One key characterstic of the cardiac function is its complexity, i.e., the multitude of different phenomena acting on various temporal and spatial scales interacting with each other. Over the past decades, many models varying in complexity describing these interactions were presented and are used in current research. Despite the incredible progress made in describing and simulating cardiac function, most of the more detailed models are not properly embedded within mathematical theory.
This work aims to give a precise and comprehensive mathematical formulation
of coupled cardiac elastodynamics, including electrophysiology, elasticity and physiological boundary conditions developed in recent years. Focussing on the analysis of dynamic elasticity, the concept of anisotropy is applied to common cardiac tissue models, such as the models of Guccione et al. and Holzapfel and Ogden. Frequently used modeling approaches, such as incompressibility and the active strain decomposition, are integrated in one overarching framework, allowing for propositions on polyconvexity of the materials and solvability of the elastic system. The equations of elastodynamics are then complemented by the monodomain equations, describing the propagation of the excitation potential in cardiac tissue, and a surrogate model to simulate cardiovascular blood pressure. The full mathematical description of this coupled model allows a detailed formulation of a discretization scheme in space and time for the electro-elastodynamical system.
The classification of the coupled model within the context of weak solutions is presented and a time-segregated numerical approximation method for the full system is derived. The formulated numerical method is then examined by application on coupled test cases, providing first convergence results in space for the displacement in coupled cardiac problems
