To better understand the reduction in volumetric variance by aGSN the variance of a brain structure can be parceled into two components, one associated with variability in native brain size and the second from other sources. Other sources of variability include boundary definition variability, natural structure variability, and variability due to interactions with other structures. Shape preserving scaling removes variance due to brain size, since each brain’s volume is rendered identical. The reduction to near zero correlation between native brain (hemispheres) and native structure volumes following mean preserving scaling confirms this (left column, Table vs. Table ). Linear regression of brain structure volumes with native brain volume as the regressor was done to obtain R2 values, which indicate the fraction of variance explained by the model. Brain-to-structure correlations were also determined using a Pearson correlation coefficient (ρ), and ρ2 values were identical to R2 values. More importantly, the fractional variance reduction following mean preserving scaling was nearly identical to that predicted by regression analysis, suggesting equivalence regarding variance reduction (Fig. ). This equivalence held up for shape standardizing as well as shape preserving scaling. However, mean preserving scaling provides variance reduction for all brain structures, while regression analysis has to be done for each structure to estimate the reduced variance (Fig. ).
Correlation of volumes for ten brain structures from the LPBA40 database after shape preserving scaling
The fraction of variance removed by mean preserving scaling is shown to be equivalent to that explained by regression analysis using brain structure as the regressor where the modeled R2 = ρ2
It is important to assess the lower limit on brain-to-structure correlation for aGSN to successfully reduce variance. The mean reduction of variance in volumes for all 56 structures was 32%. The lower limit on ρ2 for reducing volume variance was estimated to be ~7%, a level of correlation that indicates a very weak relationship between brain size and structure size. This level was set to divide the group into structures that reduced variance and those that did not. All but four structures reduced variance (the left orbital frontal gyrus, left angular gyrus, and left & right cingulate gyrus though increases were small (CV change <0.02). Visual inspection by two authors (JLL, MDC) determined that some internal boundaries of these structures were not consistently defined, so it appears that the boundary definition component of variance masked that due to brain size. While boundary definition variance was present in other structures, its fractional contribution was apparently smaller since shape preserving scaling successfully reduced total variance in all other structures. It appears that if boundary definition variance is well managed then aGSN will reduce volumetric variance.
An important finding concerning mean preserving scaling was that, when used as the reference volume, each of the ten structures preserved the volume of the other nine structures. Strikingly, even when mean preserving scale factors determined from hemispheres were randomly applied rather than matched to individual brains the mean volumes were minimally changed (~1%); however, variance was increased as much as 37%. The practical significance of this observation is that researchers should be very cautious when tabulating scaled volumes to ensure that scale factors are properly paired with brains.
Though volume was preserved for all reference volumes, average variance was only reduced when whole brain (hemispheres) was used as the reference, which was predictable based on its high structure-to-structure correlation (Table ). Additionally, as can be seen from Table left-to-right correlations remained high for caudate, putamen and hippocampus after mean preserving scaling. Finally, by removing hemispheres size effect an interesting relationship was revealed between caudate and left hippocampus as a moderate negative correlation. This relationship was assumed to fall into the other component of variance category where one structure influences another independent of brain size. The nature of this relationship is unclear and suggests the need for further investigation.
Distances and areas were preserved for both classes of scale factors, and this was achieved regardless of orientation. This finding supports making measures of mean cortical thickness and surface areas of brain structures while controlling for brain size with aGSN’s scaling. Shape preserving scale factors were applied isotropically, so they naturally preserved shape. Unlike shape preserving scale factors, shape standardizing scale factors are usually non-isotropic, but as formulated each x-, y-, and z-scale factor met the conditions for mean preserving scaling. Finally, for both classes of scale factors their area scale factors (paired products of scale factors) and volume scale factors (triple products) also met the conditions for mean preserving scaling. Correlation between the three distances for hemispheres and three distances for right hippocampus lead to nine possible first order interactions, so simple correlation analysis was not done. However, analysis of hemispheres x and z distances revealed negative correlations with the y distance in hippocampus, supporting the poor reductions in variance for distances and areas in hippocampus involving the y direction.
To support conversion between conventional GSN and aGSN we developed an analytical method to adjust conventional GSN x-y-z scale factors to mean preserving x-y-z scale factors (Appendix A.4
). FLIRT scale factors that fit each brain to the ICBM152 template served as the basis for testing. A new set of mean preserving aFLIRT scale factors was calculated from the FLIRT scale factors using Eq. (A.4.4
). Both sets of scale factors were applied to each subject’s 10 volumes of interest. The FLIRT scaled volumes (Table ) were similar to those published by Shattuck et al. 2008
, verifying proper application of the scale factors. As predicted, the aFLIRT scale factors preserved mean volumes and decreased variability. Equivalent mean volumes were seen for aFLIRT (Table ) and shape standardizing scaling (Table ), with differences <1%.
Variance in most structures was similar; however, there were several significant differences, with the shape standardizing method having lower CVs for hemispheres and all-gyri regions and the aFLIRT method having a lower CV for cerebellum. These differences were assumed to arise from differences in reference structures, where the shape standardizing method used hemispheres and the aFLIRT method used whole brain. To test this assumption we formulated a whole brain ROI by adding brainstem and cerebellum ROIs to the hemispheres’ ROI. When using this whole brain ROI as the reference structure the variance differences between shape standardizing and aFLIRT methods were practically eliminated. The CVs in hemispheres and all gyri for the shape standardizing scaling increased but remained smaller than those for aFLIRT (0.014 vs. 0.020 for hemispheres and 0.018 vs 0.023 for all gyri). Linear distance and plane area were also preserved using aFLIRT (Table ). These data indicate that the shape standardizing method based on principal axes analysis provides control of volume variability equivalent to that achievable using FLIRT.Shape preserving scale factors can be approximated from shape standardizing scale factors as si
, where sx
, and sz
are the shape standardizing scale factors and si
the resulting isotropic scale factor. While calculation of volumes using these scale factors can be done in a spreadsheet, corrections for distances and areas require more complex manipulations, as indicated in “Appendices A.2
”, since these don’t naturally align with the image’s x-y-z scale directions. Additionally, surface areas of individual structures must be determined from surface models such as that used in this study.
Volumes using FLIRT and mean preserving aFLIRT scaling (mean ± SD (CV))
Between Group Anatomical Studies When performing between group anatomical studies it is important to preserve mean measurements of “each” group while reducing variance, and aGSN does that for high-resolution brain imaging studies. This leads to improved power in between-group anatomical studies. Using the methods described above for conversion from template based GSN one can break large groups into subgroups and each independently normalized by aGSN, a strategy useful in formulating post hoc analyses for a variety of subgroups. The use of aGSN in this manner supports comparisons of distances, areas, and volumes of brain structures with transformed images.
Boundary Definition Variability
The component of variability due to boundary definition appears to vary by structure and by laboratory. For example, wide ranges of volumes have been reported for the hippocampus. Amunts et al. 2005
reported left and right hippocampus volumes as 4,713
1,007 and 4,884
10, 5 males), while Kronmuller et al. 2009
0.25 and 3.30
11 males), and for LPBA40 data (Shattuck et al. 2008
) the values were 3,907
528 and 4,120
. Amunts used post mortem
sectioned images with excellent hippocampal definition, and may have included subregions that were not possible in MR images, so this might explain why their mean values were largest. Both Kronmuller and Shattuck used MR images, but Kronmuller used a sagittal tracing method devised by Pantel et al. 2000
, and Shattuck used a coronal delineation method based on their lab’s published rules (Supplementary material from their paper). The right hippocampus was larger than left for all three groups, but standard deviations varied tremendously, covering a four-fold range from Kronmuller to Amunts. Shattuck’s mean values were closer to those of Amunts than were Kronmuller’s. While aGSN cannot resolve variability due to differences in delineation methods, we have shown that it will reduce group-wise structure variability associated with brain size, so would be useful with a consistent method of boundary definition.