A new method for mapping diffusivity profiles in tissue is presented. The Bloch‐Torrey equation is modified to include a diffusion term with an arbitrary rank Cartesian tensor. This equation is solved to give the expression for the generalized Stejskal‐Tanner formula quantifying diffusive attenuation in complicated geometries. This makes it possible to calculate the components of higher‐rank tensors without using the computationally‐difficult spherical harmonic transform. General theoretical relations between the diffusion tensor (DT) components measured by traditional (rank‐2) DT imaging (DTI) and 3D distribution of diffusivities, as measured by high angular resolution diffusion imaging (HARDI) methods, are derived. Also, the spherical tensor components from HARDI are related to the rank‐2 DT. The relationships between higher‐ and lower‐rank Cartesian DTs are also presented. The inadequacy of the traditional rank‐2 tensor model is demonstrated with simulations, and the method is applied to excised rat brain data collected in a spin‐echo HARDI experiment. Magn Reson Med 50:955–965, 2003.

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.