Minimal Surfaces in the Heart

对接近平行曲线的密集集合的建模并不明显。例如，考虑一束头发的几何体。重要的不是一根发丝弯曲的准确方式，而是一束发丝弯曲和扭曲在一起的感觉。同样的观察结果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。虽然众所周知，单个肌纤维以螺旋的形式缠绕在心脏的脑室周围，但大量这样的纤维必须顺畅和规则地排列，才能在心脏跳动时作为一个完整的功能单位运行。目前的跨心壁肌纤维取向模型表明，肌纤维成组成片或带，但肌纤维束的精确几何形状尚不清楚。我们最近已经证明，这种排列形式是一种特殊的极小曲面--广义螺旋面，它弥合了单个肌纤维和它们的集体壁结构之间的间隙。从数学上讲，这解释了肌纤维是如何在心壁中捆绑在一起的，同时节省了纤维长度，并在收缩时优化了心室射血量。这种类型的最小曲面本质上是为了优化物理资源而产生的。一个更熟悉的例子是胶片的形状，它是将封闭的线框浸入肥皂溶液中产生的。拟议的研究计划将研究最小表面模型在计算解剖学中的几个扩展和应用，包括：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相邻位置，2)开发新的数值拟合技术，使我们能够将该模型应用于比我们目前考虑的更大的人群，以及3)正常心脏中纤维的几何形状与受梗塞和相关疾病影响的心脏的几何统计特征。最小表面理论为分析心壁的纤维复合材料提供了新的基础，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.