These notes are based on (i) a series of lectures that I gave at the 14th Biennial Seminar of the Canadian Mathematical Congress held at the University of Western Ontario August 12-24, 1973 and (li) some of my lectures in a modeling course that I have cotaught in the Division of Bio-Medical Sciences at Brown during the past several years. An earlier version of these notes appeared in the Center for Dynamical Systems Lectures Notes series (CDS LN 73-1, November 1973). I have in this revised and extended version of those earlier notes incorporated a number of changes based both on classroom experience and on my research efforts with several colleagues during the intervening period. The narrow viewpoint of the present notes (use of optimization and control theory in biomedical problems) reflects more the scope of the CMC lectures given in August, 1973 than the scope of my own interests. Indeed, my real interests have included the modeling process itself as well as the contributions made by investiga­ tors who employ the techniques and ideas of control theory, systems analysis, dif­ ferential equations, and stochastic processes. Some of these contributions have quite naturally involved application of optimal control theory. But in my opinion many of the interesting efforts being made in modeling in the biomedical sciences encompass much more than the use of control theory.

1. A Brief Review of Enzyme Kinetics.- 2. Models for Enzymatically Active Membranes.- 2.1 Modeling of the physiological phenomena using Briggs-Haldane approximations.- 2.2 Control problems arising in connection with membrane models.- 2.3 Hysteresis and oscillatory phenomena in membrane models.- 3. Modeling of Enzyme Cascades.- 3.1 Physiological motivation and a review of previous modeling attempts.- 3.2 Derivation of a model based on the cascade in glycogenolysis.- 3.3 The role of optimality in model development.- 3.4 Qualitative and numerical results.- 3.5 Glycogenolytic cascade models with activators/inhibitors.- 4. Modeling and Control of Epidemics.- 5. Modeling of the Control System in Glucose Homeostasis.- 6. Modeling and Control of Tumor Growth.- 6.1 Probabilistic models.- 6.2 Deterministic models.- 6.3 Optimal fractionated therapy.- 7. A Survey of Recent Efforts.- 7.1 Biped locomotion.- 7.2 Countercurrent dialysis.- 7.3 Drug regimens.- 7.4 Insect respiration.- 7.5 Patient care and diagnostic models.- 7.6 Ecological systems and resource management.- 7.7 Parameter estimation and identification.- 7.8 Miscellaneous topics.