Geodesic Tubes for Uncertainty Quantification in Diffusion MRI

Abstract

Based on diffusion tensor imaging (DTI), one can construct a Riemannian manifold in which the dual metric is proportional to the DTI tensor. Geodesic tractography then amounts to solving a coupled system of nonlinear differential equations, either as initial value problem (given seed location and initial direction) or as boundary value problem (given seed and target location). We propose to furnish the tractography framework with an uncertainty quantification paradigm that captures the behaviour of geodesics under small perturbations in (both types of) boundary conditions. For any given geodesic this yields a coupled system of linear differential equations, for which we derive an exact solution. This solution can be used to construct a geodesic tube, a volumetric region around the fiducial geodesic that captures the behaviour of perturbed geodesics in the vicinity of the original one.KeywordsDiffusion tensor imagingUncertainty quantificationGeodesic tractographyGeodesic deviationRiemannian geometry