The Asymptotic Behavior of the Nonlinear Estimators of the Diffusion Tensor and Tensor-Derived Quantities with Implications for Group Analysis

Diffusion tensor imaging (DTI) is a quantitative magnetic resonance imaging (MRI) method that is used to study the microstructural properties of white matter in the brain. Tensor-derived quantities, such as the trace and fractional anisotropy (FA), are important for characterizing the normal, diseased, and developing brain. Consequently, determining the statistical properties of the diffusion tensor estimator is important to assess whether the method of estimation is appropriate and to describe how noise in diffusion-weighted images affects the variability in estimates of the tensor and functions of the tensor estimate. Here we derive asymptotic properties of the nonlinear least squares estimator (NLSE) of the diffusion tensor. We show that the NLSE is consistent and asymptotically normal. To illustrate and validate this framework we derive the asymptotic distributions of trace and FA. We show, with simulations, experimental designs that have asymptotic distributions that are very close to the empirical distributions. The methods described in this paper are applied to estimate the variances of trace and FA in a healthy human volunteer. The variance of the trace and FA are found to vary significantly throughout the brain. This renders many popular tests used in group analysis invalid for DTI data. Unequal variances for statistical tests with tensor-derived quantities is discussed.