Matrix moments of the diffusion tensor distribution and matrix-variate Gamma approximation

Abstract

Diffusion MRI allows for the non-invasive investigation of the microscopic architecture of biological tissues in vivo. While being sensitive to microstructural tissue changes, diffusion MRI yields poor specificity regarding the cause of these changes. This specificity can be enhanced by using (i) signal representations that retrieve global diffusion properties of the voxel content from the features of the measured diffusion signal or (ii) models that relate these features to predefined sub-voxel tissue-specific compartments. Either on the voxel scale or within diffusion compartments, tissue microstructure can be described using a diffusion tensor distribution (DTD) P(D), which captures a collection of microscopic diffusion environments. A convenient way to estimate DTDs relies on choosing a plausible parametric functional form approximating them. In particular, the mean diffusion tensor 〈D〉 and covariance tensor C associated with this choice enable derivation of measures quantifying specific microstructural diffusion properties. However, high-dimensional DTDs can be quite intractable in practice. In this tutorial paper, we present the matrix moments of P(D), which allow computation of the mean diffusion tensor, covariance tensor and higher-order statistical tensors for any arbitrary functional choice approximating P(D). In turn, these statistical tensors are related to common diffusion measures, such as the mean diffusivity, microscopic diffusion anisotropy and variance of isotropic diffusivities. Using the well-known case of the matrix-variate Gaussian distribution to validate the matrix moments, we then use these tools on the more complex non-central matrix-variate Gamma distribution, computing its covariance tensor for the first time. Finally, we utilize this covariance tensor to design a new voxel-scale signal representation, the matrix-variate Gamma approximation, which we evaluate in vivo and compare to q-space trajectory imaging (QTI) in silico. While the matrix-variate Gamma approximation possesses important limitations explained by the structure of its newly computed covariance tensor, it presents a greater accuracy than QTI in systems exhibiting high orientational order, which suggests that non-central matrix-variate Gamma distributions appropriately describe anisotropic diffusion compartments, as in the DIAMOND model. The framework of matrix moments promotes a more widespread use of matrix-variate distributions as plausible approximations of the DTD by alleviating their intractability, thereby facilitating their design and validation.