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 matrices⁶ 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 population⁷. 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 local⁸ or global⁹ 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 ODFs¹⁰ 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 sampling⁵. 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 (QA¹³). 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-score¹⁰ (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 algorithm¹¹. 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-test¹⁰ 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.