Moving Frames for Heart Fiber Geometry

Elongated cardiac muscle cells named cardiomyocytes are densely packed in an intercellular collagen matrix and are aligned to helical segments in a manner which facilitates pumping via alternate contraction and relaxation. Characterizing the geometrical variation of their groupings as cardiac fibers is central to our understanding of normal heart function. Motivated by a recent abstraction by Savadjiev et al. of heart wall fibers into generalized helicoid minimal surfaces, this paper develops an extension based on differential forms. The key idea is to use Maurer-Cartan’s method of moving frames to study the rotations of a frame field attached to the local fiber direction. This approach provides a new set of parameters that are complimentary to those of Savadjiev et al. and offers a framework for developing new models of the cardiac fiber architecture. This framework is used to compute the generalized helicoid parameters directly, without the need to formulate an optimization problem. The framework admits a straightforward numerical implementation that provides statistical measurements consistent with those previously reported. Using Diffusion MRI we demonstrate that one such specialization, the homeoid, constrains fibers to lie locally within ellipsoidal shells and yields improved fits in the rat, the dog and the human to those obtained using generalized helicoids.