Modeling and numerical simulation of implantable cardiovascular devices

This thesis, taking place in the context of the Mivana project, is devoted to the modeling and to the numerical simulation of implantable cardiovascular devices. This project is led by the start-up companies Kephalios and Epygon, conceptors of minimally invasive surgical solutions for the treatment of mitral regurgitation. The design and the simulation of such devices call for efficient and accurate numerical methods able to correctly compute cardiac hemodynamics. This is the main purpose of this thesis. In the first part, we describe the cardiovascular system and the cardiac valves before presenting some standard material for the mathematical modeling of cardiac hemodynamics. Based on the degree of complexity adopted for the modeling of the valve leaflets, two approaches are identified: the resistive immersed surfaces model and the complete fluid-structure interaction model. In the second part, we investigate the first approach which consists in combining a reduced modeling of the valves dynamics with a kinematic uncoupling of cardiac hemodynamics and electromechanics. We enhance it with external physiological data for the correct simulation of isovolumetric phases, cornerstones of the heartbeat, resulting in a relatively accurate model which avoids the complexity of fully coupled problems. Then, a series of numerical tests on 3D physiological geometries, involving mitral regurgitation and several configurations of immersed valves, illustrates the performance of the proposed model. In the third and final part, complete fluid-structure interaction models are considered. This type of modeling is necessary when investigating more complex problems where the previous approach is no longer satisfactory, such as mitral valve prolapse or the closing of a mechanical valve. From the numerical point of view, the development of accurate and efficient methods is mandatory to be able to compute such physiological cases. We then consider a complete numerical study in which several unfitted meshes methods are compared. Next, we present a new explicit coupling scheme in the context of the fictitious domain method for which the unconditional stability in the energy norm is proved. Several 2D numerical examples are provided to illustrate the properties and the performance of this scheme. Last, this method is finally used for 2D and 3D numerical simulation of implantable cardiovascular devices in a complete fluid-structure interaction framework.