Current Injection into a Two-Dimensional Anisotropic Bidomain

Twenty years ago this month, Nestor Sepulveda, John Wikswo, and I published a calculation of the transmembrane potential induced when a point electrode passes current into cardiac tissue, as might happen when pacing the heart (“Current Injection into a Two-Dimensional Anisotropic Bidomain,” Biophysical Journal, Volume 55, Pages 987–999, 1989). When we wrote the paper, Sepulveda was a Research Assistant Professor and I had just gotten my PhD and was starting a one-year post doc in Wikswo’s laboratory at Vanderbilt University. We used a mathematical model of the electrical properties of cardiac tissue called the bidomain model, which was relatively new at that time. In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe this result.

The bidomain has been used to understand the response of cardiac tissue to stimulation... [Sepulveda et al.’s] simulation explains a remarkable experimental observation. Although the speed of the wave front is greater along the fibers than perpendicular to them, if the stimulation is well above threshold, the wave front originates farther from the cathode in the direction perpendicular to the fibers—the direction in which the speed of propagation is slower. The simulations show that this is due to the anisotropy in conductivity. This is called the “dog-bone” shape of the virtual cathode. It can rotate with depth in the myocardium because the myocardial fibers change orientation. The difference in anisotropy accentuates the effect of a region of hyperpolarization surrounding the depolarization region produced by a cathode electrode. Strong point stimulation can also produce reentry waves that can interfere with the desired pacing effect.

The calculation was possible because Sepulveda had developed a finite element computer program that could solve the bidomain equations: a system of two coupled partial differential equations. Meanwhile, Wikswo was performing experiments on dog hearts with collaborators at the Vanderbilt Hospital, and had observed that the wave fronts originate from a spot farther from the electrode in the direction perpendicular to the fibers than in the direction parallel to them (“Virtual Cathode Effects during Stimulation of Cardiac-Muscle 2-Dimensional In Vivo Experiments,” Circulation Research, Volume 68, Pages 513–530, 1991). As soon as Sepulveda performed the calculation, they realized that it would explain Wikswo’s data.

I remember being so surprised hyperpolarization would be produced just a millimeter or two away from a cathode that I quietly slipped into my office and developed a Fourier method to check Sepulveda’s finite element calculation. I got the same result: regions of hyperpolarization adjacent to the cathode. After our publication, six years passed before the prediction of hyperpolarized regions was verified experimentally, by three groups simultaneously including researchers in Wikswo’s lab (“Virtual Electrodes in Cardiac Tissue: A Common Mechanism for Anodal and Cathodal Stimulation,” Biophysical Journal, Volume 69, Pages 2195–2210, 1995). During these years—when I was working at the National Institutes of Health—Josh Saypol, an undergraduate summer student, and I showed that the hyperpolarization could have an important effect: it could lead to reentry, a type of cardiac arrhythmia (“A Mechanism for Anisotropic Reentry in Electrically Active Tissue,” Journal of Cardiovascular Electrophysiology, Volume 3, Pages 558–566, 1992). For a simple, visual, and non-mathematical introduction to these ideas, see my paper in the Online Journal of Cardiology.

Our original publication in 1989 has now been cited in the literature 200 times. It remains one of my most cited papers (although, to be honest, I had less to do with the research than Sepulveda and Wikswo did), and is one of my favorites.