Minimal Surfaces in the Heart

密集的平行曲线集合的建模并不明显。例如，考虑一簇头发的几何形状。这不是一根头发弯曲的精确方式，而是一束头发弯曲和扭曲在一起的感觉。同样的观察也适用于生物学中的纤维复合材料，包括哺乳动物心脏壁中的肌肉纤维。尽管已知单个肌纤维以螺旋状缠绕在心脏的心室周围，但大量这样的纤维必须平滑且有规律地排列以在心脏跳动时作为集成功能单元来操作。目前的模型肌纤维的方向在整个心脏壁建议分组成片或带，但束的肌纤维的精确几何形状是未知的。我们最近表明，这种安排采取了一种特殊的最小表面，广义螺旋面的形式，关闭个别肌纤维和他们的集体壁结构之间的差距。从数学上讲，这解释了肌纤维如何在心脏壁中捆绑，同时在收缩时节省纤维长度并优化心室射血容量。这种类型的最小表面在自然界中出现，以优化物理资源。一个更熟悉的例子是将封闭的线框浸入肥皂溶液中所形成的薄膜形状。拟议的研究计划将调查最小表面模型的几个扩展和应用，包括计算解剖学：1）考虑适当的传输概念以允许单个最小表面的局部框架旋转到心脏中的相邻位置的位置，2）新的数值拟合技术的发展，使我们能够将模型应用于比我们目前考虑的更大的群体，和3）正常心脏中纤维几何形状与受梗塞和相关疾病发作影响的心脏中纤维几何形状的统计特征。最小曲面理论为分析心脏壁的纤维复合材料提供了一个新的基础，因此也可以更广泛地应用于心脏组织工程、计算解剖学中的形状分析以及计算机视觉中的相关问题。

The modeling of dense collections of close to parallel curves is not obvious. As an example, consider the geometry of a tuft of hair. It is not the precise manner in which a single hair strand curves that is relevant, but rather, it is the sense in which a bundle of strands bends and twists together that matters. The same observation holds for fibrous composites in biology, including muscle fibers in the mammalian heart wall. Whereas it is known that individual myofibers wind as helices around the ventricles of the heart, a multitude of such fibers must be arranged smoothly and regularly to operate as an integrated functional unit as the heart beats. Current models of myofiber orientation across the heart wall suggest groupings into sheets or bands, but the precise geometry of bundles of myofibers is unknown. We have recently shown that this arrangement takes the form of a special minimal surface, the generalized helicoid, closing the gap between individual myofibers and their collective wall structure. Mathematically this explains how myofibers are bundled in the heart wall while economizing fiber length and optimizing ventricular ejection volume as they contract. Minimal surfaces of this type arise in nature to optimize physical resources. A more familiar example is the shape taken by the film that results from dipping a closed wireframe in a solution of soap. The proposed research program shall investigate several extensions and applications of minimal surface models to computational anatomy including: 1) the consideration of an appropriate notion of transport to allow the local frame of a single minimal surface to be rotated to the location of neighboring positions in the heart, 2) the development of novel numerical fitting techniques to allow us to apply the model to larger populations than we have presently considered, and 3) the statistical characterization of the geometry of fibers in normal hearts versus that of hearts affected by infarctions and the onset of related diseases. Minimal surface theory provides a novel foundation for analyzing the fibrous composite of the heart wall, and thus could also be applied more broadly to heart tissue engineering, shape analysis in computational anatomy and related problems in computer vision.