Discrete versus syncytial tissue behavior in a model of cardiac stimulation--I: Mathematical formulation

This paper presents a model describing the steady-state response of a two-dimensional (2-D) slice of myocardium to extracellular current injection. The model incorporates a continuous representation of the multicellular, syncytial cardiac tissue based on the bidomain model. The classical bidomain model is modified by introducing periodic conductivities to better represent the electrical properties of the intracellular space. Thus, junctional discontinuity between abutting myocytes is reflected in the macroscopic representation of cardiac tissue behavior. Since a solution to the resulting coupled differential equations governing the intracellular and extracellular potentials in the tissue preparation is not computationally tractable when traditional numerical approaches, such as finite element or finite difference methods are used, spectral techniques are employed to reduce the problem to the solution of a set of algebraic equations for the transform of the bidomain potentials. Further, the solution to the "periodic" bidomain problem in the Fourier space is decomposed into two separate solutions: One for the classical-bidomain potentials where it is assumed that the intracellular conductivity values along and across cells incorporate the average contribution from cytoplasm and junction, and another for the junctional potential component. The decomposition of the total solution allows to approximately solve for the junctional component thus achieving high overall computational efficiency. The results of simulation are presented in an accompanying paper.