Resistive immersed surfaces for heart valves modeling

We propose a new reduced model for the modeling of heart valves. The model represents a "compromise" between standard lumped parameter models, which may introduce artifacts in the numerical results due to the artificial boundaries, and three-dimensional FSI models, which are often computationally expensive. Within this approach, the valves are replaced by immersed fixed three-dimensional surfaces acting as resistances on the fluid. The surfaces can be defined for example by the real geometry of the valves in their open and closed configurations. The opening and the closure states of the valves are controlled by variable resistance coefficients. The surfaces corresponding to the open position are of course equipped with a zero resistance – i.e. are "invisible" – when the valves are closed. Doing so, the geometrical domain seen by the fluid is realistic for the most part of the cardiac cycle. When the valves are closed, a non-zero resistance induces a jump of the stress across the surface. The latter is accurately captured using a mathematical formulation presented in a previous work on immersed stents: a fissure in the fluid mesh, on the valve surface, is introduced in order to allow for the pressure to be discontinuous at the interface only. Various simulations illustrate the efficiency and robustness of the method in complex fluid dynamics problems. The results obtained with this new model are also compared to the ones given by fully 3D FSI computations using the concept of Lagrangian Coherent Structures.