Finslerian diffusion and the Bloch–Torrey equation

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings.An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownian motion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal.