An Investigation of Necessary Grid Resolution for Numerical Simulations of Calcium Dynamics in Cardiac Cells

Abstract

One of the challenges related to modelling calcium dynamics in cardiac cells is the large difference in the length scales involved. The dyad, where important processes take place, is very small compared to the whole cell. Therefore, resolutions fine enough to capture the details of what happens in the dyad result in huge computational problems for whole-cell simulations, and the exact choice of resolution has a substantial effect on the problem size. In this thesis, we investigate what grid resolution is necessary to capture the details of what happens in the dyad. We study simple mathematical models of calcium dynamics in the dyad and find analytical solutions to some of these simple models. Numerical simulations of the models are carried out for different resolutions using finite difference methods in 1D and 2D and a finite volume method in 3D. The accuracy of the numerical simulations is then studied by comparing the numerical solutions to analytical solutions and fine-grid numerical solutions, and the results suggest necessary resolutions in the nanometre range.