Analytic Solution of the Anisotropic Bidomain Equations for Myocardial Tissue: The Effect of Adjoining Conductive Regions

Abstract

The anisotropic bidomain model for the propagation of electrical activation in the human myocardium H consists of coupled elliptic-parabolic partial differential equations for the transmembrane potential Vm, intracellular potential phi(i), and extracellular potential phi(e) in H, together with quasi-static equations for the potential distribution phiB in the surrounding (passive) isotropic extracardiac regions B. Four local parameters sigma((i,e) (l,t)) specify the conductivities in the longitudinal (l) and transverse (t) directions with respect to cardiac muscle fibers. Continuous current flow is required at the interface S(H) between H and B. We derive analytic formulas for Vm, phi(e), phi(i), and phiB for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity sigmaB. No assumptions are required regarding the anisotropy ratios of the conductivity coefficients. The properties of these solutions are examined with a view to providing insight into the effect of the passive region B on the propagation of Vm and phi(e) in H. We show that for a suitably chosen boundary condition, the problem can be reduced to solving the bidomain equations in H alone.