(Press-News.org) Providence, RI---Did you know that heart attacks can give you
mathematics? That statement appears on the web site of James Keener,
who works in the mathematics of cardiology. This area has many
problems that are ripe for unified attack by mathematicians,
clinicians, and biomedical engineers. In an article to appear in the
April 2011 issue of the Notices of the American Mathematical Society,
John W. Cain, a mathematician at Virginia Commonwealth University,
presents a survey of six ongoing Challenge Problems in mathematical
cardiology. Cain's article emphasizes cardiac electrophysiology,
because some of the most exciting research problems in mathematical
cardiology involve electrical wave propagation in heart tissue.
At some point in our lives, many of us will undergo an
electrocardiogram (ECG), a recording of electrical activity in the
heart. To understand where these tiny electrical currents originate,
we must zoom in to the molecular level. Bodily fluids, such as blood,
contain positively charged ions. When these ions traverse cell
membranes, they cause electrical currents, which in turn elicit
changes in the voltage V across the membrane. If a sufficiently
strong stimulus current is applied to a sufficiently well-rested cell,
then the cell experiences an "action potential": V suddenly spikes and
remains elevated for a prolonged interval. These action potentials
govern heartbeat patterns and are therefore critical to understanding
and treating disorders like arrhythmia (abnormal heart rhythms) and in
particular tachycardia (faster than normal heart rhythms).
Taking the Nobel Prize-winning work of Hodgkin and Huxley as a
starting point, researchers have created mathematical models of the
cardiac action potential by viewing the cardiac cell membrane as an
electrical circuit. A major challenge that Cain identifies is
striking a balance between feasiblity and complexity: Minimize
complications in the model, so that it is amenable to mathematical
analysis, but add sufficient detail, so that the model reproduces as
much clinically relevant data as possible. The equations that govern
the model---nonlinear partial differential equations---cannot be
solved explicitly, and solutions must be obtained through
approximation by numerical methods. Adding further complications are
the intricate geometry of the heart, with its four chambers and
connections to veins and arteries, and the fact that different types
of cardiac tissue have different conduction properties.
Cain goes on to discuss various cardiac phenomena and the mathematics
that can be used to describe them. One example is heart rhythm: The
regular, coordinated contraction of the heart muscle that pumps blood
through the body. Improving the understanding and treatment of
irregularities in that rhythm is critical in the fight against heart
disease.
A healthy heart does not beat in a perfectly regular pattern; in fact,
such a pattern would be a sign of potentially serious pathologies.
The body's autonomic nervous system uses neurotransmitters to speed up
or slow down the heart, and tiny fluctuations in those substances
induce variability in the intervals between consecutive beats. The RR
interval is the interval between consecutive heartbeats measured in an
ECG. Attempts to quantify heart rate variability (HRV) usually
involve analyzing time series of RR intervals.
Unfortunately, some ways of analyzing RR time series give the same
results for patients with healthy hearts and for those with fatal
cardiac abnormalities. One challenge for mathematicians and
statisticians is to devise quantitative methods for distinguishing
between the RR time series of people with healthy hearts and the RR
time series of those with cardiac pathologies. Cain asks, Can some
pathologies be diagnosed solely by analysis of RR time series and, if
so, which ones? To spot subtle pathologies, methods are needed for
quantifying the "regularity" of a cardiac rhythm. Also, given the
existing array of diagnostic tests that clinicians have at their
disposal, there could be advantages in the use of "automated"
mathematical/statistical methods.
INFORMATION:
Cain's article, "Taking Math to Heart: Mathematical Challenges in
Cardiac Electrophysiology", will be freely available on the Notices web
site (http://www.ams.org/notices) on March 14, 2011.
The article marks Mathematics Awareness Month (MAM,
http://www.mathaware.org), which is held each year in April. The 2011
MAM theme is complex systems, of which the human heart is a prime
example.
Specific questions about the article are best directed to the author
(his email address is in the article). General inquiries may be
directed to:
Mike Breen and Annette Emerson
AMS Public Awareness Office
Email: paoffice@ams.org
Telephone: 401-455-4000
Founded in 1888 to further mathematical research and scholarship, the
more than 30,000-member American Mathematical Society fulfills its
mission through programs and services that promote mathematical
research and its uses, strengthen mathematical education, and foster
awareness and appreciation of mathematics and its connections to other
disciplines and to everyday life.
American Mathematical Society
201 Charles Street
Providence, RI 02904
401-455-4000
Taking mathematics to heart
2011-03-15
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