Minimal Surfaces in the Heart

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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.