Modeling the fluid dynamics of the heart: from blood flows in idealized left ventricles to patient-specific aortic valve simulations

The computational fluid dynamics of the heart represents a challenging task both in terms of mathematical and numerical modeling; this is mainly due to the pulsatile nature of the blood flow, to its complex interaction with the valves, and, more in general, to the reciprocal action of the components responsible for the heart functioning. Even if one focuses on the study of the left ventricle, the blood flow patterns result to be significantly dependent on the mechanical contraction and relaxation of the muscle, on the conformation of the chamber, and on the interaction with the valves, which define a complex fluid-structure interaction problem. In this respect, also the blood flow in the aorta, and hence in the downstream circulation, is strongly affected by the aortic valve, whose behavior should be suitably mathematically and numerically modeled. In this work, we firstly focus on the study of the fluid dynamics inside the left ventricle in idealized configurations, for which we propose a mathematical model based on the Navier-Stokes equations endowed with mixed, time-dependent boundary conditions, which allow a simplified treatment of the aortic and mitral valves’ behavior. In this idealized setting, we perform numerical simulations which highlight the role and influence of modeling the valves to study and characterize the blood flow patterns inside the ventricle, as well as other parameters of clinical relevance. In addition, we consider a reduced, patient-specific fluid-structure-interaction model for the simulation of the blood flow through the aortic valve. Specifically, we propose an efficient coupled model which represents the valve dynamics by means of a zero-dimensional (0D) equation with the opening angle as primitive variable, while the blood flow by means of the full 3D Navier-Stokes equations. In this coupled model, the valve’s leaflets, which are reconstructed from MRI data of the open and closed configurations for a specific patient, influence the Navier-Stokes equations by means of resistive immersed surfaces, whose position depends on the opening angle of the valve. Moreover, the dynamics of the valve described by the 0D model is dependent on the velocity and pressure variables, specifically on the pressure jump and the flow rate through the valve itself. We per form patient-specific numerical simulations of the aortic valve based on this reduced 3D-0D model, for which we highlight its ability to correctly capture the fluid dynamics indicators expected for the patient.