An adaptive, preconditioned, electromechanical model for the simulation of cardiac arrhythmias

In this thesis a coupled model of cardiac electromechanical activity is presented, using the finite element method to model both electrophysiology and mechanics within a deforming domain. The efficiency of the electrical model was improved using adaptive mesh refinement and the mechanical system performance was improved with the addition of preconditioning. Unstructured triangular meshes were used throughout. The electrophysiology model uses the ten Tusscher-Panfilov 2006 detailed cellular model, and includes anisotropic diffusion, uses a semi-implicit time stepping scheme, stores data in an efficient sparse storage format and applies a Reverse Cuthill-McKee ordering algorithm to reduce the matrices’ bandwidths. Linear elements were used to approximate the transmembrane voltage and spatial and temporal convergence tests were undertaken. Local mesh adaptivity is added to the electrical component of the model and improvements to the performance and efficiency gained by this technique were investigated. Two different monitor functions were utilised and these demonstrated that by targeting adaptive mesh refinement at the front of the electrical wave significant efficiency and performance benefits could be achieved. The cardiac mechanical model is based on finite deformation elasticity theory, enforces the incompressibility of the tissue and incorporates anisotropic tension to simulate fibre orientation. This uses isoparametric quadratic elements for deformation, linear elements for pressure, was integrated with numerical quadrature and the resulting non-linear system solved with the iterative Newton method. Preconditioning was added to the mechanical component of the model and improvements in the performance of the solver due to this were investigated. An ILUT (Incomplete Lower Upper factorisation with drop Tolerance) preconditioner was implemented and this demonstrated performance improvements of up to 27 times on the meshes tested. The resulting cardiac electromechanical solver was then used to consider how known changes in cardiac electrophysiology, which are manifest in end-stage heart disease, affect the stability of the electrical wave. Specifically, investigations were undertaken into the introduction of fibrotic regions (with different sizes and concentrations) and electrical remodelling caused by end-stage cardiac disease. These were modelled on both static and deforming domains to consider whether deformation can alter the stability of a spiral wave. These simulations demonstrated that fibrotic regions and tissue deformation can have significant disruptive effects on the stability of a re-entrant spiral wave and that remodelling the electrophysiology stabilises the wave.