Whole brain voxel-wise analysis of single-subject serial DTI by permutation testing

Abstract

Diffusion tensor MRI (DTI) has been widely used to investigate brain microstructural changes in pathological conditions as well as for normal development and aging. In particular, longitudinal changes are vital to the understanding of progression but these studies are typically designed for specific regions of interest. To analyze changes in these regions traditional statistical methods are often employed to elucidate group differences which are measured against the variability found in a control cohort. However, in some cases, rather than collecting multiple subjects into two groups, it is necessary and more informative to analyze the data for individual subjects. There is also a need for understanding changes in a single subject without prior information regarding the spatial distribution of the pathology, but no formal statistical framework exists for these voxel-wise analyses of DTI. In this study, we present PERVADE (permutation voxel-wise analysis of diffusion estimates), a whole brain analysis method for detecting localized FA changes between two separate points in time of any given subject, without any prior hypothesis about where changes might occur. Exploiting the nature of DTI that it is calculated from multiple diffusion-weighted images of each region, permutation testing, a non-parametric hypothesis testing technique, was modified for the analysis of serial DTI data and implemented for voxel-wise hypothesis tests of diffusion metric changes, as well as for suprathreshold cluster analysis to correct for multiple comparisons. We describe PERVADE in detail and present results from Monte Carlo simulation supporting the validity of the technique as well as illustrative examples from a healthy subject and patients in the early stages of multiple sclerosis.

Figure 1

Overall schematics of permutation testing to calculate voxel-wise p-values and multiple-comparison-corrected cluster-wise p-values in a serial DTI study of a single subject. This corresponds to steps 4 and 5 in the PERVADE procedures (see human brain data analysis section on the methods). (a) DWIs (including b=0) are either kepted as observed (i=1) or permuted (i=2,... N’).For the simplicity of illustration, it is assumed that only seven images are acquired in each time point (A or B). Colored boundaries indicate the observed time point of each DWI. Permutation is done separately in each gradient direction (see figure 2). (b) Two FA maps are calculated from (original or reassigned) DWIs by diffusion tensor processing. (c) Two FA maps are subtracted to create ΔFA map. (d) All ΔFA maps (total N’) represent the map of voxel-wise permutation distribution of ΔFA. (e) Voxel-wise p-value maps are estimated by comparing ΔFA maps wit h the permutation distribution from (d). (f) Clusters are defined by thresholding p-value maps. (g) Sizes of the largest clusters in each cluster mask represent the permutation distribution of image-wise maximal cluster size. (h) All the clusters in the observed cluster mask are compared to the permutation distribution from (g) to estimate cluster-wise p-values, and only clusters p<0.05 are declared significant.

Figure 2

Detailed schematics of permutation testing to estimate voxel-wise p-values in a serial DTI study of a single subject. It is assumed that in each time point the DTI scan of seven DWIs (first DWI can be regarded as b=0) are repeated twice. In each DTI dataset (inside the black dotted box), diffusion-weighted signals S($S\left(\overrightarrow{b}\right)$) in the same row are acquired by applying the same diffusion gradients. ${\overrightarrow{b}}^A$ or ${\overrightarrow{b}}^B$ stresses that the effective gradients can be different in two time points due to the head positioning. Boxes around the signals are colored based on the diffusion gradients in the observed data while solid and dashed linestyles indicate the original time point A and B. On the right side is one possible permutation of DTI dataset. Permutation is done separately for each gradient (thus signals in the same gradient forms the exchangeability block), easily identified by the colors. For each N’-1 permuted and one observed DTI data, ΔFA from two time points are calculated and voxel-wise p-value of observed FA difference are estimated as the probability of more (or equal) extreme difference than the observed $\widehat{\theta }$ from the null distribution of N’ differences, i.e. proportion of counts outside the red dashed lines to all N’ counts in the histogram of right lower corner.

Figure 3

Qunatile-quantile plots for expected p-values following the uniform distribution between 0 and 1 versus observed p-values by permutation testing under diverse simulation conditions. Blue and green symbols indicate uncorrected and corrected (permuting the rotated gradients paired with the diffusion-weighted signals and equalizing the gain factors) permutation scheme while red correspond to perfect match between the expected and the observed. (a) Directions of diffusion encoding gradients set rotated by 20 degrees each in x, y and z in one of two time points only. (b) Gradients rotated by 10 degrees. (c) MR signal gain factor larger by 10% in one of two time points. (d) Gain factor larger by 5%.

Figure 4

Healthy volunteer. [Top row] p-value maps from the baseline (0 month) versus 0 month (same day, subject was moved in between), 1 month, and 2 month as well as FA map from the baseline. P-values are rescaled by log transformation, and the maximal intensity displayed is p=0.03. White matter regions (as well as the whole brain) used for masking before permutation procedures are shown in red contours. [Bottom left] Q-Q plot of estimated voxel-wise p-values in the whole brain white matter of baseline versus 2 month. [Bottom right] Histogram of baseline versus 2 month permutation distribution of image-wise maximal cluster size from N’=1000 permutation trials. Critical value was 11 (i.e. probability of maximal cluster size ≥ 11 voxels was around 0.05). The largest cluster in the observed p-value map was 8 (p=0.312) thus no clusters were declared significant.

Figure 5

Patient #1.[Top row] Registration of the baseline FA map to the reference FA map (10 months) by linear registration only and non-linear registration (after linear registration). Images are magnified to splenium of corpus callosum and nearby structures. Red contour denotes the border defined in the reference image. [Bottom row] P-value maps for the same anatomic structure either by linear or non-linear registration for baseline versus 10 months or 16 months. Significant clusters detected by PERVADE are displayed in the smaller embedded boxes.

Figure 6

Patient #2. [Top row] b=0 images at different time points showing lesions around the left posterior horn of the lateral ventricle emerging and fading over time. Images are smoothed to the same degree as PERVADE and constrast / level were adjusted for the best visibility of lesions. Yellow contours demarcate the regions of the new lesions at the time of the appearance. [Middle row] Detected significant clusters for baseline versus subsequent time points. [Bottom row left] Permutation distribution of ΔFA as well as observed ΔFA for three voxels (NAWM and not within a significant cluster, NAWM and within a significant cluster, and core of a new lesion). Locations of these voxels are shown as well. [Bottom row right] 3D rendering of the tracks (light green), significant clusters (magenta) and lesions visible at b=0 images (cyan). Note that significant clusters are also present within the lesions though they are obscured by rendered lesions.

Figure 7

Patient #3. [Left column] b=0 images of baseline versus 3 months as well as detected clusters. Arrowheads and arrows indicate the lesion fading over time and detected NAWM cluster each. [Middle column] ΔFA, p-values and significant clusters in a sagittal slice (viewed from left) including these clusters. [Right column] 3D rendering of the tracks (light green), cluster within the lesion (orange) and cluster in the NAWM (blue). Cluster-wise p-values are shown as well.