Steven H. Baete^{1,2}, Ying-Chia Lin^{1,2}, Jingyun Chen^{1,2,3}, Ricardo Otazo^{1,2}, and Fernando E. Boada^{1,2}

Higher dimensional diffusion protocols are now routinely acquired in large-scale studies. While these diffusion data sets contain a wealth of information about white matter architecture, this information is not fully exploited when their dimensionality is reduced to simplify statistical correlations with neurocognitive markers over the whole brain. To overcome this limitation, we analyze the full Orientation Distribution Function (ODF) at each voxel using a Low-Rank plus Sparse decomposition to identify key ODF features. We use this approach to link neurocognitive measures to brain structure in a cohort of healthy Human Connectome Project volunteers.

Recent advances in MRI acquisition$$$^{1,2}$$$ make it possible to include high quality diffusion weighted protocols (multi-shell, large number of gradient directions) in large-population explorations of human behavior$$$^{3,4}$$$. Statistical analysis of such data often reduce their dimensionality (e.g., FA-based analysis) to ease correlation with neurocognitive markers across the whole brain$$$^{5}$$$. While such simplified analysis have indeed detected more obvious group differences; they, however, do not fully capitalize on all acquired image volumes. Other, more powerful, whole brain analysis methods employ tractography to identify structurally connected fiber populations globally$$$^{6}$$$ or locally$$$^{7}$$$. Tractography, however, has known limitations$$$^{8}$$$, and since it depends on identified fiber directions, it might discard other features of the ODF.

Here, we use a statistical analysis method, dubbed ODF L+S, which operates on the full Orientation Distribution Functions (ODF) at each voxel$$$^{9}$$$. This approach takes full advantage of the available diffusion information and avoids outlier bias by identifying features of the ODF that are consistent across a subject pool while, at the same time, discarding an individual’s variation via the use of Low-Rank plus Sparse (L + S) matrix decomposition$$$^{10-12}$$$. In the Human Connectome Project (HCP) dataset we localize the correlations of brain structure with neurocognitive measures for episodic memory$$$^{13}$$$ and language tasks$$$^{14}$$$ with the ODF L+S approach.

In vivo healthy subject MRI datasets were downloaded from the HCP consortium led by Washington University, University of Minnesota and Oxford University$$$^{3}$$$: 355 subjects from the December 2015 release (S900, 180/175 female/male, 28.2±3.9y/o); preprocessed$$$^{15}$$$ diffusion MRI (monopolar gradient pulse sequence; 6b0 and 270 q-space samples on three shells, b=1000,2000,3000s/mm2, TR/TE=5500/89.50ms, 1.25mm isotropic, simultaneous multi-slice acceleration of 3; acquired on Siemens 3T Skyra with 100mT/m gradients). We further selected neurocognitive measurements from the HCP Data Dictionary$$$^{16}$$$ relating to episodic memory (NIH Toolbox Picture Sequence Memory Test PicSeq_AgeAdj$$$^{13}$$$) and to language and vocabulary comprehension and decoding (NIH Toolbox Picture Vocabulary Test PicVocab_AgeAdj and Oral Reading Recognition Test ReadEng_AgeAdj$$$^{14}$$$). Diffusion weighted images were reconstructed to diffusion ODFs and registered to the MNI-atlas$$$^{17}$$$ (as included with FSL) with the generalized q-space diffeomorphic reconstruction$$$^{18,19}$$$ (DSIStudio). Tractography was performed with a modified streamline tracking algorithm and images were generated with Matlab and DSIStudio.

For statistical analysis, the registered ODFs of each voxel are reorganized into a matrix M (1 ODF per row, Fig 1), which is decomposed into a low-rank matrix L and a sparse matrix S$$$^{10-12}$$$ as previously described$$$^{9}$$$. The L+S decomposition estimates ODF features common/different within/between groups by minimizing the rank of L, whilst separating the sparse individual variability and outliers in S (Fig 1). Here, the decomposition is solved using the Robust PCA via Nonconvex Rank Approximation algorithm$$$^{20}$$$ which employs a non-convex approximation of the true matrix rank of L rather than the more common nuclear norm ($$$\lambda=1/\sqrt{n}$$$, $$$\mu=0.9$$$). The latter over-penalizes larger singular values, a bias avoided by using a non-convex rank estimate$$$^{20}$$$. The Principal Component (PC) scores of L are used to correlate ODF features (Principal Components $$$PC_i$$$) with independent variables. Multiple comparisons in whole brain analysis are corrected with the Threshold-Free Cluster Enhancement Method (TFCE$$$^{21}$$$) while age and sex are included as nuisance variables. Significant correlations are used to calculate correlation ODFs $$$R_{ODF}$$$ (Fig. 1) which serve as input for tractography.

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Figure 1: Differences between or correlations in groups
of ODFs, taken from registered voxels, are identified by reorganizing the
ODF-values in an ODF-matrix M. Subsequently, the L+S decomposition isolates the
features of M common/different between groups in L and splits the individual
variability and outliers away in S. Significance of ODF differences is assessed
by a statistical analysis of the Principal Component
scores (t) of L. Difference ODFs Δ$$$_{ODF}$$$ can be calculated from the
significant group differences while correlation ODFs R$$$_{ODF}$$$ can be
calculated from significant correlations with a random variable.

Figure 2: a) Voxels with ODFs significantly correlated
with Episodic Memory, as measured with a Picture Sequence task, in a cohort of
healthy HCP volunteers detected after isolating key ODF-features using the ODF
L+S approach. b-d) Fiber tractography of significantly correlated R$$$_{ODF}$$$
indicated in panel a) (p < 0.05, QA$$$_{RODF}$$$ < 0.001). Voxels with FWE p-value < 0.05 are
overlaid on the MNI-atlas and tracts are displayed within a surface and on a
slice derived from the MNI-atlas.

Figure3: a) Voxels with ODFs
significantly correlated with language and vocabulary comprehension and
decoding, as measured with Picture Vocabulary and Reading Recognition tests, in
a cohort of healthy HCP volunteers detected after isolating key ODF-features
using the ODF L+S approach. b-c) Fiber tractography of significantly correlated
R$$$_{ODF}$$$ indicated in panel a) (p < 0.05, QA$$$_{RODF}$$$ <
0.001). Voxels with FWE p-value <
0.05 are overlaid on the MNI-atlas and tracts are displayed within a surface
and on a slice derived from the MNI-atlas.