Analytic solution of the anisotropic bidomain equations for myocardial tissue: The effect of adjoining conductive regions

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 V_m, intracellular potential φ_i, and extracellular potential φ_e in H, together with quasi-static equations for the potential distribution φ_B in the surrounding (passive) isotropic extracardiac regions B. Four local parameters σ_ℓ,t ^i,e 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 V_m, φ_e, φ_i, and φ_B for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity σ_B. 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 V_m and φ_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.