A New Homework Problem and Some Bidomain History

Russ Hobbie and I describe the bidomain model of cardiac tissue in Section 7.9 of Intermediate Physics for Medicine and Biology. The bidomain equations govern the intracellular and extracellular electrical potentials in the heart. As described in my unpublished paper “The Cardiac Bidomain Model in Twelve Publications,” the model was developed in the late 1970s by several researchers, including Less Tung and David Geselowitz. However, the history of the biodmain model is complex. Two other researchers working in Russia, A. Muler and V. Markin, also developed the same idea at about the same time. In addition, another group of researchers including Robert Eisenberg, Richard Mathias, and Arthur Peskoff at Rush University Medical Center, were using the bidomain model to describe the lens of the eye. 

What do the lens and the heart have in common? Both tissues are made up of cells that are coupled together by gap junctions: small channels that connect the inside of one cell to the inside of its neighboring cells. To get from one cell to another, current does not have to cross the cell membrane and enter the extracellular space. Instead, it can just pass through the junctions, always remaining inside the cell (such a tissue is called a syncytium). This key feature underlies the bidomain concept, which treats the tissue as a two-phase medium, intracellular and extracelluler, coupled by the membrane.

“Electrical Properties  of Spherical Syncytia,” by Eisenberg, Barcilon, and Mathias.

Today I introduce a new homework problem for readers of IPMB, motivated by two papers from the group at Rush:

R. S. Eisenberg, V. Barcilon, and R. T. Mathias, “Electrical properties of spherical syncytia,” Biophysical Journal, Volume. 25, Pages 151–180, 1979.

A. Peskoff, “Electric potential in three-dimensional electrically syncytial tissues,” Bulletin of Mathematical Biology, Volume 41, Pages 163–181, 1979.

Both articles imagine that a microelectrode injects a current pulse into the intracellular space of a spherical lens. The general problem is too complicated for an undergraduate homework exercise, so I’ve simplified it considerably. First, I assume the current pulse has been on long enough that the system has reached steady state. Second, I assume the microelectrode injects current into the center of a lens of radius a (at the origin of a spherical coordinate system). Third, I ignore anisotropy, and assume the lens is an isotropic tissue. Fourth, and finally, the boundary conditions at the lens surface r = a complicate the analysis, so I avoid them by considering a very large lens (essentially, an unbounded tissue). With these assumptions, the homework problem is still difficult, but not prohibitively so.

Section 7.9

Problem 31 ½. Consider a tissue represented using the bidomain model, with intracellular conductivity σ_i and extracellular conductivity σ_e. The cell membrane has a conductance per unit area G_m, and the surface area of cell membrane divided by tissue volume (the surface-to-volume ratio) is β. A current I₀ is injected into the intracellular space at the origin (r = 0).

(a) Write the bidomain equations for the intracellular and extracellular potentials V_i and V_e (Eq. 7.32) in spherical coordinates (use Appendix L). Assume that the electrical potential varies only with the radial distance r, and not with angular variables θ and φ. 

(b) The solutions to these equations are 

Substitute these solutions into the equations from part (a) and find an expression for the tissue length constant λ in terms of σ_i, σ_e, G_m, and β. 

(c) For r λ, find approximations for V_i and V_e. Determine how each behaves as r goes to zero. 

(d) For r >> λ, find approximations for V_i and V_e. They should both be the same. Find an expression for the “effective conductivity” of the tissue based on these expressions.

(e) Derive an expression for the transmembrane potential, V_m = V_i - V_e.

My advice: stop reading and solve the problem… 

OK, for those who absolutely don’t have time to solve this yourself, let me outline the solution.

(a) The bidomain equations become 

The delta function source term only appears in the intracellular equation because the current is injected into the intracellular space. Notice that the sign of the term corresponding to the membrane current (the one containing β) is different in the two equations. Current coming out of the intracellular space is current going into the extracellular space.

(b) When you substitute the solutions into the differential equations, you should find that they work if

(c) When r λ, V_i is proportional to 1/r, while V_e is constant. Therefore, V_i goes to infinity as r goes to zero, but V_e remains finite.

(d) When r >> λ, both V_i and V_e are equal. They each fall off as 1/r and the conductivity factor that appears is σ_i + σ_e. This is the parallel combination of the intracellular and extracellular resistances, and can be thought of as the “effective” conductivity of the tissue. 

(e) The first terms in the expressions for V_i and V_e, which contain the factor 1/r, cancel out when you take the difference to get V_m. Therefore, V_m falls off as e^-r/λ/r. If you are more than a few length constants from the stimulating electrode, V_m is much smaller than either V_i  or V_e. If you want to determine V_m by measuring V_i and V_e and taking their difference, you must record them very accurately, as you will be subtracting two big numbers to get a much smaller one, which is always susceptible to noise.

If you look at the derivations by Eisenberg et al. or Peskoff, you’ll find more complicated expressions than those given above, because they include time dependence, they don’t require the electrode to be at the center of the lens, and they consider a lens of finite radius rather than an unbounded tissue. I hope that this homework problem at least gives you a feel for their results.

Both Eisenberg et al.’s and Perkoff’s papers were published in 1979, just one year after Les Tung’s PhD dissertation and Walter Miller and David Geselowitz’s Circulation Research paper, two founding documents of the bidomain model. Moreover, Eisenberg, Mathias, and their collaborators had been publishing related studies for several years before the 1979 papers. They truly have a claim as critical contributors to the development of the bidomain model. To survey this body of work, see Eisenberg’s 2023 review in Modeling and Artificial Intelligence in Ophthalmology. 

Robert Eisenberg is an emeritus faculty member at Rush University Medical Center in Chicago. He has a long history of using mathematical modeling and computer simulation to address issues in bioelectricity. His graduate study was at the University College London, where he studied under Andrew Huxley (of the Hodgkin and Huxley model) among others. Richard Mathias worked with Eisenberg at Rush during the era when the bidomain model was developed, and later was on the faculty at Stony Brook University. He not only looked at the lens electrical properties, but more importantly examined its fluid flow behavior using a model analogous to the electrical bidomain model. Because the lens is transparent, it cannot contain any blood vessels, so fluid flow in the intracellular and extracellular spaces is crucial for keeping the tissue perfused. Victor Barcilon (1939–2020) was an applied mathematician who worked for many years at the University of Chicago. Arthur Peskoff is with the UCLA School of Medicine.