERTIES, AND THEIR APPLICATIONS There is a wide choice of possible biomaterials to use for any given biomedical application. The engineer must begin by selecting which general class of material to use. The four basic classes or types of materials are metals, ceramics/glasses, polymers, and composites, which are mixtures of any of the first three types of materials. Natural materials such as animal heart valves are made of proteins that have a repeating polymeric-type structure

222

5. BIOMATERIALS

and thus fall under the polymer category. Every type of biomaterial can be categorized as belonging to one of these four main classes. It is useful to know the types or classes of materials and the basic properties they possess based on their molecular structure when designing a new medical device. The current uses of the various types of biomaterials for medical devices are shown in Table 5.1, and their mechanical properties are shown in Table 5.2. TABLE 5.1 Materials and Their Medical Uses Class of Material

Current Uses

Metal Stainless steel

Joint replacements, bone fracture fixation, heart valves, electrodes

Titanium and titanium alloys

Joint replacements, dental bridges and dental implants, coronary stents

Cobalt-chrome alloys

Joint replacements, bone fracture fixation

Gold

Dental fillings and crowns, electrodes

Silver

Pacemaker wires, suture materials, dental amalgams

Platinum

Electrodes, neural stimulation devices

Ceramics Aluminum oxides

Hip implants, dental implants, cochlear replacement

Zirconia

Hip implants

Calcium phosphate

Bone graft substitutes, surface coatings on total joint replacements, cell scaffolds

Calcium sulfate

Bone graft substitutes

Carbon

Heart valve coatings, orthopedic implants

Glass

Bone graft substitutes, fillers for dental materials

Polymers Nylon

Surgical sutures, gastrointestinal segments, tracheal tubes

Silicone rubber

Finger joints, artificial skin, breast implants, intraocular lenses, catheters

Polyester

Resorbable sutures, fracture fixation, cell scaffolds, skin wound coverings, drug delivery devices

Polyethylene (PE)

Hip and knee implants, artificial tendons and ligaments, synthetic vascular grafts, dentures, and facial implants

Polymethylmethacrylate (PMMA)

Bone cement, intraocular lenses

Polyvinylchloride (PVC)

Tubing, facial prostheses

Natural Materials Collagen and gelatin

Cosmetic surgery, wound dressings, tissue engineering cell scaffold

Cellulose

Drug delivery

223

5.2 BIOMATERIALS: TYPES, PROPERTIES, AND THEIR APPLICATIONS

TABLE 5.1 Materials and Their Medical Uses—Cont’d Class of Material

Current Uses

Chitin

Wound dressings, cell scaffold, drug delivery

Ceramics or demineralized ceramics

Bone graft substitute

Alginate

Drug delivery, cell encapsulation

Hyaluronic acid

Postoperative adhesion prevention, ophthalmic and orthopedic lubricant, drug delivery, cell scaffold

TABLE 5.2 Mechanical Properties of Materials with Literature Values or Minimum Values from Standards Yield MPa

UTS MPa

Deform %

Modulus GPa

Metals High-strength carbon steel

1,600

2,000

206

F138 , annealed

170

480

200

F138, cold worked

690

860

200

F138, wire

—

1,035

200

F75 , cast

450

655

200

827

1,172

200

795

860

105

2–300

414

PEEK

3.6

PMMA Cast

45–75

1.3

2–3

Acetal (POM)

3.1

UHMWPE

200

0.5

Silicone rubber

800

0.03

Alumina

400

0.1

380

Zirconia, Mg partially stabilized

634

200

Zirconia, Yttria stabilized

900

200

F799 , forged F136 Ti64 Gold Aluminum, 2024-T4

303

Polymers

Ceramics

Carbons and composites LTI pyrolytic carbon þ 5–12% Si

600

2.0

PAN AS4 fiber

3,980

1.65

240 Continued

224 TABLE 5.2 Cont’d

5. BIOMATERIALS

Mechanical Properties of Materials with Literature Values or Minimum Values from Standards— Yield MPa

UTS MPa

Deform %

Modulus GPa

PEEK, 61% C fiber, long

2,130

1.4

125

PEEK, 61% C fiber, þ 45

300

17.2

PEEK, 30% C fiber, chopped

208

1.3

Hydroxyapatite (HA) mineral

100

0.001

114–130

Bone (cortical)

80–150

1.5

18–20

Collagen

Biologic tissues

1 2 3 4

1.2

F138, wrought stainless steel: 17–19 Cr, 13–15.5 Ni, 2–3 Mo, > det(D*eye(2)-A) ans ¼ D^2þD*K20þD*K21þK10*DþK10*K20þK10*K21þK12*DþK12*K20 >> adj¼det(D*eye(2)-A)*inv(D*eye(2)-A) adj ¼ [ DþK20þK21, K21] [ K12, DþK10þK12]

ð7:49Þ

7.6 TWO-COMPARTMENT MODELING

Substituting these values from MATLAB into Eq. (7.49) gives 2 D þ DðK10 þ K12 þ K20 þ K21 Þ þ ðK10 K20 þ K10 K21 þ K12 K20 Þ Q

K21 D þ ðK20 þ K21 Þ F ¼ K12 D þ ðK10 þ K12 Þ

393

ð7:50Þ

Returning to the time domain, we have the following independent differential equations: €q1 þ ðK10 þ K12 þ K20 þ K21 Þ_q1 þ ðK10 K20 þ K10 K21 þ K12 K20 Þq1 df1 ðtÞ þ ðK20 þ K21 Þf1 ðtÞ þ K21 f2 ðtÞ ¼ dt €q2 þ ðK10 þ K12 þ K20 þ K21 Þ_q2 þ ðK10 K20 þ K10 K21 þ K12 K20 Þq2 df2 ðtÞ ¼ K12 f1 ðtÞ þ þ ðK10 þ K12 Þf2 ðtÞ dt

ð7:51Þ

Note that the characteristic equation, det(DIA), is identical for both q1 and q2, and the form of the natural response is the same for either variable. Also note that the coefficients in the natural response are not identical for q1 and q2, and depend on the input to the compartment and the initial conditions. The roots of the characteristic equation are determined using MATLAB as >> eig(A) ans ¼ -1/2*K10-1/2*K12-1/2*K20-1/2*K21þ1/2*(K10^2þ2*K10*K122*K10*K20-2*K10*K21þK12^2-2*K12*K20þ2*K21*K12þK20^2þ 2*K20*K21þK21^2)^(1/2) -1/2*K10-1/2*K12-1/2*K20-1/2*K21-1/2*(K10^2þ2*K10*K122*K10*K20-2*K10*K21þK12^2-2*K12*K20þ2*K21*K12þK20^2þ 2*K20*K21þK21^2)^(1/2)

This expression simplifies to s1, 2 ¼

ðK10 þ K12 þ K20 þ K21 Þ 1 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK20 þ K21 K10 K12 Þ2 þ 4K21 K12

ð7:52Þ

From Eq. (7.52), we note that there can be no positive real roots and no imaginary roots if all Kij 0. If (K20 þ K21 K10 K12)2 þ 4K21K12 ¼ 0, then the roots are repeated and equal to ðK10 þ K12 þ K20 þ K21 Þ s1, 2 ¼ . For repeated roots to happen, (K20 þ K21) must equal (K10 þ 2 K12), and either K21 or K12 must be zero. If K21 and K12 are both equal to zero, then there is no movement of solute between the compartments. In the following example, we revisit Fick’s Law of diffusion using compartmental analysis and compute the concentration. The difference in analysis involves the transfer rates as

394

7. COMPARTMENTAL MODELING

functions of the volume of each compartment—that is K12 ¼

K K and K21 ¼ , where V1 V2

DA . The system in Example 7.7 is called a closed compartment because there is no output Dx to the environment in a closed system.

K¼

EXAMPLE PROBLEM 7.7 Consider the two-compartment model in Figure 7.16, with q1(0) ¼ z and q2(0) ¼ 0. Solve for the concentration in each compartment. K12 q1

q2 K21

FIGURE 7.16 Illustration for Example Problem 7.7.

Solution Conservation of mass for each compartment is q_ 1 ¼ K21 q2 K12 q1 q_ 2 ¼ K12 q1 K21 q2 Using the D-Operator method gives €q1 þ ðK12 þ K21 Þ_q1 ¼ 0 €q1 þ ðK12 þ K21 Þ_q2 ¼ 0 The roots are s1,2 ¼ 0, (K12 þ K21), which gives q1 ðtÞ ¼ B1 þ B2 eðK12 þK21 Þt q2 ðtÞ ¼ B3 þ B4 eðK12 þK21 Þt We use the initial conditions to solve for Bi as follows q1 ð0Þ ¼ z ¼ B1 þ B2 eðK12 þK21 Þt jt¼0 ¼ B1 þ B2 To find q_ 1 ð0Þ, we use the conservation of mass equation for q_ 1 at time zero q_ 1 ð0Þ ¼ K21 q2 ð0Þ K12 q1 ð0Þ ¼ K12 z and from the solution, d B1 þ B2 eðK12 þK21 Þt q_ 1 ¼ ¼ ðK12 þ K21 ÞB2 eðK12 þK21 Þt dt q_ 1 ð0Þ ¼ K12 z ¼ ðK12 þ K21 ÞB2

7.6 TWO-COMPARTMENT MODELING

395

To solve for B1 and B2, we evaluate

1 1 B1 z ¼ 0 ðK12 þ K21 Þ B2 K12 z which gives 3 zK21 6 ðK12 þ K21 Þ 7 7 6 B1 7 ¼6 7 6 zK B2 12 5 4 ðK12 þ K21 Þ 2

and q1 ðtÞ ¼

z K21 þ K12 eðK12 þK21 Þt uðtÞ ðK12 þ K21 Þ

The concentration is c1 ðtÞ ¼

z K21 þ K12 eðK12 þK21 Þt uðtÞ V1 ðK12 þ K21 Þ

Repeating the same steps for q2 as before gives q2 ðtÞ ¼

K12 z 1 eðK12 þK21 Þt uðtÞ ðK12 þ K21 Þ

and c2 ðtÞ ¼

K12 z 1 eðK12 þK21 Þt uðtÞ V2 ðK12 þ K21 Þ

or using q2 ¼ z q1 gives the same result. A more straightforward solution involves substituting q2 ¼ z q1 into q_ 1 ¼ K21 q2 K12 q1 , and solving q_ 1 ¼ K21 ðz q1 Þ K12 q1 ¼ K21 z ðK21 þ K12 Þq1 : K K If K12 ¼ and K21 ¼ , and V1 ¼ V2 ¼ V, then these results are the same as those computed V1 V2 using Fick’s Law of diffusion in Section 7.2.2: z Kt c1 ¼ e V þ 1 uðtÞ 2 and c2 ¼

Kt z 1 e V uðtÞ 2

In a two-compartment model, the half-life is defined using two terms based on the roots of the characteristic equation. The half-life associated with the smaller root is called the elimination half-life, and the distribution half-life is used for the larger root.

396

7. COMPARTMENTAL MODELING

7.6.1 Source Compartment A compartment that only outputs to other compartments, without any inputs from other compartments, is called a source compartment. A source compartment has an input f(t). This type of compartment is simply a one-compartment model that can be solved independently of the other compartments in the system. The output of the source compartment is an exponential decay, as described in Section 7.5. While in many situations, the source compartment does not send the solute to the environment, it is perfectly fine for a source compartment to do so. Using the model shown in Figure 7.15, a source compartment exists if either K12 or K21 is zero. For repeated roots in a two-compartment model, a source compartment must be one of the compartments and (K20 þ K21) must equal (K10 þ K12). In Example Problem 7.8, the digestive system is introduced as a source compartment. By including a digestive system component, the solute is not instantaneously delivered into the plasma but is slowly released from the digestive system into the plasma through a bolus input.

EXAMPLE PROBLEM 7.8 Consider the two-compartment model shown in Figure 7.17 with the ingestion of a bolus solute in the digestive system and removal of the solute via metabolism and excretion in urine. Solve for the plasma concentration.

Solution This model has a source compartment. Rather than solving the problem with a bolus input, the initial condition is changed to q2(0) with no input. The conservation of mass for each compartment is q_ 1 ¼ K21 q2 ðK1M þ K1U Þq1

ð7:53Þ

q_ 2 ¼ K21 q2

ð7:54Þ

Digestive System

K21

Metabolized

K1M

K1U

Urine

Plasma

FIGURE 7.17

A two-compartment model with realistic ingestion of solute and removal from the plasma by metabolism and excretion in urine. It normally takes about 30 minutes to pass through the digestive system.

7.6 TWO-COMPARTMENT MODELING

397

Since Eq. (7.54) involves only q2, it is easily solved as q2 ¼ q2 ð0ÞeK21 t uðtÞ: By substituting the solution for q2 into Eq. (7.53), we now have one equation, giving q_ 1 ¼ q2 ð0ÞK21 eK21 t ðK1M þ K1U Þq1 and after rearranging q_ 1 þ ðK1M þ K1U Þq1 ¼ q2 ð0ÞK21 eK21 t

ð7:55Þ

This is a first-order differential equation with a forcing function q2 ð0ÞK21 eK21 t . The natural solution is q1n ¼ B1 eðK1M þK1U Þt and the forced response is q1f ¼ B2 eK21 t . To determine B2, q1f ¼ B2 eK21 t is substituted into Eq. (7.55), which gives K21 B2 eK21 t þ ðK1M þ K1U ÞB2 eK21 t ¼ q2 ð0ÞK21 eK21 t Solving for B2 gives B2 ¼

q2 ð0ÞK21 K1M þ K1U K21

The complete response is q1 ¼ q1n þ q1f ¼ B1 eðK1M þK1U Þt þ B2 eK21 t ¼ B1 eðK1M þK1U Þt þ

q2 ð0ÞK21 eK21 t K1M þ K1U K21

ð7:56Þ

B1 is found using the initial condition q1(0) ¼ 0

q2 ð0ÞK21 q2 ð0ÞK21 eK21 t ¼ B1 þ q1 ð0Þ ¼ 0 ¼ B1 eðK1M þK1U Þt þ K1M þ K1U K21 K þ K1U K21 1M t¼0 giving B1 ¼

q2 ð0ÞK21 K1M þ K1U K21

and q2 ð0ÞK21 eK21 t eðK1M þK1U Þt uðtÞ K1M þ K1U K21

ð7:57Þ

1 q2 ð0ÞK21 eK21 t eðK1M þK1U Þt uðtÞ V1 ðK1M þ K1U K21 Þ

ð7:58Þ

q1 ¼ or in terms of concentration, c1 ¼

To determine the time when the maximum solute is in compartment 1 in Example Problem 7.8, Eq. (7.57) is differentiated with respect to t, set equal to zero, and solved as follows: 0 1 d@ q2 ð0ÞK21 q_ 1 ¼ eK21 t eðK1M þK1U Þt A dt K1M þ K1U K21 ð7:59Þ q2 ð0ÞK21 K21 t ðK1M þK1U Þt ¼ K21 e þ ðK1M þ K1U Þe K1M þ K1U K21

398

7. COMPARTMENTAL MODELING

Setting Eq. (7.59) equal to zero and t ¼ tmax gives q2 ð0ÞK3 K21 eK21 tmax þ ðK1M þ K1U ÞeðK1M þK1U Þtmax ¼ 0 K1M þ K1U K21 or K21 eK21 tmax ¼ ðK1M þ K1U ÞeðK1M þK1U Þtmax Multiplying both sides of the previous equation by eðK1M þK1U Þtmax and dividing by K21 gives eðK1M þK1U Þt eK21 tmax ¼ eðK1M þK1U K21 Þt ¼ Taking the logarithm of both sides gives ðK1M þ K1U K21 Þtmax Solving for tmax yields

tmax

K1M þ K1U K21

K1M þ K1U ¼ ln K21

K1M þ K1U ln K21 ¼ ðK1M þ K1U K21 Þ

ð7:60Þ

It should be clear from Eq. (7.59) that the smaller the term K1M þ K1U compared to K21, the more time it takes to reach the maximum concentration or quantity in the plasma.

EXAMPLE PROBLEM 7.9 Suppose 50 g of solute is ingested. Find the maximum amount of solute in the plasma if the compartmental model in Figure 7.17 is used with K1M þ K1U ¼ 0.005 min1 and K21 ¼ 0.02 min1.

Solution Using Eq. (7.60) gives

tmax

K1M þ K1U 0:005 ln ln lnð0:25Þ 0:02 K21 ¼ ¼ 92:42 min ¼ ¼ 0:015 ðK1M þ K1U K21 Þ 0:005 0:02

The maximum amount of solute in compartment 1 at tmax is therefore q2 ð0ÞK21 q1 ðtmax Þ ¼ eK21 t eðK1M þK1U Þt K1M þ K1U K21 t¼92:42 ¼

50 0:02 0:0292:42 e e0:00592:42 ¼ 31:5 g 0:005 0:02

The next example introduces an encapsulated pill input that releases a portion immediately and the remainder continuously until the pill completely dissolves. Mathematically this input is approximated as zd(t) þ (1z)(u(t)u(tt0)). To estimate the gastric transfer

7.6 TWO-COMPARTMENT MODELING

399

rate and the fraction released immediately experimentally, the pill is dissolved in a solution similar to the stomach, and the concentration is measured. From this data, parameter values can be determined.

EXAMPLE PROBLEM 7.10 Consider the two-compartment model shown in Figure 7.15 with K12 ¼ K20 ¼ 0, K21 ¼ K10 ¼ 0.2, f1(t) ¼ 0, and f2(t) ¼ 20d(t) þ 80(u(t)u(t30)). Assume that the initial conditions are zero (not including that provided by 20d(t)). Solve for the quantity in each compartment.

Solution Since K12 ¼ 0, this model has a source compartment. The solution is carried out using superposition, separating the input f2(t) into 20d(t), 80u(t), and 80u(t30), and then summing the individual responses to get the complete response. The conservation of mass for each compartment is q_ 1 ¼ 0:2q2 0:2q1

ð7:61Þ

q_ 2 ¼ f2 0:2q2

ð7:62Þ

20d(t) Input First, consider the 20d(t) input. As in Example Problem 7.8, we treat the impulse input as a change in an initial condition, yielding q2d ¼ 20e0:2t uðtÞ: Substituting this result into Eq. (7.61) gives q_ 1d ¼ 4e0:2t 0:2q1d

ð7:63Þ

The root for Eq. (7.63) is s ¼ 0.2, and has a natural solution q1n ¼ B1 e0:2t . The input in Eq. (7.63) has the same form as the natural solution (expected since (K20 þ K21) equals (K10 þ K12 ) and K12 ¼ 0), and so the forced response is q1f ¼ B2 te0:2t : Substituting q1f into Eq. (7.63) gives B2 ¼ 4. The complete response is q1d ¼ q1n þ q1f ¼ B1 e0:2t þ 4te0:2t

ð7:64Þ

B1 is found using the initial condition q1(0) ¼ 0 and

q1d ð0Þ ¼ 0 ¼ B1 e0:2t þ 4te0:2t t¼0 ¼ B1 Thus, q1d ¼ 4te0:2t uðtÞ

ð7:65Þ

80u(t) Input Next, consider the 80u(t) input. The conservation of mass equations are q_ 1u ¼ 0:2q2u 0:2q1u

ð7:66Þ

q_ 2u ¼ 80 0:2q2u

ð7:67Þ Continued

400

7. COMPARTMENTAL MODELING

Solving Eq. (7.67) gives q2u ¼ 400 1 e0:2t . Substituting q2u into Eq. (7.66) gives q_ 1u ¼ 80 1 e0:2t 0:2q1u

ð7:68Þ

The root for Eq. (7.68) is s ¼ 0.2, and has a natural solution q1n ¼ B1 e0:2t . The input in Eq. (7.66) has the same term as in the natural solution, and so the forced response is q1f ¼ B3 þ B2 te0:2t . Substituting q1f into Eq. (7.68) gives B2 ¼ 80 and B3 ¼ 400. The complete response is q1u ¼ q1n þ q1f ¼ B1 e0:2t þ 400 80te0:2t

ð7:69Þ

B1 is found using the initial condition q1(0) ¼ 0 and

q1u ð0Þ ¼ 0 ¼ B1 e0:2t þ 400 80te0:2t t¼0 ¼ B1 þ 400 and B1 ¼ 400. Thus,

q1u ¼ 400 400e0:2t 80te0:2t uðtÞ

ð7:70Þ

80u(t30) Input

Next, consider the 80u(t 30) input. By the property of a linear system, then q1u30 ¼ 400 400e0:2ðt30Þ 80ðt 30Þe0:2ðt30Þ uðt 30Þ q2u30 ¼ 400 1 e0:2ðt30Þ uðt 30Þ

Complete Solution The complete response is

q1 ¼ q1d þ q1u þ q1u30 ¼ 4te0:2t uðtÞ þ 400 400e0:2t 80te0:2t uðtÞ 400 400e0:2ðt30Þ 80ðt 30Þe0:2ðt30Þ uðt 30Þ q2 ¼ q2d þ q2u þ q2u30 ¼ 20e0:2t uðtÞ þ 400 1 e0:2t uðtÞ 400 1 e0:2ðt30Þ uðt 30Þ

which is plotted in Figure 7.18.

400 Quantity

200

0 0

FIGURE 7.18

30 Time

Plot of the solute quantities for Example Problem 7.10.

7.6 TWO-COMPARTMENT MODELING

401

When delivering anesthesia, a similar input is used as in Example Problem 7.10—that is, a bolus plus constant infusion. The reason for this type of input is to quickly raise the anesthesia to a desired level (bolus) and then to maintain the level for the operation (step).

7.6.2 Sink Compartment A sink compartment is one that has only inputs and no output. Similar to the source compartment, a sink acts like an integrator and has a zero root. Moreover, the solution of the nonsink compartment is independent of the sink compartment in the two-compartment case. Once solved, the quantity in the nonsink compartment is used to solve the quantity in the sink compartment. Using the model shown in Figure 7.15, a sink compartment exists if either K12 and K10, or K21 and K20 are zero.

EXAMPLE PROBLEM 7.11 Consider the two-compartment model shown in Figure 7.15 with K12 ¼ K10 ¼ 0, K21 ¼ 0.2, K20 ¼ 1, f1(t) ¼ 0, and f2(t) ¼ 10d(t). Assume that the initial conditions are zero. Solve for the quantity in each compartment.

Solution Since K21 ¼ 0:2 and K12 ¼ 0, this compartment model has a sink for compartment 1. As before, rather than solving the problem with a bolus input, the initial condition is changed in compartment 2 to q2 ð0Þ ¼ 10, with zero input. The conservation of mass for each compartment is q_ 1 ¼ K21 q2 ¼ 0:2q2

ð7:71Þ

q_ 2 ¼ ðK21 þ K20 Þq2 ¼ 1:2q2

ð7:72Þ

Since Eq. (7.72) involves only q2, we solve directly to get q2 ¼ 10e1:2t uðtÞ: Next, substitute q2 into (Eq. 7.71), yielding q_ 1 ¼ 0:2q2 ¼ 2e1:2t

ð7:73Þ

Eq. (7.73) gives a single root at s ¼ 0, and q1n ¼ B1 . The forced response is q1f ¼ B2 e1:2t , which when substituted into Eq. (7.71) gives B2 ¼ 1:67: The complete response is q1 ¼ q1n þ q1f ¼ B1 1:667e1:2t With q1 ð0Þ ¼ 0, we have B1 ¼ 1:667 from Eq. (7.74), and the complete response is q1 ¼ 1:667 1 e1:2t uðtÞ

ð7:74Þ

ð7:75Þ

This result indicates that more than 80 percent of the solute has moved from the system into the environment. If K20 ¼ 0, then q2 ¼ 10e0:2t uðtÞ and q1 ¼ 10 1 e0:2t uðtÞ: Here, all the solute exponentially moves from compartment 2 to 1 as expected.

402

7. COMPARTMENTAL MODELING

EXAMPLE PROBLEM 7.12 Consider a two-compartment system for the distribution of creatinine in the body illustrated in Figure 7.19. Compartment 1 represents the plasma and compartment 2 the muscle. Creatinine is a waste product of metabolism in the muscle that’s cleared from the body through the urine (transfer rate K10). Assume creatinine production in the muscle is f2(t) and is given by a step input. Find the concentration of creatinine in the plasma compartment.

Solution The differential equations describing the rate of change of creatinine in the compartments 1 and 2 are written by using the conservation of mass equation as q_ 1 ¼ K21 q2 ðK10 þ K12 Þq1

ð7:75Þ

q_ 2 ¼ K12 q1 K21 q2 þ f2 ¼ K12 q1 K21 q2 þ 1

ð7:76Þ

The D-Operator is used to remove q2, giving €q1 þ ðK10 þ K12 þ K21 Þ_q1 þ K10 K21 q1 ¼ K21

ð7:77Þ

The roots of the characteristic equation are s1, 2 ¼

ðK10 þ K12 þ K21 Þ 1 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK21 K10 K12 Þ2 4K21 K10

The natural response is an overdamped response: q1n ¼ B1 e s1 t þ B2 e s2 t The forced response is a constant (B3) and when substituted into the differential equation 1 yields B3 ¼ : The complete response is K10 1 q1 ¼ B1 e s1 t þ B2 e s2 t þ K10 and c1 ¼

1 1 B1 e s1 t þ B2 e s2 t þ V1 K10

for t 0: The constants B1 and B2 are determined using the initial conditions. f2(t)=u(t)

K12 q1

q2 K21

K10

FIGURE 7.19

Illustration for Example Problem 7.12.

403

7.7 THREE-COMPARTMENT MODELING

7.7 THREE-COMPARTMENT MODELING The general form of the three-compartment model is shown in Figure 7.20. As before, we begin with the general form of the three-compartment model and then examine special cases. To analyze the system in Figure 7.20, conservation of mass is used to write a differential equation for each compartment describing the rate of change of the quantity of solute in the compartment, given as accumulation ¼ input – output, where Compartment 1 Accumulation ¼ q_ 1 Input ¼ f1 ðtÞ þ K21 q2 þ K31 q3 Ouput ¼ ðK10 þ K12 þ K13 Þq1

Compartment 2 Accumulation ¼ q_ 2 Input ¼ f2 ðtÞ þ K12 q1 þ K32 q3 Ouput ¼ ðK20 þ K21 þ K23 Þq2

Compartment 3 Accumulation ¼ q_ 3 Input ¼ f3 ðtÞ þ K13 q1 þ K23 q2 Ouput ¼ ðK30 þ K31 þ K32 Þq3

Therefore, q_ 1 ¼ f1 ðtÞ þ K21 q2 þ K31 q3 ðK10 þ K12 þ K13 Þq1 q_ 2 ¼ f2 ðtÞ þ K12 q1 þ K32 q3 ðK20 þ K21 þ K23 Þq2 q_ 3 ¼ f3 ðtÞ þ K13 q1 þ K23 q2 ðK30 þ K31 þ K32 Þq3

ð7:78Þ

The D-Operator is used to simplify the system, where Eq. (7.78) is written in matrix form as DIQ ¼ AQ þ F

ð7:79Þ

where 2

3 2 3 2 3 f1 ðtÞ q1 ðK10 þ K12 þ K13 Þ K21 K31 5, F ¼ 4 f2 ðtÞ 5 K12 ðK20 þ K21 þ K23 Þ K32 Q ¼ 4 q2 5, A ¼ 4 f2 ðtÞ q2 K13 K23 ðK30 þ K31 þ K32 Þ

f1(t)

f2(t)

K12

K10

K20

q2 K21 K13

f3(t)

K23

K31

K32 q3

K 30 A general three-compartment model. Compartment 1 has volume V1, compartment 2 has volume V2, and compartment 3 has volume V3.

FIGURE 7.20

404

7. COMPARTMENTAL MODELING

To make the solution more readable, matrix A 2 a11 a12 A ¼ 4 a21 a22 a31 a32

is written as 3 a13 a23 5 a33

Solving Eq. (7.79) gives Q ¼ ðDI AÞ1 F ¼

1 adjðDI AÞF detðDI AÞ

ð7:80Þ

detðDI AÞQ ¼ adjðDI AÞF and using MATLAB, we have >> syms D q1 q2 q3 a11 a12 a13 a21 a22 a23 a31 a32 a33 >> A¼[a11 a12 a13; a21 a22 a23; a31 a32 a33]; >> det(D*eye(3)-A) ans ¼ D^3-D^2*a33-a22*D^2þD*a22*a33-D*a23*a32a11*D^2þa11*D*a33þa11*a22*D-a11*a22*a33þa11*a23*a32a21*a12*Dþa21*a12*a33-a21*a13*a32-a31*a12*a23a31*a13*Dþa31*a13*a22 >> adj¼det(D*eye(3)-A)*inv(D*eye(3)-A) adj ¼ [ D^2-D*a33-a22*Dþa22*a33-a23*a32, a12*D-a12*a33þa13*a32, a12*a23þa13*D-a13*a22] [ a21*D-a21*a33þa23*a31, D^2-D*a33-a11*Dþa11*a33-a13*a31, a23*D-a23*a11þa13*a21] [ a21*a32þa31*D-a31*a22, a32*D-a32*a11þa12*a31, D^2-a22*Da11*Dþa11*a22-a12*a21]

Substituting the values from MATLAB into Eq. (7.80) gives 2 6 4

D3 ða11 þ a22 þ a33 ÞD2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a13 a31 a23 a32 ÞD Q¼ a11 a22 a33 þ a11 a23 a32 þ a22 a13 a31 þ a33 a12 a21 a12 a23 a31 a13 a32 a21 D2 ða33 þ a22 ÞD þ a22 a33 a23 a32 a21 D a21 a33 þ a23 a31 a31 D þ a21 a32 a31 a22

3 a12 D a12 a33 þ a13 a32 a13 D þ a12 a23 a13 a22 7 D2 ða33 þ a11 ÞD þ a11 a33 a13 a31 a23 D a23 a11 þ a13 a21 5F a32 D a32 a11 þ a12 a31 D2 ða22 þ a11 ÞD þ a11 a22 a12 a21

ð7:81Þ Returning to the time domain gives the following independent differential equations: ___ q1 ða11 þ a22 þ a33 Þ€q1 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a13 a31 a23 a32 Þq_ 1 þ ða11 a23 a32 a11 a22 a33 þ a22 a13 a31 þ a33 a12 a21 a12 a23 a31 a13 a32 a21 Þq1 ¼ €f1 ða33 þ a22 Þf_ þ ða22 a33 a23 a32 Þf1 þ a12 f_ ða12 a33 a13 a32 Þf2 1

þ a13 f_ 3 þ ða12 a23 a13 a22 Þf3

ð7:82Þ

405

7.7 THREE-COMPARTMENT MODELING

___ q2 ða11 þ a22 þ a33 Þ€q2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a13 a31 a23 a32 Þq_ 2 þ ða11 a23 a32 a11 a22 a33 þ a22 a13 a31 þ a33 a12 a21 a12 a23 a31 a13 a32 a21 Þq2 ¼ a21 f_ ða21 a33 a23 a31 Þf1 þ €f2 ða33 þ a11 Þf_ þ ða11 a33 a13 a31 Þf2 1

ð7:83Þ

þ a23 f_ 3 ða23 a11 a13 a21 Þf3

___ q3 ða11 þ a22 þ a33 Þ€q3 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a13 a31 a23 a32 Þ_q3 þ ða11 a23 a32 a11 a22 a33 þ a22 a13 a31 þ a33 a12 a21 a12 a23 a31 a13 a32 a21 Þq3 ¼ a31 f_ þ ða21 a32 a31 a22 Þf1 þ a32 f_ ða32 a11 a12 a31 Þf2 1

ð7:84Þ

þ €f3 ða22 þ a11 Þf_ 3 þ ða11 a22 a12 a21 Þf3

The characteristic equation, det(DIA), is identical for q1, q2, and q3, as well as the form of the natural response. Note that the coefficients in the natural response are not identical and depend on the input to the compartments and the initial conditions. The characteristic equation is s3 ða11 þ a22 þ a33 Þs2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a13 a31 a23 a32 Þs a11 a22 a33 þ a11 a23 a32 þ a22 a13 a31 þ a33 a12 a21 a12 a23 a31 a13 a32 a21 ¼ 0

ð7:85Þ

The roots of the characteristic equation are determined using the MATLAB command “eig” and may be underdamped, overdamped, or critically damped, depending on the

transfer rates. The expressions for the roots are far too complex to be usable and will not be written here. With a two-compartment model, there was only one way for repeat roots to occur. With a three-compartment model, there are many more configurations for repeat roots to occur. Complex roots can occur under certain conditions, which are discussed in Section 7.7.2. In the remainder of this section and the next, we consider special cases of the threecompartment model: mammillary, catenary, and unilateral. Each model may be closed and may have sink and source compartments.

7.7.1 Mammillary Three-Compartment Model A mammillary three-compartment model is shown in Figure 7.21, which is characterized by a central compartment connected to two peripheral compartments. All exchanges of the solute are through the central compartment, and there is no direct exchange of solute f1(t)

f2(t)

K32

K12 q1

q2 K21

K10

f3(t)

K23 K20

K30

FIGURE 7.21 A mammillary three-compartment model.

406

7. COMPARTMENTAL MODELING

between compartments 1 and 3. Each compartment can have an input and an output to the environment. The mammillary three-compartment model is given by the following set of equations: q_ 1 ¼ f1 ðtÞ þ K21 q2 ðK10 þ K12 Þq1 q_ 2 ¼ f2 ðtÞ þ K12 q1 ðK20 þ K21 þ K23 Þq2 þ K32 q3

ð7:86Þ

q_ 3 ¼ f3 ðtÞ þ K23 q2 ðK30 þ K32 Þq3 With 2

a11

6 A ¼ 4 a21 a31

a12

a13

a22

7 6 a23 5 ¼ 4

a32

a33

ðK10 þ K12 Þ

K21

K12

ðK20 þ K21 þ K23 Þ

K32

K23

ðK30 þ K32 Þ

3 7 5

and Eq. (7.81), we have 3 D ða11 þ a22 þ a33 ÞD2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a23 a32 ÞD Q¼ a11 a22 a33 þ a11 a23 a32 þ a33 a12 a21 a12 a23 a31 2 6 4

D2 ða33 þ a22 ÞD þ a22 a33 a23 a32

a12 D a12 a33

a12 a23

a21 D a21 a33

D2 ða33 þ a11 ÞD þ a11 a33

a23 D a23 a11

a21 a32

a32 D a32 a11

D2 ða22 þ a11 ÞD þ a11 a22 a12 a21

7 5F

ð7:87Þ Returning to the time domain gives the following independent differential equations: ___ q1 ða11 þ a22 þ a33 Þ€q1 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a23 a32 Þq_ 1 þ ða11 a23 a32 þ a33 a12 a21 a11 a22 a33 Þq1 ¼ €f1 ða33 þ a22 Þf_ 1 þ ða22 a33 a23 a32 Þ f1 þ a12 f_ 2 a12 a33 f2

ð7:88Þ

___ q2 ða11 þ a22 þ a33 Þ€q2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a23 a32 Þ_q2 þ ða11 a23 a32 þ a33 a12 a21 a11 a22 a33 Þq2 ¼ a21 f_ ða21 a33 a23 a31 Þ f1 þ €f2 ða33 þ a11 Þ f_ þ a11 a33 f2 þ a23 f_ a23 a11 f3

ð7:89Þ

___ q3 ða11 þ a22 þ a33 Þ€q3 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a23 a32 Þ_q3 þ ða11 a23 a32 þ a33 a12 a21 a11 a22 a33 Þq3 ¼ a21 a32 f1 þ a32 f_ 2 a32 a11 f2 þ €f3 ða22 þ a11 Þ f_ 3 þ ða11 a22 a12 a21 Þf3

ð7:90Þ

The roots of a mammillary three-compartment model are all real and determined from the characteristic equation s3 ða11 þ a22 þ a33 Þs2 þ ða11 a22 þ a22 a33 þ a33 a11 a12 a21 a23 a32 Þs þ a11 a23 a32 þ a33 a12 a21 a11 a22 a33 ¼ 0

407

7.7 THREE-COMPARTMENT MODELING

EXAMPLE PROBLEM 7.13 Consider the mammillary three-compartment model shown in Figure 7.21, with a loss of solute to the environment only from compartment 1 and input only from compartment 2. Additionally, K12¼2, K21¼1.5, K10¼0.5, K23¼1.3, K32¼0.4, and f2(t)¼10d(t). Assume that the initial conditions are zero. Solve for the quantity in each compartment.

Solution With the input f2(t)¼10d(t) transformed into a change in initial condition for compartment 2 to q2(0)¼10 and no input, conservation of mass for each compartment yields q_ 1 ¼ K21 q2 ðK10 þ K12 Þq1 ¼ 2:5q1 þ 1:5q2 q_ 2 ¼ K12 q1 ðK21 þ K23 Þq2 þ K32 q3 ¼ 2q1 2:8q2 þ 0:4q3 q_ 3 ¼ K23 q2 ðK30 þ K32 Þq3 ¼ 1:3q2 0:4q3

ð7:91Þ

Using the D-Operator method with MATLAB, we get >> syms D >> A¼[-2.5 1.5 0;2 -2.8 0.4;0 1.3 -0.4]; >> det(D*eye(3)-A) ans ¼ D^3þ57/10*D^2þ28/5*Dþ3/10

and 28 q_ þ 0:3q1 ¼ 0 5 1 28 ___ q2 þ 5:7€q2 þ q_ 2 þ 0:3q2 ¼ 0 5 28 ___ q3 þ 5:7€q3 þ q_ 3 þ 0:3q3 ¼ 0 5 ___ q1 þ 5:7€q1 þ

Using the “eig(A)” command gives the roots as 4.46, 1.18, and 0.06. Thus, we have q1 ¼ B1 e4:46t þ B2 e1:18t þ B3 e0:06t q2 ¼ B4 e4:46t þ B5 e1:18t þ B6 e0:06t q3 ¼ B7 e4:46t þ B8 e1:18t þ B9 e0:06t (since the forced response is zero). Note that since there is no input, all we needed to do was define the matrix A and then use the “eig(A)” command (i.e., no need to use the “det” command). However, we shall use the “det” command because it gives the intermediate result. The initial conditions are q1(0)¼0, q2(0)¼10, and q3(0)¼0. To determine the initial conditions for the derivative terms, we use Eq. (7.91) and get q_ 1 ð0Þ ¼ 2:5q1 ð0Þ þ 1:5q2 ð0Þ ¼ 15 q_ 2 ð0Þ ¼ 2q1 ð0Þ 2:8q2 ð0Þ þ 0:4q3 ð0Þ ¼ 28 q_ 3 ð0Þ ¼ 1:3q2 ð0Þ 0:4q3 ð0Þ ¼ 13 Continued

408

7. COMPARTMENTAL MODELING

To determine the initial conditions for the second derivative, we take the derivative of Eq. (7.91) and set t ¼ 0, giving €q1 ð0Þ ¼ 2:5_q1 ð0Þ þ 1:5_q2 ð0Þ ¼ 79:5 €q2 ð0Þ ¼ 2_q1 ð0Þ 2:8_q2 ð0Þ þ 0:4_q3 ð0Þ ¼ 113:6 €q3 ð0Þ ¼ 1:3_q2 ð0Þ 0:4_q3 ð0Þ ¼ 41:6 Solution details are provided for q1 here, and a final solution for q2 and q3. Using the initial conditions, we solve for B1, B2 and B3 from q1 ð0Þ ¼ 0 ¼ B1 þ B2 þ B3 q_ 1 ð0Þ ¼ 15 ¼ 4:46B1 1:18B2 0:06B3 €q1 ð0Þ ¼ 79:5 ¼ 19:9B1 þ 1:4B2 þ 0:0036B3 giving B1 ¼ 4:219, B2 ¼ 3:1818, and B3 ¼ 1:0372: Therefore,

q1 ¼ 4:219e4:46t þ 3:1818e1:18t þ 1:0372e0:06t uðtÞ

We repeat this process for q2 and q3, yielding q2 ¼ 5:51e4:46t þ 2:8179e1:18t þ 1:6721e0:06t uðtÞ q3 ¼ 1:762e4:46t 4:6849e1:18t þ 6:4469e0:06t uðtÞ

7.7.2 The Unilateral Three-Compartment Model A unilateral three-compartment model is shown in Figure 7.22, which is characterized by a closed loop of connected compartments, whereby the solute circulates around the loop in one direction only. Each compartment can have an input and an output to the environment. f1(t)

K10

f2(t)

K12 q1

K20 q2

f3(t) K31

K23

K30

FIGURE 7.22

A unilateral three-compartment model.

7.7 THREE-COMPARTMENT MODELING

409

In general, the unilateral three-compartment model is given by the following set of equations: q_ 1 ¼ f1 ðtÞ þ K31 q3 ðK10 þ K12 Þq1 q_ 2 ¼ f2 ðtÞ þ K12 q1 ðK20 þ K23 Þq2 q_ 3 ¼ f3 ðtÞ þ K23 q2 ðK30 þ K31 Þq3

ð7:92Þ

To examine a simple unilateral three-compartment model with complex roots, consider a closed system (i.e., K10 ¼ K20 ¼ K30 ¼ 0). From Eq. (7.85), the roots are found from the characteristic equation, given as s3 þ ðK12 þ K23 þ K31 Þs2 þ ðK12 K23 þ K23 K31 þ K31 K12 Þs ¼ 0

ð7:93Þ

which are s1 ¼ 0, and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK12 þ K23 þ K31 Þ 1 ðK12 þ K23 þ K31 Þ2 4ðK12 K23 þ K23 K31 þ K31 K12 Þ 2 2 Complex roots occur when 4ðK12 K23 þ K23 K31 þ K31 K12 Þ > ðK12 þ K23 þ K31 Þ2 : Repeated roots occur when 4ðK12 K23 þ K23 K31 þ K31 K12 Þ ¼ ðK12 þ K23 þ K31 Þ2 : Consider the case of complex roots and a zero root, which gives rise to a natural solution of the form s 2, 3 ¼

qi ¼ B1 þ eat ðB2 cos od t þ B3 sin od tÞ ¼ B1 þ B4 eat cos ðod t þ fÞ where a and od are the real and imaginary part of the complex root, and the Bi terms are determined from initial conditions after the forced response is determined. We write pffiffiffiffiffiffiffiffiffiffiffiffiffi the complex roots in standardized format as s2, 3 ¼ zo0 o0 z2 1, which has a characteristic equation of s2 þ 2 zo0 s þ o20 ¼ 0

ð7:94Þ

The system is at its most oscillatory when z ¼ 0, a pure sinusoid. To get a better understanding of the system, we determine the extent of its oscillatory behavior by finding the optimal values of the transfer rates to achieve maximum oscillatory behavior (i.e., minimum z). To write an expression for z, we use the coefficients of the characteristic equation (Eq. (7.93)) and set them equal to the terms in Eq. (7.94): 2zo0 ¼ ðK12 þ K23 þ K31 Þ o20 ¼ ðK12 K23 þ K23 K31 þ K31 K12 Þ which gives o0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK12 K23 þ K23 K31 þ K31 K12 Þ

and 1 ðK12 þ K23 þ K31 Þ ð7:95Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2o0 2 ðK12 K23 þ K23 K31 þ K31 K12 Þ @z @z @z ¼ 0, ¼ 0, and ¼ 0, which allows us to To find the minimum z, we find @K12 @K23 @K31 determine the conditions that allow minimum z. First, we use the chain rule to find z¼

410

7. COMPARTMENTAL MODELING 1 2

@z 2ðK12 K23 þ K23 K31 þ K31 K12 Þ ðK23 þ K31 ÞðK12 þ K23 þ K31 ÞðK12 K23 þ K23 K31 þ K31 K12 Þ ¼ 4ðK12 K23 þ K23 K31 þ K31 K12 Þ @K12 ¼

2ðK12 K23 þ K23 K31 þ K31 K12 Þ ðK23 þ K31 ÞðK12 þ K23 þ K31 Þ 4ðK12 K23 þ K23 K31 þ K31 K12 Þ

3 2

1 2

¼0

The minimum occurs when the numerator is zero—that is, 2ðK12 K23 þ K23 K31 þ K31 K12 Þ ¼ ðK23 þ K31 ÞðK12 þ K23 þ K31 Þ 2 2 ¼ K23 K12 þ K23 þ K23 K31 þ K31 K12 þ K31 K23 þ K31

Simplifying, we have 2 2 K12 ðK23 þ K31 Þ ¼ K23 þ K31

or K12 ¼

2 2 K23 þ K31 ðK23 þ K31 Þ

2 2 @z K2 þ K31 @z K2 þ K23 , and for . ¼ 0, we get K23 ¼ 12 ¼ 0, we get K31 ¼ 12 ðK12 þ K31 Þ ðK12 þ K23 Þ @K23 @K31 The only way these relationships are valid is if K12 ¼ K23 ¼ K31 ¼ K, and from Eq. (7.95), we 3 find z ¼ pffiffiffi ¼ 0:866, which does not have a very noticeable oscillatory behavior. We will 2 3 see in the next section that a more noticeable oscillatory response is possible with more than three compartments.

Repeating for

EXAMPLE PROBLEM 7.14 Consider the unilateral three-compartment model shown in Figure 7.22 with no loss of solute to the environment from any compartments and an input for compartment 3 only. Additionally, K12 ¼ K23 ¼ K31 ¼ 2, and f3 ðtÞ ¼ 5dðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment.

Solution As before, we transform the input, f3(t) ¼ 5d(t), into a change in initial condition for compartment 3 to q3(0)¼5 and no input. The conservation of mass for each compartment yields q_ 1 ¼ K31 q3 ðK10 þ K12 Þq1 ¼ 2q1 þ 2q3 q_ 2 ¼ K12 q1 ðK20 þ K23 Þq2 ¼ 2q1 2q2 q_ 3 ¼ K23 q2 ðK30 þ K31 Þq3 ¼ 2q2 2q3

ð7:96Þ

7.7 THREE-COMPARTMENT MODELING

411

Using the D-Operator method with MATLAB, we get >> syms D >> A¼[-2 0 2;2 -2 0;0 2 -2]; >> det(D*eye(3)-A) ans ¼ D^3þ6*D^2þ12*D

and ___ q1 þ 6€q1 þ 12_q1 ¼ 0 ___ q2 þ 6€q2 þ 12_q2 ¼ 0 ___ q3 þ 6€q3 þ 12_q3 ¼ 0 The roots from the characteristic equation are 0, 3 j1:7321: The complete solution is the natural solution, since the forced response is zero, and is given by q1 ¼ B1 þ e3t ðB2 cos 1:7321t þ B3 sin 1:7321tÞ q2 ¼ B4 þ e3t ðB5 cos 1:7321t þ B6 sin 1:7321tÞ q3 ¼ B7 þ e3t ðB8 cos 1:7321t þ B9 sin 1:7321tÞ The initial conditions are q1(0)¼0, q2(0)¼0, and q3(0)¼5. To determine the initial conditions for the derivative terms, we use Eq. (7.96) and get q_ 1 ð0Þ ¼ 2q1 ð0Þ þ 2q3 ð0Þ ¼ 10 q_ 2 ð0Þ ¼ 2q1 ð0Þ 2q2 ð0Þ ¼ 0 q_ 3 ð0Þ ¼ 2q2 ð0Þ 2q3 ð0Þ ¼ 10 To determine the initial conditions for the second derivative, we take the derivative of Eq. (7.96) with t ¼ 0, giving €q1 ð0Þ ¼ 2_q1 ð0Þ þ 2_q3 ð0Þ ¼ 40 €q2 ð0Þ ¼ 2_q1 ð0Þ 2_q2 ð0Þ ¼ 20 €q3 ð0Þ ¼ 2_q2 2_q3 ¼ 20 For q1, we have q1 ð0Þ ¼ 0 ¼ B1 þ B2 q_ 1 ð0Þ ¼ 10 ¼ 3B2 þ 1:7321B3 €q1 ð0Þ ¼ 40 ¼ 6B2 10:4B3

5 5 which gives B1 ¼ , B2 ¼ , and B3 ¼ 2:9, and 3 3 5 5 q1 ¼ e3t cos 1:7321t 2:9 sin 1:7321t uðtÞ 3 3 Repeating for q2 and q3, we have 0 0 11 5 5 q2 ¼ @ e3t @ cos 1:7321t þ 2:9 sin 1:7321tAAuðtÞ 3 3 q3 ¼

5 1 þ 2e3t cos 1:7321t uðtÞ 3 Continued

412

7. COMPARTMENTAL MODELING

Illustrated in Figure 7.23 is a plot of the quantity in each compartment. While it is difficult to see the oscillations, the first peak is evident by the overshoot or undershoot. To determine the time at peak undershoot for q3, we use the technique of Section 2.9.2 by finding the time that satisfies @q3 ¼ 0, which gives Tp3 ¼ 1:21: Similarly, Tp ¼ 1:81 for both q1 and q2. @t 6

q1, q2, q3

5 4 q3 3 2

q1 q2

1 0 0

0.5

1.5 Time

2.5

FIGURE 7.23 Illustration of the quantity in each compartment in Example Problem 7.14.

7.7.3 Source Compartment In Section 7.5.1, a source compartment in a two-compartment model was described as one that only has output to other compartments, without any inputs from other compartments. A source compartment also appears in three-compartment models, whose output is solved independent of the other compartments as before. The following example involves a three-compartment mammillary model with a source compartment, as illustrated in Figure 7.24. The body is now divided into the digestive system, plasma, and the tissues to more accurately depict their behavior. f1(t)

K12 q1

K32 q3

q2 K23 K20

FIGURE 7.24 Illustration for Example Problem 7.15. Compartment 1 is the digestive system, compartment 2 is the plasma, and compartment 3 is the tissues.

413

7.7 THREE-COMPARTMENT MODELING

EXAMPLE PROBLEM 7.15 Consider the mammillary three-compartment model with the source compartment shown in Figure 7.24. The input is f1(t)¼d(t). Assume that the initial conditions are zero. Solve for the quantity in each compartment.

Solution Once again, the input is transformed into a change in initial condition for compartment 1, q1(0)¼1. The equations describing this model are q_ 1 ¼ K12 q1

ð7:97Þ

q_ 2 ¼ K12 q1 ðK20 þ K23 Þq2 þ K32 q3

ð7:98Þ

q_ 3 ¼ K23 q2 K32 q3

ð7:99Þ K12 t

K12 t

uðtÞ ¼ e uðtÞ: SubstitutSince Eq. (7.97) involves only q1, it is easily solved as q1 ¼ q1 ð0Þe ing the solution for q1 into Eqs. (7.98) and (7.99), we now have two equations as q_ 2 ¼ K12 eK12 t uðtÞ ðK20 þ K23 Þq2 þ K32 q3

ð7:100Þ

q_ 3 ¼ K23 q2 K32 q3

ð7:101Þ

The D-Operator gives the reconstructed differential equations for q2 and q3 as >> syms D K20 K23 K32 >> A¼[-(K20þK23) K32;K23 -K32]; >> det(D*eye(2)-A) ans ¼ D^2þD*K32þK20*DþK20*K32þK23*D >> adj¼det(D*eye(2)-A)*inv(D*eye(2)-A) adj ¼ [ DþK32, K32 ] [ K23, DþK20þK23]

and €q2 þ ðK32 þ K20 þ K23 Þ_q2 þ K20 K32 q2 ¼ K12 ðK32 K12 ÞeK12 t

ð7:102Þ

€q3 þ ðK32 þ K20 þ K23 Þ_q3 þ K20 K32 q3 ¼ K23 K12 eK12 t

ð7:103Þ

The roots are >> eig(A) ans ¼ -1/2*K20-1/2*K23-1/2*K32þ1/2*(K20^2þ2*K20*K232*K20*K32þK23^2þ2*K32*K23þK32^2)^(1/2) -1/2*K20-1/2*K23-1/2*K32-1/2*(K20^2þ2*K20*K232*K20*K32þK23^2þ2*K32*K23þK32^2)^(1/2)

Continued

414

7. COMPARTMENTAL MODELING

which simplifies to s1, 2 ¼

ðK20 þ K23 þ K32 Þ 1 2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK32 þ K20 þ K23 Þ2 4K32 K23

ð7:104Þ

Only real roots are possible with Kij 0: The natural response for q2 is q2n ¼ B1 e s1 t þ B2 e s2 t The forced response is q2f ¼ B3 eK12 t , which after substituting into Eq. (7.103) gives B3 ¼

2 K12

K23 K12 K12 ðK32 þ K20 þ K23 Þ þ K20 K32

The complete response is then q2 ¼ B1 e s1 t þ B2 e s2 t þ B3 eK12 t We use the initial conditions, q2 ð0Þ ¼ 0 and q_ 2 ð0Þ ¼ K12 q1 ð0Þ, to solve for B1 and B2 as follows: q2 ð0Þ ¼ B1 þ B2 þ B3 ¼ 0 and with q_ 2 ¼ s1 B1 e s1 t þ s2 B2 e s2 t K12 B3 eK12 t , we have q_ 2 ð0Þ ¼ K12 ¼ s1 B1 þ s2 B2 K12 B3 To solve for B1 and B2, we evaluate

B1 B3 1 1 ¼ K12 ð1 þ B3 Þ s1 s2 B2 which gives

3 K12 B3 þ K12 þ B3 s2 7 6 ðs1 s2 Þ 7 6 B1 7 ¼6 6 K12 B3 þ K12 þ B3 s1 7 B2 5 4 ðs1 s2 Þ 2

The final solution is

1 K12 B3 þ K12 þ B3 s2 s1 t K12 B3 þ K12 þ B3 s1 s2 t e e C B ðs1 s2 Þ ðs1 s2 Þ C B CuðtÞ q2 ðtÞ ¼ B C B K K 23 12 K12 t A @þ e 2 K ðK þ K þ K Þ þ K K K12 12 32 20 23 20 32 0

Repeating for q3, we have B3 ðK12 þ s2 Þ s1 t B3 ðK12 þ s1 Þ s2 t K23 K12 q3 ðtÞ ¼ e e þ 2 eK12 t uðtÞ ðs1 s2 Þ ðs1 s2 Þ K12 K12 ðK32 þ K20 þ K23 Þ þ K20 K32

7.7 THREE-COMPARTMENT MODELING

415

7.7.4 Sink Compartment A sink compartment in a three-compartment model gives rise to a zero root and is described as a compartment with only inputs and no output to other compartments except to the environment. The solution for the sink compartment is found as usual using the D-Operator approach from the resulting quantities in the other two compartments. To illustrate a sink compartment, Example Problem 7.16 involves a three-compartment model describing the transport of a thyroid hormone to the hepatic duct (sink compartment).12 The thyroid system was first described in Example Problem 7.5 and is extended to this example. We will then extend the model in Section 7.8.4. Before describing the current model, more background material on the thyroid system is presented. The thyroid hormones thyroxine (T4) and triiodothyronine (T3), which are produced by the thyroid gland, maintain the body temperature, regulate energy metabolism, and are important for growth and development. The thyroid hormones themselves do not exist inside the thyroid cell but are part of a large thyroglobulin molecule that consists of approximately 70 tyrosine amino acids. The metabolic rate falls to approximately 50 percent of normal without these hormones, and too much thyroid hormone can increase the metabolic rate by 100 percent above the normal rate. The pituitary gland controls the discharge of T4 and T3 through its release of the thyroid-stimulating hormone, TSH. As previously described, the pituitary gland is under the control of the hypothalamus through its release of the thyrotropin-releasing-hormone, TRH. Ingested iodine, in the form of iodide, is an essential element in the formation of thyroid hormones. Blood flow through the thyroid gland is among the highest of any organ in the body, which allows the quick uptake of iodide. Typically after ingestion, 80 percent of the iodide is rapidly excreted by the kidneys, and the other 20 percent is taken up by the thyroid gland. Once iodide is taken up by the thyroid gland, it is used in a series of enzyme reactions to create the thyroid hormones. Iodide is first transported across the membrane of the thyroid cell by a pump mechanism called the sodium-iodide symporter. The pump allows iodide concentrations in the thyroid cell to be much greater than in the plasma. Once inside the cell, iodide is oxidized by thyroidal peroxidase to iodine, which iodinates the tyrosine component of the thyroglobulin molecule to first form monoiodotyrosine (MIT) and then diiondotryosine (DIT). Thyroperoixidase then catalyzes the joining of two molecules of DIT to form T4 (a twobenzene ringed structure consisting of an inner tyrosyl ring and an outer phenolic ring, within the thyroglobulin-iodine molecule) or to a lesser extent, the joining of one molecule of MIT to DIT to form T3. Reverse T3 is also formed but is excluded from this discussion. While the process of creating the thyroglobulin-iodine molecule is quick, the thyroid gland keeps approximately a 60-day supply in reserve. Thyroglobulin itself is not released into the plasma when the thyroid is stimulated by TSH, but T4 and T3 are released through a lysosomal protease enzyme action on the thyroglobulin-iodine molecule. Almost all of the output from the thyroid gland is T4 (greater than 90 percent). In addition, MIT and DIT are released from the thyroglobulin-iodine molecule when T4 and T3 are released; however, MIT and DIT do not leave the cell but are deiodinated, allowing the release of iodine. 12

See Haddad et al., 2003, in references for original problem developement.

416

7. COMPARTMENTAL MODELING

The iodine is then reused in the cell, repeating the enzyme reactions to form the thyroid hormones. Once in the plasma, the thyroid hormones reversibly combine with proteins. The binding to proteins protects T4 and T3 from immediate metabolism and excretion as they are inactive in this mode. T4 binds primarily with thyroid binding globulin (TBG) and, to a lesser extent, to thyroxine-binding prealbumin (TBPA) and albumin. TBG in the plasma is present in low concentrations and has a high affinity for T4; it usually binds about 70 percent of the available T4. TBPA in the plasma is present in high concentrations but has a low affinity for T4. Ten percent of the plasma-bound hormone T4 is used each day, giving it a 7-day halflife. The 7-day half-life for T4 creates a stable pool of thyroid hormone in the plasma. Approximately 0.04 percent of the T4 is not bound to proteins in the plasma during normal conditions, and we call the unbound T4 free-T4. T3 has a lower binding affinity for plasma proteins. Those T3 that are bound are primarily with TBPA and albumin. T3 is rapidly cleared from the plasma with a half-life of 1 day. Approximately 10 percent of T3 is not bound to proteins in the plasma during normal conditions. While a small amount of T3 is released by the thyroid into the plasma, almost all of the T3 in the plasma is produced by deiodination enzymes in the liver and to a much lesser extent in the kidneys, where an iodine atom is removed from T4 that converts it into T3. T4 is also eliminated within the liver and to a lesser extent in the kidneys, by conjugation of sulfate or glucuronic acid with the phenolic hydroxyl group of the outer phenolic ring, turning T4 into T3. Also within the liver, T4 undergoes deamination and decarboxylase reactions that convert it into T3. Once in the plasma, free-T3 moves into the interstitial space and easily moves across the cell membranes in the tissues. When transported into the cells, free-T3 moves into the cell nucleus and binds with thyroid hormone receptors, which then synthesize new proteins through gene transcription. These new proteins are connected to energy metabolism, body temperature, body weight, and the control of growth, reproduction, and differentiation. While T4 also moves into the cell and binds with thyroid hormone receptors, it takes about 10 times more T4 than T3 to equal the effect of T3 in gene transcription. Example Problem 7.16 involves a three-compartment model describing the transport of T4 throughout the body. To model the transport of T4 in the body, a bolus of radioactive iodine-T4 is injected into the plasma. The use of radioactive iodine-T4 allows us to track the transport of T4 in the body apart from the natural T4. The pathways of interest in the following example, as described in Figure 7.25, involve the plasma, liver, and hepatic duct. f1(t)

K12 q1

q2 K21

K23

FIGURE 7.25 Illustration for Example Problem 7.16. Compartment 1 is the plasma, compartment 2 is the liver, and compartment 3 is the hepatic duct, simplified to a single compartment. The input f1(t) is a bolus of radioactive iodine-T4.

7.7 THREE-COMPARTMENT MODELING

417

Within the liver, some of the T4 is converted into T3 and I131. The I131 moves into the hepatic duct, where it is absorbed within the bile. The feedback control by the pituitary gland and the involvement of the kidneys are ignored in this example.

EXAMPLE PROBLEM 7.16 Consider a three-compartment thyroxine transport model as shown in Figure 7.25. The input is f1 ðtÞ ¼ 0:1 103 dðtÞg of radioactive iodine-T4. Additionally, K12¼0.6, K21¼0.5, and K23¼0.3. Assume that the initial conditions are zero. Solve for the quantity of radioactive iodine-T4 in the plasma compartment.

Solution Since there is no output to another compartment for compartment 3, this compartment model has a sink for compartment 3. As before, rather than solving the problem with a bolus input, the initial condition is changed for compartment 1 to q1 ð0Þ ¼ 0:1 103 , with zero input. The conservation of mass for each compartment yields q_ 1 ¼ K21 q2 K12 q1 ¼ 0:6q1 þ 0:5q2 q_ 2 ¼ K12 q1 ðK21 þ K23 Þq2 ¼ 0:6q1 0:8q2

ð7:105Þ

q_ 3 ¼ K23 q2 ¼ 0:3q2 Using the D-Operator method with MATLAB, we get >> syms D >> A¼[-0.6 0.5 0;0.6 -0.8 0;0 0.3 0]; >> det(D*eye(3)-A) ans ¼ D^3þ7/5*D^2þ9/50*D

and 7 9 ___ q1 þ €q1 þ q_ 1 ¼ 0 5 50 The “eig(A)” command gives the roots as 0, 1.26, and 0.14. Thus, we have q1 ¼ B1 þ B2 e1:26t þ B3 e0:14t (since the forced response is zero). The initial conditions are q1 ð0Þ ¼ 0:1 103 , q2 ð0Þ ¼ 0, and q3(0)¼0. To determine the initial conditions for the derivative terms, we use Eq. (7.105), which gives q_ 1 ð0Þ ¼ 0:6q1 ð0Þ þ 0:5q2 ð0Þ ¼ 0:06 103 q_ 2 ð0Þ ¼ 0:6q1 ð0Þ 0:8q2 ð0Þ ¼ 0:06 103 To determine the initial conditions for the second derivative, we take the derivative of Eq. (7.105) and set t ¼ 0, giving €q1 ð0Þ ¼ 0:6_q1 ð0Þ þ 0:5_q2 ð0Þ ¼ 6:6 105 €q2 ð0Þ ¼ 0:6_q1 ð0Þ 0:8_q2 ð0Þ ¼ 8:4 105

Continued

418

7. COMPARTMENTAL MODELING

Using the initial conditions, we solve for B1, B2 , and B3 from q1 ð0Þ ¼ 0:1 103 ¼ B1 þ B2 þ B3 q_ 1 ð0Þ ¼ 0:06 103 ¼ 1:26B2 0:14B3 €q1 ð0Þ ¼ 6:6 105 ¼ 1:4B2 þ 0:02B3 giving 4 104 , and B3 ¼ 0:05 104 , and B1 ¼ 0:48 10 , B2 ¼ 0:47 1:26t þ 0:05e0:14t 104 uðtÞ q1 ¼ 0:48 þ 0:47e

The model used in Example Problem 7.16 is too simple to capture the real transport dynamics of thyroid hormone. Some investigators have included multiple compartments for the hepatic duct and many other compartments. Some have included chemical reactions in the model. We will investigate these models in a later chapter.

7.8 MULTICOMPARTMENT MODELING Realistic models of the body typically involve more than three compartments. The concepts described in the previous sections can be applied to a compartment model of any size. Each compartment is characterized by a conservation of mass differential equation describing the rate of change of the solute. Thus, for the case of n compartments, there are n equations of the form dqi ¼ input output dt where qi is the quantity of solute in compartment i, which can be generalized for the system to DIQ ¼ AQ þ F

ð7:106Þ

where

and for the first row in A, we have

and so on for the other rows in A. Equation (7.106) is solved as before from Q ¼ ðDI AÞ1 F ¼ or

1 adjðDI AÞF detðDI AÞ

detðDI AÞQ ¼ adjðDI AÞF

ð7:107Þ

7.8 MULTICOMPARTMENT MODELING

419

MATLAB is used to reconstruct the differential equations, as before, in terms of a single variable and the inputs. The characteristic equation, det(DIA), is identical for q1, q2, . . . qn, as well as the form of the natural response. The roots of the characteristic equation are determined using the MATLAB command “eig(A)” and may be underdamped, overdamped, or critically damped, depending on the transfer rates. The expression for the roots is far too complex to be usable and will not be written here. Most models will have many elements in A as zero, which makes the solution much more tractable. In the remainder of this section, we consider special cases of the multicompartment model: mammillary, catenary, and unilateral. Each model may be closed and may have sink and source compartments.

7.8.1 Mammillary Multicompartment Model A mammillary n-compartment model is shown in Figure 7.26, which is characterized by a central compartment connected to n 1 peripheral compartments. All exchange of solute is through the central compartment, and there is no direct exchange of solute among the other compartments. Each compartment can have an input and an output to the environment. The matrix A, given in Eq. (7.106), has nonzero elements defined as a11 ¼ ðK10 þ K12 þ K13 þ þ K1n Þ aii ¼ ðKi1 þ Ki0 Þ, a1i ¼ Ki1 , ai1 ¼ K1i ,

2in 2in 2in

This system only has real roots. f2(t) q2 K20

K12

f1(t) K10

K21

fi(t)

Ki1 q1

qi K1i

K1n

Ki0

Kn1

fn(t) qn Kn0

FIGURE 7.26

A mammillary n-compartment model.

ð7:108Þ

420

7. COMPARTMENTAL MODELING

7.8.2 Catenary Multicompartment Model A catenary n-compartment model is shown in Figure 7.27, which is characterized by a chain of compartments, with each compartment exchanging solute with the two adjacent compartments, except for the first and last in the chain. Each compartment can have an input and an output to the environment. The matrix A, given in Eq. (7.106), has nonzero elements defined as a11 ¼ ðK10 þ K12 Þ ai, i1 ¼ Ki1, i , aii ¼ Ki0 þ Ki, ði1Þ þ Ki, ðiþ1Þ , ai, iþ1 ¼ Kiþ1, i , ann ¼ Kn0 þ Kn, ðn1Þ

2in 2in1 2in1

ð7:109Þ

This system only has real roots.

7.8.3 Unilateral Multicompartment Model A unilateral n-compartment model is shown in Figure 7.28, which is characterized by a closed loop of connected compartments, whereby solute circulates around the loop in one direction only. Each compartment can have an input and an output to the environment. f1(t)

f2(t)

f3(t) K32

K21 q1

K43

Kn,n–1

K34

Kn–1,n

K23

K12 K10

fn(t)

K30

K20

Kn0

FIGURE 7.27 A catenary n-compartment model. f1(t) K10

f2(t) K12

fn–1(t) K23

Kn–2,n–1

qn–1

K20 Kn1

Kn–1,n

fn–1(t)

Kn0

FIGURE 7.28

A unilateral n-compartment model.

Kn–1,0

7.8 MULTICOMPARTMENT MODELING

421

The matrix A, given in Eq. (7.106), has nonzero elements defined as aii ¼ ðKi0 þ Ki, iþ1 Þ,

1in1

ann ¼ ðKn0 þ Kn1 Þ

ð7:110Þ

a1n ¼ Kn1 ai, i1 ¼ Ki1, i ,

2in

In Section 7.7.2, we investigated the three-compartment unilateral complex roots case and determined that in the roots with the most oscillatory behavior, all the transfer rates were equal to the same value. We will continue this investigation to explore the oscillatory behavior for a closed system unilateral n-compartment model with equal transfer rates, K, and bolus input. The system matrix A is

and the determinant of (DI-A) is

As Godfrey illustrates, the roots of this system are K þ K

cos

2pm 2pm þ j sin , n n

m ¼ 1, 2, n

ð7:111Þ

and lie evenly on the unit circle of radius K, centered at (K,0) in the complex plan. For a closed system, one of the roots is 0, and for an even n, there is another root at –2K. The remaining roots are complex and given by Eq. (7.111). For m ¼ 1 and m ¼ n1, Godfrey shows that the damping ratio is equal to z ¼ sin

p n

ð7:112Þ

and as n increases to infinity, the damping ratio approaches zero. Since the quantity within a compartment can never be less than zero, as n approaches infinity, the amplitude of the sinusoid approaches 0. Our approach to solving a unilateral n-compartment model is the same as before, letting MATLAB do the work for us, as shown in the following example.

422

7. COMPARTMENTAL MODELING

EXAMPLE PROBLEM 7.17 Consider the unilateral five-compartment model with no loss of solute to the environment from any compartments and an input for compartment 3 only. Additionally, all transfer rates equal 2 and f3(t) ¼ 5d(t). Assume that the initial conditions are zero. Solve for the quantity in compartment 3.

¼ 2q1 þ 2q5 ¼ 2q1 2q2 ¼ 2q2 2q3 ¼ 2q3 2q4 ¼ 2q4 2q5

Using MATLAB, we have >> syms D >> A¼[-2 0 0 0 2;2 -2 0 0 0;0 2 -2 0 0;0 0 2 -2 0;0 0 0 2 -2]; >> det(D*eye(5)-A) ans ¼ D^5þ10*D^4þ40*D^3þ80*D^2þ80*D

and d5 q 3 d4 q3 d3 q3 d2 q3 dq3 ¼0 þ 10 4 þ 40 3 þ 80 2 þ 80 5 dt dt dt dt dt The roots from the characteristic equation (i.e., eig(A)) are 0, 3.6180 j1.1756, and 1.3820 j1.9021. The complete solution for q3 is the natural solution, since the forced response is zero, and is given by q3 ¼ B1 þ e3:618t ðB2 cos 1:1756t þ B3 sin 1:1756tÞ þ e1:382t ðB4 cos 1:9021t þ B5 sin 1:9021tÞ The initial conditions for q3 are found using the conservation of mass equations and successive derivatives, giving q3 ð0Þ ¼ 5, q_ 3 ð0Þ ¼ 10, €q3 ð0Þ ¼ 20, ___ q3 ð0Þ ¼ 40, and ____ q3 ð0Þ ¼ 80: Solving for the unknown coefficients using the initial conditions, we have q3 ð0Þ ¼ 5 ¼ B1 þ B2 þ B4 q_ 3 ð0Þ ¼ 10 ¼ 3:61B2 þ 1:1756B3 1:382B4 þ 1:9021B5 €q3 ð0Þ ¼ 20 ¼ 11:72B2 8:51B3 1:71B4 5:26B5 ___ q3 ð0Þ ¼ 40 ¼ 32:4B2 þ 44:58B3 þ 7:64B4 þ 4:02B5 ____ q3 ð0Þ ¼ 80 ¼ 64:79B2 199:4B3 18:18B4 þ 9:02B5 and using MATLAB yields B1 ¼ 1.0, B2 ¼ 2.0, B3 ¼ 0.0, B4 ¼ 2.00, and B5 ¼ 0.0. Thus, for t0, we have q3 ¼ 1 þ 2e3:618t cos 1:1756t þ 2:00e1:382t cos 1:9021t

423

7.8 MULTICOMPARTMENT MODELING

The same approach can be used to find the quantities in the other compartments, which are plotted in Figure 7.29. Note that the oscillation about the steady state of 1 is more pronounced in q4 than q3. In fact, the prominence of oscillation about steady state decreases as we move from q4 to q5 to q1 to q2 to q3. In general, we see the most prominent oscillation in the compartment that receives the solute from the compartment stimulated by the bolus. Also note that the steady-state quantity in each compartment equals one. 1.5

1.5

0.5

0.5 0

0 0

2.00

1.50

Time

–0.5

Time

1.00

2 0.50

0.00

0 0

Time

2 Time

1.50

1.00 0.50 0.00 0

Time

FIGURE 7.29 Plots of the responses for Example Problem 7.17.

As we increase the number of compartments in a closed unilateral system, in general, the more oscillatory the response becomes. In some cases, the oscillatory behavior is prominent but lasts only a very short time. In other cases, the oscillations may be less prominent but last for a longer period of time. Consider the closed system unilateral model shown in

7. COMPARTMENTAL MODELING

1.5

424

1 0.5

0.5

0 0

10 Time

20 Time

(b)

1.5

(a)

1 0.5

0 0

(c)

20 Time

0 (d)

100

150

Time

FIGURE 7.30 Plots of the response for q4 for the closed unilateral system shown in Figure 7.28 for (a) 10, (b) 20, (c) 40, and (d) 80 compartments. A bolus input of 5 to compartment 3 is the only input.

Figure 7.28 (with zero initial conditions and equal transfer rates, K ¼ 2) with input f3(t) ¼ 5d(t). The response for a model of 10, 20, 40, and 80 compartments is shown in Figure 7.30 (note the time scale changes for each model). As shown, the number of oscillations increases as the number of compartments increases, and the steady-state value decreases. The time it takes for the solute to move through the system also increases as the number of compartments increases. For 20 compartments and higher, there is essentially no solute left in the compartment after the initial oscillation until the solute flows through the system. If the system is open and solute is allowed to move into the environment, the oscillatory behavior is reduced. Consider the model used in Figure 7.28, with the exception that solute output to the environment is allowed in compartment 4, with a transfer rate of K40 ¼ 0.2 (10 percent of the transfer rate among the compartments). Shown in Figure 7.31 is the response for q4 with 40 and 80 compartments. An oscillatory response is still noted with fewer prominent oscillations as compared with no output to the environment. Also note that the peak oscillation for the first is much smaller than before. Finally, observe that the steady-state value is now zero.

425

7.8 MULTICOMPARTMENT MODELING

0.8

0.6

0.4

0.2

0 0

40 Time

(a)

100

(b)

150

Time

FIGURE 7.31 Plots of the response for q4 for the open unilateral system shown in Figure 7.27 for (a) 40 and (b) 80 compartments. The model is identical to the one shown in Figure 7.29, except K40 ¼ 0.2.

7.8.4 General Multicompartment Model Although the previous models presented in this section are important, many systems are more complex and follow the form of a general multicompartment model. Some systems are composed of subsystems, described in Sections 7.8.1–7.8.3, which are linked together with transference among subunits. For example, a model13 that describes thyroid hormone distribution and metabolism using two mammillary three-compartment models linked together is shown in Figure 7.32. Mammillary compartments 1–3 describe T3, and mammillary compartments 4–6 describe T4; compartments 1 and 4 are the same space, as are 2 and 5, and 3 and 6. The plasma is represented by compartments 2 and 5, compartments 1 and 4 represent the fast tissue (liver, kidneys, lung, heart, and gut), and compartments 3 and 6 represent the slow tissue (muscle, skin, and brain). Fast and slow indicate how quickly the hormones are synthesized via transfer rates Ki0. Transfer rates K41 and K63 are used to f2(t) K10

K12

K21

K23 K32

K41 K40

K30

K63 K45

K54

K56

K60

K65 f5(t)

FIGURE 7.32 13

Six-compartment model that describes thyroid hormone distribution and metabolism.

See DiStefano and Mori, 1977, in references for original problem development.

426

7. COMPARTMENTAL MODELING

model the transformation of T4 into T3. Input to the system is through the plasma in either compartment q2 or q5. Consider the movement of a drug in the body with the pharmacokinetic model in Figure 7.10. After ingestion, the drug moves into the blood, where it is distributed in the plasma. Drug distribution in the plasma is among water and proteins. Since drugs are relatively small molecules, they easily move through the capillaries and into most fluids and organs of the body. In addition, drugs move easily into the intracellular fluids of body tissues. Each arrow in Figure 7.10 needs to be defined with a transfer rate. Obtaining values for the transfer rates is usually very difficult or even impossible.

EXAMPLE PROBLEM 7.18 Consider the model14 illustrated in Figure 7.33 for the oral input of the thyroid hormone replacement therapy in which the body does not produce any thyroid hormone. While the model is appropriate for either T3 or T4, here we track T3 and ignore T4 for simplicity. Assume T3 exists in compartment 1 (gut) in solid form and in compartment 2 (still the gut) in liquid form. Compartment 3 represents the plasma, and compartments 4 and 5 are the slow and fast tissues, respectively. Assume that the input is bolus, f1(t) ¼ 25d(t), and that the initial conditions are zero. Further, K12 ¼ 1.1, K20 ¼ 0.01, K23 ¼ 0.9, K34 ¼ 15, K43 ¼ 30, K40 ¼ 1.0, K35 ¼ 0.5, K53 ¼ 0.4, and K50 ¼ 0.05. Note that K40 and K50 represent T3 metabolism. Solve for the quantity in compartment 3.

Solution The input is transformed into a new initial condition, q1(0). The conservation of mass for each compartment yields q_ 1 ¼ K12 q1 ¼ 1:1q1

ð7:113Þ

q_ 2 ¼ K12 q1 ðK20 þ K23 Þq2 ¼ 1:1q1 0:91q2

ð7:114Þ

q_ 3 ¼ K23 q2 ðK34 þ K35 Þq3 þ K43 q4 þ K53 q5 ¼ 0:9q2 15:5q3 þ 30q4 þ 0:4q5

ð7:115Þ

q_ 4 ¼ K34 q3 ðK40 þ K43 Þq4 ¼ 15q3 31q4

ð7:116Þ

q_ 5 ¼ K35 q3 ðK50 þ K53 Þq5 ¼ 0:5q3 0:45q5

ð7:117Þ

f1(t)

K34 q1

K12

K23

K40

K43 K35 K53

K20

K50

FIGURE 7.33 Illustration for Example Problem 7.18. Compartments 1 and 2 are the digestive system, compartment 3 is the plasma, and compartments 4 and 5 are the fast and slow tissues, respectively.

See DiStefano and Mak, 1979, in references for original problem development.

7.8 MULTICOMPARTMENT MODELING

427

Since the conservation of mass equation for q1 involves only q1, it is easily solved as q1 ¼ 25e1:1t uðtÞ: Substituting the solution for q1 into Eq. (7.114), gives q_ 2 ¼ 27:5e1:1t 0:91q2

ð7:118Þ

Equation (7.118) involves an input and only q2, which is solved independently as q2 ¼

K12 q1 ð0Þ eðK20 þK23 Þt eK12 t ¼ 144:74 e0:91t e1:1t uðtÞ K12 ðK20 þ K23 Þ

ð7:119Þ

The solution for q2 is substituted into Eq. (7.115), yielding K12 q1 ð0Þ eðK20 þK23 Þt eK12 t 15:5q3 þ 30q4 þ 0:4q5 K12 ðK20 þ K23 Þ ¼ 130:27 e0:91t e1:1t 15:5q3 þ 30q4 þ 0:4q5

q_ 3 ¼ K23

ð7:120Þ

Equations (7.120), (7.116), and (7.117) are solved using MATLAB and the D-Operator, as follows: >> >> >> >>

syms D A¼[-15.5 30 0.4;15 -31 0;0.5 0 -0.45]; det(D*eye(3)-A) adj¼det(D*eye(3)-A)*inv(D*eye(3)-A)

which gives the reconstructed differential equations for q3 as ___ q3 þ 46:95€q3 þ 51:225_q3 þ 7:525q3 ¼ 2531:8e1:1t 1803:12e0:91t

ð7:121Þ

with roots 45.84, 0.94, and 0.18. The natural response is q3n ¼ B1 e45:84t þ B2 e0:94t þ B3 e0:18t The forced response is q3f ¼ B4 e1:1t þ B5 e0:91t , which after substituting into Eq. (7.113) gives B4 ¼ 380:39 and B5 ¼ 1870:41 The complete response is then q3 ¼ B1 e45:84t þ B2 e0:94t þ B3 e0:18t þ 380:39e1:1t þ 1870:41e0:91t We use the initial conditions, q3 ð0Þ ¼ 0, q_ 3 ð0Þ ¼ 0, and €q3 ð0Þ ¼ 24:75, to solve for B1, B2, and B3 as follows: q3 ð0Þ ¼ 0 ¼ B1 þ B2 þ B3 þ 380:39 þ 1870:41 and with q_ 3 ¼ 45:84B1 e45:84t 0:94B2 e0:94t 0:18B3 e0:18t 418:39e1:1t 1701:97e0:91t we have q_ 3 ð0Þ ¼ 0 ¼ 45:84B1 0:94B2 0:18B3 418:39 1701:97 Continued

428

7. COMPARTMENTAL MODELING

which with €q3 ¼ 2101B1 e45:84t þ 0:88B2 e0:94t þ 0:031B3 e0:18t þ 460:23e1:1t þ 1548:80e0:91t gives €q3 ð0Þ ¼ 24:75 ¼ 2101B1 þ 0:88B2 þ 0:031B3 þ 460:23 þ 1548:80 To solve for the unknown constants, we evaluate 2

6 4 45:84 2101

0:94 0:88

2250:7

7 7 6 76 0:18 54 B2 5 ¼ 4 2120:3 5 0:031

1984:3

which gives 2

:004

7 6B 7 6 4 2 5 ¼ 4 2259:51 5 B3

8:844

Thus, q3 ¼ 0:004e45:84t 2259:51e0:94t þ 8:84e0:18t þ 380:39e1:1t þ 1870:41e0:91t uðtÞ

ð7:122Þ

which is plotted in Figure 7.34. Note that if the oral dose involved T4 instead of T3, the model would need to be changed by adding three more compartments for T4 (lower part of Figure 7.31). We still need the three T3 compartments, since T4 transforms into T3. Another way to solve for the response is to directly analyze the system using the D-Operator matrix approach on Eqs. (7.113)–(7.117), which appears easier, since the input is 0 and there is no forced response. However, considerable additional work is required to calculate the two extra initial conditions (___ q1 ð0Þ and ____ q1 ð0Þ) needed to solve for the extra two terms in the natural response, which is not trivial. Thus, we have 6 5

4 3 2 1 0 0

15 Time

FIGURE 7.34 Illustration of the quantity in compartment 3 in Example Problem 7.18.

429

7.8 MULTICOMPARTMENT MODELING

>> syms D >> A¼[-1.1 0 0 0 0;1.1 -.91 0 0 0;0.9 -15 .5 30 .4;0 0 15 -31 0;0 0.5 0 -.45]; >> det(D*eye(5)-A) ans ¼ D^5 þ (1224*D^4)/25 þ (293191*D^3)/2000 þ (787421*D^2)/5000 þ (2656059*D)/40000 þ 301301/40000 >> eig(A) ans ¼ 0.1748 0.9392 45.8360 0.9100 1.1000

The reconstructed differential equation for q3 is _____ q3 þ 48:92____ q3 þ 146:6___ q3 þ 157:5€q3 þ 66:4_q3 þ 7:53q3 ¼ 0 From the roots, the response is written as q3 ¼ B1 e0:18t þ B2 e0:94t þ B3 e45:84t þ B4 e0:91t þ B5 e1:1t

ð7:123Þ

which is the same form as Eq. (7.122). To calculate B1 through B5, we use the initial conditions, q1(0) ¼ 25, q2(0) ¼ 0, q3(0) ¼ 0, q4(0) ¼ 0, and q5(0) ¼ 0, to find, after considerable effort, that q3 ð0Þ ¼ 0, q_ 3 ð0Þ ¼ 0, €q3 ð0Þ ¼ 24:75, ___ q3 ð0Þ ¼ 433:3752, and ____ q1 ð0Þ ¼ 17, 935. Using the initial conditions and Eq. (7.123), we have q3 ð0Þ ¼ 0 ¼ B1 þ B2 þ B3 þ B4 þ B5 q_ 3 ð0Þ ¼ 0 ¼ 0:18B1 0:94B2 45:84B3 0:91B4 1:1B5 €q3 ð0Þ ¼ 24:75 ¼ ð0:18Þ2 B1 þ ð0:94Þ2 B2 þ ð45:84Þ2 B3 þ ð0:91Þ2 B4 þ ð1:1Þ2 B5 ___ q3 ð0Þ ¼ 433:3725 ¼ ð0:18Þ3 B1 þ ð0:94Þ3 B2 þ ð45:84Þ3 B3 þ ð0:91Þ3 B4 þ ð1:1Þ3 B5 ___ q3 ð0Þ ¼ 17, 935 ¼ ð0:18Þ4 B1 þ ð0:94Þ4 B2 þ ð45:84Þ4 B3 þ ð0:91Þ4 B4 þ ð1:1Þ4 B5 To solve for the unknown constants, we evaluate the unknown coefficients using MatLab: 3 32 3 2 2 B1 0 1 1 1 1 1 7 76 7 6 6 0 45:84 0:91 1:1 76 B2 7 6 7 6 0:18 0:94 7 76 7 6 6 76 B3 7 ¼ 6 24:75 7 6 0:031 0:88 2101 0:83 1:21 7 76 7 6 6 7 76 7 6 6 96, 298 :75 1:33 54 B4 5 4 433:3725 5 4 0:005 0:83 17, 934 0:0009 0:78 4, 4139, 948 0:69 1:46 B5 Continued

430

7. COMPARTMENTAL MODELING

which gives

3 3 2 8:84 B1 6 B2 7 6 2259:15 7 7 6 7 6 6 B3 7 ¼ 6 0:004 7 7 6 7 6 4 B4 5 4 1870 5 380:3 B5 2

and q3 ¼ 0:004e45:84t 2259:15e0:94t þ 8:84e0:18t þ 380:3e1:1t þ 1870e0:91t uðtÞ

7.9 EXERCISES 1. Determine the number of Naþ and Kþ ions inside a cell with a volume of 1 nL and concentrations given in Table 7.2. 2. Suppose the concentrations of Naþ , Cl , and Kþ are 20, 52, and 158 mM/L, respectively. Determine the number of ions in a cell of 2 nL. 3. A cell with a volume of 1.5 nL contains 2 1014 molecules of Kþ and 1:5 1013 molecules of Naþ. What is the concentration for each ion? 4. Two compartments, with volumes V1 and V2, are separated by a thin membrane, and solute moves from one compartment to the other by diffusion. If an amount z of solute is dumped into compartment 1 at t ¼ 0, then find the concentration in each compartment. 5. Two compartments, with volumes V1 and V2, are separated by a thin membrane, and solute moves from one compartment to the other by diffusion. If an amount z of solute is dumped into compartment 2 at t ¼ 0, then find the concentration in each compartment. 6. Two compartments, with equal volumes of 0:0572 cm3 , are separated by a thin membrane, and solute moves from one compartment to the other by diffusion. If all of the solute is initially dumped into one compartment, then find the transfer rate if the time constant equals 27 103 s1 : 7. A system is given by two compartments separated by a thin membrane, and solute moves from one compartment to the other by diffusion. The volume of compartment 1 is 0:1cm3 and compartment 2 is 0:3 cm3 . The transfer rate is 2:0 103 s1 . Suppose 3 g of solute is dumped into compartment 2. Solve for the concentration in both compartments. 8. A system is given by two compartments separated by a thin membrane, and solute moves from one compartment to the other by diffusion. Suppose the volume of compartment 1 equals 0:0572 cm3 and is twice as large as compartment 2. If 100 g of solute is dumped into compartment 2, then solve for the concentration in both compartments for an arbitrary transfer rate K (where the solution is expressed in terms of K). 9. A system is given by two compartments separated by a thin membrane, and solute moves from one compartment to the other by diffusion. The volume of compartment 1 is 0:0572 cm3 and compartment 2 is 0:0286 cm3 . Suppose 100 M of solute is dumped into compartment 2, M and the concentration response for compartment 2 is c2 ðtÞ ¼ 1165:67 þ 2331:3e0:015t uðtÞ 3 . cm (a) Find the transfer rate. (b) Solve for the concentration in compartment 1.

7.9 EXERCISES

431

10. A system is given by two compartments separated by a thin membrane, and solute moves from one compartment to the other by diffusion. The volume of compartment 1 is 0:03 cm3 and compartment 2 is 0:01 cm3 . Suppose 50 M of solute is dumped into compartment 2, and M the concentration response for compartment 1 is c1 ðtÞ ¼ 1250 1 e0:01t uðtÞ 3 . (a) Find the cm transfer rate. (b) Solve for the concentration in compartment 2. 11. Find the initial osmotic pressure at room temperature for a cell if the only ions present are KCl on either side of the membrane. Assume the concentrations for Kþ and Cl from Table 7.2 and that the ions cannot cross the membrane. The cell volume is 2 nL. Determine the final cell volume. 12. Find the initial osmotic pressure at room temperature for a cell if the only ions present are CaCl2 on either side of the membrane. Assume the concentrations for Caþ2 and Cl from Table 7.2 and that the ions cannot cross the membrane. The cell volume is 2 nL. Determine the final cell volume. 13. Find the initial osmotic pressure at room temperature for a cell if all the ions present are listed in Table 7.2. Assume that the ions cannot cross the membrane. The cell volume is 2 nL. Determine the final cell volume. 14. Find the initial osmotic pressure at room temperature for a cell if the only ions present are KCl and NaCl on either side of the membrane. Assume the concentrations for K þ , Naþ , and Cl from Table 7.2, and that only Kþ can cross the membrane. The cell volume is 2 nL. Describe what happens to the ions. Determine the final cell volume. 15. Find the initial osmotic pressure at room temperature for a cell if the only ions present are KCl and NaCl on either side of the membrane, and 0:2 109 M of a protein inside the cell. Assume the concentrations for Kþ , Naþ , and Cl from Table 7.2, and that only Kþ can cross the membrane. The cell volume is 2 nL. Describe what happens to the ions. Determine the final cell volume. 16. Find the osmolarity and osmotic pressure of 2 mM Na2 SO4 at room temperature. 17. Find the osmolarity and osmotic pressure of a 9% solution of NaCl at room temperature. 18. Find the osmotic pressure at room temperature for a cell if the ions in Table 7.2 are present. 19. Consider a cell with an internal osmolarity of 300 mOsm and volume of 2 nL in a 30 nL solution of 300 mOsm. A 3 nL, 5% NaCl by weight solution is added to the extracellular space. Assuming that NaCl is impermeable and that the moles inside the cell do not change, describe the events that take place until steady state is achieved. What is the final osmolarity of the cell? What is the volume of the cell at steady state? 20. Consider a cell with an internal osmolarity of 300 mOsm and volume of 2 nL in a 30 nL solution of 300 mOsm. Three mM of CaCl2 is added to the extracellular space. Assuming that CaCl2 is impermeable and that the moles inside the cell do not change, describe the events that take place until steady state is achieved. What is the final osmolarity of the cell? What is the volume of the cell at steady state? 21. Consider a cell with an internal osmolarity of 300 mOsm and volume of 2 nL in a 30 nL solution of 300 mOsm. Five mM of urea is added to the extracellular space. Assuming that urea is permeable and that the moles originally inside the cell are impermeable, describe the events that take place until steady state is achieved. Continued

432

7. COMPARTMENTAL MODELING

22. Given the cell described in Figure 7.7 with a ¼ 500 mM, PK ¼ 1:0, and PNa ¼ 0:04, at steady state, plot the relationship between ~ Jp and V: 23. Suppose 500 mg of dye was introduced into the plasma compartment. After reaching steady state, mg . Find the volume of the plasma compartment. the concentration in the blood is 0.0893 cm3 24. Suppose 1 g bolus of solute is injected into a plasma compartment of 3 L. The transfer rate out of the compartment equals 0:7 hr1 . Solve for the solute concentration. What is the half-life of the solute in the plasma compartment? 25. An unknown quantity of radioactive iodine (I131) is instantaneously passed into the plasma. The time dependence of the quantity of I131 in the plasma exhibits an exponential decay from 100 mg with a time constant of 1 day, while the urine shows an exponential rise from zero to 75 mg with a time constant of 1 day. Assuming the compartment model in Example Problem 7.5, determine the transfer rates and the half-life. 26. Suppose a patient ingested a small quantity of radioactive Iodine (I131). A simple model describing the removal of I131 from the bloodstream into the urine and thyroid is given in Example Problem 7.5. (a) Sketch the response of the system. (b) Suppose the thyroid is not functioning and does not take up any I131. Sketch the response of the abnormal system and compare to the result from (a). 27. A radioactive bolus of I131 is injected into a plasma compartment. The time dependence of the mg concentration of I131 in the plasma is c1 ¼ 143e1:6t . The amount of I131 is 10 K mg. 100 mL Assuming the compartmental model in Example Problem 7.5, find (a) the volume of the plasma compartment, (b) K ¼ K1 þ K2 , and (c) the half-life. 28. Find the half-life for the model given in Eq. (7.33) and Figure 7.8. 29. Use SIMULINK to simulate the model given in Eq. (7.33) and Figure 7.8. Use the parameters given in Figure 7.11. 30. Demonstrate for the one-compartment pharmacokinetic model given in Section 7.5.3 with Eq. (7.33) and Figure 7.8 that as g increases, both tmax and q1 ðtmax Þ decrease. 31. An antibiotic is exponentially administered into the body, with f ðtÞ ¼ 75e2t uðtÞ: Assume the model given in Figure 7.8 with K10 ¼ 0:3: (a) Analytically solve for the quantity of the antibiotic in the plasma. (b) Simulate the quantity of the antibiotic in the plasma using SIMULINK. (c) What is the time to maximum concentration, and what is the quantity in the plasma at that time? 32. For the one-compartment repeat dosage in Section 7.5.4, derive Eq. (7.42) from (7.41) and Eq. (7.44) from (7.33). 33. A 2 g bolus of antibiotic is administered to a person with a plasma volume of 3 L. The average impulse response for this drug is shown in Figure 7.35. Assuming a one-compartment model, determine the transfer rate. If the concentration of the drug is not to fall below 10 percent of the initial dosage at steady state, how often does the drug need to be given to maintain this minimum level? 34. A 4 g bolus of antibiotic is administered to a person with a plasma volume of 3 L. The average washout response for this drug in a plasma volume of 3 L is shown in Figure 7.36. Assuming a one-compartment model, determine the transfer rate. If the concentration of the drug is not to fall below 25 percent of the initial dosage at steady state, how often does the drug need to be given to maintain this minimum level?

433

7.9 EXERCISES

1 0.9 Concentration, g/L

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

4 6 Time, hours

FIGURE 7.35 Illustration for Exercise 33. 1 0.9 Concentration, g/L

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

FIGURE 7.36

4 6 Time, hours

Illustration for Exercise 34.

35. Given the two-compartment model in Figure 7.16 and q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ z, solve for the concentration in compartment 2. 36. Given the two-compartment model in Figure 7.16 and q1 ð0Þ ¼ a and q2 ð0Þ ¼ z, solve for the concentration in each compartment. 37. Given the two-compartment model shown in Figure 7.17 with a pulse ingestion of solute in the digestive system and removal of solute via metabolism and excretion in urine, solve for the plasma concentration. Continued

434

7. COMPARTMENTAL MODELING

38. Given the two-compartment model shown in Figure 7.17 with a zdðtÞ þ ð1 zÞðuðtÞ uðt t0 ÞÞ ingestion of solute in the digestive system and removal of solute via metabolism and excretion in urine, solve for the plasma concentration. 39. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0, K10 ¼ 0:5, K21 ¼ 0:3, K20 ¼ 0:9, f1 ðtÞ ¼ 0, and f2 ðtÞ ¼ 5dðtÞ: Assume that the initial conditions are zero. (a) Solve for the quantity in each compartment. (b) Find the maximum amount of solute in compartment 1. 40. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:2, K10 ¼ 0:7, K21 ¼ 0, K20 ¼ 1, f1 ðtÞ ¼ 2uðtÞ, and f2 ðtÞ ¼ 0: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 41. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0, K10 ¼ 0:6, K21 ¼ 0:1, K20 ¼ 0:8, f1 ðtÞ ¼ 0, and f2 ðtÞ ¼ dðtÞ þ 5uðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 42. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:3, K10 ¼ 0:2, K21 ¼ 0, K20 ¼ 0:4, f1 ðtÞ ¼ 4uðtÞ, and f2 ðtÞ ¼ 5dðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 43. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:3, K10 ¼ 0:5, K21 ¼ 0, K20 ¼ 1, f1 ðtÞ ¼ 2uðtÞ, and f2 ðtÞ ¼ 3dðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 44. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0, K10 ¼ 0:6, K21 ¼ 0:1, K20 ¼ 0:5, f1 ðtÞ ¼ 0, and f2 ðtÞ ¼ dðtÞ þ 5uðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 45. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:3, K10 ¼ 0:7, K21 ¼ 0, K20 ¼ 0:1, f1 ðtÞ ¼ 3dðtÞ, and f2 ðtÞ ¼ 0: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 46. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:4, K10 ¼ 1:0, K21 ¼ 0, K20 ¼ 0:3, f1 ðtÞ ¼ 2uðtÞ, and f2 ðtÞ ¼ 0: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 47. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:2, K10 ¼ 0:8, K21 ¼ 0, K20 ¼ 0:3, f1 ðtÞ ¼ dðtÞ þ 2uðtÞ, and f2 ðtÞ ¼ 0: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 48. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:3, K10 ¼ 0:5, K21 ¼ 0:1, K20 ¼ 0:4, f1 ðtÞ ¼ 2uðtÞ, and f2 ðtÞ ¼ 5dðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 49. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:3, K10 ¼ 0:7, K21 ¼ 3, K20 ¼ 1, f1 ðtÞ ¼ 2dðtÞ, and f2 ðtÞ ¼ 5uðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 50. Consider the two-compartment model shown in Figure 7.15 with K12 ¼ 0:2, K10 ¼ 0:6, K21 ¼ 0:1, K20 ¼ 0:5, f1 ðtÞ ¼ 3dðtÞ, and f2 ðtÞ ¼ dðtÞ þ 5uðtÞ: Assume that the initial conditions are zero. Solve for the quantity in each compartment. 51. Suppose 1 g of solute is ingested into the digestive system that has a transfer rate of 1:4 hr1 into the plasma. The plasma compartment is 3 L and has transfer rate of 0:7 hr1 into the environment. (a) Solve for the solute concentration in the plasma. (b) When is the maximum solute concentration observed in the plasma compartment? (c) What is the maximum solute in the plasma compartment?

435

7.9 EXERCISES

52. Consider the model in Figure 7.37. The time dependence of the concentration of a radioactively labeled solute in the plasma is c1 ðtÞ ¼ 143e1:6t þ 57e2:8t

mg 100 mL

after injecting a bolus of 10 g into the plasma. (a) Find the volume of the plasma compartment. mg (b) Find the transfer rates K12 , K21 , and K13 : (c) Suppose the input is changed to 5uðtÞ , 100 mL mg , and solve for c1 ðtÞ and c2 ðtÞ: (d) Suppose the input is changed to 5ðuðtÞ uðt 2ÞÞ 100 mL and solve for c1 ðtÞ and c2 ðtÞ: 53. A unit step input is applied to the compartmental system in Figure 7.38. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 2 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 54. A unit step input is applied to the compartmental system in Figure 7.38. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 2 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. f1(t)

K12 Plasma

Unknown

K21 K10 Urine

FIGURE 7.37

Illustration for Exercise 52.

f1(t)

K12 q1

q2 K21 K20

FIGURE 7.38

Illustration for Exercises 53–68.

Continued

436

7. COMPARTMENTAL MODELING

55. The input to the compartmental system in Figure 7.38 is 2uðtÞ 2uðt 1Þ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 2 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 56. The input to the compartmental system in Figure 7.38 is 2uðt 1Þ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 57. The input to the compartmental system in Figure 7.38 is 2e0:5562t uðtÞ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 58. The input to the compartmental system in Figure 7.38 is 2e0:5562t uðtÞ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 1: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 59. The input to the compartmental system in Figure 7.38 is 2e0:1438t uðtÞ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 60. The input to the compartmental system in Figure 7.38 is 2e0:1438t uðtÞ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 2 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 61. The input to the compartmental system in Figure 7.38 is 2e0:1438t uðtÞ 2e0:1438ðt10Þt uðt 10Þ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 62. The input to the compartmental system in Figure 7.38 is 3et uðtÞ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 2 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. 63. The input to the compartmental system in Figure 7.38 is 3et uðtÞ 3eðt3Þ uðt 3Þ. The transfer rates are K20 ¼ 0:2, K21 ¼ 0:1, and K12 ¼ 0:4: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1;

7.9 EXERCISES

64.

65.

66.

67.

68.

69.

70.

71.

437

(b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. The input to the compartmental system in Figure 7.38 is 0:5e2t uðtÞ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 4: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. The input to the compartmental system in Figure 7.38 is 0:5e2ðt2Þ uðt 2Þ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. The input to the compartmental system in Figure 7.38 is 0:5e2t uðtÞ 0:5e2ðt1:5Þ uðt 1:5Þ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. The input to the compartmental system in Figure 7.38 is 3 cos 4tuðtÞ. The transfer rates are K20 ¼ 0:3, K21 ¼ 1:0, and K12 ¼ 0:6: The initial conditions are q1 ð0Þ ¼ 0 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. The input to the compartmental system in Figure 7.38 is 3 sin 2tuðtÞ. The transfer rates are K20 ¼ 3, K21 ¼ 5, and K12 ¼ 7: The initial conditions are q1 ð0Þ ¼ 1 and q2 ð0Þ ¼ 0: Write a single differential equation involving the input and only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. (e) Using SIMULINK, simulate the system from the original set of differential equations and graph q1 and q2. For the compartmental system in Figure 7.39, a bolus of solute ð f3 ðtÞ ¼ dðtÞÞ is ingested into the digestive system (compartment 3). Assume that the initial conditions are zero. Write a single differential equation involving only variable (a) q1; (b) q2. For t > 0, solve the system for (c) q1; (d) q2. Consider the three-compartment model shown in Figure 7.20 with nonzero parameters and inputs K12 ¼ 0:4, K10 ¼ 0:5, K21 ¼ 0:6, K31 ¼ 0:9, K32 ¼ 0:7, K23 ¼ 0:2, K13 ¼ 0:8, f1 ðtÞ ¼ 3uðtÞ, and f2 ðtÞ ¼ 5dðtÞ: Assume that the initial conditions are zero. Write a single differential equation involving the input and only variable (a) q1; (b) q2; (c) q3. For t > 0, solve the system for (d) q1; (e) q2; (f) q3. (g) Using SIMULINK, simulate the system from the original set of differential equations and graph q1, q2, and q3. Consider the three-compartment model shown in Figure 7.20 with nonzero parameters and inputs K12 ¼ 0:5, K10 ¼ 0:3, K21 ¼ 0:6, K31 ¼ 0:9, K32 ¼ 0:7, K23 ¼ 0:2, K13 ¼ 0:8, and f3 ðtÞ ¼ 3dðtÞ: Assume that the initial conditions are zero. Write a single differential equation involving the input and only variable (a) q1; (b) q2; (c) q3. For t > 0, solve the system for (d) q1; (e) q2; (f) q3. (g) Using SIMULINK, simulate the system from the original set of differential equations and graph q1, q2, and q3. Continued

438

7. COMPARTMENTAL MODELING

f1(t)

K31 K12 q1

Suggested Readings

508

8.6 467

Diffusion, Biochemical Reactions, and Enzyme Kinetics 473

8.7

AT THE CONCLUSION OF THIS CHAPTER, STUDENTS WILL BE ABLE TO: • Using the law of mass action, quantitatively describe a chemical reaction using differential equations.

• Use the quasi-steady-state approximation to describe metabolism in a compartmental model.

• Using the law of mass action, quantitatively describe enzyme kinetics using differential equations.

• Model biochemical reactions, enzyme kinetics, diffusion, carrier-mediated transport and active transport in a physiological system.

• Describe and calculate the quasi-steady-state approximation for chemical reactions.

• Quantitatively describe the Na-K pump.

• Simulate chemical reactions and enzyme kinetics using SIMULINK.

• Explain and model activators and inhibitors in a chemical reaction.

Introduction to Biomedical Engineering, Third Edition

• Qunatitatively describe cellular respiration.

447

448

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

This chapter focuses on chemical reactions that occur inside and outside a cell following the law of mass action—that is, the rate of accumulation is proportional to the product of the reactants. These chemical reactions support all functions needed to support life and involve such activities as the synthesis of hormones and proteins, muscle contraction, respiration, reproduction, neural signaling, and many other reactions. While the law of mass action is useful, it is not appropriate for all chemical reactions. In some cases, the exact chemical reaction mechanism is not known, and the law of mass action doesn’t work. The law of mass action creates models that are nonlinear and possibly time-varying. In addition, the rate of accumulation is profoundly impacted by temperature, such that an increase in temperature increases the reaction rate. The reactants are assumed to be uniformly distributed in the compartment, the probability of a collision depends on the concentration of the reactants, and such collisions are sufficient to create the products. Since the models here are nonlinear, we typically use SIMULINK to solve these problems. However, we will present analytical solutions to some special nonlinear cases in this chapter. Catalysts are substances that dramatically change reaction rates. These substances are generally present in small amounts and are not consumed in the reaction. Their function is to decrease the amount of time to reach steady state in a chemical reaction. Consider two reactants that spontaneously create a product at room temperature but do so at a very low rate. In the presence of an appropriate catalyst, the speed of the reaction is dramatically increased, and the time to reach steady state is decreased. Enzymes are protein catalysts used in biological reactions that regulate and control most processes in the body. Enzymes increase the reaction rate by thousands or even trillions and are reactant specific. Typically, the enzyme concentration is rather small in relation to the reactant. While the reaction rate is profoundly increased in the presence of an enzyme, the overall energy used to form the product does not change. In this chapter, we first examine simple chemical reactions and then enzyme kinetics. Next, we look at quasi-steady-state approximations, the Michaelis-Menten elimination, and enzyme regulation. We finish by examining important processes, such as the transport of oxygen and carbon dioxide through the circulatory system to the cell, the Na-K pump, and cellular respiration.

8.1 CHEMICAL REACTIONS Consider the following single-stage chemical reaction K

A þ B ! P

ð8:1Þ

in which chemicals A and B react to form the product P, with reaction rate constant K. Equation (8.1) is known as a stoichiometric equation that lists the number of reactants on the left side necessary to form the product on the right side. Conservation of mass requires that the total quantity of reactant A must equal the quantity of A in the reactant and the product, and likewise, the same is true with reactant B. The stoichiometric equation does not describe the dynamics or kinetics of the chemical reaction—that is, the time course of the reaction that may be very fast or very slow. The kinetics of the reaction is written

8.1 CHEMICAL REACTIONS

449

according to the law of mass action. The arrow in Eq. (8.1) shows the direction of the reaction that occurs spontaneously, and the reaction rate constant describes how quickly the reaction occurs. The reaction rate constant is a function of temperature, whereby an increase in temperature generally increases K. For our purposes, temperature is constant and so is K.

8.1.1 Chemical Bonds Individual atoms rarely appear in nature, but they appear within a molecule. Molecules are collections of atoms that are held together by strong chemical bonds in which electron(s) are shared or electron(s) are transferred from one atom to another. Forces between molecules are relatively weak, allowing molecules to act independently of one another. A compound is a molecule that contains at least two different atoms, whereas a molecule can contain just one type of atom. Molecules such as hydrogen, H2 , and oxygen, O2 , are not compounds because they consist of a single type of atom. For simplicity in presentation, we will use the term molecules to describe both molecules and compounds. An atom is an electrically neutral particle that consists of an equal number of protons and electrons. An atom with an unbalanced number of electrons or protons is called an ion, which makes it a positively or negatively charged particle. Examples of positively charged ions, called cations, include Naþ , Kþ , and Caþ2 ; Naþ and Kþ have each lost one electron, and Caþ2 has lost two electrons. An example of a negatively charged ion, called an anion, is Cl , which has gained an extra electron. There are many types of chemical bonds that join atoms and molecules together. They vary in strength, with ionic and covalent bonds displaying the strongest bonds, and the hydrogen bond displaying the weakest bond. Many molecules are composed of positively and negatively charged ions that are bound by an ionic bond. These bonds are extremely strong due to the electrostatic attraction between the oppositely charged ions. Ionic bonds usually involve an atom that has few electrons in the outer shell, with another atom that has an almost complete set of electrons in the outer shell. Ionic bonds involve a transfer of electrons from one atom to another atom. Consider sodium chloride, where sodium has one electron in the outer shell and chloride has seven electrons in its outer shell. The bond is formed by sodium transferring its electron to chloride, resulting in a molecule consisting of Naþ Cl ðusually written as NaClÞ: Consider magnesium and oxygen atoms forming an ionic bond. Magnesium has two electrons in its outer field that are transferred to oxygen, forming Mgþ2 O2 : Covalent bonds involve two atoms sharing an electron pair that increases the stability of the molecule. The atoms do not have to be the same type, but they must be the same electronegativity. For instance, H2 is formed by a covalent bond. Consider carbon dioxide, a molecule that consists of one carbon and two oxygen atoms. These atoms share the electrons, with carbon having four electrons in its outer shell, and oxygen having six electrons in its outer shell. The four electrons in carbon are used in the outer shell of the two oxygen atoms. A hydrogen bond involves the force between a hydrogen atom in one molecule and an electronegative atom in another molecule, typically with oxygen in H2 O, nitrogen in NH3 , and fluorine in HF: Note that the force experienced in a hydrogen bond is not within a molecule but between molecules. The small size of hydrogen allows it to become very close to another molecule with a small atomic radii like the ones previously mentioned.

450

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

An example of a hydrogen bond is found among water molecules. The oxygen in water, which is negatively charged, binds to two positively charged hydrogens from two other water molecules. These bonds connect the water molecules together and give it some unusual properties, such as high surface tension and viscosity. Keep in mind that this bond is a relatively weak bond compared to that holding the water molecule together. As water is heated to the boiling point, all of the hydrogen bonds are dissolved as it becomes a gas. When water freezes, the hydrogen bonds form a structured, less dense orientation. Between freezing and boiling, water is oriented in very dense, random configuration. Hydrogen also plays an an important role within proteins and nucleic acids, which allows bonding within the molecule to achieve different shapes. A chemical reaction in which a molecule loses electrons is called oxidation, and a gain of electrons is called reduction. Some multistage chemical reactions involve both oxidation and reduction. Energy is released when a chemical bond is formed, which is the same amount of energy necessary to break that chemical bond. Thus, chemical bonds store energy. The first step in a chemical reaction requires energy to break the bonds holding the reactant molecules together so they can be rearranged to form an activated complex, an unstable high-energy intermediate. From the activated complex, the molecules join together to form the product and typically release energy. In chemical reactions, if the energy stored in the chemical bonds of the product is less than the total energy stored in the reactants, then net energy is released. The amount of energy required in a chemical reaction is inversely proportional to the reaction rate constant.

8.1.2 Kinetics of a Single-Stage Chemical Reaction To understand the law of mass action, a compartment model visualization of Eq. (8.1) is shown in Figure 8.1, with the understanding that the volumes for the two compartments are identical (i.e., the reaction occurs in a single container of constant volume). The differential equation describing Eq. (8.1) or Figure 8.1 is q_ A ¼ KqA qB

ð8:2Þ

where qA and qB are the quantities of reactants A and B, and K is the transfer rate or reaction rate constant. The transfer rate includes the volume in converting the concentrations to quantities in Eq. (8.2). Note that the right-hand side of Eq. (8.2) involves the product qA qB , instead of a single variable as before. In addition, we could have written Eq. (8.2) in terms of qB , q_ B ¼ KqA qB or in terms of qP , q_ P ¼ KqA qB . Since drawing the compartmental model is not essential in writing the differential equation that describes the kinetics of the

K qAqB

FIGURE 8.1 Compartmental model of the single-stage chemical reaction in Eq. (8.1). Another way to describe this system is to remove one of the reactants in the left box and include it as part of the reaction rate—that is, KqB :

451

8.1 CHEMICAL REACTIONS

chemical reaction, we will write the differential equations directly from the stoichiometric equation. Of course, Eq. (8.2) can be written in terms of concentrations by substituting (concentrationvolume) for quantity. Eq. (8.2) is a nonlinear differential equation. In general, nonlinear equations cannot be solved directly, but they must be simulated using a program like SIMULINK. Keep in mind that there are some special nonlinear differential equations that can be solved, but these are few in number, with a few presented in this chapter. Since q_ A ¼ q_ B , we can integrate Rt Rt both sides of dqA ¼ dqb and have qA qA ð0Þ ¼ qB qB ð0Þ, which, when substituted into 0

Eq. (8.2), gives q_ A ¼ KqA qB ¼ KqA ðqA qA ð0Þ þ qB ð0ÞÞ

ð8:3Þ

Rearranging Eq. (8.3) yields dqA ¼ Kdt qA ðqA qA ð0Þ þ qB ð0ÞÞ

ð8:4Þ

The left-hand side of Eq. (8.4) is rewritten using a technique called partial fraction expansion (details are briefly described here for the unique roots case)1 as 0 1 B1 dqA B2 dqA A @ ¼ Kdt þ ðqA qA ð0Þ þ qB ð0ÞÞ qA 1 0 10 ð8:5Þ 1 dq dq A A A ¼ Kdt @ A@ ðqA qA ð0Þ þ qB ð0ÞÞ qA qA ð0Þ qB ð0Þ where B1 ¼ ðqA qA ð0ÞþqB ð0ÞÞ

1 1 1 ¼ ¼ qA ðqA qA ð0ÞþqB ð0ÞÞ qA ¼qA ð0ÞqB ð0Þ qA qA ¼qA ð0ÞqB ð0Þ qA ð0ÞqB ð0Þ

In general, if 1 1 ¼ sn þ an1 sn1 þ a1 s þ a0 ðs s1 Þ ðs sn Þ

and the roots are real, then 1 B1 Bn þ ¼ ðs s1 Þ ðs sn Þ s s1 s sn where Bi ¼ ðs si Þ

1 ðs s1 Þ ðs sn Þ

s¼si

It should be clear when evaluating the coefficient Bi in the previous equation that the factor ðs si Þ always cancels with the same term in the denominator before calculating Bi :

452

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

1 1 1 B2 ¼ qA ¼ ¼ ðqA qA ð0ÞþqB ð0ÞÞ qA ¼0 qA ðqA qA ð0ÞþqB ð0ÞÞ qA ¼0 qA ð0ÞqB ð0Þ

Integrating both sides of Eq. (8.5) gives

1 qA ð0Þ qB ð0Þ

qA qA ð0Þ þ qB ð0Þ qA ln ln ¼ Kt qA ð0Þ qB ð0Þ

ð8:6Þ

and after algebraically manipulating, we have qA ¼

ðqA ð0Þ qB ð0ÞÞ qB ð0ÞeKðqA ð0ÞqB ð0ÞÞt 1 qA ð0Þ

ð8:7Þ

One final note regarding the solution of Eq. (8.2) is to consider the case in which qB ð0Þ qA ð0Þ. Here, the change in qB is small compared to qB ð0Þ, and qB can be treated essentially as a constant—that is, qB ¼ qB ð0Þ. Thus, an approximation to Eq. (8.2) is q_ A ¼ KqB qA ¼ KqB ð0ÞqA

ð8:8Þ

which can be straightforwardly solved as qA ¼ qA ð0ÞeKqB ð0Þt uðtÞ

ð8:9Þ

Substituting Eq. (8.9) into q_ P ¼ KqB qA ¼ KqB ð0ÞqA ¼ KqB ð0ÞqA ð0ÞeKqB ð0Þt and integrating gives ð8:10Þ qP ¼ qA ð0Þ 1 eKqB ð0Þt uðtÞ It should be clear that the steady-state value of qP equals qA ð0Þ: Let us take another look at the solution in Eq. (8.7) for qA , and compare it to that of Eq. (8.9) by assuming that qB ð0Þ qA ð0Þ, which gives the approximation qB ¼ qB ð0Þ, and thus qA qA ð0Þ þ qB ð0Þ qB ð0Þ ¼ ln ¼ 0, and qA qA ð0Þ ¼ qB qB ð0Þ ¼ 0. Hence, we have ln qB ð0Þ qB ð0Þ therefore Eq. (8.7) becomes ð8:11Þ Rearranging and taking the exponential of both sides gives qA ¼ qA ð0ÞeKqB ð0Þt uðtÞ

ð8:12Þ

which is the result we found in Eq. (8.9). In terms of the body, many small molecules like glucose and other nutrients are available in large quantities as compared with other reactants. Thus, this simplification is often quite appropriate.

453

8.1 CHEMICAL REACTIONS

8.1.3 Single-Stage Reversible Chemical Reaction Next, consider the single-stage reversible chemical reaction as given in Eq. (8.13): ð8:13Þ Here, the chemicals A and B react to form the product P, and P has a reverse reaction to form A and B. The law of mass action describing this system is given by q_ P ¼ K1 qA qB K1 qP q_ A ¼ K1 qA qB þ K1 qP q_ B ¼ K1 qA qB þ K1 qP

ð8:14Þ

Equation (8.14) is nonlinear, which can be solved using SIMULINK or also mathematically, as shown in the following example.

EXAMPLE PROBLEM 8.1 Consider the reaction given in Eq. (8.14), with qA ð0Þ ¼ 10, qB ð0Þ ¼ 15, and qP ð0Þ ¼ 0, K1 ¼ 2, and K1 ¼ 3: Solve for qP :

Solution From Eq. (8.14), it is clear that q_ P ¼ _qA ¼ _qB and after integrating, we have qP qP ð0Þ ¼ qA ð0Þ qA ¼ qB ð0Þ qB

ð8:15Þ

To solve for qP , we eliminate qa and qB using Eq. (8.15) by substituting qA ¼ qA ð0Þ þ qP ð0Þ qP and qB ¼ qB ð0Þ þ qP ð0Þ qP into Eq. (8.14), giving q_ P ¼ K1 qA qB K1 qP ¼ K1 ðqP ð0Þ þ qA ð0Þ qP ÞðqP ð0Þ þ qB ð0Þ qP Þ K1 qP

ð8:16Þ

Since qP ð0Þ ¼ 0, we have q_ P ¼ K1 ðqA ð0Þ qP ÞðqB ð0Þ qP Þ K1 qP ¼ 2ð10 qP Þð15 qP Þ 3qP 0 1 53 ¼ 2@q2P qP þ 150A 2

ð8:17Þ

and after rearranging terms dqP dqP ¼ ¼ 2dt 53 ð q 18:3 ÞðqP 8:2Þ P q2P qP þ 150 2 Once again, partial fraction expansion is used to rewrite Eq. (8.18) as dqP dqP 0:0989 ¼ 2dt ðqP 18:3Þ ðqP 8:2Þ

ð8:18Þ

ð8:19Þ

Continued

454 where

and

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

B1 ¼ ðqP 18:3Þ

B2 ¼ ðqP 8:2Þ

1 1 ¼ ¼ 0:0989 ðqP 18:3ÞðqP 8:2Þ qP ¼18:3 ðqP 8:2Þ qP ¼18:3

1 1 ¼ ¼ 0:0989 ðqP 18:3ÞðqP 8:2Þ qP ¼8:2 ðqP 18:3Þ qP ¼8:2

Integrating Eq. (8.19) and rearranging yields ðqP 8:2Þ ln 2:2 ¼ 20:2t ðqP 18:3Þ and after solving for qP , we have

8:2 1 e20:2t uðtÞ 1 0:45e20:2t The solution for the reactants are found from qA ¼ qA ð0Þ þ qP ð0Þ qP and qB ¼ qB ð0Þþ qP ð0Þ qP , yielding 1:8059 þ 3:7e20:2t qA ¼ uðtÞ 1 0:45e20:2t and 6:8059 þ 1:4798e20:2t uðtÞ qB ¼ 1 0:4476e20:2t qP ¼

ð8:20Þ

These results are shown in Figure 8.2. Note that conservation of mass is still maintained, since qP contains both qA and qB , so qA þ qB þ 2 qP ¼ 25:

16 14

qA, qB, qP

10 8

6 4 2

0 0

0.05

0.1

0.15

0.2

0.25

Time

FIGURE 8.2

Illustration of the amount of product, qP , in Example 8.1.

455

8.1 CHEMICAL REACTIONS

In general, it can be shown that the solution for qP in Eq. (8.14) is 0 1

K1 ðabÞt Bb 1 e C qP ¼ @ AuðtÞ b K1 ðabÞt 1 e a where

ð8:21Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s K2 2 l¼ ðqA ð0Þ þ qB ð0Þ þ 2qP ð0ÞÞ þ 4ðqA ð0Þ þ qP ð0ÞÞðqB ð0Þ þ qP ð0ÞÞ K1 qA ð0Þ þ qB ð0Þ þ 2qP ð0Þ þ a¼

K2 þl K1

2 qA ð0Þ þ qB ð0Þ þ 2qP ð0Þ þ

b¼

K2 l K1

Next, consider the case in which qB ð0Þ qA ð0Þ, where the change in qB is small and can be treated as a constant qB ð0Þ: Thus, an approximation for Eq. (8.14) is q_ P ¼ K1 qB ð0ÞqA K1 qP

ð8:22Þ

Next, we remove qA in Eq. (8.22) by letting qA ¼ qP ð0Þ þ qA ð0Þ qP , giving q_ P ¼ K1 qB ð0ÞðqA ð0Þ þ qP ð0ÞÞ ðK1 qB ð0Þ þ K1 ÞqP which can be straightforwardly solved as ðK1 qP ð0Þ K1 qB ð0ÞqA ð0ÞÞ ðK1 qB ð0ÞþK1 Þt ðK1 qB ð0ÞðqA ð0Þ þ qP ð0ÞÞÞ þ e uðtÞ qP ¼ ðK1 qB ð0Þ þ K1 Þ ðK1 qB ð0Þ þ K1 Þ

ð8:23Þ

ð8:24Þ

8.1.4 Higher-Order Chemical Reactions and Sequential Reactions Consider the higher-order chemical reaction in Eq. (8.25): ð8:25Þ in which aA of reactant A and bB of reactant B form product P, and P has a reverse reaction to form aA þ bB. The law of mass action for Eq. (8.25) is q_ P ¼ K1 qaA qbB K1 qP q_ A ¼ K1 qaA qbB þ K1 qP q_ B ¼

K1 qaA qbB

þ K1 qP

ð8:26Þ

456

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Note that when there are a molecules of A reacting, the qA term in Eq. (8.26) is raised to the a power, and similarly, when b molecules of B react, then the qB term in Eq. (8.26) is raised to the b power. In problems like this, we resort to using SIMULINK for the solution. Now consider the sequential reaction in Eq. (8.27) K1

A ! B ! P

ð8:27Þ

in which reactant A produces reactant B, which then produces reactant P. The law of mass action for Eq. (8.27) is q_ A ¼ K1 qA q_ B ¼ K1 qA K2 qB q_ P ¼ K2 qB

ð8:28Þ

Equation (8.28) is easily solved, assuming nonzero initial condition for reactant A, qA ð0Þ, and zero for the other two reactants, as qA ¼ qA ð0ÞeK1 t uðtÞ qB ¼

K1 qA ð0Þ K1 t e eK2 t uðtÞ K 2 K1

0 1 K1 K2 qA ð0Þ @1 eK1 t 1 eK2 t A uðtÞ qP ¼ K1 K2 K2 K1

ð8:29Þ

Sequential reactions are typical of those occurring in the body and can be quite complex, as described in Sections 8.4–8.6. As we will see, enzymes play a role in sequential reactions, and reactants can be of higher orders. Some sequential reactions involve many sequences in which some branch backward to previous reactions. When there are profound differences in the transfer rates in a sequential reaction, simplifications occur that make analysis much easier. For instance, suppose K2 K1 . Here, the second reaction is much faster than the first reaction, and it appears that reactant A immediately produces reactant P. The term rate limiting is often used to describe this behavior— that is, the first reaction slows the creation of the product. For a rate limiting reaction, an approximation to Eq. (8.29) for qB and qP is to eliminate the eK2 t term, since it goes to zero almost immediately, as compared to the eK1 t term, giving

ð8:30Þ

8.1.5 Quasi-Steady-State Another way to look at a rate limiting reaction is to assume that reactant B is in a quasisteady-state mode—that is q_ B ¼ 0. From q_ B ¼ 0 and Eq. (8.28), we have that 0 ¼ K1 qA K2 qB ,

457

8.1 CHEMICAL REACTIONS

and therefore qB ¼ KK12 qA : Since q_ P ¼ K1 qB , we eliminate qB by substituting qB ¼ KK12 qA , which

gives q_ P ¼ K1 qA and qP ¼ qA ð0Þ 1 eK1 t : While quasi-steady-state assumes that reactant B is immediately in steady state and reac

tant A creates product P directly, there is a period of time,

, in which qB moves from 0 to K1 K1 K2 qA : Note also that steady state for reactant B is quite small and equals K2 qA : 5 K2

Figure 8.3 illustrates the approximation in Eq. (8.30), with the true solution for Eq. (8.29) given with qA ð0Þ ¼ 10, qB ð0Þ ¼ 0 and qP ð0Þ ¼ 0, K1 ¼ 2, and K2 ¼ 500: For qB , the approximation is quite accurate after it reaches quasi-steady-state, t ¼ K52 ¼ 0:01: For qP , the approximation is quite accurate for the entire duration. Now suppose K1 K2 : By a similar rational, the second reaction is now slower as compared to the first reaction and is rate limiting. For the rate limiting second reaction, an approximation to Eq. (8.29) for qB and qP is to eliminate the eK1 t term, since it goes to zero almost immediately, as compared to the eK2 t term, giving

ð8:31Þ

Here, reactant A disappears almost immediately and reactant B increases almost immediately to qA ð0Þ: Another way to look at this rate limiting reaction is to assume that reactant A is in a quasi-steady-state mode—that is q_ A ¼ 0. From q_ A ¼ 0 and Eq. (8.28), we have qA ¼ 0 and

0.05

10 qB (Approx.)

0.04

0.03

6 qB (True)

0.02

0.01

0 0

0.05

0.1 Time

0.15

0.2

Time

FIGURE 8.3 A rate limiting sequential reaction for K1 . Note that both the approximation and true solution for qP are drawn in the figure on the right.

458 qB (Approx.)

6 qB (True)

6 qP

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

2 0

0 0

0.04

0.08

0.12 Time

0.16

0.2

Time

A rate limiting sequential reaction for K2 . Note that both the approximation and true solution for qP are drawn in the figure on the right.

FIGURE 8.4

q_ B ¼ K2 qB , which has the solution q B ¼ qA ð0ÞeK2 t : Next, q_ P ¼ K2 qB ¼ qA ð0ÞK2 eK2 t , which has the solution qP ¼ qA ð0Þ 1 eK2 t : While quasi-steady-state assumes that reactant A is immediately in steady state at time 0 and reactant B rises to qA ð0Þ immediately, there is a period of time, K51 , in which qA falls to 0 and qB rises. Figure 8.4 illustrates the approximation in Eq. (8.31), with the true solution in Eq. (8.29) given with qA ð0Þ ¼ 10, qB ð0Þ ¼ 0 and qP ð0Þ ¼ 0, K1 ¼ 500, and K2 ¼ 2: For qB , the approximation is quite accurate after it reaches quasi-steady-state, t ¼ K51 ¼ 0:01: For qP , the approximation is quite accurate the entire duration.

8.2 ENZYME KINETICS As described earlier, catalysts are substances that accelerate reactions but are not consumed or changed by the reaction. Most chemical reactions in the body can occur without the presence of catalysts, but they occur at a very low rate. An enzyme is a large protein that catalyzes biochemical reactions in the body. These reactions convert a reactant, now called a substrate because it involves an enzyme, into a product by lowering the free energy of activation. Enzymes can increase the rate of the reaction by an order of thousands to trillions. Each enzyme is highly specific and only allows a particular substrate to bind to its active site. Many enzymes are used in the control and regulation of functions of the body. In general, an enzyme reaction involves a series of reactions. Binding the enzyme with the substrate is the first step in creating an intermediate complex, which increases the ability of the substrate to react with other molecules. The next step is when the substrateenzyme complex breaks down to form the free enzyme and product. Both the first step and the second step are reversible, but in the second step, the reverse reaction is so small that it is often omitted.

8.2 ENZYME KINETICS

459

The overall rate of the enzyme catalyzed reaction with a single substrate is a function of the amount of enzyme and substrate and is given as Reaction Rate ¼

K2 Enzyme Substrate KM þ Substrate

ð8:32Þ

where K2 and KM are reaction rate constants (KM is called the Michaelis constant). We will derive and discuss this equation in more detail later in this section. In an enzyme catalyzed reaction with much more substrate than enzyme, the reaction rate depends linearly on enzyme concentration according to Eq. (8.32), since the substrate concentration is essentially a constant. When there is much more enzyme than substrate, only a small portion of the enzyme is combined with the substrate, and the reaction rate is determined by both the enzyme and substrate levels. The typical enzyme catalyzed reaction consists of a series of reactions, each step with its own reaction rate. As we will see, the overall reaction rate is determined by the slowest reaction in the chain of reactions. The slowest reaction is called the capacity-limited reaction. Enzyme catalyzed reactions serve a regulatory role, as well as accelerating biochemical reactions. Consider the relationship between adenosine diphosphate (ADP) and ATP inside the cell. ATP is created in the mitochondria where oxidation of nutrients (carbohydrates, proteins, and fats) produces carbon dioxide, water, and energy. The energy from the oxidation of nutrients converts ADP into ATP. ATP is used as fuel for almost all activities of the body, such as the Na K pump, action potentials, synthesis of molecules, creation of hormones, and contractions of muscles. At steady state, the concentration of ADP is very low in the cell, and thus the creation of ATP in the mitochondria is at a low rate. During periods of high cell activity, ATP is consumed, releasing energy through the loss of one phosphate radical, leaving ADP. The increased concentration of ADP causes an increase in the oxidation of nutrients in the mitochondria, producing more ATP. Thus, the ADP-ATP cycle is balanced and based on the needs of the cell. The reactions necessary to synthesize ATP are described in Section 8.5. Enzyme reactions do not appear to follow the law of mass action; that is, as the substrate increases, the reaction rate does not increase without bound but reaches a saturation level (that is, it is capacity-limited). Capacity-limited reactions are quite prevalent and describe most metabolic reactions and functions of the body, such as the movement of molecules across the cell membrane and how substrates are removed from the body through the kidneys.

8.2.1 Michaelis-Menten Kinetics and the Quasi-Steady-State Approximation In 1903, Victor Henri first described the relationship between a substrate and an enzyme, followed by Michaelis and Menten in 1913, and Briggs and Haldane in 1925, with a capacity-limited elimination rate for the chemical reaction. The first assumption is that the enzyme and substrate quickly reach steady state in the formation of the complex, and then the complex more slowly dissociates into the product and enzyme. We also assume that the amount of enzyme is much smaller than the substrate. We refer to the combined work here

460

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

as Michaelis-Menten kinetics. The models presented here assume that temperature and other conditions remain constant unless otherwise indicated. This section introduces a standard approximation that greatly simplifies the analysis of biochemical reactions and predictions of how fast a reaction will occur. As shown, the quasi-steady-state approximation provides an excellent representation of the system’s differential equations, which are problematic, since they involve stiff differential equations. By assuming a quasi-steady-state approximation, the set of differential equations is reduced to a set of algebraic equations. When the early pioneers developed the theory of enzyme reactions, the solution of the differential equations was not possible by either simulation or direct solution. Thus, the quasi-steady-state approximation allowed a rather complete description of the enzyme reaction except for the initial stage involving the formation of the complex. The set of algebraic equations also provides a mechanism to measure the parameters of the reaction. Stiff differential equations require simulation solutions with an extremely small step size using the standard integrators. Consider Figure 8.4 and the graph for qB : From 0 to .02 s, qB changes quickly, and after that, it changes slowly. The step size needs to be very small, after which a small step size is not needed, since qB is slowly changing. Thus, simulations take a very long time to run using the standard integrator. If the step size is too large, the simulation solution is incorrect because small errors amplify as it reaches steady state. The default integrator ode45 in SIMULINK is not a good choice for stiff problems because it is inefficient. The integrator ode23tb is a better choice for stiff problems, which manipulates the step size using an efficient algorithm. We will explore this issue later in this section. The capacity-limited reaction model uses a two-step process given by Eq. (8.33): ð8:33Þ The enzyme mediated reaction first has substrate S combining with enzyme E to form the unstable complex ES. Then the complex ES breaks down into the product P and E. The law of mass action equation for Eq. (8.33) is q_ S ¼ K1 qS qE þ K1 qES

q_ ES ¼ K1 qS qE ðK1 þ K2 ÞqES q_ P ¼ K2 qES

ð8:34Þ

Since qE ð0Þ ¼ qE þ qES or qE ¼ qE ð0Þ qES , we have q_ E ¼ q_ ES : Thus, q_ E ¼ K1 qS qE þ ðK1 þ K2 ÞqES

ð8:35Þ

To begin with the classical description, we assume that the system is closed—that is, qS ð0Þ ¼ qS þ qES þ qP and qE ð0Þ ¼ qE þ qES . Note that E is not consumed in the reaction. Moreover, it is assumed that the complex ES increases quickly to a maximum, qESmax and then changes very slowly after that. Therefore, we assume complex ES is in a quasisteady-state mode shortly after the reaction starts—that is, q_ ES ¼ 0. We also assume that

461

8.2 ENZYME KINETICS

the amount of enzyme is very small in comparison to the substrate and product and that K2 K1 : From q_ ES ¼ 0 and Eq. (8.34), we have 0 ¼ K1 qS qE ðK1 þ K2 ÞqES

ð8:36Þ

Since qE ð0Þ ¼ qE þ qES , we substitute qE ¼ qE ð0Þ qES into Eq. (8.36) to give 0 ¼ K1 qS ðqE ð0Þ qES Þ ðK1 þ K2 ÞqES

ð8:37Þ

and after rearranging, qES ¼

K1 qS qE ð0Þ qS qE ð0Þ q q ð0Þ qE ð0Þ ¼ S E ¼ ¼ K1 þK2 q ðK1 qS þ K1 þ K2 Þ ð þ K Þ S M qS þ 1 þ KM K1

ð8:38Þ

2 . The constant, KM , is called the Michaelis constant, as mentioned in the where KM ¼ K1KþK 1

beginning of this chapter. The maximum complex ES from Eq. (8.38), with qs ¼ qs ð0Þ, is approximately qE ð0Þ qE ð0Þ qESmax ¼ KM M 1 þ qS qS ¼qS ð0Þ 1 þ qKS ð0Þ

ð8:39Þ

Further, since qE ¼ qE ð0Þ qES , we have qE ð0Þ q ð0Þ ¼ E qE ¼ qE ð0Þ KM 1 þ qS 1 þ KqMS

ð8:40Þ

To find a quasi-steady-state approximation for qS , we use Eq. (8.34); q_ S ¼ K1 qS qE þ

_ _ K1 qES ¼ qES þ K2 qES : With qES ¼ 0 and qES from Eq. (8.38), we have K2 qE ð0Þ q_ S ¼ K2 qES ¼ 1 þ KqMS

ð8:41Þ

Now, we have only two parameters that describe the change in substrate in Eq. (8.41), which takes the place of the set of differential equations in Eq. (8.34). Moreover, we will show that these two parameters can be estimated directly from data. Rearranging Eq. (8.41) gives KM dqS ¼ K2 qE ð0Þdt ð8:42Þ 1þ qS and after integrating both sides of Eq. (8.42) yields 0 1 q ð0Þ S A ¼ K2 qE ð0Þt ðqS ð0Þ qS Þ þ KM ln@ qS or

0 t¼

1 @ðqS ð0Þ qS Þ þ KM K2 qE ð0Þ

0 11 q ð0Þ S AA ln@ qS

ð8:43Þ

462

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Note that Eq. (8.43) provides a nonlinear expression for t as a function of qS and that, until recently, it was thought impossible to write an expression of qS as a function of t. In 1997, Schnell and Mendoza used computer algebra to solve Eq. (8.43) for qS as a function of t using the Lambert W function. While Eq. (8.43) can be used to solve for t by substituting values of qS from qS ð0Þ to zero, it is far easier to simulate Eq. (8.41) for qS : The product is given by K2 qE ð0Þ q_ P ¼ K2 qES ¼ 1 þ KqMS

ð8:44Þ

To eliminate qS , we have qS ¼ qS ð0Þ qES qP : Since qES is small compared to the other quantities after the initial phase of the response, we have qS ¼ qS ð0Þ qP and K2 qE ð0Þ K q ð0Þ ¼ 2 E q_ P ¼ KM 1 þ KqMS 1 þ qS ð0Þq P

ð8:45Þ

Solving Eq. (8.45) using the same approach to find Eq. (8.43), we have 1 qS ð0Þ qP qP KM ln t¼ qS ð0Þ K2 qE ð0Þ

ð8:46Þ

This approximation works well for the reaction, except during the initial quick phase when both qES and qP are small. Biochemists define the reaction rate as either the rate of disappearance of the substrate or the appearance of the product. Using the Michaelis-Menten approximation, the reaction rates are equal and are given by V ¼ q_ S ¼ q_ P : However, based on the true differential equations in Eq. (8.34), these two definitions are not equal. Using the approximation in Eq. (8.41), we have V ¼ q_ S ¼

K2 qS qE ð0Þ Vmax qS Vmax ¼ ¼ ðqS þ KM Þ ðqS þ KM Þ 1 þ KM

ð8:47Þ

where Vmax ¼ K2 qE ð0Þ is the maximum velocity of the substrate disappearance. Equation (8.47) is plotted in Figure 8.5. Note that at high substrate levels, V approaches Vmax , since all of the enzyme is engaged and the velocity saturates. In this region, the reaction rate is independent of qs : At low substrate levels, the reaction rate is approximately linearly depenqs : dent on substrate level, V ¼ VKmax M Now consider when V ¼ Vmax 2 : Substituting into Eq. (8.47), we have Vmax Vmax qS ¼ 2 ð qS þ KM Þ

463

8.2 ENZYME KINETICS

50 40

30 20 10 0 0

FIGURE 8.5 Velocity of substrate disappearance. Note that the reaction velocity is for t ¼ 0, and qS is the intial quantity of substrate. V max ¼ 50 and KM ¼ 3:

and after dividing both sides by Vmax , we have 1 qS ¼ 2 ðqS þ KM Þ Simplifying the previous equation gives qS ¼ KM when V ¼ Vmax 2 : Given the lack of computer simulation capability in the early 1900s, the quasi-steady-state approximation gave an excellent and efficient solution to enzyme kinetics. However, with the current computer power and the availability of stiff differential equation simulators, it is far easier to simulate enzyme kinetics using the computer rather than solving a set of algebraic equations.

EXAMPLE PROBLEM 8.2 Simulate the reaction given in Eq. 8.34 and compare with the quasi-steady-state approximation for qS , qE , qES , and qP: Assume that K1 ¼ 8, K1 ¼ 0:01, K2 ¼ 5, qS ð0Þ ¼ 1, qE ð0Þ ¼ 0:08, qES ð0Þ ¼ 0, and qP ð0Þ ¼ 0:

Solution The SIMULINK model is shown in Figures 8.6 and 8.7, both executed using the ode23tb integrator. Because of the ease in solution, the quasi-steady-state approximation for qS , qE , qES , and qP is carried out using the differential equation for q_ S , Eq. (8.41), and the algebraic equations, Continued

464

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

t Clock

To Workspace qS To Workspace1 qSD –

1 s

Integrator

qE To Workspace8 Constant

Product

Gain

Gain3

qE0

–

K2 qES qESD + –

1 s

To Workspace2

qES

Integrator1 Gain1

qES

-KGain2 K3

qPD

1 s Integrator2

qP qP To Workspace3

FIGURE 8.6

SIMULINK model for Eq. (8.34) in Example Problem 8.2.

Eqs. (8.38) and (8.40), and qP ¼ qS ð0Þ qS qES : Figure 8.8 presents the simulation solution with both models. It is clear that there is an excellent agreement between the two solutions after the quick phase is completed at 0.5 s. Notice the expected error between qE and qEMM , and qES and qESMM : Also note that the initial product predicted by the quasi-steady-state approximation is negative.

465

8.2 ENZYME KINETICS

-C-

qPMM

-K3qE0 ×

1 Constant1

To Workspace4

+ +

Divide1

qSDMM

qSMM 1 ÷ s Integrator3 Divide

qE0

qESMM

qS0

To Workspace5

qS0 –

qE0

qPMM

–

qSMM

qESMM

To Workspace6

qSMM

qEMM Divide2

To Workspace7 – qEMM

qE0 qE0

FIGURE 8.7 SIMULINK model in Example Problem 8.2 using the quasi-steady-state approximation.

0.1 0.08

qS qSMM

0.6

Quantity

0.8

0.4

0.06 qE

0.04

qEMM 0.2

0.02

0 0

0.05

Time

Time qESMM 0.9 0.04 0.7 Quantity

Quantity

qES 0.03 0.02

0.5 0.3

0.01 0.1

qP qPMM

0 0

–0.1 0

4 Time

Time

FIGURE 8.8

Simulations for Example Problem 8.2.

466

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

8.2.2 Estimation of the Michaelis-Menten Parameters The reaction rate given by Eq. (8.47) contains two parameters, Vmax and KM , and is given by V¼

Vmax qS Vmax ¼ ðqS þ KM Þ 1 þ KM

ð8:48Þ

One can estimate Vmax and KM by first taking the reciprocal of Eq. (8.48), giving 1 KM 1 1 þ ¼ Vmax qS V Vmax

ð8:49Þ

which is called the Lineweaver-Burk equation. By plotting V1 against q1S , a straight line results 1 and the slope as KM : Since the as shown in Figure 8.9, with the ordinate-intercept as Vmax substrate changes as a function of time, the Lineweaver-Burk equation is usually carried out with the initial value of substrate, qS ð0Þ—that is,

1 KM 1 1 þ ¼ Vð0Þ Vmax qS ð0Þ Vmax

ð8:50Þ

The method to generate data for the Lineweaver-Burk equation is to perform a series of experiments that increase the quantity of substrate, while keeping the amount of enzyme constant. Recall at V ¼ Vmax 2 , qS ¼ KM : If qS KM , then V ¼ Vmax : While estimation of the parameters from the Lineweaver-Burk plot looks attractive, any measurement error is amplified by the transformation, thus giving poor estimates of the parameters.

1 V(0)

KM Vmax 1 Vmax

–

1 KM

FIGURE 8.9

1 qS(0) A Lineweaver-Burk plot.

8.3 ADDITIONAL MODELS USING THE QUASI-STEADY-STATE APPROXIMATION

467

8.3 ADDITIONAL MODELS USING THE QUASI-STEADY-STATE APPROXIMATION The quasi-steady-state approximation for Michaelis-Menten kinetics can be used for more than just enzyme reactions.2 The quasi-steady-state approximation is useful when describing the elimination of substances from the body with capacity-limited rates, such as renal excretion and metabolism, and even for linearized models of muscle using the Hill equation. While the quasi-steady-state approximation was developed for enzyme reactions with variables qS , qE , qES , and qP , we will use it as a nonlinear transfer rate in a compartment model, where the substrate, qS , in Eq. (8.47) is the quantity in a compartment. Here, we first consider a one-compartment model with a variety of inputs and then a two-compartment model.

8.3.1 One-Compartment Model Consider a one-compartment model in which the elimination is characterized by the quasi-steady-state approximation q_ S ¼

Vmax qS þ f ðtÞ ðqS þ KM Þ

ð8:51Þ

Impulse Input Consider first an impulse input. Previous examples have used f ðtÞ ¼ zdðtÞ, where z is the strength of the impulse function—a problem that is handled most simply by a change in the initial condition. To solve Eq. (8.51) with an impulse input, we have q_ S ¼

Vmax qS ð qS þ KM Þ

ð8:52Þ

with qS ð0Þ ¼ z: As before, Eq. (8.52) is rearranged to give KM dqS ¼ Vmax dt 1þ qS and after integrating, we have t¼

z z qS þ KM ln Vmax qS 1

The same comments about the solution of Eq. (8.43) apply to Eq. (8.53).

Material in this section based on Godfrey.

ð8:53Þ

468

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Step Input We now draw our attention to the case in which the input to the system in Eq. (8.51) is a step input, f ðtÞ ¼ zuðtÞ: With initial condition qS ð0Þ, we have for t 0 q_ S ¼

Vmax ðz Vmax ÞqS þ zKM qS þ z ¼ ðqS þ KM Þ ðqS þ KM Þ

ð8:54Þ

Separating qS and t in Eq. (8.54) gives dt ¼

KM dqS qS dqS þ ðz Vmax ÞqS þ zKM ðz Vmax ÞqS þ zKM 0

B C KM dqS 1 qS dqS C AB þ@ B C zKM A ðz Vmax ÞqS þ zKM ðz Vmax Þ @ qS þ ðz Vmax Þ 0

B B KM dqS 1 AB 1 þ@ ðz Vmax ÞqS þ zKM ðz Vmax Þ B @

zKM ðz Vmax Þ qS þ

zKM ðz Vmax Þ

ð8:55Þ 1 C C CdqS C A

Then we integrate Eq. (8.55) Z

Z dt ¼

B KM dqS 1 B @ðz Vmax ÞqS þ zKM þ z Vmax qS ð0Þ qS

11 zKM B CC B1 ðz Vmax Þ CCdqS @ zKM AA qS þ ðz Vmax Þ

ð8:56Þ

which gives 0 11 KM Aln@ ðz Vmax ÞqS þ zKM A þ @qS qS ð0ÞA C B@ B ðz Vmax Þ ðz Vmax ÞqS ð0Þ þ zKM z Vmax C C B C B C B 1 0 C B zKM t¼B C qS þ C B C B ð z V Þ C B max C B zKM C B C B ln C B zKM C A @ ðz Vmax Þ2 B A @qS ð0Þ þ ðz Vmax Þ 00

1 0

0 1 ð Þq qS qS ð0Þ Vmax KM z V þ zK max S M A ln@ ¼ ðz Vmax ÞqS ð0Þ þ zKM z Vmax ðz Vmax Þ2

ð8:57Þ

8.3 ADDITIONAL MODELS USING THE QUASI-STEADY-STATE APPROXIMATION

469

To determine qS at steady state for this one-compartment system, we set q_ S ¼ 0 in Eq. (8.54), which gives q S ð1Þ ¼

zKM Vmax z

ð8:58Þ

Since qS needs to be positive, this requires Vmax > z. Thus, the maximum removal rate must be larger than the input for a bound solution for qS . To examine the case when Vmax ¼ z, we substitute these values into Eq. (8.54), giving q_ S ¼

zKM ð q S þ KM Þ

ð8:59Þ

Separating qS and t in Eq. (8.59) gives dt ¼

1 ðqS þ KM ÞdqS zKM

and after integrating t¼

1 2 qS q2S ð0Þ þ qS qS ð0Þ 2zKM z

ð8:60Þ

which increases without bound, qS ! 1: If z Vmax , then the substrate is not eliminated quickly enough, continuously increasing, and qS ! 1: Exponential Input Next, consider an exponential input to a one-compartment model of Eq. (8.51), a type of input previously observed in Example Problem 7.8, when a substrate is digested and moves into the plasma. For t 0, the model is q_ S ¼

Vmax qS þ q2 ð0ÞK21 eK21 t ðqS þ KM Þ

ð8:61Þ

where q2 ð0Þ is the initial amount of substrate in the digestive system and K21 is the transfer rate from the digestive system to the compartment. Simulating Eq. (8.61) is the only way to solve this problem. Before simulation became an easy solution method, the quasi-steady-state approximation was linearized with a lower and upper bound, with the actual solution falling in between these two bounds. For the lower bound, the quasi-steady-state approximation is VKmax , where M the quasi-steady-state approximation is a constant. As we will see, the lower bound is a good approximation for small values of qS . For the upper bound, the quasi-steady-state Vmax , where qSmax is the maximum qS . As before with the lower approximation is qSmax þ KM bound, the quasi-steady-state approximation is a constant for the upper bound. The upper bound is a good approximation for large values of qS : This approach works for all inputs.

470

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

For example, consider the exponential input with qS ð0Þ ¼ 0, where the equation for the lower bound is q_ S ¼ with a solution of

Vmax qS þ q2 ð0ÞK21 eK21 t KM

0 B q2 ð0ÞK21 qS ¼ B @Vmax K21 KM

ð8:62Þ

1 Vmax C K t C e 21 e KM t A

ð8:63Þ

and the equation for the upper bound is q_ S ¼ with a solution of

Vmax qS þ q2 ð0ÞK21 eK21 t ðqSmax þ KM Þ

0 B qS ¼ B @

1 C K t q2 ð0ÞK21 C e 21 e A Vmax K21 qSmax þ KM

Vmax qS þKM max

ð8:64Þ

! t

ð8:65Þ

To illustrate the approximations, Figure 8.10 plots the simulation of Eq. (8.61), with the analytic solution of Eqs. (8.63) and (8.65). For small values of qS , the lower bound

qS, qSLower, qSUpper

qSUpper

qSLower

0 0

Time Responses for a one-compartment model with quasi-steady-state approximation ðqS Þ, lower

bound approximation ðqSLower Þ, and upper bound approximation qSUpper , with an exponential input. Parameters are V max ¼ 3, KM ¼ 2:5, K21 ¼ 1:0, qS ð0Þ ¼ 0, qS max ¼ 4:0, and q2 ð0Þ ¼ 10:

FIGURE 8.10

8.3 ADDITIONAL MODELS USING THE QUASI-STEADY-STATE APPROXIMATION

f2(t)

f1(t) Vmax (q1 + KM)

471

K12 q1

q2 K21

K10

K20

FIGURE 8.11 A two-compartment model with a quasi-steady-state approximation transfer rate.

approximation is quite good, as is the upper bound for large values of qS : The accuracy of the lower and upper bound approximations depends quite heavily on the closeness of KM to qSmax : The farther apart the two are, the poorer the approximation.

8.3.2 Two-Compartment Model Next, consider a two-compartment model with a quasi-steady-state approximation transfer rate as shown in Figure 8.11. Such models are useful in describing some capacity-limited biochemical reactions that are more complex than a one-compartment model. Note that the quasi-steady-state approximation is for compartment 1 only, and the other transfer rates are the usual constants. The equations describing the system are 0 1 V max Aq1 þ K21 q2 þ f1 ðtÞ q_ 1 ¼ @K10 þ K12 þ ðq1 þ KM Þ ð8:66Þ q_ 2 ¼ K12 q1 ðK20 þ K21 Þq2 þ f2 ðtÞ

EXAMPLE PROBLEM 8.3 Simulate the model in Eq. (8.66), given that K12 ¼ 1, K21 ¼ 2, Vmax ¼ 3, KM ¼ 0:5, q1 ð0Þ ¼ 0, q2 ð0Þ ¼ 0, K10 ¼ 0, and K20 ð0Þ ¼ 0: The inputs are f1 ðtÞ ¼ uðtÞ uðt 4Þ, and f2(t) ¼ 0.

Solution The equations describing this system are 1 q_ 1 ¼ 3 þ q1 þ 2q2 þ uðtÞ uðt 4Þ ðq1 þ 0:5Þ q_ 2 ¼ 3q1 2q2 The SIMULINK model shown in Figure 8.12 was executed using the ode23tb integrator. The solution is shown in Figure 8.13. Continued

472

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

To Workspace1 q2

q2D + –

q2 1 s Integrator1

t Clock

To Workspace

-KK20+K21 -K-

K12

-K-

Vmax

K21

KM q1Dot

+ Step

Vmax

1 s

q1+KM

Vmax -------------q1+KM +

-C-

KM +

Divide1

K12+K10

Integrator2

–

To Workspace6

–

Step1 q1

Product1 ×

FIGURE 8.12

SIMULINK model for Example Problem 8.3.

0.3

0.2 Quantity

0.1 q2

0 0

Time

FIGURE 8.13

Response for Example Problem 8.3.

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

f1(t)

473

f2(t) K12 q1

Vmax (q1 + KM) q1

q2 K21 K10

K20q2

FIGURE 8.14 A two-compartment model with a quasi-steady-state approximation and biochemical reaction transfer rates. In addition, a constant input from compartment 1 into compartment 2 is used, with Kq121 as the transfer rate.

Consider the model shown in Figure 8.14 that contains a quasi-steady-state approximation and a biochemical reaction as transfer rates. We also have a constant transfer of substrate from compartment 1 into compartment 2 by using Kq121 as the transfer rate. The equations that describe this system are 0 1 Vmax A q1 þ K21 q2 þ f1 ðtÞ q_ 1 ¼ K12 @K10 þ ðq1 þ KM Þ ð8:67Þ q_ 2 ¼ K12 K20 q22 þ f2 ðtÞ Note that the constant loss from compartment 1 and input to compartment 2 could describe an active pump. Also note that a chemical reaction in compartment 2 consumes the substrate with rate K20 . We will expand on these concepts with multicompartment models in the next section that includes diffusion.

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS In Chapter 7, we discussed diffusion as a flow of ions down the concentration gradient. Up to now, we have approached biochemical reactions and enzyme kinetics occurring in a homogeneous volume. Now, we include diffusion from another compartment, biochemical reactions, and enzyme kinetics in our analysis. As we will see, the movement of a substrate or an enzyme into the cell by diffusion allows a product to be created. This product then can be used inside the cell or diffused out of the cell to be used by another cell or tissue. Additionally, the same situation occurs within the organelles of the cell. These reactions can serve a regulatory role as well as accelerating biochemical reactions; recall the reaction involving ADP and ATP in the mitochondria, where the availability of ADP regulates the production of ATP.

474

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

8.4.1 Diffusion and Biochemical Reactions Consider the movement of a substrate A into a cell by diffusion, which then reacts with B to form product P, as shown in Figure 8.15. Product P then leaves the cell by diffusion. Assume that the quantity of Bi is regulated by another system. Let subscript i denote inside the cell, and let o denote outside the cell. The chemical reaction is q_ Ai ¼ K1 qAi qBi þ K1 qPi q_ Pi ¼ K1 qAi qBi K1 qPi

ð8:68Þ

where K1 and K1 are the reaction rates, qAo and qAi are the quantities of substrate A, qBi is the quantity of substrate Bi , and qPo and qPi are the quantities of product P. Diffusion across the membrane is given by q_ Ai ¼ Coi qAo Cio qAi q_ Pi ¼ Doi qPo Dio qPi

ð8:69Þ

where Coi , Cio , Doi , and Dio are the diffusion transfer rates that depend on the volume, as described in Example Problem 7.5. The equations describing the complete system (biochemical reaction and diffusion) are q_ Ao ¼ Coi qAo þ Cio qAi q_ Ai ¼ K1 qAi qBi þ K1 qPi þ Coi qAo Cio qAi q_ Po ¼ Doi qPo þ Dio qPi

ð8:70Þ

q_ Pi ¼ K1 qAi qBi K1 qPi þ Doi qPo Dio qPi Transport of Oxygen into a Slow Muscle Fiber Consider the delivery of oxygen into a slow muscle fiber. Let’s begin at the lung where oxygen first diffuses through the alveoli membrane into the capillaries and from the capillaries into the red blood cell, as shown in Figure 8.16. Let qOA be the quantity of O2 in the

Outside

Inside

Po Doi Dio

Ao Coi Cio

Pi K1 ® Ai + Bi Pi K–1 Bi

FIGURE 8.15

Diffusion and a biochemical reaction.

475

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

BCR

BAC qOH A BCA

BRC

Alveoli

FIGURE 8.16

qOR

qOC

Capillaries

Red Blood Cell

The diffusion of oxygen from the alveoli into the red blood cell.

alveoli, qOC be the quantity of O2 in the capillaries, and qOR be the quantity of O2 in the red blood cells. The equation that describes the movement of oxygen is given by q_ OA ¼ BCA qOC BAC qOA q_ OC ¼ BAC qOA þ BRC qOR BCA qOC BCR qOC

ð8:71Þ

q_ OR ¼ BCR qOC BRC qOR Once inside the red blood cell, oxygen then binds with hemoglobin ðHbÞ, forming oxyhemoglobin ðHbO8 Þ. This is a reversible reaction that allows oxygen to be taken up by the red blood cell and released into the tissues. The binding of O2 with hemoglobin allows 100 times more oxygen in the blood than if it had just dissociated into the blood. The overall chemical reaction is Hemoglobin has four polypeptide subunits (proteins), with each polypeptide subunit attached to a heme group. Each heme group can bind with a molecule of O2 : The four molecules of O2 that bind to Hb do not simultaneously react with heme groups but occur in four steps, with each step facilitating the next step. Figure 8.17 illustrates the five states of hemoglobin based on the number of O2 molecules bound to it, ranging from 0 to 4. Let qH0 be the quantity of Hb, qH1 be the quantity of HbO2 , and so on, up to qH4 be the quantity of HbO8 : Equation (8.72) describes the chemical reactions that take place to create the oxyhemoglobin: q_ H0 ¼ K10 qH1 K01 qH0 qOR q_ H1 ¼ K01 qH0 qOR þ K21 qH2 K10 qH1 K12 qH1 qOR

ð8:72Þ

q_ H2 ¼ K12 qH1 qOR þ K32 qH3 K21 qH2 K23 qH2 qOR q_ H3 ¼ K23 qH2 qOR þ K43 qH4 K32 qH3 K34 qH3 qOR q_ H4 ¼ K34 qH3 qOR K43 qH4

K01qOR qHH0

K12qOR

K10

K23qOR qH2

qH1 K21

FIGURE 8.17

K34qOR qH3

K32 The five states of hemoglobin.

qH4 K43

476

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

BRI qOR

BIM

BIR Red Blood Cell

FIGURE 8.18

qOM

q OI BMI Interstitial Fluid/ Capillaries

Cytosol of Slow Muscle Fiber

The diffusion of oxygen from the red blood cell into the cytosol of the slow muscle fiber.

Note that we have assumed that the reverse reactions do not involve oxygen and that oxygen is treated as a molecule and not two oxygen atoms, which would have introduced a q2O term in Eq. (8.72). Oxygen is transported through the arterial system to the capillaries, where it diffuses out of the red blood cell into the interstitial fluid. It then moves into the cytosol (the liquid part of the cytoplasm that does not contain any organelles) of slow muscle fibers, as shown in Figure 8.18, where qOI is the quantity of O2 in the interstitial fluid, qOM is the quantity of O2 in the cytosol of the slow muscle fiber, and the other quantities are defined as before. The equation describing the diffusion process and reactions with Hb is given by q_ OR ¼ BIR qOC BRI qOR þ K10 qH1 K01 qH0 qOR þ K21 qH2 K12 qH1 qOR q_ OI

þ K32 qH3 K23 qH2 qOR þ K43 qH4 K34 qH3 qOR ¼ BRI qOR þ BMI qOM BIR qOI BIM qOI

ð8:73Þ

q_ OM ¼ BIM qOI BMI qOM On the arterioles side of the capillary membrane, PO2 is approximately 100 mm Hg and 98 percent saturated. On the venule side of the capillary membrane, PO2 is approximately 40 mm Hg and 75 percent saturated. When PO2 is high, oxygen quickly binds with hemoglobin, and when PO2 is low, oxygen is quickly released from hemoglobin. As the red blood cell moves through the capillary, the PO2 gradient causes oxygen to be released into the interstitial fluid. Once inside the slow muscle fiber, oxygen is moved to the mitochondria using a different mechanism than that used in other cells. After oxygen diffuses across the cell membrane, it quickly binds to myoglobin (Mb), a protein-like hemoglobin whose function is to transport and store oxygen, and forms oxymyoglobin ðMbO2 Þ. By storing oxygen in oxymyoglobin, oxygen is driven into the cell by a large concentration gradient until it binds with all available myoglobin. At this point, the oxygen concentration on either side of the membrane equilibrates. Slow muscle fibers also have many more mitochondria than other cells, which allows higher levels of oxidative metabolism. Thus, the muscle fiber is able to store a large quantity of oxygen in oxymyoglobin, and when needed, it is readily available to the mitochondria to create ATP. This greatly increases oxygen transport to the mitochondria than if the cell just depended on oxygen diffusion in the absence of myoglobin. When oxygen is bound to myoglobin to create oxymyoglobin in the cytosol, oxymyoglobin then diffuses from the cytosol into the mitochondria, and once in the mitochondria, a

477

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

reverse reaction occurs, releasing O2 and Mb. Oxygen in the mitochondria is consumed with sugar to create ATP during cell respiration, as described in the next section. The myoglobin then diffuses back to the cytosol, where the process repeats itself. The first reaction in the cytosol is given by D1

O2 þ Mb ! MbO2 and in the mitochondria D2

! Mb þ O2 MbO2 which is described by the following equations that includes diffusion: q_ OM ¼ D1 qOM qMbM þ BIM qOI q_ MbOM ¼ D1 qOM qMbM KMT qMbOM

ð8:74Þ

q_ MbM ¼ D1 qOM qMbM þ KTM qMbT q_ MbT ¼ D2 qMbOT KTM qMbT q_ MbOT ¼ D2 qMbOT þ KMT qMbOM where KMT and KTM are the diffusion transfer rates from the cytosol into the mitochondria and the mitochondria into the cytosol, respectively; qOM is the quantity of O2 inside the cytosol; qMbM is the quantity of Mb in the cytosol; qMbOM is the quantity of MbO2 in the cytosol; qMbOT is the quantity of MbO2 inside the mitochondria; and qMbT is the quantity of Mb inside the mitochondria. We assume that none of the MbO2 or Mb leaves the cell. We assume that no O2 leaves the cell and no Mbi diffuses into the mitochondria. Carbon dioxide is created in the cells during cell respiration, as discussed shortly. It then diffuses out of the cell into the interstitial fluid and then moves to the capillaries. Once inside the capillaries, 90 percent of the carbon dioxide moves into the red blood cell, and 10 percent dissolves into the fluid of the blood. Carbon dioxide is then transported to the lungs. When inside the red blood cell, carbon dioxide almost instantaneously goes through the following reactions: 1. Approximately 70 percent of the carbon dioxide reacts with water to form carbonic acid þ ðH2 CO3 Þ, using the enzyme

carbonic anhydrase, and then dissociates into hydrogen ðH Þ and biocarbonate HCO3 ions. These reactions occur within a fraction of a second and are given by carbonic anhydrase

þ CO2 þ H2 O ! H2 CO3 ! HCO 3 þH

2. The hydrogen then binds with the hemoglobin in the red blood cell. Hþ þ HbH ! ðHÞHb 3. The biocarbonate diffuses out of the red blood cell, replaced by chloride ions via a biocarbonate-chloride carrier protein in the cell membrane.

478

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

4. The remaining 20 percent of the carbon dioxide in the red blood cell combines with hemoglobin to form carbaminohemoglobin. This is a weak bond that is easily broken: Hb þ CO2 ! HbCO2 After the red blood cell reaches the lungs, the oxygen that diffused across the alveoli membrane displaces the carbon dioxide in the blood and binds with the hemoglobin. Carbon dioxide then diffuses through the alveoli membrane and is then exhaled. The entire process then repeats itself.

8.4.2 Diffusion and Enzyme Kinetics Consider the movement of a substrate into a cell by diffusion, which then reacts with an enzyme to ultimately form a product that leaves the cell by diffusion, as shown in Figure 8.19. The chemical reaction is ð8:75Þ and diffusion by q_ Si ¼ Boi qSo Bio qSi

ð8:76Þ

q_ Pi ¼ Doi qPo Dio qPi

where Boi , Bio , Doi , and Dio are the diffusion transfer rates. The equations describing the complete system are q_ Si ¼ K1 qSi qE þ K1 qESi þ Boi qSo Bio qSi q_ ESi ¼ K1 qSi qE ðK1 þ K2 ÞqESi

ð8:77Þ

q_ Pi ¼ K2 qESi þ Doi qPo Dio qPi

Inside

Boi

So Bio

Si + E

FIGURE 8.19

Po Doi

Dio

Si K1 ® ES i K–1

Outside

K2 ® E + Pi

Diffusion and enzyme kinetics.

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

479

To remove qSo in Eq. (8.77), we assume a constant total substrate qST ¼ qSi þ qSo , and with qSo ¼ qST qSi , we have q_ Si ¼ K1 qSi qE þ K1 qESi þ Boi ðqST qSi Þ Bio qSi q_ ESi

¼ K1 qSi qE þ K1 qESi ðBoi þ Bio ÞqSi þ Boi qST ¼ K1 qSi qE ðK1 þ K2 ÞqESi

ð8:78Þ

q_ Pi ¼ K2 qESi þ Doi qPo Dio qPi max q We can substitute the quasi-steady-state approximation, K1 qSi qE þ K1 qESi ¼ q VþK ð Si M Þ S i (based on Eq. (8.47)), into Eq. (8.78) and get

Vmax qS ðBoi þ Bio ÞqSi þ Boi qST ðqSi þ KM Þ i 0 1 V max ¼ @ þ Boi þ Bio AqSi þ Boi qST ðqSi þ KM Þ

q_ Si ¼

ð8:79Þ

where qESi ¼ qE ð0Þ and qPi ¼ qSi ð0Þ qSi qESi : K

1þ q M Si

8.4.3 Carrier-Mediated Transport Now consider carrier-mediated transport, where an enzyme carrier in the cell membrane has a selective binding site for a substrate, which, when bound, transports the substrate through the membrane to be released inside the cell. Many also refer to this process as facilitated diffusion. Carrier-mediated transport does not use energy to transport the substrate but depends on the concentration gradient. Without carrier-mediated transport, the substrate cannot pass through the membrane. Carrier-mediated transport differs from diffusion, since it is capacity-limited and diffusion is not. That is, as the quantity of the substrate increases, the carrier-mediated transport reaction rate increases and then saturates, where regular diffusion increases linearly without bound, as shown in Figure 8.20. Figure 8.21 illustrates carrier-mediated transport, described by

ð8:80Þ where So and Si are the substrate outside and inside the cell, Co is the carrier on the outside of the membrane, Ci is the carrier on the inside of the membrane, Po is the bound substrate and carrier complex on the outside of the membrane, and Pi is the bound substrate and

480

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Diffusion

Rate

Vmax

Carrier-Mediated Transport

Substrate

FIGURE 8.20 Comparison of the effect of the amount of substrate and the rate of diffusion and carriermediated transport.

carrier complex on the inside of the membrane. As shown in Figure 8.21, the substrate first moves to the binding site on the outside of the membrane and binds with the carrier to form the carrier-substrate complex, Po (left). Next, the carrier-substrate complex moves from the outside to the inside of the membrane, given by the carrier-substrate complex Pi (right). It is not known how the carrier-substrate complex moves through the membrane, but we are fairly certain it happens. The last step is when the carrier-substrate complex, Pi , dissociates into the substrate, Si , and carrier, Ci : We have assumed that the reaction rates are the same for creation and the dissociation of the complex, . In addition, we assume that the reaction of

FIGURE 8.21 Carrier-mediated transport, with the carrier within the membrane. On the left side is the binding of the carrier with the substrate. On the right side is the substrate released from the binding site into the cell.

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

carrier-substrate complex from outside to inside has the same reaction rate, law of mass action, we get

481 . Using the

q_ So ¼ K1 qSo qCo þ K1 qPo q_ Co ¼ K1 qSo qCo þ K1 qPo þ K2 qCi K2 qCo q_ Po ¼ K1 qSo qCo ðK1 þ K2 ÞqPo þ K2 qPi q_ Pi ¼ ðK1 þ K2 ÞqPi þ K2 qPo þ K1 qSi qCi

ð8:81Þ

q_ Si ¼ K1 qSi qCi þ K1 qPi q_ Ci ¼ K1 qSi qCi þ K1 qPi þ K2 qCo K2 qCi Since the carrier is not consumed in the reaction, then the total carrier is a constant, given as qCo þ qCi þ qPo þ qPi ¼ z: Naturally, we can add an input to the system in Eq. (8.81) or simplify using the quasisteady-state approximation as before. In addition, the substrate can be involved in other reactions inside the cell, such as moving into an organelle (mitochondria) via diffusion and then experiencing an enzyme reaction. Glucose Transport Consider the transport of glucose across the cell membrane. We know that glucose does not diffuse across the cell membrane but is transported across the cell membrane by a carrier-mediated transport process. Glucose binds to a protein that transports it across the membrane, allowing it to pass into the cytosol. This process does not use any energy. Using the model illustrated in Figure 8.21, we have ð8:82Þ where Go and Gi is glucose outside and inside the cell, respectively; Co is the carrier on the outside of the membrane; Ci is the carrier on the inside of the membrane; Po is the bound substrate and carrier complex on the outside of the membrane; and Pi is the bound substrate and carrier complex on the inside of the membrane. Note that there is no reverse reaction for glucose in Eq. (8.82), since glucose does not leave the cell. We assume that glucose is consumed at a constant rate Ji inside the cell during cell respiration (described in the next section) and that glucose is available in the interstitial fluid at a rate of Jo . The equations that describe this system are given by q_ Go ¼ K1 qGo qCo þ Jo q_ Co ¼ K1 qGo qCo þ K2 qCi K2 qCo q_ Po ¼ K1 qGo qCo K2 qPo þ K2 qPi q_ Gi ¼ K1 qPi Ji q_ Ci ¼ K1 qPi þ K2 qCo K2 qCi q_ Pi ¼ K2 qPo K1 qPi

ð8:83Þ

482

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

As described earlier, transport of glucose through the cell membrane is a capacitylimited reaction because of the enzyme carrier. Glucose concentration in the blood varies from a typical value of 90 100mgmL , up to 200 100mgmL after eating, and down to 40 100mgmL three hours after eating. Thus, the input, Jo , is a function of eating. The body uses the following two mechanisms to control glucose concentration: 1. Automatic feedback involving insulin secretion by the pancreas 2. The liver The pancreas, in addition to digestive functions, secretes insulin directly into the blood. Insulin facilitates diffusion of glucose across the cell membrane. The rate of insulin secretion is regulated so glucose is maintained at a constant level. The liver acts as a storage vault for glucose. When excess amounts of glucose are present, almost immediately two-thirds is stored in the liver. Conversely, when the glucose level in the blood falls, the stored glucose in the liver replenishes the blood glucose. Insulin has a moderating effect on the function of the liver. The binding of the carrier enzyme with glucose is a function of the transfer rate, K1 , and is a function of the insulin level, which can increase the transport of glucose by as much 20 times the base rate without insulin. We will ignore the aspects of insulin and liver storage in our model.

8.4.4 Active Transport Now consider active transport, which is similar to carrier-mediated transport but operates against the concentration gradient and uses energy to move the substrate across the cell membrane. Active transport uses an enzyme carrier in the cell membrane that has a selective binding site for a substrate, which, when bound, transports the substrate through the membrane to be released inside the cell. Active transport uses energy to run, typically by the hydrolysis of ATP. Here, the energy from 2ATP

is used in the transport of the substrate, leaving ADP and an inorganic phosphate PO4 in the cytosol. The ADP is then recycled in the mitochondria to create more ATP using glucose as described in the next section. Active transport is capacity-limited like carrier-mediated diffusion: as the quantity of the substrate increases, the transport reaction rate increases and then saturates, as shown in Figure 8.20. The Na-K pump is the most important active transport process, which pumps Naþ out of the cell against the concentration gradient and replaces it inside the cell with Kþ against the concentration gradient. This pump is used to maintain the ion gradients and resting membrane potential, as described in Chapter 12, and is also required to maintain cell volume, as described in the previous chapter. Another important active transport process is the Na-Ca ATP-ase pump that keeps Caþ2 levels low inside the cell. It is vitally important that the concentration of Caþ2 be kept

7 low inside the cell approximately 10 M as compared to the outside concentration approximately 103 M . The concentration gradient drives Caþ2 into the cell, and the Na-Ca ATP-ase pump drives Caþ2 out of the cell. Na-K Pump The Na-K pump is an integral part of the cell membrane that exists in all cells in the body. Approximately 70 percent of all ATP in the neuron and 25 percent of all ATP in all other cells

8.4 DIFFUSION, BIOCHEMICAL REACTIONS, AND ENZYME KINETICS

483

is used to fuel the Na-K pump at rest. This pump is vital to maintain the cell’s resting membrane potential. In this section, we focus on the enzyme reactions. The Na-K pump was discovered by Jens Skou in 1957, who subsequently received a Nobel Prize for his work in 1997. Using radioactive ions, Skou showed that the concentrations of the ions are interdependent, implying the involvement of a common mechanism using an ATP-ase carrier. The overall reaction for the Na-K pump is given by þ þ þ ATP þ 3Naþ i þ 2Ko ! ADP þ Pi þ 3Nao þ 2Ki ATPase

where the subscripts i for inside the cell and o for outside the cell are used as before. The pump has 3 Naþ binding sites and 2 Kþ binding sites in its two conformations. There is a higher concentration of Naþ outside the cell than inside and a higher concentration of Kþ inside the cell than outside; left unchecked by the pump, this gradient would drive Naþ into the cell and Kþ out of the cell, and thus change the resting membrane potential. Any change in the concentration gradient of Kþ and Naþ is prevented by the Na-K pump. The pump transports a steady stream of Naþ out of the cell and Kþ into the cell. The Na-K pump uses six steps to move 3 Naþ ions out of the cell and 2 Kþ ions into the cell at a total cost of 1 ATP molecule. Figure 8.22 illustrates the six steps that are continually repeated at a rate of 100 s1 : 1. Three Naþ i ions in the cytosol move into and bind to the carrier in the pump, ðNa3 CÞi : Note that the pump has a bound ATP molecule. 2. ATP is hydrolyzed. ADP is released, and the inorganic phosphate, P, binds with ðNa3 CÞi to create ðNa3 CPÞi . 3. Using the energy gained by the hydrolyzation, a conformational change occurs in the pump that moves ðNa3 CPÞo to the outside of the cell membrane, exposing Naþ o ions to the outside. The 3 Naþ ions exit the pump. o 4. On the outside of the cell, 2 Koþ ions then bind to the carrier and inorganic phosphate in the pump, creating ðK2 CPÞo . 5. Dephosphorylation of the pump occurs, releasing the inorganic phosphate. Following this, a conformational change occurs in the pump that moves ðK2 CÞi , exposing 2 Kiþ ions to the inside of the cell. 6. ATP binds to the pump, and the 2 Kiþ ions are released into the cell. The following equations list the reactions that describe the Na-K pump, where C is the carrier and P is the inorganic phosphate. Note that we have eliminated the reverse reactions as they are relatively small. K1

! ATPi K2 3Naþ ! ðNa3 CÞi i þ Ci K3

ðNa3 CÞi þ ATPi ! ðNa3 CPÞi þ ADPi K4

ðNa3 CPÞi ! ðNa3 CPÞo ! 3Naþ o þ ðCPÞo K6

ðCPÞo þ 2Koþ ! ðK2 CPÞo ! Pi þ ðK2 CÞi K8

ðK2 CÞi ! 2Kiþ þ Ci K9

ADPi !

484

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Na3C

Na3CP

Na+

P ATP

Inside

Outside

ADP

Outside

4 Na3CP

Inside

K+

Na+ K2CP

Inside

Outside

K2CP

Inside

K+

Inside ATP

Outside

FIGURE 8.22 The six steps that characterize the Na-K pump.

Using the law of mass action, the Na-K pump is characterized by the following set of differential equations: q_ Naþ ¼ JNaD K2 q3Naþ qCi i

q_ ðNa3 CÞ ¼ K2 q3Naþ qCi K3 qðNa3 CÞi qATPi i

q_ ATPi ¼ K1 K3 qðNa3 CÞi qATPi q_ ðNa3 CPÞ ¼ K3 qðNa3 CÞi qATPi K4 qðNa3 CPÞi i

q_ ðNa3 CPÞ ¼ K4 qðNa3 CPÞi K5 qðNa3 CPÞo o

q_ Naþo ¼ K5 qðNa3 CPÞo JNaD q_ ðCPÞ ¼ K5 qðNa3 CPÞo K6 q2Koþ qðCPÞo o

q_ Koþ ¼ JKD K6 q2Koþ qðCPÞo

ð8:84Þ

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP

485

q_ ðK2 CPÞ ¼ K7 qðK2 CPÞo o

q_ Pi ¼ K7 qðK2 CPÞo q_ ðK2 CÞ ¼ K7 qðK2 CPÞo K8 qðK2 CÞi i

q_ ADPi ¼ K9 qADPi q_ Ci ¼ K8 qðK2 CÞi K2 qCi q3Naþ i

q_ Kþ ¼ K8 qðK2 CÞi JKD i

where we assume a flow of K out of the cytosol at a rate of JKD due to diffusion, a flow of Naþ into the cell at a rate of JNaD from diffusion, ATP into the cytosol from the mitochondria at a rate of K1 , and ADP into the mitochondria from the cytosol at a rate of K9 : Note that there are other models that describe the Na-K pump based on different assumptions.

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP At this time we wish to discuss cellular respiration involving glucose metabolism in more detail by describing the chemical processes that enable the body to create energy in the form of ATP. As previously noted, ATP is the fuel that supports life’s processes, where energy is stored in the inorganic phosphate bonds. Energy is required to form these bonds in ATP, and energy is released when the bonds are broken. Cellular respiration is among the best-known metabolic pathways, with detailed models going back to the 1960s. While much progress has been made, much uncertainty exists, and the system is still under considerable research. Here we will provide sufficient coverage based on the techniques in this chapter and focus on aerobic respiration that uses oxygen to create ATP in the mitochondria. Cellular respiration consists of three major steps: glycolysis, Krebs cycle, and the electron transport chain, as illustrated in Figure 8.23. The first step occurs in the cytosol, and the two other steps occur in the mitochondria. Each step is very complex, with an overall reaction given by C6 H12 O6 þ 6O2 þ 36ADP þ 36P ! 6CO2 þ 6H2 O þ 36ATP

ð8:85Þ

which consists of the oxidation of glucose ðC6 H12 O6 Þ to CO2 and O2 , and a reduction of O2 to H2 O: Once inside the cell, 1 M of glucose is transformed into 36 M of ATP by adding an inorganic phosphate P to each ADP. While there is some adenosine monophosphate (AMP) that, when combined with two inorganic phosphates, creates ATP, we will ignore this part of the reaction in this section. Mitochondria are the powerhouses of the cell that produces the ATP used by the cell. A mitochondrion has two lipid bilayer membranes. The outer membrane is smooth and acts like a typical membrane. The inner membrane has many infoldings that contain the oxidative enzymes. Within the inner membrane is a cavity called the matrix that contains a nucleus with DNA, ribosomes, and dissolved enzymes. The matrix enzymes work with the oxidative enzymes to create ATP. The mitochondria’s nucleus allows it to self-replicate

486

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Glycolysis Glucose ⇒ 2 Pyruvate

Cytosol Electron Transport Chain

Outside 2FADHt2

Glucose

2(NADH + H +)

2Acetyl CoA

Glucose

2NADH + H ↓

6(NADH + H +)+ 2FADH2 Krebs Cycle

Mitochondria

+2 ATP

+32 ATP

2FADH2

+2 ATP

FIGURE 8.23

Cellular respiration consisting of three steps: glycolysis, Krebs cycle, and the electron transport chain. The amount of ATP created or lost in each step is indicated at the bottom of the figure. A total of 36 molecules of ATP are created from 1 molecule of glucose.

based on the needs of the cell for ATP. If there is an increased need for ATP, more mitochondria are produced. We call the process of adding an inorganic phosphate to a molecule phosphorylation. We call the use of O2 in the phosphorylation process oxidative phosphorylation. While the overall chemical reaction in Eq. (8.85) looks simple enough, it consists of many enzyme catalyzed reactions. A carrier-mediated transport process is the mechanism postulated to move glucose into the cell from the interstitial fluid. Ten enzyme catalyzed reactions take place in the cytosol during glycolysis,3 where a glucose molecule is broken down into two pyruvate ðC3 H4 O3 Þ molecules in which energy is consumed and produced. The overall reaction in glycolysis is given by C6 H12 O6 þ 2NADþ þ 2ADP þ 2P ! 2C3 H4 O3 þ 2ðNADH þ Hþ Þ þ 2ATP þ 2H2 O

ð8:86Þ

where NADþ is nicotinamide adenine dinucleotide and when combined with two hydrogen, forms NADH þ Hþ : During the electron transport chain, each molecule of NADH þ Hþ is used to create three molecules of ATP. The next two steps occur in the mitochondria. The second step begins with the conversion of pyruvate, C3 H4 O3 , into acetyl coenzyme A, followed by eight more major enzyme catalyzed reactions called the Krebs cycle.4 The overall result of the Krebs cycle is the

Glycolysis is also known as the Embden-Meyerhoff pathway.

The Krebs cycle is also known as the citric acid and tricarboxylic acid cycle.

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP

487

conversion of pyruvate into CO2 and NADH þ Hþ : The overall reaction for the Krebs cycle is given by Acetyl-CoA þ 3NADþ þ FAD þ 2H2 O þ ADP þ P! CoA SH þ 2CO2 þ FADH2 þ ATP þ 3ðNADH þ Hþ Þ

ð8:87Þ

where FAD is flavin adenine dinucleotide, and when combined with two hydrogen forms FADH2 : During the electron transport chain, each molecule of FADH2 is used to create two molecules of ATP. The last step in cellular respiration is the electron transport chain.5 Each of the large number of enzyme catalyzed reactions leading up to the electron transport chain contributes only two molecules of ATP, four molecules of FADH2 , and eight molecules of NADH þ Hþ : During the electron transport chain, NADH Hþ and FADH2 are used to create ATP by the oxidation of hydrogen atoms. Almost all of the energy created in cellular respiration occurs in the electron transport chain in the metabolism of glucose (32 molecules of ATP from one molecule of glucose). The overall reaction for the electron transport chain is given by 1 ðNADH þ Hþ Þ þ O2 þ 3ADP þ 3P ! NADþ þ 4H2 O þ 3ATP 2 and 1 FADH2 þ O2 þ 2ADP þ 2P ! FAD þ 3H2 O þ 2ATP 2

ð8:88Þ

The analysis in this section illustrates some of the major steps in converting glucose into ATP. To maximize the amount of energy created from glucose, many reactions occur, rather than a direct reaction of glucose into one ATP, water, and carbon dioxide. Using a series of reactions in the cytosol and the mitochondria, a total of 36 molecules of ATP are created from one molecule of glucose. We focus more detail on glycolysis and the Krebs cycle and a broader treatment of the electron transport chain. We assume that the process of transporting glucose into the cell occurs via a carrier-mediated transport and begins the glycolysis of glucose in the cytosol.

8.5.1 Glycolysis Glycolysis involves 10 enzyme catalyzed reactions in the cytosol, which transform one glucose molecule into two molecules of pyruvic acid, as shown in Figure 8.24. The intermediate complex reactions of the substrates and enzymes are not shown in Figure 8.24 for simplicity. However, they are included in the differential equations using a reaction rate constant for the intermediate complex given by Bi that corresponds to the reaction rate Ki : As shown in Figure 8.23, a net gain of 2 molecules of ATP and 2ðNADH þ HþÞ for each glucose molecule is seen during glycolysis. The major output of glycolysis is the creation of 2 pyruvate molecules from one molecule of glucose.

The electron transport chain is also known as the chemiosmotic mechanism.

488

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

(1)

ATP + Hexokinase (Hx) +

(2)

Phosphoglucose Isomerase (PI) +

Glucose (G)

Glucose-6-Phosphate (G6P)

+ + ADP + H +Hx

Fructose-6-Phosphate (F6P)

+ PI

Fructose-1,6-Bisphosphate (F6BP)

+ ADP + H ++PF

K2 Glucose-6-Phosphate K–2 (3)

ATP + Phosphofructokinase (PF) +

(4)

Aldolase (ALD) +

Fructose-6-Phosphate

K4 Fructose-1,6-Bisphosphate K–4

Dihydroxyacetone-Phosphate Glyceraldehyde-3-Phosphate + ALD + (DP) (G3P)

K5 (5) Triose Phosphate Isomerases (TPI) + Dihydroxyacetone-Phosphate

Glyceraldehyde-3-Phosphate

+ TPI

K–5 (6)

2(Glyceraldehyde 3-Phosphate Dehydrogenase) (GPD) + 2P + 2NAD++

K6 2(Glyceraldehyde-3-Phosphate)

K–6

(7)

2(Phosphoglycerate Kinase) (PGK) + 2ADP +

(8)

2(Phosphoglycerate Mutase) (PM) +

2(3-Phosphoglycerate)

2Enolase (EN) +

2(2-Phosphoglycerate)

2(1,3-Bisphosphoglycerate)

2(1,3-Bisphosphoglycerate) (BP)

+ + 2(NADH+H ) + 2GPD

2(3-Phosphoglycerate) (P3G) + 2ATP+2PGK

K8 2(2-Phosphoglycerate) (P2G) + 2PM K–8 K9

(9)

2(Phosphoenolpyruvate) (PP)

+ 2EN + H2O

2(Pyruvate) (PV)

+ 2ATP + 2PK

K–9 (10)

2Pyruvate Kinase (PK) + 2ADP +

2(Phosphoenolpyruvate)

K10

FIGURE 8.24 An overview of the 10 steps of glycolysis. Note that in steps 6–10, there are two molecules of everything, since 2 molecules of glyceraldehyde-3-phosphate are created for each glucose molecule. Also note that the intermediate complex reactions are not included in the illustration but are included in the differential equations describing glycolysis. Abreviations are listed next to each molecule.

Step 1 of glycolysis begins with the binding of glucose to a phosphate enzyme called hexokinase, resulting in glucose-6-phosphate. The immediate phosphorylation of glucose is required so glucose does not leave the cell. This reaction requires ATP and is given by B1

G þ Hx ! GHx K1 GHx þ ATP ! G6P þ ADP þ Hþ þ Hx q_ G ¼ B1 qG qHx þ JG q_ GHx ¼ B1 qG qHx K1 qATP qGHx

ð8:89Þ

where JG is the flow of glucose into the cell, as described in Section 8.4.3. The second step involves the reaction of glucose-6-phosphate with the enzyme phosphoglucose isomerase enzyme to create fructose-6-phosphate. This reaction is given by

ð8:90Þ

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP

489

The third step involves the reaction of fructose-6-phosphate and ATP with the enzyme phosphofructokinase to create fructose-1,6-bisphosphate and is given by B3

F6P þ PF ! F6PPF K3

F6PPF þ ATP ! F6BP þ ADP þ Hþ þ PF q_ F6P ¼ K2 qG6PPI K2 qF6P qPI B3 qF6P qPF q_ F6PPF ¼ B3 qF6P qPF K3 qF6PPF qATP

ð8:91Þ

The fourth step involves the reaction of fructose-1,6-bisphosphate with the enzyme aldolase to create dihydroxyacetone-phosphate and glyceraldehyde-3-phosphate and is given by

ð8:92Þ

The fifth step involves the reaction of dihydroxyacetone-phosphate with the enzyme triose phosphate isomerases to create glyceraldehyde-3-phosphate and is given by ð8:93Þ

The sixth step involves the reaction of glyceraldehyde-3-phosphate with the enzyme glyceraldehyde-3-phosphate dehydrogenase, an inorganic phosphate, and NADþ to create 1,3-bisphosphoglycerate and is given by

ð8:94Þ

The seventh step involves the reaction of 1,3-bisphosphoglycerate with the enzyme phosphoglycerate kinase and ADP to create 3-phosphoglycerate and ATP and is given by B7

BP þ PGK ! BPPGK K7

BPPGK þ ADP ! P3G þ PGK þ ATP q_ BP ¼ K6 qG3PD K6 qBP qNADHHþ qGPD B7 qBP qPGK q_ BPPGK ¼ B7 qBP qPGK K7 qBPPGK qADP Note that we have assumed only a forward reaction in this step.

ð8:95Þ

490

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

The eighth step involves the reaction of 3-phosphoglycerate with the enzyme phosphoglycerate mutase to create 2-phosphoglycerate and is given by

ð8:96Þ

The ninth step involves the reaction of 2-phosphoglycerate with the enzyme enolase to create phosphoenolpyruvate and water, and is given by

ð8:97Þ

The last step involves the reaction of phosphoenolpyruvate, ADP, and Hþ with the enzyme pyruvate kinase to create pyruvate and ATP and is given by B10

PP þ PK ! PPPK K10

PPPK þ ADP ! PV þ ATP þ PK q_ PP ¼ K9 qP2GEN K9 qPP qEN qH2 O B10 qPP qPK

ð8:98Þ

q_ PPPK ¼ B10 qPP qPK K10 qPPPK qADP q_ PV¼ K10 qPPPK qADP The equations for ATP and ADP in the cytosol are q_ ATP ¼ K1 qATP qGHx K3 qF6PPF qATP þ K7 qBPPGK qADP þ K10 qPPPK qADP þ JATP q_ ADP ¼ K1 qATP qGHx þ K3 qF6PPF qATP K7 qBPPGK qADP K10 qPPPK qADP þ JADP

ð8:99Þ

where JATP and JADP are the net usage of ATP and ADP from other processes, such as the reactions in Sections 8.5.2 and 8.5.3 in the mitochondria and the Na-K pump described in Section 8.4.4. ADP and ATP Movement in/out of the Mitochondria Let’s assume that the movement of ADP and ATP through the mitochondria’s membranes is by diffusion from the cytosol through the outer membrane and carrier-mediated diffusion through the inner membrane. Further, we assume that ADP only enters the mitochondria and that ATP only leaves the mitochondria for simplicity. The equations that describe this transport are given in Eqs. (8.100) and (8.101).

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP

491

ð8:100Þ

where the subscript c is for the cytosol, o is for the outer membrane, and i is for the matrix; Ds are the diffusivity constants and Bs are the reaction rates; Cs and Es are enzymes; and Rs and Ts are the intermediate complexes. The equations that describe this system are q_ ADPc ¼ D1 qADPc þ JADPc q_ ADPo ¼ D1 qADPc B1 qADPo qCo q_ ADPi ¼ B1 Ti JADPi q_ To ¼ B1 qADPo qCo B2 qTo þ B2 qTi q_ Ti ¼ B2 qTo ðB2 þ B1 ÞqTi

ð8:101Þ

q_ Co ¼ B1 qADP qCo þ B2 qCi B2 qCo q_ Ci ¼ B1 qTi þ B2 qCo B2 qCi q_ ATPi ¼ B3 qATPi qEi þ JATPi q_ Ri ¼ B3 qATPi qEi þ B4 Ro B4 Ri q_ Ro ¼ B4 qRi ðB4 þ B3 ÞqRo q_ ATPo ¼ B3 qRo D2 qATPo q_ ATPc ¼ D2 qATPo JATPc q_ Ei ¼ B3 qATPi qEi þ B5 qEo B5 qEi q_ Eo ¼ B3 qRo þ B5 qEi B5 qEo where JATPc and JATPi are the consumption/production of ATP in the cytosol and matrix, and JADPc and JADPi are the consumption/production of ADP in the cytosol and matrix.

492

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Conversion of Pyruvate to Acetyl CoA Pyruvate moves into the mitochondria by diffusion. Once inside the mitochondria, an enzyme catalyzed reaction occurs that converts pyruvate (PV) into acetyl coenzyme A (AC) as follows: B0

PV þ CoA SH ! PVCoA K0 PVCoA þ NADþ ! AC þ CoA SH þ CO2 þ ðNADH þ Hþ Þ

ð8:102Þ

q_ PV ¼ JPV B0 qPV qCoA q_ CoA ¼ B0 qPV qCoA þ K0 qPVCoA qNADþ q_ PVCoA ¼ B0 qPV qCoA K0 qPVCoA qNADþ q_ AC ¼ K0 qPVCoA qNADþ where CoA-SH (CoA) is coenzyme A, PVCoA is the complex pyruvate dehydrogenase, and JPV is the production of pyruvate given by Eq. (8.98). Keep in mind that from one glucose molecule, two pyruvate molecules are created that pass into the mitochondria. Also note that 2ðNADH þ Hþ Þ are converted into 2FADH2 to transfer acetyl coenzyme A across the mitochondrial membrane, thus costing 2 ATP.

8.5.2 Krebs Cycle The Krebs cycle involves a series of enzyme catalyzed reactions that reduce the acetyl portion of acetyl coenzyme A in the mitochondrial matrix, as shown in Figure 8.25. The Krebs cycle continuously recycles, reusing the substrates and enzymes with an overall reaction given by ! Acetyl-CoA þ 3NADþ þ FAD þ 2H2 O þ ADP þ P CoA SH þ 2CO2 þ FADH2 þ ATP þ 3ðNADH þ Hþ Þ

ð8:103Þ

The reaction begins with the joining of acetyl-coenzyme A with oxaloacetate and water to form citrate and is given by B1

AC þ H2 O þ CS þ O ! ACOCS ! CT þ CoA SH þ CS q_ AC ¼ B1 qAC qH2 O qCS qO þ JAC q_ ACOCS ¼ B1 qAC qH2 O qCS qO K1 qACOCS

ð8:104Þ

where JAC is the flow of acetyl coenzyme A based on Eq. (8.102). The next step involves the reaction of citrate with the enzyme aconitase to create isocitrate, which is given by

ð8:105Þ

493

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP (1)

H2O + Citrate Synthase (CS) + Acetyl CoA (AC)

Oxaloacetate (O)

Citrate (CT)

+ CoA-SH + AC

(2)

Citrate

Aconitase (AT) +

Isocitrate (IT)

+ AT

K–2

(3)

NAD + + Isocitrate Dehydrogenase (ID) +

Isocitrate

(4)

a-Ketoglutarate Dehydrogenase (KGH) + NAD + + CoA-SH +

(5)

Succinyl-CoA Synthetase (SCS) + ADP + P +

Succinyl-CoA

Succinate Dehydrogenase (STD) + FAD +

Succinate

Fumarase (FS) + H2O +

Fumarate

(6)

a-Ketoglutarate (KG)

K–3

a-Ketoglutarate

Succinyl-CoA (SC)

+ (NADH+H +) + CO2 + KGH

Succinate (ST)

+ CoA-SH + ATP

Fumarate (FT)

+ FADH2 + STD

K–5 K6 K–6 K7

(7)

+ (NADH+H +) + CO2 + ID

Malate (MT)

+ FS

Oxaloacetate

+ MTD+ (NADH+H +)

K–7

(8)

Malate Dehydrogenase (MTD) + NAD++

Malate K–8

FIGURE 8.25

Overview of the Krebs cycle.

The third step involves the reaction of isocitrate and NADþ with the enzyme isocitrate dehydrogenase to create a ketoglutarate, ðNADH þ Hþ Þ and CO2 , which is given by

ð8:106Þ

The fourth step involves the reaction of a ketoglutarate, NADþ and CoA with the enzyme a ketoglutarate dehydrogenase to create succinyl CoA, ðNADH þ Hþ Þ, and CO2 : This reaction is given by B4

KG þ KGH ! KGKGH K4

KGKGH þ NADþ þ CoA SH ! SC þ KGH þ ðNADH þ Hþ Þ þ CO2 q_ KG ¼ K3 qITID qNADþ B4 qKG qKGH K3 qKG qID qNADHþHþ qCO2 q_ KGKGH ¼ B4 qKG qKGH K4 qKGKGH qNADþ qCoA

ð8:107Þ

494

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

The fifth step involves the reaction of succinyl CoA, ADP, and P with the enzyme succinyl CoA synthetase, to create succinate, CoA and ATP: This reaction is given by

ð8:108Þ

The sixth step involves the reaction of succinate and FAD with the enzyme succinate dehydrogenase to create fumarate and FADH2. This reaction is given by

ð8:109Þ

The seventh step involves the reaction of fumarate and H2 O with the enzyme fumarase to create malate. This reaction is given by

ð8:110Þ

The last step involves the reaction of malate and NADþ with the enzyme malate dehydrogenase to create oxaloacetate and (NADH þ Hþ). This reaction is given by

ð8:111Þ

The final equation combines the oxaloacetate from the first reaction with that in the last reaction: q_ O ¼ K8 qMTMTD qNADþ K8 qO qMTD qNADHþHþ B1 qO

ð8:112Þ

It should be clear that this reaction continually recycles and that it requires two cycles to process each glucose molecule.

495

8.5 CELLULAR RESPIRATION: GLUCOSE METABOLISM AND THE CREATION OF ATP

8.5.3 Electron Transport Chain The electron transport chain is the last step in the conversion of glucose into ATP, as illustrated in Figure 8.26. It involves a series of enzyme catalyzed chemical reactions that transfer electrons from ðNADH þ Hþ Þ and FADH2 (donor molecules) to acceptor molecules. Ultimately the electron transport chain produces 32 molecules of ATP from one molecule of glucose through hydrogen oxidation, and also regenerates NAD and FAD for reuse in glycolysis. The overall reaction is given by 1 ðNADH þ Hþ Þ þ O2 þ 3ADP þ 3P ! NADþ þ 4H2 O þ 3ATP 2 and 1 FADH2 þ O2 þ 2ADP þ 2P ! FAD þ 3H2 O þ 2ATP 2

ð8:113Þ

The electron transport chain activity takes place in the inner membrane and the space between the inner and outer membrane, called the intermembrane space. In addition to one molecule of ATP created during each Krebs cycle, three pairs of hydrogen are released and bound to 3NADþ to create 3ðNADH þ Hþ Þ, and one pair of hydrogen is bound to FAD to form FADH2 within the mitochondrial matrix. As described before, two cycles through the Krebs cycle are needed to fully oxidize one molecule of glucose, and thus 6ðNADH þ Hþ Þ and 2FADH2 molecules are created.

Mitochondrion Outer Membrane

Inner Membrane H+ H+

ADP

e –e – 1 Q

H+ H+ Cytosol

e –e –

ATP C

Intermembrane Space

e –e – 3

H+ H+

NAD + NADH + H + FAD FADH2 H2 O 1 2H + + O2 2

ATP

4 H+

Matrix

ADP + P H+ ATP

5 ADP

FIGURE 8.26

A simplified illustration of the mitochondrion electric transport chain. Hydrogen pumps are labeled 1 (NADH dehydrogenase), 2 (cytochrome bc1 complex), and 3 (cytochrome c oxidase complex). Electron carriers are labeled Q (Coenzyme Q) and C (cytochrome c). The conversion of ADPþP to ATP is accomplished in the protein channel 4 (ATP synthetase), which also moves hydrogen ions back into the matrix, where they are used again in sites 1–3. Carrier mediated diffusion exchanges ATP and ADP between the matrix and the intermembrane space. Then ATP and ADP are exchanged between the intermembrane space and the cytosol by diffusion.

496

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

The energy stored in these molecules of ðNADH þ Hþ Þ and FADH2 is used to create ATP by the release of hydrogen ions through the inner membrane and electrons within the inner membrane. The energy released by the transfer of each pair of electrons from ðNADH þ H þ Þ and FADH2 is used to pump a pair of hydrogen ions into the intermembrane space. The transfer of a pair of electrons is through a chain of acceptors from one to another, with each transfer providing the energy to move another pair of hydrogen ions through the membrane. At the end of the acceptor chain, the two electrons reduce an oxygen atom to form an oxygen ion, which is then combined with a pair of hydrogen ions to form H2 O: The movement of the hydrogen ions creates a large concentration of positively charged ions in the intermembrane space and a large concentration of negatively charged ions in the matrix, which sets up a large electrical potential. This potential is used by the enzyme ATP synthase to transfer hydrogen ions into the matrix and to create ATP. The ATP produced in this process is transported out of the mitochondrial matrix through the inner membrane using carrier facilitated diffusion and diffusion through the outer membrane. In the following description, we assume all of the hydrogen and electrons are available from these reactions. In reality, some are lost and not used to create ATP. Other descriptions of the electron transport chain have additional sites and are omitted here for simplicity. We first consider the use of ðNADH þ H þ Þ in the electron transport chain. During the first step, a pair of electrons from NADH þ Hþ are transferred to the electron carrier coenzyme Q by NADH dehydrogenase (site 1 and Q in Figure 8.26), and using the energy released, a pair of hydrogen ions are pumped into the intermembrane space. Next, the coenzyme Q carries the pair of electrons to the cytochrome bc1 complex (site 2 in Figure 8.26). When the pair of electrons are transfered from the cytochrome bc1 complex to cytochrome c (site C in Figure 8.26), the energy released is used to pump another pair of hydrogen ions into the intermembrane space through the cytochrome bc1 complex. In the third step, cytochrome c transfers electrons to the cytochrome c oxidase complex (site 3 in Figure 8.26), and another pair of hydrogen ions are pumped through the cytochrome c oxidase complex into the intermembrane space. A total of 6 hydrogen ions have now been pumped into the intermembrane space, which will allow the subseqent creation of 3 molecules of ATP. Also occuring in this step, the cytochrome oxidase complex transfers the pair of electrons within the inner membrane from the cytochrome c to oxygen in the matrix. Oxygen then combines with a pair of hydrogen ions to form water. As described previously, the transfer of hydrogen ions into the intermembrane space creates a large concentration of positive charges and a large concentration of negative charges in the matrix, creating a large electrical potential across the inner membrane. The energy from this potential is used in this step by the enzyme ATP synthase (site 4 in Figure 8.26) to move hydrogen ions in the intermembrane space into the matrix and to synthesize ATP from ADP and P. The ATP in the matrix is then transported into the intermembrane space and ADP is transported into the matrix using a carrier-mediated transport process (site 5 in Figure 8.26). From the intermembrane space, ATP diffuses through the outer membrane into the cytosol, and ADP diffuses from the cytosol into the intermembrane space.

8.6 ENZYME INHIBITION, ALLOSTERIC MODIFIERS, AND COOPERATIVE REACTIONS

497

In parallel with ðNADH þ Hþ Þ, FADH2 goes through a similar process but starts at coenzyme Q, where it directly provides a pair of electrons. Thus, FADH2 provides two fewer hydrogen ions than ðNADH þ Hþ Þ. The focus of this section has been the synthesis of ATP. Glycolysis and the Krebs cycle are also important in the synthesis of small molecules such as amino acids and nucleotides, and large molecules such as proteins, DNA, and RNA. There are other metabolic pathways to store and release energy that were not covered here. The interested reader can learn more about these pathways using the references at the end of this chapter and the website http://www.genome.jp.

8.6 ENZYME INHIBITION, ALLOSTERIC MODIFIERS, AND COOPERATIVE REACTIONS Up to this point, we have considered the case of an enzyme binding with one substrate to make a product. Here, we examine the case in which the enzyme is free to bind with more than one molecule and form a product. As we will see, these reactions can regulate the amount of product synthesized and change the overall reaction rate in forming the product. In this section, we begin with enzyme inhibitors that are either competitive or allosteric. A competitive enzyme inhibitor, referred to as the inhibitor, binds to the active site on the enzyme and prevents the substrate from binding with the enzyme. Thus, the inhibitor competes with the substrate to bind with the substrate and reduces the synthesis of the product and its overall reaction rate. Some enzymes have more than one binding site and are called allosteric enzymes. The site that binds with the substrate is called the active site. The other site, called the allosteric or regulatory site, binds with another molecule called a modifier or effector. A modifier binds to the regulatory site on the enzyme and doesn’t directly block the binding of the substrate with the enzyme at the active site. The effector role is to either increase (allosteric activator) or decrease (allosteric inhibitor) the activity of the enzyme. Some refer to allosteric inhibition as noncompetitive inhibition. Finally, we examine reactions that are cooperative. These reactions are sequential and have a sigmoidal reaction velocity.

8.6.1 Competitive Enzyme Inhibitors Consider an enzyme catalyzed reaction between a substrate, S, and enzyme, E, synthesizing product, P1 , as ð8:114Þ and another enzyme reaction, where enzyme inhibitor I reacts with E to form product P2 as ð8:115Þ where C1 and C2 are the intermediate complexes. Note that we have eliminated the reverse reaction from the product to the intermediate complex. Another form of a competitive enzyme inhibitor eliminates the formation of product P2 ði:e ., K4 ¼ 0Þ:

498

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

The equations that describe this system in Eqs. (8.114) and (8.115) are q_ S ¼ K1 qC1 K1 qS qE q_ C1 ¼ K1 qS qE ðK1 þ K2 ÞqC1

ð8:116Þ

q_ I ¼ K3 qC2 K2 qI qE q_ C2 ¼ K3 qI qE ðK3 þ K4 ÞqC2 We eliminate qE from Eq. (8.116) by using qE ¼ E0 qC1 qC2 , giving q_ S ¼ ðK1 qS þ K1 ÞqC1 þ K1 qS qC2 K1 E0 qS q_ C1 ¼ K1 E0 qS ðK1 qS þ K1 þ K2 ÞqC1 K1 qS qC2 q_ I ¼ ðK3 qI þ K3 ÞqC2 þ K3 qI qC1 K3 E0 qI

ð8:117Þ

q_ C2 ¼ K3 E0 qI ðK3 qI þ K3 þ K4 ÞqC2 K3 qI qC1 with nonzero initial conditions of qS ð0Þ ¼ S0 , qI ð0Þ ¼ I0 and qE ð0Þ ¼ E0 : The quasi-steady-state approximation is found from Eq. (8.117) with q_ C1 ¼ q_ C2 ¼ 0, yielding 0 ¼ K1 E0 qS ðK1 qS þ K1 þ K2 ÞqC1 K1 qS qC2 0 ¼ K3 E0 qI ðK3 qI þ K3 þ K4 ÞqC2 K3 qI qC1

ð8:118Þ

Next, we solve for qC2 , which gives qC2 ¼

K3 qI ðE0 qC1 Þ K3 qI þ K3 þ K4

ð8:119Þ

qC1 ¼

E0 qS q s qS þ KM 1 þ KiI

ð8:120Þ

and then we solve for qC1 , giving

where

s KM

K1 þK2 K1

and

i KM

K3 þK4 K3

V ¼ K2 q C 1 ¼

: The velocity of the reaction is given by

K2 E0 qS Vmax ¼ s K q q I s 1þ qS þ KM 1 þ qMS 1 þ KiI Ki M

ð8:121Þ

where Vmax ¼ K2 E0 : Comparing Eq. (8.121) with Eq. (8.47), we see that Vmax does not change with the inclusion of an enzyme inhibitor. In this case the term KM in Eq. (8.47) has been replaced by the q s term KM 1 þ KiI

in Eq. (8.121), which reduces the reaction rate. The left side of Figure 8.27

shows a plot of the reaction rate versus the substrate with increasing quantities of the

499

8.6 ENZYME INHIBITION, ALLOSTERIC MODIFIERS, AND COOPERATIVE REACTIONS

1 V(0)

qI(0) = b

qI = 0 qI = 10

qI(0) = a

qI = 30

30 V

qI(0) = 0

20 10 0 0

10 20 30 40 50 60 70 80 90 100 qs

1 qS(0)

FIGURE 8.27 (Left) Velocity of product appearance for a substrate, enzyme, and enzyme inhibitor using s i Eq. (8.121). V max ¼ 50, KM ¼ 3, and KM ¼ 3: (Right) Lineweaver-Burk plot, where b > a.

inhibitor enzyme. As shown, increasing the quantity of the enzyme inhibitor shows a slower synthesis of the product, P1 : Keep in mind that all of the substrate will eventually be synthesized into the P1 but it does so more slowly. To obtain the Lineweaver-Burk equation for this system, we take the reciprocal of Eq. (8.121), giving 1 1 1 qI s K ð8:122Þ 1þ 1þ i ¼ V Vmax qS M KM and at t ¼ 0, we have 1 1 1 qI ð0Þ s KM ¼ 1þ 1þ i Vð0Þ Vmax qS ð0Þ KM

ð8:123Þ

The plot of the Lineweaver-Burk equation is shown on the right side of Figure 8.27 for three values of the quantity of the enzyme inhibitor. One again, the relationship between 1 1 1 Vð0Þ vs: qs ð0Þ is a straight line. Further, we note that the intercept of the Vð0Þ axis is a constant as the quantity of the enzyme inhibitor is increased. Additionally, the slope of the line increases as the quantity of the enzyme inhibitor is increased, which is indicative of a slowing reaction rate.

8.6.2 Allosteric Activators and Inhibitors Next, consider the noncompetitive allosteric modifier that binds with an enzyme on a regulatory site and a substrate that binds with the enzyme on the active site. The effect of a modifier on the reaction is to either increase or decrease the activity of the enzyme.

500

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS

Consider an enzyme catalyzed reaction between a substrate, S, modifier, M, and enzyme, E, synthesizing product, P, as

ð8:124Þ

where C1 ¼ SE, C2 ¼ ME, and C3 ¼ SME are the intermediate complexes.6 Note that we have eliminated the reverse reaction from the product to the intermediate complex. Another variation of allosteric reaction eliminates the product from forming through C3 : The equations that describe the system in Eq. (8.124) are q_ S ¼ K1 qC1 þ K4 qC3 K1 qS qE K4 qS qC2 q_ M ¼ K3 qC2 þ K6 qC3 K3 qM qE K6 qM qC1 q_ C1 ¼ K1 qS qE þ K6 qC3 ðK1 þ K2 ÞqC1 K6 qM qC1 q_ C2 ¼ K3 qM qE þ ðK4 þ K5 ÞqC3 K3 qC2 K4 qS qC2

ð8:125Þ

q_ C3 ¼ K4 qS qC2 þ K6 qM qC1 ðK4 þ K5 þ K6 ÞqC3 q_ P ¼ K2 qC1 þ K5 qC3 We eliminate qE from Eq. (8.125) by using qE ¼ E0 qC1 qC2 qC3 , giving q_ S ¼ ðK1 þ K1 qS ÞqC1 þ ðK4 þ K1 qS ÞqC3 K1 qS E0 ðK4 K1 ÞqS qC2 q_ M ¼ ðK3 þ K3 qM ÞqC2 þ ðK6 þ K3 qM ÞqC3 K3 qM E0 ðK6 K3 ÞqM qC1 q_ C1 ¼ K1 qS E0 K1 qS qC2 þ ðK6 K1 qS ÞqC3 ðK1 þ K2 þ K6 qM þ K1 qS ÞqC1 q_ C2 ¼ K3 qM E0 K3 qM qC1 þ ðK4 þ K5 K3 qM ÞqC3 ðK3 qM þ K3 þ K4 qS ÞqC2 q_ C3 ¼ K4 qS qC2 þ K6 qM qC1 ðK4 þ K5 þ K6 ÞqC3 q_ P ¼ K2 qC1 þ K5 qC3 with nonzero initial conditions of qS ð0Þ ¼ S0 , qM ð0Þ ¼ M0 , and qE ð0Þ ¼ E0 :

This model is adapted from Rubinow, page 89.

ð8:126Þ

8.6 ENZYME INHIBITION, ALLOSTERIC MODIFIERS, AND COOPERATIVE REACTIONS

501

Quasi-Steady-State Approximation to the Reaction Rate The quasi-steady-state approximation is found from Eq. (8.126) with q_ C1 ¼ q_ C2 ¼ q_ C3 ¼ 0, yielding 0 ¼ K1 qS E0 K1 qS qC2 þ ðK6 K1 qS ÞqC3 ðK1 þ K2 þ K6 qM þ K1 qS ÞqC1 0 ¼ K3 qM E0 K3 qM qC1 þ ðK4 þ K5 K3 qM ÞqC3 ðK3 qM þ K3 þ K4 qS ÞqC2

ð8:127Þ

0 ¼ K4 qS qC2 þ K6 qM qC1 ðK4 þ K5 þ K6 ÞqC3 Equation (8.127) can be solved for qC1 and qC3 using the D-Operator method or the graph theory method from Rubinow, giving

qC1 ¼

9 8 0 1 =

K K K > > 1 1 1 s i A > > > > K6i KM þ K3i @ K6i þ KM > > > > K K K 4 6 > > = < 3 þ qS 0 1 > > > > > > > > K K K 1 1 1 > i i s i 2 @ A > > þqM K6 þ KM þ K3 þ KM þ qM > > > ; : K3 K4 K6 0

K1 s KM K3i @ K4

K6i

1 K1 i A þ KM K6

9 8 0 1 = < K K 1 s @ 1 i i K6 þ K3i A þ K3i KM þ qM K M ; : K4 K6 i 2 qM þ KM

ð8:128Þ

502

8. BIOCHEMICAL REACTIONS AND ENZYME KINETICS s KM ¼

K1 þ K2 K1

i KM ¼

K4 þ K5 K4

K3i ¼

K3 K3

K6i ¼

K6 K6

The reaction rate is V ¼ q_ P ¼ K2 qC1 þ K5 qC3 93 8 0 1 2 = A ¼ [7 2;2 6]; >> F ¼ [10;3]; >> V ¼ A\F V¼ 1.4211 0.0263

Thus, V1 ¼ 1.4211 V.

EXAMPLE PROBLEM 9.6 For the following circuit, find V3 using the node-voltage method. 1/4 Ω

2Id

1/4 Ω

1/2 Ω

1/3 Ω

1Ω

Continued

534

9. BIOINSTRUMENTATION

Solution Notice that this circuit has three essential nodes and a dependent current source. We label the essential nodes 1, 2, and 3 in the redrawn circuit, with the reference node at the bottom of the circuit and three node voltages V1, V2, and V3, as indicated. 1/4 Ω

2Id 1

1/4 Ω

2 + 1/2 Ω

3 +

1/3 Ω

1Ω

Note that Id ¼ V3 according to Ohm’s law. Summing the currents leaving node 1 gives 5 þ 2ðV1 V2 Þ þ 2Id þ 4ðV1 V3 Þ ¼ 0 which reduces to 6V1 2V2 2V3 ¼ 5 Summing the currents leaving node 2 gives 2Id þ 2ðV2 V1 Þ þ 3V2 þ 4ðV2 V3 Þ ¼ 0 which simplifies to 2V1 þ 9V2 6V3 ¼ 0 Summing the currents leaving node 3 gives 4ðV3 V2 Þ 3 þ V3 þ 4ðV3 V1 Þ ¼ 0 reducing to 4V1 4V2 þ 9V3 ¼ 3 The three node equations are written in matrix format as 2 3 2 3 2 3 6 2 2 V1 5 4 2 5 4 5 4 9 6 05 V2 ¼ 4 4 9 3 V3

9.5 LINEAR NETWORK ANALYSIS

535

Notice that the system matrix is no longer symmetrical because of the dependent current source, and two of the three nodes have a current source, giving rise to a nonzero term on the right-hand side of the matrix equation. Solving with MATLAB gives >> A ¼ [6 2 2;2 9 6;4 4 9]; >> F ¼ [5; 0; 3]; >> V ¼ A\F V¼ 1.1471 0.5294 0.4118

Thus, V3 ¼ – 0.4118 V.

If one of the branches has an independent or controlled voltage source located between two essential nodes, as shown in Figure 9.21, the current through the source is not easily expressed in terms of node voltages. In this situation, we form a supernode by combining the two nodes. The supernode technique requires only one node equation in which the current, IA, is passed through the source and written in terms of currents leaving node 2. Specifically, we replace IA with IB þ IC þ ID in terms of node voltages. Because we have two unknowns and one supernode equation, we write a second equation by applying KVL for the two node voltages 1 and 2 and the source as V1 VD þ V2 ¼ 0 or VD ¼ V1 V2

+ -

VΔ

+ 2

FIGURE 9.21 A dependent voltage source is located between nodes 1 and 2.

536

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.7 For the following circuit, find V3.

+ −

1/5 Ω

1/4 Ω

1/2 Ω

+ 1V

1/3 Ω

1/2 Ω

Solution The circuit has three essential nodes, two of which are connected to an independent voltage source and form a supernode. We label the essential nodes as 1, 2, and 3 in the redrawn circuit, with the reference node at the bottom of the circuit and three node voltages, V1, V2, and V3 as indicated. 1/5 Ω

2 +

1/4 Ω

3 +

+ 1V

1/2 Ω

1/3 Ω

Summing the currents leaving node 1 gives 2 þ 2V1 þ 5ðV1 V3 Þ þ 4ðV1 V2 Þ ¼ 0 Simplifying gives 11V1 4V2 5V3 ¼ 2

1/2 Ω

9.6 LINEARITY AND SUPERPOSITION

537

Nodes 2 and 3 are connected by an independent voltage source, so we form a supernode 2þ3. Summing the currents leaving the supernode 2þ3 gives 4ðV2 V1 Þ þ 3V2 1 þ 2V3 þ 5ðV3 V1 Þ ¼ 0 Simplifying yields 9V1 þ 7V2 þ 7V3 ¼ 1 The second supernode equation is KVL through the node voltages and the independent source, giving V2 þ 1 þ V3 ¼ 0 or V2 þ V3 ¼ 1 The two node and KVL equations are written in matrix format as 3 2 3 2 3 2 2 11 4 5 V1 4 9 7 7 5 4 V2 5 ¼ 4 1 5 V3 1 0 1 1 Solving with MATLAB gives A ¼ [11 4 5;9 7 7; 0 1 1]; F ¼ [2;1;1]; V ¼ A\F V¼ 0.4110 0.8356 0.1644

Thus, V3 ¼ – 0.1644.

9.6 LINEARITY AND SUPERPOSITION If a linear system is excited by two or more independent sources, then the total response is the sum of the separate individual responses to each input. This property is called the principle of superposition. Specifically for circuits, the response to several independent sources is the sum of responses to each independent source with the other independent sources dead, where • A dead voltage source is a short circuit. • A dead current source is an open circuit. In linear circuits with multiple independent sources, the total response is the sum of each independent source taken one at a time. This analysis is carried out by removing all of

538

9. BIOINSTRUMENTATION

the sources except one and assuming the other sources are dead. After the circuit is analyzed with the first source, it is set equal to a dead source, and the next source is applied with the remaining sources dead. When each of the sources has been analyzed, the total response is obtained by summing the individual responses. Note carefully that this principle holds true solely for independent sources. Dependent sources must remain in the circuit when applying this technique, and they must be analyzed based on the current or voltage for which it is defined. It should be apparent that voltages and currents in one circuit differ among circuits and that we cannot mix and match voltages and currents from one circuit with another. Generally, superposition provides a simpler solution than is obtained by evaluating the total response with all of the applied sources. This property is especially valuable when dealing with an input consisting of a pulse or delays. These are considered in future sections.

EXAMPLE PROBLEM 9.8 Using superposition, find V0 as shown in the following figure. 2Ω

3Ω +

10 V

+ −

5Ω

Solution Start by analyzing the circuit with just the 10 V source active and the two current sources dead, as shown in the following figure. 2Ω

3 Ω +

10 V

+ −

V010

5Ω

539

9.6 LINEARITY AND SUPERPOSITION

The voltage divider rule easily gives the response, V010 , due to the 10 V source 8 V010 ¼ 10 ¼8V 2þ8 Next consider the 2 A source active and the other two sources dead, as shown in the following circuit. 2Ω

3Ω +

V02

5Ω

Combining the resistors in an equivalent resistance, REQ ¼ 2 k ð3 þ 5Þ ¼

28 ¼ 1:6 O, and then 2þ8

applying Ohm’s law gives V02 ¼ 2 1:6 ¼ 3:2 V. Finally, consider the response,V03 , to the 3 A source as shown in the following figure.

2Ω

3Ω +

V03

5Ω

To find V03 , note that the 3 A current splits into 1.5 A through each branch (2 þ 3 O and 5 O), and V03 ¼ 1:5 2 ¼ 3 V: The total response is given by the sum of the individual responses as V0 ¼ V010 þ V02 þ V03 ¼ 8 þ 3:2 3 ¼ 8:2 V This is the same result we would have found if we analyzed the original circuit directly using the node-voltage method.

540

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.9 Find the voltage across the 5 A current source, V5 , in the following figure using superposition. 3V0

3Ω 5Ω

10V

+ −

2Ω

−

Solution First consider finding the response, V010 , due to the 10 V source only, with the 5 A source dead, as shown in the following figure. As required during the analysis, the dependent current source is kept in the modified circuit and should not be set dead. 3V010

3Ω

I=0A

A 5Ω

10 V

+ −

V010

3V010

2Ω

V510

Notice that no current flows through the open circuit created by the dead current source and that the current flowing through the 5 O resistor is 3V0 . Therefore, applying KCL at node A gives V010 10 V010 þ þ 3V010 3V010 ¼ 0 3 2 which gives V010 ¼ 4V. KVL gives V010 5 3V010 þ V510 ¼ 0, and therefore V510 ¼ 64 V:

9.7 THE´VENIN’S THEOREM

541

Next, consider finding the response, V05 , due to the 5 A source, with the 10 V source dead.

3V05

3Ω 5Ω

V05

B +

2Ω

V55

First combine the two resistors in parallel (3O k 2O), giving 1.2 O. V05 is easily calculated by Ohm’s law as V05 ¼ 5 1:2 ¼ 6V: KCL is then applied at node B to find I5 , giving 3V05 þ I5 5 ¼ 0 With V05 ¼ 6 V, I5 ¼ 3 6 þ 5 ¼ 23 A: Finally, apply KVL around the closed path V05 5I5 þ V55 ¼ 0 or V55 ¼ V05 þ 5I5 ¼ 6 þ 5 23 ¼ 121 V: The total response is given by the sum of the individual responses as V5 ¼ V510 þ V55 ¼ 64 þ 121 ¼ 185 V

9.7 THE´VENIN’S THEOREM Any combination of resistances, controlled sources, and independent sources with two external terminals (A and B, denoted A,B) can be replaced by a single resistance and an independent source, as shown in Figure 9.22. A The´venin equivalent circuit reduces the original circuit into a voltage source in series with a resistor. This theorem helps reduce complex circuits into simpler circuits. We refer to the circuit elements connected across the terminals A,B (that are not shown) as the load. The The´venin equivalent circuit is

542

9. BIOINSTRUMENTATION

A REQ

Independent and Dependent Sources, and Resistances

Replaced by

VOC

B B

FIGURE 9.22

A general circuit consisting of independent and dependent sources can be replaced by a voltage source (Voc) in series with a resistor (REQ).

equivalent to the original circuit in that the same voltage and current are observed across any load. Usually the load is not included in the simplification because it is important for other analysis, such as maximum power expended by the load. Although we focus here on sources and resistors, this theorem can be extended to any circuit composed of linear elements with two terminals. The´venin’s Theorem states that an equivalent circuit consisting of an ideal voltage source, VOC , in series with an equivalent resistance, REQ, can be used to replace any circuit that consists of independent and dependent voltage and current sources and resistors. VOC is equal to the open circuit voltage across terminals A,B, as shown in Figure 9.23, and calculated using standard techniques such as the node-voltage method. The resistor REQ is the resistance seen across the terminals A,B when all sources are dead. Recall that a dead voltage source is a short circuit, and a dead current source is an open circuit.

Independent and Dependent Sources, and Resistances

VOC

B The open circuit voltage, Voc, is calculated across the terminals A,B using standard techniques such as the node-voltage method.

FIGURE 9.23

9.7 THE´VENIN’S THEOREM

543

EXAMPLE PROBLEM 9.10 Find the The´venin equivalent circuit with respect to terminals A,B for the following circuit. 2Ω A

10 V

+ −

2Ω

Solution The solution to finding the The´venin equivalent circuit is done in two parts: first finding VOC and then solving for REQ. The open circuit voltage, VOC, is easily found using the node-voltage method, as shown in the following circuit. 2Ω A +

10 V

2Ω

−

VOC

The sum of currents leaving the node is VOC 10 VOC 4¼0 þ 2 2 and VOC ¼ 9 V: Next, REQ is found by first setting all sources dead (the current source is an open circuit and the voltage source is a short circuit) and then finding the resistance seen from the terminals A,B, as shown in the following figure. Continued

544

9. BIOINSTRUMENTATION

2Ω A

2Ω

B REQ

From the previous circuit, it is clear that REQ is equal to 1O (that is, 2O k 2O). Thus, the The´venin equivalent circuit is 1Ω A

+ −

It is important to note that the circuit used in finding VOC is not to be used in finding REQ as not all voltages and currents are relevant in the other circuit and one cannot simply mix and match.

If the terminals A,B are shorted as shown in Figure 9.24, the current that flows is denoted ISC , and the following relationship holds: REQ ¼

VOC ISC

ð9:18Þ

9.8 INDUCTORS In the previous sections of this chapter, we considered circuits involving sources and resistors that are described with algebraic equations. Any changes in the source are instantaneously observed in the response. In this section we examine the inductor, a passive element that relates the voltage-current relationship with a differential equation. Circuits that

545

9.8 INDUCTORS

A ISC Independent and Dependent Sources, and Resistances

FIGURE 9.24 The short circuit current, Isc, is calculated by placing a short across the terminals A,B and finding the current through the short using standard techniques such as the node-voltage method.

contain inductors are written in terms of derivatives and integrals. Any changes in the source with circuits that contain inductors—that is, a step input—have a response that is not instantaneous but have a natural response that changes exponentially and a forced response that is the same form as the source. An inductor is a passive element that is able to store energy in a magnetic field and is made by winding a coil of wire around a core that is an insulator or a ferromagnetic material. A magnetic field is established when current flows through the coil. We use the symbol to represent the inductor in a circuit; the unit of measure for inductance is the henry or henries (H), where 1 H ¼ 1 V – s/A. The relationship between voltage and current for an inductor is given by v¼L

di dt

ð9:19Þ

The convention for writing the voltage drop across an inductor is similar to that of a resistor, as shown in Figure 9.25. Physically, current cannot change instantaneously through an inductor, since an infinite voltage is required, according to Eq. (9.19) (i.e., the derivative of current at the time of the instantaneous change is infinity). Mathematically, a step change in current through an inductor is possible by applying a voltage that is a Dirac delta function. For convenience, when a circuit has just DC currents (or voltages), the inductors can be replaced by short circuits, since the voltage drops across the inductors are zero.

FIGURE 9.25

An inductor.

546

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.11 Find v in the following circuit. i (A)

+ v -

t (s)

Solution The solution to this problem is best approached by breaking it up into time intervals consistent with the changes in input current. Clearly, for t < 0 and t > 2, the current is zero and therefore v ¼ 0. We use Eq. (9.19) to determine the voltage in the other two intervals as follows.

For 0 < t < 1

In this interval, the input is i ¼ t, and v¼L

di dðtÞ ¼2 ¼2V dt dt

For 1 t 2

In this interval, the input is i ¼ ðt 2Þ, and v¼L

di dððt 2ÞÞ ¼2 ¼ 2 V dt dt

Equation (9.19) defines the voltage across an inductor for a given current. Suppose one is given a voltage across an inductor and asked to find the current. We start from Eq. (9.19) by multiplying both sides by dt, giving vðtÞdt ¼ Ldi Integrating both sides yields Zt

ZiðtÞ vðlÞdl ¼ L da

iðt0 Þ

or iðtÞ ¼

1 L

Zt vðlÞdl þ iðt0 Þ t0

ð9:20Þ

547

9.8 INDUCTORS

For t0 ¼ 0, as is often the case in solving circuit problems, Eq. (9.20) reduces to 1 iðtÞ ¼ L

Zt vðlÞdl þ ið0Þ

ð9:21Þ

and for t0 ¼ 1, the initial current is by definition equal to zero, and therefore Eq. (9.20) reduces to 1 iðtÞ ¼ L

Zt vðlÞdl

ð9:22Þ

The initial current in Eq. (9.20), iðt0 Þ, is usually defined in the same direction as i, which means iðt0 Þ is a positive quantity. If the direction of iðt0 Þ is in the opposite direction of i (as will happen when we write node equations), then iðt0 Þ is negative.

EXAMPLE PROBLEM 9.12 Find i for t 0 if i(0) ¼ 2 A and vðtÞ ¼ 4e3t uðtÞ in the following circuit.

+ −

Solution From Eq. (9.20), we have 1 iðtÞ ¼ L

Zt t0

1 vdl þ iðt0 Þ ¼ 2

e3l t ¼2 þ2 3 l¼0 ¼

2 4 e3t uðtÞ V 3

Zt 0

4e3l dl þ 2

548

9. BIOINSTRUMENTATION

9.9 CAPACITORS A capacitor is a device that stores energy in an electric field by charge separation when appropriately polarized by a voltage. Simple capacitors consist of parallel plates of conducting material that are separated by a gap filled with a dielectric material. Dielectric materials—that is, air, mica, or Teflon—contain a large number of electric dipoles that become polarized in the presence of an electric field. The charge separation caused by the polarization of the dielectric is proportional to the external voltage and given by qðtÞ ¼ C vðtÞ

ð9:23Þ

where C represents the capacitance of the element. The unit of measure for capacitance is the farad or farads (F), where 1 F ¼ 1 C/V. We use the symbol

to denote a capacitor;

most capacitors are measured in terms of microfarads (1 mF ¼ 10 F) or picofarads (1 pF ¼ 1012 F). Figure 9.26 illustrates a capacitor in a circuit. Using the relationship between current and charge, Eq. (9.23) is written in a more useful form for circuit analysis problems as dq dv ¼C ð9:24Þ i¼ dt dt The capacitance of a capacitor is determined by the permittivity of the dielectric F (e ¼ 8:854 1012 for air) that fills the gap between the parallel plates, the size of the m gap between the plates, d, and the cross-sectional area of the plates, A, as eA C¼ ð9:25Þ d As described, the capacitor physically consists of two conducting surfaces that store charge, separated by a thin insulating material that has a very large resistance. In actuality, current does not flow through the capacitor plates. Rather, as James Clerk Maxwell hypothesized when he described the unified electromagnetic theory, a displacement current flows internally between capacitor plates, and this current equals the current flowing into the capacitor and out of the capacitor. Thus, KCL is maintained. It should be clear from Eq. (9.29) that dielectric materials do not conduct DC currents; capacitors act as open circuits when DC currents are present. i

+ −

FIGURE 9.26

Circuit with a capacitor.

549

9.9 CAPACITORS

EXAMPLE PROBLEM 9.13 Find i for the following circuit. i v (V)

+ −

t (s)

Solution For t < 0 and t > 2, v ¼ 0 V, and therefore i ¼ 0 in this interval: For nonzero values, the voltage waveform is described with two different functions: v ¼ t V for 0 t 1, and v ¼ ðt 2ÞV for 1 < t 2: Equation (9.24) is used to determine the current for each interval as follows.

For 0 < t < 1 i¼C

dv d ¼ 2 ðt Þ ¼ 2 A dt dt

For 1 t 2 i¼C

dv d ¼ 2 ððt 2ÞÞ ¼ 2 A dt dt

Voltage cannot change instantaneously across a capacitor. To have a step change in voltage across a capacitor, an infinite current must flow through the capacitor, and that is not physically possible. Of course, this is mathematically possible using a Dirac delta function. Equation (9.24) defines the current through a capacitor for a given voltage. Suppose one is given a current through a capacitor and asked to find the voltage. To find the voltage, we start from Eq. (9.24) by multiplying both sides by dt, giving iðtÞdt ¼ C dv Integrating both sides yields Zt to

ZvðtÞ iðlÞdl ¼ C dv vðto Þ

550

9. BIOINSTRUMENTATION

or 1 vðtÞ ¼ C

Zt idt þ vðto Þ

ð9:26Þ

For t0 ¼ 0, Eq. (9.26) reduces to 1 vðtÞ ¼ C

Zt idt þ vð0Þ

ð9:27Þ

and for t0 ¼ 1, Eq. (9.27) reduces to 1 vðtÞ ¼ C

Zt iðlÞdl

ð9:28Þ

The initial voltage in Eq. (9.26), vðt0 Þ, is usually defined with the same polarity as v, which means vðt0 Þ is a positive quantity. If the polarity of vðt0 Þ is in the opposite direction of v, then vðt0 Þ is negative.

EXAMPLE PROBLEM 9.14 Find v for the circuit that follows. is

(A) 2 is

t (s)

Solution The current waveform is described with three different functions: for the interval t 0, for the interval 0 < t 2, and for t > 2. To find the voltage, we apply Eq. (9.28) for each interval as follows.

For t < 0 vðtÞ ¼

1 C

idt ¼

1 2

Z0 0dt ¼ 0 V 1

551

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS

For 0 t 2 vðtÞ ¼

1 C

Zt idt þ vð0Þ 0

and with v(0) ¼ 0, we have 1 vðtÞ ¼ 2

Zt l dl ¼ 0

t 1 l2 t2 ¼ V 2 2 0 4

The voltage at t ¼ 2 needed for the initial condition in the next part is t2 vð2Þ ¼ ¼ 1 V 4 t¼2

For t > 2 1 vðtÞ ¼ C

1 idt þ vð2Þ ¼ 2

Zt 0dt þ vð2Þ ¼ 1 V 2

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS INVOLVING RESISTORS, CAPACITORS, AND INDUCTORS Sometimes a circuit consisting of resistors, inductors, and capacitors cannot be simplified by bringing together like elements in series and parallel combinations. Consider the circuit shown in Figure 9.27. In this case, the absence of parallel or series combinations of resistors, inductors, or capacitors prevents us from simplifying the circuit for ease in solution. In this section, the node-voltage method is applied to write equations involving integrals and differentials using element relationships for resistors, inductors, and capacitors. From these equations, any unknown currents and voltages of interest can be solved using the standard differential equation approach. 5H

3Ω 2H

2Ω

FIGURE 9.27 A circuit that cannot be simplified.

552

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.15 Write the node equations for the following circuit for t 0 if the initial conditions are zero. L1

C1 + −

vs(t)

Solution With the reference node at the bottom of the circuit, we have two essential nodes, as shown in the following redrawn circuit. Recall that the node involving the voltage source is a known voltage and that we do not write a node equation for it. When writing the node-voltage equations, _ where Dv_ is the derivative of the voltage across the the current through a capacitor is ic ¼ C Dv, Zt 1 Dvdl þ iL ð0Þ, where Dv is the voltage capacitor, and the current through an inductor is iL ¼ L 0

across the inductor. Since the initial conditions are zero, the term iL ð0Þ ¼ 0: L1 2

1 + C1

vs(t)

+ −

−

Summing the currents leaving node 1 gives C1 ðv_ 1 v_ s Þ þ

v1 v1 v2 þ ¼0 R1 R2

−

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS

which simplifies to

C1 v_ 1 þ

553

1 1 1 þ v1 v2 ¼ C1 v_ s R1 R2 R2

Summing the currents leaving node 2 gives v2 v1 1 þ C2 v_ 2 þ R2 L1

Zt ðv2 vs Þ dl ¼ 0 0

Typically we eliminate integrals in the node equations by differentiating. When applied to the previous expression, this gives 1 1 1 1 v_ 2 v_ 1 þ C2 € v2 þ v2 vs ¼ 0 R2 R2 L1 L1 and after rearranging yields € v2 þ

1 1 1 1 v_ 2 þ v_ 1 ¼ v2 vs C2 R2 C2 L1 C2 R2 C2 L1

When applying the node-voltage method, we generate one equation for each essential node. To write a single differential equation involving just one node voltage and the inputs, we use the other node equations and substitute into the node equation of the desired node voltage. Sometimes this involves differentiation as well as substitution. The easiest case involves a node equation containing an undesired node voltage without its derivatives. Another method for creating a single differential equation is to use the D operator or the Laplace transform. Consider the node equations for Example Problem 9.15, and assume that we are interested in obtaining a single differential equation involving node voltage v1 and its derivatives, and the input. For ease in analysis, let us assume that the values for the circuit elements are R1 ¼ R2 ¼ 1 O, C1 ¼ C2 ¼ 1 F, and L1 ¼ 1 H, giving us v_ 1 þ 2v1 v2 ¼ v_ s and € v2 þ v_ 2 þ v2 v_ 1 ¼ vs Using the first equation, we solve for v2, calculate v_ 2 and €v2 , and then substitute into the second equation as follows. v2 ¼ v_ 1 þ 2v1 v_ s €1 þ 2v_ 1 €vs v_ 2 ¼ v € v2 ¼ ___ v1 þ 2€v1 ___ vs After substituting into the second node equation, we have ___ v1 þ 2€ v1 ___ vs þ € v1 þ 2v_ 1 €vs þ v_ 1 þ 2v1 v_ s v_ 1 ¼ vs and after simplifying v1 þ 2v_ 1 þ 2v1 ¼ ___ vs þ €vs þ v_ s vs ___ v1 þ 3€

554

9. BIOINSTRUMENTATION

In general, the order of the differential equation relating a single output variable and the inputs is equal to the number of energy storing elements in the circuit (capacitors and inductors). In some circuits, the order of the differential equation is less than the number of capacitors and inductors in the circuit. This occurs when capacitor voltages and inductor currents are not independent; that is, there is an algebraic relationship between the capacitor—specifically, voltages and the inputs, or the inductor currents and the inputs. This occurs when capacitors are connected directly to a voltage source or when inductors are connected directly to a current source. Example Problem 9.15 involved a circuit with zero initial conditions. When circuits involve nonzero initial conditions, our approach remains the same as before except that the initial inductor currents are included when writing the node-voltage equations.

EXAMPLE PROBLEM 9.16 Write the node equations for the following circuit for t 0 assuming the initial conditions are iL1 ð0Þ ¼ 8 A and iL2 ð0Þ ¼ 4 A: iL1 1H

10 Ω

iL2

+ −

Solution With the reference node at the bottom of the circuit, there are three essential nodes, as shown in the redrawn circuit that follows. iL1

1 10 Ω

+ −

−

iL2

−

3 +

−

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS

555

Summing the currents leaving node 1 gives ð v1 vs Þ þ v_ 1 þ 10

Zt ðv1 v2 Þ dl þ 8 ¼ 0 0

where iL1 ð0Þ ¼ 8 A: Summing the currents leaving node 2 gives Zt

1 ðv2 v1 Þ dl 8 þ 2v_ 2 þ 2

Zt ðv2 v3 Þ dl 4 ¼ 0 0

where iL2 ð0Þ ¼ 4 A: Notice that the sign for the initial inductor current is negative because the direction is from right to left and the current is defined on the circuit diagram in the opposite direction for the node 2 equation. Summing the currents leaving node 3 gives 1 2

Zt ðv3 v2 Þ dl þ 4 þ 3v_ 3 ¼ 0 0

In this example, the node equations were not simplified by differentiating to remove the integral, which would have eliminated the initial inductor currents from the node equations. If we were to write a single differential equation involving just one node voltage and the input, a fifth-order differential equation would result because there are five energy storing elements in the circuit. To solve the differential equation, we would need five initial conditions, the initial node voltage for the variable selected, and the first through fourth derivatives at time zero.

9.10.1 Discontinuities and Initial Conditions in a Circuit Discontinuities in voltage and current occur when an input such as a unit step is applied or a switch is thrown in a circuit. As we have seen, when solving an nth order differential equation, one must know n initial conditions, typically the output variable and its (n – 1) derivatives at the time the input is applied or the switch thrown. As we will see, if the inputs to a circuit are known for all time, we can solve for initial conditions directly based on energy considerations and not have to depend on being provided with them in the problem statement. Almost all of our problems involve the input applied at time zero, so our discussion here is focused on time zero, but it may be easily extended to any time an input is applied. Energy cannot change instantaneously for elements that store energy. Thus, there are no discontinuities allowed in current through an inductor or voltage across a capacitor at any time—specifically, the value of the variable remains the same at t ¼ 0 and t ¼ 0þ : In the previous problem when we were given initial conditions for the inductors and capacitors, this implied, iL1 ð0 Þ ¼ iL1 ð0þ Þ and iL2 ð0 Þ ¼ iL2 ð0þ Þ, and v1 ð0 Þ ¼ v1 ð0þ Þ, v2 ð0 Þ ¼ v2 ð0þ Þ, and v3 ð0 Þ ¼ v3 ð0þ Þ: With the exception of variables associated with current through an inductor and voltage across a capacitor, other variables can have discontinuities, especially at a time when a unit step is applied or when a switch is thrown; however, these variables must obey KVL and KCL.

556

9. BIOINSTRUMENTATION

While it may not seem obvious at first, a discontinuity is allowed for the derivative of the current through an inductor and voltage across a capacitor at t ¼ 0 and t ¼ 0þ , since diL ð0 þÞ vL ð0 þÞ dvC ð0 þÞ iC ð0 þÞ ¼ and ¼ dt L dt L as discontinuities are allowed in vL ð0 þÞ and iC ð0 þÞ: Keep in mind that the derivatives in the previous expression are evaluated at zero after differentiation—that is, diL ð0 þÞ diL ðtÞ dvC ð0 þÞ dvC ðtÞ ¼ ¼ and dt dt t¼0þ dt dt t¼0þ In calculations to determine the derivatives of variables not associated with current through an inductor and voltage across a capacitor, the derivative of a unit step input may be needed. Here we assume the derivative of a unit step input is zero at t ¼ 0þ . The initial conditions for variables not associated with current through an inductor and voltage across a capacitor at times of a discontinuity are determined only from the initial conditions from variables associated with current through an inductor and voltage across a capacitor and any applicable sources. The analysis is done in two steps involving KCL and KVL or using the node-voltage method. 1. First, we analyze the circuit at t ¼ 0 . Recall that when a circuit is at steady state, an inductor acts as a short circuit and a capacitor acts as an open circuit. Thus, at steady-state at t ¼ 0 , we replace all inductors by short circuits and capacitors by open circuits in the circuit. We then solve for the appropriate currents and voltages in the circuit to find the currents through the inductors (actually the shorts connecting the sources and resistors) and voltages across the capacitors (actually the open circuits among the sources and resistors). 2. Second, we analyze the circuit at t ¼ 0þ . Since the inductor current cannot change in going from t ¼ 0 to t ¼ 0þ , we replace the inductors with current sources whose values are the currents at t ¼ 0 : Moreover, since the capacitor voltage cannot change in going from t ¼ 0 to t ¼ 0þ , we replace the capacitors with voltage sources whose values are the voltages at t ¼ 0 : From this circuit we solve for all desired initial conditions necessary to solve the differential equation.

EXAMPLE PROBLEM 9.17 Use the node-voltage method to find vc for the following circuit for t 0. iR1

iR2 400 Ω

5u(t) V

+ − +

iR3 100 Ω

5μF

10 V −

−

iL 10 mH

500 Ω

557

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS

Solution For t 0, the circuit is redrawn for analysis in the following figure. C 400 Ω

L 100 Ω

iC + 15 V

−

5mF

−

10 mH

500 Ω

−

Summing the currents leaving node C gives vC 15 vC vL ¼0 þ 5 106 v_ C þ 100 400 which simplifies to v_ C þ 2500vC 2000vL ¼ 7500 Summing the currents leaving node L gives vL vC 1 þ 100 10 103

vL dl þ iL ð0þ Þ þ

vL ¼0 500

which, after multiplying by 500 and differentiating, simplifies to 6v_ L þ 50 103 vL 5v_ C ¼ 0 Using the D operator method, the two differential equations are written as DvC þ 2500vC 2000vL ¼ 7500 or ðD þ 2500ÞvC 2000vL ¼ 7500 6DvL þ 50 103 vL 5DvC ¼ 0 or 6D þ 50 103 vL 5DvC ¼ 0 We then solve for vL from the first equation, vL ¼ 0:5 103 D þ 1:25 vC 3:75 and then substitute vL into the second equation, giving 6D þ 50 103 vL 5DvC ¼ 6D þ 50 103 0:5 103 D þ 1:25 vC 3:750 5DvC ¼ 0 Reducing this expression yields D2 vC þ 10:417 103 DvC þ 20:83 106 vC ¼ 62:5 106

Continued

558

9. BIOINSTRUMENTATION

Returning to the time domain gives € vC þ 10:417 103 v_ C þ 20:83 106 vC ¼ 62:5 106 The characteristic equation for the previous differential equation is s2 þ 10:417 103 s þ 20:833 106 ¼ 0 with roots 7:718 103 and 2:7 103 and the natural solution vCn ðtÞ ¼ K1 e7:718 10 t þ K2 e2:7 10 t V 3

Next, we solve for the forced response, assuming that vCf ðtÞ ¼ K3 . After substituting into the differential equation, this gives 20:833 106 K3 ¼ 62:5 106 or K3 ¼ 3. Thus, our solution is now vC ðtÞ ¼ vCn ðtÞ þ vCf ðtÞ ¼ K1 e7:718 10 t þ K2 e2:7 10 t þ 3 V 3

Initial conditions for vC ð0þ Þ and v_ C ð0þ Þ are necessary to solve for K1 and K2 : For t ¼ 0 , the capacitor is replaced by an open circuit and the inductor by a short circuit as shown in the following circuit. iR

iR1

iR3

400 Ω

100 Ω

iL(0 −)

+ vC(0 −)

10 V

− −

Notice vL ð0 Þ ¼ 0 V because the inductor is a short circuit. Also note that the 500 O resistor is not shown in the circuit, since it is shorted out by the inductor, and so iR3 ð0 Þ ¼ 0A: Using the voltage divider rule, we have vC ð0 Þ ¼ 10

100 ¼2V 400 þ 100

and by Ohm’s law iL ð0 Þ ¼

10 ¼ 0:02 A 100 þ 400

It follows that iR1 ð0 Þ ¼ iR2 ð0 Þ ¼ iL ð0 Þ ¼ 0:02 A: Because voltage across a capacitor and current through an inductor are not allowed to change from t ¼ 0 to t ¼ 0þ we have vC ð0þ Þ ¼ vC ð0 Þ ¼ 2V and iL ð0þ Þ ¼ iL ð0 Þ ¼ 0:02 A:

559

9.10 A GENERAL APPROACH TO SOLVING CIRCUITS

The circuit for t ¼ 0þ is drawn by replacing the inductors in the original circuit with current sources whose values equal the inductor currents at t ¼ 0 and the capacitors with voltage sources whose values equal the capacitor voltages at t ¼ 0 , as shown in the following figure with nodes C and L and reference. Note also that the input is now 10 þ 5uðtÞ ¼ 15 V. iR1

iR3

iR2 C

400 Ω

15 V

+ −

+ 100 Ω

+ −

+ iL

0.02 A

500 Ω

−

To find vL ð0þ Þ, we sum the currents leaving node L, yielding vL 2 vL ¼0 þ 0:02 þ 500 100 which gives vL ð0þ Þ ¼ 0 V: Now iR3 ð0þ Þ ¼

vL ð 0þ Þ ¼ 0 A, iR2 ð0þ Þ ¼ 0:02 þ iR3 ð0þ Þ ¼ 0:02 A, and 500

15 2 ¼ 0:0325 A: 400 þ To find iC ð0 Þ, we write KCL at node C, giving

iR1 ð0þ Þ ¼

iR1 ð0þ Þ þ iC ð0þ Þ þ iR2 ð0þ Þ ¼ 0 or iC ð0þ Þ ¼ iR1 ð0þ Þ iR2 ð0þ Þ ¼ 0:0325 0:02 ¼ 0:125 A To find v_ C ð0þ Þ, note that iC ð0þ Þ ¼ Cv_ C ð0þ Þ or v_ C ð0þ Þ ¼

iC ð0þ Þ 0:0125 V ¼ 2:5 103 : ¼ C s 5 106

With the initial conditions, the constants K1 and K2 are solved as vC ð0Þ ¼ 2 ¼ K1 þ K2 þ 3 Next, v_ C ðtÞ ¼ 7:718 103 K1 e7:71810 t 2:7 103 K2 e2:710 t 3

Continued

560

9. BIOINSTRUMENTATION

and at t ¼ 0, v_ C ð0Þ ¼ 2:5 103 ¼ 7:718 103 K1 2:7 103 K2 Solving gives K1 ¼ 0:04 and K2 ¼ 1:04: Substituting these values into the solution gives vC ðtÞ ¼ 0:04e7:71810 t 1:04e2:710 t þ 3 V 3

for t 0:

9.11 OPERATIONAL AMPLIFIERS Section 9.3 introduced controlled voltage and current sources that are dependent on a voltage or current elsewhere in a circuit. These devices were modeled as a two-terminal device. In this section, we look at the operational amplifier, also known as an op amp, which is a multiterminal device. An operational amplifier is an electronic device that consists of large numbers of transistors, resistors, and capacitors. Fully understanding its operation requires knowledge of diodes and transistors—topics that are not covered in this book. However, fully understanding how an operational amplifier operates in a circuit involves a topic already covered: the controlled voltage source. Circuits involving operational amplifiers form the cornerstone for any bioinstrumentation, from amplifiers to filters. Amplifiers used in biomedical applications have very high-input impedance to keep the current drawn from the system being measured low. Most body signals have very small magnitudes. For example, an ECG has a magnitude in the millivolts, and the EEG has a magnitude in the microvolts. Analog filters are often used to remove noise from a signal, typically through frequency domain analysis to design the filter. As the name implies, the operation amplifier is an amplifier, but as we will see, when it is combined with other circuit elements, it integrates, differentiates, sums, and subtracts. One of the first operational amplifiers appeared as an eight-lead dual-in-line package (DIP), shown in Figure 9.28. Differing from previous circuit elements, this device has two input and one output terminals. Rather than draw the operational amplifier using Figure 9.28, the operational amplifier is drawn with the symbol in Figure 9.29. The input terminals are labeled the noninverting input (þ) and the inverting input (–). The power supply terminals are labeled Vþ and V–, which are frequently omitted, since they do not affect the circuit behavior except in saturation conditions, as will be described. Most people shorten the name of the operational amplifier to the “op amp.” Figure 9.30 shows a model of the op amp, focusing on the internal behavior of the input and output terminals. The input-output relationship is v o ¼ A vp v n ð9:29Þ Since the internal resistance is very large, we will replace it with an open circuit to simplify analysis, leaving us with the op amp model show in Figure 9.31. With the replacement of the internal resistance with an open circuit, the currents in ¼ ip ¼ 0 A. In addition, current iA, the current flowing out of the op amp, is not zero. Because iA is unknown, seldom is KCL applied at the output junction. In solving op amp problems, KCL is almost always applied at input terminals.

561

9.11 OPERATIONAL AMPLIFIERS

Offset Null

Inverting Input

Eight Terminal Operational Amplifier

Noninverting Input

V+

Output

Offset Null

V-

FIGURE 9.28 An eight-terminal operational amplifier. The terminal NC is not connected, and the two terminal offset nulls are used to correct imperfections (typically not connected). Vþ and V– are terminal power to provide energy to the circuit. Keep in mind that a ground exists for both Vþ and V–, a ground that is shared by other elements in the circuit. Modern operational amplifiers have ten or more terminals.

V+ Inverting Input Output Noninverting Input V−

FIGURE 9.29

Circuit element symbol for the operational amplifier.

in vn

−

A (vp−vn)

− +

vo iA

ip vp

FIGURE 9.30

An internal model of the op amp. The internal resistance between the input terminals, R, is very large, exceeding 1 MΩ. The gain of the amplifier, A, is also large, exceeding 104. Power supply terminals are omitted for simplicity.

562

9. BIOINSTRUMENTATION

−

vo= A (vp−vn) iA

ip +

FIGURE 9.31 Idealized model of the op amp with the internal resistance, R, replaced by an open circuit.

EXAMPLE PROBLEM 9.18 Find v0 for the following circuit. i2 R2 i1

−

+ R1 + VS

+ −

vn vp −

−

Solution Using the op amp model of Figure 9.31, we apply KCL at the inverting terminal giving i1 i2 ¼ 0 since no current flows into the op amp’s input terminals. Replacing the current using Ohm’s law gives vs vn vo vn þ ¼0 R1 R2 Multiplying by R1 R2 and collecting like terms, we have R2 vs ¼ ðR1 þ R2 Þvn R1 vo

9.11 OPERATIONAL AMPLIFIERS

563

Now vo ¼ A vp vn , and since the noninverting terminal is connected to ground, vp ¼ 0, vo ¼ Avn or vo A Substituting vn into the KCL inverting input equation gives 0 1 v o Rs vs ¼ ðR1 þ R2 Þ@ A R1 vo A 0 1 R 1 þ R2 @ þ R1 Avo ¼ A vn ¼

or R2 vs vo ¼ R1 þ R2 R1 þ A As A goes to infinity, the previous equation goes to vo ¼

R2 vs R1

Interestingly, with A going to infinity, v0 remains finite due to the resistor R2. This happens because a negative feedback path exists between the output and the inverting input terminal R2 through R2. This circuit is called an inverting amplifier with an overall gain of . R1

An operational amplifier with a gain of infinity is known as an ideal op amp. Because of the infinite gain, there must be a feedback path between the output and input, and we cannot connect a voltage source directly between the inverting and noninverting input terminals. When analyzing an ideal op amp circuit, we simplify the analysis by letting vn ¼ vp Consider the previous example. Because vp ¼ 0, vn ¼ 0. Applying KCL at the inverting input gives

vs vo þ ¼0 R1 R2

or vo ¼

R2 vs R1

Notice how simple the analysis becomes when we assume vn ¼ vp . Keep in mind that this approximation is valid as long as A is very large (infinity) and a feedback is included.

564

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.19 Find the overall gain for the following circuit.

+ vp

−

+ vn

+ −

R2 i1

R1 −

−

Solution Assuming the op amp is ideal, we start with vn ¼ vp . Then, since the op amp’s noninverting terminal is connected to the source, vn ¼ vp ¼ vs . Because no current flows into the op amp, by KCL we have i1 þ i2 ¼ 0 and vs vs vo þ ¼0 R1 R2 or vo ¼

R1 þ R2 vs R1

The overall gain is vo R1 þ R2 ¼ vs R1 This circuit is a noninverting op amp circuit used to amplify the source input. Amplifiers are used in almost all clinical instrumentation for ECG, EEG, EOG, and so on.

565

9.11 OPERATIONAL AMPLIFIERS

Example Problem 9.20 describes a summing op amp circuit.

EXAMPLE PROBLEM 9.20 Find the overall gain for the following circuit.

Ra Rb

−

+ +

+ −

vn vp −

−

+ v0 −

Solution As before, we start the solution with vn ¼ vp and note that the noninverting input is connected to ground, yielding vn ¼ vp ¼ 0 V. Applying KCL at the inverting input node gives

Va Vb vo ¼0 Ra Rb R2

or R2 R2 Va þ Vb vo ¼ Ra Rb This circuit is a weighted summation of the input voltages. We can add additional source resistor inputs so that in general R2 R2 R2 vo ¼ Va þ Vb þ . . . þ Vm Ra Rb Rm

566

9. BIOINSTRUMENTATION

The op amp circuit in Example Problem 9.21 provides an output proportional to the difference of two input voltages. This op amp is often referred to as a differential amplifier.

EXAMPLE PROBLEM 9.21 Find the overall gain for the following circuit. i2

ib R1 Va

+ −

ia + vn

+ vp

+ − −

− +

−

Solution Assuming an ideal op amp, we note no current flows into the input terminals and that vn ¼ vp . Apply KCL at the inverting input terminal gives ia ¼ i2 or vn Va vn vo þ ¼0 R1 R2 and ðR1 þ R2 Þvn R2 Va ¼ R1 vo The previous equation involves two unknowns, so we need another equation easily found by applying the voltage divider at the noninverting input. R2 v b ¼ vn R1 þ R2 Substituting this result for vn into the KCL equation at the inverting terminal gives vp ¼

R2 Vb R2 Va ¼ R1 vo or vo ¼

R2 ðVb Va Þ R1

As shown, this op amp circuit, also known as the differential amplifier, subtracts the weighted input signals. This amplifier is used for bipolar measurements involving ECG and EEG, since the typical recording is obtained between two bipolar input terminals. Ideally, the measurement

9.11 OPERATIONAL AMPLIFIERS

567

contains only the signal of interest uncontaminated by noise from the environment. The noise is typically called a common-mode signal. A common-mode signal comes from lighting, 60-Hz power line signals, inadequate grounding, and power supply leakage. A differential amplifier with appropriate filtering can reduce the impact of a common-mode signal.

The response of a differential amplifier can be decomposed into differential-mode and common-mode components: vdm ¼ vb va and vcm ¼

ðva þ vb Þ 2

As described, the common-mode signal is the average of the input voltages. Using the two previous equations, one can solve va and vb in terms of vdm and vcm as va ¼ vcm

vdm 2

vb ¼ vcm þ

vdm 2

and

When substituted into the response in Example Problem 9.21, we get R1 R2 R1 R2 R2 ðR1 þ R2 Þ þ R2 ðR1 þ R2 Þ v0 ¼ vcm þ vdm ¼ Acm vcm þ Adm vdm 2R1 ðR1 þ R2 Þ R1 ðR1 þ R2 Þ Notice the term multiplying vcm, Acm, is zero, characteristic of the ideal op amp that amplifies only the differential-mode of the signal. Since real amplifiers are not ideal and resistors are not truly exact, the common-mode gain is not zero. So when one designs a differential amplifier, the goal is to keep Acm as small as possible and Adm as large as possible. The rejection of the common-mode signal is called common-mode rejection, and the measure of how ideal the differential amplifier is called the common-mode rejection ratio, given as Adm CMRR ¼ 20 log10 Acm where the larger the value of CMRR, the better. Values of CMRR for a differential amplifier for EEG, ECG, and EMG are 100 to 120 db. The general approach to solving op amp circuits is to first assume that the op amp is ideal and vp ¼ vn . Next, we apply KCL or KVL at the two input terminals. In more complex circuits, we continue to apply our circuit analysis tools to solve the problem, as Example Problem 9.22 illustrates.

568

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.22 Find v0 for the following circuit. R2 −

+ vn + vp R2

+ −

1 +

− −

v0 R2

−

Solution With vn ¼ vp , we apply KCL at the inverting input vn v1 vn vo þ ¼0 R2 R2 and 2vn v1 vo ¼ 0 Next, we apply KVL from ground to node 1 to the noninverting input and back to ground, giving v1 Vs þ vp ¼ 0 and with vn ¼ vp , we have vn v1 ¼ Vs : Now we apply KCL at node 1, noting no current flows into the noninverting input terminal: v1 v1 vo v1 vn þ þ ¼0 R1 R2 R2 Combining like terms in the previous equation gives R1 vn þ ð2R1 þ R2 Þv1 R1 vo ¼ 0 With three equations and three unknowns, we first eliminate v1 by subtracting the inverting input KCL equation by the KVL equation, giving v1 ¼ vo 2Vs

569

9.11 OPERATIONAL AMPLIFIERS

Next, we eliminate vn by substituting v1 into the inverting input KCL equation, as follows: 1 v n ¼ ð v1 þ v o Þ 2 1 ¼ ðvo 2Vs þ vo Þ 2 ¼ vo Vs Finally, we substitute the solutions for v1 and vn into the node 1 KCL equation, giving R1 vn þ ð2R1 þ R2 Þv1 R1 vo ¼ 0 R1 ðvo Vs Þ þ ð2R1 þ R2 Þðvo 2Vs Þ R1 vo ¼ 0 After simplification, we have vo ¼

ð3R1 þ 2R2 Þ Vs R2

Example Problems 9.23 and 9.24 illustrate an op amp circuit that differentiates and integrates by using a capacitor.

EXAMPLE PROBLEM 9.23 Find v0 for the following circuit. iR

+ vC − −

+ C + VS

+ −

−

vp −

v0 −

Solution With the noninverting input connected to ground, we have vp ¼ 0 ¼ vn . From KVL vC ¼ Vs Continued

570

9. BIOINSTRUMENTATION

and it follows that iC ¼ C

dvC dVs ¼C dt dt

Since no current flows into the op amp, iC ¼ iR . With iR ¼

vn vo vo ¼ R R

and iC ¼ C

dVs vo ¼ iR ¼ dt R

we have vo ¼ RC If R ¼

dVs dt

1 dVs . , the circuit in this example differentiates the input, vo ¼ dt C

EXAMPLE PROBLEM 9.24 Find v0 for the following circuit. + vC −

− R

+ VS

+ −

Solution It follows that vn ¼ vp ¼ 0 and iC ¼ iR ¼

Vs R

571

9.11 OPERATIONAL AMPLIFIERS

Therefore, vC ¼

1 C

Zt iC dl ¼

1 C

Zt 1

Vs dl R

From KVL, we have vC þ vo ¼ 0 and vo ¼ With R ¼

1 RC

Zt Vs dl 1

1 , the circuit operates as an integrator C Zt Vs dl vo ¼ 1

9.11.1 Voltage Characteristics of the Op Amp In the preceding examples involving the op amp, we did not consider the supply voltage (shown in Figure 9.29) and that the output voltage of an ideal op amp is constrained to operate between the supply voltages Vþ and V. If analysis determines v0 is greater than Vþ, v0 saturates at Vþ. If analysis determines v0 is less than V, v0 saturates at V. The output voltage characteristics are shown in Figure 9.32. v0 V+

V+

V−

FIGURE 9.32

Voltage characteristics of an op amp.

A (vp-vn)

572

9. BIOINSTRUMENTATION

EXAMPLE PROBLEM 9.25 For the circuit shown in Example Problem 9.22, let V þ ¼ þ10 V and V ¼ 10 V. Graph the output voltage characteristics of the circuit.

Solution The solution for Example Problem 9.22 is 3R1 þ 2R2 Vs vo ¼ R2 which saturates whenever v0 is less than V and greater than Vþ, as shown in the following graph. v0 V+ 3R + 2R2 slope is 1 R2

V−

9.12 TIME-VARYING SIGNALS An alternating current (a-c) or sinusoidal source of 50 or 60 Hz is common throughout the world as a power source supplying energy for most equipment and other devices. While most of this chapter has focused on the transient response, when dealing with sinusoidal sources, attention is now focused on the steady-state or forced response. In bioinstrumentation, analysis in the steady-state simplifies the design by focusing only on the steadystate response, which is where the device actually operates. A sinusoidal voltage source is a time-varying signal given by vs ¼ Vm cosðot þ fÞ

ð9:30Þ

where the voltage is defined by angular frequency (o in radians/s), phase angle (f in radians or degrees), and peak magnitude (Vm). The period of the sinusoid T is related to frequency f (Hz or cycles/s) and angular frequency by o ¼ 2Pf ¼

2P T

ð9:31Þ

573

9.12 TIME-VARYING SIGNALS

An important metric of a sinusoid is its rms value (square root of the mean value of the squared function), given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ZT u u1 2 cos2 ðot þ fÞdt Vm ð9:32Þ Vrms ¼ t T 0

Vm which reduces to Vrms ¼ pffiffiffi : 2 To appreciate the response to a time-varying input, vs ¼ Vm cos ðot þ fÞ, consider the circuit shown in Figure 9.33, in which the switch is closed at t ¼ 0 and there is no initial energy stored in the inductor. Applying KVL to the circuit gives di þ iR ¼ Vm cos ðot þ fÞ dt and after some work, the solution is L

i ¼ in þ if

Vm oL ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos@f Ae R R2 þ o2 L 2

R t L

Vm oL þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos@ot þ f A R R 2 þ o 2 L2

The first term is the natural response that goes to zero as t goes to infinity. The second term is the forced response that has the same form as the input (i.e., a sinusoid with the same frequency o, but a different phase angle and maximum amplitude). If all you are interested in is the steady-state response, as in most bioinstrumentation applications, then the only unknowns are the response amplitude and phase angle. The remainder of this section deals with techniques involving the phasor to efficiently find these unknowns.

9.12.1 Phasors The phasor is a complex number that contains amplitude and phase angle information of a sinusoid and for the signal in Eq. (9.30) is expressed as ð9:33Þ t=0 R

−

FIGURE 9.33

An RL circuit with sinusoidal input.

574

9. BIOINSTRUMENTATION

In Eq. (9.33), by practice, the angle in the exponential is written in radians, and in the notation is written in degrees. Work in the phasor domain involves the use of complex algebra in moving between the time and phasor domain, so the rectangular form of the phasor is also used, given as V ¼ Vm ð cos f þ j sin fÞ

ð9:34Þ

9.12.2 Passive Circuit Elements in the Phasor Domain To use phasors with passive circuit elements for steady-state solutions, the relationship between voltage and current is needed for the resistor, inductor, and capacitor. Assume that

For a resistor, v ¼ IR ¼ RIm cos ðot þ yÞ and the phasor of v is ð9:35Þ where . Note that there is no phase shift for the relationship between the phasor current and voltage for a resistor. For an inductor, v¼L

di ¼ oLIm sin ðot þ yÞ ¼ oLIm cos ðot þ y 90 Þ dt

and the phasor of v is

ð9:36Þ

Note that inductor current and voltage are out of phase by 90 —that is, current lags behind voltage by 90 . For a capacitor, define v ¼ Vm cos ðot þ yÞ and . Now i¼C

dv d ¼ C ðVm cos ðot þ yÞÞ dt dt

¼ CVm o sin ðot þ yÞ ¼ CVm o cos ðot þ y 90 Þ

575

9.12 TIME-VARYING SIGNALS

and the phasor for i is

or V¼

1 j I¼ I joC oC

ð9:37Þ

Note that capacitor current and voltage are out of phase by 90 —that is, voltage lags behind current by 90 . Equations (9.35)–(9.37) all have the form of V¼ZI, where Z represents the impedance of the circuit element and is, in general, a complex number, with units of ohms. The impedj . The impedance is a ance for the resistor is R, the inductor, joL, and the capacitor, oC complex number and not a phasor even though it may look like one. The imaginary part of the impendence is called reactance. The final part to working in the phasor domain is to transform a circuit diagram from the time to phasor domain. For example, the circuit shown in Figure 9.34 is transformed into the phasor domain shown in Figure 9.35 by replacing each circuit element with their impedance equivalent and sources by their phasor. For the voltage source, we have For the capacitor, we have 0:5 mF

j ¼ j4000 O oC

For the resistor, we have 1000 O

0.5 μF

= 100 sin 500t mV

1000 O

1000 Ω

+ −

FIGURE 9.34

200 mH

A circuit diagram.

576

9. BIOINSTRUMENTATION

− j4000 Ω

500 − 90⬚ mV

1000 Ω

+ −

j100 Ω

FIGURE 9.35 Phasor and impedance equivalent circuit for Figure 9.34.

For the inductor, we have 200 mH

joL ¼ j100 O

Each of the elements is replaced by its phasor and impedance equivalents, as shown in Figure 9.35.

9.12.3 Kirchhoff’s Laws and Other Techniques in the Phasor Domain It is fortunate that all of the material presented before in this chapter involves Kirchhoff’s current and voltage laws, and all of the other techniques apply to phasors. That is, for KVL, the sum of phasor voltages around any closed path is zero X Vi ¼ 0 ð9:38Þ and for KCL, the sum of phasor currents leaving any node is zero X Ii ¼ 0

ð9:39Þ

Impedances in series are given by Z ¼ Z1 þ þ Zn

ð9:40Þ

1 1 1 þ þ Z1 Zn

ð9:41Þ

Impedances in parallel are given by Z¼

The node-voltage method, superposition and The´venin equivalent circuits are also applicable in the phasor domain. Example Problems 9.26 and 9.27 illustrate the process, with the most difficult aspect involving complex algebra.

577

9.12 TIME-VARYING SIGNALS

EXAMPLE PROBLEM 9.26 For the circuit shown in Figure 9.35, find the steady-state response i.

Solution The impedance for the circuit is Z ¼ j4000 þ 1000 þ j100 ¼ 1000 j3900 O Using Ohm’s law,

Returning to the time domain, the steady-state current is i ¼ 124 cosð500t 14 Þ mA

EXAMPLE PROBLEM 9.27 Find the steady-state response v using the node-voltage method for the following circuit. +

1Ω

vs = 50 sin 10 w t V

1 2

is = 20 cos (10 w t + 20⬚) A

−

Solution The first step is to transform the circuit elements into their impedances, which for the capacitor and inductor are 1 F 10

1 H 5

j ¼ j O oC joL ¼ j2 O

The phasors for the two sources are

Continued

578

9. BIOINSTRUMENTATION

Since the two resistors retain their values, the phasor drawing of the circuit is shown in the following figure with the ground at the lower node. 1 +

1Ω

Vs = 50 −90⬚ V

+ −

−j Ω

j2 Ω

1 2

Is = 20 20⬚

−

Writing the node-voltage equation for node 1 gives

Collecting like terms, converting to rectangular form, and converting to polar form gives

The steady-state solution is v ¼ 15:6 cos ð10t 76 Þ V

9.13 ACTIVE ANALOG FILTERS This section presents several active analog filters involving the op amp. Passive analog filters use passive circuit elements: resistors, capacitors, and inductors. To improve performance in a passive analog filter, the resistive load at the output of the filter is usually increased. By using the op amp, fine control of the performance is achieved without increasing the load at the output of the filter. Filters are used to modify the measured signal by removing noise. A filter is designed in the frequency domain so the measured signal to be retained is passed through and noise is rejected.

579

9.13 ACTIVE ANALOG FILTERS

Figure 9.36 shows the frequency characteristics of four filters: low-pass, high-pass, bandpass, and notch filters. The signal that is passed through the filter is indicated by the frequency interval called the passband. The signal that is removed by the filter is indicated by the frequency interval called the stopband. The magnitude of the filter, jHðjoÞj, is one in the passband and zero in the stopband. The low-pass filter allows slowly changing signals with frequency less than o1 to pass through the filter and eliminates any signal or noise above o1. The high-pass filter allows quickly changing signals with frequency greater than o2 to pass through the filter and eliminates any signal or noise with frequency less than o2. The band-pass filter allows signals in the frequency band greater than o1 and less than o2 to pass through the filter and eliminates any signal or noise outside this interval. The notch filter allows signals in the frequency band less than o1 and greater than o2 to pass through

H ( jω )

Passband Stopband ω

ω1 H ( jω )

Passband Stopband ω

ω2 H ( jω )

Passband Stopband ω1

ω2

Stopband ω

Passband

Stopband

H ( jω )

ω1 ω2

FIGURE 9.36

Passband

Ideal magnitude-frequency response for four filters, from top to bottom: low-pass, high-pass, band-pass, and notch.

580

9. BIOINSTRUMENTATION

H (jω ) M M 2

Stopband

Passband

Stopband

FIGURE 9.37 A realistic magnitude-frequency response for a band-pass filter. Note that the magnitude M does not necessarily need to be one. The passband is defined as the frequency interval when the magnitude is greater than pMﬃﬃ2.

the filter and eliminates any signal or noise outside this interval. The frequencies o1 and o2 are typically called cutoff frequencies for the low-pass and high-pass filters. In reality, any real filter cannot possibly have these ideal characteristics but instead has a smooth transition from the passband to the stopband, as shown, for example, in Figure 9.37 (the reason for this behavior is discussed in Chapter 11). Further, it is sometimes convenient to include both amplification and filtering in the same circuit, so the maximum of the magnitude does not need to be one, but it can be a value of M specified by the needs of the application. To determine the filter’s performance, the filter is driven by a sinusoidal input. One varies the input over the entire spectrum of interest (at discrete frequencies) and records M the output magnitude. The critical frequencies are when jHðjoÞj ¼ pffiffiffi . 2

EXAMPLE PROBLEM 9.28 Using the low-pass filter in the following circuit, design the filter to have a gain of 5 and a rad cutoff frequency of 500 . s

Rb − Ra VS

+ −

+ v0 −

581

9.13 ACTIVE ANALOG FILTERS

Solution By treating the op amp as ideal, note that the noninverting input is connected to ground and, therefore, the inverting input is also connected to ground. The operation of this filter is readily apparent because at low frequencies, the capacitor acts like an open circuit, reducing the circuit to an inverting amplifier that passes low-frequency signals. At high frequencies, the capacitor acts like a short circuit, which connects the output terminal to the inverting input and ground. The phasor method will be used to solve this problem by first transforming the circuit into the phasor domain, as shown in the following figure.

1 jwC Rb

V0 −

Summing the currents leaving the inverting input gives

Vs V0 V0 ¼0 1 Ra Rb joC

Collecting like terms and rearranging yields 0

1C B 1 þ C ¼ Vs V0 B @ 1 R b A Ra joC After further manipulation,

1 0 C B C B C B C B C B C V0 1 B 1 1 C C¼ 1 B ¼ B C B C 1 1 1 Vs Ra B R a @joC þ A C B þ C B 1 Rb A Rb @ joC 1 0 B V0 1 B B ¼ Vs Ra C B @

C C C 1 C A jo þ Rb C 1

Continued

582

9. BIOINSTRUMENTATION

Similar to the reasoning for the characteristic equation for a differential equation, the cutoff 1 1 frequency is defined as oc ¼ , (i.e., the denominator term, jo þ is set equal to zero). Thus, Rb C Rb C rad 1 , then ¼ 500: The cutoff frequency is also defined with the cutoff frequency set at oc ¼ 500 s Rb C M V0 is given by as when jHðjoÞj ¼ pffiffiffi , where M ¼ 5: The magnitude of Vs 2 1 V0 Ra C ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ffi s 1 2 2 o þ Rb C and at the cutoff frequency, oc ¼ 500

rad , s 1 5 Ra C pffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 1 o2c þ Rb C

With

1 ¼ 500, the magnitude is Rb C 1 1 1 5 Ra C Ra C Ra C ffi¼ pffiffiffi pffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 5002 þ 5002 500 2 2 oc þ Rb C

which gives Ra C ¼

1 2500

1 1 and ¼ 500), there are an 2500 Rb C infinite number of solutions. Therefore, one can select a convenient value for one of the elements—say, Ra ¼ 20 kO—and the other two elements are determined as Since we have three unknowns and two equations (Ra C ¼

C¼

1 1 ¼ ¼ 20 nF 2500 Ra 2500 20000

and Rb ¼

1 1 ¼ 100 kO ¼ 500 C 500 20 109

A plot of the magnitude versus frequency is shown in the following figure. As can be seen, the cutoff frequency gives a value of magnitude equal to 3.53 at 100 Hz, which is the design goal.

583

9.13 ACTIVE ANALOG FILTERS

Magnitude

0 0

200

400

600 800 Frequency (rad/s)

1000

1200

1400

EXAMPLE PROBLEM 9.29 Using the high-pass filter in the following circuit, design the filter to have a gain of 5 and a rad . cutoff frequency of 100 s Rb − Ra VS

+ −

+ v0 −

Solution Since the op amp is assumed ideal and the noninverting input is connected to ground, the inverting input is also connected to ground. The operation of this filter is readily apparent because at low frequencies, the capacitor acts like an open circuit, so no input voltage is seen at the noninverting input. Since there is no input, then the output is zero. At high frequencies, the capacitor acts like a short circuit, which reduces the circuit to an inverting amplifier that passes through high-frequency signals. Continued

584

9. BIOINSTRUMENTATION

As before, the phasor method will be used to solve this problem by first transforming the circuit into the phasor domain, as shown in the following figure. Rb

jwC

V0 −

Summing the currents leaving the inverting input gives

Vs 1 Ra þ joC

V0 ¼0 Rb

Rearranging yields V0 ¼ Vs At cutoff frequency oc ¼ 100

Rb 1 Ra þ joC

Rb Ra

jo jo þ

1 Ra C

rad 1 V0 . The magnitude of ¼ is given by Vs s Ra C V0 Rb o ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V R ffi s a 1 2 2 o þ Ra C

and at the cutoff frequency, 5 R oc pffiffiffi ¼ b sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi R 2 a 1 2 2 oc þ Ra C With

1 rad ¼ 100 and oc ¼ 100 , gives Ra C s

1 5 Rb oc Rb R 100 Rb Ra C ffi¼ b pffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffi 2ffi ¼ Ra qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 100 2 2 2 Ra 2Ra a 1 100 þ 100 o2c þ Ra C

Rb ¼ 5. Since we have three unknowns and two equations, one can select a convenient Ra value for one of the elements—say, Rb ¼ 20 kO—and the other two elements are determined as Thus,

Ra ¼

Rb 20000 ¼ 4 kO ¼ 5 5

585

9.13 ACTIVE ANALOG FILTERS

and C¼

1 1 ¼ 2:5 mF ¼ 100Ra 100 4000

A plot of the magnitude versus frequency is shown in the following figure. As can be seen, the cutoff frequency gives a value of magnitude equal to 3.53 at 100 Hz, which is the design goal. 6

Magnitude

0 0

200

400

600 800 1000 Frequency (rad/s)

1200

1400

Example Problem 9.30 demonstrates the technique to create band-pass filters (which require two cutoff frequencies).

EXAMPLE PROBLEM 9.30 Using the band-pass filter in the following circuit, design the filter to have a gain of 5 and pass rad through frequencies from 100 to 500 . s CL RbL

RbH

− VS

+ −

RaL

+ vL −

RaH

+ v0 −

Continued

586

9. BIOINSTRUMENTATION

Solution As usual, the design of the filter is done in the phasor domain and uses the work done in the previous two examples. Note that the elements around the op amp on the left are the low-pass filter circuit elements, and those on the right are the high-pass filter circuit elements. In fact, when working with op amps, filters can be cascaded together to form other filters, so a low-pass and high-pass filter cascaded together will form a band-pass. The phasor domain circuit is given in the next figure.

1 jwCL RbL

RbH

− RaL VS

+ −

−

RaH

jwCH

−

+ V0 −

As before the noninverting input to the op amps is connected to ground, which means that the inverting input is also connected to ground. Summing the currents leaving the inverting input for each op amp gives

Vs VL VL ¼0 1 RaL RbL joCL VL

RaH

1 þ joCH

Solving the first equation for VL gives VL ¼

V0 ¼0 RbH

C 1 B 1 CV B 1 A s RaL CL @ jo þ RbL CL

Solving the second equation for V0 gives V0 ¼

RbH RaH

jo jo þ

1 RaH CH

587

9.13 ACTIVE ANALOG FILTERS

Substituting VL into the previous equation yields V0 ¼

RbH RaH

jo 1 jo þ RaH CH

1 B B RaL CL @

C CV 1 A s jo þ RbL C L 1

The form of the solution is simply the product of each filter. The magnitude of the filter is 1 V0 RbH o R aL CL ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 s aH 1 1 2 2 o þ o þ RaH CH RbL CL Since there are two cutoff frequencies, two equations evolve: ocH ¼

1 rad ¼ 100 RaH CH s

and 1 rad ¼ 500 RbL CL s 5 rad At either cutoff frequency, the magnitude is pffiffiffi , such that at ocH ¼ 100 s 2 1 RaL CL 5 R ocH vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ bH vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 0 12ffi u u 2 2 RaH u u u 2 1 A u 2 1 A tocH þ @ toC þ @ H RaH CH RbL CL ocL ¼

1 RaL CL Rb 100 ¼ H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 RaH 100 þ 100 1002 þ 5002 Therefore,

pffiffiffiffiffi 500 26 ¼

RbH RaH RaL CL

The other cutoff frequency gives the same result as the previous equation. There are now ! pffiffiffiffiffi 1 1 RbH three equations , and six unknowns. For ¼ 100, ¼ 500, and 500 26 ¼ RaH CH RbL CL RaH RaL CL convenience, set RbL ¼ 100 kO and RaH ¼ 100 kO, which gives CL ¼ CH ¼

pffiffiffiffiffi Rb 1 1 ¼ 0:1 mF. Now from 500 26 ¼ H , RaH RaL CL 100RaH

1 ¼ 20 nF and 500RbL

pffiffiffiffiffi RbH ¼ 500 26CL RaH ¼ 5:099 RaL Once again, one can specify one of the resistors—say, RaL ¼ 10 kO—giving RbH ¼ 50:099 kO. Continued

588

9. BIOINSTRUMENTATION

A plot of the magnitude versus frequency is shown in the following figure. As can be seen, the cutoff frequency gives a value of magnitude equal to 3.53 at 500 Hz, which is the design goal. 4.5 4 3.5

Magnitude

3 2.5 2 1.5 1 0.5 0 0

200

400

600

800

1000

1200

1400

Frequency (rad/s)

None of the filters in Example Problems 9.28–9.30 have the ideal characteristics of Figure 9.36. To improve the performance from the pass-band to stopband in a low-pass filter with a sharper transition, one can cascade identical filters together—that is, connect the output of the first filter to the input of the next filter and so on. The more cascaded filters, the better the performance. The magnitude of the overall filter is the product of the individual filter magnitudes. While this approach is appealing for improving the performance of the filter, the overall magnitude of the filter does not remain a constant in the pass-band. Better filters are available with superior performance, such as a Butterworth filter. Two Butterworth filters are shown in Figure 9.38. Analysis of these filters is carried out in Exercises 53 and 55.

9.14 BIOINSTRUMENTATION DESIGN Figure 9.2 described the various elements needed in a biomedical instrumentation system. The purpose of this type of instrument is to monitor the output of a sensor or sensors and to extract information from the signals that are produced by the sensors. Acquiring a discrete-time signal and storing this signal in computer memory from a continuous-time signal is accomplished with an analog-to-digital (A/D) converter. The A/D converter uniformly samples the continuous-time waveform and transforms it into a sequence of numbers, one every tk seconds. The A/D converter also transforms the

589

9.14 BIOINSTRUMENTATION DESIGN

FIGURE 9.38

(Top) Second-order Butterworth low-pass filter. (Bottom) Third-order Butterworth low-pass

filter.

continuous-time waveform into a digital signal (i.e., the amplitude takes one of 2n discrete values), which is converted into computer words and stored in computer memory. To adequately capture the continuous-time signal, the sampling instants tk must be selected carefully so information is not lost. The minimum sampling rate is twice the highest frequency content of the signal (based on the sampling theorem from communication theory). Realistically, we often sample at five to ten times the highest frequency content of the signal so as to achieve better accuracy by reducing aliasing error.

9.14.1 Noise Measurement signals are always corrupted by noise in a biomedical instrumentation system. Interference noise occurs when unwanted signals are introduced into the system by outside sources, such as power lines and transmitted radio and television electromagnetic waves. This kind of noise is effectively reduced by careful attention to the circuit’s wiring configuration to minimize coupling effects. Interference noise is introduced by power lines (50 or 60 Hz), fluorescent lights, AM/FM radio broadcasts, computer clock oscillators, laboratory equipment, and cellular phones.

590

9. BIOINSTRUMENTATION

Electromagnetic energy radiating from noise sources is injected into the amplifier circuit or into the patient by capacitive and/or inductive coupling. Even the action potentials from nerve conduction in the patient generate noise at the sensor/amplifier interface. Filters are used to reduce the noise and to maximize the signal-to-noise (S/N) ratio at the input of the A/D converter. Low-frequency noise (amplifier d.c. offsets, sensor drift, temperature fluctuations, etc.) is eliminated by a high-pass filter with the cutoff frequency set above the noise frequencies and below the biological signal frequencies. High-frequency noise (nerve conduction, radio broadcasts, computers, cellular phones, etc.) is reduced by a low-pass filter with the cutoff set below the noise frequencies and above the frequencies of the biological signal that is being monitored. Power line noise is a very difficult problem in biological monitoring, since the 50- or 60-Hz frequency is usually within the frequency range of the biological signal that is being measured. Band-stop filters are commonly used to reduce power line noise. The notch frequency in these band-stop filters is set to the power line frequency of 50 or 60 Hz with the cutoff frequencies located a few Hertz to either side. The second type of corrupting signal is called inherent noise. Inherent noise arises from random processes that are fundamental to the operation of the circuit’s elements and thus is reduced by good circuit design practice. While inherent noise can be reduced, it can never be eliminated. Low-pass filters can be used to reduce high-frequency components. However, noise signals within the frequency range of the biosignal being amplified cannot be eliminated by this filtering approach.

9.14.2 Computers Computers consist of three basic units: the central processing unit (CPU), the arithmetic and logic unit (ALU), and memory. The CPU directs the functioning of all other units and controls the flow of information among the units during processing procedures. It is controlled by program instructions. The ALU performs all arithmetic calculations (add, subtract, multiply, and divide) as well as logical operations (AND, OR, NOT) that compare one set of information to another. Computer memory consists of read only memory (ROM) and random access memory (RAM). ROM is permanently programmed into the integrated circuit that forms the basis of the CPU and cannot be changed by the user. RAM stores information temporarily and can be changed by the user. RAM is where user-generated programs, input data, and processed data are stored. Computers are binary devices that use the presence of an electrical signal to represent 1 and the absence of an electrical pulse to represent 0. The signals are combined in groups of 8 bits, a byte, to code information. A word is made up of 2 bytes. Most desktop computers that are used today are 32-bit systems, which means they can address 4.295 109 locations in memory. Most new computers today are 64-bit systems that can address 1.8447 1019 locations in memory. The first microcomputers were 8-bit devices that could interact with only 256 memory locations. Programming languages relate instructions and data to a fixed array of binary bits so the specific arrangement has only one meaning. Letters of the alphabet and other symbols such as punctuation marks are represented by special codes. ASCII stands for American Standard Code for Information Exchange. ASCII provides a common standard that allows

591

9.15 EXERCISES

different types of computers to exchange information. When word processing files are saved as text files, they are saved in ASCII format. Ordinarily, word processing files are saved in special program-specific binary formats, but almost all data analysis programs can import and export data in ASCII files. The lowest level of computer languages is machine language and consists of the 0s and 1s that the computer interprets. Machine language represents the natural language of a particular computer. At the next level, assembly languages use English-like abbreviations for binary equivalents. Programs written in assembly language can manipulate memory locations directly. These programs run very quickly and are often used in data acquisition systems that must rapidly acquire a large number of samples, perhaps from an array of sensors, at a very high sampling rate. Higher-level languages such as FORTRAN, PERL, and Cþþ contain statements that accomplish tasks that require many machine or assembly language statements. Instructions in these languages often resemble English and contain commonly used mathematical notations. Higher-level languages are easier to learn than machine and assembly languages. Program instructions are designed to tell computers when and how to use various hardware components to solve specific problems. These instructions must be delivered to the CPU of a computer in the correct sequence in order to give the desired result. Newer programing languages such as MATLAB and LabView are easier to use and more user friendly. When computers are used to acquire physiological data, programming instructions tell the computer when data acquisition should begin, how often samples should be taken from how many sensors, how long data acquisition should continue, and where the digitized data should be stored. The rate at which a system can acquire samples depends on the speed of the computer’s clock—233 MHz—and the number of computer instructions that must be completed in order to take a sample. Some computers can also control the gain on the input amplifiers so signals can be adjusted during data acquisition. In other systems, the gain of the input amplifiers must be manually adjusted.

9.15 EXERCISES 1. Find the power absorbed for the circuit element in Figure 9.7 if v (a) (V) 2 i (A) 1

t (s)

(a)

Continued

592

9. BIOINSTRUMENTATION

i (A)

(b)

v (V) 1

t (s) 0

t (s)

(b) i (A) 2

t (s)

v (V)

(d) 1

t (s)

(d) i (A)

(e)

2 v (V) 1

(e)

t (s)

593

9.15 EXERCISES

2. The voltage and current at the terminals in Figure 9.7 are v ¼ te10, 000t uðtÞ V i ¼ ðt þ 10Þe10, 000t uðtÞ A (a) (b) (c) (d)

Find the time when the power is at its maximum. Find the maximum power. Find the energy delivered to the circuit at t ¼ 1 104 s. Find the total energy delivered to the circuit element.

3. For the following circuit find (a) v1, (b) the power absorbed and delivered. 2A

+ 5V −

+ v1 − 2v1

−

− + + −

3A − + 6V

4. For the following circuit, find the power in each circuit element. 2V

1Ω

− + 4Ω

+ − + −

3Ω

5. Find i2 in the following circuit. i2 i1

1 Ω 3

1 Ω 8

3i1

1 Ω 4

Continued

594

9. BIOINSTRUMENTATION

6. Find ib for the following circuit.

3Ω

6Ω

8Ω

2Va

2Ω

7. Find the equivalent resistance Rab for the following circuit.

8Ω

2Ω

8Ω

6Ω 6Ω

4Ω

3Ω

8. Find the equivalent resistance Rab for the following circuit. 6Ω 2Ω

4Ω 9Ω

2Ω

4Ω

6Ω

595

9.15 EXERCISES

9. Find i1 and v1 for the following circuit. 2Ω

5Ω i1

16 Ω

20 A

80 Ω

+ v1 -

3Ω

12 Ω

10. Use the node-voltage method to determine v1 and v2. 5Ω

35 V

+ −

2Ω

3Ω

−

10 Ω

3Ω

+ −

50 V

−

11. Use the node-voltage method to determine v1 and v2. +

2Ω

16 Ω

−

10 Ω

−

12. Use the node-voltage method to determine v1 and v2. i1 +

−

2Ω

16 Ω

10 Ω

2i1

−

Continued

596

9. BIOINSTRUMENTATION

13. Use the node-voltage method to determine v1 and v2. 5Ω

3Ω

2Ω

−

3Ω

10 Ω v1

10 V

6Ω + −

5Ω

3ia

2 Vb

−

14. Use the node-voltage method to determine v1 and v2. 4Ω 5V

3Ω

+ −

10 V

+ −

5Ω

2Ω +

−

2Ω

+ −

3v1

−

15. Use the node-voltage method to determine v1 and v2. 5Ω 3ia

ia 3Ω

15 V

+ −

− +

2Ω

− 2Ω

− 1Ω

5Ω

597

9.15 EXERCISES

16. Use the superposition method to find vo. +

4Ω

+ −

3Ω

− +

2Ω

10 V

−

17. Use the superposition method to find vo. 0.5 Ω

3Ω

2Ω +

5Ω 3A

2Ω

2ia

+ −

− +

4Ω

−

18. Find the The´venin equivalent with respect to terminals a and b. a

3Ω

2Ω

5Ω

19. Find the The´venin equivalent with respect to terminals a and b. i1 a 2Ω

16 Ω

10 Ω

2i1

Continued

598

9. BIOINSTRUMENTATION

20. A current pulse given by iðtÞ ¼ 2 þ 10e2t uðtÞ is applied through a 10-mH inductor. (a) Find the voltage across the inductor. (b) Sketch the current and voltage. (c) Find the power as a function of time. 21. The voltage across an inductor is given by the following figure. If L ¼ 30 mH and ið0Þ ¼ 0 A, find iðtÞ for t 0. v (mv) 4

t (s)

22. The voltage across a 4 mF capacitor is vðtÞ ¼ ð200,000t 50,000Þe2000t uðtÞ V. Find (a) the current through the capacitor, (b) power as a function of time, and (c) energy. 23. The current through a 5 mF capacitor is 8 t < 0 ms < 0 mA 0 t < 1 ms iðtÞ ¼ 5t2 mA : 5 2-t2 mA t 1 ms Find the voltage across the capacitor. 24. The switch has been in position a for a long time. At t ¼ 0, the switch instantaneously moves to position b. Find iL and v1 for t > 0. 3Ω

a iL

t=0 b 10 Ω

+ v1 −

25. The switch has been in position a for a long time. At t ¼ 0, the switch instantaneously moves to position b. Find iL , i1, and v1 for t > 0. a 2Ω

t=0

b 10 V

4Ω

5i1

+ v1 −

12 Ω

599

9.15 EXERCISES

26. The switch has been in position a for a long time. At t ¼ 0, the switch instantaneously moves to position b. Find vc and i1 for t > 0. a i1

t=0

5Ω

b + vC −

10 Ω

10 V

27. The switch has been in position a for a long time. At t ¼ 0, the switch instantaneously moves to position b. Find vc and i1 for t > 0. a i1

t=0 b 40 Ω

40 Ω

12 Ω 10 Ω

+ vC −

10 F

28. The switch has been in position a for a long time. At t ¼ 0, the switch instantaneously moves to position b. Find vc and i1 for t > 0. a i1

t=0

5Ω

b 20 V

+ −

4Ω

+ vC −

2i1

12 Ω

29. Find iL and vc for t > 0 for the following circuit if (a) is ¼ 3uðtÞ A; (b) is ¼ 1 þ 3uðtÞ A. iL

5Ω

+ vC −

Continued

600

9. BIOINSTRUMENTATION

30. Find iL and vc for t > 0 for the following circuit if (a) vs ¼ 5uðtÞ V; (b) vs ¼ 5uðtÞ þ 3 V. + vC − 2Ω

1.5 F

+ −

31. Find iL and vc for t > 0 for the following circuit if (a) is ¼ 3uðtÞ A; (b) is ¼ 3uðtÞ 1 A.

+ vC −

1F is

4Ω

10 H

32. For the following circuit we are given that iL1 ð0Þ ¼ 2 A, iL2 ð0Þ ¼ 5 A, vc1 ð0Þ ¼ 2 V, vc2 ð0Þ ¼ 3 V, and is ¼ 2e2t uðtÞ A. Use the node-voltage method to find Vb for t > 0.

+ vC1 − 1Ω

2F +

iL1 3H

3Ω

+ vC 2 − 1Ω

iL2 2F 2H

601

9.15 EXERCISES

33. Use the node-voltage method to find vc1 for t > 0 for the following circuit if (a) vs ¼ 2e3t uðtÞ V; (b) vs ¼ 3 cos ð2tÞuðtÞ V; (c) vs ¼ 3uðtÞ 1 V. 3H 4F

+ −

2Ω

6Ω

+ vC1 −

3Ω

34. The operational amplifier shown in the following figure is ideal. Find vo and io. 100 kΩ

+12 V

25 kΩ

−12V

5 kΩ 2V

20 kΩ

35. The operational amplifier shown in the following figure is ideal. Find vo. 90 kΩ 100 kΩ

+12 V

20 kΩ

5 kΩ 10 V 5V

80 kΩ

−12 V 20 kΩ

Continued

602

9. BIOINSTRUMENTATION

36. Find the overall gain for the following circuit if the operational amplifier is ideal. Draw a graph of vo versus Vs if Vs varies from 0 to 10 V. +12 V − +

+ −12 V + −

20 kΩ 10 kΩ

5 kΩ −

37. Find vo in the following circuit if the operational amplifier is ideal. 5 kΩ

2 kΩ

+9 V + 10 kΩ 3 5V

+ − 4V

+ −

− 6 kΩ

+ −

+ −9 V v0

3 kΩ

−

38. Find io in the following circuit if the operational amplifiers are ideal. 10 kΩ

+12 V

3 kΩ

4 kΩ

−12 V

20 kΩ

i0 5 kΩ

+12 V

2 kΩ

−12 V

5V 3V

603

9.15 EXERCISES

39. Suppose the input Vs is given as a triangular waveform as shown in the following figure. If there is no stored energy in the following circuit with an ideal operational amplifier, find vo. 1 Ω 2 2F

+12 V

−12 V VS

Vs (V) 1

0 1

t (s)

–1

40. Suppose the input Vs is given in the following figure. If there is no stored energy in the following circuit with an ideal operational amplifier, find vo. Vs (V) 10

0 1

t (s)

−10

Continued

604

9. BIOINSTRUMENTATION

1MΩ 0.6 μF

+12 V − +

+ −12 V VS

+ −

−

41. Suppose the input Vs is given in the following figure. If there is no stored energy in the following circuit with an ideal operational amplifier, find vo. Vs 2 (V)

t (s)

1 μF

250 kΩ

+12 V

−12 V VS

605

9.15 EXERCISES

42. Suppose the input Vs is given in the following figure. If there is no stored energy in the following circuit with an ideal operational amplifier, find vo.

Vs (V) 0.2 .5

.25

t (s)

.75

−0.2 0.25 μF

+12 V

50 kΩ

−12 V VS

43. The following circuit is operating in the sinusoidal steady state. Find the steady-state expression for iL if is ¼ 30 cos 20t A.

5Ω

+ vC −

Continued

606

9. BIOINSTRUMENTATION

44. The following circuit is operating in the sinusoidal steady state. Find the steady-state expression for vc if vs ¼ 10 sin 1000t V. iL

5Ω + −

20 Ω

+ vC −

45. The following circuit is operating in the sinusoidal steady state. Find the steady-state expression for iL if is ¼ 5 cos 500t A. iL

10 Ω

+ vC −

15 F

46. The following circuit is operating in the sinusoidal steady state. Find the steady-state expression for vc if is ¼ 25 cos 4000t V. iL

4Ω

16 H

+ vC −

rad . s rad 48. Design a high-pass filter with a magnitude of 20 and a cutoff frequency of 300 . s 47. Design a low-pass filter with a magnitude of 10 and a cutoff frequency of 250

49. Design a band-pass filter with a gain of 15 and passthrough frequencies from 50 to 200

rad . s

rad . s rad . 51. Design a high-pass filter with a magnitude of 10 and a cutoff frequency of 500 s rad 52. Design a band-pass filter with a gain of 10 and passthrough frequencies from 20 to 100 . s 50. Design a low-pass filter with a magnitude of 5 and a cutoff frequency of 200

607

9.15 EXERCISES

53. Suppose the operational amplifier in the following circuit is ideal. (The circuit is a low-pass first-order Butterworth filter.) Find the magnitude of the output vo as a function of frequency. − R

+ −

−

54. With an ideal operational amplifier, the following circuit is a second-order Butterworth low-pass filter. Find the magnitude of the output vo as a function of frequency. C1

55. A third-order Butterworth low-pass filter is shown in the following circuit with an ideal operational amplifier. Find the magnitude of the output vo as a function of frequency. C2 R2

608

9. BIOINSTRUMENTATION

Suggested Readings R. Aston, Principles of Biomedical Instrumentation and Measurement, Macmillan Publishing Co., New York, 1990. J.D. Bronzino, Biomedical Engineering and Instrumentation: Basic Concepts and Applications, PWS Engineering, Boston, 1986. J.D. Bronzino, V.H. Smith, M.L. Wade, Medical Technology and Society: An Interdisciplinary Perspective, The MIT Press, Cambridge, MA, 1990. J.J. Carr, J.M. Brown, Introduction to Biomedical Equipment Technology, fourth ed., Prentice Hall, Upper Saddle River, NJ, 2000. J. Dempster, Computer Analysis of Electrophysiological Signals, Academic Press, London, 1993. D.A. Johns, K. Martin, Analog Integrated Circuit Design, John Wiley & Sons, New York, 1997. J.W. Nilsson, S. Riedel, Electric Circuits, seventh ed., Prentice Hall, Upper Saddle River, NJ, 2004. R.B. Northrop, Noninvasive Instrumentation and Measurement in Medical Diagnosis, CRC Press, Boca Raton, 2001. R.B. Northrop, Introduction to Instrumentation and Measurements, CRC Press, Boca Raton, 1997. R. Perez, Design of Medical Electronic Devices, Academic Press, San Diego, 2002. A. Rosen, H.D. Rosen (Eds.), New Frontiers in Medical Device Technology, John Wiley & Sons, Inc., New York, 1995. J.G. Webster (Ed.), Bioinstrumentation, John Wiley & Sons, New York, 2003. J.G. Webster (Ed.), Medical Instrumentation: Application and Design, third ed., John Wiley & Sons, New York, 1997. W. Welkowitz, S. Deutsch, M. Akay, Biomedical Instruments—Theory and Design, second ed., Academic Press, Inc., San Diego, 1992. D.E. Wise (Ed.), Bioinstrumentation and Biosensors, Marcel Dekker Inc., New York, 1991. D.E. Wise (Ed.), Bioinstrumentation: Research, Developments, and Applications, Butterworth Publishers, Stoneham, MA, 1990.

C H A P T E R

10 Biomedical Sensors Yitzhak Mendelson, PhD O U T L I N E 10.1

Introduction

610

10.5

Bioanalytical Sensors

647

10.2

Biopotential Measurements

616

10.6

Optical Sensors

651

10.3

Physical Measurements

621

10.7

Exercises

662

10.4

Blood Gas Sensors

639

Suggested Readings

666

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : • Describe the different classifications of biomedical sensors. • Describe the characteristics that are important for packaging materials associated with biomedical sensors. • Calculate the half-cell potentials generated by different electrodes immersed in an electrolyte solution.

• Describe how displacement transducers, airflow transducers, and thermistors are used to make physical measurements. • Describe how blood gases are measured. • Describe how enzyme-based and microbial biosensors work and some of their uses. • Explain how optical biosensors work and describe some of their uses.

• Describe the electrodes that are used to record the ECG, EEG, and EMG and those that are used for intracellular recordings.

Introduction to Biomedical Engineering, Third Edition

609

610

10. BIOMEDICAL SENSORS

10.1 INTRODUCTION Diagnostic bioinstrumentation is used routinely in clinical medicine and biological research for measuring a wide range of physiological variables. Generally, the measurement is derived from sensors or transducers and further processed by the instrument to provide valuable diagnostic information. Biomedical sensors or transducers are the main building blocks of diagnostic medical instrumentation found in many physician offices, clinical laboratories, and hospitals. They are routinely used in vivo to perform continuous invasive and noninvasive monitoring of critical physiological variables, as well as in vitro to help clinicians in various diagnostic procedures. Similar devices are also used in nonmedical applications such as in environmental monitoring, agriculture, bioprocessing, food processing, and the petrochemical and pharmacological industries. Increasing pressures to lower health care costs, optimize efficiency, and provide better care in less expensive settings without compromising patient care are shaping the future of clinical medicine. As part of this ongoing trend, clinical testing is rapidly being transformed by the introduction of new tests that will revolutionize the way physicians will diagnose and treat diseases in the future. Among these changes, patient self-testing and physician office screening are the two most rapidly expanding areas. This trend is driven by the desire of patients and physicians alike to have the ability to perform some types of instantaneous diagnosis right next to the patient and to move the testing apparatus from an outside central clinical laboratory closer to the point of care. Generally, medical diagnostic instruments derive their information from sensors, electrodes, or transducers. Medical instrumentation relies on analog electrical signals for an input. These signals can be acquired directly by biopotential electrodes—for example, in monitoring the electrical signals generated by the heart, muscles or brain, or indirectly by transducers that convert a nonelectrical physical variable such as pressure, flow, or temperature, or biochemical variables, such as partial pressures of gases or ionic concentrations, to an electrical signal. Since the process of measuring a biological variable is commonly referred to as sensing, electrodes and transducers are often grouped together and are termed sensors. Biomedical sensors play an important role in a wide range of diagnostic medical applications. Depending on the specific needs, some sensors are used primarily in clinical laboratories to measure in vitro physiological quantities such as electrolytes, enzymes, and other biochemical metabolites in blood. Other biomedical sensors for measuring pressure, flow, and the concentrations of gases, such as oxygen and carbon dioxide, are used in vivo to follow continuously (monitor) the condition of a patient. For real-time continuous in vivo sensing to be worthwhile, the target analytes must vary rapidly and, most often, unpredictably.

10.1.1 Sensor Classifications Biomedical sensors are usually classified according to the quantity to be measured and are typically categorized as physical, electrical, or chemical, depending on their specific applications. Biosensors, which can be considered a special subclassification of biomedical sensors, are a group of sensors that have two distinct components: a biological recognition element, such as a purified enzyme, antibody, or receptor, that functions as a mediator and provides the selectivity that is needed to sense the chemical component (usually referred to as the

10.1 INTRODUCTION

611

analyte) of interest, and a supporting structure that also acts as a transducer and is in intimate contact with the biological sensing sensed by the biological recognition element into a quantifiable measurement, typically in the element. The purpose of the transducer is to convert the biochemical reaction into the form of an optical, electrical, or physical signal that is proportional to the concentration of a specific chemical. Thus, a blood pH sensor is not considered a biosensor according to this classification, although it measures a biologically important variable. It is simply a chemical sensor that can be used to measure a biological quantity.

10.1.2 Sensor Packaging Packaging of certain biomedical sensors, primarily sensors for in vivo applications, is an important consideration during the design, fabrication, and use of the device. Obviously, the sensor must be safe and remain functionally reliable. In the development of implantable biosensors, an additional key issue is the long operational lifetime and biocompatibility of the sensor. Whenever a sensor comes into contact with body fluids, the host itself may affect the function of the sensor, or the sensor may affect the site in which it is implanted. For example, protein absorption and cellular deposits can alter the permeability of the sensor packaging that is designed to both protect the sensor and allow free chemical diffusion of certain analytes between the body fluids and the biosensor. Improper packaging of implantable biomedical sensors could lead to drift and a gradual loss of sensor sensitivity and stability over time. Furthermore, inflammation of tissue, infection, or clotting in a vascular site may produce harmful adverse effects. Hence, the materials used in the construction of the sensor’s outer body must be biocompatible, since they play a critical role in determining the overall performance and longevity of an implantable sensor. One convenient strategy is to utilize various polymeric covering materials and barrier layers to minimize leaching of potentially toxic sensor components into the body. It is also important to keep in mind that once the sensor is manufactured, common sterilization practices by steam, ethylene oxide, or gamma radiation must not alter the chemical diffusion properties of the sensor packaging material.

10.1.3 Sensor Specifications The need for accurate medical diagnostic procedures places stringent requirements on the design and use of biomedical sensors. Depending on the intended application, the performance specifications of a biomedical sensor may be evaluated in vitro and in vivo to ensure that the measurement meets the design specifications. To understand sensor performance characteristics, it is important first to understand some of the common terminology associated with sensor specifications. The following definitions are commonly used to describe sensor characteristics and selecting sensors for particular applications. Sensitivity Sensitivity is typically defined as the ratio of output change for a given change in input. Another way to define sensitivity is by finding the slope of the calibration line relating the input to the output (i.e., DOutput/DInput), as illustrated in Figure 10.1. A high sensitivity implies that a small change in input quantity causes a large change in its output.

612

10. BIOMEDICAL SENSORS

FIGURE 10.1 Input versus output calibration curve of a typical sensor.

For example, a temperature sensor may have a sensitivity of 20 mV/ C; that is, the output of this sensor will change by 20 mV for 1 C change in input temperature. Note that if the calibration line is linear, the sensitivity is constant, whereas the sensitivity will vary with the input when the calibration is nonlinear, as illustrated in Figure 10.1. Alternatively, sensitivity can also be defined as the smallest change in the input quantity that will result in a detectable change in sensor output. Range The range of a sensor corresponds to the minimum and maximum operating limits that the sensor is expected to measure accurately. For example, a temperature sensor may have a nominal performance over an operating range of 200 to þ500 C. Accuracy Accuracy refers to the difference between the true value and the actual value measured by the sensor. Typically, accuracy is expressed as a ratio between the preceding difference and the true value and is specified as a percent of full-scale reading. Note that the true value should be traceable to a primary reference standard.1 Precision Precision refers to the degree of measurement reproducibility. Very reproducible readings indicate a high precision. Precision should not be confused with accuracy. For example, measurements may be highly precise but not necessary accurate. Resolution When the input quantity is increased from some arbitrary nonzero value, the output of a sensor may not change until a certain input increment is exceeded. Accordingly, resolution is defined as the smallest distinguishable input change that can be detected with certainty. 1

An independently calibrated reference obtained by an absolute measurement of the highest quality that is subsequently used in the calibration of similar measured quantities.

10.1 INTRODUCTION

613

Reproducibility Reproducibility describes how close the measurements are when the same input is measured repeatedly over time. When the range of measurements is small, the reproducibility is high. For example, a temperature sensor may have a reproducibility of 0.1 C for a measurement range of 20 C to 80 C. Note that reproducibility can vary depending on the measurement range. In other words, readings may be highly reproducible over one range and less reproducible over a different operating range. Offset Offset refers to the output value when the input is zero, as illustrated in Figure 10.1. Linearity Linearity is a measure of the maximum deviation of any reading from a straight calibration line. The calibration line is typically defined by the least-square regression fit of the input versus output relationship. Typically, sensor linearity is expressed as either a percent of the actual reading or a percent of the full-scale reading. The conversion of an unknown quantity to a scaled output reading by a sensor is most convenient if the input-output calibration equation follows a linear relationship. This simplifies the measurement, since we can multiply the measurement of any input value by a constant factor rather than using a “lookup table” to find a different multiplication factor that depends on the input quantity when the calibration equation follows a nonlinear relation. Note that although a linear response is sometimes desired, accurate measurements are possible even if the response is nonlinear as long as the input-output relation is fully characterized. Response Time The response time indicates the time it takes a sensor to reach a certain percent (e.g., 95 percent) of its final steady-state value when the input is changed. For example, it may take 20 seconds for a temperature sensor to reach 95 percent of its maximum value when a change in temperature of 1 C is measured. Ideally, a short response time indicates the ability of a sensor to respond quickly to changes in input quantities. Drift Drift refers to the change in sensor reading when the input remains constant. Drift can be quantified by running multiple calibration tests over time and determining the corresponding changes in the intercept and slope of the calibration line. Sometimes, the input-output relation may vary over time or may depend on another independent variable that can also change the output reading. This can lead to a zero (or offset) drift or a sensitivity drift, as illustrated in Figure 10.2. To determine zero drift, the input is held at zero while the output reading is recorded. For example, the output of a pressure transducer may depend not only on pressure but also on temperature. Therefore, variations in temperature can produce changes in output readings even if the input pressure remains zero. Sensitivity drift may be found by measuring changes in output readings for different nonzero constant inputs. For example, for a pressure transducer, repeating the measurements over a range of temperatures will reveal how much the slope of the input-output calibration line varies with temperature. In practice, both zero

614

10. BIOMEDICAL SENSORS

FIGURE 10.2 Changes in input versus output response caused by (a) sensitivity errors and (b) offset errors.

and sensitivity drifts specify the total error due to drift. Knowing the values of these drifts can help to compensate and correct sensor readings. Hysteresis In some sensors, the input-output characteristic follows a different nonlinear trend, depending on whether the input quantity increases or decreases, as illustrated in Figure 10.3. For example, a certain pressure gauge may produce a different output voltage when the input pressure varies from zero to full scale and then back to zero. When the measurement is not perfectly reversible, the sensor is said to exhibit hysteresis. If a sensor exhibits hysteresis, the input-output relation is not unique but depends on the direction change in the input quantity. The following sections will examine the operation principles of different types of biomedical sensors, including examples of invasive and noninvasive sensors for measuring biopotentials and other physical and biochemical variables encountered in different clinical and research applications.

FIGURE 10.3

Input versus output response of a sensor with hysteresis.

615

10.1 INTRODUCTION

EXAMPLE PROBLEM 10.1 A new temperature sensor produced the readings in Table 10.1. TABLE 10.1 Sample Calibration Data for a Temperature Sensor Temperature ( C)

Reading (mV)

12.3

18.2

25.4

37.0

43.6

55.8

62.0

67.8

70.4

72.1

100

73.0

1. Plot the input-output calibration for this sensor. 2. Find the offset and sensitivity for readings between 0 to 70 C. 3. Estimate the average sensitivity for readings ranging between 70 C to 100 C.

Solution 1.

Reading (mV)

70 60 50 40 30 20 10 0 0

FIGURE 10.4

40 60 80 Temperature (°C)

100

120

Input-Output Calibration for a Temperature Sensor.

Continued

616

10. BIOMEDICAL SENSORS

2. The equation of the linear regression line describing the input-output calibration data can be written as Reading ¼ a ðReference temperatureÞ þ b where, a ¼ is the slope and b is the y-intercept of the regression line. Accordingly, the offset (b) and sensitivity (a) are equal to 10.87 mV and 0.84 mV/ C, respectively. 3. From the equation of the linear regression line, the average sensitivity for readings between 70 C to 100 C is 0.17 mV/ C.

10.2 BIOPOTENTIAL MEASUREMENTS Biopotential measurements are made using different kinds of specialized electrodes. The function of these recording electrodes is to couple the ionic potentials generated inside the body to an electronic instrument. Biopotential electrodes are classified either as noninvasive (skin surface) or invasive (e.g., microelectrodes or wire electrodes). Biopotential measurements must be carried out using high-quality electrodes to minimize motion artifacts and ensure that the measured signal is accurate, stable, and undistorted. Body fluids are very corrosive to metals, so not all metals are acceptable for biopotential sensing. Furthermore, some materials are toxic to living tissues. For implantable applications, we typically use relatively strong metal electrodes made, for example, from stainless steel or noble materials such as gold, or from various alloys such as platinum-tungsten, platinumiridium, titanium-nitride, or iridium-oxide. These electrodes do not react chemically with tissue electrolytes and therefore minimize tissue toxicity. Unfortunately, they give rise to large interface impedances and unstable potentials. External monitoring electrodes can use nonnoble materials such as silver with lesser concerns of biocompatibility, but they must address the large skin interface impedance and the unstable biopotential. Other considerations in the design and selection of biopotential electrodes are cost, shelf life, and mechanical characteristics.

10.2.1 The Electrolyte/Metal Electrode Interface When a metal is placed in an electrolyte (i.e., an ionizable) solution, a charge distribution is created next to the metal/electrolyte interface, as illustrated in Figure 10.5. This localized charge distribution causes an electric potential, called a half-cell potential, to be developed across the interface between the metal and the electrolyte solution. The half-cell potentials of several important metals are listed in Table 10.2. Note that the hydrogen electrode is considered to be the standard electrode against which the half-cell potentials of other metal electrodes are measured.

10.2 BIOPOTENTIAL MEASUREMENTS

FIGURE 10.5

Distribution of charges at a metal/electrolyte interface.

Half-Cell Potentials of Important Metals

TABLE 10.2

Primary metal and chemical reaction Al ! Al

3þ

þ 3e

Cr ! Cr

Cd ! Cd

2þ

Zn ! Zn

2þ

Fe ! Fe

2þ

Ni ! Ni

2þ

1.706

0.744

0.401

þ 2e

0.763

0.409

0.230

0.126

þ 2e

Pb ! Pb

2þ

þ 2e

H2 ! 2H þ 2e þ

0.000 (standard by definition)

Ag ! Ag þ e Au ! Au Cu ! Cu

3þ

2þ

Half-cell potential (V)

þ 3e

3þ

617

þ0.799

þ 3e

þ1.420

þ 2e

þ0.340

Ag þ Cl ! AgCl þ 2e

þ0.223

EXAMPLE PROBLEM 10.2 Silver and zinc electrodes are immersed in an electrolyte solution. Calculate the potential drop between these two electrodes.

Solution From Table 10.2, the half-cell potentials for the silver and zinc electrodes are 0.799 V and 0.763 V, respectively. Therefore, the potential drop between these two metal electrodes is equal to 0:799V ð0:763VÞ ¼ 1:562V

618

10. BIOMEDICAL SENSORS

Typically, biopotential measurements are made by utilizing two similar electrodes composed of the same metal. Therefore, the two half-cell potentials for these electrodes would be equal in magnitude. For example, two similar biopotential electrodes can be taped to the chest near the heart to measure the electrical potentials generated by the heart (electrocardiogram, or ECG). Ideally, assuming that the skin-to-electrode interfaces are electrically identical, the differential amplifier attached to these two electrodes would amplify the biopotential (ECG) signal, but the half-cell potentials would be cancelled out. In practice, however, disparity in electrode material or skin contact resistance could cause a significant DC offset voltage that would cause a current to flow through the two electrodes. This current will produce a voltage drop across the body. The offset voltage will appear superimposed at the output of the amplifier and may cause instability or base line drift in the recorded biopotential.

EXAMPLE PROBLEM 10.3 Silver and aluminum electrodes are placed in an electrolyte solution. Calculate the current that will flow through the electrodes if the equivalent resistance of the solution is equal to 2kO.

Solution 0:799 V ð1:706 VÞ ¼ 2:505 V 2:505 V=2 kO ¼ 1:252 mA

10.2.2 ECG Electrodes Examples of different types of noninvasive biopotential electrodes used primarily for ECG recording are shown in Figure 10.6. A typical flexible biopotential electrode for ECG recording is composed of certain types of polymers or elastomers that are made electrically conductive by the addition of a fine carbon or metal powder. These electrodes (Figure 10.6a) are available with prepasted AgCl gel for quick and easy application to the skin using a double-sided peel-off adhesive tape.

FIGURE 10.6 Biopotential skin surface ECG electrodes: (a) flexible Mylar electrode and (b) disposable snaptype Ag/AgCl electrode.

10.2 BIOPOTENTIAL MEASUREMENTS

619

The most common type of biopotential electrode is the silver/silver chloride electrode (Ag/AgCl), which is formed by electrochemically depositing a very thin layer of silver chloride onto a silver electrode (Figure 10.6b). These electrodes are recessed from the surface of the skin and imbedded in foam that has been soaked with an electrolyte paste to provide good electrical contact with the skin. The electrolyte-saturated foam is also known to reduce motion artifacts that could be produced, for example, during stress testing when the layer of the skin moves relative to the surface of the Ag/AgCl electrode. This motion artifact could cause large interference in the recorded biopotential and, in extreme cases, could severely degrade the measurement.

10.2.3 EMG Electrodes A number of different types of biopotential electrodes are used in recording electromyographic (EMG) signals from different muscles in the body. The shape and size of the recorded EMG signals depend on the electrical property of these electrodes and the recording location. For noninvasive recordings, proper skin preparation, which normally involves cleansing the skin with alcohol or the application of a small amount of an electrolyte paste, helps to minimize the impedance of the skin-electrode interface and improve the quality of the recorded signal considerably. The most common electrodes used for surface EMG recording and nerve conduction studies are circular discs, about 1 cm in diameter, that are made of silver or platinum. For direct recording of electrical signals from nerves and muscle fibers, a variety of percutaneous needle electrodes are available, as illustrated in Figure 10.7. The most common type of needle electrode is the concentric bipolar electrode shown in Figure 10.7a. This electrode is made from thin metallic wires encased inside a larger canula or hypodermic needle. The two wires serve as the recording and reference electrodes. Another type of percutaneous EMG electrode is the unipolar needle electrode (Figure 10.7b). This electrode is made of a thin wire that is mostly insulated by a thin layer of Teflon, except about 300 mm near the distal tip. Unlike a bipolar electrode, this electrode requires a second unipolar reference electrode to form a closed electrical circuit. The second recording electrode is normally placed either adjacent to the recording electrode or attached to the surface of the skin.

FIGURE 10.7 Intramascular biopotential electrodes: (a) bipolar and (b) unipolar configuration.

620

10. BIOMEDICAL SENSORS

10.2.4 EEG Electrodes The most commonly used electrodes for recording electroencephalographic signals from the brain (EEG) are cup electrodes and subdermal needle electrodes. Cup electrodes are made of platinum or tin approximately 5–10 mm in diameter. These cup electrodes are filled with a conducting electrolyte gel and can be attached to the scalp with an adhesive tape. Recording of electrical potentials from the scalp is difficult because hair and oily skin impede good electrical contact. Therefore, clinicians sometimes prefer to use subdermal EEG electrodes instead of metal surface electrodes for EEG recording. These are basically fine platinum or stainless-steel needle electrodes about 10 mm long by 0.5 mm wide, which are inserted under the skin to provide a better electrical contact.

10.2.5 Microelectrodes Microelectrodes are biopotential electrodes with an ultrafine tapered tip that can be inserted into individual biological cells. These electrodes serve an important role in recording action potentials from single cells and are commonly used in neurophysiological studies. The tip of these electrodes must be small with respect to the dimensions of the biological cell to avoid cell damage and at the same time sufficiently strong to penetrate the cell wall. Figure 10.8 illustrates the construction of three typical types of microelectrodes: glass micropipettes, metal microelectrodes, and solid-state microprobes. In Figure 10.8a, a hollow glass capillary tube, typically 1 mm in diameter, is heated and softened in the middle inside a small furnace and then quickly pulled apart from both ends. This process creates two similar microelectrodes with an open tip that has a diameter on the

FIGURE 10.8 Biopotential microelectrodes: (a) a capillary glass microelectrode, (b) an insulated metal microelectrode, and (c) a solid-state multisite recording microelectrode.

10.3 PHYSICAL MEASUREMENTS

621

order of 0.1 to 10 mm. The larger end of the glass tube (the stem) is then filled typically with a 3M KCl electrolyte solution. A short piece of Ag/AgCl wire is inserted through the stem to provide an electrical contact with the electrolyte solution. When the tip of the microelectrode is inserted into an electrolyte solution, such as the intracellular cytoplasm of a biological cell, ionic current can flow through the fluid junction at the tip of the microelectrode. This establishes a closed electrical circuit between the Ag/AgCl wire inside the microelectrode and the biological cell. A different kind of microelectrode made from a small-diameter strong metal wire (e.g., tungsten or stainless steel) is illustrated in Figure 10.8b. The tip of this microelectrode is usually sharpened down to a diameter of a few micrometers by an electrochemical etching process. The wire is then insulated up to its tip. Solid-state microfabrication techniques commonly used in the production of integrated circuits can be used to produce microprobes for multichannel recordings of biopotentials or for electrical stimulation of neurons in the brain or spinal cord. An example of such a microsensor is shown in Figure 10.8c. The probe consists of a precisely micromachined silicon substrate with four exposed recording sites. One of the major advantages of this fabrication technique is the ability to mass-produce very small and highly sophisticated microsensors with highly reproducible electrical and physical properties.

10.3 PHYSICAL MEASUREMENTS 10.3.1 Displacement Transducers Displacement transducers are typically used to measure physical changes in the position of an object or medium. They are commonly employed in detecting changes in length, pressure, or force. Variations in these parameters can be used to quantify and diagnose abnormal physiological functions. In this section, we will describe inductive types of displacement transducers that can be used to measure blood pressure, electromagnetic transducers to measure blood flow, potentiometer transducers to measure linear or angular changes in position, and other types of elastic, strain gauge, capacitive, and piezoelectric type transducers. Inductive Displacement Transducers Inductive displacement transducers are based on the inductance L of a coil given by L ¼ m n2 l A

ð10:1Þ

where m is the permeability of the magnetically susceptible medium inside the coil (in henry per meter), n is the number of coil turns (in turns per meter), l is the coil length (in meters), and A is the cross-sectional area of the coil (in square meters). These types of transducers measure displacement by changing either the self-inductance of a single coil or the mutual inductance coupling between two or more stationary coils, typically by the displacement of a ferrite or iron core in the bore of the coil assembly. A widely used inductive displacement transducer is the linear variable differential transformer (LVDT) shown in Figure 10.9. This device is essentially a three-coil mutual inductance transducer that is composed of a primary coil (P) and two secondary coils (S1 and S2) connected in series but opposite in

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S Ferrite core (a)

FIGURE 10.9 LVDT transducer: (a) an electric diagram and (b) a cross-section view.

polarity in order to achieve a wider linear output range. The mutual inductance coupled between the coils is changed by the motion of a high-permeability slug. The primary coil is usually excited by passing an AC current. When the slug is centered symmetrically with respect to the two secondary coils, the primary coil induces an alternating magnetic field in the secondary coils. This produces equal voltages (but of opposite polarities) across the two secondary coils. Therefore, the positive voltage excursions from one secondary coil will cancel out the negative voltage excursions from the other secondary coil, resulting in a zero net output voltage. When the core moves toward one coil, the voltage induced in that coil is increased proportionally to the displacement of the core, while the voltage induced in the other coil is decreased proportionally, leading to a typical voltage-displacement diagram, as illustrated in Figure 10.10. Since the voltages induced in the two secondary coils are out of phase, special phase-sensitive electronic circuits must be used to detect both the position and the direction of the core’s displacement. Electromagnetic Flow Transducer Blood flow through an exposed vessel can be measured by means of an electromagnetic flow transducer. It can be used in research studies to measure blood flow in major blood vessels near the heart, including the aorta at the point where it exits from the heart.

10.3 PHYSICAL MEASUREMENTS

623

FIGURE 10.10 Output voltage versus core displacement of a typical LVDT transducer.

Consider a blood vessel of diameter, l, filled with blood flowing with a uniform velocity, ~ u. If the blood vessel is placed in a uniform magnetic flux, ~ B (in weber), that is perpendicular to the direction of blood flow, the negatively charged anion and positively charged cation particles in the blood will experience a force, ~ F (in newton), which is normal to both the magnetic field and blood flow directions and is given by ~ F ¼ qð~ u~ BÞ

ð10:2Þ

where q is the elementary charge (1.6 1019 C). As a result, these charged particles will be deflected in opposite directions and will move along the diameter of the blood vessels according to the direction of the force vector, ~ F. This movement will produce an opposing force, ~ Fo, which is equal to V ~ Fo ¼ q ~ E¼q l

ð10:3Þ

where ~ E is the net electrical field produced by the displacement of the charged particles and V is the potential produced across the blood vessel. At equilibrium, these two forces will be equal. Therefore, the potential difference, V, is given by V ¼Blu and is proportional to the velocity of blood through the vessel.

ð10:4Þ

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EXAMPLE PROBLEM 10.4 Calculate the voltage induced in a magnetic flow probe if the probe is applied across a blood vessel with a diameter of 5 103 m and the velocity of blood is 5 102 m/s. Assume that the magnitude of the magnetic field, B, is equal to 1.5 105 Wb/m2.

Solution From Eq. (10.4), V ¼ B l u ¼ ð1:5 105 Wb=m2 Þ ð5 103 mÞ ð5 102 m=sÞ ¼ 37:5 1010 V (Note: [Wb] ¼ [V S])

Practically, this device consists of a clip-on probe that fits snugly around the blood vessel, as illustrated in Figure 10.11. The probe contains electrical coils to produce an electromagnetic field that is transverse to the direction of blood flow. The coil is usually excited by an AC current. A pair of very small biopotential electrodes are attached to the housing and rest against the wall of the blood vessel to pick up the induced potential. The flowinduced voltage is an AC voltage at the same frequency as the excitation voltage. Using an AC method instead of DC excitation helps to remove any offset potential error due to the contact between the vessel wall and the biopotential electrodes. Potentiometer Transducers A potentiometer is a resistive-type transducer that converts either linear or angular displacement into an output voltage by moving a sliding contact along the surface of a resistive element. Figure 10.12 illustrates linear and angular-type potentiometric transducers. A voltage, Vi, is applied across the resistor, R. The output voltage, Vo, between the sliding contact and one terminal of the resistor is linearly proportional to the displacement. Typically, a constant current source is passed through the variable resistor, and the small change in output voltage is measured by a sensitive voltmeter using Ohm’s law (i.e., I ¼ V/R).

FIGURE 10.11 Electromagnetic blood flow transducer.

10.3 PHYSICAL MEASUREMENTS

FIGURE 10.12

625

Linear translational (a) and angular (b) displacement transducers.

EXAMPLE PROBLEM 10.5 Calculate the change in output voltage of a linear potentiometer transducer that undergoes a 20 percent change in displacement.

Solution Assuming that the current flowing through the transducer is constant, from Ohm’s law, DV ¼ I DR Hence, since the resistance between the sliding contact and one terminal of the resistor is linearly proportional to the displacement, a 20 percent change in displacement will produce a 20 percent change in the output voltage of the transducer.

Elastic Resistive Transducers In certain clinical situations, it is desirable to measure changes in the peripheral volume of a leg when the venous outflow of blood from the leg is temporarily occluded by a blood pressure cuff. This volume-measuring method is called plethysmography and can indicate the presence of large venous clots in the legs. The measurement can be performed by wrapping an elastic resistive transducer around the leg and measuring the rate of change in resistance of the transducer as a function of time. This change corresponds to relative changes in the blood volume of the leg. If a clot is present, it will take more time for the blood stored in the leg to flow out through the veins after the temporary occlusion is removed. A similar transducer can be used to follow a patient’s breathing pattern by wrapping the elastic band around the chest. An elastic resistive transducer consists of a thin elastic tube filled with an electrically conductive material, as illustrated in Figure 10.13. The resistance of the conductor inside the flexible tubing is given by R¼r

I A

ð10:5Þ

where r is the resistivity of the electrically conductive material (in O m), l is the length, and A is the cross-sectional area of the conductor.

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FIGURE 10.13

Elastic resistive transducer.

EXAMPLE PROBLEM 10.6 A 0.1 m long by 0.005 m diameter elastic resistive transducer has a resistance of 1 kO. (1) Calculate the resistivity of the electrically conductive material inside the transducer, and (2) calculate the resistance of the transducer after it has been wrapped around a patient’s chest having a circumference of 1.2 m. Assume that the cross-sectional area of the transducer remains unchanged.

Solution 1. The cross-sectional area of the transducer (A) is equal to p (0.0025)2 m2 ¼ 1.96 105 m2. From RA 1 103 O 1:96 105 m2 ¼ 0:196 O m Eq. (10.5), r ¼ ¼ 0:1 m l 2. 1:2 m Rstretched ¼ 0:196 O m ¼ 12 kO 1:96 105 m2

EXAMPLE PROBLEM 10.7 Calculate the change in voltage that is induced across the elastic transducer in Example Problem 10.6 assuming that normal breathing produces a 10 percent change in chest circumference and a constant current of 0.5 mA is passed through the transducer.

Solution From Ohm’s law (V ¼ I R), V ¼ 0.5 mA 12 kO ¼ 6 V. If R changes by 10 percent, DV ¼ 0.6 V.

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10.3 PHYSICAL MEASUREMENTS

Strain Gauge Transducers Strain gauges are displacement-type transducers that measure changes in the length of an object as a result of an applied force. These transducers produce a resistance change that is proportional to the fractional change in the length of the object, also called strain, S, which is defined as S¼

Dl l

ð10:6Þ

where Dl is the fractional change in length, and l is the initial length of the object. Examples include resistive wire elements and certain semiconductor materials. To understand how a strain gauge works, consider a fine wire conductor of length, l, cross-sectional area, A, and resistivity, r. The resistance of the unstretched wire is given by Eq. (10.5). Now suppose that the wire is stretched within its elastic limit by a small amount, Dl, such that its new length becomes (l þ Dl). Because the volume of the stretched wire must remain constant, the increase in the wire length results in a smaller cross-sectional area, Astretched. Thus, lA ¼ ðl þ DlÞ Astretched

ð10:7Þ

The resistance of the stretched wire is given by Rstretched ¼ r

l þ Dl Astretched

ð10:8Þ

The increase in the resistance of the stretched wire DR is DR ¼ Rstretched r

l A

ð10:9Þ

Substituting Eq. (10.8) and the value for Astretched from Eq. (10.7) into Eq. (10.9) gives DR ¼ r

ðl þ DlÞ2 l r ðl2 þ 2lDl þ Dl2 l2 Þ r ¼ lA A lA

ð10:10Þ

Assume that for small changes in length, Dl n2), according to Snell’s law, n1 sin f1 ¼ n2 sin f2

ð10:24Þ

where f is the angle of incidence, as shown in Figure 10.37. Accordingly, any light passing from a lower refractive index to a higher refractive index is bent toward the line perpendicular to the interface of the two materials. For small incident angles, f1, the light ray enters the fiber core and bends inward at the first core/cladding interface. For larger incident angles, f2, the ray exceeds a minimum angle required to bend it back into the core when it reaches the core/cladding boundary. Consequently, the light escapes into the cladding. By setting sinf2 ¼ 1.0, the critical angle, fcr, is given by sin fcr ¼

n2 n1

ð10:25Þ

Any light rays entering the optical fiber with incidence angles greater than fcr are internally reflected inside the core of the fiber by the surrounding cladding. Conversely, any entering light rays with incidence angles smaller than fcr escape through the cladding and are therefore not transmitted by the core.

FIGURE 10.37

Optical fiber illustrating the incident and refracted light rays. The solid line shows the light ray escaping from the core into the cladding. The dashed line shows the ray undergoing total internal reflection inside the core.

10.6 OPTICAL SENSORS

653

EXAMPLE PROBLEM 10.14 Assume that a beam of light passes from a layer of glass with a refractive index n1 ¼ 1.47 into a second layer of glass with a refractive index of n2 ¼ 1.44. Using Snell’s law, calculate the critical angle for the boundary between these two glass layers.

Solution fcr ¼ arcsin

n2 ¼ arcsinð0:9796Þ n1

fcr ¼ 78:4 Therefore, light that strikes the boundary between these two glasses at an angle greater than 78.4 will be reflected back into the first layer.

The propagation of light along an optical fiber is not confined to the core region. Instead, the light penetrates a characteristic short distance (on the order of one wavelength) beyond the core surface into the less optically dense cladding medium. This effect causes the excitation of an electromagnetic field, called the “evanescent-wave,” that depends on the angle of incidence and the incident wavelength. The intensity of the evanescent-wave decays exponentially with distance, according to Beer-Lambert’s law. It starts at the interface and extends into the cladding medium.

10.6.2 Sensing Mechanisms Optical sensors are typically interfaced with an optical module, as shown in Figure 10.38. The module supplies the excitation light, which may be from a monochromatic source such as a diode laser or from a broadband source (e.g., quartz-halogen) that is filtered to provide a narrow bandwidth of excitation. Typically, two wavelengths of light are used: one that is sensitive to changes in the species to be measured and one that is unaffected by changes in the analyte concentration. This wavelength serves as a reference and is used to compensate for fluctuations in source output and detector stability. The light output from the optic module is coupled into a fiber optic cable through appropriate lenses and an optical connector. Several optical techniques are commonly used to sense the optical change across a biosensor interface. These are usually based on evanescent wave spectroscopy, which plays a major role in fiber optic sensors, and a surface plasmon resonance principle. In fluorescence-based sensors, the incident light excites fluorescence emission, which changes in intensity as a function of the concentration of the analyte to be measured. The emitted light travels back down the fiber to the monitor, where the light intensity is measured by a photodetector. In other types of fiber optic sensors, the light-absorbing properties of the sensor chemistry change as a function of analyte chemistry. In the absorption-based design, a reflective surface near the tip or some scattering material within the sensing chemistry itself is usually used to return the light back through the same optical

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FIGURE 10.38

General principle of a fiber optic–based sensor.

fiber. Other sensing mechanisms exploit the evanescent-wave interaction with molecules that are present within the penetration depth distance and lead to attenuation in reflectance related to the concentration of the molecules. Because of the short penetration depth and the exponentially decaying intensity, the evanescent-wave is absorbed by compounds that must be present very close to the surface. The principle has been used to characterize interactions between receptors that are attached to the surface and ligands that are present in solution above the surface. The key component in the successful implementation of evanescent-wave spectroscopy is the interface between the sensor surface and the biological medium. Receptors must retain their native conformation and binding activity, and sufficient binding sites must be present for multiple interactions with the analyte. In the case of particularly weak absorbing analytes, sensitivity can be enhanced by combining the evanescent-wave principle with multiple internal reflections along the sides of an unclad portion of a fiber optic tip. Alternatively, instead of an absorbing species, a fluorophore2 can also be used. Light that is absorbed by the fluorophore emits detectable fluorescent light at a higher wavelength, thus providing improved sensitivity.

10.6.3 Intravascular Fiber Optic Blood Gas Sensors Considerable effort has been devoted over the last three decades to develop disposable extracorporeal sensors (for ex vivo applications) or intra-arterial fiber optic sensors that 2

A compound that produces a fluorescent signal in response to light.

10.6 OPTICAL SENSORS

655

can be placed in the arterial line (for in vivo applications) to enable continuous trending of arterial blood gases. With the advent of continuous arterial blood gas monitoring, treatment modalities can be proactive rather than reactive, which is vital for therapeutic interventions in ICU patients who may experience spontaneous and often unexpected changes in acidbase status. Intra-arterial blood gas sensors typically employ a single- or a double-fiber configuration. Typically, the matrix containing the indicator is attached to the end of the optical fiber. Since the solubility of O2 and CO2 gases, as well as the optical properties of the sensing chemistry itself, is affected by temperature variations, fiber optic intravascular sensors include a thermocouple or thermistor wire running alongside the fiber optic cable to monitor and correct for temperature fluctuations near the sensor tip. A nonlinear response is characteristic of most chemical indicator sensors. Therefore, the operating range of these sensors is typically optimized to match the range of concentrations according to the intended application. Intra-arterial fiber optic blood gas sensors are normally placed inside a standard 20-gauge arterial cannula that is sufficiently small, thus allowing adequate spacing between the sensor and the catheter wall. The resulting lumen is large enough to permit the withdrawal of blood samples, introduction of a continuous or intermittent anticoagulant (e.g., heparin) flush, and the recording of a blood pressure waveform. In addition, the optical fibers are encased in a protective tubing to contain any fiber fragments in case they break off. The material in contact with the blood is typically treated with a covalently bonded layer of heparin, resulting in low susceptibility to fibrin deposition. Despite excellent accuracy of indwelling intra-arterial catheters in vitro compared to blood gas analyzers, when these multiparameter probes were first introduced into the vascular system, it quickly became evident that the readings (primarily pO2) vary frequently and unpredictably, mainly due to the sensor tip intermittently coming in contact with the wall of the arterial blood vessel and intermittent reductions in blood flow due to arterial vasospasm. A more advanced multiparameter disposable probe (Figure 10.39) consisting of pO2, pCO2, and pH sensors was developed by Diametrics Medical, Inc. The sensor has a diameter of 0.5 mm and can be inserted intravascularly through a 20-gauge indwelling cannula. Clinical studies confirmed that the system is adequate for trend monitoring, eliminating invasive blood sampling, potential errors in analysis, and significant delays in obtaining results, which may affect treatment. The device has been evaluated in neurosurgical patients for continuous monitoring in the brain and in critically ill pediatric patients. Fiber Optic pO2 Sensors Various fiber optic sensors were developed to measure pO2 in blood based on the principle of fluorescence quenching. Quenching reduces the intensity of the emitted fluorescence light and is related to the concentration of the quenching molecules. For example, quenching can result from collisions encountered between the fluorophore (a fluorescent substance) and the quencher. For quenching to occur, the fluorophore and quencher must be in contact. When light is absorbed by a molecule, the absorbed energy is held as an excited electronic state of the molecule. It is then lost by coupling to the mechanical movement of the molecule (heat), reradiated from the molecule in a mean time of about 10 ns (fluorescence), or converted to another excited state with a much longer mean lifetime

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Section through Sensor Tip Exterior microporous polyethylene membrane + Bio-compatible + Outside diameter less than 0.5 mm (nominal average) + Indwelling sensor length: approximately 15–20 mm

Temperature thermocouple + Type T (Copper and constantan) + Used to report blood gas values at 37°C or patient temperature

Carmeda™ covalently bonded heparin surface coating + Improves bio-compatibility of the sensor, inhibits protein and thrombin formation on the sensor

pH, PCO2, PO2 optical cell miniature No sensing elements at the tip sensing elements + Prevents elements from detecting + Each is 0.175 mm diameter erroneous data from tissue + Acrylic optical fiber construction + pH: Phenol red in polyacrylamide gel + PCO2: Phenol red in bicarbonate solution + PO2: Ruthenium dye in silicone matrix + Covers 360° spiral along sensor’s length (rather than merely at the tip) to eliminate “wall effect” + Mirror encapsulated within the tip of the optical fiber to return light back along the same fiber for detection.

FIGURE 10.39

Principle of an indwelling arterial optical blood gas catheter. A heparin-coated porous polyethylene membrane encapsulates the optical fibers and a thermistor. Courtesy of Diametrics, Inc., St. Paul, MN.

(phosphorescence). A wide variety of substances act as fluorescence quenchers. One of the best-known quenchers is molecular oxygen. A typical fiber optic sensor for measuring pO2 using the principle of fluorescence quenching consists of a dye that is excited at 470 nm (blue) and fluoresces at 515 nm (green) with an emitted intensity that depends on the pO2. If the excited dye encounters an oxygen molecule, the excess energy will be transferred to the oxygen molecule, decreasing the fluorescence signal. The degree of quenching depends on the concentration of oxygen. The optical information is derived from the ratio of light intensities measured from the green fluorescence and the blue excitation light, which serves as an internal reference signal. The ratio of green to blue intensity is described by the Stern-Volmer equation Io =I ¼ 1 þ KpO2

ð10:26Þ

where Io and I are the fluorescence emission intensities in the absence (i.e., pO2 ¼ 0) and presence of the oxygen quencher, respectively. K is the Stern-Volmer quenching coefficient, which is dependent on temperature. The method provides a nearly linear readout of pO2 over the range of 0–150 mmHg (0–20 kPa), with a precision of about 1 mmHg (0.13 kPa). The slope of the plot described by Eq. (10.26) is a measure of the oxygen sensitivity of the sensor. Note that the sensor will be most sensitive to low levels of oxygen.

10.6 OPTICAL SENSORS

657

Fiber Optic pH Sensors A fiber optic pH sensor can be designed by placing a reversible color-changing dye as an indicator at the end of a pair of optical fibers. For example, the popular indicator phenol red can be used because this dye changes its absorption properties from the green to the blue part of the spectrum as the acidity is increased. The dye can be covalently bound to a hydrophilic polymer in the form of water-permeable microbeads to stabilize the indicator concentration. The indicator beads are contained in a sealed hydrogen ion-permeable envelope made out of hollow cellulose tubing, forming a miniature spectrophotometric cell at the end of the optical fibers. The phenol red dye indicator is a weak organic acid, and its unionized acid and base forms are present in a concentration ratio that is determined by the ionization constant of the acid and the pH of the medium according to the familiar Henderson-Hasselbach equation.3 The two forms of the dye have different optical absorption spectra. Hence, the relative concentration of one form, which varies as a function of pH, can be measured optically and related to variations in pH. In the pH sensor, green and red lights that emerge from the distal end of one fiber pass through the dye, where it is backscattered into the other fiber by the light-scattering beads. The base form of the indicator absorbs the green light. The red light is not absorbed by the indicator and is used as an optical reference. The ratio of green to red light is related to the pH of the medium. A similar principle can also be used with a reversible fluorescent indicator where the concentration of one indicator form is measured by its fluorescence rather than by the absorbance intensity. Light, typically in the blue or UV wavelength region, excites the fluorescent dye to emit longer-wavelength light. The concept is based on the fluorescence of weak acid dyes that have different excitation wavelengths for the basic and acidic forms but the same emitted fluorescent wavelength. The dye is encapsulated in a sample chamber that is permeable to hydrogen ions. When the dye is illuminated with the two different excitation wavelengths, the ratio of the emitted fluorescent intensities can be used to calculate the pH of the solution that is in contact with the encapsulated dye. Fiber Optic pCO2 Sensors The pCO2 of a sample is typically determined by measuring changes in the pH of a bicarbonate solution that is isolated from the sample by a CO2-permeable membrane but remains in equilibrium with the CO2 gas. The bicarbonate and CO2, as carbonic acid, form a pH buffer system. By the Henderson-Hasselbach equation, the hydrogen ion concentration is proportional to the pCO2 of the sample. This measurement can be done with either a pH electrode or a dye indicator. Mixed Venous Oxygen Saturation Sensors Fiber optic catheters can be used in vivo to measure mixed venous oxygen saturation (SvO2) inside the pulmonary artery, which represents the blood outflow from all tissue 3

pH ¼ 6:1 þ log

HCO 3 CO2

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beds. Under normal conditions, oxygen saturation in the pulmonary artery is normally around 75 percent, and oxygen consumption is less than or equal to the amount of oxygen delivered. However, in critically ill patients, oxygen delivery is often insufficient for the increased tissue demands because many such patients have compromised compensatory mechanisms. If tissue oxygen demands increase and the body’s compensatory mechanisms are overwhelmed, the venous oxygen reserve will be tapped, and that change will be reflected as a decrease in SvO2. For this reason, SvO2 is regarded as a reliable indicator of tissue oxygenation and, therefore, can be used to indicate the effectiveness of the cardiopulmonary system during cardiac surgery and in the ICU. The fiber optic SvO2 catheter consists of two separate optical fibers; one fiber is used for transmitting incident light to the flowing blood, and the other directs the backscattered light to a photodetector. The catheter is introduced into the vena cava and further advanced through the heart into the pulmonary artery by inflating a small balloon located at the distal end. The flow-directed catheter also contains a small thermistor for measuring cardiac output by thermodilution. The principle is based on the relationship between SvO2 and the ratio of the infrared-tored (IR/R) light backscattered from the red blood cell in blood SvO2 ¼ A BðIR=RÞ

ð10:27Þ

where, A and B are empirically derived calibration coefficients. Several problems limit the wide clinical application of intravascular fiber optic oximeters. These include the dependence of the optical readings on motion artifacts due to catheter tip “whipping” against the blood vessel wall. Additionally, the introduction of the catheter into the heart requires an invasive procedure and can sometimes cause arrhythmias.

10.6.4 Intravascular Fiber Optic Pressure Sensors Pressure measurements provide important diagnostic information. For example, pressure measurements inside the heart, cranium, kidneys, and bladder can be used to diagnose abnormal physiological conditions that are otherwise not feasible to ascertain from imaging or other diagnostic modalities. In addition, intracranial hypertension resulting from injury or other causes can be monitored to assess the need for therapy and its efficacy. Likewise, dynamic changes of pressure measured inside the heart, cranial cavities, uterus, and bladder can help to assess the efficiency of these organs during contractions. Several approaches can be used to measure pressure using minimally invasive sensors. The most common technique involves the use of a fiber optic catheter. Fiber optic pressure sensors have been known and widely investigated since the early 1960s. The major challenge is to develop a small enough sensor with high sensitivity, high fidelity, and adequate dynamic response that can be inserted either through a hypodermic needle or in the form of a catheter. Additionally, for routine clinical use, the device must be cost-effective and disposable. A variety of ideas have been exploited for varying a light signal in a fiber optic probe with pressure. Most designs utilize either an interferometer principle or measure changes in light intensity. Interferometric-based pressure sensors are known to have a high sensitivity, but they involve complex calibration and require complicated fabrication. On the other

10.6 OPTICAL SENSORS

659

FIGURE 10.40 Fiber optic in vivo pressure sensor. Courtesy of Fiso Technologies, Quebec, Canada.

hand, fiber optic pressure sensors based on light intensity modulation have a lower sensitivity but involve simpler construction. The basic operating principle of a fiber optic pressure sensor is based on light intensity modulation. Typically, white light or light produced by a light emitting diode (LED) is carried by an optical fiber to a flexible mirrored surface located inside a pressure-sensing element. The mirror is part of a movable membrane partition that separates the fiber end from the fluid chamber. Changes in the hydrostatic fluid pressure cause a proportional displacement of the membrane relative to the distal end of the optical fiber. This in turn modulates the amount of light coupled back into the optical fiber. The reflected light is measured by a sensitive photodetector and converted to a pressure reading. A fiber optic pressure transducer for in vivo application based on optical interferometry using white light was developed by Fiso Technologies (Figure 10.40). The sensing element is based on a Fabry-Pe´rot principle. A miniaturized Fabry-Pe´rot cavity is defined on one end by a micromachined silicon diaphragm membrane that acts as the pressure sensing element and is bonded on a cup-shaped glass base attached to the opposite side of the optical fiber. When external pressure is applied to the transducer, the deflection of the diaphragm causes variation of the cavity length that in turn is converted to a pressure reading. Due to its extremely small size (dia: 550 mm), the sensor can be inserted through a hypodermic needle.

10.6.5 Intravascular Fiber Optic Temperature Sensors Miniature fiber optic temperature sensors, also commercialized by Fiso Technologies, are based on a similar Fabry-Pe´rot principle utilized in the construction of a miniaturized fiber optic pressure transducer. The Fabry-Pe´rot cavity in these designs is formed by two optical fibers assembled into a glass capillary tube or a transparent semiconductor material. The cavity length changes with temperature variations due to differences in the thermal expansion coefficient between the glass capillary and optical fibers. Due to their miniature construction (dia: 210–800 mm), the thermal inertia is close to zero, allowing ultrafast temperature response. The miniature size of the sensor allows the integration into minimally invasive medical devices for direct in situ measurement in space-restricted cavities.

10.6.6 Indicator-Mediated Fiber Optic Sensors Since only a limited number of biochemical substances have an intrinsic optical absorption or fluorescence property that can be measured directly with sufficient selectivity by standard spectroscopic methods, indicator-mediated sensors have been developed to use specific reagents that are immobilized either on the surface or near the tip of an optical

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fiber. In these sensors, light travels from a light source to the end of the optical fiber, where it interacts with a specific chemical or biological recognition element. These transducers may include indicators and ion-binding compounds (ionophores), as well as a wide variety of selective polymeric materials. After the light interacts with the biological sample, it returns either through the same optical fiber (in a single-fiber configuration) or a separate optical fiber (in a dual-fiber configuration) to a detector, which correlates the degree of light attenuation with the concentration of the analyte. Typical indicator-mediated sensor configurations are shown schematically in Figure 10.41. The transducing element is a thin layer of chemical material that is placed near the sensor tip and is separated from the blood medium by a selective membrane. The chemical-sensing material transforms the incident light into a return light signal with a magnitude that is proportional to the concentration of the species to be measured. The stability of the sensor is determined by the stability of the photosensitive material that is used and also by how effective the sensing material is protected from leaching out of the probe. In Figure 10.41a, the indicator is immobilized directly on a membrane that is positioned at the end of the fiber. An indicator in the form of a powder can also be physically retained in position at the end of the fiber by a special permeable membrane, as shown in Figure 10.41b, or a hollow capillary tube, as shown in Figure 10.41c.

10.6.7 Immunoassay Sensors The development of immunosensors is based on the observation of ligand-binding reaction products between a target analyte and a highly specific binding reagent. The key

FIGURE 10.41

Different indicator-mediated fiber optic sensor configurations.

10.6 OPTICAL SENSORS

FIGURE 10.42

661

Principle of a fiber optic immunoassay biosensor.

component of an immunosensor is the biological recognition element, which typically consists of antibodies or antibody fragments. Immunological techniques offer outstanding selectivity4 and sensitivity through the process of antibody-antigen interaction. This is the primary recognition mechanism by which the immune system detects and fights foreign matter and has therefore allowed the measurement of many important compounds at micromolar and even picomolar concentrations in complex biological samples. Evanescent-type biosensors can be used in immunological diagnostics to detect antibody-antigen binding. Figure 10.42 shows a conceptual diagram of an immunoassay biosensor. The immobilized antibody on the surface of the unclad portion of the fiber captures the antigen from the sample solution, which is normally introduced into a small flow through a chamber where the fiber tip is located. The sample solution is then removed and a labeled antibody is added into the flow chamber. A fluorescent signal is excited and measured when the labeled antibody binds to the antigen that is already immobilized by the antibody.

10.6.8 Surface Plasmon Resonance Sensors When monochromatic polarized light (e.g., from a laser source) impinges on a transparent medium having a conducting metallized surface (e.g., Ag or Au), there is a charge density oscillation at the interface. When light at an appropriate wavelength interacts with the dielectric-metal interface at a defined angle, called the resonance angle, there is a match of resonance between the energy of the photons and the electrons at the metal interface. As a result, the photon energy is transferred to the surface of the metal as packets of electrons, called plasmons, and the light reflection from the metal layer will be attenuated. This results in a phenomenon known as surface plasmon resonance (SPR) and is shown schematically in Figure 10.43. The resonance is observed as a sharp dip in the reflected light intensity when the incident angle is varied. The resonance angle depends on the incident wavelength, the type of metal, the polarization state of the incident light, and the nature of the medium in contact with the surface. Any change in the refractive index of the medium will produce a shift in the resonance angle and thus provide a highly sensitive means of monitoring surface interactions.

The sensor’s ability to detect a specific substance in a mixture containing other substances.

662

FIGURE 10.43 Sweden.

10. BIOMEDICAL SENSORS

Principle of a surface plasmon resonance (SPR) detection system. Courtesy of Biacore AB, Uppsala,

SPR is generally used for sensitive measurement of variations in the refractive index of the medium immediately surrounding the metal film. For example, if an antibody is bound to or absorbed into the metal surface, a noticeable change in the resonance angle can be readily observed because of the change of the refraction index at the surface if all other parameters are kept constant. The advantage of this concept is the improved ability to detect the direct interaction between antibody and antigen as an interfacial measurement.

10.7 EXERCISES 1. Give an example of a biomedical transducer that is used to monitor patients in the intensive care unit. 2. Discuss the important considerations in the selection of materials for packaging of an implantable biosensor. 3. Estimate the response time of an airflow transducer to monitor changes in breathing rate. 4. Explain why low drift is an important specification for implantable sensors. 5. The calibration tests of a new pressure transducer produced the readings in Table 10.4. (a) Plot the input-output calibration for this transducer. (b) Find the offset for readings between 0 to 200 mmHg. (c) Find the sensitivity for readings between 0 to 200 mmHg. (d) Estimate the average sensitivity for readings ranging between 200 to 300 mmHg. (e) State whether the response of this transducer over the entire measurement range is linear or nonlinear. 6. Suggest a method to measure hysteresis in a blood flow transducer. 7. Discuss the problem of using a pressure transducer with hysteresis to monitor blood pressure. 8. Explain how the accuracy of a new temperature sensor can be determined. 9. Two identical silver electrodes are placed in an electrolyte solution. Calculate the potential drop between the two electrodes.

663

10.7 EXERCISES

TABLE 10.4 Sample Calibration Data for a Pressure Sensor Pressure (mmHg)

Reading (mV)

100

120

100

140

120

160

135

180

150

200

165

220

180

240

190

260

200

280

210

300

220

320

225

340

230

360

235

380

237

400

239

420

240

440

240

10. Cadmium and zinc electrodes are placed in an electrolyte solution. Calculate the current that will flow through the electrodes if the equivalent resistance of the solution is equal to 14 kO. 11. Explain what will happen when the Ag/AgCl gel of an ECG electrode used to monitor a patient in the ICU dries out over time. 12. By how much would the inductance of an inductive displacement transducer coil change if the number of coil turns is decreased by a factor of 6? Continued

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10. BIOMEDICAL SENSORS

13. Determine the ratio between the cross-sectional areas of two blood vessels assuming that the voltage ratio induced in identical magnetic flow probes is equal to 2:3 and the ratio of blood flows through these vessels is 1:5. 14. A 4.5 kO linear rotary transducer is used to measure the angular displacement of the knee joint. Calculate the change in output voltage for a 165 change in the angle of the knee. Assume that a constant current of 14 mA is supplied to the transducer. 15. Provide a step-by-step derivation of Eq. (10.11). 16. An elastic resistive transducer with an initial resistance, Ro, and length, lo, is stretched to a new length. Assuming that the cross-sectional area of the transducer changes during stretching, derive a mathematical relationship for the change in resistance DR as a function of the initial length, lo; the change in length, Dl; the volume of the transducer, V; and the resistivity, r. 17. The area of each plate in a differential capacitor sensor is equal to 5.6 cm2. Calculate the equilibrium capacitance in air for each capacitor assuming that the equilibrium displacement for each capacitor is equal to 3 mm. 18. Plot the capacitance (y-axis) versus displacement (x-axis) characteristics of a capacitance transducer. 19. Calculate the sensitivity of a capacitive transducer (i.e., DC/Dd) for small changes in displacements. 20. A capacitive transducer is used in a mattress to measure changes in breathing patterns of an infant. During inspiration and expiration, the rate of change (i.e., dV/dt) in voltage across the capacitor is equal to 1V/s, and this change can be modeled by a triangular waveform. Plot the corresponding changes in current flow through this transducer. 21. Derive the relationship for the current through the capacitor-equivalent piezoelectric crystal as a function of V and C. 22. Two identical ultrasonic transducers are positioned across a blood vessel, as shown in Figure 10.44. Calculate the diameter of the blood vessel if it takes 380 ns for the ultrasonic sound wave to propagate from one transducer to the other. 23. Discuss the advantages of MEMS-type sensors.

FIGURE 10.44

Two identical ultrasonic transducers positioned across a blood vessel.

10.7 EXERCISES

665

24. Calculate the resistance of a thermistor at 98 F assuming that the resistance of this thermistor at 12 C is equal to 7.0 kO and b ¼ 4,600. 25. The resistance of a thermistor with a b ¼ 5,500 measured at 18 C is equal to 250 O. Find the temperature of the thermistor when the resistance is doubled. 26. Calculate the b of a thermistor assuming that it has a resistance of 4.4 kO at 21 C (room temperature) and a resistance of 2.85 kO when the room temperature increases by 20 percent. 27. A Chromel/Constantan thermocouple has the following empirical coefficients: C0 ¼ 2.340 102 C1 ¼ 4.221 102 C2 ¼ 3.284 105 Find the EMF generated by this thermocouple at a temperature of 250 C. 28. Find the Seebeck coefficient for the Chromel/Alumel thermocouple at a temperature of 200 C. 29. Explain why the temporal artery thermometer is not used to measure core body temperature over the radial artery. 30. Compare and contrast the temporal artery and tympanic thermometers. 31. Compare and contrast a temperature pill with a temporal artery thermometer. 32. Sketch the current (y-axis) versus pO2 (x-axis) characteristics of a typical polarographic Clark electrode. 33. Explain why the value of the normalized ratio (R) in a pulse oximeter is independent of the volume of arterial blood entering the tissue during systole. 34. Explain why the value of the normalized ratio (R) in a pulse oximeter is independent of skin pigmentation. 35. Explain the difference between a potentiometric and amperometric sensor. 36. Explain the difference between intravascular fiber optic pO2 and SvO2 sensors. 37. A pH electrode is attached to a sensitive voltmeter that reads 0.652 V when the electrode is immersed in a buffer solution with a pH of 6.7. After the pH electrode is moved to an unknown buffer solution, the reading of the voltmeter is decreased by 20 percent. Calculate the pH of the unknown buffer solution. 38. Plot the optical density, OD, of an absorbing solution (y-axis) as a function of the concentration of this solution (x-axis). What is the slope of this curve? 39. An unknown sample solution whose concentration is 1.55 103 g/L is placed in a 1 cm clear holder and found to have a transmittance of 44 percent. The concentration of this sample is changed such that its transmittance has increased to 57 percent. Calculate the new concentration of the sample. 40. Calculate the angle of the refracted light ray if an incident light ray passing from air into water has a 75-degree angle with respect to the normal. 41. Explain why fiber optic sensors typically require simultaneous measurements using two wavelengths of light. 42. Plot the fluorescence intensity of a fiber optic pO2 sensor (y-axis) as a function of oxygen concentration (x-axis). Continued

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10. BIOMEDICAL SENSORS

43. A chemical sensor is used to measure the pH of a dye with an absorbance spectrum shown in Figure 10.45. Assume that the absorbance of each form of the dye is linearly related to its pH. Devise a method to measure the pH of the dye.

1.0 pH=8.0 Absorbance

0.8 0.6 0.4

pH=5.0

0.2

400

FIGURE 10.45

500 600 700 Wavelength [nm]

Optical absorbance spectra of a dye in its acid (pH ¼ 5.0) and base (pH ¼ 8.0) forms.

44. Explain the difference between absorption-based and fluorescence-based measurements.

Suggested Readings J.A. Allocca, A. Stuart, Transducers: Theory and Applications, Reston Publishing, Reston, VA, 1984. R. Aston, Principles of Biomedical Instrumentation and Measurement, Macmillan, New York, 1990. D. Buerk, Biosensors: Theory and Applications, CRC Press, Boca Raton, FL, 1995. R.S.C. Cobbold, Transducers for Biomedical Measurement: Principles and Applications, Wiley, New York, 1974. L. Cromwell, F.J. Weibell, E.J. Pfeiffer, Biomedical Instrumentation and Measurements, Prentice Hall, Englewood Cliffs, NJ, 1980. B. Eggins, Biosensors: An Introduction, Wiley, New York, 1997. B.R. Eggins, Chemical Sensors and Biosensors for Medical and Biological Applications, John Wiley, New York, 2002. L.A. Geddes, L.E. Baker, Principles of Applied Biomedical Instrumentation, third ed., Wiley-Interscience, New York, 1989. E.A.H. Hall, Biosensors, Prentice Hall, Englewood Cliffs, NJ, 1991. G. Harsanyi, Sensors in Biomedical Applications: Fundamental, Technology and Applications, CRC Press, Boca Raton, FL, 2000. M.R. Neuman, Biomedical Sensors, in: J.D. Bronzino (Ed.), The Biomedical Engineering Handbook, second ed., CRC/IEEE Press, Boca Raton, FL, 1999. T. Togawa, T. Tamura, P.A. Oberg, Biomedical Transducers and Instruments, CRC Press, Boca Raton, FL, 1997. J.G. Webster, Encyclopedia of Medical Devices and Instrumentation, John Wiley, New York, 1988. J.G. Webster, Medical Instrumentation: Application and Design, third ed., John Wiley, New York, 1998. D.L. Wise, Bioinstrumentation and Biosensors, Marcel Dekker, New York, 1991. J. Cooper, A.E.G. Cass, Biosensors, Oxford University Press, 2004.

C H A P T E R

11 Biosignal Processing Monty Escabı´, PhD O U T L I N E 11.1

Introduction

668

11.2

Physiological Origins of Biosignals

668

11.3

Characteristics of Biosignals

671

11.4

Signal Acquisition

674

11.5

Frequency Domain Representation of Biological Signals 679

11.6

Linear Systems

11.7

Signal Averaging

11.8

The Wavelet Transform and the Short-Time Fourier Transform 727

11.9

Artificial Intelligence Techniques

11.10

721

Exercises

732 741

Suggested Readings

745

700

A T T HE C O NC LU SI O N O F T H IS C HA P T E R , S T UD EN T S WI LL B E A BL E T O : • Describe the different origins and types of biosignals.

• Describe the basic properties of a linear system.

• Distinguish between deterministic, periodic, transient, and random signals.

• Describe the concepts of filtering and signal averaging.

• Explain the process of A/D conversion. • Define the sampling theorem.

• Explain the basic concepts and advantages of fuzzy logic.

• Describe the main purposes and uses of the Fourier transforms.

• Describe the basic concepts of artificial neural networks.

• Define the Z-transform.

Introduction to Biomedical Engineering, Third Edition

667

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11. BIOSIGNAL PROCESSING

11.1 INTRODUCTION Biological signals, or biosignals, are space, time, or space-time records of a biological event such as a beating heart or a contracting muscle. The electrical, chemical, and mechanical activity that occurs during this biological event often produces signals that can be measured and analyzed. Biosignals, therefore, contain useful information that can be used to understand the underlying physiological mechanisms of a specific biological event or system and that may be useful for medical diagnosis. Biological signals can be acquired in a variety of ways—for example, by a physician who uses a stethoscope to listen to a patient’s heart sounds or with the aid of technologically advanced biomedical instruments. Following data acquisition, biological signals are analyzed in order to retrieve useful information. Basic methods of signal analysis, such as amplification, filtering, digitization, processing, and storage, can be applied to many biological signals. These techniques are generally accomplished with simple electronic circuits or with digital computers. In addition to these common procedures, sophisticated digital processing methods are quite common and can significantly improve the quality of the retrieved data. These include signal averaging, wavelet analysis, and artificial intelligence techniques.

11.2 PHYSIOLOGICAL ORIGINS OF BIOSIGNALS 11.2.1 Bioelectric Signals Nerve and muscle cells generate bioelectric signals that are the result of electrochemical changes within and between cells (see Chapter 5). If a nerve or muscle cell is stimulated by a stimulus that is strong enough to reach a necessary threshold, the cell will generate an action potential. The action potential, which represents a brief flow of ions across the cell membrane, can be measured with intracellular or extracellular electrodes. Action potentials generated by an excited cell can be transmitted from one cell to adjacent cells via its axon. When many cells become activated, an electric field is generated that propagates through the biological tissue. These changes in extracellular potential can be measured on the surface of the tissue or organism by using surface electrodes. The electrocardiogram (ECG), electrogastrogram (EGG), electroencephalogram (EEG), and electromyogram (EMG) are all examples of this phenomenon (Figure 11.1).

11.2.2 Biomagnetic Signals Different organs, including the heart, brain, and lungs, also generate weak magnetic fields that can be measured with magnetic sensors. Typically, the strength of the magnetic field is much weaker than the corresponding physiological bioelectric signals. Biomagnetism is the measurement of the magnetic signals that are associated with specific physiological activity and that are typically linked to an accompanying electric field from a specific tissue or organ. With the aid of very precise magnetic sensors or SQUID (superconducting quantum interference device) magnetometers, it is possible to directly monitor

11.2 PHYSIOLOGICAL ORIGINS OF BIOSIGNALS

669

VOLTAGE (mV)

−5

−10

−15 (a)

TIME (ms)

30 25 20

VOLTAGE (mV)

15 10 5 0 −5 −10 −15 −20

TIME (ms)

(b)

FIGURE 11.1 (a) Electrogram recorded from the surface of a pig’s heart during normal sinus rhythm. (b) Electrogram recorded from the surface of the same pig’s heart during ventricular fibrillation (VF). (Sampled at 1,000 samples/s.)

magnetic activity from the brain (magnetoencephalography, MEG), peripheral nerves (magnetoneurography, MNG), gastrointestinal tract (magnetogastrography, MGG), and the heart (magnetocardiography, MCG).

11.2.3 Biochemical Signals Biochemical signals contain information about changes in concentration of various chemical agents in the body. The concentration of various ions, such as calcium and potassium, in cells can be measured and recorded. Changes in the partial pressures of oxygen (PO2) and

670

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carbon dioxide (PCO2) in the respiratory system or blood are often measured to evaluate normal levels of blood oxygen concentration. All of these constitute biochemical signals. These biochemical signals can be used for a variety of purposes, such as determining levels of glucose, lactate, and metabolites and providing information about the function of various physiological systems.

11.2.4 Biomechanical Signals Mechanical functions of biological systems, which include motion, displacement, tension, force, pressure, and flow, also produce measurable biological signals. Blood pressure, for example, is a measurement of the force that blood exerts against the walls of blood vessels. Changes in blood pressure can be recorded as a waveform (Figure 11.2). The upstrokes in the waveform represent the contraction of the ventricles of the heart as blood is ejected from the heart into the body and blood pressure increases to the systolic pressure, the maximum blood pressure (see Chapter 3). The downward portion of the waveform depicts ventricular relaxation as the blood pressure drops to the minimum value, better known as the diastolic pressure.

AORTIC PRESSURE (mmHG) 100 90 80 70 60 50 40 30 20 10 0

0.5

1.5

2.5 TIME (secs)

3.5

4.5

FIGURE 11.2 Blood pressure waveform recorded from the aortic arch of a 4-year-old child. (Sampled at 200 samples/s.)

11.3 CHARACTERISTICS OF BIOSIGNALS

671

11.2.5 Bioacoustic Signals Bioacoustic signals are a special subset of biomechanical signals that involve vibrations (motion). Many biological events produce acoustic noise. For instance, the flow of blood through the valves in the heart has a distinctive sound. Measurements of the bioacoustic signal of a heart valve can be used to determine whether it is operating properly. The respiratory system, joints, and muscles also generate bioacoustic signals that propagate through the biological medium and can often be measured at the skin surface by using acoustic transducers such as microphones and accelerometers.

11.2.6 Biooptical Signals Biooptical signals are generated by the optical or light induced attributes of biological systems. Biooptical signals can occur naturally, or in some cases, the signals may be introduced to measure a biological parameter with an external light medium. For example, information about the health of a fetus may be obtained by measuring the fluorescence characteristics of the amniotic fluid. Estimates of cardiac output can be made by using the dye dilution method that involves monitoring the concentration of a dye as it recirculates through the bloodstream. Finally, red and infrared light are used in various applications, such as to obtain precise measurements of blood oxygen levels by measuring the light absorption across the skin or a particular tissue.

EXAMPLE PROBLEM 11.1 What types of biosignals would the muscles in your lower legs produce if you were to sprint across a paved street?

Solution Motion of the muscles and the external forces imposed as your feet hit the pavement produce biomechanical signals. Muscle stimulation by nerves and the contraction of muscle cells produce bioelectric signals. Metabolic processes in the muscle tissue could be measured as biochemical signals.

11.3 CHARACTERISTICS OF BIOSIGNALS Biological signals can be classified according to various characteristics of the signal, including the waveform shape, statistical structure, and temporal properties. Two broad classes of signals that are commonly encountered include continuous and discrete signals. Continuous signals are defined over a continuum of time or space and are described by continuous variables. The notation x(t) is used to represent a signal, x, that varies as a function of continuous time, t. Signals that are produced by biological phenomena are almost always continuous signals. Some examples include voltage measurements from the heart (see Figure 11.1), arterial blood pressure measurements (see Figure 11.2), and measurements of electrical activity from the brain.

672

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Discrete signals are also commonly encountered in today’s clinical setting. Unlike continuous signals, which are defined along a continuum of points in space or time, discrete signals are defined only at a subset of regularly spaced points in time and/or space. Discrete signals are therefore represented by arrays or sequences of numbers. The notation, x(n), is used to represent a discrete sequence, x, that exists only at a subset of points in discrete time, n. Here, n ¼ 0, 1, 2, 3 . . . is always an integer that represents the nth element of the discrete sequence. Although most biological signals are not discrete per se, discrete signals play an important role due to today’s advancements in digital technology. Sophisticated medical instruments are commonly used to convert continuous signals from the human body to discrete digital sequences (see Chapter 7) that can be analyzed and interpreted with a computer. Computer axial tomography (CAT) scans, for instance, take digital samples from continuous x-ray images of a patient that are obtained from different perspective angles (see Chapter 15). These digitized or discrete image slices are then digitally enhanced, manipulated, and processed to generate a full three-dimensional computer model of a patient’s internal organs. Such technologies are indispensable tools for clinical diagnosis. Biological signals can also be classified as being either deterministic or random. Deterministic signals can be described by mathematical functions or rules. Periodic and transient signals make up a subset of all deterministic signals. Periodic signals are usually composed of the sum of different sine waves or sinusoid components and can be expressed as xðtÞ ¼ xðt þ kTÞ

ð11:1Þ

where x(t) is the signal, k is an integer, and T is the period. The period represents the distance along the time axis between successive copies of the periodic signal. Periodic signals have a basic waveshape with a duration of T units that repeats indefinitely. Transient signals are nonzero or vary only over a finite time interval and subsequently decay to a constant value as time progresses. The sine wave, shown in Figure 11.3a, is a simple example of a periodic signal, since it repeats indefinitely with a repetition interval of 1 second. The product of a decaying exponential and a sine wave, as shown in Figure 11.3b, is a transient signal, since the signal amplitude approaches zero as time progresses. Real biological signals almost always have some unpredictable noise or change in parameters and, therefore, are not entirely deterministic. The ECG of a normal beating heart at rest is an example of a signal that appears to be almost periodic but has a subtle unpredictable component. The basic waveshape consists of the P wave, QRS complex, and T wave and repeats (see Figure 3.22). However, the precise shapes of the P waves, QRS complexes, and T are somewhat irregular from one heartbeat to another. The length of time between QRS complexes, which is known as the R-R interval, also changes over time as a result of heart rate variability (HRV). HRV is used as a diagnostic tool to predict the health of a heart that has experienced a heart attack. The extended outlook for patients with low HRV is generally worse than it is for patients with high HRV. Random signals, also called stochastic signals, contain uncertainty in the parameters that describe them. Because of this uncertainty, mathematical functions cannot be used to precisely describe random signals. Instead, random signals are most often analyzed using statistical techniques that require the treatment of the random parameters of the signal with probability distributions or simple statistical measures such as the mean and standard deviation. The electromyogram (EMG), an electrical recording of electrical activity in skeletal muscle that is

673

11.3 CHARACTERISTICS OF BIOSIGNALS

1.5

VOLTAGE (V)

0.5

0 0

0.5

1.5

2.5

3.5

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−0.5

−1

−1.5

TIME (s)

(a) 1 0.8

VOLTAGE (V)

0.6 0.4 0.2 0

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−0.2 −0.4 −0.6 −0.8

TIME (s)

(b) (a) Periodic sine wave signal x(t) ¼ sin(ot) with period of 1 Hz. (b) Transient signal y(t) ¼ e0.75tsin (ot) for the same 1 Hz sine wave.

FIGURE 11.3

used for the diagnosis of neuromuscular disorders, is a random signal. Stationary random signals have statistical properties, such as a mean and variance, that remain constant over time. Conversely, nonstationary random signals have statistical properties that vary with time. In many instances, the identification of stationary segments of random signals is important for proper signal processing, pattern analysis, and clinical diagnosis.

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11. BIOSIGNAL PROCESSING

EXAMPLE PROBLEM 11.2 Ventricular fibrillation (VF) is a cardiac arrhythmia in which there are no regular QRS complexes, T waves, or rhythmic contractions of the heart muscle (see Figure 11.1b). VF often leads to sudden cardiac death, which is one of the leading causes of death in the United States. What type of biosignal would most probably be recorded by an ECG when a heart goes into VF?

Solution An ECG recording of a heart in ventricular fibrillation will be a random, continuous, bioelectric signal.

11.4 SIGNAL ACQUISITION 11.4.1 Overview of Biosignal Data Acquisition Biological signals are often very small and typically contain unwanted interference or noise. Such interference has the detrimental effect of obscuring relevant information that may be available in the measured signal. Noise can be extraneous in nature, arising from sources outside the body, such as thermal noise in sensors or 60-cycle noise in the electronic components of the acquisition system. Noise can also be intrinsic to the biological media, meaning it can arise from adjacent tissues or organs. ECG measurements from the heart, for instance, can be affected by bioelectric activity from adjacent muscles. In order to extract meaningful information from biological signals sophisticated data acquisition techniques and equipment are commonly used. High-precision low-noise equipment is often necessary to minimize the effects of unwanted noise. Figure 11.4 shows the basic components in a bioinstrumentation system. Throughout the data acquisition procedure, it is critical that the information and structure of the original biological signal of interest be faithfully preserved. Since these signals are often used to aid the diagnosis of pathological disorders, the procedures of amplification, analog filtering, and A/D conversion should not generate misleading or untraceable distortions. Distortions in a signal measurement could lead to an improper diagnosis.

11.4.2 Sensors, Amplifiers, and Analog Filters Signals are first detected in the biological medium, such as a cell or on the skin’s surface, by using a sensor (see Chapter 6). A sensor converts a physical measurand into an electric output and provides an interface between biological systems and electrical recording instruments. The type of biosignal determines what type of sensor will be used. ECGs, for example, are measured with electrodes that have a silver-silver chloride (Ag-AgCl) interface attached to the body that detects the movement of ions. Arterial blood pressure is measured with a sensor that detects changes in pressure. It is very important that the sensor used to detect the biological signal of interest does not adversely affect the properties and characteristics of the signal it is measuring.

675

11.4 SIGNAL ACQUISITION

SENSORS

ANALOG SIGNAL CONDITIONER

PARAMETER TO BE OBSERVED

DATA ACQUISITION SYSTEM

DATA STORAGE AND DISPLAY

DIGITAL SIGNAL PROCESSING

FIGURE 11.4 Sensors adapt the signal that is being observed into an electrical analog signal that can be measured with a data acquisition system. The data acquisition system converts the analog signal into a calibrated digital signal that can be stored. Digital signal processing techniques are applied to the stored signal to reduce noise and extract additional information that can improve understanding of the physiological meaning of the original parameter.

After the biosignal has been detected with an appropriate sensor, it is usually amplified and filtered. Operational amplifiers are electronic circuits that are used primarily to increase the amplitude or size of a biosignal. Bioelectric signals, for instance, are often faint and require up to a thousand-fold boosting of their amplitude with such amplifiers. An analog filter may then be used to remove noise or to compensate for distortions caused by the sensor. Amplification and filtering of the biosignal may also be necessary to meet the hardware specifications of the data acquisition system. Continuous signals may need to be limited to a certain band of frequencies before the signal can be digitized with an analog-to-digital converter, prior to storing in a digital computer.

11.4.3 A/D Conversion Analog-to-digital (A/D) converters are used to transform biological signals from continuous analog waveforms to digital sequences. An A/D converter is a computer-controlled voltmeter, which measures an input analog signal and gives a numeric representation of the signal as its output. Figure 11.5a shows an analog signal, and Figure 11.5b shows a digital version of the same signal. The analog waveform, originally detected by the sensor and subsequently amplified and filtered, is a continuous signal. The A/D converter transforms the continuous, analog signal into a discrete, digital signal. The discrete signal consists of a sequence of numbers that can easily be stored and processed on a digital computer. A/D conversion is particularly important because storage and analysis of biosignals are becoming increasingly computer based. The digital conversion of an analog biological signal does not produce an exact replica of the original signal. The discrete, digital signal is a digital approximation of the original, analog signal that is generated by repeatedly sampling the amplitude level of the original signal at fixed time intervals. As a result, the original, analog signal is represented as a sequence of numbers: the digital signal.

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11. BIOSIGNAL PROCESSING

2.5 2

VOLTAGE (V)

1.5 1 0.5 0

0.5

1.5

2.5

3.5

4.5

−0.5 −1 −1.5

TIME (s)

(a) 2.5 2

VOLTAGE (V)

1.5 1 0.5 0

0.5

1.5

2.5

3.5

4.5

−0.5 −1 −1.5

TIME (s)

(b)

FIGURE 11.5

(a) Analog version of a periodic signal. (b) Digital version of the analog signal.

The two main processes involved in A/D conversion are sampling and quantization. Sampling is the process by which a continuous signal is first converted into a discrete sequence in time. If x(t) is an analog signal, sampling involves recording the amplitude value of x(t) every T seconds. The amplitude value is denoted as x(kT) where k ¼ 0, 1, 2, 3, . . . is an integer that denotes the position or the sample number from the sample set or data sequence.

11.4 SIGNAL ACQUISITION

677

T represents the sampling interval or the time between adjacent samples. In real applications, finite data sequences are generally used in digital signal processing. Therefore, the range of a data points is k ¼ 0, 1, . . . N-1, where N is the total number of discrete samples. The sampling frequency, fs, or the sampling rate, is equal to the inverse of the sampling period, 1/T, and is measured in units of Hertz (s1). The following are digital sequences that are of particular importance: The unit-sample of impulse sequence: d(k) ¼ 1 if k ¼ 0 0 if k 6¼ 0 The unit-step sequence: u(k) ¼ 1 if k > 0 0 if k < 0 The exponential sequence: aku(k) ¼ ak if k > 0 0 if k < 0 The sampling rate used to discretize a continuous signal is critical for the generation of an accurate digital approximation. If the sampling rate is too low, distortions will occur in the digital signal. Nyquist’s theorem states that the minimum sampling rate used, fs, should be at least twice the maximum frequency of the original signal in order to preserve all of the information of the analog signal. The Nyquist rate is calculated as fnyquist ¼ 2 fmax

ð11:2Þ

where fmax is the highest frequency present in the analog signal. The Nyquist theorem therefore states that fs must be greater than or equal to 2 fmax in order to fully represent the analog signal by a digital sequence. Practically, sampling is usually done at five to ten times the highest frequency, fMax. The second step in the A/D conversion process involves signal quantization. Quantization is the process by which the continuous amplitudes of the discrete signal are digitized by a computer. In theory, the amplitudes of a continuous signal can be any of an infinite number of possibilities. This makes it impossible to store all the values, given the limited memory in computer chips. Quantization overcomes this by reducing the number of available amplitudes to a finite number of possibilities that the computer can handle. Since digitized samples are usually stored and analyzed as binary numbers on computers, every sample generated by the sampling process must be quantized. During quantization, the series of samples from the discretized sequence are transformed into binary numbers. The resolution of the A/D converter determines the number of bits that are available for storage. Typically, most A/D converters approximate the discrete samples with 8, 12, or 16 bits. If the number of bits is not sufficiently large, significant errors may be incurred in the digital approximation. A/D converters are characterized by the number of bits that they use to generate the numbers of the digital approximation. A quantizer with N bits is capable of representing a total of 2N possible amplitude values. Therefore, the resolution of an A/D converter

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increases as the number of bits increases. A 16-bit A/D converter has better resolution than an 8-bit A/D converter, since it is capable of representing a total of 65,536 amplitude levels, compared to 256 for the 8-bit converter. The resolution of an A/D converter is determined by the voltage range of the input analog signal divided by the numeric range (the possible number of amplitude values) of the A/D converter.

EXAMPLE PROBLEM 11.3 Find the resolution of an 8-bit A/D converter when an input signal with a 10 V range is digitized.

Solution input voltage range 10 V ¼ 0:0391 V=bit ¼ 39:1 mV=bit ¼ 256 2N

EXAMPLE PROBLEM 11.4 The frequency content of an analog EEG signal is 0.5–100 Hz. What is the lowest rate at which the signal can be sampled to produce an accurate digital signal?

Solution Highest frequency in analog signal ¼ 100 Hz. fnyquist ¼ 2 fmax ¼ 2 100 Hz ¼ 200 samples/second.

Another problem often encountered is determining what happens if a signal is not sampled at a rate high enough to produce an accurate representation of the signal. A direct result of the sampling theorem is that all frequencies of the form [f – kfs], where 1 k 1 and fs ¼ 1/T, look the same once they are sampled.

EXAMPLE PROBLEM 11.5 A 360 Hz signal is sampled at 200 samples/second. What frequency will the “aliased” digital signal contain?

Solution According to the preceding formula, fs ¼ 200, and the pertinent set of frequencies that look alike is in the form of [360 – k 200] ¼ [. . . 360 160 40 240 . . . .]. The only signal in this group that will be accurately sampled is 40 Hz, since the sampling rate is more than twice this value. Note that for real signals 40 Hz and þ40 Hz are equivalent—that is, cos(ot) ¼ cos(ot) and sin(ot) ¼ sin(ot). Thus, the sampled signal will exhibit a period of 40 Hz. The process is shown in Figure 11.6.

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11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

360 Hz Signal,T=.005 s 1 0.5 0 −0.5 −1 0

0.005

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0.015

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40 Hz Signal, T=.005 s 1 0.5 0 −0.5 −1 0

0.005

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0.015 TIME (s)

FIGURE 11.6

A 360 Hz sine wave is sampled every 5 ms—that is, at 200 samples/s. This sampling rate will adequately sample a 40 Hz sine wave but not a 360 Hz sine wave.

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS In the early nineteenth century, Joseph Fourier laid out one of the most important theories on the field of function approximation. At the time, his result was applied toward the problem of heat transfer in solids, but it has since gained a much broader appeal. Today, Fourier’s findings provide a general theory for approximating complex waveforms with simpler functions that has numerous applications in mathematics, physics, and engineering. This section summarizes the Fourier transform and variants of this technique that play an important role in the analysis and interpretation of biological signals.

11.5.1 Periodic Signal Representation: The Trigonometric Fourier Series As an artist mixes oil paints on a canvas, a scenic landscape is meticulously recreated by combining various colors on a palate. It is well known that all shades of the color spectrum

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11. BIOSIGNAL PROCESSING

can be recreated by simply mixing primary colors (red, green, and blue, or RGB) in the correct proportions. Television and computer displays often transmit signals as RGB, and these signals are collated together to create colors much as a master painter would on a canvas. In fact, the human visual system takes exactly the opposite approach. The retina decomposes images and scenery from the outside world into purely red-green-blue signals that are independently analyzed and processed by our brains. Despite this, we perceive a multitude of colors and shades. This simple color analogy is at the heart of Fourier’s theory, which states that a complex waveform can be approximated to any degree of accuracy with simpler functions. In 1807, Fourier showed that an arbitrary periodic signal of period, T, can be represented mathematically as a sum of trigonometric functions. Conceptually, this is achieved by summing or mixing sinusoids while simultaneously adjusting their amplitudes and frequency, as shown for a square wave function in Figure 11.7.

(a)

0 −2

−1

0 Time (sec)

(b)

0 −2 5

(c)

0 −2 5

Amplitude

(d)

(e)

0 −2 5

0 −2

FIGURE 11.7 A square wave signal (a) is approximated by adding sinusoids (B–E). (b) 1 sinusoid, (c) 2 sinusoids, (d) 3 sinusoids, (e) 4 sinusoids. Increasing the number of sinusoids improves the quality of the approximation.

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

681

If the amplitudes and frequencies are chosen appropriately, the trigonometric signals add constructively, thus recreating an arbitrary periodic signal. This is akin to combining prime colors in precise ratios to recreate an arbitrary color and shade. RGB are the building blocks for more elaborate colors, much as sinusoids of different frequencies serve as the building blocks for more complex signals. All of these elements (the color and the required proportions; the frequencies and their amplitudes) have to be precisely adjusted to achieve a desired result. For example, a first-order approximation of the square wave is achieved by fitting the square wave to a single sinusoid of appropriate frequency and amplitude. Successive improvements in the approximation are obtained by adding higher-frequency sinusoid components, or harmonics, to the first-order approximation. If this procedure is repeated indefinitely, it is possible to approximate the square wave signal with infinite accuracy. The Fourier series summarizes this result as xðtÞ ¼ a0 þ

1 X

ðam cos moo t þ bm sin moo tÞ

ð11:3aÞ

m¼1

where x(t) is the periodic signal to be approximated, o0 ¼ 2p=T is the fundamental frequency of x(t) in units of radians/s, and the coefficients am and bm determine the amplitude of each cosine and sine term at a specified frequency om ¼ mo0 . Equation (11.3a) tells us that the periodic signal, x(t), is precisely replicated by summing an infinite number of sinusoids. The frequencies of the sinusoid functions always occur at integer multiples of oo and are referred to as “harmonics” of the fundamental frequency. If we know the coefficients am and bm for each of the corresponding sine or cosine terms, we can completely recover the signal x(t) by evaluating the Fourier series. How do we determine am and bm for an arbitrary signal? The coefficients of the Fourier series correspond to the amplitude of each sine and cosine. These are determined as ð 1 a0 ¼ xðtÞdt ð11:3bÞ T T

am ¼

2 xðtÞ cosðmoo tÞdt T

ð11:3cÞ

ð 2 xðtÞ sinðmoo tÞdt bm ¼ T

ð11:3dÞ

where the integrals are evaluated over a single period, T, of the waveform.

EXAMPLE PROBLEM 11.6 Find the trigonometric Fourier series of the square wave signal shown in Figure 11.7A, and implement the result in MATLAB for the first ten components. Plot the time waveform and the Fourier coefficients. Continued

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11. BIOSIGNAL PROCESSING

Solution First note that T ¼ 2 and

o0 ¼

2p ¼p T

To simplify the analysis, integration for am and bm is carried out over the first period of the waveform (from –1 to 1) ð ð 1 1 1 1=2 5 xðtÞdt ¼ 5dt ¼ a0 ¼ T 1 2 1=2 2 am ¼

2 T

ð1 1

xðtÞ cosðmoo tÞdt ¼

ð 1=2 1=2

5 cosðmptÞdt

1=2 sinðmptÞ sinðmp=2Þ ¼ 5 ¼ 5 sincðmp=2Þ ¼ 5 mp mp=2 1=2

2 bm ¼ T

1=2 cosðmptÞ xðtÞ sinðmoo tÞdt ¼ 5 sinðmptÞdt ¼ 5 ¼0 mp 1 1=2

ð1

ð 1=2

1=2

where by definition sincðxÞ ¼ sinðxÞ=x. Substituting the values for a0, am, and bm into Eq. (11.3a) gives 1 X 5 sinðmp=2Þ xðtÞ ¼ þ 5 cosðmptÞ 2 mp=2 m¼1

MATLAB implementation: %Plotting Fourier Series Approximation subplot(211) time¼-2:0.01:2; %Time Axis x¼5/2; %Initializing Signal for m¼1:10 x¼xþ5*sin(m*pi/2)/m/pi*2*cos(m*pi*time); end plot(time,x,’k’) %Plotting and Labels xlabel(’Time (sec)’) ylabel(’Amplitude’) set(gca,’Xtick’,[2:2]) set(gca,’Ytick’,[0 5]) set(gca,’Box’,’off’) %Plotting Fourier Magnitudes subplot(212) m¼1:10;

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11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

Am¼[5/2 5*sin(m*pi/2)./m/pi*2]; %Fourier Magnitudes Faxis¼(0:10)*.5; %Frequency Axis plot(Faxis,Am,’k.’) %Plotting axis([0 5 2 4]) set(gca,’Box’,’off’) xlabel(’Frequency (Hz)’) ylabel(’Fourier Amplitudes’)

Note that the approximation of summing the first ten harmonics (Figure 11.8a) closely resembles the desired square wave. The Fourier coefficients, am, for the first ten harmonics are shown as a function of the harmonic frequency in Figure 11.8b. To fully replicate the sharp transitions of the square wave, an infinite number of harmonics are required.

Amplitude

0 −2

−1

(a)

Time (sec)

Fourier Amplitudes

4 3 2 1 0 −1 −2

0.5

1.5

(b)

2 2.5 3 Frequency (Hz)

3.5

4.5

FIGURE 11.8

(a) MATLAB result showing the first ten terms of Fourier series approximation for the square wave. (b) The Fourier coefficients are shown as a function of the harmonic frequency.

11.5.2 Compact Fourier Series The trigonometric Fourier series provides a direct approach for fitting and analyzing various types of biological signals, such as the repetitive beating of a heart or the cyclic oscillations produced by the vocal folds as one speaks. Despite its utility, alternate forms of the Fourier series are sometimes more appealing because they are easier to work with

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11. BIOSIGNAL PROCESSING

and because signal measurements can often be interpreted more readily. The most widely used counterparts for approximating and modeling biological signals are the exponential and compact Fourier series. The compact Fourier series is a close cousin of the standard Fourier series. This version of the Fourier series is obtained by noting that the sum of sinusoids and cosines can be rewritten by a single cosine term with the addition of a phase constant am cosmoo t þ bm sinmoo t ¼ Am cosðmoo t þ fm Þ, which leads to the compact form of the Fourier series: xðtÞ ¼

1 A0 X Am cosðmoo t þ fm Þ: þ 2 m¼1

The amplitude for each cosine, Am, is related to the Fourier coefficients through qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Am ¼ a2m þ b2m and the cosine phase is obtained from am and bm as 1 bm : fm ¼ tan am

ð11:4aÞ

ð11:4bÞ

ð11:4cÞ

EXAMPLE PROBLEM 11.7 Convert the standard Fourier series for the square pulse function of Example Problem 11.5 to compact form and implement in MATLAB.

Solution We first need to determine the magnitude, Am, and phase, fm, for the compact Fourier series. The magnitude is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j sinðmp=2Þj Am ¼ a2m þ b2m ¼ ð5 sincðmp=2ÞÞ2 þ ð0Þ2 ¼ 5 pm=2 Since

j sinðmp=2Þj ¼

we have

Am ¼

1 m ¼ odd 0 m ¼ even

10=mp m ¼ odd : 0 m ¼ even

Unlike am or bm in the standard Fourier series, note that Am is strictly a positive quantity for all m. The phase term is determined as 0 m ¼ 0, 1, 4, 5, 8, 9 . . . 0 1 bm 1 ¼ for ¼ tan fm ¼ tan 5 sincðmp=2Þ am p m ¼ 2, 3, 6, 7, 10, 11 . . .

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

685

Combining results 1 5 X 10 xðtÞ ¼ þ cosðmoo t þ fm Þ 2 m¼1 mp

where fm is as just defined. An interesting point regards the similarity of standard and compact versions of the Fourier series for this square wave example. In the standard form, the coefficient am alternates between positive and negative values, while for the compact form the Fourier coefficient, Am, is identical in magnitude to am, but it is always a positive quantity. The sign (þ or –) of the standard Fourier coefficient is now consumed in the phase term, which alternates between 0 and p . This forces the cosine to alternate in its external sign because cosðxÞ ¼ cosðx þ pÞ. The two equations are therefore mathematically identical, differing only in the way that the trigonometric functions are written out. MATLAB implementation: %Plotting Fourier Series Approximation time¼-2:0.01:2; %Time Axis x¼5/2; %Initializing Signal m¼1:10; A¼(10*sin(m*pi/2)./m/pi); %Fourier Coefficients P¼angle(A); %Phase Angle A¼abs(A); %Fourier Magnitude for m¼1:10 x¼xþA(m)*cos(m*pi*timeþP(m)); end subplot(211) plot(time,x,’k’) %Plotting and Labels xlabel(’Time (sec)’) ylabel(’Amplitude’) set(gca,’Xtick’,[-2:2]) set(gca,’Ytick’,[0 5]) set(gca,’Box’,’off’) %Plotting Fourier Magnitudes subplot(212) m¼1:10; A¼[5/2 A]; %Fourier Magnitudes Faxis¼(0:10)*.5; %Frequency Axis plot(Faxis,A,’k.’) %Plotting axis([0 5 -2 4]) set(gca,’Box’,’off’) xlabel(’Frequency (Hz)’) ylabel(’Fourier Amplitudes’)

The results are identical to those shown in Figures 11.8a and b.

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11. BIOSIGNAL PROCESSING

11.5.3 Exponential Fourier Series The main result from the Fourier series analysis is that an arbitrary periodic signal can approximate by summing individual cosine terms with specified amplitudes and phases. This result serves as much of the conceptual and theoretical framework for the field of signal analysis. In practice, the Fourier series is a useful tool for modeling various types of quasi-periodic signals. An alternative and somewhat more convenient form of this result is obtained by noting that complex exponential functions are and cosinespthrough ffiffiffiffiffiffiffi directly relatedto sinusoids Euler’s identities: cosðyÞ ¼ e jy þ ejy =2 and sinðyÞ ¼ e jy ejy =2j, where j ¼ 1. By applying Euler’s identity to the compact trigonometric Fourier series, an arbitrary periodic signal can be expressed as a sum of complex exponential functions: xðtÞ ¼

þ1 X m¼1

cm e jkoo t

ð11:5aÞ

This equation represents the exponential Fourier series of a periodic signal. The coefficients cm are complex numbers that are related to the trigonometric Fourier coefficients cm ¼

am jbm Am jfm ¼ e 2 2

ð11:5bÞ

The proof for this result is beyond the scope of this text, but it is important to realize that the trigonometric and exponential Fourier series are intimately related, as can be seen by comparing their coefficients. The exponential coefficients can also be obtained directly by integrating x(t), ð 1 xðtÞejmoo t dt ð11:5cÞ cm ¼ T T

over one cycle of the periodic signal. As for the trigonometric Fourier series, the exponential form allows us to approximate a periodic signal to any degree of accuracy by adding a sufficient number of complex exponential functions. A distinct advantage of the exponential Fourier series, however, is that it requires only a single integral (Eq. (11.5c)), compared to the trigonometric form, which requires three separate integrations.

EXAMPLE PROBLEM 11.8 Find the exponential Fourier series for the square wave of Figure 11.7a and implement in MATLAB for the first ten terms. Plot the time waveform and the Fourier series coefficients.

Solution Like Example Problem 11.6, the Fourier coefficients are obtained by integrating from 1 to 1. Because a single cycle of the square wave signal has nonzero values between 1/2 and þ1/2, the integral can be simplified by evaluating it between these limits:

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

687

1=2 ð ð 1 1 1=2 5 ejmpt jmoo t jmpt cm ¼ xðtÞe dt ¼ 5e dt ¼ T 2 1=2 2 jmp

1=2

5 eþjmp=2 ejmp=2 5 sinðmp=2Þ : ¼ 2 2 mp=2 jmp

Therefore, xðtÞ ¼

þ1 X m¼1

cm e jkoo t ¼

1 X 5 sinðmp=2Þ e jmpt 2 mp=2 m¼1

MATLAB implementation: %Plotting Fourier Series Approximation subplot(211) time¼-2:0.01:2; %Time Axis x¼0; %Initialize Signal for m¼-10:10 if m¼¼0 x¼xþ5/2; %Term for m¼0 else x¼xþ5/2*sin(m*pi/2)/m/pi*2*exp(j*m*pi*time); end end plot(time,x,’k’) %Plotting and Labels xlabel(’Time (sec)’) ylabel(’Amplitude’) set(gca,’Xtick’,[-2:2]) set(gca,’Ytick’,[0 5]) set(gca,’Box’,’off’) %Plotting Fourier Magnitudes subplot(212) m¼(-10:10)þ1E-10; A¼[5/2*sin(m*pi/2)./m/pi*2]; %Fourier Magnitudes Faxis¼(-10:10)*.5; %Frequency Axis plot(Faxis,A,’k.’) %Plotting axis([-5 5 -2 4]) set(gca,’Box’,’off’) xlabel(’Frequency (Hz)’) ylabel(’Fourier Amplitudes’)

Note that we now require positive and negative frequencies in the approximation. Results showing the MATLAB output are shown in Figure 11.9. Continued

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11. BIOSIGNAL PROCESSING

Amplitude

0 −2

−1

0 Time (sec)

(a)

Fourier Amplitudes

4 3 2 1 0 −1 −2 −5

−4

−3

(b)

−2

1 −1 0 Frequency (Hz)

FIGURE 11.9 (a) MATLAB result showing the first ten terms of exponential Fourier series approximation for the square wave. (b) The compact Fourier coefficients are shown as a function of the harmonic frequency. Note that both negative and positive frequencies are now necessary to approximate the square wave signal.

In practice, many periodic or quasi-periodic biological signals can be accurately approximated with only a few harmonic components. Figures 11.10 and 11.11 illustrate a harmonic reconstruction of an aortic pressure waveform obtained by applying a Fourier series approximation. Figure 11.10 plots the coefficients for the cosine series representation as a function of the harmonic number. Note that the low-frequency coefficients are large in amplitude, whereas the high-frequency coefficients contain little energy and do not contribute substantially to the reconstruction. The amplitude coefficients, Am, are plotted on a log10 scale so the smaller values are magnified and are therefore visible. Figure 11.11 shows several levels of harmonic reconstruction. The mean plus the first and second harmonics provide the basis for the general systolic and diastolic shape, since the amplitudes of these harmonics are large and contribute substantially to the reconstructed waveform. Additional harmonics add fine details but do not contribute significantly to the raw waveform.

11.5.4 Fourier Transform In many instances, conceptualizing a signal in terms of its contributing cosine or sine functions has various advantages. The concept of frequency domain is an abstraction that

689

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

A(m) (mmHg)

100

.01

Æ(m) (radians/s)

3 2 1 0 −1 −2 −3

10 m

FIGURE 11.10 Harmonic coefficients of the aortic pressure waveform shown in Figure 11.2.

is borne out of the Fourier series representation for a periodic signal. A signal can be expressed either in the “time-domain” by the signal’s time function, x(t), or alternatively in the “frequency-domain” by specifying the Fourier coefficient and phase, Am and fm, as a function of the signal’s harmonic frequencies, om ¼ moo . Thus, if we know the Fourier coefficients and the frequency components that make up the signal, we can fully recover the periodic signal x(t). One of the disadvantages of the Fourier series is that it applies only to periodic signals, and many biological signals are not periodic. In fact, a broad class of biological signals includes signals that are continuous functions of time but that never repeat in time. Luckily, the concept of Fourier series can also be extended for signals that are not periodic. The Fourier integral, also referred to as the Fourier transform, is used to decompose a continuous aperiodic signal into its constituent frequency components 1 ð

XðoÞ ¼ 1

xðtÞejot dt

ð11:6Þ

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11. BIOSIGNAL PROCESSING

ORIGINAL

MEAN + 1st & 2nd

80 60

40 40 20 20

0 −20

0 0

0.1

0.2 TIME

0.3

0.4

5 HARMONIC

0 0.1

0.2 TIME

0.3

0.4

0.1

0.2 TIME

0.3

0.4

0.2 TIME

0.3

0.4

10 HARMONIC

0.1

FIGURE 11.11 Harmonic reconstruction of the aortic pressure waveform shown in Figure 11.2.

much as the Fourier series decomposes a periodic signal into its corresponding trigonometric components. X(o) is a complex valued function of the continuous frequency, o, and is analogous to the coefficients of the complex Fourier series, cm. A rigorous proof for this relationship is beyond the scope of this text, but it is useful to note that the Fourier integral is derived directly from the exponential Fourier series by allowing the period, T, to approach infinity. The coefficients cm of the trigonometric series approach X(o) as T ! 1. Conceptually, a function that repeats at infinity can be considered as aperiodic, since you will never observe it repeating. Tables of Fourier transforms for many common signals can be found in most signals and systems or signal processing textbooks. As for the Fourier series, a procedure for converting the frequency-domain version of the signal, X(o), to its time-domain expression is desired. The time-domain signal, x(t), can be completely recovered from the Fourier transform with the inverse Fourier transform (IFT) 1 xðtÞ ¼ 2p

1 ð

XðoÞe jot do:

ð11:7Þ

These two representations of a signal are interchangeable, meaning that we can always go back and forth between the time-domain version of the signal, x(t), and the frequency-domain

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

691

version obtained with the Fourier transform, X(o). The frequency domain expression therefore provides all of the necessary information for the signal and allows one to analyze and manipulate biological signals from a different perspective.

EXAMPLE PROBLEM 11.9 Find the Fourier Transform (FT) of the rectangular pulse signal xðtÞ ¼ 1, jtj < a 0, jtj > a

Solution Equation (11.6) is used. ða

jot

XðoÞ ¼ a

a ejot 2 sin oa dt ¼ ¼ jo o a

As for the Fourier series representation of a signal, the magnitude and the phase are important attributes of the Fourier transform. As stated previously, X(o) is a complex valued function, meaning that it has a real, Re{X(o)}, and imaginary, Im{X(o)}, component and can be expressed as XðoÞ ¼ RefXðoÞg þ j ImfXðoÞg:

ð11:8Þ

As for the Fourier series, the magnitude determines the amplitude of each complex exponential function (or equivalent cosine) required to reconstruct the desired signal, x(t), from its Fourier transform qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11:9Þ jXðoÞj ¼ RefXðoÞg2 þ ImfXðoÞg2 In contrast, the phase determines the time shift of each cosine signal relative to a reference of time zero. It is determined as 1 ImfXðoÞg yðoÞ ¼ tan : ð11:10Þ RefXðoÞg Note the close similarity for determining the magnitude and phase from the trigonometric and compact forms of the Fourier series (Eqs. (11.4a, b, c)). The magnitude of the Fourier transform, jXðoÞj, is analogous to Am, whereas am and bm are analogous to Re{X(o)} and Im{X(o)}, respectively. The equations are identical in all other respects.

EXAMPLE PROBLEM 11.10 Find the magnitude and phase of the signal with the Fourier transform XðoÞ ¼

1 1 þ jo Continued

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11. BIOSIGNAL PROCESSING

Solution The signal has to be put in a recognizable form similar to Eq. (11.8). To achieve this, XðoÞ ¼

1 1 jo 1 jo 1 o ¼ j : ¼ 1 þ jo 1 jo 1 þ o2 1 þ o2 1 þ o2

Therefore RefXðoÞg ¼

1 1 þ o2

and

ImfXðoÞg ¼

o : 1 þ o2

Using Eqs. (11.9) and (11.10), the magnitude is jXðoÞj ¼

1 1 þ o2

and the phase yðoÞ ¼ tan1 ðoÞ ¼ tan1 ðoÞ:

11.5.5 Properties of the Fourier Transform In practice, computing Fourier transforms for complex signals may be somewhat tedious and time consuming. When working with real-world problems, it is therefore useful to have tools available that help simplify calculations. The FT has several properties that help simplify frequency domain transformations. Some of these are summarized following. Let x1(t) and x2(t) be two signals in the time domain. The FTs of x1(t) and x2(t) are represented as X1(o) ¼ F{x1(t)} and X2(o) ¼ F{x2(t)}. Linearity The Fourier transform is a linear operator. Therefore, for any constants a1 and a2, Ffa1 x1 ðtÞ þ a2 x2 ðtÞg ¼ a1 X1 ðoÞ þ a2 X2 ðoÞ

ð11:11Þ

This result demonstrates that the scaling and superposition properties defined for a liner system also hold for the Fourier transform. Time Shifting/Delay If x1(t t0) is a signal in the time domain that is shifted in time, the Fourier transform can be represented as Ffx1 ðt t0 Þg ¼ XðoÞ ejo t0

ð11:12Þ

In other words, shifting a signal in time corresponds to multiplying its Fourier transform by a phase factor, ejot0. Frequency Shifting If X1(oo0) is the Fourier transform of a signal, shifted in frequency, the inverse Fourier transform is F1 fX1 ðo o0 Þg ¼ xðtÞ ejo0 t

ð11:13Þ

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693

Convolution Theorem The convolution between two signals, x1(t) and x2(t), in the time domain is defined as 1 ð

cðtÞ ¼

x1 ðtÞx2 ðt tÞdt ¼ x1 ðtÞ * x2 ðtÞ

ð11:14Þ

where * is shorthand for the convolution operator. The convolution has an equivalent expression in the frequency domain CðoÞ ¼ FfcðtÞg ¼ Ffx1 ðtÞ * x2 ðtÞg ¼ X1 ðoÞ X2 ðoÞ:

ð11:15Þ

Convolution in the time domain, which is relatively difficult to compute, is a straightforward multiplication in the frequency domain. Next, consider the convolution of two signals, X1(o) and X2(o), in the frequency domain. The convolution integral in the frequency domain is expressed as 1 ð

XðoÞ ¼

X1 ðnÞX2 ðo nÞdn ¼ X1 ðoÞ * X2 ðoÞ

ð11:16Þ

It can be shown that the inverse Fourier transform (IFT) of X(o) is xðtÞ ¼ F1 fXðoÞg ¼ F1 fX1 ðoÞ * X2 ðoÞg ¼ 2p x1 ðtÞ x2 ðtÞ

ð11:17Þ

Consequently, the convolution of two signals in the frequency domain is 2p times the product of the two signals in the time domain. As we will see subsequently for linear systems, convolution is an important mathematical operator that fully describes the relationship between the input and output of a linear system.

EXAMPLE PROBLEM 11.11 What is the FT of 3 sin (25t) þ 4 cos (50t)? Express your answer only in a symbolic equation. Do not evaluate the result.

Solution Ff3 sin ð25tÞ þ 4 cos ð50tÞg ¼ 3Ff sin ð25tÞg þ 4Ff cos ð50tÞg

11.5.6 Discrete Fourier Transform In digital signal applications, continuous biological signals are first sampled by an analog-to-digital converter (see Figure 11.4) and then transferred to a computer, where they can be further analyzed and processed. Since the Fourier transform applies only to continuous signals of time, analyzing discrete signals in the frequency domain requires that we first modify the Fourier transform equations so they are structurally compatible with the digital samples of a continuous signal.

694

11. BIOSIGNAL PROCESSING

The discrete Fourier transform (DFT) XðmÞ ¼

N1 X

xðkÞej

2pmk N ;m

¼ 0, 1, . . . , N=2

ð11:18Þ

k¼0

provides the tool necessary to analyze and represent discrete signals in the frequency domain. The DFT is essentially the digital version of the Fourier transform. The index m represents the digital frequency index, x(k) is the sampled approximation of x(t), k is the discrete time variable, N is an even number that represents the number of samples for x(k), and X(m) is the DFT of x(k). The inverse discrete Fourier transform (IDFT) is the discrete-time version of the inverse Fourier transform. The inverse discrete Fourier transform (IDFT) is represented as xðkÞ ¼

X 2pmk 1 N1 XðmÞe j N ; k ¼ 0, 1, . . . , N 1 N m¼0

ð11:19Þ

As for the FT and IFT, the DFT and IFT represent a Fourier transform pair in the discrete domain. The DFT allows one to convert a set of digital time samples to its frequency domain representation. In contrast, the IDFT can be used to invert the DFT samples, allowing one to reconstruct the signal samples x(k) directly from its frequency domain form, X(m). These two equations are thus interchangeable, since either conveys all of the signal information.

EXAMPLE PROBLEM 11.12 Find the discrete Fourier transform of the signal xðkÞ ¼ 0:25k for k ¼ 0:15 k X N 1 15 15 15 X X X 2pmk 2pmk 2pm xðkÞej N ¼ 0:25k ej 16 ¼ 0:25 ej N ¼ ak XðmÞ ¼ k¼0

k¼0

k¼0 j2pm N

Note that the preceding is a geometric sum in which a ¼ 0:25 e N X k¼M

ak ¼

. Since for a geometric sum

aNþ1 aM a1

we obtain 32mp

XðmÞ ¼

a16 a0 0:2516 ej N 1 ¼ 2mp a1 0:25ej N 1

An efficient computer algorithm for calculating the DFT is the fast Fourier transform (FFT). The output of the FFT and DFT algorithms are the same, but the FFT has a much faster execution time than the DFT (proportional to N log2 ðNÞ versus N2 operations). The ratio of computing time for the DFT and FFT is therefore DFT computing time N2 N ¼ ¼ FFT computing time N log2 N log2 N

ð11:20Þ

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

695

In order for the FFT to be efficient, the number of data samples, N, must be a power of two. If N ¼ 1024 signal samples, the FFT algorithm is approximately 1024= log2 ð1024Þ ¼ 10 times faster than the direct DFT implementation. If N is not a power of two, alternate DFT algorithms are usually used.

Figures 11.12 and 11.13 show two signals and the corresponding DFT, which was calculated using the FFT algorithm. The signal shown in Figure 11.12a is a sine wave 6

Amplitude

2 0 −2 −4 −6 0

0.05

(a)

0.1 Time (sec)

0.15

0.2

Magnitude

150

100

0 (b)

FIGURE 11.12

100

200

300

400

500

Frequency (Hz) (a) 100 Hz sine wave. (b) Fast Fourier transform (FFT) of 100 Hz sine wave.

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11. BIOSIGNAL PROCESSING

Amplitude

2 0 −2 −4 −6 0

0.05

(a)

0.1 Time (sec)

0.15

0.2

150

Magnitude

100

0 (b)

FIGURE 11.13

100

200 300 Frequency (Hz)

400

500

(a) 100 Hz sine wave corrupted with noise. (b) Fast Fourier transform (FFT) of noisy 100 Hz sine

wave.

of frequency of 100 Hz. Figure 11.12b shows the FFT of the 100 Hz sine wave. Notice that the peak of the FFT occurs at 100 Hz frequency, indicating that all of the energy is confined to this frequency. Figure 11.13a shows a 100 Hz sine wave corrupted with random noise that was added to the waveform. The frequency of the signal is not distinct in the time domain. After transforming this signal to the frequency domain, the signal (Figure 11.13b) reveals a definite component at 100 Hz frequency, which is marked by the large peak in the FFT.

697

11.5 FREQUENCY DOMAIN REPRESENTATION OF BIOLOGICAL SIGNALS

EXAMPLE PROBLEM 11.13 Find and plot the magnitude of the discrete Fourier transform of the signal xðnÞ ¼ sinðp=4 nÞ þ 2 cosðp=3 nÞ in MATLAB.

Solution n¼1:1024; %Discrete Time Axis x¼sin(pi/4*n)þ2*cos(pi/3*n); %Generating the signal X¼fft(x,1024*16)/1024; %Computing 16k point Fast Fourier Transform Freq¼(1:1024*16)/(1024*16)*2*pi; %Normalizing Frequencies between 0-2*pi plot(Freq,abs(X),’k’) %Plotting axis([0.7 1.15 0 1.2]) xlabel(’Frequency (rad/s)’) ylabel(’Fourier Magnitude’)

Results are shown in Figure 11.14. 1.2

Fourier Magnitude

0.8

0.6

0.4

0.2

0 0.7

0.75

0.8

0.85

0.9 0.95 1 Frequency (rad/s)

1.05

1.1

1.15

FIGURE 11.14

Fast Fourier transform magnitude for the sum of two sinusoids. Dominant energy peaks are located at the signal frequencies p/3 and p/4 rad/s.

11.5.7 The z-Transform The z-transform provides an alternative tool for analyzing discrete signals in the frequency domain. This transform is essentially a variant of the DFT, where we allow z ¼ ej

2pm N .

698

11. BIOSIGNAL PROCESSING

In most applications, the z-transform is somewhat easier to work with than the DFT because it does not require the use of complex numbers directly. The z-transform plays a similar role for digital signals as the Laplace transform does for the analysis of continuous signals. If a discrete sequence x(k) is represented by xk , the (one-sided) z-transform of the discrete sequence is expressed by XðzÞ ¼

1 X

xk zk ¼ x0 þ x1 z1 þ x2 z2 þ K þ xk zk

ð11:21Þ

k¼0

Note that the z-transform can be obtained directly from the DFT by allowing N ! 1 and 2pm

replacing z ¼ ej N in Eq. (11.18). In most practical applications, sampled biological signals are represented by a data sequence with N samples so the z-transform is estimated for k ¼ 0 . . . N-1 only. Tables of common z-transfo

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