An electro-fluid-structure model for cardiac numerical simulations based on an embedded strategy

In this Thesis, we propose a loosely coupled model for numerical cardiac simulations representing the main physical processes involved in heart physiology by means of an electro-fluid-structure interaction (EFSI) problem. Firstly, we develop computational methods for simulating cardiac electrophysiology, specifically focusing on the propagation of electrical signals through the cardiac tissue. We employ the anisotropic Monodomain model, which combines a reaction-diffusion equation with various membrane models representing ionic currents. To accurately capture the behavior of these models, we propose a second-order scheme based on Strang splitting, where the nonlinear subproblem is solved by an explicit second-order predictor-corrector scheme. This scheme effectively handles the stiff gating variables and the rapid gradient increase of the action potential, resulting in improved computational efficiency. Furthermore, we extend our computational framework by incorporating the Eikonal model. This model allows us to compute the time activation, considering the reduced computational costs associated with its implementation. Secondly, a generalized formulation for the fluid-structure interaction (FSI) problem, which incorporates an active term expressed as active stress, is developed. This formulation draws inspiration from the Immersed Boundary Method and combines various mathematical tools. Specifically, we utilize elastodynamics equations to model the solid behavior, a high-order Navier-Stokes solver for the fluid flow, and an L^2-projection method for the transfer of velocities and forces between the fluid grid and the solid mesh. By incorporating the active term, our formulation accurately accounts for the active mechanical contribution of the cardiac muscle. We validate the effectiveness and reliability of our approach through extensive testing, including the use of the Turek-Horn benchmark. Additionally, we simulate the filling phase of a two-dimensional idealized ventricle, where the deformed mesh obtained serves as the initial condition for simulating the active contraction stage. Throughout these simulations, our proposed formulation consistently demonstrates its ability to extend the capabilities of FSI modeling in the context of cardiac simulations. Lastly, we achieve the coupling between the generalized fluid-structure interaction (FSI) framework, incorporating active stress, and the electrophysiological environment by evaluating a time activation map using two different models: the reduced Eikonal model and the Monodomain model coupled with the LuoRudy ionic model. We conduct these evaluations on a three-dimensional idealized ventricle, enabling us to observe the effects of the coupling on the cardiac system. Specifically, we compare the results obtained from a uniform contraction scenario with those obtained considering the deformations generated by the activation maps. This comparative analysis provides valuable insights into the impact of electromechanical couplings on cardiac modeling. These advancements open up possibilities for advanced electromechanical couplings in cardiac modeling, providing a more comprehensive understanding of heart physiology.