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
