Adaptive multiscale model for simulating cardiac conduction.

We present a multiscale model and an adaptive numerical scheme for simulating cardiac action potential propagation along a linear strand of heart muscle cells. This model couples macroscale partial differential equations posed over the tissue to microscale equations posed over discrete cellular geometry. The microscopic equations are used only near action potential wave fronts, and the macroscopic equations are used everywhere else. We study the effects of gap-junctional and ephaptic coupling on conduction in the multiscale model and its fully macroscale and fully microscale analogues. Our simulations reveal that the adaptive multiscale model accurately reproduces the action potential wave forms and wave speeds of the fully microscale model. They also demonstrate that, at low gap-junctional conductivities, the accuracy of fully macroscale simulations is sensitive to numerical grid spacing. Moreover, adaptive multiscale simulations capture the effect of ephaptic coupling, whereas fully macroscale simulations do not. We propose two ways of generalizing our multiscale model to higher dimensions, and we argue that such generalizations may be necessary to obtain accurate three-dimensional simulations of cardiac conduction in certain pathophysiological parameter regimes.