Mathematical modeling, analysis and numerical simulation, combined with imaging and experimental validation, provide a powerful tool for studying various aspects of cardiovascular treatment and diagnosis. This talk will address two examples where such a synergy led to novel results. The first example concerns a mathematical study of fluid-structure interaction (FSI) in blood flow with clinical application to 2D and 3D Doppler assessment of mitral regurgitation (MR). Our computational studies, performed in collaboration with several experts in echocardiography, addressed current imaging challenges in Doppler assessment of MR, which led to refinement and reinforcement of the emerging 3D echocardiographic applications. The second example concerns a novel dimension reduction/multi-scale approach to modeling of endovascular stents as 3D meshes of 1D curved rods forming a 3D network of 1D hyperbolic conservation laws. Our computational studies, motivated by the questions posed to us by cardiologists at the Texas Heart Institute, provided novel insight into the mechanical properties of 4 currently available coronary stents on the US market, and suggested optimal stent design for a novel application of stents in transcatheter aortic valve replacement.
The applications discussed above gave rise to new mathematical problems whose solutions required a development of sophisticated mathematical ideas. They include a design of a novel unconditionally stable, loosely coupled partitioned scheme for numerical simulation of solutions to FSI in blood flow, and the development of the theory and numerics for nonlinear hyperbolic nets and networks arising in dimension reduction of the stent problem. An overview of the basic mathematical ideas associated with this research, and application to the two related problems in cardiovascular diagnosis and treatment, will be presented. This talk will be accessible to a wide scientific audience.