Diffusion and Geometry in Modeling Biological Tissue

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I aim to use computer vision and mathematical modeling to better understand the role of geometry in biological composites, such as the mammalian heart wall, and its consequences for mechanical and elecrophysiological function. Myofibers are known to have a helical shape, with the constituent muscle cells, contracting and relaxing in concert to pump blood from the chambers of the heart. This local geometry causes the wringing and upward twisting motion familiar to anyone who has viewed a heart ultrasound scan. However, moving beyond the scale of individual myocytes, much less is known about the collective geometry of heart wall fibers. A deeper theoretical understanding of the properties conferred by myofiber arrangement will improve not only our understanding of heart function and disease caused by injury or other cardiomyopathies, but also that of other biological and man-made composites with helical fibers. ******With empirical fits to mammalian hearts imaged using diffusion Magnetic Resonance Imaging (dMRI), we have shown that heart wall myofibers appear to lie on a special type of minimal surface, the generalized helicoid (Savadjiev et al., PNAS 2012). Minimal surfaces arise in nature due to physical considerations, such as the shape taken on by a film of soap when one dips a wireframe curve into concentrated soap solution. We have posited that helicoidal arrangements of myocytes in the heart wall give it strength while optimizing mechanical function, and we have developed algorithms for fitting minimal surface based models to orientation data obtained from diffusion imaging.******In the current research proposal I consider the hypothesis that helicoidal arrangements of fibers optimize diffusion. Our generalized helicoid model provides parameters by which to describe the curvature of the space of myofibers in which diffusion occurs, following which stochastic diffusion models can be applied to the analysis of heart wall fiber orientation data. I shall examine electophysiological properties related to the contraction wave, as well as the management of potentially dangerous irregular wave patterns in such biological tissues. In parallel I shall look at the more challenging question of understanding the mechanical consequences of helicoidal fiber patterns in biological tissue. Examples from nature, such as the cuticle of an insect, suggest that helicoidal arrangements help dissipate forces from surface contact very efficiently. However, little has been done in terms of mathematical analysis to determine how this occurs, and further, unlike rigid composites, the heart wall is a dynamic deforming structure. I conjecture that the helicoidal arrangement offers optimality properties related to the efficacy of contraction while also dissipating forces within the wall to minimize wear and tear. The results of this research will improve our understanding of biological and man-made fibrous composites.