Mathematics Awareness Week 1994

Mathematics and Medicine

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Mathematical Models of the Human Immune System
(One in a series of six articles on Mathematics and Medicine being distributed by the Joint Policy Board for Mathematics in celebration of Mathematics Awareness Week 1994.)

At Los Alamos National Laboratory in New Mexico, Alan Perelson and his colleagues are trying to explain why the virus that causes AIDS is so deadly. The researchers do not study AIDS patients, nor do they follow the effects of the virus (human immunodeficiency virus, or HIV) in animals such as mice. They use mathematics.

Perelson is part of a burgeoning branch of science known as mathematical biology. Researchers working in this field replace clinical and laboratory experiments with mathematical models, producing fast, safe routes to new theories. These computer- based models can imitate natural events -- such as the slow development of viral diseases -- instantaneously, in marked contrast to the lengthy times needed to conduct experimental trials. Ultimately, these mathematical models may suggest ways to prevent or treat deadly diseases such as AIDS.

Perelson's models have little to do with traditional models of the spread of diseases in large populations -- the business of epidemiology. "My work is very different. It looks at how the virus spreads within an individual," Perelson says. Among other things, he models aspects of the human immune system, which is responsible for fighting foreign organisms that invade the body.

In one case he has investigated how HIV is able to deplete certain white blood cells of the immune system while appearing to infect few of those cells. The cells, called T-helper cells, are among the many immune cells that act in concert to thwart invaders. Perelson and his colleagues found that by making two assumptions in their models, they could approximate the dynamics of HIV and T-cells found in some patients. The assumptions are that HIV can affect precursors of T-cells -- that is, cells that manufacture T-cells -- and that HIV can mutate over time, becoming more infectious.

The mathematics of the models are similar to those used in population dynamics, and include ordinary differential equations. Nevertheless, the models are coming up short of an accurate picture of the attack of HIV. One reason is that HIV can hide in specific locations of the body. "To explain this requires another generation of models," Perelson notes. He and colleagues are now developing these improved models.

Working with Thomas Kepler of North Carolina State University in Raleigh, Perelson has used models to investigate mutations in the human immune system itself during an infection. When a foreign organism like HIV (called an antigen) invades the body, white blood cells (known as B cells) produce antibodies, which bind to the antigen, rendering it harmless. This binding is at first weak, but gets stronger over time. The increase in binding is due in part to the ability of B cells to mutate, that is, to create alternative versions, with better-binding versions surviving to attack the invading virus.

To model this process, the researchers employed the statistical mathematics known as optimal control theory. This resulted in a model in which the process of mutation turns on and off a number of times at specific points, leading to the survival of proper numbers of better-binding antibodies and to the kind of overall binding improvement observed in the immune system.

After mathematical biologists develop such models, they offer them to experimentalists who can test the results using living systems. "We give them to immunologists, who search on to see if we are right," says Perelson. The success of modeling depends on the efforts of many scientists, and many different specialists work in the area: mathematicians, physicists, computer scientists, biologists, and others. Although the roots of the kind of modeling he does go back over two decades, "in the last three or four years there's been an enormous interest in this type of activity, including large numbers of students," Perelson notes.

Back to Math Awareness Week 1994

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