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Mathematics In the Heart
Dr. James Paul Keener, University of Utah

During the course of a typical human life, the heart beats about 2 billion times. Yet, in spite of the obvious reliability of the heart, nearly half of the deaths in the United States are due to the failure of the heart to beat properly. Mathematics is at the "heart" of trying to understand how the heart beats and why it fails to beat properly.

The heart is comprised of millions of individual cells which are capable of contracting. The contraction of these cells is coordinated by an electrical signal that spreads from one cell to another, much in the way that a forest fire spreads from one tree to the next. However, because the cells are packed together in a three dimensional region, there is no way to measure what cells on the inside of the heart are doing. Some people stick electrodes into the heart to measure what cells are doing, but this is not a very practical method to measure from a large number of cells or for the person at a ski resort who notices that her high altitude exercise is accompanied by some skipped heart beats.

Invariably any attempt to understand the heartbeat involves mathematics. To follow what individual cells do requires tracking how ions move and how voltages change. To determine how cells communicate to each other requires a description of currents between cells. To follow how a signal propagates requires a description of the geometry of the medium, the orientation of fibers, the thickness of walls, etc. And all of these descriptions use the language of mathematics.

Understanding this description requires the tools of mathematics, whereby equations are solved and conclusions are drawn.

In the final stage, mathematics is used to visualize the answer, and instructions are given to a computer so that it displays each pixel on a computer screen in a way that represents some feature of the numbers of the solution. The result is that we can use mathematics to see things that cannot be seen in seen in any other way.

The 1999 Math Awareness theme poster depicts the electrical activity of a normal heart if it were stimulated at a single point. Color is used to represent the time of arrival of the electrical signal as it propagates through the heart. The rectangular domain represents a large slab of the ventricular wall, and all domains use geometry and fiber orientation that are taken from anatomical data.

The study of cardiology has a rich mathematical history reaching back to the early part of this century. In the 1920's, Dutch engineer van der Pol designed an electrical circuit that oscillated spontaneously. At the time in the engineering community, there was little interest in oscillations, so van der Pol proposed his circuit as a model for the cardiac pacemaker. The equation describing his system is today described in most books on differential equations and is known as the van der Pol equation. Even though he knew little of the true physiology, the model gives important insight into how pacemakers work.

The breakthrough discovery was made by Hodgkin and Huxley in the early 1950's. Even though they worked on nerve axons, their work provided the paradigm for essentially all of the subsequent mathematical work on electrophysiology. In their work, they accomplished two fundamental things. First, they showed how to write mathematical expressions for how the voltage potential of a nerve cell changes in time. Second, they showed that with spatial coupling their equations supported wave behavior. Perhaps the most impressive aspect of their work is that it involved lots of computation, much of it done on a 50's era desk top calculator. Their work was rewarded with a Nobel Prize in 1963.

Work on cardiac cells was hampered by their small size, and it was not until the 1970's that the necessary technology was developed. After that, however, the use of mathematics in cardiology burgeoned. Today, investigators use chaos theory to study cardiac arrhythmias, fractals to study the geometry of the specialized conduction system, finite element methods to describe the fiber structure of the heart wall, differential equations to describe the electrical activity of the tissue, large scale computer simulations to see if it all works, and computer graphics to see what it looks like.

The results have been impressive, and we are now able to see things that cannot be seen any other way. But a word of caution is in order. It is easy with modern computer graphics to make stunningly beautiful pictures and movies. But simply because it is strikingly beautiful does not make it correct. It is easier to produce science fiction than science. The difference is in the mathematics that lies, unseen, in the background.