Cardiac tissue primarily consists of interconnected cardiac cells which contract in a synchronized manner as the heart beats. Most computational models of cardiac tissue, however, homogenize out the individual cells and their surroundings. This approach has been immensely useful for describing cardiac mechanics on an overall level, but gives very limited understanding of the interaction between individual cells and their intermediate surroundings. Several models have been developed for single cells, see e.g. [1, 2]. In this work, we extend the mechanical part of these frameworks to a domain representing multiple cells, allowing us to investigate cell-matrix and cell-cell interactions. We present a mechanical model in which each cell and the extracellular matrix have an explicit geometrical representation, similar to the electrophysiological model presented in [3]. The strain energy functions are defined separately for each of the intracellular and extracellular subdomains, while we assume continuity of displacement and stresses along the membrane. Active tension is only assigned to the intracellular subdomain. For each state, we find an equilibrium solution using the finite element method. We explore passive and active mechanics for a single cell surrounded by an extracellular matrix and for small collections of cells combined into tissue blocks. The explicit geometric representation gives rise to highly varying strain and stress patterns. We show that the extracellular matrix stiffness highly influences the cardiomyocyte stresses during contraction. Through large-scale simulations enabled by high-performance computing, we also demonstrate that our model can be scaled to small collections of cells, resembling small cardiac tissue samples.

[1] Tracqui, T. and Ohayon, J. An integrated formulation of anisotropic force–calcium relations driving spatio-temporal contractions of cardiac myocytes. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2009).
[2] Ruiz-Baier, R. Gizzi, A., Rossi, S. Cherubini, C. Laadhari, A. Filippi, S. and Quarteroni, A. Mathematical modelling of active contraction in isolated cardiomyocytes. Mathematical Medicine and Biology (2014).
[3] Tveito, A., Jæger, KH. Kuchta, M. Mardal, K-A. and Rognes, ME. A cell-based framework for numerical modeling of electrical conduction in cardiac tissue. Frontiers in Physics (2017).
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