PREFACE The new frontiers of mathematics are the domains of biology and medicine. The gradual invasion of thesefieldsby mathematicians is already yielding a number of benefits as a result of the application of known techniques in tandem with digital computers. However, more important from the standpoint of the mathematician is the fact that these two areas represent a cornucopia of challenging problems. Mathematics is no exception to the law that all cultural activities require the con- tinued infusion of new ideas for their growth, and even existence. Let us briefly discuss some of the types of questions arising in biomedical re- search which are susceptible to mathematical techniques which we now possess. To begin with, there is the conceptually straightforward task of describing behavior over time of a physiological system. The difficulties encountered are standard in mathematical physics: recognition of state variables, determination of local cause and effect relations, with or without hereditary effects, and the analytic and compu- tational obstacles resulting from high dimensionality. In human systems, there are further obstacles due to the fact that certain essential state variables cannot be measured without irreversibly affecting the entire system. We then face the usual inverse problems of ascertaining the validity of various hypotheses on the basis of comparisons of theoretical and experimental results. Sensitivity analyses of results can be used to guide experimentation. In all this, there are the new opportunities for mathematical experimentation afforded by the electronic computer. Starting with these descriptive processes, which may be deterministic or sto- chastic, we turn to control processes. How do these systems react to the admini- stration of drugs and to exposure to radiation? Here, of course, one principal goal is that of improving treatment of cancer based upon radiation therapy and chemotherapy. A very intriguing mathematical problem is that of determining the degree of control which may be affected in situations in which we know very little about basic mechanisms. Here adaptive control theory may play a role. Problems of medical diagnosis and treatment are those of recognizing when a complex system is out of kilter and of restoring it to equilibrium condition as soon as possible. Interesting stochastic and control processes are encountered here, in addition to "bookkeeping" difficulties of such magnitude as to represent genuinely new problems. Turning from the study of large-scale processes to small-scale phenomena, con- sider the reproduction of cells. How do we explain the incredible accuracy with which this process takes place, generation after generation? Current ideas use intricate algebraic and topological analysis, with research proceeding at such a pace that it is difficult to predict what final form theories will assume. For those with a penchant for cryptoanalysis, here is a cipher well worth breaking. We encounter a number of basic problems of communication and control in the study vii
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