Using math to solve cardiac mysteries

by Jackie Klopp

Rhythms are an intrinsic part of nature. Our own bodies are awash in complex and diverse patterns of beats. The thumping heart is perhaps the most obvious source of body rhythms, although many others pervade our physiology, including our respiratory and nervous systems.

Understanding the dynamics of these rhythms is of crucial importance in medicine because abnormal patterns, or arrhythmias, may cause a serious body malfunction such as cardiac arrest.

McGill's Department of Physiology is home to a number of researchers studying body rhythms and one interdisciplinary team is making important headway towards a theoretical understanding of cardiac rhythms. At the heart of these efforts are Leon Glass, Michael Guevara and Alvin Shrier, who bring a creative mixture of skills to the task.

Glass, with a PhD in chemistry and a deep interest in applying mathematics to biology, came to McGill in 1975. With Michael Mackey, he taught courses on dynamics in biological systems that captured the imagination of Michael Guevara, then a graduate student.

Guevara was interested in understanding cardiac rhythms. He had previously trained as a paramedic and worked in emergency rooms. "A lot of the explanations given to me in the hospital about cardiac rhythms seemed like nonsense," recalls Guevara.

The electrocardiogram measurements of heartbeats he witnessed in hospitals always struck Guevara as having a mathematical flavour about them, a notion Glass and Mackey's courses reinforced. Guevara decided he wanted to apply the mathematical approach he learned in class to real heart cells.

Enter Alvin Shrier, currently chair of the Department of Physiology, who came to McGill with just the critical knowledge needed to prepare heart cells. He welcomed Guevara into his laboratories and a fruitful collaboration began.

"Groups such as ours are somewhat unique," says Glass. "Dr. Shrier has continued the policy of former chairs of the department and deans of medicine in encouraging interdisciplinary studies."

Their fortuitous combination of expertise, experience and interest gave birth to experiments that broke new ground in science. The experiments were the first to dramatically reveal a previously unknown mathematical order in biology.

In a nutshell, the experiments involved preparing chicken heart cells in the laboratory. These cells, under proper conditions, pulse without outside help. This is possible because of specialized cells that generate an electrical wave, exciting other heart cells as it travels through them. It is this same kind of mechanism that governs the considerably more complex rhythms in human hearts.

Shrier, Glass and Guevara were able to observe the pulse patterns of heart cells in their laboratory. Further, by applying their own electrical pulses to the cells they could alter the electrical wave and trace how the rhythms changed accordingly.

To Glass, there are two particularly striking aspects of these experiments. The first is the discovery of erratic behaviour known as "chaos." The second is that the complex rhythms observed in the laboratory followed a relatively simple mathematical model.

This model, drawn from the field of mathematics called nonlinear dynamics, involves a set of equations with properties that appear remarkably well-suited to describing actual heart behaviour.

One property of these equations is that a change in parameters may make a whole new set of behaviours come into being. This enables these equations to describe the wide range of behaviours in actual hearts, from simple rhythms to the erratic "chaotic" ones. These equations can thus capture transitions in the heart from normal rhythms to the irregularities of arrhythmias.

"The equations are also relatively insensitive to small details," says Glass. "What this means in practice is that, although each person's heart may be described by slightly different equations, there are limited and classifiable sets of heart dynamics for all people." This brings considerable order to the diverse heartbeat symphonies taking place inside each one of us.

The leap from these observations and models to an understanding of heart rhythms in actual people is, however, not a simple one. It requires building a bridge between cardiologists with in-depth knowledge of heart behaviour and biological theoreticians like Guevara, now an associate professor, Shrier and Glass.

In 1994, the Guggenheim Memorial Foundation recognized the importance of such a project by awarding Glass a highly competitive fellowship. This allowed him to spend a year at the Beth Israel hospital in Boston, where he joined a cardiac electrophysiology group.

This group applies electric stimulation to hearts in human patients. The resulting heartbeats are recorded as peaks on an electrocardiogram. Glass used such heartbeat "maps" to guide his mathematical exploration of the underlying mechanisms at work in hearts with arrhythmias.

"There are a couple of examples where I think that we understand reasonably well what some of the dynamics are in real people's hearts," says Glass.

Unfortunately, while of great theoretical interest, the well-explained arrhythmias "are not the most dangerous." For the last three years, Glass and his colleagues have been exploring dangerous arrhythmias called re-entrant tachycardias. In these arrhythmias the heart is excited by an electrical wave that moves in a circular pathway.

"The frequency of this rhythm is much more rapid than a normal heartbeat," explains Glass, "And this may lead to an inadequate pumping of blood to the heart, brain and body." Understanding why and how this happens is important for finding ways to treat people with these arrhythmias.

Ultimately the hope is that an understanding of the mathematical order behind the complex rhythms in the body will assist doctors in providing better diagnosis and treatment of diseases where rhythms go wrong.

Although direct applications have not yet entered the practice of medicine, nonlinear dynamics theory, spurred on by the unique McGill experiments, is inspiring many efforts. For example, there are, according to Glass, proposals for analyzing people's normal heart rhythms to assess the risk of sudden cardiac death.

Another potential application involves designing better devices that can be implanted into the body to monitor heart activity. Such devices would apply appropriate electrical jolts when the heart's beats go awry, knocking it back into a normal rhythm.

The work of Glass, Guevara and Schrier is no doubt bringing us a step closer to keeping all our body rhythms healthily in tune.

Jackie Klopp, a physics graduate currently pursuing a PhD in political science, is a science writing intern for the Reporter. The internship program is funded by the Natural Sciences and Engineering Research Council.