Applied Mathematics Seminar

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Cardiac fluid dynamics fundamentally involves interactions between complex blood flows and the structural deformations of the muscular heart walls and the thin, flexible valve leaflets. This talk will provide an overview of modern immersed boundary type methods for modeling such fluid structure interactions along with their application to cardiac flow problems. I will focus on numerical methods for these interactions that treat deformable immersed structures whose mechanical responses are described by constitutive models that can be parameterized using experimental or clinical data. I will then present models of in vitro pulse duplicator systems commonly used in the development and regulation of prosthetic heart valves, which enable detailed comparisons between experimental measurements and computational predictions but rely on highly simplified representations of cardiac anatomy and physiology. Using these models, I will describe investigations of the key determinants of prosthetic heart valve dynamics and studies of intracardiac flow dynamics associated with different forms of mitral valve disease. Moving beyond these simplified settings, I will introduce recent in vivo models, including a patient-specific model of transcatheter aortic valve replacement and a comprehensive model of the human heart. This heart model incorporates fully three-dimensional descriptions of all major cardiac structures together with biomechanical descriptions parameterized using experimental tensile test data obtained exclusively from human tissue specimens. Simulation results demonstrate that the model produces physiological stroke volumes, pressure-volume loops, valvular pressure-flow relationships, and vortex formation times, illustrating its potential for predicting cardiac function in both health and disease. Time permitting, I will conclude by describing ongoing extensions of this framework to incorporate detailed cardiac electrophysiology and electromechanical coupling.

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The structure of microbial communities inherently reflects the mechanics of their environment. In particular, mechanical feedback between growth, stress, and transport can drive instabilities and patterning across scales. In this talk, I present a mathematical framework for understanding how mechanical feedback shapes the morphology of growing microbial communities. We first consider a model bacterial colony growing atop a frictional substrate in which individual cell growth and division is inhibited by collective growth-induced stresses. Particle simulations show the spontaneous emergence of concentric ring patterns in cell size, and a multiscale continuum theory linking single-cell stress responses to collective mechanics shows this patterning arises from stress accumulated over many cell cycles. Next, we consider a colony growing atop a viscous fluid and fueled by the consumption of a suspended nutrient. Here fluid flow is generated by both growth-induced stresses at the surface and nutrient-dependent density gradients in the bulk. Reformulating this problem as an integro-differential equation, we establish a condition for the morphological instability of axisymmetric colonies which reveals a competition between stabilizing growth stresses and destabilizing buoyant flows. These results show how mechanical feedback between growth and stress governs the structure and morphology of growing biological systems.