Minimal Surfaces in the Heart
Speaker: Kaleem Siddiqi, School of Computer Science, McGill University
Dense collections of close to parallel curves or streamlines are ubiquitous in nature. Examples include hair patterns, zebra stripes and fiber tracts in the human brain. Abstract representations which capture their collective geometric structure are non-trivial to construct. Using Cartan’s method of moving frames we show how this can be done, and we describe algorithms to apply differential geometric constructions to the modeling of muscle fibers in the mammalian heart. Here we show that this fiber arrangement takes the form of a special minimal surface, the generalized helicoid, closing the gap between individual myofibers and their collective wall structure. The model holds across species, with a smooth variation in its three curvature parameters within the myocardial wall providing tight fits to diffusion magnetic resonance images from the rat, the dog and the human. Mathematically it explains how myofibers are bundled in the heart wall while economizing fiber length and optimizing ventricular ejection volume as they contract. Our analysis provides a novel foundation for analyzing the fibrous composite of the heart wall and should therefore find applications in heart tissue engineering and in the study of heart muscle diseases.
Related Publications:
Heart Wall Myofibers are Arranged in Minimal Surfaces to Optimize Organ Function. P. Savadjiev, G. J. Strijkers, A. J. Bakermans, E. Piuze, S. W. Zucker & K. Siddiqi. Proceedings of the National Academy of Science, 109(24):9248-9253, 2012.
Maurer-Cartan Forms for Fields on Surfaces: Applications to Heart Fiber Geometry. E. Piuze, J. Sporring, K. Siddiqi. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI). 2015
Bio:
Kaleem Siddiqi received the BS degree from Lafayette College in 1988 and the MS and PhD degrees from Brown University in 1990 and 1995, respectively, all in the field of electrical engineering. He is currently Professor, William Dawson scholar, and an associate director research of the School of Computer Science at McGill University. He is also a member of McGill’s Centre for Intelligent Machines. Before moving to McGill in 1998, he was a postdoctoral associate in the Department of Computer Science at Yale University (1996-1998) and held a visiting position in the Department of Electrical Engineering at McGill University (1995-1996). His research interests include the areas of computer vision and medical imaging. He is a member of the Phi Beta Kappa, Tau Beta Pi, and Eta Kappa Nu. He is a senior member of the IEEE.
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