A polyconvex computational formulation for electro-activation in cardiac mechanics

Cardiovascular diseases (CVD) represent the main cause of death in the world with more significant prevalence in developed countries. With a constantly increasing ageing population, the burden that CVD patients put on the healthcare system has been recognised as an urgent matter in need of immediate attention. With the aim of addressing this menace, the computational modelling of the complex physical phenomena occurring in the human heart has become an area of increasing scientific interest over the last decades. The support that computational mechanics can pro-vide through the design of in-silica diagnostic tools is nowadays well-acknowledged by experts in the field, especially in challenging cardiopathies such as heart infarction or dysrhythmia. Even more, great effort has been devoted to attempting the com-putational modelling of CVD with several objectives in mind: provide augmented simulation-based diagnosis, experiment innovative surgery techniques, offer patients personalised treatments and to better understand the very complex electro-chemo-bio-mechanical phenomena underpinning the behaviour of the heart. Unfortunately, most of the computational frameworks considered in the field are unable to account for the material complexity of the myocardium (heterogeneity, anisotropy, nearly-incompressibility, etc.), exhibiting spurious pressure oscillations, shear locking and volumetric locking. The main objective of this thesis is twofold: on the one hand, a novel computa-tional framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle is presented; on the other hand, this novel framework is used as a reference scenario to explore alternative strategies and estab-lish a balance between their applicability and efficiency. First, taking as a reference the mixed formulations introduced by Bonet et al. in the context of nonlinear elasticity, two new mixed formulations designed for polyconvex materials have been developed and customised for active stress and active strain coupling approaches. These proposed formulations add considerable flexibility to the modelling of the heart by including the geometry and the transmembrane potential (and possibly a Lagrange multiplier enforcing weakly the incompressibility constraint) as unknown fields as well as the deformation gradient tensor, its cofactor, its determinant, the gradient of the transmembrane potential and their respective work conjugates. Their Finite Element implementation is explained, where a static condensation procedure is presented in order to yield an extremely competitive computational approach. The superiority in accuracy of these formulations with respect to classical low order implementations is enhanced by a monolithic fully-implicit scheme for the electrome-chanical problem. The remarkable robustness and accuracy of these top-notch mixed formulations creates an ideal scenario to carry out a comprehensive and rigorous study of different ionic models and electromechanical activation couplings. For first time, large scale simulations have been performed in order to set up a four-way comparison among the ionic models proposed by Bueno-Orovio and Ten Tusscher, and the active strain and active stress coupling approaches under the same mesh and boundary condi-tions. In addition, these simulations constitute a suitable benchmark to analyse the possible loss of ellipticity and polyconvexity of the Holzapfel-Ogden constitutive model, widely used in the context of cardiac mechanics, putting forward possible polyconvexifications. Moreover, an invariant representation of Guccione's constitu-tive model is proposed. Secondly, despite the correct mathematical characterisation of the myocardium, the complex multi-scale interaction that takes place between the electrophysiology and electrochemistry at cellular level and the macro-scale response of the heart mus-cle customarily require immense computational resources. This poses an extreme challenge to the scalability of electro-mechanical solvers due to the size and con-ditioning of the system of equations required to obtain accurate solutions, both in terms of wall deformation and transmembrane potential propagation. In order to undertake this challenge, the results obtained by the polyconvex mixed formulations represent an ideal reference scenario to evaluate the applicability of more compu-tationally efficient techniques and cheaper Finite Element discretisations. In the search towards an efficient modelling of electro-activation, this manuscript presents a coupled electromechanical computational framework whereby, first, the use of an efficient stabilised low order tetrahedral Finite Element methodology is explored and compared against the very accurate super enhanced mixed formulation previously introduced and, second, the use of tailor-made staggered and staggered linearised solvers is implemented in order to assess their feasibility against a fully monolithic approach. These comparisons take the global displacements and intermembrane po-tential as accuracy indicators as well as considering the overall computational time. Through a comprehensive set of examples, culminating in a realistic ventricular ge-ometry, we aim to put forward some suggestions regarding the level of discretisation and coupling required to ensure sufficiently reliable results yet with an affordable computational time. Additionally, the possible locking phenomena due to shear de-formations and anisotropic behaviour is assessed for our enriched mixed formulation and compared against less expensive methodologies.