The Bidomain Model of Cardiac Tissue: Predictions and Experimental Verification

“The Bidomain Model of Cardiac Tissue:
Predictions and Experimental Verification”

In the early 1990s, I was asked to write a chapter for a book titled Neural Engineering. My chapter had nothing to do with nerves, but instead was about cardiac tissue analyzed with the bidomain model. (You can learn more about the bidomain model in Chapter 7 of Intermediate Physics for Medicine and Biology.)

“The Bidomain Model of Cardiac Tissue: Predictions and Experimental Verification” was submitted to the editors in January, 1993. Alas, the book was never published. However, I still have a copy of the chapter, and you can download it here. Now-after nearly thirty years-it’s obsolete, but provides a glimpse into the pressing issues of that time.

I was a impudent young buck back in those days. Three times in the chapter I recast the arguments of other scientists (my competitors) as syllogisms. Then, I asserted that their premise was false, so their conclusion was invalid (I’m sure this endeared me to them). All three syllogisms dealt with whether or not cardiac tissue could be treated as a continuous tissue, as opposed to a discrete collection of cells.

The Spach Experiment

The first example had to do with the claim by Madison Spach that the rate of rise of the cardiac action potential, and time constant of the action potential foot, varied with direction.

I then argued that their first premise is incorrect. In one-dimensional cable theory, the time course of the action potential doesn’t depend on axial resistance, as Spach claimed. But in a three-dimensional slab of tissue superfused by a bath, the time course of the action potential depends on the direction of propagation. Therefore, I contended, their conclusion didn’t hold; their experiment did not prove that cardiac tissue isn’t continuous. To this day the issue is unresolved.


A second example considered the question of defibrillation. When a large shock is applied to the heart, can its response be predicted using a continuous model, or are discrete effects essential for describing the behavior?

I argued that the problem is again with the first premise, which is true for tissue having “equal anisotropy ratios” (the same ratio of conductivity parallel and perpendicular to the fibers, in both the intracellular and extracellular spaces), but is not true for “unequal anisotropy ratios.” (Homework Problem 50 in Chapter 7 of IPMB examines unequal anisotropy ratios in more detail). If the premise is false, the conclusion is not proven. This issue is not definitively resolved even today, although the sophisticated simulations of realistically shaped hearts with their curving fiber geometry, performed by Natalia Trayanova and others, suggest that I was right.

Reentry Induction

The final example deals with the induction of reentry by successive stimulation through a point electrode. As usual, I condensed the existing dogma to a syllogism.

Once again, that pesky first premise is the problem. In tissue with equal anisotropy ratios you can remove anisotropy by a coordinate transformation, so reentry is impossible. However, if the tissue has unequal anisotropy ratios the symmetry is broken, and reentry is possible. Therefore, you can’t conclude that the observed induction of reentry by successive stimulation through a point electrode implies the tissue is discrete.

I always liked this book chapter, in part because of the syllogisms, in part because of its emphasis on predictions and experiments, but mainly because it provides a devastating counterargument to claims that cardiac tissue acts discretely. Although it was never published, I did send preprints around to some of my friends, and the chapter took on a life of its own. This unpublished manuscript has been cited 13 times!

I’ll end with the closing paragraph of the chapter.

Originally published at

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.

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