Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging

Abstract

This article presents two related advancements to the diffusional kurtosis imaging estimation framework to increase its robustness to noise, motion, and imaging artifacts. The first advancement substantially improves the estimation of diffusion and kurtosis tensors parameterizing the diffusional kurtosis imaging model. Rather than utilizing conventional unconstrained least squares methods, the tensor estimation problem is formulated as linearly constrained linear least squares, where the constraints ensure physically and/or biologically plausible tensor estimates. The exact solution to the constrained problem is found via convex quadratic programming methods or, alternatively, an approximate solution is determined through a fast heuristic algorithm. The computationally more demanding quadratic programming-based method is more flexible, allowing for an arbitrary number of diffusion weightings and different gradient sets for each diffusion weighting. The heuristic algorithm is suitable for real-time settings such as on clinical scanners, where run time is crucial. The advantage offered by the proposed constrained algorithms is demonstrated using in vivo human brain images. The proposed constrained methods allow for shorter scan times and/or higher spatial resolution for a given fidelity of the diffusional kurtosis imaging parametric maps. The second advancement increases the efficiency and accuracy of the estimation of mean and radial kurtoses by applying exact closed-form formulae.

Figure 1

MK maps obtained using the standard protocol and (a) UNLS; (b) ULLS; (c) CLLS-H; (d) CLLS-QP; and (e) the reference protocol and ULLS.

Figure 2

MD maps obtained using the standard protocol and (a) UNLS; (b) ULLS; (c) CLLS-H; (d) CLLS-QP; and (e) the reference protocol and ULLS.

Figure 3

FA maps obtained using the standard protocol and (a) UNLS; (b) ULLS; (c) CLLS-H; (d) CLLS-QP; and (e) the reference protocol and ULLS.

Figure 4

Voxels (indicated in red) for which the diffusion and kurtosis tensors estimated using standard (rows 1, 3, and 5) and reference (rows 2, 4, and 6) protocols and ULLS violate the constraints on (a) directional diffusivities; (b) minimum directional kurtoses; and (c) maximum directional kurtoses. The shade of red for each voxel indicates the fraction of constraints that are violated.

Figure 5

MK maps obtained using the fast protocol and (a) UNLS; (b) ULLS; (c) CLLS-QP; and (d) the reference protocol and ULLS. Note that due to the different numbers of gradient directions, the CLLS-H algorithm is not applicable.

Figure 6

MD maps obtained using the fast protocol and (a) UNLS; (b) ULLS; (c) CLLS-QP; and (d) the reference protocol and ULLS.

Figure 7

FA maps obtained using the fast protocol and (a) UNLS; (b) ULLS; (c) CLLS-QP; and (d) the reference protocol and ULLS.

Figure 8

Voxels (indicated in red) for which the diffusion and kurtosis tensors estimated using the fast protocol and ULLS violate the constraints on (a) directional diffusivities; (b) minimum directional kurtoses; and (c) maximum directional kurtoses. The shade of red for each voxel indicates the fraction of constraints that are violated.

Figure 9

Voxels in the (a) corpus callosum and (b) prefrontal white matter from Dataset 1 and their corresponding directional diffusivities (top right), kurtoses (bottom left), and the product b_maxD(n)K(n) (bottom right), estimated using different unconstrained and constrained methods. Parameters controlling the minimum and maximum kurtoses were set to K_min = 0 and C = 3. Note that the directional kurtoses must satisfy b_maxD(n)K(n) < C. Note that the lines connecting the data points are intended to aid with visualizing the points that fall outside of the plotted range, and do not imply any ordering among the gradient directions.

Figure 10

Kurtosis maps for Dataset 2 estimated using the standard protocol, utilizing the CLLS-QP algorithm and the proposed exact expressions; (a) MK; (b) K_║; and (c) K_⊥. Note that the maps are displayed with different scales.