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.

Abstract: I'll talk about the recently discovered strong negative dependence properties of the symmetric exclusion process, a model of non-intersecting random walkers. The negative dependence theory gives a simple way to show central limit theorems for the bulk motion of particles. Our results are general enough to deal with non-equilibrium systems of particles with inhomogeneous hopping rates.

Tchoukaillon is a member of the sowing family of board games that originated in Africa and Asia, of which mancala is the most commonly known. Tchoukaillon is a solitaire sowing game and sowing occurs when a player picks up stones in a particular bin and distributes them in adjacent bins. We will analyze interesting patterns, uncover multiple ways to find a winning board, apply linear algebra, discuss commutative diagrams, connect this game with Erdos, and conclude with the Chinese Remainder Theorem. You should also be slightly intrigued how game is defined!

Pizza and soda will be served!

Several recent coupled-physics medical imaging modalities aim to combine a high-contrast, low-resolution, modality with a high-resolution, low-contrast, modality and ideally offer high-contrast, high-resolution, reconstructions. Mathematically, these modalities involve the reconstruction of constitutive parameters in partial differential equations from knowledge of internal functionals of the parameters and solutions to said equations. This recent field of research is often referred to as Hybrid Inverse Problems.

This talk presents recent theoretical results of uniqueness, stability and explicit reconstructions for several hybrid inverse problems. We provide an explicit characterization of what can (and cannot) be reconstructed in coupled-physics imaging modalities such as Magnetic Resonance Elastography, Transient Elastography, Photo-Acoustic Tomography, and Ultrasound Modulation Tomography. Numerical simulations confirm the high-resolution, high-contrast, potential of these novel modalities.