# An analytical representation of the leading non-Gaussian corrections for a class

An analytical representation of the leading non-Gaussian corrections for a class of diffusion orientation distribution functions (dODFs) is presented. TH-302 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 usually illustrated with numerical simulations for a two-compartment model of fiber crossings and for human brain data. These results indicate that this 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. is a normalization constant. The power affects the radial weighting of the dODF with larger 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. 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 Such a Gaussian dODF however has a significant shortcoming in that it does not reliably predict fiber bundle directions for voxels having two or IKBKE antibody 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). 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 this 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 usually that it is compatible with diffusional TH-302 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. METHODS Gaussian dODF The diffusional average of an arbitrary function indicating the components of s. The Gaussian approximation for the dPDF is usually defined by > -1. Now let us define a dimensionless tensor U ≡ is the mean diffusivity and set the normalization constant to be = ±ê1 where ê1 is the principal DT eigenvector. Note that the locations of these maxima are independent of the radial weighting power for the sake of notational simplicity. Because of the normalization condition of Eq. [3] we must have dependent” dODF so that provides a natural means for interpolating between the Gaussian and exact dODFs. Corrections for the Gaussian dODF may TH-302 then be systematically derived in terms of the Taylor series TH-302 for in powers of about = 0. If one makes the standard assumption that indicates the components of U are the components of the DKT (19-21 24 and the sums around the indices (= 1 in Eq. [16] yields the kurtosis dODF 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 expansion let us first consider an example with is a length that controls the diffusion wave vector.