The well-known bidomain and monodomain equations model a cardiac tissue at the macroscopic scale. They represent an average of the electrical function of individual cells assumed to organize into a regular, periodic network. However, it is important to understand the effect of dysfunction and disorganization of the tissue at the cellular level on the propagation of action potential at the tissue scale. For instance, in order to understand how and in which conditions conduction pathways at the cellular scale may provide a substrate for arrhythmias at the tissue scale. In order to understand the impact of such alterations, we have designed a realistic microscopic-scale model that solves a non-homogenized bidomain model with time-based transmission conditions on the transmembrane voltage, including gap junctions.
Considering a cell network in 2D or 3D, included in a connected extracellular matrix, we implemented a time-based coupling model between these two media (i.e. on the cell membrane), and either a linear, non-linear, or geometric gap junction model at each border between the cells. The geometry was discretized with a P1-Lagrange finite element method, coupled with an implicit-explicit time-stepping method. High-performance parallel computation was required due to the number of vertices (more than 100) a single 2D cell requires to be meshed.
In the resulting simulations, we studied the impact of the methods to model gap junctions:
- for a non-linear gap-junction model, the impact of the Cx43:Cx45 ratio on the propagation velocity;
- the influence of choosing a non-linear gap-junction model versus a linear one on the propagation of a second action potential wave that arrives early in the repolarization phase;
- the impact of the intra/extra-cellular conductivity ratio and the gap-junction conductance on the propagation velocity.