Coupled lattice Boltzmann simulation method for bidomain type models in cardiac electrophysiology with multiple time-delays

In this work, we propose a mathematical model of the cardiac electrophysiology which take into account time delays in signal transmission, in order to capture the whole activities of macro- to micro-scale transport processes, and use this model to analyze the propagation of electrophysiological waves in the heart by using a developed coupling Lattice Boltzmann Method (LBM). The propagation of electrical activity in the heart is mathematically modeled by a modified bidomain system. As transmembrane potential evolves, the domain has anisotropical properties which are transposed into intracellular and extracellular conductivity. The new bidomain system is a multi-scale, stiff and strongly nonlinear coupled reaction-diffusion model in the shape of a set of ordinary differential equations coupled with a set of partial differential equations with multiple time delays. Due to delays, dynamic and geometry complexity, numerical simulation and implementation of this type of coupled systems are very ambitious mathematical and computational problems but are crucial in several biomedical applications. We introduce a modified LBM scheme, reliable, efficient, stable and easy to implement in the context of such bidomain systems with multiple time delays. Numerical tests to confirm effectiveness and accuracy of our approach are provided and, the influence and impact of delays to restore normal heart rhythm are analyzed.