Formal characterization and extension of the linearized diffusion tensor model.

We analyzed the properties of the logarithm of the Rician distribution leading to a full characterization of the probability law of the errors in the linearized diffusion tensor model. An almost complete lack of bias, a simple relation between the variance and the signal-to-noise ratio in the original complex data, and a close approximation to normality facilitated estimation of the tensor components by an iterative weighted least squares algorithm. The theory of the linear model has also been used to derive the distribution of mean diffusivity, to develop an informative statistical test for relative lack of fit of the ellipsoidal (or spherical) model compared to an unrestricted linear model in which no specific shape is assumed for the diffusion process, and to estimate the signal-to-noise ratios in the original data. The false discovery rate (FDR) has been used to control thresholds for statistical significance in the context of multiple comparisons at voxel level. The methods are illustrated by application to three diffusion tensor imaging (DTI) datasets of clinical interest: a healthy volunteer, a patient with acute brain injury, and a patient with hydrocephalus. Interestingly, some salient features, such as a region normally comprising the basal ganglia and internal capsule, and areas of edema in patients with brain injury and hydrocephalus, had patterns of error largely independent from their mean diffusivities. These observations were made in brain regions with sufficiently large signal-to-noise ratios (>2) to justify the assumptions of the log Rician probability model. The combination of diffusivity and its error may provide added value in diagnostic DTI of acute pathologic expansion of the extracellular fluid compartment in brain parenchymal tissue.