Bidomain modeling of electrical and mechanical properties of cardiac tissue

Abstract

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

I. INTRODUCTION

This review discusses the bidomain model, a mathematical description of cardiac tissue. Most of the review covers the electrical bidomain model used to study pacing and defibrillation of the heart. For a book-length analysis of this topic, consult the recently published second edition of Cardiac Bioelectric Therapy.¹ In particular, one chapter in that book complements this review: it contains a table listing many bidomain predictions and their experimental confirmation, includes many original figures from earlier publications, and cites additional references.² Near the end, the review covers the mechanical bidomain model, which describes mechanotransduction and the resulting growth and remodeling of cardiac tissue.

The review has several aims: to (1) introduce the bidomain model to younger investigators who are bringing new technologies from outside biophysics into cardiac physiology, (2) examine the interaction of theory and experiment in biological physics, (3) emphasize intuitive understanding by focusing on simple models and qualitative explanations of mechanisms, and (4) highlight unresolved controversies and open questions. The overall goal is to enable technologists entering the field to more effectively contribute to some of the pressing scientific questions facing physiologists.

II. A BIDOMAIN MODEL FOR THE EXTRACELLULAR POTENTIAL AND MAGNETIC FIELD OF CARDIAC TISSUE

The idea of representing a syncytial tissue like cardiac muscle as “interpenetrating domains” was first conceptualized by Otto Schmitt³ in 1969, but the modern bidomain model was not developed until the late 1970s. Several areas of research led to the bidomain formulation. Investigators sought to understand the electrocardiogram,^4,5 injury currents in the heart,⁶ and the lens of the eye.^7,8 Russian scientists analyzed the bidomain model theoretically.^9–11

Another path leading to the bidomain model was biomagnetism. Wikswo was interested in the magnetic field produced by biological tissue, and specifically the relationship between the electrocardiogram and the magnetocardiogram¹² and their relative information content.¹³ In order to better understand the source of biomagnetic signals, he measured the magnetic field of a nerve,¹⁴ and with his graduate student Roth recorded the magnetic field of a single nerve axon.¹⁵ Wikswo then turned his attention back to the heart, and he and Roth studied a cylindrical strand of cardiac tissue, such as a papillary muscle.

To calculate the magnetic field, you need to know where the current flows. In an axon, the current is either intracellular or in the surrounding saline bath. The distribution of current can be determined from the intracellular and bath potentials, and equations governing these potentials were derived by Plonsey and Clark.¹⁶ Their analysis amounted to a Jackson E&M problem:¹⁷ Given the voltage across the axon membrane, solve Laplace's equation for the intracellular and bath potentials in terms of Fourier transforms and Bessel functions.

In a papillary muscle, the distribution of current is more complicated (Fig. 1). You still have current in the intracellular space and the surrounding bath, but it also flows in the interstitial space (the extracellular space between the cells). Moreover, cardiac tissue is anisotropic; the electrical conductivity is greater along the myocardial fibers than across them. Clark and his colleagues modified the nerve axon equations to include anisotropy,¹⁸ but Wikswo and Roth did not know how to derive an equation describing the interstitial potential. Without it, they could not calculate the magnetic field and therefore could not interpret their data.

FIG. 1.

The magnetic field produced by a cylindrical strand of cardiac tissue perfused by a saline bath. An action potential wave front propagates from left to right. The intracellular current is green, the interstitial and bath currents are blue, and the magnetic field is the purple loop.

The key to resolving this problem was in the 1978 Ph.D. dissertation by Tung,⁶ who derived the bidomain model by representing cardiac tissue as two spaces (or domains)—intracellular and interstitial—that can exchange current through the cell membrane¹⁹ (Fig. 2). The intracellular space of each cell is coupled to neighboring cells through gap junctions,²⁰ resulting in a tissue that acts as a syncytium. The bidomain model automatically predicts the interstitial current. The equations derived by Roth and Wikswo²¹ for the electrical potential produced by a strand of cardiac muscle were similar to Plonsey and Clark's calculation for an axon,¹⁶ with modifications based on Tung's dissertation.⁶

FIG. 2.

The two-dimensional bidomain model, illustrated as grids of resistors. The green grid represents the electrical properties of the intracellular space, the blue grid the interstitial space, and the red grid the cell membrane.

The bidomain model provided an interpretation of the magnetic field recordings from a papillary muscle,²² helped explain other biomagnetic phenomena,^23,24 and allowed Wikswo to achieve his goal of understanding the information content of the electrocardiogram and magnetocardiogram.^25,26 However, most cardiac electrophysiologists do not care about magnetic fields. They do, however, care about the interstitial and bath potentials, which they can measure easily. Several articles from this era^27–30 use the bidomain model to determine these potentials from a known transmembrane potential (the voltage across the cell membrane). Many of these studies were performed at Duke University, where Plonsey then worked.^31–33

A critical issue in these calculations is how to treat the interface between cardiac tissue and a saline bath. The boundary conditions are subtle, and Krassowska³⁴ and Roth³⁵ each analyzed them. The bidomain model predicts a roughly uniform potential deep in the tissue, with a rapid drop at the tissue-bath interface, and then a continuing falloff with distance in the bath²⁹ (Fig. 3). Knisley and his co-workers performed an experiment that confirmed the model predictions.³⁶

FIG. 3.

The interstitial and bath potentials produced by an action potential propagating to the left in a cylindrical strand of cardiac tissue. The potential as a function of distance along the strand (z) is shown at several radial distances from the strand center; the thicker trace is from the strand surface.

Another example of predicting the extracellular potential from the transmembrane potential is the calculation of the electrocardiogram. Anisotropy, which is accounted for naturally by the bidomain model, plays a role in analyzing extracellular potentials.^5,37,38 One of the first uses of the bidomain model was Geselowitz and Miller's calculation of the ECG,^4,39 and this continues to be an important application of the model to this day.^40–42 Interestingly, the bidomain formulation is not essential for predicting wave front propagation through the heart; the simpler monodomain model suffices.⁴⁰

By the late 1980s, several groups were using the bidomain model to predict the electrical potential produced by propagating wave fronts in cardiac tissue. The next step was to use the model to analyze electrical stimulation.

III. CURRENT INJECTION INTO A TWO-DIMENSIONAL ANISOTROPIC BIDOMAIN

In the 1980s, Sepulveda went to Vanderbilt University to work for Wikswo as a research scientist. Sepulveda had written his own finite element code and was using it to predict the distribution of current during defibrillation.⁴³ He then extended his code to solve the equations of the bidomain model. One of his first bidomain calculations was of the magnetic field produced by a circular wave front propagating outward in a sheet of cardiac tissue,⁴⁴ a situation Plonsey and his colleague Barr had examined previously.⁴⁵

Next, Sepulveda, Roth, and Wikswo turned their attention to a numerical analysis of electrical stimulation. Instead of assuming a transmembrane potential distribution and then calculating the intracellular and interstitial potentials, Sepulveda calculated all the potentials, including the transmembrane potential, produced when current is injected through a point (unipolar) electrode into a passive (constant membrane resistance), two-dimensional sheet of cardiac tissue, similar to what happens when pacing the heart.⁴⁶ Cardiac tissue was depolarized (positive transmembrane potential) under a unipolar cathode, as expected. A surprise was the dog-bone shape of the depolarized region (the virtual cathode), extending further transverse to the fibers than parallel to them⁴⁶ (Fig. 4). Wikswo and his co-workers verified this prediction during experiments to determine the site of initial excitation following a stimulus⁴⁷ (Fig. 5).

FIG. 4.

The transmembrane potential produced in a passive, two-dimensional sheet of cardiac tissue during stimulation by a unipolar cathode, calculated using the bidomain model and unequal anisotropy ratios. The fibers are horizontal. The regions of hyperpolarization (negative transmembrane potential) to the left and right of the cathode (black dot) are the virtual anodes. Contours are drawn every ten millivolts.

FIG. 5.

The site of initial excitation (the virtual cathode) during point cathodal stimulation of a dog heart. The fibers are horizontal, the black dot shows the location of the extracellular stimulating electrode, and the four shades correspond to different current strengths.

In addition to the dog bone-shaped depolarized region, the tissue was hyperpolarized (negative transmembrane potential) adjacent to the cathode in the direction parallel to the fibers⁴⁶ (Fig. 4). Such hyperpolarized regions are known as virtual anodes (hyperpolarization with no anodal electrode nearby). The presence of virtual anodes is an unanticipated and nonintuitive prediction of the bidomain model. They are not some artifact of the numerical simulation; they can be derived using an approximate analytical method based on perturbation theory.⁴⁸

Virtual anodes appeared only when the tissue had unequal anisotropy ratios. Anisotropy is characterized by the ratio of conductivity parallel to the fibers to that perpendicular to the fibers, and unequal anisotropy ratios mean that this ratio is not the same in the intracellular space as in the interstitial space.^38,49–51 Why is unequal anisotropy ratios so important? In many materials, you can remove anisotropy by rescaling. Instead of measuring distance in millimeters, define dimensionless coordinates with different scaling factors parallel and perpendicular to the fibers. If you choose the scaling factors wisely, you can render the tissue isotropic in these new coordinates. For a bidomain, however, in general you cannot remove the anisotropy in both the intracellular and interstitial spaces at the same time. An argument based on the principle of simultaneous diagonalization of two quadratic forms shows that you can make the intracellular space isotropic by rescaling, but the rescaled interstitial space will also be isotropic only if the anisotropy ratios are equal¹⁹ (Fig. 6). Unequal anisotropy ratios mean there exists a preferred direction in space that cannot be removed by merely stretching the distance scale in one direction.

FIG. 6.

The method of simultaneous diagonalization of two quadratic forms applied to the bidomain model. The ellipses represent the conductivity tensors, with the semi-major axes aligned with the directions of highest conductivity. Step 1: Select coordinates aligned with the semi-major and semi-minor axes of the interstitial conductivity; step 2: Rescale the coordinates so that the interstitial space is isotropic; step 3: Rotate the rescaled coordinates so they align with the semi-major and semi-minor axes of the intracellular conductivity. Usually the semi-major axes are parallel in the intracellular and interstitial spaces, but that is not necessary for this process. You cannot make both the intracellular and interstitial spaces isotropic unless the tissue has equal anisotropy ratios (the two ellipses have the same eccentricity).

To test the bidomain prediction of virtual anodes adjacent to a cathode, you need to measure the transmembrane potential. In the 1990s, a new optical method started to be used to map transmembrane potential in cardiac tissue. The development of this technique is described in Optical Mapping of Cardiac Excitation and Arrhythmias,⁵² edited by Rosenbaum and Jalife. Cardiac tissue is perfused with a fluorescent dye that is absorbed into the cell membrane. Then, the tissue is excited with short wavelength light and observed using light fluoresced at a longer wavelength (Fig. 7). The key is that the fluorescence depends on the transmembrane potential, allowing electrical measurements to be made using optical techniques. This method lets you measure not only the rapid depolarization of the action potential but also the slower repolarization. In addition, it enables you to record transmembrane potential during strong electrical shocks. Optical mapping quickly became the preferred method for testing bidomain predictions.

FIG. 7.

Optical mapping. The tissue is perfused with a fluorescent dye that is absorbed into the cell membrane, so the fluorescence depends on transmembrane potential. The excitation light (green) has a short wavelength, and the fluorescent light (red) has a long wavelength. A camera images the fluorescent light. This technique allows the use of optical methods to make electrical measurements of transmembrane potential.

Sepulveda, Roth, and Wikswo⁴⁶ made their prediction of the distribution of transmembrane potential during unipolar stimulation in 1989. Through the early 1990s, several studies examined the implications of the virtual anodes,^19,53,54 but there was not yet any experimental data confirming that they exist. Finally, in 1995 three groups used optical mapping to measure the transmembrane potential produced during unipolar stimulation: Knisley,⁵⁵ Tung and his student Neunlist,⁵⁶ and Wikswo and his team consisting of Lin and Abbas.⁵⁷ All three experiments confirmed the bidomain prediction. This success spurred interest in using the bidomain model to study pacing and defibrillation. But first, the limitation of a passive membrane had to be overcome.

IV. ACTION POTENTIAL PROPAGATION IN A THICK STRAND OF CARDIAC MUSCLE

In early bidomain calculations, researchers had either assumed a transmembrane potential distribution and then calculated the intracellular and interstitial potentials,^21,44,45 or calculated the transmembrane potential using a passive membrane.⁴⁶ To reach its full potential, the bidomain model needed numerical methods so one could calculate how action potentials propagate through active bidomain tissue.

The bidomain model can be recast as a pair of coupled partial differential equations: A reaction-diffusion equation for the transmembrane potential, and a boundary value problem for the interstitial potential. The equation governing the transmembrane potential is nonlinear (active) because of the opening and closing of membrane ion channels. It can be solved using a finite difference technique and Euler's method to step forward in time. The equation governing the interstitial potential is linear, but because it depends on the transmembrane potential it has to be solved at every time step, which increases the computation time. Overrelaxation—an iterative method—is one way to solve the interstitial potential. One advantage of an iterative technique is that the result from the previous time step can be used as the initial guess for the new time step, drastically reducing the number of iterations needed.

Roth applied this technique to calculate the transmembrane action potential propagating along a cylindrical strand of cardiac tissue immersed in a perfusing bath⁵⁸ (Fig. 8). He found that the wave front was not flat across the strand cross section, but instead was curved, a result consistent with experiment.⁵⁹

FIG. 8.

The transmembrane potential of an action potential wave front propagating from left to right along a cylindrical strand of cardiac tissue, calculated using the active bidomain model. The wave front is curved, with the action potential at the tissue-bath surface propagating ahead of the action potential at the center of the strand.

Roth's calculation was not the first attempt to solve the active bidomain model using a numerical method. In 1984, Barr and Plonsey had developed a preliminary algorithm to calculate action potential propagation in a sheet of cardiac tissue.⁶⁰ Simultaneous with Roth's work, Henriquez and Plonsey were examining propagation in a perfused strand of cardiac tissue.^61,62 For the next several years, Henriquez continued to improve bidomain computational methods with his collaborators and students at Duke.^63–66 His 1993 article published in Critical Reviews of Biomedical Engineering remains the definitive summary of the bidomain model.⁶⁷

Over time, other researchers developed even more sophisticated ways to solve the active bidomain equations numerically. For example, Keener and Bogar derived an operator-splitting scheme.⁶⁸ Because solving the two bidomain equations is so time-consuming, this problem continues to attract attention from applied mathematicians and computer scientists interested in novel numerical methods and high-performance computing applications.^69–76

V. A MECHANISM FOR ANISOTROPIC REENTRY IN ELECTRICALLY ACTIVE TISSUE

Once the computational tools needed to simulate an active bidomain existed, many problems in cardiac electrophysiology could be addressed. One of the first to be studied was reentry, when a wave front propagates around a loop and returns to a region it has already passed through. In a chapter in the influential textbook Cardiac Electrophysiology: From Cell to Bedside, Winfree predicted that stimulating cardiac tissue with unequal anisotropy ratios could produce an arrhythmia.⁷⁷ Winfree was a theoretical biologist who studied cardiac electrophysiology.^78,79 His book When Time Breaks Down⁸⁰ provides much insight into spiral waves, reentry, and arrhythmias in the heart.

Saypol and Roth tested Winfree's prediction using a numerical calculation.^53,81 In their simulation, they used the Hodgkin and Huxley model of a nerve axon to represent the active cardiac membrane. A nerve action potential is much shorter than one in cardiac tissue, so the simulation was not too realistic. Nevertheless, the model captured many features that are now known to be crucial for understanding reentry in the heart.

Their plan was to stimulate a sheet of tissue twice through a unipolar electrode. The weak first stimulus (S1) excited an initial action potential that propagated outward. When the refractory period of the first action potential was almost finished, they stimulated again (S2) with a stronger current. Recall that during unipolar stimulation (Fig. 4) virtual anodes cause hyperpolarization in direction parallel to the fibers (x), but not in the direction perpendicular to them (y).⁴⁶ This hyperpolarization is key for shortening the refractory period and initiating reentry. They wrote⁵³

“When the tissue is stimulated, the refractory period is shortened in the area of hyperpolarization along the x axis… If the second stimulus is timed just right, it can take the tissue along the x direction out of the refractory period, while along the y direction the tissue remains unexcitable. Thus, the action potential elicited by the large depolarization directly below the electrode can propagate only in the x direction.”

Their mechanism of shortening the refractory period (deexcitation) was different from that proposed by Winfree, but the result was the same. Two successive stimuli caused unidirectional propagation, which is a prerequisite for initiating reentry. Saypol and Roth used symmetry to reduce the calculation to only one quadrant of the x-y plane, but when you consider the entire plane you find four reentrant loops (Fig. 9),⁸¹ a situation Wikswo later named quatrefoil reentry.⁸²

FIG. 9.

Quatrefoil reentry in a two-dimensional sheet of cardiac tissue, calculated using the bidomain model. (a) Immediately after the S2 stimulus, two wave fronts propagate outward from the cathode (black dot) through the virtual anodes. (b) The wave fronts continue propagating parallel to the fibers (horizontal), and in addition, they propagate outward and back toward each other. (c) The wave fronts collide, forming a closed elliptical wave front; in addition, wave fronts propagate perpendicular to the fibers back toward the cathode. (d) The inwardly propagating wave fronts collide and then begin propagating outward parallel to the fibers, reentering tissue through which they had propagated previously.

Testing this mechanism experimentally required having a way to measure the transmembrane potential during the S2 stimulus. Optical mapping could accomplish this. Wikswo and Lin used their optical mapping system at Vanderbilt to confirm the mechanism for quatrefoil reentry.⁸²

Spiral waves were a hot topic in the mid-1990s, and a group led by Jalife at SUNY Health Science Center was using optical mapping to observe them.^83–86 A spiral wave is a type of reentry in which the wave front rotates around a central core, constantly chasing its refractory tail. A crucial question was: how does an electric shock trigger a spiral wave? One answer was proposed by Ideker, whose laboratory at Duke recorded cardiac arrhythmias in dogs using arrays of extracellular electrodes. He tested Winfree's critical point hypothesis: In a S1–S2 stimulus protocol, a spiral wave is initiated at a critical point where a contour of critical S2 stimulus strength intersects a contour of critical S1 refractoriness (Fig. 10).⁸⁰ Ideker and his team confirmed this hypothesis in experiments.^87,88 In fact, they could move the core of the spiral wave around wherever they wanted by merely adjusting the timing and strength of the S2 stimulus.

FIG. 10.

The critical point hypothesis. An S1 wave front propagates from left to right. A critical contour of S1 refractoriness (red) separates recovered and excitable tissue on the left from refractory tissue on the right. Then an S2 stimulus is applied through a line electrode at the bottom edge of the tissue. A critical contour of S2 stimulus strength (blue) separates tissue at the bottom that is stimulated with a strength above resting threshold from tissue at the top where the stimulus is below threshold. The point where these two contours intersect is the critical point, around which a spiral wave rotates (purple).

In When Time Breaks Down, Winfree described how the critical point hypothesis could be illustrated using a simple cellular automaton.⁸⁰ The tissue is divided into cells (often represented as hexagons) that can exist in one of three states: excited, refractory, or resting. The dynamics of reentry can be reproduced by a few simple rules: an excited cell turns into a refractory cell at the next time step, a refractory cell turns into a resting cell, and a resting cell remains at rest unless one of its nearest neighbors was excited in the previous time step, in which case that cell becomes excited at the next time step. In this model, a spiral wave will rotate about a critical point where all three states (excited, refractory, and resting) meet at one point. Two critical points result in two spiral waves propagating in opposite directions, called figure-of-eight reentry. The beauty of cellular automata is that even a child can follow their rules and understand cardiac reentry.

Cellular automata can be extended to include virtual anodes. Figure 11 shows a refractory tissue that is stimulated through a unipolar cathode.⁸⁹ The tissue under and near the cathode is depolarized (moved to the excited state) but the refractory tissue at the virtual anodes is forced back to rest (deexcited). Four phase singularities are present in Fig. 11, resulting in quatrefoil reentry. The crucial feature of this example is that before the S2 stimulus the tissue was uniformly refractory; there was no refractory gradient. Thus, the bidomain/virtual anode mechanism for reentry induction is fundamentally different than the critical point mechanism; in the bidomain mechanism, a refractory gradient is not necessary.⁹⁰

FIG. 11.

A cellular automaton resulting in quatrefoil reentry. Each cell (hexagon) can be in one of three states: excited (yellow), refractory (red), or resting (purple). When a stimulus is applied to a uniformly refractory tissue through a unipolar cathode (bold center hexagon), it will be excited under and around the cathode and deexcited at the virtual anodes adjacent to the cathode in the direction parallel to the fibers (horizontal). Four critical points are produced where yellow, red, and purple cells meet, resulting in quatrefoil reentry.

Roth collaborated with Trayanova, from Tulane University, and her student Lindblom to simulate the S1–S2 experiment using the bidomain model with a planar S1 wave front (the pinwheel experiment). They found that both the critical point hypothesis and the hypothesis of deexcitation at a virtual anode were necessary to fully describe the wave front dynamics⁹¹ (Fig. 12). The bidomain/virtual anode mechanism dominated during the first few milliseconds after excitation, but the critical point mechanism became important later as reentry developed. Their article in the Journal of Cardiovascular Electrophysiology was accompanied by an editorial from Winfree.⁹²

FIG. 12.

The results of the pinwheel experiment, as simulated using the bidomain model. The S1 wave front is planar and propagates either parallel to (longitudinal, L) or perpendicular to (transverse, T) the fibers. The S2 stimulus is through a unipolar electrode, either a cathode (C) or an anode (A). Hyperpolarized (deexcited) tissue is red. The initial excitation launches two wave fronts with four critical points. These wave fronts then develop into either figure-of-eight reentry with two surviving critical points or quatrefoil reentry with four.

Quatrefoil reentry and the pinwheel experiment focused on unipolar stimulation, but Efimov showed that these same concepts play a role during defibrillation. In the late 1990s, Efimov established a laboratory at the Cleveland Clinic, where he used optical mapping to study the response of the heart to strong shocks.^93,94 He realized that the crucial question was not why do defibrillation shocks succeed, but why do they sometimes fail? His answer was that although the shock may have stopped the fibrillating wave fronts, it fails if it restarts fibrillation by carving out excitable pathways with hyperpolarization at the virtual anodes, thereby reinitiating reentry.⁹⁵ Efimov's team showed that the response to a defibrillation shock was independent of the direction of the pre-shock wave front, supporting the bidomain/virtual-electrode mechanism over the critical point hypothesis.⁹⁶

Gray and his student Banville observed behavior similar to what Efimov saw.⁹⁷ Gray and Ideker concluded that defibrillation was caused by many mechanisms.⁹⁸ In 2000, Efimov, Gray, and Roth collaborated on a review article focusing on the roles that virtual electrodes and deexcitation play during defibrillation.⁹⁹

VI. THE RESPONSE OF A SPHERICAL HEART TO A UNIFORM ELECTRIC FIELD

Section V examined the role that the bidomain model played in understanding the induction of cardiac arrhythmias. That story began in 1992 and continued to the end of the century. This section returns to the early 1990s and reexamines the same period, but with a focus on defibrillation.

In late 1991, Trayanova (then at Duke), her student Malden, and Roth began collaborating to calculate the transmembrane potential produced in the heart by a uniform electric field, such as during defibrillation. They modeled the heart as a spherical shell of bidomain tissue surrounding a blood cavity and immersed it in a saline bath¹⁰⁰ (Fig. 13). The tissue was anisotropic, with the fiber direction following lines of constant longitude (using an analogy with a globe). The membrane was passive, so they calculated only the initial shock response and not the subsequent propagation of wave fronts.

FIG. 13.

A spherical heart. The cardiac tissue forms a spherical shell, surrounded by a bath and surrounding a cavity filled with blood. The red circles show the direction of the curving fibers. The black arrows indicate the direction of a uniform electric field.

When the tissue had equal anisotropy ratios, the heart was depolarized over a thin layer on the inner surface and hyperpolarized over a thin layer on the outer surface (or vice versa), with negligible polarization deep inside the heart wall. The thickness of the surface layer is set by the electrical length constant, which is typically a millimeter or less. The restriction of polarization to the surface made understanding defibrillation difficult, because the shock needs to affect a large fraction of the heart (a critical mass) in order to extinguish all fibrillating wave fronts.^43,101 For unequal anisotropy ratios, however, the polarization extended throughout the heart wall¹⁰⁰ (Fig. 14). The curvature of the fibers, when combined with unequal anisotropy ratios, allowed the electric field to influence the entire heart.

FIG. 14.

The transmembrane potential produced when a spherical heart is shocked by an electric field. The dashed curve is for equal anisotropy ratios; throughout much of the heart wall, the polarization is zero. The solid curve is for unequal anisotropy ratios; the polarization is significant deep within the wall.

Otani—a former plasma physicist then working at Cornell—showed that fiber curvature was not essential for deep polarization. He found that straight fibers with a nonuniform electric field had the same effect, again only for tissue having unequal anisotropy ratios.¹⁰² Knisley, along with Trayanova and her student Aguel, began the effort to confirm experimentally the relationship between fiber geometry and the transmembrane potential,¹⁰³ a process Trayanova has continued in collaboration with Efimov and others.¹⁰⁴ Later, Langrill Beaudoin and Roth used a perturbation technique to gain insight into how a specific fiber geometry was related to the resulting transmembrane potential distribution.¹⁰⁵ Tung and his co-workers at Johns Hopkins University also provided insight using an activating function approach.¹⁰⁶ They were able to grow cultured, two-dimensional sheets of tissue with any desired fiber geometry and found excellent agreement between theory and experiment.^107,108

The bidomain model is based on the assumption that cardiac tissue can be treated as a continuum. An alternative hypothesis is that the individual cells behave discretely. Plonsey and Barr¹⁰⁹ and Krassowska, Pilkington and Ideker¹¹⁰ each published a study in which the gap junctions coupling adjacent cells have a high resistance, leading to a saw-tooth distribution of transmembrane potential (Fig. 15). In an electric field, current enters each cell at one end (hyperpolarizing it) and then leaves each cell at the other end (depolarizing it), thereby avoiding the high-resistance gap junction between cells. In this case, the magnitude of the membrane polarization is proportional to the electric field strength. Experimentalists using optical mapping have searched for the saw-tooth effect, but have not yet found it.^111,112 Optical techniques have limited spatial resolution, primarily caused by blurring resulting from averaging over depth, and it is not clear if they can resolve such polarization changes at the cellular level.

FIG. 15.

The saw-tooth effect, as calculated using the bidomain model. Current is injected into a one-dimensional fiber, creating an exponential decay of the transmembrane potential from the stimulus site. Because the fiber is periodically interrupted by high-resistance gap junctions, a saw-tooth-like oscillation is superimposed on the transmembrane potential, allowing the fiber to be polarized even many length constants from the stimulus.

Krassowska helped clarify if cardiac tissue should be modeled as continuous or discrete. Working with mathematician Neu, she developed a model of cardiac tissue that postulated a discrete representation of coupled cells. Then, by using a homogenization procedure, she was able to derive the bidomain equations as the spatial average of this discrete model. The saw-tooth effect represented the first-order perturbation.¹¹³ This elegant analysis not only clarified the issue of continuous vs discrete tissue, but also helped explain the complex relationship between macroscopic and microscopic conductivities, and provided a better understanding of why cardiac tissue has unequal anisotropy ratios. Hooks and his co-workers¹¹⁴ and Stinstra and his colleagues¹¹⁵ extended this work to include a more irregular and realistic microscopic representation of tissue and explored the implications for the macroscopic bidomain conductivities, which can influence many important phenomena such as the ST segment shift caused by subendocardial ischemia.¹¹⁶

In addition to the continuous bidomain model and the discrete saw-tooth model, Krassowska—working at Duke with her student Kumar—showed that intermediate heterogeneities at the level of small bundles of cells could also cause polarization of cardiac tissue.¹¹⁷ In addition, the cardiac vascular system creates heterogeneities that could be important during electrical excitation.^118–120 Such heterogeneities might be responsible for the prompt response of the heart to a shock.¹²¹ Krassowska and Roth teamed up to write a review article about how the transmembrane potential is related to the electric field.¹²² It appeared in a 1998 focus issue of the journal Chaos, edited by Winfree, that examined fibrillation in normal ventricular myocardium. At about the same time, Ideker and his colleagues wrote a complementary review about the mechanisms by which electrical stimulation alters the transmembrane potential.¹²³

Another factor that affects how the heart responds to an electric field is the boundary conditions at the heart surface. Experimentalists often use a Langendorff apparatus, in which a heart is kept alive by perfusing it with oxygenated saline through its vascular system. Experiments can be performed in two ways: (1) with the heart suspended in air so the epicardial, or outer, surface is insulated, or (2) with the heart immersed in saline. Several groups analyzed these two boundary conditions, including Trayanova,¹²⁴ Entcheva working with a group that included Efimov,¹²⁵ and Roth and his student Latimer.¹²⁶ A bath adjacent to the epicardial surface changes the transmembrane potential dramatically.

After analyzing the spherical heart, Trayanova extended the model to a realistically shaped heart, solving the bidomain equations using the finite element method. She also added an active membrane so she could study defibrillation shock responses.¹²⁷ Trayanova has mentored a remarkable group of students, postdocs, and young researchers, including Eason,¹²⁸ Rodriguez,¹²⁹ Bishop,¹¹⁸ Plank,⁴¹ Vigmond,⁷⁰ and others. She and her team have helped push bidomain calculations toward translational applications in medicine, such as low-voltage defibrillation¹³⁰ and the placement of implantable cardioverter-defibrillators in congenital heart defect patients.¹³¹ They are moving the field toward personalized medicine.¹³²

VII. A MATHEMATICAL MODEL OF MAKE AND BREAK ELECTRICAL STIMULATION OF CARDIAC TISSUE BY A UNIPOLAR ANODE OR CATHODE

Sections V and VI examined arrhythmia induction and defibrillation. This section again spirals back to the 1990s and examines another part of the story: pacing.

In 1970 Dekker, a Dutch medical doctor, observed four mechanisms for stimulating the heart: cathode make, anode make, cathode break, and anode break.¹³³ In 1995, Roth showed that the bidomain model predicts all four^134,135 (Fig. 16).

FIG. 16.

The four mechanisms for stimulating the heart: cathode make, anode make, cathode break, and anode break. The black arrows indicate the direction of propagation. The dashed arrows indicate where depolarization (D) diffused into adjacent hyperpolarized (H) and therefore excitable tissue, causing break excitation.

Cathode make is the traditional way to stimulate: depolarization under the cathode raises the transmembrane potential, exciting an action potential. It has the lowest current threshold of the four mechanisms. Anode make occurs when you stimulate with an anode, which hyperpolarizes the tissue under the electrode but depolarizes it at virtual cathodes along the fiber direction. This depolarization can excite an action potential, albeit with a higher current threshold.

The more interesting mechanisms involve break stimulation, which means excitation does not occur until the stimulus pulse turns off. Break excitation typically happens when you stimulate refractory tissue. During cathode break stimulation, the tissue under the cathode is depolarized but excitation does not occur because the tissue is refractory. The virtual anode, however, is hyperpolarized, which forces the transmembrane potential there back toward resting potential, shortening the refractory period and carving out an excitable pathway (as discussed previously; see Sec. V and Fig. 9). When the stimulus pulse ends, the depolarization under the cathode diffuses into the excitable tissue at the anode, exciting a wave front.

Anode break stimulation works by a similar mechanism: hyperpolarization occurs under the anode and depolarization at the virtual cathode. After the end of the stimulus pulse, depolarization diffuses into the hyperpolarized and therefore excitable tissue and excites it. Anode break excitation in cardiac tissue differs from anode break excitation observed in the squid nerve axon.¹³⁶ In a nerve, break excitation is caused by removal of inactivation of the sodium channel and occurs even in a space-clamped axon. In the heart, break excitation is a spatial effect and requires adjacent regions of depolarization and hyperpolarization.

Not only did the bidomain model predict all four mechanisms of excitation but also the order of current threshold was the same as Dekker observed in his experiment.¹³³ Wikswo, Lin, and Abbas used optical mapping to confirm the bidomain mechanisms.⁵⁷ Soon Trayanova and her student Skouibine,¹³⁷ as well as Efimov and his team,⁹⁵ were studying the role of break excitation during defibrillation.

Dekker also measured the strength-interval curve.¹³³ Using an S1–S2 stimulus protocol, he plotted the S2 stimulus threshold vs the S1–S2 interval (the time between S1 and S2). Usually the strength-interval curve has negative slope because the S2 threshold decreases as the interval increases and the tissue recovers from refractoriness. However, sometimes the strength-interval curve contains a dip: it reaches a minimum and then has a section of positive slope, when the threshold paradoxically rises as the interval increases. The dip appears during anodal stimulation, not cathodal. Also, it is associated with break excitation, not make. Bidomain simulations predict the anodal dip in the strength-interval curve and suggest a mechanism¹³⁵ (Fig. 17). Break excitation requires adjacent regions of depolarization and hyperpolarization. The limiting factor is often the presence of depolarization, and the amount of depolarization present decreases as the S1 action potential repolarizes, making break excitation more difficult. Sidorov and his colleagues, working at Vanderbilt with Wikswo, used optical mapping (a technique not available to Dekker) to confirm the predicted mechanism.¹³⁸

FIG. 17.

The strength-interval curve for anodal (purple) and cathodal (red) stimulation, calculated using the bidomain model. The anodal curve contains a dip at about 290 ms and has a positive slope between 290 and 300 ms.

Several investigators have extended the analysis of the strength-interval curve. The bidomain model predicts the change in shape of the strength-interval curve that occurs within a few weeks after a pacemaker is implanted.^139,140 The dip disappears when post-repolarization refractoriness is present,¹²⁹ because if the tissue remains refractory after the S1 action potential has completely repolarized (which often happens in ischemic tissue) then shortening the S1–S2 interval will not result in additional S1 depolarization to aid break excitation.¹⁴¹ The dip in the strength-interval curve could possibly be due to ion channel kinetics,¹⁴² or caused by intracellular calcium,¹⁴³ but these mechanisms are unlikely. Excitation begins near the edge of the virtual anode rather than at its center where hyperpolarization is greatest, so the most probable hypothesis is diffusive interaction of depolarization and hyperpolarization, as predicted by the bidomain model.¹⁴⁴ The dip in the anodal strength-interval curve is predicted even for a minimal model of the ion channel kinetics, suggesting that the mechanism is a generic bidomain behavior.¹⁴⁵ The three-dimensional structure of the heart, including transmural fiber rotation and unequal orthotropic anisotropy, influence make and break stimulation and the strength-interval curve.^146,147

Another curiosity observed for decades in experiments but never explained is the no response phenomenon.¹⁴⁸ Sometimes you can excite an action potential with a strong anodal stimulus, but an even stronger stimulus fails to cause excitation. The bidomain model explains this behavior^149,150 (Fig. 18). During anode break excitation, if the hyperpolarization at the virtual anode is stronger and more extensive, the tissue will be more excitable, the break wave front will propagate more quickly, and it will reach the edge of the virtual anode sooner. If the surrounding tissue is still refractory, the wave front will then die away, resulting in no response. A weaker stimulus would have resulted in slower propagation through the virtual anode, giving the surrounding tissue more time to recover from refractoriness. The no response phenomenon is related to the upper limit of vulnerability,¹⁵¹ and may therefore play a role in defibrillation.¹⁵²

FIG. 18.

The no response phenomenon. During stimulation with a unipolar anode, a weak stimulus is subthreshold and a stronger stimulus causes anode break excitation. However, for some intervals (272–278 ms), an even stronger stimulus results in no response: a break wave front begins, but then decays when it reaches the edge of the hyperpolarized tissue.

VIII. FREQUENCY LOCKING OF MEANDERING SPIRAL WAVES IN CARDIAC TISSUE

Researchers have long known that the core of a spiral wave is not always stationary. Sometimes it meanders: it traces out a flower-like pattern resembling what you might create using the game Spirograph. The analogy is a good one: Spirograph patterns arise from the coupling of rotational motion with two periods, and meandering arises when the period of the spiral rotation does not match the period of meander. The meander path, however, depends on how you locate the spiral wave core.¹⁵³ Winfree showed that you could create all sorts of meander patterns, and even chaos, by varying the parameters in a numerical simulation of a spiral wave.¹⁵⁴

Roth^155,156 applied the bidomain model to study meander in a two-dimensional sheet of cardiac tissue, using the two-variable FitzHugh-Nagumo model for the membrane behavior, as Winfree had.¹⁵⁴ When the tissue had equal anisotropy ratios, he reproduced Winfree's results. However, for unequal anisotropy ratios, the spiral wave core would sometimes have a slow drift superimposed on the meander flower. In other cases the two frequencies would lock, causing a repeating pattern rather than the many-petal Spirograph flower that occurs when the two frequencies are slightly different¹⁵⁵ (Fig. 19). Canadian mathematician LeBlanc proved several mathematical theorems that predict the results of Roth's computer simulations.¹⁵⁷

FIG. 19.

The meander path for a spiral wave, as predicted by the bidomain model. For equal anisotropy ratios, the periods of spiral wave rotation and meander are independent, and the core of the spiral wave traces a Spirograph-like pattern (red). For unequal anisotropy ratios, the periods of spiral wave rotation and meander lock, and the core repeatedly traces a four-petal flower (purple).

The bidomain meander studies are summarized in a conference paper for a biomathematics meeting at Vanderbilt,¹⁵⁸ and in a chapter for the book Cardiac Electrophysiology: From Cell to Bedside.¹⁵⁹ Although the results are interesting, they have not had a large impact. The meander pattern is a subtle effect, and any sort of heterogeneity in the tissue would probably overwhelm it (although this remains an open question). Moreover, the bidomain seems to have no influence on the decay of meander to chaos. In the end, the project confirmed that the bidomain model is more important for studying the response of cardiac tissue to electrical stimulation than it is for studying propagation.⁴⁰

IX. INFLUENCE OF A PERFUSING BATH ON THE FOOT OF THE CARDIAC ACTION POTENTIAL

One exception to the rule that the bidomain model does not have much effect on propagation is when there is a boundary between cardiac tissue and a perfusing bath. Spach and his team at Duke measured the transmembrane potential in a slab of cardiac tissue that was superfused by a saline bath (optical mapping was not yet available, so they used electrodes).¹⁶⁰ They found that the rate of rise of the action potential and the time constant of the action potential foot depended on if the wave front propagated parallel or perpendicular to the cardiac fibers. One-dimensional continuous cable theory implies that the shape of the action potential wave front does not depend on the electrical conductivity; the wave front propagation may speed up in highly conductive tissue, but its time dependence does not change. They concluded that because they observed a change in shape of the action potential when propagating in different directions (and therefore for different conductivities), the tissue must be discontinuous.^160,161

The original calculations of action potential propagation in a continuous bidomain strand perfused by a bath^58,61,62 hinted at different interpretations of Spach's data. As discussed earlier, the wave front is not one-dimensional because its profile varies with depth below the strand surface (Fig. 8). The same effect occurs during propagation through a perfused planar slab, more closely resembling Spach's experiment.⁶⁴ The conductivity of the bath is higher than the conductivity of the interstitial space, so the wave front propagates ahead on the surface of the tissue and drags along the wave front deeper below the surface, resulting in a curved front. The extra electrotonic load experienced at the surface slows the rate of rise and the time constant of the action potential foot¹⁶² (Fig. 20). Plonsey, Henriquez, and Trayanova analyzed this effect,¹⁶³ and subsequently so did Henriquez and his collaborators^62,64,66 and Roth.^162,164

FIG. 20.

The action potential at the surface of a planar slab of cardiac tissue, plotted in the phase plane. The dashed curve is when no bath is present, The solid curves are when a bath is present, for propagation parallel to (longitudinal, L) and perpendicular to (transverse, T) the fibers.

The debate about whether propagation is continuous or discontinuous has never been settled, and many researchers today cite Spach's data as evidence that cardiac tissue behaves discontinuously. In order to resolve this issue, the key experiment that must be performed is to measure the rate of rise of the action potential with and without the perfusing bath present. Without the bath, the wave front should behave one-dimensionally and Spach's argument is valid. With the bath present, the wave front is curved which changes the action potential upstroke. This experiment is difficult, because optical mapping averages over depth and may not measure accurately the transmembrane potential at the tissue surface. Moreover, electrodes are susceptible to artifacts arising from capacitive coupling to the bath, so removing the bath may change their response. Perhaps the best way to do the experiment is the method developed by Sharifov and Fast: use optical mapping, but apply the dye by superfusing the tissue so it is taken up only at the tissue surface, thereby reducing any averaging over depth.¹⁶⁵ The Spach controversy reinforces the need to examine models and experiments for potential sources of error.¹⁶⁶

X. VIRTUAL ELECTRODE EFFECTS AROUND AN ARTIFICIAL HETEROGENEITY DURING FIELD STIMULATION OF CARDIAC TISSUE

Langrill Beaudoin and Roth examined what happens if a cylindrical insulator is present in cardiac tissue exposed to an electric field. Their project was motivated by experiments using plunge electrodes to record during and after defibrillation shocks.¹⁶⁷ Could the insulated shafts of the plunge electrodes be causing an artifact? They found that when an electric field is present an insulator would polarize the surrounding tissue, but only if the tissue had unequal anisotropy ratios (Fig. 21).¹⁶⁸ Woods—a graduate student in Wikswo's lab—tested this prediction using optical mapping, and found results consistent with the bidomain model.¹⁶⁹ This behavior may unpin spiral waves from obstacles during defibrillation.¹⁷⁰

FIG. 21.

An explanation of why an insulator (black circle) in cardiac tissue with unequal anisotropy ratios causes regions of depolarization (D) and hyperpolarization (H). An electric field is applied from left to right. The intracellular current density is represented by green arrows, and the interstitial current density by blue arrows. The red lines indicate the fiber direction.

A similar situation arises for a silver recording electrode placed on the tissue surface. The bidomain model predicts that during a shock current leaves the tissue on one side of the electrode, passes through the low-resistance metal, and then returns to the tissue on the other side.¹⁷¹ This prediction is difficult to study using optical mapping, because the electrode blocks the view of the tissue underneath it. However, Knisley developed translucent recording electrodes, and his student Liau was able to observe the effect experimentally.¹⁷²

Optical techniques are useful for testing bidomain predictions, but they are not perfect.¹⁷³ In particular, optical recordings are known to average over depth.^55,174–176 Janks and Roth observed that the optical signal decays with depth in the tissue, but the transmembrane potential caused by a shock can decay with depth too.¹⁷⁷ If the optical decay length is larger than the electrical decay length, the optical signal will average over tissue that is little affected by the shock, so the signal underestimates the surface polarization. Prior and Roth¹⁷⁸ used this idea to simulate the optical signal recorded during unipolar stimulation and compared it to the actual polarization at the tissue surface. They are not the same, because of optical averaging over depth.

The accuracy of optical mapping continues to be an interesting question. Roth collaborated with Pertsov—a biophysicist from SUNY Upstate Medical University who helped make some of the first optical recordings of spiral waves^83–86—to examine optical mapping techniques and controversies.¹⁷⁹ What is needed is a way to map transmembrane potential that does not suffer from the limitations of optical methods. One proposal is that the biomagnetic field produced by action currents in the heart could be used as the gradient field during magnetic resonance imaging.¹⁸⁰ It is a fascinating idea, and would allow the power of MRI to be applied to cardiac electrophysiology, but the magnetic field is too small for present technology. Some entirely new technique may be needed, such as a molecule taken up by the membrane whose magnetic resonance response, rather than its optical fluorescence, depends on the transmembrane potential. Another possibility is to track wave front propagation using high resolution ultrasound-based strain imaging.^181,182 New methods like these have the potential to revolutionize the field.

Many effects predicted by the bidomain model depend on the tissue having unequal anisotropy ratios.^{23,44,45,53,100,105,155} Often these effects have simple qualitative explanations.¹⁸³ For instance, current in cardiac tissue distributes between the intracellular and interstitial spaces according to their relative conductivities. When flowing parallel to the fibers, about half the current is intracellular and half interstitial. But when flowing perpendicular to the fibers, most of the current must be interstitial because of the low conductivity of the intracellular space perpendicular to the fibers (an implication of unequal anisotropy ratios).

Consider an insulator in cardiac tissue exposed to an electric field,¹⁸³ the case examined in Woods's experiment¹⁶⁹ (Fig. 21). To the left of the insulator, the current is parallel to the fibers, in the direction of the applied field. Half is intracellular and half interstitial. As it approaches the insulator, the current must turn to go around it, and now flows perpendicular to the fibers. A small amount is intracellular and most interstitial. Where the current changed directions, some of the current must have left the intracellular space and entered the interstitial space, depolarizing the tissue. When the current then turns back parallel to the fibers, current enters the intracellular space, hyperpolarizing the fibers. The net result is a complicated distribution of transmembrane potential, having three regions of depolarization (virtual cathodes) and three of hyperpolarization (virtual anodes) surrounding the insulator.

The value of qualitative explanations like in Fig. 21 is that they build intuition. Certainly a detailed calculation provides a more accurate representation of the tissue, but simple models that promote insight are important too. A cellular automaton,⁸⁹ spherical heart,¹⁰⁰ or hand-waving argument¹⁸³ helps researchers understand their more complex and realistic simulations.

XI. MECHANICAL BIDOMAIN MODEL OF CARDIAC TISSUE

The mechanical bidomain model^184–187 is a mechanical version of the electrical bidomain model discussed so far in this review. It can be illustrated using springs¹⁸⁷ (Fig. 22), just as the electrical model is illustrated using resistors (Fig. 2).

FIG. 22.

The two-dimensional mechanical bidomain model, illustrated as grids of springs. The green grid represents the mechanical properties of the intracellular space, the blue grid the interstitial space, and the red grid the cell membrane.

Many analogies exist between the electrical and mechanical bidomain models. The electrical bidomain model focuses on how current distributes between the intracellular and interstitial spaces; the mechanical bidomain model focuses on how forces distribute between the cytoskeleton and the extracellular matrix. The electrical model relates current and potential using Ohm's law; the mechanical model relates stress and strain using Hooke's law. The electrical model predicts the intracellular and interstitial potentials, and their difference the transmembrane potential; the mechanical model predicts intracellular and interstitial displacements, and their difference. In the electrical model, current passes from one space to the other through ion channels in the membrane; in the mechanical model, force passes from one space to the other via integrins in the membrane (integrins are proteins that mechanically couple the inside and outside of cells¹⁸⁸) The electrical model determines active membrane currents from measurement of intracellular or extracellular potentials;¹⁸⁹ the mechanical model might determine active stresses from measurement of intracellular or extracellular displacements.¹⁸² The electrical model contains a length constant that varies with the electrical conductivities of the two spaces and the membrane resistance; the mechanical model contains a length constant that varies with the mechanical moduli of the two spaces and the integrin spring constant. Activation of ion channels produces action potentials; activation of integrins produces mechanotransduction¹⁹⁰—the growth and remodeling of tissue in response to mechanical forces.

The behavior of the mechanical bidomain model is illustrated by a slab of cardiac tissue undergoing shear¹⁸⁶ (Fig. 23). Most mechanical models are monodomains, representing behavior averaged over the intracellular and interstitial spaces. The monodomain displacement varies linearly across the slab cross section, so the monodomain shear strain is constant. The bidomain displacement—the difference between the intracellular and interstitial displacements—is restricted to a region a few length constants from the slab surface. If your hypothesis is that strain drives mechanotransduction, you expect growth and remodeling to occur throughout the tissue. If your hypothesis is that the bidomain displacement drives mechanotransduction via forces acting on integrins, you expect growth and remodeling to occur in thin layers near the upper and lower surfaces.

FIG. 23.

A slab of cardiac tissue being sheared, as predicted by the mechanical bidomain model. The top surface is pulled to the right, and the bottom surface (not visible) to the left. The monodomain displacement varies linearly across the slab thickness, but the bidomain displacement falls exponentially with depth and is large only near the slab surface. The length of the bidomain arrows is exaggerated in the figure; in general, the bidomain displacement is much smaller than the monodomain displacement.

Perturbation theory allows one to separate the monodomain (sometimes called the “common mode”) behavior from the bidomain (sometimes called the “differential mode”) behavior;¹⁹¹ a separation that harkens back to Tung's dissertation.⁶ Numerical algorithms have been developed to solve the coupled partial differential equations of the mechanical bidomain model,¹⁹² and have been applied to study remodeling of cardiac tissue in the border zone around a region of ischemia.¹⁹³ The mechanical bidomain model provides insight into development. It predicts that in a growing colony of stem cells, differentiation occurs within a few length constants of the edge of the colony¹⁹⁴ (Fig. 24), consistent with experiment.¹⁹⁵ The model also has implications for tissue engineering. It implies that if an engineered sheet of cardiac tissue is stretched, cells grow preferentially at the edge of the sheet,¹⁹⁶ again consistent with experiment.¹⁹⁷ Unequal anisotropy ratios for the mechanical moduli have a profound effect on mechanotransduction,¹⁹⁸ just as unequal anisotropy ratios for the electrical conductivities have a profound effect on the response to an electric shock.

FIG. 24.

The intracellular and interstitial displacements, and their difference the bidomain displacement, in a growing circular stem cell colony.

Many of the predictions of the mechanical bidomain model have not yet been confirmed by experiment, so it is too early to say if the mechanical bidomain model will do for mechanotransduction what the electrical bidomain model has done for defibrillation: provide a valuable, if perhaps incomplete, theoretical description that makes predictions, guides experiments, and provides insight into underlying mechanisms.

XII. CONCLUSION

The electrical bidomain model has been an important description of cardiac tissue for over forty years (for a history of the bidomain model from another point of view, see Ref. 199). It has elucidated phenomena important during pacing and defibrillation, such as break excitation and virtual electrode induced phase singularities during defibrillation. It has not yet had a major impact on pacemaker and defibrillator design, perhaps because engineers have optimized these devices despite having an incomplete understanding of the underlying biophysics. However, the development of low-energy defibrillators may hinge on bidomain concepts.^119,130

Many open questions remain. For instance, a reoccurring debate has been the competing roles of continuous and discrete properties of cardiac tissue.²⁰⁰ At the moment, the continuous model is dominant, but the issue is not completely resolved. A key experiment would be to test Spach's hypothesis of discontinuous propagation by measuring action potential upstrokes with and without a perfusing bath present. Other controversies are the observation by Pertsov and his collaborators of stimulus responses that challenge the continuous bidomain model,²⁰¹ the importance of small-scale anatomical structures, such as the vascular system during defibrillation,²⁰² and ephaptic propagation.²⁰³ To get a glimpse of current issues in the field, examine the 2017 focus issue of the journal Chaos, edited by Cherry and her colleagues (the third such focus issue in that journal, all of which have helped summarize and direct research in cardiac electrophysiology).^204–208

The mechanical bidomain model does not have the long history of theoretical predictions and experimental verifications that the electrical bidomain model has. It is at an earlier stage of its development, and its importance and significance remain to be seen.²⁰⁹

AUTHOR DECLARATIONS

Conflict of Interest

I have no conflicts of interest to disclose.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.