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/sub m/, intracellular potential /spl phi//sub i/, and extracellular potential /spl phi//sub e/ in H, together with quasi-static equations for the potential distribution /spl phi//sub B/ in the surrounding (passive) isotropic extracardiac regions B. Four local parameters /spl sigma//sup i,e//sub /spl lscr/,t/ specify the conductivities in the longitudinal (/spl lscr/) and transverse (t) directions with respect to cardiac muscle fibers. Continuous current flow is required at the interface S/sub H/ between H and B. We derive analytic formulas for V/sub m/, /spl phi//sub e/, /spl phi//sub i/, and /spl phi//sub B/ for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity /spl sigma//sub 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/sub m/ and /spl phi//sub 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.