Bayesian uncertainty quantification in linear models for diffusion MRI

TL;DR

This paper introduces a Bayesian framework for linear diffusion MRI models, enabling accurate uncertainty quantification of derived features, validated through simulations and in vivo data, enhancing group analysis reliability.

Contribution

It recasts popular dMRI models as Bayesian models, providing closed-form posterior distributions for affine functions of coefficients, and validates uncertainty estimates with simulations and real data.

Findings

Bayesian interpretation allows uncertainty quantification in dMRI models.

Theoretical quantiles match empirical results, validating the approach.

Uncertainty maps improve group analysis by downweighting uncertain subjects.

Abstract

Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue microstructure. By fitting a model to the dMRI signal it is possible to derive various quantitative features. Several of the most popular dMRI signal models are expansions in an appropriately chosen basis, where the coefficients are determined using some variation of least-squares. However, such approaches lack any notion of uncertainty, which could be valuable in e.g. group analyses. In this work, we use a probabilistic interpretation of linear least-squares methods to recast popular dMRI models as Bayesian ones. This makes it possible to quantify the uncertainty of any derived quantity. In particular, for quantities that are affine functions of the coefficients, the posterior distribution can be expressed in closed-form. We simulated measurements from single- and double-tensor models where the correct values of several quantities are known, to validate that the theoretically derived quantiles agree with those observed empirically. We included results from residual bootstrap for comparison and found good agreement. The validation employed several different models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI) and Constrained Spherical Deconvolution (CSD). We also used in vivo data to visualize maps of quantitative features and corresponding uncertainties, and to show how our approach can be used in a group analysis to downweight subjects with high uncertainty. In summary, we convert successful linear models for dMRI signal estimation to probabilistic models, capable of accurate uncertainty quantification.