General tools for diffusion tensor distributions, and matrix-variate Gamma approximation for multidimensional diffusion MRI data

Either on the voxel scale or within sub-voxel diffusion compartments, tissue microstructure can be described using a diffusion tensor distribution $$$\mathcal{P}(\mathbf{D})$$$. One way to resolve microstructural heterogeneity relies on choosing a plausible parametric functional form to approximate $$$\mathcal{P}(\mathbf{D})$$$. However, such a high-dimensional mathematical object is usually intractable. Here, we define matrix moments enabling the computation of diffusion metrics for any arbitrary functional choice approximating $$$\mathcal{P}(\mathbf{D})$$$. Applying these general tools to the matrix-variate Gamma distribution on the voxel scale, we obtain a new signal representation, the matrix-variate Gamma approximation, that we validate in vivo and in silico.