Diffusion and Geometry in Modeling Biological Tissue

我的目标是使用计算机视觉和数学建模来更好地理解几何在生物复合材料中的作用，例如哺乳动物的心壁，以及它对机械和电生理功能的影响。众所周知，肌纤维具有螺旋形状，其组成的肌肉细胞同时收缩和放松，从心腔中泵出血液。这种局部几何形状会导致任何看过心脏超声扫描的人都熟悉的扭曲和向上扭曲的运动。然而，超越单个心肌细胞的规模，人们对心壁纤维的集体几何形状知之甚少。从理论上更深入地了解肌纤维排列的特性，不仅可以提高我们对损伤或其他心肌疾病引起的心脏功能和疾病的理解，而且还可以加深我们对其他生物和人造螺旋纤维复合材料的理解。通过对使用扩散磁共振成像(DMRI)的哺乳动物心脏进行经验拟合，我们发现心壁肌纤维似乎位于一种特殊类型的极小表面上，即广义螺旋面(Savadjiev等人，PNAS 2012)。自然界中最小的表面是由于物理考虑而产生的，例如当一个人将线框曲线浸入浓缩肥皂溶液时，肥皂膜所呈现的形状。我们假设心肌细胞的螺旋排列在优化机械功能的同时给予它力量，我们开发了基于最小曲面的模型来拟合从扩散成像获得的取向数据的算法。在当前的研究方案中，我考虑了纤维的螺旋排列优化扩散的假设。我们的广义螺旋面模型提供了参数来描述发生扩散的肌纤维空间的曲率，随后随机扩散模型可以应用于心壁纤维取向数据的分析。我将研究与收缩波有关的电生理学特性，以及此类生物组织中潜在危险的不规则波的处理。同时，我将研究更具挑战性的问题，即了解生物组织中螺旋状纤维图案的力学后果。自然界的例子，如昆虫的角质层，表明螺旋排列有助于非常有效地消散表面接触的力。然而，在数学分析方面几乎没有做什么来确定这是如何发生的，而且，与刚性复合材料不同，心壁是一种动态变形结构。我推测，螺旋排列提供了与收缩效果相关的最佳特性，同时也分散了墙内的力，以将磨损降至最低。这项研究的结果将提高我们对生物和人造纤维复合材料的理解。

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.