A Mathematical Model of Make and Break Electrical Stimulation of Cardiac Tissue by a Unipolar Anode or Cathode

“A Mathematical Model of Make and Break
Electrical Stimulation of Cardiac Tissue
by a Unipolar Anode or Cathode.”

Suppose I was going to die tomorrow and I could choose only one paper to cite on my tombstone. Which would I pick? I’d select

B. J. Roth, 1995, “A Mathematical Model of Make and Break Electrical Stimulation of Cardiac Tissue by a Unipolar Anode or Cathode,” IEEE Transactions on Biomedical Engineering, Volume 42, Pages 1174–1184.

Below is the introduction, with references removed. I like the way it starts with a question.

What is the mechanism by which an electrical current, passed through a unipolar electrode, excites cardiac tissue? This simple question appears to have a straightforward answer: The stimulus current depolarizes the tissue under the electrode until the transmembrane potential reaches threshold, triggering an action potential wave front. Excitation of cardiac tissue, however, is more complicated than one might initially expect. Stimulation with a cathode might be explained by depolarization of the tissue under the electrode, but how does one explain stimulation with an anode? Even more intriguing, excitation is elicited by turning a stimulus off (break) as well as by turning it on (make). Why should turning off the stimulus excite the tissue? Indeed, four distinct mechanisms are responsible for stimulation of cardiac tissue—cathode make, anode make, cathode break, and anode break—and only cathode-make stimulation can be explained by depolarization under the electrode. To understand the other three mechanisms, we make detailed calculations of the transmembrane potential distribution induced by current through a unipolar electrode. We have three goals: to explain the mechanisms of excitation qualitatively; to predict stimulation thresholds quantitatively; and to determine how the threshold varies with electrode size and with stimulus pulse duration and frequency.

Our calculations are based on the bidomain model of cardiac tissue, which is useful for predicting the transmembrane potential induced by an extracellularly applied electric field. The bidomain model is a two- or three-dimensional cable model that accounts for the resistance of both the intracellular and the extracellular spaces. Many of the most interesting and nonintuitive predictions of the bidomain model occur when the ratios of the electrical conductivities parallel to and perpendicular to the myocardial fibers in the intracellular and extracellular spaces differ. For instance, current that is passed through a point extracellular electrode into a two-dimensional bidomain with unequal anisotropy ratios induces adjacent areas of depolarization and hyperpolarization. Such a region of hyperpolarization near a cathode is called a virtual anode; a region of depolarization near an anode is called a virtual cathode. The existence of virtual anodes and cathodes is predicted by the bidomain model and is essential for three of the four mechanisms of stimulation. Recently, virtual anodes and cathodes were observed experimentally in cardiac tissue.

My use of the royal “we” seemed reasonable when I wrote the paper, but now it grates on my ear. According to Google Scholar, in the twenty years since I published this article it has been cited 169 times. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I turned the prediction of break excitation of cardiac tissue into a homework problem (Chapter 7, Problem 48).

I did this research while working at the National Institutes of Health in Bethesda, Maryland. Sometimes on a slow afternoon I would sneak away from my desk and browse the stacks of the NIH library. One day I found a fascinating paper by Egbart Dekker, who measured the threshold for each of the four mechanisms of excitation (E. Dekker, 1970, “Direct Current Make and Break Thresholds for Pacemaker Electrodes on the Canine Ventricle,” Circulation Research, Volume 27, Pages 811–823.) Once I read Dekker’s article, I knew I could simulate this behavior using the then-new bidomain model and perhaps gain insight about mechanisms. At the time I was not well versed in mathematical models of the cardiac membrane kinetics with all their different ion currents, so I just used the Hodgkin-Huxley model of a nerve axon. A paper describing that study was unpublishable because who in their right mind would use a squid nerve axon model to represent a cardiac action potential? After the manuscript using the Hodgkin-Huxley model was rejected, I set to work learning about cardiac ion channel dynamics. I chose the Beeler-Reuter model, and the paper using the BR model (no, I did not choose that model because of my initials) was ultimately accepted for publication.

I sent a draft of my article to my PhD advisor, John Wikswo. He and his post doc Marc Lin immediately verified the model predictions experimentally (see their lovely paper: J. P. Wikswo, S. F. Lin, and R. A. Abbas, 1995, “Virtual Electrodes in Cardiac Tissue: A Common Mechanism for Anodal and Cathodal Stimulation,” Biophysical Journal, Volume 69, Pages 2195–2210). I remember the day Wikswo emailed me asking something like “what would you say if I told you the cathode make, cathode break, and anode make mechanisms all behave exactly as you predicted, but your anode break mechanism is totally wrong?” I began to panic, wondering how in the world I messed up, and sent Wikswo a frantic email asking for more details. His response was along the lines of “I asked ‘what would you say?’ I didn’t claim your prediction was actually wrong.” Ha, ha, ha; all four mechanisms were verified. Their paper was published the same month as mine and now has 300 citations. Your can read a layman’s account of this work in an article published in the Vanderbilt Register.

The figures in my original article were all black-and-white contour plots of action potential wave fronts propagating through the tissue. Wikswo had beautiful color figures in his paper. So, a few years later I “colorized” the figures, including them in a review article (B. J. Roth, S.-F. Lin and J. P. Wikswo, Jr., 1998, “Unipolar Stimulation of Cardiac Tissue,” Journal of Electrocardiology, Volume 31, Supplement, Pages 6–12). This always reminds me of how some of the classic old black-and-white movies have been colorized to look modern.

One reason I like publishing in the IEEE TBME is that they provide a short biographical sketch of the author. Below is my bio from 20 years ago. My how time flies.

Bradley J. Roth was raised in Morrison, Illinois. He received the B.S. degree from the University of Kansas in 1982, where he was a Summerfield Scholar and received the Stranathan Award from the Department of Physics and Astronomy. He received the Ph.D. degree in physics from Vanderbilt University.

From 1988-1995, he worked in the Biomedical Engineering and Instrumentation Program at the National Institutes of Health. One of his primary accomplishments while at NIH was the study of the bidomain model and its application to solving fundamental problems solving the interaction of applied electric fields with cardiac muscle. Using the results of numerical simulations, he has formulated mechanisms for stimulation, defibrillation, and the initiation of arrhythmias in the heart In September, 1995, he became the Robert T. Lagemann Assistant Professor of Living State Physics at Vanderbilt University.