Bidomain model

Post-publication activity

Dr. Bradley J. Roth, Department of Physics, Oakland University,

The bidomain model is a set of mathematical equations that govern the electrical properties of cardiac tissue. It was developed in the late 1970s, and is now used extensively in numerical simulations of the electrical behavior of the heart.

Anisotropic cable model

The bidomain model is a two- or three-dimensional cable model. It is a continuum model, in the sense that it predicts the electrical behavior averaged over many cells. The model accounts for the different electrical conductivities of the intracellular and extracellular spaces (Figure 1).

Figure 1: An electrical circuit that approximates the two-dimensional bidomain model. The intracellular and extracellular spaces are both represented by resistor grids. The resistors are larger in the direction perpendicular to the fibers (y) than in the direction parallel to them (x). The two spaces are coupled by the cell membrane, represented as a parallel combination of a resistor and a capacitor. Adapted from Roth (1992).

Both of these spaces are anisotropic: they have a different electrical conductivity in the direction parallel to the myocardial fibers than in the direction perpendicular to them. Moreover, the degree of anisotropy is different in the two spaces. In the intracellular space the conductivity parallel to the fibers is about ten times greater than the conductivity perpendicular to the fibers (10:1), whereas in the extracellular space the ratio is only about 5:2. This condition of unequal anisotropy ratios leads to many of the interesting predictions of the bidomain model.

Typical values for the bidomain conductivities (Roth, 1997)
\(g_{ix}\)	intracellular conductivity parallel to the fibers	0.20 S/m
\(g_{iy}\)	intracellular conductivity perpendicular to the fibers	0.02 S/m
\(g_{ex}\)	extracellular conductivity parallel to the fibers	0.20 S/m
\(g_{ey}\)	extracellular conductivity perpendicular to the fibers	0.08 S/m

The bidomain equations

The bidomain equations are a set to two coupled partial differential equations governing the intracellular potential, \(V_i\ ,\) and extracellular potential, \(V_e\) \[\tag{1} g_{ix}\frac{\partial^2 V_i}{\partial x^2}+g_{iy}\frac{\partial^2 V_i}{\partial y^2} =\beta\left[ C_m \frac{\partial \left(V_i - V_e\right)}{\partial t}+G_m\left(V_i - V_e\right) \right] \]

\[\tag{2} g_{ex}\frac{\partial^2 V_e}{\partial x^2}+g_{ey}\frac{\partial^2 V_e}{\partial y^2} =-\beta\left[ C_m \frac{\partial \left(V_i - V_e\right)}{\partial t}+G_m\left(V_i - V_e\right) \right] \]

where \(C_m\) and \(G_m\) are the capacitance per unit area and conductance per unit area of the cell membrane, and \(\beta\) is the ratio of membrane surface area to tissue volume. This formulation of the bidomain equations is applicable if the myocardial fibers are straight and parallel to the \(x\) axis, and if the membrane behaves passively (the membrane current is linearly related to the membrane potential). For active tissue capable of supporting propagation of action potential wave fronts, the term containing \(G_m\) should be replaced by a more complex, nonlinear term that specifies how the ion channels in the cell membrane respond to changes in the transmembrane potential, \(V_m=V_i-V_e\ .\) If electrodes are present that inject current into or out of the cardiac tissue, additional source terms are also necessary.

When solving the bidomain equations numerically, equations (1) and (2) are often rewritten as (Roth, 1991) \[\tag{3} C_m \frac{\partial V_m}{\partial t}=-J_{ion}-\frac{1}{\beta}\left[g_{ex}\frac{\partial^2 V_e}{\partial x^2}+g_{ey}\frac{\partial^2 V_e}{\partial y^2}\right] \]

\[\tag{4} \left(g_{ix}+g_{ex}\right)\frac{\partial^2 V_e}{\partial x^2}+\left(g_{iy}+g_{ey}\right)\frac{\partial^2 V_e}{\partial y^2} =-g_{ix}\frac{\partial^2 V_m}{\partial x^2}-g_{iy}\frac{\partial^2 V_m}{\partial y^2} \]

The intracellular potential is suppressed in favor of the transmembrane potential, and the membrane conductance term is replaced by \(J_{ion}\ ,\) a nonlinear function of the transmembrane potential based on a model of the active properties of the membrane (such as the Beeler-Reuter model). Equation (3) depends explicitly on time, is nonlinear because of \(J_{ion}\ ,\) and has properties similar to a parabolic partial differential equation (a reaction-diffusion system). Equation (4) has no explicit time dependence, and is an elliptic partial differential equation (a boundary value problem). These equations are often approximated with finite differences or finite elements. At each time step, equation (3) is used to update \(V_m\,\) and then equation (4) is solved for \(V_e\) using the updated \(V_m\) in the source term on the right-hand-side. Note that there are other linear transformations of the bidomain equations that may be solved more efficiently for given problems or properties as discussed by Hooke et al. (1994).

Boundary conditions

At the surface of the heart, \(V_i\) and \(V_e\) obey three boundary conditions (Krassowska and Neu, 1994) \[\tag{5} \frac{\partial V_i}{\partial n} =0 \]

\[\tag{6} \sigma_{en}\frac{\partial V_e}{\partial n} =\sigma_{bath}\frac{\partial V_{bath}}{\partial n} \]

\[\tag{7} V_e =V_{bath} \]

where \(V_{bath}\) is the potential in the volume conductor (or tissue bath) with conductivity \(\sigma_{bath}\) adjacent to the cardiac tissue, and \(n\) denotes the direction perpendicular to the surface. The first condition implies that the intracellular current density perpendicular to the surface is zero, the second specifies that the normal components of the extracellular and bath current densities are continuous, and the third stipulates that the extracellular and bath potentials are continuous. The tissue-bath interface can affect the propagation of action potentials in cardiac tissue by making the wave fronts tend to propagate fastest near the tissue surface (Roth, 1991). The boundary conditions also have implications during electrical stimulation. When an electric field is applied to the heart, these boundary conditions typically result in a large transmembrane potential at the heart surface, that falls off approximately exponentially with depth into the tissue.

Electrical stimulation

Figure 2: The transmembrane potential surrounding a cathodal electrode (center dot), calculated using the bidomain model. Yellow and red indicate depolarization, blue and magenta indicate hyperpolarization, and purple indicates rest. The myocardial fibers are horizontal, and box is 4 mm by 4 mm. The details of the calculation are described in Sepulveda et al. (1989).

One early prediction of the bidomain model was the transmembrane potential distribution around an extracellular electrode injecting current, such as in a pacemaker ( Figure 2). If the electrode is negative (a cathode), then the transmembrane potential becomes positive (is depolarized) directly under the electrode. However, when the tissue has unequal anisotropy ratios, there also exist regions of negative transmembrane potential (hyperpolarization) adjacent to the electrode along the fiber direction.

Figure 3: Pseudocolor isochronal map showing the position of the action potential wavefront at subsequent times after a strong shock applied to the a cathode (center dot). Each color indicates the time that the action potential arrives at that location. This experimental data from Lin et al. (1999), measured in a rabbit heart, confirms the prediction of quatrefoil reentry that was first proposed by simulations using the bidomain model.

During a strong stimulus, the regions of hyperpolarization affect both the mechanisms of action potential wave front excitation and the induction of reentrant wave fronts (a cause of cardiac arrhythmias). For instance, if the tissue is refractory at the time of the stimulus, the hyperpolarized regions may cause the tissue at the virtual anode to recover excitability (deexcite) thereby providing an excitable path that can support wave propagation along the fiber direction. A strong and well-timed stimulus can result in a type of arrhythmia called quatrefoil reentry ( Figure 3).

Fiber curvature

Figure 4: The transmembrane potential induced when an electric field is applied horizontally (arrow). The fiber direction is indicated by the line segments. Red indicates depolarization, blue hyperpolarization, and purple rest. The box is 20 mm by 20 mm. The calculation was performed by Roth and Langrill Beaudoin (2003).

The bidomain model is particularly useful when analyzing the mechanisms by which an electric field interacts with cardiac tissue. One such mechanism is fiber curvature. If the direction of the myocardial fibers varies throughout the tissue, then an electric field will cause a transmembrane potential distribution: both depolarization and hyperpolarization (sometimes called a virtual cathode and virtual anode). This mechanism only operates if the tissue has unequal anisotropy ratios. In Figure 4, the electric field is applied horizontally and the color shows the resulting transmembrane potential.

Defibrillation

When a heart experiences ventricular fibrillation (a lethal cardiac arrhythmia), the only way to restore a normal rhythm is to apply a strong electrical shock, a process called defibrillation. Perhaps the most important application of the bidomain model is simulating defibrillation (Rodriguez et al., 2006). The video below shows a bidomain simulation of defibrillation, using a realistic heart geometry, curving fiber geometry, and nonlinear membrane model. The transmembrane potential is plotted on the surface of the left and right ventricles of the heart. Initially, the heart is in fibrillation, characterized by a chaotic distribution of wave fronts. About two-thirds of the way through the video, a strong shock is applied through the electrodes in the surrounding bath. Immediately, the boundary conditions (equations (5), (6), and (7)) cause the surface of the heart to be depolarized on the right and hyperpolarized on the left. Although not apparent from the this video, which shows only the transmembrane potential on the heart surface, the curving fiber geometry and other heterogeneities, together with the unequal anisotropy ratios, induce a weaker transmembrane potential distribution (virtual cathodes and anodes) deep within the heart wall. These virtual electrodes interact with the fibrillation wave fronts, with the result that fibrillation terminates: successful defibrillation. For other stimulus strengths and timing, the shock may fail to defibrillate because these same virtual electrodes reduce reentry (Efimov et al., 1998). The video was supplied by Dr. James Eason, with assistance from Courtenay Glisson, and the simulation was based on the model described by Rodriguez et al. (2006)

Media:FullShock30 SI.mov

References

Efimov I.R., Cheng Y., van Wagoner D.R., Mazgalev T. and Tchou P.J. (1998) Virtual electrode-induced phase singularity: A basic mechanism of defibrillation failure. Circ. Res., 82:918-925. doi:10.1161/01.RES.82.8.918.
Hooke N, Henriquez, C.S., Lanzkron P., Rose D. (1994) Linear algebraic transformations of the bidomain equations: implications for numerical methods. Math Biosci., 120:127-45. doi:10.1016/0025-5564(94)90049-3.
Krassowska W. and Neu J.C. (1994) Effective boundary conditions for syncytial tissues. IEEE Trans. Biomed. Eng., 41:143-150.
Lin S.-F., Roth B.J. and Wikswo J.P. (1999) Quatrefoil reentry in myocardium: An optical imaging study of the induction mechanism. J. Cardiovasc. Electrophysiol., 10:574-586.
Rodriguez B., Eason J. and Trayanova N. (2006) Differences between left and right ventricular anatomy determine the types of reentrant circuits induced by an external electric shock: A rabbit heart simulation study. Prog. Biophys. Mol. Biol., 90:399-413.
Roth B.J. (1991) Action potential propagation in a thick strand of cardiac muscle. Circ. Res., 68:162-173. doi:10.1161/01.RES.68.1.162.
Roth B.J. (1992) How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J. Math. Biol., 30:633-646. doi:10.1007/BF00948895.
Roth B.J. (1997) Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng., 44:326-328.
Roth B.J. and Langrill Beaudoin D. (2003) Approximate analytical solutions of the bidomain equations for electrical stimulation of cardiac tissue with curving fibers. Phys. Rev. E, 67:051925. doi:10.1103/PhysRevE.67.051925.
Sepulveda N.G., Roth B.J. and Wikswo J.P. (1989) Current injection into a two-dimensional anisotropic bidomain. Biophys. J., 55:987-999 doi:10.1016/S0006-3495(89)82897-8.

Internal references

Ian Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853. doi:10.4249/scholarpedia.2853.

Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475. doi:10.4249/scholarpedia.1475.

Carsten Wolters and Jan C de Munck (2007) Volume conduction. Scholarpedia, 2(3):1738. doi:10.4249/scholarpedia.1738.

External links

Brad Roth's website
A simple tutorial on how virtual electrodes induce cardiac arrhythmias, based on a cellular automata model.
Website of the Virtual Heart Lab led by Dr. James Eason, who produced the video.

Bidomain model

Contents

Anisotropic cable model

The bidomain equations

Boundary conditions

Electrical stimulation

Fiber curvature

Defibrillation

References

Recommended reading

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