Mathematics of pacemakers

Summary: Artificial pacemakers send a signal to the heart to keep it pumping and mathematicians develop models to determine when and how often to do so.

While a pacemaker is often thought of as a regulator for the heart, a variety of natural pacemakers are responsible for regulating numerous bodily functions including circadian rhythms and menstruation. The actions of natural pacemakers can be modeled as coupled oscillators, where, for example, the behavior of the natural pacemaker influences the function of the heart and vice versa. Square waves or sine waves are often useful in understanding the theory of coupled oscillators, which dates back to 1665 when Christiaan Huygens noticed synchronization in pendulum clocks.

Scientists and mathematicians have shown that chaotic oscillation or amplitude death can also occur in coupled scenarios. A change in the rhythm or in the way they are coupled can result in a change in function, such as in irregular menstrual periods or menopause. Dynamical systems model the interactions between coupled oscillators and allow for theoretical predictions. Using these models, mathematicians, biologists, and medical professionals have made significant advances in understanding natural pacemakers and in designing effective artificial pacemakers. Some of the related mathematical theory is taught to undergraduate mathematics students.

Heart Rhythms and Pacemakers

The sinoatrial node (SA node) is thought to act as the heart’s natural pacemaker via electrical impulses. The typical rate for a resting heart is 60 to 70 beats per minute. The pacemaker cells keep the heart pumping at a steady rate, but medical problems can lead to chaotic behavior and cardiac arrest.

Defibrillation may reset the rhythm in some cases but an artificial cardiac pacemaker may be required if the rhythm remains chaotic. Wavelet transforms have been used to effectively model cardiac signals but implementation is difficult because of high power consumption. Australian anesthesiologist Mark Lidwell and physicist Edgar Booth are believed to have designed the first artificial pacemaker in 1928.

American physiologist Albert Hyman also developed an early pacemaker. Many designers of artificial pacemakers have assumed that regular impulses from a pacemaker should be used to stabilize the heartbeat. However, a periodic signal may lead to chaos in some mathematical models, so scientists are developing pacemakers that send impulses based on chaos control theory.

Body Clock and Jet Lag

Jet lag is thought to to result from a desynchronization of the suprachiasmatic nucleus (SCN) pacemaker cells in the hypothalamus of mammals. Experimental studies suggest that the SCN may synchronize within one week. Scientists and mathematicians have mathematically modeled the system as a network with connections between the cells, which are called nodes in the language of graph theory.

For example, mathematicians Channa Navaratna and Menaka Navaratna have adapted a model of neuroscientist Peter Achermann and bioinformaticist Hanspeter Kunz. The hypothalamus is thought to have 16,000 pacemaker cells, so they analyzed computer data from a model with this many pacemaker cells and found that the number of long-distance connections in the network determined the synchronicity time. They examined the types of network connections that are needed between the nodes in order to make the model synchronize in a week, and they designed a model that consistently synchronized in close to seven days.

Scientists and mathematicians have also studied many other issues related to pacemakers, such as interference and power issues. There is controversy and conflicting evidence on whether devices such as cell phones or iPods affect pacemakers. Many medical professionals presume an association until clearer evidence to the contrary is found and recommend keeping the devices at least a few inches away from a pacemaker to err on the side of caution. Scientists have developed what some call “origami batteries” made of carbon nanotubes and cellulose that may power the next generation of pacemakers. The batteries can be cut into many shapes.

Bibliography

Barold, S. Serge, et al. Cardiac Pacemakers Step by Step: An Illustrated Guide. Hoboken, NJ: Wiley, 2003.

Glantz, Stanton. Mathematics for Biomedical Applications. Berkeley: University of California Press, 1979.

Strogatz, Steven. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Boulder, CO: Westview Press, 2001.