Generalized scalar measures for diffusion MRI using trace, variance, and entropy

This paper details the derivation of rotationally invariant scalar measures from higher-rank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and even to functions whose domains are the unit sphere. It is shown that the mean diffusivity is invariant to the selection of tensor rank for the model used. However, this rank invariance does not apply to the anisotropy measures. Therefore, a variance-based, general anisotropy measure is proposed. Also an information theoretical parametrization of anisotropy is introduced that is frequently more consistent with the meaning attributed to anisotropy. We accomplished this by associating anisotropy with the amount of orientational information present in the data, regardless of the imaging technique used. Using a simplified model of fibrous tissue, we simulated anisotropy values with varying orientational complexity and tensor models. Simulations suggested that a lower-rank tensor model may produce artificially low anisotropy values in voxels with complex structure. This was confirmed with a spin-echo experiment performed on an excised rat brain. Magn Reson Med 53:866-876, 2005. (c) 2005 Wiley-Liss, Inc.