A 3D reaction–diffusion system describing calcium dynamics in cardiac cell^★

¹ Institut de mathématique de Bordeaux (IMB) et l’institut de rythmologie et modélisation cardiaque (Liryc), université de Bordeaux et INRIA-Carmen Bordeaux Sud-Ouest, Bordeaux, France.
² Ecole Supérieure de Technologie d’Essaouira, Université Cadi Ayyad, B.P. 383 Essaouira El Jadida, Essaouira, Morocco, Morocco.

^* Corresponding author: mostafa.bendahmane@u-bordeaux.fr

Received: 9 October 2018
Accepted: 10 October 2018

Abstract

We are interested in modeling the interaction of calcium dynamics in a medium including sarcolemma and sarcoplasmic reticulum. The governing equations consist of a nonlinear reaction–diffusion system representing the various calcium fluxes and theirs buffers in the two media. We address the question of existence of weak solutions by using a fixed-point approach. We propose a finite element method for this system, we establish the existence of the discrete solution, and we show that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. Finally, we give some 2D and 3D numerical examples to our model.