Statistical Analysis of Diffusion Tensors in Diffusion-Weighted Magnetic Resonance Imaging Data

Abstract

Diffusion tensor imaging has been widely used to reconstruct the structure and orientation of fibers in biological tissues, particularly in the white matter of the brain, because it can track the effective diffusion of water along those fibers. The raw diffusion-weighted images from which diffusion tensors are estimated, however, inherently contain noise. Noise in the images produces uncertainty in the estimation of the tensors (which are 3 × 3 positive-definite matrices) and of their derived quantities, including eigenvalues, eigenvectors, and the fiber pathways that are reconstructed based on those tensor elements. The aim of this article is to provide a comprehensive theoretical framework of statistical inference for quantifying the effects of noise on diffusion tensors, on their eigenvalues and eigenvectors, and on their morphological classification. We propose a semiparametric model to account for noise in diffusion-weighted images. We then develop a one-step, weighted least squares estimate of the tensors and justify use of the one-step estimates based on our theoretical framework and computational results. We also quantify the effects of noise on the eigenvalues and eigenvectors of the estimated tensors by establishing their limiting distributions. We construct pseudo-likelihood ratio statistics to classify tensor morphologies. Simulation studies show that our theoretical results can accurately predict the stochastic behavior of the estimated eigenvalues and eigenvectors, as well as the bias that is introduced by sorting the eigenvalues by their magnitudes. Implementation of these methods is illustrated in a diffusion-weighted dataset from seven healthy human subjects.