Functioning of the Mitral Valve

The mitral valve is of fundamental importance for controlling the flow in and out of the left ventricle in the heart. Modeling of the mitral valve is a multi-physics, multi-scale problem of extreme complexity. The valve itself is a thin, anisotropic elastic (or viscoelastic) structure, subject to active stress regulated by electrophysiological processes, and coupled to the oscillating blood flow and the mechanical deformation of the heart. The closing of the valve, in a fluid-structure interaction (FSI) setting, is an extremely ambitious numerical problem. Outflow boundary conditions constitute another considerable challenge, which is best met by coupling the flow in the heart to 1D circulatory models. Despite the many difficulties, several research components within CBC are soon in place to allow building a more complete model for the mitral valve and its interaction with the surroundings. Such a model is likely to establish new knowledge on how the mitral valve functions and thereby how surgery and artificial valves can be improved. Moreover, this type of model is an important step toward a full model of the heart that links multi-scale electro-mechanics with valves and blood flow, and thereby enables totally new applications to cardiovascular surgery and interventional cardiology.

Through the long-term, solid work by CBC@NTNU over several years, we have already established a validated soft tissue model of the mitral valve. The CBC@NTNU group is now approaching the full 3D FSI problem of the valve moving in blood flow through different paths, using mathematical models of diverse complexity. Examples are models with a rigid valve, or a beam-like valve, combined with image-deduced (prescribed) boundary motion of the heart. Activity on 1D circulatory models has recently been started. These models also have many exciting clinical applications on their own, e.g., for planning of by-pass operations. CBC@NTNU has very well developed collaborations with medical doctors at St. Olavs Hospital in Trondheim and world-leading scientists at NTNU in medical ultrasound imaging.

CBC@Simula has implemented validated models of left ventricular electro-mechanics for healthy and infarcted hearts, and we are in the process of implementing more efficient and flexible models in FEniCS. Along with the flexible hyperelasticity solver in FEniCS and new developments on overlapping and non-matching finite element meshes, we have computational developments that can ease the implementation of the challenging models described above. The experience with exploring localized turbulence in blood and CSF flow will be of particularly high relevance for the blood flow in the heart. In addition, our tools for error control and uncertainty quantification have a significant potential for the clinical applications of the models.