NIH Public Access Author Manuscript NMR Biomed. Author manuscript; available in PMC 2015 February 01.

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Published in final edited form as: NMR Biomed. 2014 February ; 27(2): 202–211.

Leading Non-Gaussian Corrections for Diffusion Orientation Distribution Function Jens H. Jensena,b,*, Joseph A. Helperna,b,c, and Ali Tabesha,b aDepartment

of Radiology and Radiological Science, Medical University of South Carolina, Charleston, South Carolina, USA.

bCenter

for Biomedical Imaging, Medical University of South Carolina, Charleston, South Carolina, USA. cDepartment

of Neurosciences, Medical University of South Carolina, Charleston, South Carolina,

USA.

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Abstract An analytical representation of the leading non-Gaussian corrections for a class of diffusion orientation distribution functions (dODFs) is presented. This formula is constructed out of the diffusion and diffusional kurtosis tensors, both of which may be estimated with diffusional kurtosis imaging (DKI). By incorporating model-independent non-Gaussian diffusion effects, it improves upon the Gaussian approximation used in diffusion tensor imaging (DTI). This analytical representation therefore provides a natural foundation for DKI-based white matter fiber tractography, which has potential advantages over conventional DTI-based fiber tractography in generating more accurate predictions for the orientations of fiber bundles and in being able to directly resolve intra-voxel fiber crossings. The formula is illustrated with numerical simulations for a two-compartment model of fiber crossings and for human brain data. These results indicate that the inclusion of the leading non-Gaussian corrections can significantly affect fiber tractography in white matter regions, such as the centrum semiovale, where fiber crossings are common.

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Keywords orientation distribution function; kurtosis; diffusion; tensor; fiber tractography; non-Gaussian; white matter; MRI

INTRODUCTION A convenient method for extracting directional information from diffusion MRI (dMRI) data of brain is to calculate a diffusion orientation distribution function (dODF) (1-7). In particular, the local maxima of dODFs are often interpreted as indicating the directions of axonal fiber bundles and can be utilized in white matter fiber tractrography (8-12). An *

Corresponding Author: Jens H. Jensen, Ph.D. Center for Biomedical Imaging Department of Radiology and Radiological Science Medical University of South Carolina 96 Jonathan Lucas Street, MSC 323 Charleston, SC 29425-0323 Tel: (843) 876-2467 [email protected].

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important advantage of many dODFs is that their calculation is not dependent on a specific model of tissue microstructure.

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A class of dODFs may be defined by the radial integral [1]

where ψα(n̂) is the dODF in a direction given by a unit vector n̂, P(s,t) is the water diffusion displacement probability density function (dPDF) for a molecular displacement s over a diffusion time t, and Z is a normalization constant. The power α affects the radial weighting of the dODF, with larger α corresponding to a greater sensitivity to long diffusion displacements. Note that the dODF of Eq. [1] does not make any explicit assumptions about tissue microstructure.

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If one approximates the dPDF by a Gaussian function, as is done for diffusion tensor imaging (DTI) (13), then the local maxima of the dODF are completely determined by the diffusion tensor (DT) and correspond to the direction of the principal DT eigenvector. As a consequence, using this Gaussian approximation of the dODF for fiber tractography is equivalent to commonly used DTI-based algorithms that rely primarily on the principal DT eigenvector to determine the fiber track orientation in each voxel (4,14-17). Such a Gaussian dODF, however, has a significant shortcoming in that it does not reliably predict fiber bundle directions for voxels having two or more intersecting bundles, which is sometimes referred to as the “fiber crossing problem” (4,7,17). For this reason, more refined approximations for the dODF are often employed, such as in Q-ball imaging (1,2) and in diffusion spectrum imaging (DSI) (3).

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The purpose of this paper is to present an approximation for the dODF derived by systematically calculating the leading non-Gaussian corrections. We show that the maxima for this approximation depend only on the DT and the diffusional kurtosis tensor (DKT). In comparison to the Gaussian dODF, this approximation, which we term the kurtosis dODF, allows the direction of fiber bundles to be estimated with substantially improved accuracy. We illustrate the kurtosis dODF for both a simple numerical model and for human brain data. Our results both extend and simplify those of a previous report (18). A key feature of the kurtosis dODF is that it is compatible with diffusional kurtosis imaging (DKI) in that DKI yields estimates for both the DT and DKT (19-23). Thus if a DKI dataset is available, employing the kurtosis dODF may be a practical means of generating fiber tractography that improves upon DTI-based approaches. The kurtosis dODF could also be helpful in assessing and elucidating other dODF approximations by giving rigorous results for a specific limiting case.

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METHODS NIH-PA Author Manuscript

Gaussian dODF The diffusional average of an arbitrary function F(s) of the diffusion displacement s is given by (21) [2]

where we assume that the dPDF is normalized so that [3]

The components of the DT may then be expressed as [4]

with si indicating the components of s. The Gaussian approximation for the dPDF is defined by

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[5]

where D represents the DT (assumed to be positive definite). It is this approximation that forms the basis of DTI (13). By combining Eqs. [1] and [5], one finds the Gaussian dODF to be

[6]

where Γ(x) is Euler's gamma function and a “G” has been added to the dODF symbol subscript to indicate that it is for the Gaussian approximation. Convergence of the Eq. [1] integral requires α > -1.

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Now let us define a dimensionless tensor U ≡ D̄D-1, where D̄ is the mean diffusivity, and set the normalization constant to be

[7]

The Gaussian dODF then takes the simple from [8]

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The eigenvectors of U coincide with those of the DT, and the maxima for the Gaussian dODF occur at n̂ = ±ê1, where ê1 is the principal DT eigenvector. Note that the locations of these maxima are independent of the radial weighting power α. Non-Gaussian Corrections In order to systematically define non-Gaussian corrections for the dODF, it is helpful to introduce the Fourier transform of the dPDF: [9]

where we have suppressed the explicit reference to the diffusion time in the arguments of P̃ for the sake of notational simplicity. Because of the normalization condition of Eq. [3], we must have [10]

In terms of P̃, the dODF may be written as

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[11]

The corresponding Fourier transform for the Gaussian dODF is [12]

which is equivalent to the DTI approximation for the dMRI signal as a function of the diffusion wave vector q (13). One may construct an “η dependent” dODF so that [13]

Clearly, one sees that

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[14]

and [15]

Hence, the scaling parameter η provides a natural means for interpolating between the Gaussian and exact dODFs. Corrections for the Gaussian dODF may then be systematically derived in terms of the Taylor series for ψα,η in powers of η about η = 0. If one makes the standard assumption that P̃(q) = P̃(–q), the leading non-Gaussian corrections are of order η4, and a direct calculation shows

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[16]

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where Uij indicates the components of U, [17]

Wijkl are the components of the DKT (19-21,24), and the sums on the indices (i,j,l,k) are carried out from 1 to 3. Dropping the terms of order η6 and higher and setting η = 1 in Eq. [16] yields the kurtosis dODF

[18]

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which is the central result of this paper that extends the Gaussian approximation by including the leading non-Gaussian corrections. The expansion in the parameter η has been a formal device for organizing the non-Gaussian corrections, but for most specific models this would correspond, in essence, to an expansion for a physically well-defined parameter, often the ratio of a characteristic length scale for the microstructure to the diffusion length. In order to better understand the physical meaning of the η expansion, let us first consider an example with [19]

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where F(x) is an unspecified analytic function consistent with Eqs. [10] and [12], and a is a length that controls the diffusion wave vector scale at which non-Gaussian diffusion effects become apparent. Note that Eqs. [10] and [12] imply that F(0) = 1. For this case, the expansion in η is equivalent to an expansion in a divided by the diffusion length (2D̄t)1/2, with D/D̄ being kept fixed. The diffusion length enters here as the characteristic length scale for Gaussian diffusion. As a second example, suppose

[20]

which corresponds to a model with N Gaussian compartments. Here δDm represents the difference between the DT of the mth compartment and the total DT. Consistency with Eq. [10] implies that compartment weights, fm, sum to unity. We may quantify the spread in compartmental diffusivities by

[21]

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The expansion in η can then be regarded as an expansion in (ΔD/D̄)1/2 with D/D̄, δDm/ΔD and fm being kept fixed.

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Because of the symmetry ψα,K(n̂) = ψα,K(–n̂), all isolated local maxima for the kurtosis dODF come in pairs related by point reflection through the origin. However, unlike the Gaussian dODF, the number of maxima pairs can exceed one, allowing fiber crossings to be resolved. The kurtosis dODF also differs from the Gaussian dODF in that the location of the maxima depend on the choice of α, and so the optimization of this parameter becomes of interest. For α = 0, the kurtosis dODF is numerically identical to the result given in Ref. 18, but Eq. [18] is much easier for practical calculations in being an explicit algebraic formula rather than an integral. For other α values, Eq. [18] is a new result. Although a rigorous upper limit on the number of maxima pairs allowed for the kurtosis dODF is not self-evident from the structure of Eq. [18], one may easily construct models with up to 4 distinct maxima pairs. Examples with 2 and 3 maxima pairs are presented in Ref. 18 for α = 0. Two-Compartment Model

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In order to illustrate and investigate the kurtosis approximation for the dODF, we consider a simple two-compartment model of crossing fibers defined by [22]

where DA and DB are the compartmental DTs and f is the relative water fraction for compartment A. These compartments may be regarded as representing two intersecting axonal fiber bundles within a single voxel. For the DTs, we choose

[23]

and

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[24]

The DT for compartment A models a fiber bundle oriented parallel to the x-axis with a principal eigenvalue λ∥, while the DT for compartment B models a similar fiber bundle rotated in the xy-plane by an angle ξ. For numerical calculations, we use the values of λ∥ = 2.0μm2/ms and λ⊥ = 0.4μm2/ms, which are typical for white matter with high fractional anisotropy (FA). The total DT for the model is [25]

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and the components of the DKT are (18)

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The Gaussian and kurtosis dODFs may then be calculated by using Eqs. [25] and [26] together with Eqs. [8] and [18]. The two-compartment DKT of Eq. [26] will always yield a nonzero kurtosis for at least some directions unless DA = DB, f = 0, or f = 1. Our model thus represents non-Gaussian diffusion for most parameter choices. As illustrations, two main cases are examined. For the first case, we take f = 0.8 so that compartment A represents a dominant fiber bundle and compartment B represents a small admixture of a subdominant component. Ideally, the global maxima would correspond to the direction of the dominant bundle, and we calculate the departure from this (i.e., the angular error) given by the exact, Gaussian, and kurtosis dODFs for a range of ξ and α values.

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For the second case, we take f = 0.5 so that compartments A and B represent an intra-voxel crossing of two similar fiber bundles. While the Gaussian dODF is not able to resolve this crossing, the kurtosis dODF can if the crossing angle is not too small. For a range of ξ and α values, we determine the accuracy of the predicted direction for compartment A given by the maxima of the respective dODFs. Human Experiments

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In order to illustrate the kurtosis dODF for normal brain, we obtained DKI data from one healthy volunteer (male, age 52). Our DKI scanning protocol has been described in detail elsewhere (21,24). Briefly, we used a twice-refocused dMRI sequence (25) with the imaging parameters: echo time = 99 ms, repetition time = 5300 ms, field of view = 222×222 mm2, acquisition matrix = 74×74, voxel size = 3×3×3 mm3, number of slices = 40, inter-slice gap = 0 mm, bandwidth = 1352 Hz/pixel, parallel imaging factor = 2, b-values = 0, 1000, 2000 s/mm2, number of zero b-value images = 6, number of gradient directions = 30, number of averages = 1, and total acquisition time = 6 min, 46 s. This protocol was performed three times within the same scan session to test reproducibility and to allow for better signal averaging. The scans were obtained with a 3T MRI system (Tim Trio, Siemens Medical, Erlangen, Germany) using a 32-channel head coil under a protocol approved by the institutional review board of the Medical University of South Carolina. Post-processing of raw diffusion-weighted images followed an approach very similar to that described in Ref. 24. To summarize, the diffusion-weighted images were skull-stripped, smoothed, co-registered and then globally fit to a standard DKI signal formula by means of constrained weighted linear least squares. Constraints were imposed on the diffusion parameters to keep them within prescribed limits as dictated by general physical properties (e.g., positive semidefiniteness of diffusion tensor eigenvalues), which helped to reduce the impact of signal noise. This data analysis utilized freely available in-house software, referred to as Diffusional Kurtosis Estimator (24, 26), and included calculation of parametric maps for the mean diffusivity (MD), FA, and mean kurtosis (MK). The precise correspondence between the dMRI signal and DKI parameters relies on an assumption of

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short gradient pulse durations (19). Although this condition is not strictly met for the dMRI sequence used is this study, as is typically the case for human dMRI, the effects of a long gradient pulse duration on DKI parameter estimates are likely to be small in brain (27). Kurtosis dODFs were calculated on a voxel-by-voxel basis for α = 0, 2, and 4 by using Eq. [18]. The local maxima of these dODFs were determined by applying a quasi-Newton method utilizing the angular derivatives of the dODF, as provided by Eq. [29] of the Appendix, with 100 initial points distributed on a uniform half-sphere grid. Distinct local maxima were then identified as the convergent solutions that were more than 1° apart from other local maxima. Locations of the maxima for the Gaussian dODFs were determined from the DT eigenvectors.

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For the full DKI dataset (i.e., using the combined data from all three trials), the angular difference between the principal directions for the two dODFs, as indicated by the global maxima, was determined in each voxel, as well as the number of distinct fiber bundle directions identified in each voxel by the kurtosis dODF. For the voxels with 2 or more directions, the angles between the largest and second largest maxima of the kurtosis dODF were calculated, as an estimate of the fiber crossing angles. In order to test reproducibility, angular differences for the two dODFs and the number of directions detected by the kurtosis dODF were also calculated separately for the three individual trials. To suppress the confounding effects of cerebral spinal fluid, voxels with MD > 1.5 μm2/ms were excluded from analysis. The Euler method (16) was used to construct an elementary example of DKI-based fiber tractography (i.e., corresponding to the kurtosis dODF) for a single dataset. This was compared to DTI-based fiber tractography (i.e., corresponding to the Gaussian dODF) employing the same data, algorithm and seed region. To highlight the effect of voxels with crossing fibers, a seed region was selected within the centrum semiovale, a fiber crossingrich white matter area where the superior longitudinal fasciculus, corpus callosum, and corona radiata intersect. A step size of 1 mm was used for the Euler method. Fiber termination criteria were FA 60°/mm. The radial weighting power α was set to 4 for the kurtosis dODF.

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RESULTS The exact, kurtosis, and Gaussian dODFs are shown in Fig. 1 for the two compartment model with α = 2 and ξ = 60° for the two cases of f = 0.8 and f = 0.5. In both examples, the exact dODF accurately predicts the true directions of the fiber bundles. The Gaussian dODF yields only a single direction, which deviates substantially from either of the true directions. When f = 0.8, the kurtosis dODF gives a good approximation for the direction of the dominant fiber bundle (angular error = –0.96°). When f = 0.5, the kurtosis dODF detects the directions for both fiber bundles with an angular error of 4.78°. The angular errors for the predicted direction of fiber bundle A, as obtained from the maxima of the three α = 2 dODFs, are plotted in Fig. 2 as a function of the crossing angle ξ. When f = 0.8, the exact dODF has a maximum error of 2.01° for ξ = 16.1 °, the Gaussian

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dODF has a maximum error of 7.24° for ξ = 52.3°, and the kurtosis dODF has a maximum error of 2.11° for ξ = 15.0°. When f = 0.5, the exact dODF has a maximum error of 13.9° for ξ = 27.8°, the Gaussian dODF has a maximum error of 45.0° for ξ = 90.0°, and the kurtosis dODF has a maximum error of 12.8° for ξ = 25.6°. In the case of f = 0.5, the angular errors for all the dODFs are identical up to ξ = 25.6°, as none of them are able to resolve the fiber crossing for these small intersection angles. Overall, the kurtosis dODF substantially improves upon the predictions of the Gaussian dODF. The dependence of the angular error as a function of the radial weighting power α is illustrated in Fig. 3 for the two-compartment model with ξ = 45°. The exact dODF becomes more accurate as α is increased, while the accuracy of the Gaussian dODF is independent of α. The kurtosis dODF depends on α, but its accuracy does not necessarily improve monotonically. This suggests that there may be an optimal choice of α for the kurtosis dODF. Based on these results, we set the maximum α to 4 in the analysis of human data.

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The distributions of angular differences between the Gaussian and kurtosis dODF global maxima, as obtained from the human data, are plotted in Fig. 4 for α = 0, 2, and 4. Also shown in Fig. 4 are the angular differences as a function of the FA. For voxels with FA > 0.2, the mean angular differences are 4.2° for α = 0 , 6.0° for α = 2, and 7.2° for α = 4. For the same DKI dataset and radial weighting powers, the fractions of voxels in which the kurtosis dODF identifies 1, 2, or 3 directions are given by Fig. 5. Most of the voxels have only one direction, but a substantial fraction has 2 or 3 suggesting the presence of intravoxel fiber crossings. The fraction of voxels with detected crossings grows with increasing α. Also shown in Fig. 5 are the distributions of the fiber crossing angles for voxels with 2 or more directions and with FA ≥ 0.2. The crossing angles were estimated from the angular differences between the directions identified by the largest and second largest maxima of the kurtosis dODF. Interestingly, similar distributions are found for all three radial weighting powers. These distributions are well fit by a Gaussian function centered at a crossing angle of 90° and having a standard deviation of 22°.

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For a single axial slice, maps of the number of fiber crossings identified by the kurtosis dODF are displayed in Fig. 6, again demonstrating the greater sensitivity to fiber crossings for larger α. For the same slice, Fig. 6 also gives maps of the angular difference between directions corresponding to the global maxima of the Gaussian and kurtosis dODFs. Substantial differences are seen in much of the white matter, indicating that the nonGaussian corrections for the dODF are significant. Maps for the number of detected fiber crossings and the Gaussian/kurtosis dODF angular differences are presented by Fig. 7, as calculated separately for the three individual trials with α = 4. Only voxels from white matter regions, defined as MK ≥ 0.9, are displayed, since the results for gray matter largely reflect noise due to gray matter's low degree of diffusional anisotropy. That similar maps are obtained in each case supports the reproducibility of dODF calculations. White matter segmentation used the MK rather than the FA, since the FA is low in both gray matter and in white matter regions with fiber crossings. The MK threshold of 0.9 was based on the data of Falangola and coworkers (28).

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Figure 8 illustrates fiber tractography obtained with the Gaussian and kurtosis dODFs. The kurtosis dODF (DKI-based) tractography utilized a radial weighting power of α = 4 and generates a richer set of tracks than the Gaussian dODF (DTI-based) tractography. This improvement is a consequence of the ability of the kurtosis dODF to resolve intra-voxel fiber crossings.

DISCUSSION

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The analytical representation of Eq. [18] for the kurtosis dODF gives the leading nonGaussian corrections for the exact dODF of Eq. [1]. It is constructed out of the DT and DKT and thus provides a natural foundation for DKI-based fiber tractography. That this kurtosis dODF is given by a relatively simple formula greatly facilitates its implementation. Potential advantages of DKI-based tractography over a DTI-based approach are greater accuracy in the estimation of fiber bundle directions and the ability to resolve intra-voxel fiber crossings. Since efficient acquisition and post-processing of DKI are now well-established, this provides a convenient method for combining advanced (non-Gaussian) dMRI with fiber tractography. Moreover, as DKI is being applied to investigate a variety of neuropathologies (29-48), DKI-based tractography may be useful when DKI data is already being acquired for other purposes. For example, DKI-based tractography could be applied to the identification of tract-based regions of interest (49). Our results (Figs. 4-8) for human brain indicate that the non-Gaussian corrections provided by Eq. [18] are substantial for white matter.

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In utilizing the kurtosis dODF, the choice of the radial weighting power α is a key consideration. By putting greater weight on longer diffusion displacements, larger α tends to increase the sensitivity to fiber crossings, as diffusion tends to be higher in directions parallel to the fiber axes. On the other hand, the quality of the approximation obtained by using the leading non-Gaussian approximation depends on α so that the accuracy with which the directions of fiber bundles are estimated may, in some cases, decrease if α becomes too large. For example, in Fig. 3, the minimal angular error with f = 0.5 occurs at α = 4.37. Thus the optimization of α would be an important extension of this work, with the range of 0 ≤ α ≤ 5 being suggested by our preliminary results. We note in passing that α = 2 has sometimes been referred to as the “correct” radial weighting power for the dODF (5-7). However from our perspective this choice is, rather, a definition that cannot be correct or incorrect, but may be more or less useful. Indeed, Fig. 3 shows that the choice α = 2 does not necessarily give the most accurate predictions, even for the exact dODF. In Ref. 18, it is demonstrated that the kurtosis dODF for α = 0 can be expressed as a FunkRadon transform of a function of the diffusivity, D(θ,φ), and diffusional kurtosis, K(θ,φ), in directions given by spherical angles (θ,φ). More specifically, [27]

where the spherical angles are taken with respect to a coordinate system having n̂ as the polar axis. A direct numerical evaluation of Eq. [27] requires a separate one-dimensional integration for each distinct orientation of n̂. These integrals may be completely avoided by

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utilizing Eq. [18], which substantially reduces the algorithmic complexity of calculating the kurtosis dODF. Hence Eq. [18] should be regarded as supplanting Eq. [27] for practical applications. Also in Ref. 18, a “non-Gaussian” dODF is defined as the difference between the kurtosis and Gaussian dODFs, which led to an increased sensitivity to fiber crossings. However, here we have found that the sensitivity to fiber crossings may also be increased by adjusting the radial weighting power α, as shown by Figs. 5 and 6. We regard this latter approach as more flexible and well-founded.

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The dODF considered here is of the same type used for DSI (3). However, DSI employs the exact dODF of Eq. [1] directly (usually with α = 2) and so is potentially more accurate than the kurtosis dODF. Nonetheless, obtaining the DKI data necessary to calculate the kurtosis dODF is typically less demanding in terms of acquisition time and the required maximum bvalues. In particular, a full brain DKI acquisition can be obtained in less than 10 min using maximum b-values of about 2000 s/mm2 (21,24), while a full brain DSI acquisition typically requires about 30 min to 1 hr and maximum b-values of about 4000 s/mm2 or more (50). With the advent of multiband echo planar sequences for dMRI (51,52), the data acquisition times for DSI may be substantially reduced from these reference values, but the same holds true for DKI so that the relative time disparity would be unchanged. Another advanced dMRI method based on the same type of dODF is Q-ball imaging (1,2). This technique also uses an approximate expression for the dODF, although a different one than is given by Eq. [18]. In particular, the Q-ball dODF has, unlike the kurtosis dODF, an explicit dependence on the b-value with the accuracy improving with increasing b-value. Most often, Q-ball imaging uses a radial weighting power of α = 0, as this leads to the simplest post-processing procedure, and employs a maximum b-value of at least 2500 s/mm2 (2,53). The comparative accuracies of the kurtosis and Q-ball dODF approximations would be a valuable topic for further investigation. In particular, it would be of interest to investigate the relative performance of these two techniques for comparable datasets.

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An alternative method of estimating the direction of white matter fiber bundles is the use of a fiber orientation density function (fODF) (7,54,55). This method differs from the dODF approach in that it makes specific assumptions about white matter microstructure. To the extent that these are assumptions are accurate, this may be advantageous in terms of accuracy and imaging requirements. However, fODFs are susceptible to producing spurious fiber crossings (7,56), can be sensitive to the computational details (56), and may be less robust for pathological tissues than dODFs which are based purely on the dPDF. Recently, Wedeen and coworkers (57) gave evidence, derived with DSI-based fiber tractography, that axonal fibers are arranged in a grid-like pattern. This picture suggests that intra-voxel fiber crossings predominately occur at near right angles. This is qualitatively consistent with our results of Fig. 5, which indeed shows a peak for the fiber crossing angle distribution at near 90°. This distribution does, however, have a width of about 22°, indicating a fairly large range of detected crossing angles. Nonetheless, this may still be consistent with a grid arrangement for the fiber pathways, since the measured distribution

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may be broadened by noise, systematic errors, and the finite spatial resolution of the diffusion images.

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CONCLUSION We have demonstrated how to rigorously calculate the leading non-Gaussian corrections for the dODF in terms of a compact, analytical formula. This result forms a natural foundation for DKI-based fiber tractography, which is expected to be more accurate than conventional DTI-based approaches. Our results for human brain indicate that the non-Gaussian corrections are substantial and that their inclusion may significantly alter fiber tractography. The formula also provides an explicit benchmark for evaluating other dODF approximations.

Acknowledgments Grant sponsor: National Institutes of Health Grant numbers: 1R01AG027852 and 1R01EB007656

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Grant sponsor: NSF/EPSCoR Grant number: EPS-0919440

APPENDIX In order to find the kurtosis dODF maxima, it is helpful to have explicit formulae for the derivatives of the dODF with respect to the spherical angles, as these can be used to substantially improve the efficiency and robustness of the numerical algorithm. For the kurtosis dODF, these are straightforward to calculate from the algebraic expression of Eq. [18]. In terms of the spherical angles θ and φ, the direction vector n̂ can be written [28]

where x̂, ŷ and ẑ are the Cartesian coordinate unit vectors. The derivatives of the kurtosis dODF are then

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[29]

with [30]

[31]

and

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[32]

In Eq. [32], we have introduced [33]

The sums over the indices in Eqs. [29] and [32] are all taken from 1 to 3.

Abbreviations used

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DKI

diffusional kurtosis imaging

DKT

diffusional kurtosis tensor

dMRI

diffusion MRI

dODF

diffusion orientation distribution function

dPDF

displacement probability density function

DSI

diffusion spectrum imaging

DT

diffusion tensor

DTI

diffusion tensor imaging

FA

fractional anisotropy

fODF

fiber orientation density function

MD

mean diffusivity

MK

mean kurtosis

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NIH-PA Author Manuscript FIG 1.

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The exact (solid line), kurtosis (dashed line), and Gaussian (dotted line) dODF contours plotted in the xy-plane for the two-compartment model with f = 0.8 and with f = 0.5. In both cases, the crossing angle is ξ = 60° and the radial weighting power is α = 2. The arrows give the true directions for fiber bundles A and B, while the dODF maxima are indicated by circles. The exact dODF predicts the fiber bundle directions for both cases with good accuracy. The Gaussian dODF identifies a single direction, which does not closely correspond to that of either bundle A or B. For f = 0.8, the kurtosis dODF predicts the direction of bundle A with an angular error of –0.96° but fails to resolve the subdominant bundle B, while for f = 0.5, it identifies both fiber bundles with an angular error of 4.78°.

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NIH-PA Author Manuscript FIG 2.

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Angular errors as a function of the crossing angle ξ for the α = 2 exact, kurtosis, and Gaussian dODFs for the two-compartment model with f = 0.8 and with f = 0.5. For most crossing angles, the accuracy of the kurtosis dODF is comparable to that of the exact dODF and substantially better than that of the Gaussian dODF. However, for small crossing angles, the angular errors for three dODFs are similar, reflecting a low sensitivity to the fiber crossing in this case.

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NIH-PA Author Manuscript FIG 3.

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Angular errors as a function of the radial weighting power α for the exact, kurtosis, and Gaussian dODFs for the two-compartment model with f = 0.8 and with f 0.5. In both cases, the crossing angle is set to ξ = 45°. The accuracy of the exact dODF improves with increasing α, while that of the Gaussian dODF is independent of α. The accuracy of the kurtosis dODF depends on α, but not necessarily in a monotonic manner. For f = 0.5, the angular error of the kurtosis dODF has a minimum for α = 4.37.

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NIH-PA Author Manuscript FIG 4.

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Distribution of the absolute angular difference between the principal fiber bundle direction identified with the Gaussian dODF and the primary direction estimated with the kurtosis dODF for human data (left plot). The principal direction corresponds to the global maximum of the dODF. The vertical axis shows the fraction of brain parenchyma voxels based on angular bins of 1°. The distribution is broader for higher α values. The difference exceeds 5° for 36% of the voxels with α = 0, for 49% of the voxels with α 2, and for 58% of the voxels with α = 4. The right plot shows the angular differences as a function of the FA values. These tend to decrease with increasing FA. The offset from zero for high FA values may reflect, in part, a positive noise bias that is a consequence of the FA being a positive semidefinite quantity by construction.

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NIH-PA Author Manuscript FIG 5.

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Fraction of brain parenchyma voxels in which the kurtosis dODF detects 1, 2, or 3 fiber directions (left plot). The sensitivity to fiber crossings (i.e., voxels with 2 or more directions) increases with the radial weighting power α. The right plot shows distributions, for α = 0, 2, and 4, of the angular differences (with a bin size of 1°) between the directions identified by the largest maxima and the second largest maxima for all voxels with FA ≥ 0.2 and in which 2 or more fiber directions are identified. Similar distributions are observed for all three α values. These are well fit by a Gaussian function centered on 90° with a standard deviation of 22° (calculated from the α = 4 data).

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FIG 6.

Maps showing the number of directions detected by the kurtosis dODF for α = 0, 2, and 4 from a single axial brain slice (first row). Voxels rendered in color have MK ≥ 0.9 and are largely composed of white matter. Gray matter and cerebral spinal fluid voxels are in gray scale; diffusion in these voxels is relatively isotropic and the number of detected directions may be strongly affected by noise. The second row shows the angular differences between the principal directions, as identified by the global maxima of the Gaussian and kurtosis dODFs, with the scale bar being calibrated in degrees (colored voxels are again for MK ≥

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0.9 to highlight white matter regions). The third row gives the corresponding maps of the MD, FA, and MK. The calibration bar for the MD is in units of μm2/ms, while the calibration bars for the FA and MK are dimensionless.

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FIG 7.

To test reproducibility, the number of directions detected by the kurtosis dODF (α = 4) was calculated separately for the three trials (first row) using the same slice as for Fig. 6 (which combined data from all three trials). Only white matter (MK ≥ 0.9) voxels are displayed, and similar results are obtained in each case. The second row gives the corresponding maps for the angular differences between the directions identified by the global maxima of the Gaussian and kurtosis dODFs (scale bar is calibrated in degrees).

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NIH-PA Author Manuscript FIG 8.

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Comparison of fiber tractography obtained with the Gaussian (left) and kurtosis (right) dODFs. Identical seed regions within the centrum semiovale were used in both cases, and the kurtosis tractography used α = 4. The richer set of fiber tracks obtained with the kurtosis dODF is consistent with the known anatomy of the centrum semiovale and reflects the kurtosis dODF's ability to resolve fiber crossings. The Gaussian tractography is identical to a standard DTI-based approach, while the kurtosis tractography shows the improvement made possible by utilizing the information obtainable with DKI.

NIH-PA Author Manuscript NMR Biomed. Author manuscript; available in PMC 2015 February 01.

Leading non-Gaussian corrections for diffusion orientation distribution function.

An analytical representation of the leading non-Gaussian corrections for a class of diffusion orientation distribution functions (dODFs) is presented...
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