1 The Heart and Circulation.- 1.1 Plan of the Circulation.- 1.2 Volume, Flow, and Pressure.- 1.3 Resistance and Compliance Vessels.- 1.4 The Heart as a Pair of Pumps.- 1.5 Mathematical Model of the Uncontrolled Circulation.- 1.6 Balancing the Two Sides of the Heart and the Two Circulations.- 1.7 The Need for External Circulatory Control Mechanisms.- 1.8 Neural Control: The Baroreceptor Loop.- 1.9 Autoregulation.- 2 Gas Exchange in the Lungs.- 2.1 The Ideal Gas Law and the Solubility of Gases.- 2.2 The Equations of Gas Transport in One Alveolus.- 2.3 Gas Transport in the Lung.- 2.4 Optimal Gas Transport.- 2.5 Mean Alveolar and Arterial Partial Pressures.- 2.6 Transport of O2.- 2.7 Computer Solution of the Equations for O2 Transport in the Lung.- 2.8 Computing Projects Concerning Oxygen Transport by the Lung.- 2.9 Annotated References.- Exercises.- 3 Control of Cell Volume and Electrical Properties of Cell Membranes.- 3.1 Osmotic Pressure and the Work of Concentration.- 3.2 A Simple Model of Cell Volume Control.- 3.3 The Movement of Ions Across Cell Membranes.- 3.4 The Interaction of Electrical and Osmotic Effects.- 3.5 The Hodgkin-Huxley Equations for the Nerve Action Potential.- 3.6 Computer Simulation of the Nerve Action Potential.- 3.7 Suggestions for Computing Projects Concerning the Nerve Impulse.- 3.8 Annotated References.- Exercises.- 4 The Renal Countercurrent Mechanism.- 4.1 The Nephron.- 4.2 Dynamics of Na+ and H2O: Transport along the Renal Tubules.- 4.3 The Loop of Henle.- 4.4 The Juxtaglomerular Apparatus and the Renin-Angiotensin System.- 4.5 The Distal Tubule and Collecting Duct: Concentrating and Diluting Modes.- 4.6 Remarks on the Significance of the Juxtaglomerular Apparatus.- 4.7 How Nephrons Do Better Than a Factor of e.- 4.8 Computing Project on theInteracting Nephron Population Model.- 4.9 Annotated References.- Exercises.- 5 Muscle Mechanics.- 5.1 The Force-Velocity Curve.- 5.2 Crossbridge Dynamics.- 5.3 Computer Simulation of Crossbridge Attachment and Detachment.- 5.4 Suggested Computing Projects on Crossbridge Dynamics.- 5.5 Annotated References.- Exercises.- 6 Neural Systems.- 6.1 Guttman's Experiments on Phase Locking.- 6.2 Biological Rhythms.- 6.3 Model Neural Networks.- 6.4 Annotated References.- Exercises.- 7 Population Dynamics.- 7.1 Bacterial Cultures.- 7.2 Age Structures.- 7.3 Microbial Ecology.- 7.4 Nonlinear Reproduction Curves.- 7.5 Controlling Populations.- 7.6 Annotated References.- Exercises.- 8 Genetics.- 8.1 Population Genetics.- 8.2 Biotechnology.- 8.3 Annotated References.- Exercises.- 9 A Theory of Epidemics.- 9.1 Spread of Infection Within a Family.- 9.2 The Threshold of an Epidemic.- 9.3 Predicting the Severity of an Epidemic.- 9.4 Annotated References.- Exercises.- 10 Patterns of Population Growth and Dispersal.- 10.1 Random Walks and the Process of Diffusion.- 10.2 Bacterial Growth on a Petri Plate.- 10.3 Concluding Remarks.- 10.4 Annotated References.- Exercises.- Appendix A Getting Started with Matrices and Matlab.- Appendix B Background on Random Processes.
Mathematics in Medicine and the Life Sciences grew from lectures given by the authors at New York University, the University of Utah, and Michigan State University. The material is written for students who have had but one term of calculus, but it contains material that can be used in modeling courses in applied mathematics at all levels through early graduate courses. Numerous exercises are given as well, and solutions to selected exercises are included. Numerous illustrations depict physiological processes, population biology phenomena, models of them, and the results of computer simulations.
Mathematical models and methods are becoming increasingly important in medicine and the life sciences. This book provides an introduction to a wide diversity of problems ranging from population phenomena to demographics, genetics, epidemics and dispersal; in physiological processes, including the circulation, gas exchange in the lungs, control of cell volume, the renal counter-current multiplier mechanism, and muscle mechanics; to mechanisms of neural control. Each chapter is graded in difficulty, so a reading of the first parts of each provides an elementary introduction to the processes and their models. Materials that deal with the same topics but in greater depth are included later. Finally, exercises and some solutions are given to test the reader on important parts of the material in the text, or to lead the reader to the discovery of interesting extensions of that material.
