The Analytic Distribution of Fractional Anisotropy in Diffusion Mri

INTRODUCTION: Fractional anisotropy (FA) is a popular scalar measure of voxelwise diffusion characteristics, computed by a nonlinear transformation of the tensor eigenvalues [1]. Despite its ubiquitous use in diffusion MRI studies, the statistical distribution of FA has remained unknown; statistical inference on FA images has either involved a Gaussianity assumption, known to be invalid given the finite support of FA, or relied on computationally expensive non-parametric resampling approaches [2,3], that can suffer from inaccurate uncertainty estimates [4,5]. In the analytic realm, perturbation analysis has been used to derive approximate uncertainties [6], while FA has been examined with respect to relative anisotropy [7], a chi-square distributed measure [2]. We present an analytic solution for the FA distribution. Under the assumption of Gaussian distributed eigenvalues, FA is shown to follow a transformed doubly non-central beta distribution. Generalisation to arbitrary eigenvalue distributions results in a mixture of transformed doubly non-central beta distributions. Inferential statistical tests, valid for arbitrary experimental conditions, can now be derived from these analytic expressions. THEORY: The relationship between tensor eigenvalues and FA geometrically describes a cone with axis [1,1,1] and aperture set by the FA value. We derive the analytic expression for the distribution of FA, p(f), under the assumption of Gaussian distributed tensor eigenvalues, λj ~ N(μj,σ), j=1,2,3, through rotation of the cone to align with the z-axis, equivalent to a change of variables, and marginalisation of the joint probability density: