Influence of the deformation on the propagation of waves of excitation in the heart tissue

A model is constructed for changing the intracellular conductivity of the myocardium with its deformation on the basis of an analysis of the microstructural model of P.E. Hand, B.E. Griffith, C.S. Peski (Bull Math. Boil (2009)) model. The cardiac tissue was considered as a periodic lattice, where the cells are rectangular prisms filled with isotropic electrolyte, and the conductivity of the gap junctions is taken into account through the boundary conditions on the sides of these prisms and is assumed to be constant. Using the homogenization method in the form proposed in the paper G. Richardson and S.J. Chapman (SIAM Journal Appl. Math (2011)), the conductivity values are analytically expressed in terms of cell sizes, lattice periodicity parameters, electrical properties of myoplasm and gap junctions. On the basis of these relationships, the dependence of the tissue conductivity on its deformation is determined. A comparison is made with the model proposed in the book F.B. Sachse. Computational Cardiology (Springer (2004)). It is shown that both models can be well matched for elongations in the range from 0.8 to 1.2. A numerical algorithm based on the splitting method and its software implementation based on the finite element library FEniCS are developed. Conductivities calculated for different deformation dependences are compared. Corresponding profiles of the excitation waves in a rectangular two-dimensional region are considered. The effect of deformation is strongly "diluted" by extracellular conductivity. The appearance of depolarization and hyperpolarization regions (virtual electrodes) is also considered when an electric current is applied to the myocardium in a small domain. In this case, the effect of deformation is more significant.