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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.  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.
Generalization of the Trace Operator
The trace of a rank-2 tensor is given by the sum of diagonal elements of the tensor, when it is represented by a matrix. It is straightforward to show that the trace is rotationally invariant. The trace can also be expressed as the integral of the quadratic forms of the tensor (19):
where u is a unit vector, A is any rank-2 tensor, and S is the unit sphere. Note that since the integrand in the above expression has antipodal symmetry, Eq.  is also valid when the integral is evaluated on the unit hemisphere that is denoted by Ω. In this case, the coefficient before the integration should be replaced by 3/2π:
It is possible to generalize this operation to functions defined on the unit sphere because the quadratic form uTAu is a function on the unit sphere. We will denote this generalized trace operation as “gentr.” For functions f(u), with antipodal symmetry on the unit sphere, this operation is given by
The mean diffusivity index has been proposed as an orientation-independent measure of diffusion, in the context of traditional (rank-2) DTI (16). According to its original definition, it is given by the mean of the eigenvalues of the rank-2 DT, which is just the trace of the tensor divided by 3. Since diffusivity along a direction u, assumed by traditional DTI, is (20):
it is clear from Eq.  that the original definition of mean diffusivity is just
This shows that the mean diffusivity index is not just the mean value of the three eigenvalues or that of the diffusivities along three orthogonal directions, but is also the mean value of diffusivities along all directions implied by rank-2 DTI. Note that a comparison of Eqs.  and  implies that the mean value of a function defined on the unit sphere is just one-third of its generalized trace:
When an arbitrary rank-l Cartesian tensor is used, the diffusivities are given by (7):
and the corresponding mean diffusivity value becomes
where u is the ik-th component of the vector u as given in Eq. . The integral in the above expression can be evaluated analytically. The resulting 〈D〉 values for ranks up to 6 are provided in Table 1.
Table 1. Mean Diffusivity Values in Terms of the Components of the Higher Rank Tensors Through Rank Six
Note that one can also derive these expressions by incrementally using the expressions relating the components of a higher-rank tensor to those of lower-rank tensors, as presented in Table 2 of Ref. 7. This is done by using the expressions in this table from the bottom toward the top. Therefore, the mean diffusivity value is just the diffusion coefficient that can be calculated from the fitting of the HARDI data to the isotropic Stejskal-Tanner equation. Furthermore, since the same result can be obtained by starting from any of the rank-l tensors, the mean diffusivity is a model-independent measure of diffusivity.
Anisotropy as the Variance of the Normalized Diffusivities
Despite the inflation in the number of anisotropy indices already proposed, two of them (FA and RA) are the ones most widely used (16). These measures can be expressed in the following new forms:
where R is the “normalized” and unitless rank-2 DT (21):
Therefore, FA and RA are simply functions of the trace of the square of this tensor. When D is diagonalized, this quantity is related to the variance of the eigenvalues of the normalized rank-2 tensor. Unlike the case of mean diffusivity, this expression is not model-independent, i.e., trace(R2) is not equal to the average of the square of the diffusivities along all directions. In this work, we propose to use the variance of the normalized diffusivities, and show how this measure can be easily generalized to higher-rank tensors and to functions defined on the unit sphere.
We start by defining the normalized diffusivity function as
This step can be thought of as the generalization of the transition from D to R in Eq. . Once this is done, instead of trace(R2), we can use the quantity gentr(DN(u)2). When a rank-l tensor model is used, this quantity can be shown to be given by
where Nl = (l + 1)(l + 2)/2 is the number of unique elements of the rank-l tensor, μ=l!/n1x!n1y!n1z! is the multiplicity of the k1-th unique element of the DT Dk1, and uk1(p1) is the component of the unit vector specified by the p1-th index of the k1-th unique element of the DT (7). Note that in the expression for μ, n1x, n1y, and n1z are respectively the number of x, y and z indices in the full sequence of subscripts defining the component of the tensor. Definitions of μ and D follow the same lines for the k2-th component of the tensor. The gamma values defined in Eq.  can be evaluated analytically, and the resulting expressions are listed in Table 2 for tensors up to rank-6. In this table, N>, N0, and N< are respectively the maximum, median, and minimum values of the array (n1x + n2x, n1y + n2y, n1z + n2z). γ−1 values resulting from other possibilities of N>, N0, and N< are 0, and do not contribute to the gentr(DN(u)2).
Table 2. The Gamma Values That Are Needed for the Calculation of the Gentr(DN(u)2) Values for Tensor Models up to Rank Six
l = 0
l = 2
l = 4
l = 6
Next, we introduce the variance of the normalized diffusivites as a measure of anisotropy. This is done by using Eq. :
The variance of the normalized diffusivity takes its minimum value of 0 only when diffusivities along all directions are equal. When a rank-l tensor model is used, this constant diffusivity profile is achieved when all terms except l = 0 in its irreducible representation (Laplace series) are zero (22). Table 3 shows the conditions under which a constant diffusivity profile is achieved for Cartesian tensors up to rank-6. Independently of the choice of tensor rank, the minimum value for the variance is zero, as expected.
Table 3. The Conditions on the Components of the Diffusion Tensors That Yield Isotropic Diffusivity Profiles
D is finite
Dxx = Dyy = Dzz, all other components are 0
Dxxxx = Dyyyy = Dzzzz = 3Dxxyy = 3Dxxzz = 3Dyyzz, all other components are 0
Dxxxxxx = Dyyyyyy = Dzzzzzz = 5Dxxxxyy = 5Dxxxxzz = 5Dyyyyxx = 5Dyyyyzz = 5Dzzzzxx = 5Dzzzzyy = 15Dxxyyzz, all other components are 0
Under the condition that all diffusivities implied by a rank-l tensor (via Eq. ) are nonnegative, the supremum value of the variance is achieved when the tensor is given by a pure outer product of the same l vectors, i.e., when the components of the tensor are given by
where u′ is the unit vector specifying the direction of greatest diffusion coefficient where D is this maximal diffusivity. Note that although a real generalized DT may come arbitrarily close to this, it can never reach this form, since it would imply zero diffusivities along directions perpendicular to u′, as can be seen from Eq. . Because the value of zero for diffusivities is nonphysical, we refer to the variance associated with the tensor given in Eq.  as the supremum, rather than the maximum value. After performing some algebra, it is possible to show that this supremum value corresponding to a rank-l tensor is given by
This result indicates that the supremum value depends on the rank of the tensor model selected. This surprising result implies that there is an intrinsic limit to the anisotropy that can be quantified with a lower-rank tensor model. Moreover, for functions whose domains are the unit hemisphere, in general the rank of the tensor model required for an exact representation of these functions is infinity. It is clear from Eq.  that as l → ∞, the supremum value goes to infinity. Therefore, a reasonable choice of function that will quantify anisotropy in terms of the variance of the diffusivities will be a monotonic function that will map the interval [0,∞) to [0,1). Of the many functions that meet this description, we choose to use an expression that is most suitable for typical data sets. After some scaling is performed, we arrive at the definition for the generalized anisotropy:
where the exponent ϵ(V) is defined as
The logarithmic plot of GA as a function of the variance of the normalized diffusivities is shown in Fig. 2. The vertical lines in this plot show the locations of the maximal anisotropy cases that can be quantified by tensors of ranks 2, 4, and 6, respectively. These supremum values for GA are .957, .980, and .987.
The form of the expression for GA as given in Eq.  is not arbitrary, and takes into account the sensitivity of the calculated values to the variations in the variance. The behavior of this function can be best understood by examining the derivative of this expression with respect to V. This plot is also presented in Fig. 2 (dashed lines). It is apparent from the figure that the formulation of anisotropy as given in Eq.  yields low contrast in voxels that are already anisotropic, such as those in white matter. This behavior is in accordance with the previously proposed anisotropy indices. However, unlike the previously introduced indices, GA also has low contrast among voxels with very low anisotropy values, such as those in free water. As a result, the intensity differences in the GA values are concentrated in voxels within gray matter and the transition from gray matter to white matter, while high intensity is retained in the white matter. If one is interested in changing the contrast according to the values in a different kind of data set, one can easily adjust the constants in the definition of GA.
Isotropy as the Entropy of the Normalized Diffusivity Profile
In the previous section we showed the quantification of anisotropy in terms of the variance of the diffusivity values. In general, the variance in a random variable x is given by
where P(x) is the PDF. The parametrization of anisotropy in terms of the variance of diffusivities in the previous section effectively treats the diffusivity as a random variable, and assumes a PDF according to the frequencies of diffusivities that occur in the diffusivity profile. An alternative way to approach the same problem is to take the orientation as a random variable and use DN(u) as the PDF. This can be done because DN(u) is positive definite and integrates to one. In this case, the quantity we referred to as the “variance” of the normalized diffusivity is related to the integral of the square of a PDF. In this context, it is not clear why one would choose to do this rather than take the integral of any other power of the PDF. Perhaps a more appropriate choice would be to take the derivative of the integral with respect to the power of the distribution and set this power equal to one. We will refer to the negative of this quantity as σ:
Therefore, σ is a measure of how the integral varies when the power of the PDF is varied around one (note that at m = 1 the integral is just the generalized trace) and is equal to the differential entropy of the distribution. This is a well-known quantity in several disciplines: in statistics, it quantifies the uniformity of the distribution (23); in statistical mechanics, it yields the uncertainty level in a given phase space (24); and in information theory, it gives the information content of the PDF (23). When the entropy is higher the distribution is more uniform, the orientation is less certain, and we have less orientational information. Therefore, when anisotropy is viewed as a measure of certainty or information, it is natural to parametrize it in terms of entropy.
It is straightforward to show that for an isotropic diffusivity profile, σ will take its maximum value of ln3. On the other hand, for a rank-l tensor model, as the diffusivities approach the form given in Eq. , entropy approaches its infimum value given by
Obviously, for an arbitrary function which can in general be represented by a rank-∞ tensor, this infimum value is −∞. Therefore, a general anisotropy index based on entropy has to be a monotonically decreasing function of entropy that maps the interval (−∞,ln3] to [0,1). We employ a function similar to that in Eq. , and introduce the SE index:
Here the exponent function ϵ is the same as that given in Eq. .
The exponential plot of SE and its derivative as a function of the entropy are shown in Fig. 3. The vertical lines in this plot show the locations of the maximum anisotropy profiles that can be quantified by tensors of ranks 2, 4, and 6, respectively. These supremum values are .963, .980, and .985.
Expressions for Arbitrary Functions Defined on the Sphere
Thus far, we have treated the functions defined on the unit sphere as the rank → ∞ limit of the higher-rank tensors. For completeness, we present some of the results obtained for arbitrary positive definite and integrable functions below:
Note that when an even rank tensor is used, the antipodal symmetry in the diffusivity profile is automatically achieved (7). Therefore, in our previous derivations it was sufficient to evaluate the integrals on the hemisphere. In Eqs. –, we do not assume the antipodal symmetry of the function. Therefore, the integrations are on the whole sphere, S. Once the 〈f(u)〉, V, and σ values are calculated, the scaled anisotropy indices GA and SE can be calculated with Eqs. , , and  as before.
MATERIALS AND METHODS
To demonstrate the images of the scalar measures derived, we acquired a series of diffusion-weighted images at 17.6 T (750 MHz) using a Bruker Avance imaging system. The sample was an excised rat brain that was fixed in 4% paraformaldehyde and imaged in phosphate-buffered saline (PBS). These experiments were performed with the approval of the University of Florida Institutional Animal Care and Use Committee. The imaging protocol employed a diffusion-weighted spin-echo pulse sequence with the following parameters: TR = 2500 ms, TE = 28 ms, Δ = 17.8 ms, δ = 2.2 ms, resolution = 150 μm × 150 μm × 300 μm, and NA = 6. Diffusion-weighted images were acquired along 81 directions specified by the tessellations of an icosahedron on the hemisphere with a b-value of 1500 s/mm2, along with a single image acquired without diffusion sensitizing gradients (b ≈ 0).
We fitted the images to the generalized Stejskal-Tanner relation (Eq. ) to yield the components of Cartesian tensors up to rank-6, using methods described in Ref. 7. We evaluated 〈D〉 values using the expressions tabulated in Table 1. Similarly, we calculated the exact variance(DN(u)) using the expressions in Table 2. From these expressions, GA values were calculated with Eq. . We calculated the entropy numerically using iterated Gaussian quadrature employing 96 transformation points. Then SE values were obtained from Eq. . All programs used for these purposes were written in IDL (Research Systems, Inc., Boulder, CO).
RESULTS AND DISCUSSION
Figure 4 shows images calculated from a rat brain at a selected slice. As shown in the first column, 〈D〉 is constant regardless of the rank of the tensor model used. It is not possible to use a rank-0 model to quantify anisotropy. Although images of variance, GA, σ, and SE from different rank models (rank ≥ 2) look similar at first glance, it is possible to find differences between images from different rank models. These issues are discussed below.
Fiber Crossings and Anisotropy
Overcoming the inability of traditional (rank-2) DTI to resolve multiple fiber orientations within a voxel was our primary motivation to generalize the DTI technique to employ tensors of higher ranks. Therefore, an appropriate test for the application of generalized DTI should be the illustration of changes in the anisotropy values when models with different rank tensors are used. More specifically, one can expect that rank-2 DTI will yield artificially low anisotropy values in pixels with complicated fiber structure. To test this hypothesis, we performed simulations using the exact form of the diffusional signal attenuation, given in Ref. 25, under the assumptions that the fibers were perfect cylinders, and that water molecules were confined to the interior of these cylinders (i.e., the membranes were not permeable). In the presence of more than one cylinder, the signal attenuations from multiple cylinders become additive if the signals are equal when no diffusion gradient is applied.
We simulated the MR signal attenuation with the following parameters: cylinder length = 5 mm, cylinder radius = 5 μm, diffusion coefficient of water = 2.0 × 10−3 mm2/s, Δ = 17.8 ms, δ = 2.2 ms, b = 1500s/mm2. These parameters are similar to those used in our MRI experiment on the excised rat brain. The gradient directions were chosen to be identical to those used in the experiments. In this way, a number of signal attenuations were computed. We then computed the components of the tensors up to rank-6, and the GA values for one, two, and three fiber cases in which the angle between any two distinct fiber directions was 90°. The results are tabulated in Table 4. We used this simulation scheme to produce the image in Fig. 1. This simulation image contains two fiber bundles: one is linear, and one is circular. Thus, a distribution of angles is obtained in different pixels of the image. In Table 4, we also give the mean GA values and their standard deviations (SDs) calculated from all regions of this image containing fibers. We would like to point out that the calculated anisotropy values are dependent on the amount of diffusion weighting introduced. Note that although the small q limit of the expression used in the simulations has angular dependence (25), as demonstrated in Refs. 17 and26, the diffusivity profile is typically very smooth in low b-values, and can be reasonably well approximated by the quadratic forms of a rank-2 tensor. In this case, one should not expect large differences in the anisotropy values from tensors of different ranks. Our simulations suggest that a b-value of 1500s/mm2 is high enough to create a significant change in the diffusivity profiles, and hence the anisotropy values, when different fiber orientations are present.
Table 4. Simulated GA Values From Different Rank Tensor Models for One, Two, and Three Fiber Orientations*
Also included are the mean and SDs of the GA values that were calculated from any region of the simulated image presented in Fig. 1 containing fibers.
It is clear from the first row of Table 4 that in a voxel containing a single fiber orientation, the measured anisotropy is almost independent of the rank of the model being used. However, when there are two fiber orientations, the difference in the calculated GA values is significantly higher in rank-4 and -6 models compared to a rank-2 model. This change is even more dramatic in the case of three fiber orientations. In this case, the GA value implied by the rank-2 tensor is so small that one can easily label the geometry as nonfibrous, where higher-rank tensors make the anisotropy apparent. The values from the simulated image shown in Fig. 1 (presented in the last line of Table 4) are also consistent with these findings. Just as one would specify a region of interest (ROI) in a real data set to calculate the anisotropy values from a white-matter structure, the section of the simulation image that contains one or two fibers are included in the calculation of the GA values. It should be noted that in addition to the elevation of the mean values, there is also a decrease in the SDs of the GA values when higher-rank tensor models are used. This is because as the rank of the tensor model is increased, the anisotropy values in the region with crossing fibers approach GA values from those voxels with single orientations, decreasing the overall spread in the values. For example, as can be seen from the second line of Table 4, in the case of two fibers, the GA value rises from 0.56 to 0.63 when the tensor rank is increased, driving the GA values toward the GA value from voxels with a single fiber (i.e., 0.89). When we look at only the section with crossing fibers in Fig. 1, we see that the enhancement in the mean of the anisotropy values is larger than 7%, which is quite significant. These findings can also be seen in Fig. 5, where we present the GA images from rank-2 and -6 tensor fits. Also included in this figure are the differences in variance and GA values from these two tensor models.
The sensitivity of the anisotropy values to a change in ranks provides a potential tool for identifying voxels with heterogeneity in fiber orientations. However, one should keep in mind that anisotropy indices such as GA and SE are functions of some other quantities (variance and entropy), and the changes in the values of these indices depend not only on the structural complexity within the voxel, but also on how these anisotropy measures are defined. For example, since the derivative of GA has a peak around the variance value of about 10−3, small changes in the voxels, with variance values in this range, may be exaggerated in the GA difference maps calculated with different tensor ranks. For this reason, one may think that the difference in the variance or entropy maps (or some other functions of these) may be a more suitable choice when one is interested in identifying regions with complex fiber architecture. However, note that when variance or entropy is used for this purpose, the supremum or infimum values differ with a change in ranks because, as mentioned above, there is a limit to the anisotropy that can be quantified by a given model. Therefore, one may obtain highly different values of variance or entropy from different tensor models in voxels with very anisotropic structure. As a result, in this case one should be interested in voxels with high variance (entropy) differences but not high anisotropy values, because the limits of both variance and entropy values at high anisotropy depend on the rank (see Figs. 2 and 3).
Based on these findings from simulations, we investigated how the GA values in tissue change with changing tensor ranks. For this purpose, we show images of GA values from rank-2 and -6 tensors in Fig. 6 along with the difference maps of the variance and GA values when these two tensor models are employed. It is clear from the difference maps that there are certain regions where these differences are significant. In the first two rows, these regions are within the pons, and in the bottom row the GA values in the brain stem are enhanced. These regions are known to contain structures with crossing fibers. The difference images present a significant level of symmetry, implying that the anisotropy and variance changes are most likely not due to noise. Rather, they stem from local structure (such as crossing fibers) and partial-volume effects. Another point to note is that, as observed in the simulations, the GA images from higher-rank tensor models are smoother than those from lower-rank tensors. Although the diffusivity profiles from individual pixels become sharper, the resulting anisotropy maps become smoother as a result of the more accurate quantification of anisotropy when the rank of the tensor model is increased.
Simple Implementation of Entropy-Based Anisotropy for Rank-2 Tensors
Our generalization of the indices, such as FA and RA, included the extension of the variance to diffusivities along all directions, and generalization of the trace operator. If we look at those rank-2 measures from the perspective of our generalization, scalar measures like FA and RA can be thought of as a simplification to the GA values for the case of rank-2. We now consider the formulation of SE index and ask the question: Is there a simplified formulation of entropy for rank-2 tensors? This would be particularly useful in the case of SE index, because the numerical implementation of the integral given in Eq.  is a slow computational operation.
Using Eq. , the normalized diffusivity is given by
where R is defined in Eq. . Therefore, entropy of a rank-2 DT becomes (from Eq. )
Now, we use an approximation that is well-known in the field of mathematical physics that establishes the relation between classical and quantum-mechanical entropy (27):
Here, the ln operation of the matrix R is defined in terms of its Taylor expansion. From Eq. , it is easy to see that
The right-hand side of the above equation is the same expression as the “von Neumann entropy” of a density matrix (that will be denoted by σvN), which is well known in quantum mechanics. The evaluation of the infinite summation in Taylor expansion in Eq.  can be avoided, since upon diagonalization of R, σvN is given by
where ρi are the eigenvalues of R, or eigenvalues of the original DT D divided by the trace of the tensor.
The “approximation” used in Eq.  is an unusual one that is very accurate for reasonably isotropic tensors. However, as the tensor approaches the most anisotropic limit, σvN deviates from σ, as is obvious from the infimum values of 0 for σvN and 2/3 for σ in the case of rank-2 (27). Note that a similar situation occurs in the comparison of trace(R2) with a supremum value of 1 and gentr(DN(u)2) with a supremum value of 3/5 for rank-2. Therefore, just as FA values require a scaling different from that of GA values, scaling of σvN to an anisotropy measure will be different from that of σ. Following a scaling similar to the one used in the formulation of FA, we arrive at the definition for the anisotropy measure that we called “visual anisotropy”2 in an earlier work (28):
These formulations enable one to visualize the behavior of these indices for given eigenvalues of R. If we order the eigenvalues such that ρ1 ≥ ρ2 ≥ ρ3, then this condition, along with trace (R) = ρ1 + ρ2 + ρ3 = 1, and the positive definiteness of the eigenvalues, limits the allowed values of ρ1 and ρ2 to the triangular zone in the ρ1ρ2 plane given by ρ1 − ρ2 ≥ 0, ρ1 + 2ρ2 ≥ 1, ρ1 + ρ2 ≤1. (For a similar visualization see Ref. 29.) These figures along with images of FA, GA, and VA indices are shown in Fig. 7. Note that the difference in the brightness of these images are due to the different scalings that were used in the computations of the indices.
Variance or Entropy-Based Anisotropy?
In this work we have presented two different ways to quantify anisotropy, by defining it in terms of variance and entropy. Although these two formulations provide similar-looking images, it is possible to show that they differ when one tries to order pixels according to their anisotropy. In other words, when one compares the anisotropy values from two pixels, the GA values may suggest that the first pixel is more anisotropic than the second, while the SE index may suggest otherwise. This disagreement stems from the difference in the meanings attached to anisotropy. Variance is a measure of dispersion, while entropy is a measure of the lack of information, or uncertainty. The choice between the two depends on the utilization of the anisotropy index. For example, using the drop in anisotropy as a termination criterion in fiber-tract mapping algorithms assumes that the fiber direction is too uncertain in these regions, implying that anisotropy index being used is a measure of uncertainty. However, the entropy-based indices (like the variance-based ones) differ from an index like Amajor (11), which quantifies the deviation of the diffusivity along the presumed fiber direction from the diffusivities along other directions. This is most apparent when one realizes that the integrations in the formulation of anisotropy have uniform weightings. As a result, the same value may be obtained if one redistributes the same diffusivities in a different way.
Because of the logarithm in the definition of entropy, it is advantageous to use GA from a practical point of view. As shown in Eq.  and Table 2, it is possible to analytically evaluate the GA values for higher-rank tensors. The analytical evaluation of integrals in the calculation of entropy is not straightforward. Therefore, we adopted a numerical approach. As a result, the calculation of GA values is faster and more exact compared to the calculation of SE values. For model-independent functions, it will probably be necessary to evaluate the variance numerically as well.
The results presented for the scalar measures derived in this paper were obtained with the diffusivity profile. However, the formulations also remain valid for other applications such as q-space imaging in which one calculates the average propagator for water molecules if an orientation distribution function (ODF) is constructed, as in Ref. 30, by projecting the average propagator radially onto the surface of a sphere. This is because both the ODF and normalized diffusivity profiles are defined on the unit sphere, they are positive definite and integrate to one. Note that our analysis started with a non-normalized diffusivity profile and we normalized it using the generalization of the trace operation. Therefore, the formulations are applicable to all kinds of positive-valued data acquired on or projected onto the surface of a sphere.
Using functions as distributions defined on the surface of a sphere makes it possible to come up with different measures that are known from information theory. As an example, relative entropy or the Kullback-Leibler (KL) divergence is a measure of closeness of two distributions (24). There are also several distance measures between two distributions, such as the Kolmogorov distance and fidelity. A detailed analysis of these measures and their quantum analogs (in which case a density matrix analogy of the normalized rank-2 DT can be used) can be found in Ref. 31. These scalars may be useful for certain applications, such as image smoothing and registration.
SUMMARY AND CONCLUSIONS
We have presented rotationally invariant indices of mean diffusivity (〈D〉) and anisotropy based on variance and entropy (GA and SE). Our formulations can be used to quantify anisotropy from higher-rank DTs and arbitrary functions defined on the unit sphere. In the calculation of 〈D〉, one can use the relevant expression in Table 1 for generalized DTI data. In the calculation of the variance-based GA index, one can use the expressions given in Table 2 or Eq. , and Eqs. , , , and  for higher-rank tensors. Finally, in the case of the entropy-based SE index, one can use expressions given in Eqs. , , and . For general positive-valued integrable functions whose domains are the unit sphere, the indices can be calculated from Eqs. – in conjunction with Eqs. , , and .
Our results indicate that for the quantification of 〈D〉, it is appropriate to use DTs of any rank. However, lower-rank tensors may undesirably suppress the anisotropy information that is available from HARDI experiments. This fact was predicted by means of simulations and confirmed with experiments.
We acknowledge Dr. Tim M. Shepherd of the Department of Neuroscience, University of Florida, for his help in interpreting the images.
Our approach should not be confused with a recent approach by the same name in which traditional DTI was generalized using the Kramers-Moyal expansion, which employs a series of tensors with increasing ranks (10).
The word “visual” was chosen to emphasize that it was scaled nonlinearly to make the images look more appealing than the linearly scaled index ln3-σvN. The GA and SE indices we introduced in this work have an even more sophisticated scaling.