Three-dimensional pseudospectral modelling of cardiac propagation in an inhomogeneous anisotropic tissue

Various investigators have used the monodomain model to study cardiac propagation behaviour. In many cases, the governing non-linear parabolic equation is solved using the finite-difference method. An adequate discretisation of cardiac tissue with realistic dimensions, however, often leads to a large model size that is computationally demanding. Recently, it has been demonstrated, for a two-dimensional homogeneous monodomain, that the Chebyshev pseudospectral method can offer higher computational efficiency than the finite-difference technique. Here, an extension of the pseudospectral approach to a three-dimensional inhomogeneous case with fibre rotation is presented. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions, and the monodomain equation is enforced at the Gauss-Lobatto node points. The forward Euler technique is used to advance the solution in time. Numerical results are presented that demonstrate that the Chebyshev pseudospectral method offered an even larger improvement in computational performance over the finite-difference method in the three-dimensional case. Specifically, the pseudospectral method allowed the number of nodes to be reduced by approximately 85 times, while the same solution accuracy was maintained. Depending on the model size, simulations were performed with approximately 18-41 times less memory and approximately 99-169 times less CPU time.