Matrix moments of the diffusion tensor distribution

TL;DR

This paper develops mathematical tools based on matrix moments to better approximate the diffusion tensor distribution in diffusion MRI, introducing a new matrix-variate Gamma model and evaluating its effectiveness.

Contribution

It establishes practical methods for computing matrix moments of the DTD, applies them to the nc-mv-Gamma distribution, and introduces a new matrix-variate Gamma approximation for diffusion heterogeneity.

Findings

The matrix-variate Gamma approximation captures some heterogeneity but not all, such as orientation dispersion.

The covariance tensor structure explains limitations in modeling certain diffusion features.

Matrix moments facilitate the use of matrix-variate distributions in diffusion MRI analysis.

Abstract

Purpose: To facilitate the implementation/validation of signal representations and models using parametric matrix-variate distributions to approximate the diffusion tensor distribution (DTD) $\mathcal{P}(\mathbf{D})$. Theory: We establish practical mathematical tools, the matrix moments of the DTD, enabling to compute the mean diffusion tensor and covariance tensor associated with any parametric matrix-variate DTD whose moment-generating function is known. As a proof of concept, we apply these tools to the non-central matrix-variate Gamma (nc-mv-Gamma) distribution, whose covariance tensor was so far unknown, and design a new signal representation capturing intra-voxel heterogeneity via a single nc-mv-Gamma distribution: the matrix-variate Gamma approximation. Methods: Furthering this proof of concept, we evaluate the matrix-variate Gamma approximation in silico and in vivo, in a human-brain 'tensor-valued' diffusion MRI dataset. Results: The matrix-variate Gamma approximation fails to capture the heterogeneity arising from orientation dispersion and from simultaneous variances in the trace (size) and anisotropy (shape) of the underlying diffusion tensors, which is explained by the structure of the covariance tensor associated with the nc-mv-Gamma distribution. Conclusion: The matrix moments promote a more widespread use of matrix-variate distributions as plausible approximations of the DTD by alleviating their intractability, thereby facilitating the design/validation of matrix-variate microstructural techniques.