Biological tissue mechanics with fibres modelled as one-dimensional Cosserat continua. Applications to cardiac tissue

Abstract

• Cosserat-fibre continuum-based transversely isotropic material law for biological soft tissue. • Investigation of the impact of higher-order stress and strain contributions in modelling of heart muscle tissue. • Demonstration of the improved robustness of the Cosserat-fibre approach as compared with a classical continuum formulation when applied to highly heterogeneous and non-uniform material anisotropy distributions. • Computational modelling of the rat left ventricle during passive filling. Classically, the elastic behaviour of cardiac tissue mechanics is modelled using anisotropic strain energy functions capturing the averaged behaviour of its fibrous micro-structure. The strain energy function can be derived via representation theorems for anisotropic functions where a suitable non-linear strain tensor, e.g. the Green strain tensor, describes locally the current state of strain ( Holzapfel, Ogden, 2009 , Zheng, 1994 ). These approaches are usually of phenomenological nature and do not elucidate on the complex heterogeneous material composition of cardiac tissue (Rosenberg and Cimrman, 2003). In this research the fibrous characteristics of the myocardium are modelled by one-dimensional Cosserat continua. This additionally allows for the inclusion of non-local effects due to the heterogeneous material composition at smaller scales. Specifically, the non-local material response is linked to higher-order deformation modes associated with twisting and bending of an assembly of muscle fibres arising from hierarchical multi-scale features within a representative volume element (RVE). In this sense, a scaling parameter characteristic for the tissue’s underlying micro-structure, becomes a material parameter of the formulation. As the anisotropic material composition of the myocardium throughout the heart is highly non-uniform, the ability to implicitly account for scale-dependent torsion and bending effects in the constitutive law gives this approach an advanced material description over classical formulations. The assumed hyperelastic material behaviour of myocardial tissue is represented by a non-linear strain energy function which includes contributions linked to the Cosserat -fibre continuum and complementary terms which refer to the extra-cellular matrix. Utilising the Element-free Galerkin method, simulations of specimen shear cubes and the left ventricle undergoing passive filling are introduced to investigate ventricular tissue mechanics.