# The significant effect of the choice of ionic current integration method in cardiac electrophysiological simulations

*P. Pathmanathan G.R. Mirams J.A. Southern and J.P. Whiteley*

### Abstract

Finite element (FE) cardiac electro-physiology solvers commonly have ionic current determined at mesh nodes but required element interiors. We consider two interpolation approaches: (i) ionic current interpolation (ICI), where nodal ionic currents are linearly interpolated into the element and (ii) state variable interpolation (SVI), where cell model state variables are interpolated instead, from which the ionic current is evaluated. We explain why SVI leads to a method which is massively more computationally demanding than ICI (more than might originally be expected), and then demonstrate that the difference in results can be surprisingly large even on what are generally considered suitably fine meshes. We explain why the conduction velocity in ICI simulations is generally too large, identify how ICI can give ‘accidentally’ accurate conduction velocities through two particular sources of error balancing, and illustrate how the difference between ICI and SVI can be huge in anisotropic problems. We also characterize the ICI/SVI difference over a range of cell models, in terms of model upstroke-velocity and formulation of the fast sodium current. Finally, we propose and evaluate a hybrid method which provides the accuracy of SVI, while retaining the efficiency of ICI.