Development of a Micromorphic (Multiscale) Material Model aimed at Cardiac Tissue Mechanics

Dollery, Devin

Development of a Micromorphic (Multiscale) Material Model aimed at Cardiac Tissue Mechanics

Master Thesis

2020

Abstract

Computational cardiac mechanics has historically relied on classical continuum models; however, classical models amalgamate the behaviour of a material's micro-constituents, and thus only approximate the macroscopically observable material behaviour as a purely averaged response that originated on micro-structural levels. As such, classical models do not directly and independently address the response of the cardiac tissue (myocardium) components, such as the muscle fibres (myocytes) or the hierarchically organized cytoskeleton. Multiscale continuum models have developed over time to account for some of the micro-architecture of a material, and allow for additional degrees of freedom in the continuum over classical models. The micromorphic continuum [15] is a multiscale model that contains additional degrees of freedom which lend themselves to the description of fibres, referred to as micro-directors. The micromorphic model has great potential to replicate certain characteristics of the myocardium in more detail. Specifically, the micromorphic micro-directors can represent the myocytes, thus allowing for non-affine relative deformations of the myocytes and the extracellular matrix (ECM) of tissue constraining the myocytes, which is not directly possible with classical models. A generalized micromorphic approach of Sansour [73, 74, 75] is explored in this study. Firstly, numerical examples are investigated and several novel proofs are devised to understand the behaviour of the micromorphic model with regards to numerical instabilities, micro-director displacements, and macro-traction vector contributions. An alternative micromorphic model is developed by the author for comparison against Sansour's model regarding the handling of micro-boundary conditions and other numerical artifacts. Secondly, Sansour's model is applied to cardiac modelling, whereby a macro-scale strain measure represents the deformation of the ECM of the tissue, a micro-scale strain measure represents the muscle fibres, and a third strain measure describes of the interaction of both constituents. Separate constitutive equations are developed to give unique stiffness responses to both the ECM and the myocytes. The micromorphic model is calibrated for cardiac tissue, first using triaxial shear experiments [80], and subsequently, to a pressure-volume relationship. The contribution of the micromorphic additional degrees of freedom to the various triaxial shear modes is quantified, and an analytical explanation is provided for differences in contributions. The passive filling phase of the heart cycle is investigated using a patient-specific left ventricle geometry supplied by the Cape Universities Body Imaging Centre (CUBIC) [38].

Keywords

Computational and Applied Mechanics

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