Computational cardiology: A modified Hill model to describe the electro-visco-elasticity of the myocardium

This contribution presents a novel three-dimensional constitutive model which describes the orthotropic electro-visco-elastic response of the myocardium. The model can be regarded as the viscoelastic extension of the recently proposed generalized Hill model for orthotropic active muscle cells. The formulation extends the recent contributions of Cansiz et al. (CMBBE 18: 1160-1172, 2015) and Goktepe et al. (IMPS 72: 20-39, 2014) in a novel rheological description which incorporates the active (electrical) and mechanical (viscous and elastic) deformations in a multiplicative format. To this end, the stress response is additively decomposed into passive (purely mechanical) and visco-active (electro-visco-elastic) contributions in line with a rheological model in which the passive part is connected parallel to a branch that consists of an elastic spring, a dashpot and a contractile element in serial. The former branch is assumed to be a function of the total deformation gradient while the formulation of the latter one is based on a multiplicative decomposition of the total deformation gradient into a mechanical and an active part. The active deformation gradient is devised by means of the prescribed active stretch which arises from the electrical excitation of the myocardial tissue and is governed by the intracellular calcium concentration. Thanks to the proposed rheology, marked differences are observed in isometric and isotonic tests between viscoelastic and elastic cases that are performed on material level. Moreover, novel evolution equations for the description of active stretch and intracellular calcium concentration having superiorities over the existing approaches have been proposed. On the numerical side, a fully implicit finite element formulation along with surface elements accounting for blood pressure evolution in the ventricles during successive phases of the cardiac cycle is presented. The argument of the constitutive equation describing the ventricular blood pressure is specified as the associated ventricular cavity volume which leads to the mutual interaction of the non-adjacent surface elements. The performance of the theory and algorithms are demonstrated by means of representative multi-field initial boundary value problems. The results indicate that viscous effects significantly alter the electromechanical response of the cardiac tissue and is thought to be crucial in the virtual assessment of the cardiac function