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.