Mathematics Awareness Week 1994

Mathematics and Medicine

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### Theme Essay

 The mathematical sciences are making many contributions to medicine, from population biology to physiology. For example, mathematicians are building realistic three-dimensional models of the heart. Practically this means imitating the muscle fibers of the heart by hundreds of closed curves along which the forces of elasticity act to contract the heart and move blood via the equations of fluid dynamics. The heart may also be "seen" through reconstructive mathematical techniques that build medical images through computerized axial tomography (CAT), magnetic resonance imaging (MRI), or positron emission tomography (PET). Using these techniques, thousands of separate measurements are mathematically combined to create a single image that can show tumors and other abnormalities in, for example, the brain, lungs, and kidneys, as well as the heart. Mathematics researchers, along with biologists and physicians, suspect that many heart attacks can be predicted. Or, put another way, that the heart might be acting as if it were a deterministically chaotic system - an example of nonlinear dynamics theory. Mathematical work is underway to show whether high risk patients can be more easily identified from electrocardiograms if suspect patterns are more readily detected and recognized. Another way that mathematics increases understanding the heart is through fluid flow dynamics, although solutions to the equations are only approachable by computer approximation. The motion of the heart walls is among the unknowns that must be solved for in blood flow. The analysis of complex hierarchical systems is another important area for investigation in modern medicine. Mathematical modeling has been extremely helpful in neuroscience where network theory, information synthesis, and random graphs are fundamental tools. Such systems concepts have also proved invaluable to immunology, where extremely large numbers of cells and their interactions must be observed and analyzed. In this area mathematical applications involving ordinary differential equations and branching processes are used, as is control theory. Understanding the dynamics of HIV infection and its effects on the immune system is another research focus in mathematics. In addition to quantitative analysis and probability estimates of infection, epidemiological models are needed to develop vaccine strategies. Mathematical simulation and modeling are also key to visualizing and understanding recombinant DNA technology. Examples may be seen on the 1994 Mathematics Awareness Week theme poster. Strands of DNA are examined through techniques of topology and differential geometry. Databases of human genome information are so extensive and complex that mathematical approaches such as combinatorics, pattern recognition, and sequence comparisons are required. In the pharmaceutical industry, computational models of molecular structures are being developed. New drugs are being designed as new mathematical algorithms are created. Health statistics have long been collected and analyzed for a variety of purposes, including cost control, public policy research, demography, and disease trends correlated with other variables such as environmental factors. Other mathematical contributions to medicine range from the very concrete -- designing materials that go into medical products -- to the very abstract -- information management. Mathematical models assist in the design and processing of advanced materials, including: shape-memory metals, high-strength ceramics, polymeric systems, and nonlinear optical materials. And, statistical analysis synthesizes data from clinical trials in more useful and meaningful forms. Mathematical modeling reduces analytical problems to quantitative relations and equations suitable for attack by algorithmic methods. Mathematical algorithms, then, express quantitative relations and equations in a format suitable for computational solution. Back to Math Awareness Week 1994
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