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. 2016 Jul 15;135:345-62.
doi: 10.1016/j.neuroimage.2016.02.039. Epub 2016 Feb 23.

Q-space Trajectory Imaging for Multidimensional Diffusion MRI of the Human Brain

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Free PMC article

Q-space Trajectory Imaging for Multidimensional Diffusion MRI of the Human Brain

Carl-Fredrik Westin et al. Neuroimage. .
Free PMC article

Abstract

This work describes a new diffusion MR framework for imaging and modeling of microstructure that we call q-space trajectory imaging (QTI). The QTI framework consists of two parts: encoding and modeling. First we propose q-space trajectory encoding, which uses time-varying gradients to probe a trajectory in q-space, in contrast to traditional pulsed field gradient sequences that attempt to probe a point in q-space. Then we propose a microstructure model, the diffusion tensor distribution (DTD) model, which takes advantage of additional information provided by QTI to estimate a distributional model over diffusion tensors. We show that the QTI framework enables microstructure modeling that is not possible with the traditional pulsed gradient encoding as introduced by Stejskal and Tanner. In our analysis of QTI, we find that the well-known scalar b-value naturally extends to a tensor-valued entity, i.e., a diffusion measurement tensor, which we call the b-tensor. We show that b-tensors of rank 2 or 3 enable estimation of the mean and covariance of the DTD model in terms of a second order tensor (the diffusion tensor) and a fourth order tensor. The QTI framework has been designed to improve discrimination of the sizes, shapes, and orientations of diffusion microenvironments within tissue. We derive rotationally invariant scalar quantities describing intuitive microstructural features including size, shape, and orientation coherence measures. To demonstrate the feasibility of QTI on a clinical scanner, we performed a small pilot study comparing a group of five healthy controls with five patients with schizophrenia. The parameter maps derived from QTI were compared between the groups, and 9 out of the 14 parameters investigated showed differences between groups. The ability to measure and model the distribution of diffusion tensors, rather than a quantity that has already been averaged within a voxel, has the potential to provide a powerful paradigm for the study of complex tissue architecture.

Keywords: DDE; DTI; Diffusion MRI; Diffusion tensor distribution; Microscopic anisotropy; Microscopic fractional anisotropy μFA; QTI; SDE; TDE; q-space; q-space trajectory.

Figures

Fig. 1
Fig. 1
Synthetic examples of diffusion tensor distributions that cannot be differentiated using conventional SDE-based dMRI sequences such as DTI, HARDI, DSI, and DKI, but can be distinguished using the proposed QTI framework. The first row shows synthetic examples of diffusion tensor distributions within a voxel that yield an isotropic average diffusion tensor 〈D〉. As a consequence, these different structures are indistinguishable with conventional DTI. The second row shows the DTD model corresponding to each synthetic example. The DTD model includes average diffusion tensors, shown by 3×3 matrices, and covariance tensors, shown as 6×6 matrices, along with the scalar invariants from Eqs. 14, 19 and 22. In these graphical representations, green is positive, black is zero, and red is negative. Although the three examples have identical average diffusion tensors they have different covariance tensors ℂ. See the Diffusion and modeling and estimation section (2.3) for details on model estimation. The corresponding fourth-order tensors as observed by diffusional kurtosis imaging using only the linear b-tensors obtained with SDE encoding are equal in these three examples. The ability to measure and model the distribution over diffusion tensors, rather than a quantity that has already been averaged within a voxel, has the potential to provide a powerful new paradigm for the study of complex tissue architecture.
Fig. 2
Fig. 2
Examples of gradient waveforms and trajectories that produce “linear” and “planar” b-tensor shapes, from the SDE and DDE experiments. Color coding is as follows: in the gradients waveform column, red-green-blue defines the x-y-z gradient directions; in the gradient trajectory column, slewrate, slow to fast, is encoded by red-yellow-green-blue from zero to max; in the q-space trajectory column, trajectory speed, slow to fast, is encoded red-yellow-green-blue; in the b-tensor column, the diagonal elements of the b-tensor are mapped to red-green-blue. In the rank-1 case, this b-tensor color mapping corresponds to the principal eigenvector direction, analogous to the standard color coding for diffusion tensors. In the planar case, rather than relying on a randomly oriented major eigenvector for color information, the effect is to sum the colors in the plane (as in the bottom row where red plus green gives yellow in the RGB sense).
Fig. 3
Fig. 3
TDE and QTE encoding schemes can be used to produce isotropic diffusion encoding, with a spherical b-tensor. Color coding is as follows: in column 1, red-green-blue defines the x-y-z gradient directions; in columns 2 and 3, slewrate and trajectory speed are encoded by red-yellow-green-blue, where red is slow and blue is fast; in column 4 the diagonal elements of the b-tensor are mapped to red-green-blue (and thus gray indicates these elements of the isotropic b-tensor are the same). The encoding scheme in the second row was generated by transforming the TDE encoding from the top row: first, pulses were shifted closer together so the green positive pulse aligned with the negative red pulse etc., and then a transform was applied to make the encoding isotropic, as in Eq. 5. In the bottom row, the isotropic diffusion encoding from qMAS produces a lasso-like q-space trajectory.
Fig. 4
Fig. 4
QTE encoding: an example family of b-tensors, with shapes ranging from linear to spherical to planar. Gradient waveforms and trajectories are shown along with the resulting q-space trajectories and b-tensors. The curves were produced by transforming the curve with efficient isotropic encoding (C) to yield linear encoding (A), prolate encoding (B), oblate encoding (D), and planar encoding (E). We have used b-tensors from this family in our proof-of-concept clinical study, where multiple rotated versions of this family were applied.
Fig. 5
Fig. 5
This visualization of four proposed measures demonstrates how the measures would change in eight illustrative synthetic macrodomains (voxels). Note that these measures intuitively separate size, shape, and orientation coherence, as well as providing the traditional macroscopic anisotropy.
Fig. 6
Fig. 6
The signal plots show the MR signal versus b-value measured in four synthetic datasets using b-tensors of three different ranks, where ranks are identified with shapes such that rank 1 is “linear,” rank 2 is “planar,” and rank 3 is “spherical.” The four synthetic datasets represent four distinct scenarios with different distributions of microenvironments (spheres of one size, spheres of many sizes, sticks of one size and multiple orientations, and ellipsoids of multiple sizes and orientations). Notice the similarities of the light green signal curves in the second, third and fourth plots. This illustrates that when using traditional SDE (linear b-tensor), multiple different microenvironments can produce similar signal responses. In fact, they may even be identical. This shows that using simple model-based estimation of the distribution using SDE data and a predefined model of specific shape, a distribution over size and orientation of that shape that fits the data can always be found, but the parameters found will of course be meaningless if the predefined model does not reflect the underlying tissue architecture.
Fig. 7
Fig. 7
In the proposed sampling scheme, Platonic solids are used to ensure even sampling in all directions [55]. Number of distinct sample directions are listed below each solid. Note the dual nature of the icosahedron and dodecahedron: the center of the each face of the icosahedron corresponds to the vertex of the dodecahedron, and vice versa. Further, the center of each face of the truncated icosahedron corresponds to a corner in either the icosahedron or dodecahedron. When the icosahedron, dodecahedron, and truncated icosahedron nest, their vertices (right) give evenly spaced sampling directions. To employ this scheme, a gradient waveform is rotated so that the symmetry axis of its b-tensor aligns with the desired direction (Table 2).
Fig. 8
Fig. 8
In vivo data from the preliminary data set. A. MR images diffusion encoded with b = 2000 s/mm2 and b-tensors of five shapes, i.e., linear, prolate, isotropic, oblate, and planar. Data is displayed from two different b-tensor rotations (directions) sampled from the icosaheadron (icosa 1 and icosa 2, top and bottom rows). Note that the linear and the planar measurements are dual, and thus, where the linear measurement is bright the planar is dark; see yellow circles. This can be compared to the Funk-Radon transform that is performed in q-ball imaging, where the diffusion signal is the result of integration on a great circle [61]. The planar measurement inherently does this integration. B. Signal-versus-b curves averaged across all directions for corpus callosum (red), crossing white matter (green) and gray matter (blue). Curves are encoded with increasing color brightness from linear (dark, I), through prolate (II), isotropic (III), oblate (IV) to planar encoding (full color, V). As expected, the linear encoding attenuates the signal the least, while the isotropic encoding attenuates it the most, and the attenuations from the other encodings are in between. This in vivo data can be compared to the synthetic curves from Figure 6, motivating the need for a model, such as the DTD, that can decipher this data.
Fig. 9
Fig. 9
Examples of parameter maps obtained in QTI, calculated from data acquired with the clinical protocol. A. Top row shows the mean diffusivity MD (Eq. 14), the bulk VMD and shear variances Vshear (Eqs. 19, 22), and their sum Viso. B. Middle row shows examples of normalized variance measures. C. Lower row shows different kurtosis measures derived from QTI: Total mean kurtosis (MK, Eq. 37) separated into two components, bulk and mean kurtosis represented by Kbulk and Kshear(Eq. 41) and (Eq. 42). The anisotropy-related kurtosis Kμ (Eq. 43) is shown in the rightmost panel.
Fig. 10
Fig. 10
The anisotropy measure FA and the microscopic anisotropy μFA, which is known from previous DDE and qMAS studies [9, 51], are straightforwardly calculated from CM and Cμ.
Fig. 11
Fig. 11
Comparison of normalized measures in schizophrenia patients (SZ) and healthy controls (CTR). The CMD, CM and Cμ averaged across the white matter were all three significantly reduced in the schizophrenia group. Changes in Cc was not found to be significant between the groups. Significance was tested using the Wilcoxon rank-sum U-test.
Fig. 12
Fig. 12
Comparison of measured data to simulations of two hypothesized pathologies in schizophrenia. Scatter plots show Vshear versus FA (left) and VMD versus MD. Data points represent an average across the cerebral white matter. The two parameters VMD and Vshear are related to variances of the distribution, corresponding to changes in size, and changes in orientation/shape respectively. FA and Vshear were weakly correlated. MD and VMD were strongly correlated. Solid lines show results from analysis of data simulated to represent a demyelination hypothesis, i.e. increasing radial diffusivity (red), and a free-water hypothesis, i.e. replacing WM with isotropic freely diffusing water (blue). Example voxels representing these hypotheses is shown to the right.

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