Synopsis

Recent advances in data acquisition make it possible to use high quality diffusion data for routine in vivo study of white matter architecture. The dimensionality of these data sets requires a more robust methodology for their statistical analyses than currently available. Here we propose a apply Low-Rank plus Sparse (L+S) matrix decomposition to reliably detect voxelwise group differences in the Orientation Distribution Function that are robust against the effects of noise and outliers. We demonstrate the performance of this approach to replicate the established negative association between global white matter integrity and physical obesity in the Human Connectome dataset.

Purpose

Higher Angular Resolution Diffusion Imaging (HARDI) methods, such as Diffusion Spectrum Imaging (DSI[1]) and multishell Q-ball imaging[2] are robust tools for non-invasive imaging of in vivo white matter architecture. These methods capture the complex intravoxel crossings[1,3] through the synthesis of Orientation Distribution Functions (ODFs). Recent improvements in sequence acquisition times[4,5,6] have made HARDI 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 datasets. Often, whole brain statistical analysis focuses on connectivity matrices, either locally[7] or globally[8,9]. In some approaches, the tractograms also inform tract-specific smoothing[9] and enhancement of statistical maps along the tracts[7,9] using Threshold-Free Cluster Enhancement approaches (TFCE[10]). Whilst these tractography based methods are powerful, they suffer from problems related to imperfections in the tractography approach[11] and, by focusing on the identified fiber directions, they might discard subtle differences captured in the ODFs.

A promising ODF statistical analysis method is the voxelwise whole brain analysis of ODFs[12] based on Principal Component Analysis (PCA), though this method suffers from outliers and individual variability. In this work, we present the application of a novel approach for isolating ODF features that are common/different between subject cohorts from subject-specific variance via the Low-Rank plus Sparse (L+S) Matrix Decomposition[13,14,15]. We illustrate this approach by replicating the established association between global white matter integrity and physical obesity[7,16] in the Human Connectome Project dataset.

Methods

Structural (MPRAGE, 1mm isotropic, TR/TE=2400/2.14ms, 2xGRAPPA) and Diffusion (monopolar gradient pulse sequence, 270 q-space samples on three shells, b=1000,2000,3000 s/mm$$$2$$$, TR/TE=5500/89.50ms, 1.25mm isotropic, simultaneous multi-slice acceleration of 3[5]) MRI datasets were downloaded in preprocessed format from the HCP consortium led by Washington University, University of Minnesota, and Oxford University[17] (acquisition on a Siemens 3T Skyra with 100mT/m gradients). We selected the first 144 subjects from the December 2015 release for group analysis (64/80 male/female, age 28.5±4.0, BMI26.3±5.0). Diffusion reconstructions are performed in native subject space using Generalized Q-sampling Imaging reconstruction (GQI[18], implemented in Matlab). Spatial normalization transformations were calculated by the stepwise registration of the subject MPRAGE to the $$$T_1$$$-weighted MNI-152-atlas[19], as included in FSL, and the registration of the subjects diffusion images to the subjects MPRAGE. The calculated transformations were then later applied to the reconstructed diffusion datasets. Both registrations and transformations are performed using elastix and transformix[20]. Images were generated using Matlab and DSI Studio[18]. For tractography, a modified streamline tracking algorithm was used (DSI Studio).
For statistical analysis the registered ODFs of each voxel are reorganized in a matrix M (1 ODF per row, Fig 1) and M is decomposed in a Low-Rank matrix L and a Sparse matrix S[13,14,15]$$\text{minimize}\,\,||L||_*+\lambda||S||_1\\\text{subject to}\,\,L+S=M$$using the Alternating Directions algorithm with $$$||\cdot||_*$$$ the nuclear norm, $$$||\cdot||_1$$$ the $$$l_1$$$-norm and $$$\lambda = 1/\sqrt(n)$$$ a universal choice for the trade-off between L and S (with n the size of the largest dimension of the matrix M). This 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). The Principal Component (PC) scores of L are then used as input for statistical tests to evaluate group ODF differences or relationships of ODF features (Principal Components $$$PC_i$$$) with independent variables (in this abstract, BMI, age and sex). For whole-brain statistics, multiple comparisons are corrected with the Threshold-Free Cluster Enhancement Method (TFCE[10]).
When observing significant group differences between PC-scores $$$t_{i,j}$$$ of groups A and B or relations ($$$r_i$$$ with $$$p_i<p_{thres}$$$) of ODF features $$$PC_i$$$ with independent variables, we can calculate difference ODFs $$\Delta_{ODF}=\sum\limits_{i,p_i<p_{thres}}PC_i\left(\frac{1}{n_A}\sum_{j \in A}{t_{i,j}}-\frac{1}{n_B}\sum_{j\in B}{t_{i,j}}\right)$$ of correlation ODFs $$$R_{ODF}=\sum\limits_{i,p_i<p_{thres}}PC_ir_i$$$ respectively.

Results and Discussion

Conclusion

Acknowledgements

References

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