Diffusional attenuation of the magnetic resonance (MR) signal as a result of the mixing of phase incoherent spins has been known since Hahn (1) first introduced spin echoes in the early days of MR. When the diffusion process is non-Fickian (i.e., the molecular flux density is not oriented opposite to the concentration gradient), the diffusivity is better quantified with a symmetric, rank-2, positive definite tensor (2). At typical resolutions for MRI and microscopy, it has been shown that a macroscopic effective diffusion tensor (DT; henceforth referred to as the “traditional” rank-2 DT) can be calculated that is assumed to have properties similar to those of the true DT (3). The traditional DT has six distinct components, implying that six independent numbers are needed to fully describe this tensor. These six numbers can be chosen to be quantities that are more meaningful than the tensor components. As an example, the principal eigenvector, which can be expressed in terms of two numbers (such as the azimuthal and polar angles that specify a direction in three-dimensional space), has been hypothesized to give the local fiber orientation within the tissue (4). Two other numbers, the mean diffusivity and a measure of anisotropy, were found to be useful in quantitative studies in which comparative analyses were performed (5).

The inability of traditional DT imaging (DTI) to resolve more than one fiber direction in a voxel has prompted recent interest in formulating more sophisticated techniques. Tuch et al. (6) developed a clinically feasible approach called high-angular-resolution diffusion imaging (HARDI), in which apparent diffusion coefficients are measured along many directions distributed almost isotropically on the surface of a sphere. In a recent publication (7), we expressed the diffusivities in terms of Cartesian tensors of rank higher than 2 that enabled a straightforward generalization of the Bloch-Torrey result (8) and led to the formulation of a generalized Stejskal-Tanner (9) equation:

where **u** is a unit vector that specifies the direction of the diffusion gradients, whose components are given by

where θ and ϕ are the polar and azimuthal angles, respectively. In this approach, a model independent diffusivity profile is obtained when *l* = ∞, and when *l* = 2, one recovers the signal attenuation relation for traditional DTI. Consequently, we term this approach “generalized DTI”1. To utilize Eq. [1] for a rank-*l* tensor, it is sufficient to have a HARDI-type data set in which the number of directions should be greater than (*l* + 1)(*l* + 2)/2. Alternatively, similarly to traditional DTI, one may sample several concentric spheres (corresponding to different *b*-values). This formalism provides a particularly simple methodology for constructing generalized indices because functions with increasing complexity are generated by the increase in the rank of the selected tensor model. Therefore, an index that is formulated in terms of an arbitrary rank-*l* tensor can be applied to traditional DTI (with the substitution *l* = 2), generalized DTI (*l* >2, *l* even), and arbitrary functions (in the *l* → ∞ limit).

Many of the previously introduced scalar indices assume that the model being used is traditional (rank-2) DTI (4, 11–15). The failure of traditional DTI in the presence of orientational heterogeneities may create problems not only in the determination of fiber orientations, but also in the calculated scalar measures. Figure 1 shows simulations of a region with crossing fibers. It is evident from the second image that an anisotropy index based on rank-2 DTI, such as fractional anisotropy (FA) (16), produces significantly low values when there are fiber crossings. This is partly because there is less orientational variation in the diffusivities in these regions. However, traditional DTI suffers from a further reduction of anisotropy values as a result of the excessive smoothing introduced by the employment of the rank-2 tensor. This is apparent in the diffusivity profiles implied by the rank-2 and rank-6 DTs, as presented in the two rightmost images in Fig. 1. Since most clinical studies quantify DTI by the mean diffusivity and a measure of anisotropy, a reexamination of the derivation of these indices is of great importance.

The variance of diffusivities, measured along different directions with the HARDI method, was previously proposed as an anisotropy index (17) that does not assume the rank-2 tensor model. This approach has a number of problems. First, the calculated “anisotropy” maps have the same units as diffusivity, and the images produced are diffusivity-weighted. This may create problems in the interpretation of contrast (and lack of contrast) seen in the images. Second, the range of values this index can take is unclear, which makes it difficult to scale the images in a consistent way. Also, since this approach uses only the discrete samples of the diffusivity profile, the computed values depend on the distributions of the gradient vectors on the unit sphere. Note that this distribution is never truly isotropic except when the directions are specified by Platonic solids (18). However, the formulation of the variance does not take the sampling scheme into account, and treats diffusivities calculated along each direction in the same way. Imperfections in the distribution of points on the sphere is less of a concern in model-based approaches because the sampling strategy is taken into consideration in the fitting step. Similarly, since this measure of anisotropy is derived from discrete samples with no functional fit, one can expect this index to be very sensitive to noise.

In this work, we revisit the problem of quantification of mean diffusivity and anisotropy with the purpose of formulating measures generalized to higher-rank tensors and to functions whose domains are the unit sphere. We show that the commonly used expression for mean diffusivity 〈*D*〉 is a model-independent measure and is simply equal to the rank-0 tensor. This is not true for the anisotropy measures (e.g., FA and relative anisotropy (RA)) that are functions of the variance of the eigenvalues of the DT, which is not equal to the variance of the diffusivities along all directions. Therefore, we construct a generalized anisotropy (GA) index that is based on the variance of the normalized diffusion coefficients, and a scaled entropy (SE) index that treats the function as a probability distribution function (PDF). The normalization step utilizes a generalized expression for the trace operation and removes the undesired diffusion weighting from the resulting images. The construction of both SE and GA indices ensure that the resulting values are in the interval [0,1), 0 corresponding to the isotropic profile. We provide exact expressions for 〈*D*〉 and GA indices for tensors up to rank-6, and present images of these measures for a HARDI data set from an excised rat brain.

Using simulations of a simplified model of fibrous tissue, we show that anisotropy measures calculated from a rank-2 tensor model may be significantly smaller than the anisotropy values calculated from higher-rank tensors, when there is more than one fiber direction. We show several regions in our data in which this effect is significant.

We discuss whether a simple calculation of an information theoretical anisotropy index (for the case of a rank-2 tensor) is possible, and formulate an index called visual anisotropy (VA) that is defined in terms of the von Neumann entropy. Just as RA and FA can be thought of as the simple implementation of a variance-based anisotropy index in the case of a rank-2 tensor, VA is the corresponding simple implementation of an entropy-based index for a rank-2 tensor. Finally, we discuss several information theoretical scalar measures that may be useful in the comparison of two functions whose domains are the unit sphere. Although we present our results using diffusivity profiles, the formulations remain valid for any other positive-valued data acquired on or projected onto the surface of a unit sphere.