Minimal Surfaces in the Heart

接近平行曲线的密集集合的建模不明显。举个例子，考虑一簇头发的几何形状。重要的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的肌肉纤维。虽然我们知道单个肌纤维像螺旋一样绕着心脏的心室旋转，但大量的肌纤维必须平滑而有规律地排列，才能作为一个完整的功能单元在心脏跳动时运行。目前的心壁肌纤维定向模型表明，肌纤维分为片状或带状，但肌纤维束的精确几何形状尚不清楚。我们最近的研究表明，这种排列采用了一个特殊的最小表面的形式，即广义螺旋面，关闭了单个肌纤维与其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捆绑在心壁上的，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后形成的薄膜形状。建议的研究计划应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)考虑一个合适的运输的概念,允许一个极小曲面的局部框架旋转到心中的位置邻近的位置,2)新颖的数值拟合技术的发展允许我们将模型应用到更大的人口比我们目前认为,和3)的统计特征的几何纤维在正常的心与心受到梗死和相关疾病的发病的影响。最小表面理论为分析心脏壁纤维复合材料提供了新的基础，因此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.