Low Rank plus Sparse Decomposition of ODF Distributions for Improved Detection of Group Differences in Diffusion Spectrum Imaging

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

Diffusion
Spectrum Imaging (DSI)^{1,2} is a robust tool for non-invasive imaging
of in vivo white matter tract architecture. DSI's performance results
from its model-independent determination of the Orientation
Distribution Function (ODF)^{1}, which allows it to capture complex
intravoxel fiber crossings^{2,3}. Recent improvements in sequence
design^{4,5} have led to data acquisition times that, for the first
time, make DSI a routine viable and practical tool for clinical
applications and neuroscience research. This evolution has
highlighted the need for a robust methodology for statistical
analysis of group ODF data sets. Previously proposed methods for
Diffusion Tensor matrices^{6} are not well suited for the much higher
dimensionality of ODFs. The information contained in the ODF could
allow contrasting subject ODF values to that of a normal population^{7}.
However, the methodology currently available for such assessments is
currently limited to pre-determined skeletons of fiber directions
which do not capture all the information contained in the ODF. Other
methods focus on the connectome level, evaluating differences in
structural connections on a local^{8} or global^{9} level, thus possibly
missing more subtle differences in diffusion behavior captured in the
ODF.

A promising
ODF-analysis method is the voxelwise whole brain group analysis of
ODFs^{10} based on Principal Component Analysis (PCA). Here, we apply
PCA to the reduced noise ODFs as estimated by the L-component of a
Low-Rank plus Sparse (L+S) Matrix Decomposition^{11,12} of the ODF
distributions.

**ODF Generation** Groups of RDSI datasets of two crossing fiber bundles (60°,
$$$\lambda_1$$$/$$$\lambda_2$$$/$$$\lambda_3$$$
1.00/0.10/0.10$$$\mu$$$m$$$^2$$$/ms) and a water pool (10%) are
simulated with Radial (59 radial lines, 4 shells) q-space sampling^{5}.
Rician noise (SNR 30 in non-diffusion-attenuated signal) and
group-outliers (10%, SNR 5%) are added to the simulated diffusion
signals before reconstructing the ODFs. Each group contains 100 ODFs,
simulating a study with 100 co-registered cases per group. Group
differences are simulated by changing Axial diffusivity ($$$D_{ax}
=\lambda_1$$$), Radial diffusivity
($$$D_{rad}=(\lambda_1+\lambda_2)/2$$$) of one of the two fibers
fiber or crossing fiber angle of one group. Note that both an
increase in $$$D_{ax}$$$ and a decrease in $$$D_{rad}$$$ of a fiber
lead to an increase in Quantitative Anisotropy (QA^{13}). RDSI
reconstructions5, incorporating variable sample density correction,
were performed using custom-made software (Matlab, Mathworks).

**Conventional
PCA** To test for voxel-wise group differences, the ODF-values of
both groups are reorganized in a matrix M (1 ODF per row, Fig. 1) and
Principal Component Analysis (PCA, using Singular Value Decomposition
(SVD)) is performed on M. Since the first PC captures the most
significant variance in the ODFs, regardless of the group membership,
the group difference can be evaluated by a two-sample t-test of the
first PC-score^{10} (i.e. the projections of ODFs on the first PC).

**Low-Rank-Decomposition-Enhanced
PCA** In a second approach, the ODF-matrix M is decomposed as the
sum of a Low-Rank matrix L and a Sparse matrix S ($$$M=L+S$$$)^{11,12},
using the Alternating Directions algorithm^{11}. This decomposition,
estimates the underlying reduced noise ODFs by minimizing the rank of
L, whilst separating the sparse noise and outliers in S (Fig. 1).
Subsequently, the first PC-scores of the L-matrix can be used the
evaluate group differences (two-sample t-test^{10} as above).

ODF group comparisons (Fig. 2) where one group has reduced $$$D_{rad}$$$ show that statistical tests of the PC-scores for the ODF-matrix M ($$$p_{M,PCA}$$$) and the Low-Rank matrix L ($$$p_{L,PCA}$$$) can detect large differences (50%) in $$$D_{rad}$$$. A similar analysis demonstrates that smaller differences, in the range of 10% to 20%, in $$$D_{rad}$$$ are more robustly identified through the PCA analysis of L than that of M. In other words, by separating noise and outliers in S, the PCA analysis of L greatly improves the detection of ODF group differences. This advantage improves detection of differences in both $$$D_{ax}$$$ and $$$D_{rad}$$$ (Fig. 3a-b). Not surprisingly, changes in crossing angle (Fig. 3c) are detected equally well by both approaches as these correspond to a shift in the peaks of the ODF rather than a change in their heights. There is a linear relationship between test statistics and changes in diffusion parameters (Fig. 3).

The risk of
over-regularizing when estimating the reduced noise ODFs in L, is
avoided by choosing the regularization parameter
$$$\lambda=1/\sqrt{max(n_{vertices},n_{subjects})}$$$^{11}.

Differences between two groups of ODFs in a registered voxel are identified by reorganizing ODF-values and performing a 2-sample t-test on the scores of the first Principal Component of the resulting matrix M. An intermediate L+S-decomposition splits noise and outliers (S) from the signal (L) for improved group difference detection.

Detection of reductions in Radial Diffusivity ($$$D_{rad} \downarrow$$$ leads to QA $$$\uparrow$$$) of one fiber in Group B. Large reductions of $$$D_{rad}$$$ (50%) are easily detected by direct Principal Component Analysis (PCA) of the ODFs, though smaller differences (10% drop) are better identified by PCA of the Low-Rank L-matrix.

Simulations of percentual changes in Axial Diffusion ($$$D_{ax}$$$, a), Radial Diffusion ($$$D_{rad}$$$, b) and crossing angle (c) of ODFs of two crossing fibers. Two-sided t-test test statistics and p-value are plotted relative to average ODF group RMSE and percentual changes in fiber characteristics.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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