Approximate solutions for certain bidomain problems in electrocardiography

The simulation of problems in electrocardiography using the bidomain model for cardiac tissue often creates issues with satisfaction of the boundary conditions required to obtain a solution. Recent studies have proposed approximate methods for solving such problems by satisfying the boundary conditions only approximately. This paper presents an analysis of their approximations using a similar method, but one which ensures that the boundary conditions are satisfied during the whole solution process. Also considered are additional functional forms, used in the approximate solutions, which are more appropriate to specific boundary conditions. The analysis shows that the approximations introduced by Patel and Roth [Phys. Rev. E 72, 051931 (2005)] generally give accurate results. However, there are certain situations where functional forms based on the geometry of the problem under consideration can give improved approximations. It is also demonstrated that the recent methods are equivalent to different approaches to solving the same problems introduced $20\text{years}$ earlier.

Figure 1

Defibrillation of a strand of cardiac tissue. A cylindrical strand of radius $a$ placed in a bath with a uniform electric field of strength $E_0$ applied perpendicular to the strand.

Figure 2

Various approximate solutions and the exact solution for the transmembrane potentials plotted against radial distance for a cylindrical strand of cardiac tissue with $a=0.5\mathrm{mm}$, $\alpha =0.25$, $\lambda =0.174\mathrm{mm}$, ${\sigma }_T=0.0931\mathrm{S}∕\mathrm{m}$, ${\sigma }_b=2\mathrm{S}∕\mathrm{m}$ and $E_0=100\mathrm{V}∕\mathrm{m}$.

Figure 3

A semi-infinite block of cardiac tissue, perfused by a bath of conductivity ${\sigma }_b$ containing a planar wave front propagating in the $x$ direction. The horizontal lines represent the fiber directions and the shaded area represents the action potential.

Figure 4

A slab of cardiac tissue of thickness $2a$, perfused by a bath of conductivity ${\sigma }_b$ on both sides (for $\mid y\mid >a$) with planar wave front propagating along the $z$ direction. The format is the same as in Fig. 3.

Figure 5

Relative error between surface potentials for two approximate solutions for propagation in a slab of cardiac tissue of finite thickness.

Figure 6

A slab of cardiac tissue, insulated on the epicardium (at $z=0$) and in contact with a blood mass on the endocardium (at $z=1$). The hatched region represents a region of subendocardial ischemia.

Figure 7

Epicardial potential distributions on the surface of a slab of cardiac tissue with no fiber rotation. Negative potentials are indicated by dashed lines and the zero of the potential is denoted by the thick solid line. The contour interval is $0.2\mathrm{mV}$. (a) shows the solution obtained from [6] (with a minimum potential of $-0.8279\mathrm{mV}$ and a maximum potential of $0.0276\mathrm{mV}$) and (b) is the solution obtained using the method described here (with a minimum potential of $-0.8205\mathrm{mV}$ and a maximum potential of $0.0292\mathrm{mV}$).

Figure 8

Epicardial potential distributions on the surface of a slab of cardiac tissue with 120° of fiber rotation. Negative potentials are indicated by dashed lines and the zero of the potential is denoted by the thick solid line. The contour interval is $0.2\mathrm{mV}$. (a) is the solution obtained from [6] (with a minimum potential of $-0.9149\mathrm{mV}$ and maximum potential of $0.0099\mathrm{mV}$) and (b) is the solution obtained using the method described here (with a minimum potential of $-0.9084\mathrm{mV}$ and a maximum potential of $0.0113\mathrm{mV}$).