Minimal Surfaces in the Heart

心脏的最小表面

• 批准号: 183831-2013
• 负责人: Siddiqi, Kaleem
• 金额: $ 2.62万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=634326
• 关键词: Minimal; Surfaces; Heart

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.

接近平行曲线的密集收集的建模并不明显。例如，考虑一簇头发的几何形状。与之相关的单个头发曲线曲线不是一种确切的方式，而是重要的是，一束链弯曲和曲折的意义很重要。对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。众所周知，单个肌纤维作为心脏心室周围的螺旋螺旋，而许多此类纤维必须平稳地布置，以作为心脏跳动时作为集成功能单元作为一个集成的功能单元。当前的肌纤维取向的模型横跨心脏壁，表明将分组分成板或乐队，但是肌纤维束的精确几何形状尚不清楚。我们最近表明，这种布置采用了特殊的最小表面（广义螺旋）的形式，缩小了单个肌纤维及其集体壁结构之间的间隙。从数学上讲，这解释了如何将肌纤维捆绑在心脏壁上，同时节省纤维长度并在收缩时优化心室射血量。这种类型的最小表面在本质上是出现的，以优化物理资源。一个更熟悉的例子是胶片所取的形状，该形状是将封闭的线框浸入肥皂溶液中。拟议的研究计划应调查最小表面模型到计算解剖结构的几个扩展和应用，包括：1）考虑适当的运输概念，以允许将单个最小表面的局部框架旋转到心脏中相邻位置的位置，2）2）与我们在数值拟合技术中的开发相比，与我们在较大的数字拟合技术中的开发相比，我们的统计范围与较大的统计范围相比，以及3）的特征，以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），以及3），3）正常的心与受梗死影响和相关疾病发作的心脏的心脏与心脏的心脏。最小的表面理论为分析心脏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，计算解剖结构的形状分析以及计算机视觉中的相关问题。