Structural MR images of all subjects were interpreted by a board-certified neuroradiologist as being free of any morphological abnormalities of the brain. DTI tractography was successfully performed for all 12 white matter pathways in all 44 subjects, with the exception of the right arcuate fasciculus in four subjects. The means and standard deviations for the FA, MD, AD, and RD values for each tract are provided in . Left hemispheric tracts tended to have greater FA than their right hemispheric counterparts; this asymmetry was most pronounced for the dorsal cingulate bundle, as has been previously reported (Gong et al., 2005
; Wakana et al., 2007
). The generally lower RD values in the left-sided tracts also reflect their higher FA compared to the right cerebral hemisphere.
Mean and standard deviation of tract-based DTI parameters. Mean diffusivity (MD), axial diffusivity (AD), and radial diffusivity (RD) are in units of 10−3 mm2/sec. Fractional anisotropy (FA) is dimensionless.
Spatial overlap between tracts in the same hemisphere was computed as the percentage of shared voxels, as specified in the Methods. The only significant overlaps were found between the IFO and the ILF and between the IFO and the UF. The mean percent shared voxels between left IFO and left ILF was 10.7% ± 3.3% and that between left IFO and left UF was 11.0% ± 3.7%. The mean percent shared voxels between right IFO and right ILF was 11.4% ± 3.7% and that between right IFO and right UF was 7.3% ± 3.1%. Negligible degrees of overlap were seen between the AF and the ILF and between the ILF and the UF in each hemisphere, with 1% or fewer shared voxels.
Matrices for the Spearman’s rank correlation coefficient ρ for each of the four DTI parameters are given in –, with the trivial values of unity along the main diagonal excluded. All of the computed correlation values were positive; there were no inverse correlations between white matter tracts for any of the DTI parameters. Each of the four correlation matrices was evaluated for equivalence to the identity matrix using the test statistic uI, as detailed in the Methods. The null hypothesis was rejected for all four matrices. The four subjects for whom the right arcuate fasciculus could not be tracked were excluded from this particular analysis since the test statistic cannot be used with missing values. This is valid uI because, if a subset of a matrix differs from identity, then the entire matrix also differs from identity. The value of for the FA correlation matrix was 310.6, corresponding to p < 0.0001 uI for rejecting the null hypothesis. The value of for the MD correlation matrix was 431.4, uI corresponding to p < 0.0001. The value of for the AD correlation matrix was 285.5, uI corresponding to p < 0.0001. The value of for the RD correlation matrix was 365.6, uI corresponding to p < 0.0001. Hence, it is concluded that statistically significant correlations between tracts exist for all four DTI parameters.
Matrix of Spearman’s ρ values for correlation of FA between white matter tracts.
Matrix of Spearman’s ρ values for correlation of RD between white matter tracts.
Each of the four correlation matrices was evaluated for homogenous structure, implying equal correlations throughout the off-diagonal elements in the correlation matrix, using the test statistic as uH detailed in the Methods. The null hypothesis was rejected for all four matrices. The four subjects for whom the right arcuate fasciculus could not be tracked were excluded from this particular analysis since the test statistic uH cannot be used with missing uH values. This is valid because, if a subset of a matrix deviates from homogenous structure, then the entire matrix also deviates from homogenous structure. The value of uH for the FA correlation matrix was 141.6, corresponding to p < 0.0001 for rejecting the null hypothesis. The value of uH for the MD correlation matrix was 121.9, corresponding to p < 0.001. The value of uH for the AD correlation matrix was 109.5, corresponding to p < 0.01. The value of for the uH RD correlation matrix was 127.5, corresponding to p < 0.001. Hence, it is concluded that there are statistically significant variations in inter-tract correlations for all four DTI parameters. Based on the test statistic uH, FA was the DTI parameter with the greatest variation in inter-tract correlations and AD was the DTI parameter with the least variation. The range of inter-tract FA correlations was also the greatest among the four DTI parameters, with the lowest value of ρ = 0.15 found between the right UF and the right CST and the highest value of ρ = 0.88 observed between the left and right IFO.
After establishing that microstructural correlations between white matter tracts exist and have statistically significant variation, specific patterns of inter-tract relationships were sought in the correlation matrices. For all of the correlation analyses described below, missing data points from the four subjects in whom the right AF could not be tracked were excluded on a case-wise basis, i.e. correlations between the right AF and other tracts were limited to those 40 subjects in which the right AF was tracked, but the full 44 subjects were still used to perform correlations between tracts other than the right AF. We begin with the FA correlation matrix because FA is widely accepted as the most sensitive measure of white matter microstructural integrity among the four DTI parameters examined herein. Unsurprisingly, homologous pathways of the left and right cerebral hemispheres had high correlation coefficients for FA; however, there was wide disparity in the strengths of coupling (). The strongest correlation among homologous tracts was found between FA of the left and right IFO (ρ = 0.88), whereas the weakest correlation was between the left and right AF (ρ = 0.50). The relationship between FA of left and right CB was also relatively weak (ρ = 0.57), with an intermediate correlation coefficient found between left and right CST(ρ = 0.62), and relatively stronger associations between left and right UF (ρ = 0.70) and left and right ILF (ρ = 0.73). As expected, FA correlation coefficients between non-homologous tracts were generally lower than that between homologous tracts. Nevertheless, some of these non-homologous FA correlation coefficients exceeded those of the more weakly linked homologous tracts (, ). There were five pairwise FA correlation coefficients between non-homologous pathways that were greater than or equal to 0.7: left ILF - left IFO (ρ = 0.75); left ILF - right IFO (ρ = 0.73); left UF - left IFO (ρ = 0.70); right UF - left IFO (ρ = 0.70); and right UF - right IFO (ρ = 0.71). The weakest FA correlation coefficients were almost uniformly found between the CST on either side and the five pairs of association fiber pathways (, ). All values of ρ between CST and association tracts were less than 0.3 except for right CST - right ILF (ρ = 0.35) and right CST - left AF (ρ = 0.34). In contradistinction, all values of ρ among the association pathways were greater than 0.3. Similar to the findings in the FA correlation matrix, it can also be ascertained from the other three correlation matrices that homologous tracts had generally greater degrees of correlation than non-homologous tracts; however, for each DTI parameter, there were specific examples of non-homologous axonal pathways that were more strongly correlated than many, or even all, of the homologous ones.
Figure 2 Scatterplots of FA values show wide variation of inter-tract correlations between homologous pairs (A, B) and non-homologous pairs (C, D), as measured by the Spearman rank correlation coefficient ρ. The left and right IFO (A) are the most strongly (more ...)
To more systematically investigate the specific patterns of covariation among white matter tracts, agglomerative hierarchical clustering using average distance linkage was applied to each of the four correlation matrices. The degree of uncertainty for each of the linkages was calculated from multiscale bootstrapping, with statistical significance at a 95% confidence level threshold which is equivalent to α =0.05. The resulting dendrogram for FA () shows that there are three statistically significant clusters at a confidence level of 95% or greater. Since FA in the left and right IFO had the strongest correlation of any two tracts (ρ = 0.88), they represent the first linkage in the hierarchical clustering, with a statistical significance level approaching 100%. The left and right CST also represented a significant cluster at a virtually 100% confidence level. Furthermore, the ten association tracts formed a statistically significant grouping distinct from the bilateral CST at a 99% confidence level. Hence, the paired corticospinal tracts represent an outgroup that was found to be the most dissimilar of the twelve white matter pathways based on the FA correlation matrix. This partition between the association and projection tracts resulted naturally from the large FA correlational distances between the bilateral CST and the other ten axonal pathways. Of the ten association tracts, the cingulum bundles were the most distant outgroup; this pair of limbic association tracts clustered separately from the eight neocortical association pathways at an 85% confidence level. Homologous tracts of the left and right cerebral hemispheres had shorter FA correlational distances than those between most non-homologous tracts, with certain exceptions such as for the left ILF, which was more closely related to the left IFO and the right IFO than to the right ILF with at least a 91% confidence level.
Figure 3 Hierarchical clustering of FA correlational distances displayed as a dendrogram. The distance measure is 1 - ρ, where ρ is the Spearman rank correlation coefficient. The statistical confidence level of each linkage in the dendrogram is (more ...)
The dendrogram of MD correlational distances () showed some similarities to the FA dendrogram, but also many important differences. Again, homologous tracts tended to form pairs, with the bilateral CST forming a statistically significant cluster at a 97% confidence level. However, the other two significant clusters were of mostly non-homologous tracts, specifically the left IFO with the left AF and the left ILF, as well as the bilateral CST with the aforementioned three pathways. The bilateral CB constituted the most distant outgroup by MD correlational distances, with the bilateral UF as the second most distant outgroup. The bilateral IFO, which was so tightly coupled by FA values, showed a relatively large MD correlational distance.
Hierarchical clustering of MD correlational distances displayed as a dendrogram. All conventions are as in .
The dendrogram of AD correlational distances () did not have any statistically significant clusters at the 95% confidence level, likely due to a more homogenous correlation matrix structure than for FA or MD, as also reflected by the uH test statistic results reported above. The configuration of the AD dendrogram resembled that of the MD dendrogram more than the FA dendrogram, with the bilateral CB and the bilateral UF comprising the first and second most distant outgroups, respectively. The RD dendrogram () also did not show any significant clusters; however, the bilateral CST approached significance as a cluster at the 94% confidence level and the ten association tracts also formed a nearly significant cluster at the 94% confidence level. This partitioning of the projection tracts from the association tracts in the RD dendrogram was similar to the FA dendrogram. Another point of correspondence between the FA and RD dendrograms was that the left and right IFO were linked by the first edge since they had the shortest correlational distance among all tracts for both these DTI parameters.
Hierarchical clustering of AD correlational distances displayed as a dendrogram. All conventions are as in .
Hierarchical clustering of RD correlational distances displayed as a dendrogram. All conventions are as in .