Quantifying error in estimates of human brain fiber directions using Earth Mover’s Distance

Diffusion-weighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber orientation distribution function (fODF) in different parts of the brain. These functions describe the relative abundance of populations of axonal fibers crossing each other in each location. Multiple models exist for estimating fODFs. The quality of the resulting estimates can be quantified by means of a suitable measure of distance on the space of fODFs. However, there are multiple distance metrics that can be applied for this purpose, including smoothed Lp distances and the Wasserstein metrics. Here, we give four reasons for the use of the Earth Mover’s Distance (EMD) equipped with the arc-length, as a distance metric. First, the EMD is an extension of the intuitive angular error metric, often used in the DWI literature. Second, the EMD is equally applicable to continuous fODFs or fODFs containing mixtures of Dirac deltas. Third, the EMD does not require specifying smoothing parameters. Finally, the EMD is useful in practice, as well as in simulations. This is because the error of an estimated fODF, as quantified by the EMD of this fODF from the ground truth is correlated with the replicate error: the EMD between the fODFs calculated on two repeated measurements. Though we cannot calculate the error of the estimate directly in experimental data measured in vivo (in contrast to simulation in which ground truth is known), we can use the replicate error, computed using repeated measurements, as a surrogate for the error. We demonstrate the application of computing the EMD-based replicate error in MRI data, creating anatomical contrast that is not observed with an estimate of model prediction error.