Computationally efficient methods for solving the bidomain equations in 3D

The bidomain equations represent the most complete description of cardiac electrical activity. However, the equations prove computationally burdensome as the resulting system of equations has two entries per spatial node. This paper examines the computational performance obtained by decoupling the bidomain equations into two separate systems of equations, an elliptic equation for the extracellular potential, and a parabolic equation for the transmembrane voltage. Each set of equations was solved on different grids with different time steps. For the elliptic problem, the performances of direct and iterative solvers were compared. For the parabolic equation, the interconnected cable method (ICCM) was compared to the finite element method (FEM). Results were obtained by simulating activity in a 3D slab of cardiac tissue whose ionic currents were described by modified Beeler-Reuter equations. It was found that the elliptic equation solution dominated the calculation. Reducing the frequency of solution and/or halving the spatial resolution resulted in considerable speed up while maintaining a reasonable error. Direct solvers were faster by a factor of 2-3 and the ICCM was about twice as fast in solving the parabolic equation as compared to the FEM. Both the elliptic and parabolic equations scaled linearly with the number of nodes.