Texts in Applied Mathematics 10. Editors F. John J.E. Marsden L. Sirovich M. Golubitsky W.Jiiger

Transcription

1 Texts in Applied Mathematics 10 Editors F. John J.E. Marsden L. Sirovich M. Golubitsky W.Jiiger

2 Texts in Applied Mathematics 1. Sirovich: Introduction to Applied Mathematics. 2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. 3. Hale/KOfak: Dynamics and Bifurcations. 4. ChorinjMarsden: A Mathematical Introduction to Fluid Mechanics, 2nd ed. 5. Hubbard/West: Differential Equations: A Dynamical Systems Approach, Part 1: Ordinary Differential Equations. 6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems. 7. Perko: Differential Equations and Dynamical Systems. 8. Seaborn: Hypergeometric Functions and Their Applications. 9. Pipkin: A Course on Integral Equations. 10. Hoppensteadt/Peskin: Mathematics in Medicine and the Life Sciences.

3 Frank C. Hoppensteadt Charles S. Peskin Mathematics in Medicine and the Life Sciences With 73 Illustrations Springer Science+Business Media, LLC

4 Frank C. Hoppensteadt College of Natural Science Michigan State University East Lansing, MI Editors F.John Courant Institute of Mathematical Sciences New York University New York, NY M. Golubitsky Department of Mathematics U niversity of Houston Houston, TX Charles S. Peskin Courant Institute of Mathematical Sciences New York University New York, NY J.E. Marsden Department of Mathematics University of California Berkeley, CA W.Jager Department of Applied Mathematics Universtitat Heidelberg Im Neuenheimer Feld Heidelberg, FRG L. Sirovich Division of Applied Mathematics Brown University Providence, RI Cover art by Charles S. Peskin and Peter H. Carrington. Mathematics Subject Classification: 92-01, 92B05, 92D99 Library ofcongress Cataloging-in-Publication Data Hoppensteadt, F. C. Mathematics in medicine and the life sciences / Frank C. Hoppensteadt, Charles S. Peskin. p. cm. - (Texts in applied mathematics) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Biomathematics. 2. Population biology-mathematics. 3. Physiology-Mathematics. I. Peskin, Charles S. II. Title. III. Series. [DNLM 1. Biology-methods. 2. Mathematics. 3. Models. Biological. 4. Physiology-methods. QT m] OH323.5.H '.01 '51-dc20 DNLMIDLC for Library of Congress Printed on acid-free paper Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1992 Softcover reprint ofthe hardcover Ist edition 1992 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Photocomposed copy prepared using LaTEX ISBN

5 Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM}. The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS} series, which will focus on advanced textbooks and research level monographs.

6 Preface Mathematical Biology is the study of medicine and the life sciences that uses mathematical models to help predict and interpret what we observe. This book describes several major contributions that have been made to population biology and to physiology by such theoretical work. We have tried to keep the presentation brief to keep the price of the book as reasonable as possible, and to ensure that the topics are presented at a level that is accessible to a wide audience. Each topic could serve as a launching point for more advanced study, and suitable references are suggested to help with this. If the underlying mathematics is understood for these basic examples, then mathematical aspects of more advanced life science problems will be within reach. The techniques presented here range in mathematical difficulty up to calculus and matrix theory. The material is presented in general order of increasing mathematical difficulty. Some exercises deal with material in preceding sections, others are projects that extend preceding material. Our purpose in this book is not the systematic presentation of mathematical material, although there are important threads that run through several chapters. Instead, we hope to illustrate how mathematics can be used. In particular, our goal is to make available to students, having at least one term of calculus, topics in the life sciences and medicine that have benefited from mathematical modeling and analysis. In addition to exposing students to current ideas, the material is intended to reinforce their mathematics education by presenting familiar mathematical topics from novel points of view. Finally, enabling students to think in terms of models early in their academic experience should motivate them to develop and apply modeling skills further. While hoping this interdisciplinary book will be useful to a wide variety of individuals, we believe that it can have special significance for the premedical student, who will find a mathematical introduction to a host of phenomena that are central to the practice of medicine. These include genetics and epidemics as well as the functions of the heart, lungs, and kidneys. It is our hope that the mathematical study of these topics will give the student a depth of understanding and insight that could not have been achieved through traditional, descriptive education in the medical sciences. The mix of topics, taken largely from population biology and from physiology, includes important phenomena that are within reach of the students described above. The population part of the book draws its material from the areas of demographics, genetics, epidemics, and biogeography, while the

7 viii Preface physiological part surveys cardiovascular, pulmonary, renal, and muscle physiology. The final chapter is intended to introduce students to models of nerve cells and some neural circuits as a basis for studying how the brain works. We are on the rise of a wave of understanding of brain function, and mathematical modeling can be useful in understanding this complex organ. We thank Anneli Lax for her early interest in the course that led to this book and for helpful discussion during the preparation of the lecture notes on which this book is based. Besides the authors, the course has been taught by Stephen Childress, H. Michael Lacker, and Daniel Tranchina, and we are indebted to them for their comments and advice. Frank C. Hoppensteadt Charles S. Peskin November, 1990

8 Contents Series Preface... v Preface... vii Introduction The Mathematics of Populations: Demographics Geometric Population Growth Growth of Bacterial Cultures Least-Squares Estimation of the Growth Rate Growth of Human Populations Infinitesimal Sampling Intervals and Doubling Times Geometric Growth in a Population Stratified by Age Fibonacci's Rabbit Population Euler's Renewal Equations Age Structure in Human Populations The Limits of Growth Verhulst's Model Predator Satiation Chaos Infinitesimal Sampling Intervals in a Limiting Environment Age Structure of Populations near the Limits of Growth Harvesting Summary Annotated References Exercises Inheritance Mendel's Laws Bacterial Genetics: Plasmids Genetics in Small Populations of Humans The Hardy-Weinberg Equilibrium Summary Annotated References Exercises... 63

9 x Contents 3 A Theory of Epidemics Spread of Infection within a Family The Threshold of an Epidemic Calculation of the Severity of an Epidemic Summary Annotated References Exercises Biogeography The Game of Life Random Walks The Diffusion Approximation The Growth of Bacteria on Plates Another View of Random Walks Summary Annotated References Exercises The Heart and Circulation Plan of the Circulation Volume, Flow, and Pressure Resistance and Compliance Vessels The Heart as a Pair of Pumps Mathematical Model of the Uncontrolled Circulation Balancing the Two Sides of the Heart and the Two Circulations Cardiac Output and Arterial Blood Pressure: The Need for External Circulatory Control Mechanisms Neural Control: The Baroreceptor Loop Autoregulation Changes in the Circulation Occurring at Birth Dynamics of the Arterial Pulse Annotated References Exercises Gas Exchange in the Lungs The Ideal Gas Law and the Solubility of Gases The Equations of Gas Transport in One Alveolus Gas Transport in the Lung Optimal Gas Transport Mean Alveolar and Arterial Partial Pressures Transport of Annotated References Exercises

10 7 Control of Cell Volume and Contents xi the Electrical Properties of Cell Membranes Osmotic Pressure and the Work of Concentration A Simple Model of Cell Volume Control The Movement of Ions across Cell Membranes Control of Cell Volume: The Interaction of Electrical and Osmotic Effects Transient Changes in Membrane Potential: A Signaling Mechanism in Nerve and Muscle Annotated References Exercises The Renal Countercurrent Mechanism The Nephron Differential Equations of Na+ and H 20 Transport along the Renal Thbules The Loop of Henle The Juxtaglomerular Apparatus and the Renin-Angiotensin System The Distal Thbule and Collecting Duct: Concentrating and Diluting Modes Remarks on the Significance of the Juxtaglomerular Apparatus Annotated References Exercises Muscle Mechanics The Force-Velocity Curve Cross-Bridge Dynamics Annotated References Exercises Biological Clocks and Mechanisms of Neural Control A Theory of Clocks l.l.The Clock on the Wall Phase Resetting: A Rubber Handed Clock Modulated Clocks Nerve Cell Membranes Cell Membrane Potential Guttman's Experiments VON: A Voltage Controlled Oscillator Neuron Voltage Controlled Oscillators Phase Comparators and a Model Synapse VCON: A Model Spike Generator

11 xii Contents Phase Locking Properties of a VCON Neural Control Networks Network Notation von Euler's Respiration Control Mechanism Summary Annotated References Exercises Answers for Selected Exercises Index