Allometric scaling of brain regions to intra-cranial volume: An epidemiological MRI study

General Design of the AGES-Reykjavik Study

The general design and demographics of the AGES-Reykjavik have been described elsewhere [Harris et al., 2007]. The population-based sample of the AGES-Reykjavik consisted of 5,764 men and women, born between 1907 and 1935. Participants underwent extensive clinical evaluation, including cognitive function testing and brain MRI. All participants signed an informed consent. The AGES-Reykjavik was approved by the Intra-mural Research Program of the National Institute on Aging, the National Bioethics Committee in Iceland (VSN00-063), the Icelandic Data Protection Authority, and the institutional review board of the U.S. National Institute on Aging, National Institutes of Health.

AGES-Reykjavik: MRI Acquisition and Automated MRI Segmentation

MRI was performed at the Icelandic Heart Association on a single study dedicated 1.5-T GE Signa Twinspeed EXCITE system MRI scanner. The image protocol, described previously [Sigurdsson et al., 2012], included a T1-weighted 3D spoiled gradient echo (TE 8 ms; TR 21 ms; FA 30°, FoV 240 mm; matrix 256 × 256; 110 slices; slice thickness 1.5 mm), a FSE PD/T2 (TE1 22 ms; TE2 90 ms; TR 3,220 ms; echo train length 8; FA 90°, FoV 220 mm; matrix 256 × 256; slice thickness 3.0 mm), and a FLAIR (TE 100 ms; TR 8,000 ms; inversion time 2,000 ms; FA 90°, FoV 220 mm; matrix 256 × 256; slice thickness 3.0 mm).

A fully automated segmentation pipeline was developed based on the Montreal Neurological Institute processing pipeline [Sigurdsson et al., 2012; Zijdenbos et al., 2002]. The pipeline used a multispectral approach to segment voxels into global tissue classes [cerebrospinal fluid (CSF), GM, WM, and white matter hyperintensities (WMH)]. Following this, a regional parcellation pipeline—atlas-based segmentation method—was developed to obtain volumes of different sub-structures of the brain.

Determination of VOI

The regional tissue segmentation pipeline parcellated the brain in 56 different regions (Supporting Information, Appendix A). However, for the present study, we combined regions into a limited amount of 8 VOI known to differ in gross cyto-architectural features. We separately assessed scaling of neo-cortical GM and WM to investigate in further detail the previously reported proportional changes as function of TBV. Three regions of neo-cortical GM were investigated: frontal (comprising of orbito-frontal and pre-frontal GM, precentral gyrus, cingulated gyrus, insula, and fornix), temporal (comprising of lateral temporal GM, parahippocampal, and fusiform gyrus), and parieto-occipital GM. Cortical WM volume was studied in total and included all lobar WM, corpus callosum, internal and external capsule, and WMH. The medial temporal lobe (MTL), striatum, thalamus, and cerebellum were separately studied because of their importance in many studies to neurodegenerative processes. MTL included amygdala and hippocampus (including CA regions I-IV, fimbria, and subiculum of the hippocampus). Striatum included the nucleus accumbens, caudate nucleus, putamen, and globus pallidus. The thalamus included also the hypothalamus. The cerebellum included cerebellar GM and WM. Left and right hemispheres of each structure were combined. Total brain volume (TBV) was calculated as the sum of the neo-cortical GM and WM, MTL, striatum, thalamus, brainstem, and cerebellum. ICV was defined as the sum of TBV and CSF.

Quality Control of Tissue Classification and Validation of VOI

The quality of the segmentation of the eight composite VOI was mostly dependent on the performance of the global tissue segmentation into GM, WM, WMH, and CSF, and for a small part dependent on the definition of topographical borders by the regional parcellation pipeline. Performance of both global tissue segmentation and regional tissue parcellation was evaluated. The quality control of global tissue classification consisted of three steps described in [Sigurdsson et al., 2012]. In summary these were: (1) Visual inspection of the segmentation of 14 a priori selected slices of each subject (N = 4,356), which led to additional manual editing in 43 cases and rejection of 53 cases. (2) Comparison of automated versus manual global tissue segmentation of 5 preselected slices across the brain (including a slice located at the junction of the thalamus and subthalamic structures for reviewing segmentation of the deep gray matter nuclei) in 20 randomly selected cases. Resulting dice similarity index scores [Zijdenbos et al., 1994] were 0.82, 0.82, and 0.83 for GM, WM, and CSF respectively. (3) Reproducibility of the entire process of MRI acquisition and postprocessing was evaluated by repeated scanning and segmentation (four times in total) of 32 participants. Excellent intra-class correlation for all global tissue was found (r > 0.98, for all). Because the present study relies for an important part on good quality of ICV segmentation, the performance of the automated pipeline was further evaluated specifically on ICV. ICV was manually segmented on the same 20 brain scans used for step 2 of the quality control. Two researchers with extensive neuroradiological experience and blinded for the results of the automated segmentation, segmented ICV on axial 3D T1 weighted images, with correction and editing in sagittal and coronal planes. Resulting ICV were correlated to ICV obtained by the automated pipeline. Pearson's correlation was 0.97 (0.93–0.99) and Bland–Altmann plot showed a small overestimation of ICV of 31 cm³ on average by the automated segmentation, but no proportional error (Supporting Information, Appendix B and C).

Performance of regional parcellation pipeline was validated against four complete manually labeled scans. Dice similarity index scores per studied region were; frontal GM: 0.83, temporal GM: 0.83, parieto-occipital GM: 0.81, striatum: 0.83, MTL: 0.80, thalamus: 0.92, cerebellum: 0.92, white matter: 0.86.

Statistical Analysis

All statistical analyses were performed with SAS v 9.13 (SAS Institute Inc., Cary, NC, USA) and all graphs were generated with R v 3.1.2 (R Core Team [2014]. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/).

Estimation of scaling coefficients of different VOI

Allometric coefficients of VOI with ICV were calculated using the general equation for allometric analyses, log(y) = log(b) + α log(x), where x is ICV, y VOI, log(b) intercept, and α represents the allometric coefficient [Harvey, 1982], that is, the slope of the regression between log (ICV) and log (VOI). A coefficient greater than 1.0 is considered a positive allometric coefficient, that is, VOI increased with a power greater than 1 relative to ICV. A coefficient smaller than 1.0 is seen as a negative allometric coefficient, that is, VOI increased with a power less than 1 relative to ICV.

We chose ICV, instead of TBV, as measure of brain volume to avoid a possible bias toward isometry in estimating allometric coefficients of large VOI. Large structures occupy large volumes in TBV making the range of possible deviations from isometry smaller; this may produce an overestimate of coefficients toward 1 and reduce the ability to estimate allometric coefficients deviant from 1 [Deacon, 1990]. With the use of ICV none of the structures studied comprised more than 24% (WM) of ICV. Another important reason was that ICV is regarded as a marker for brain volume at its maximum size and therefore a marker of “pre-morbid” brain size. At time of scanning, brains of most study participants experienced more or less atrophy due to aging or pathological processes. These are factors we can largely control for in our statistical analyses, whereas it is more difficult to control for differences between current TBV without taking into account the original size of the brain at maximum. Log-transformed VOI were plotted against log transformed ICV (Fig. 1). For each VOI, allometric coefficients with ICV were calculated adjusted for age and sex, (log (VOI) = intercept + α log (ICV) + β_age × age + β_sex × sex) and tested against the isometric scaling law of 1:1.

Characteristics AGES-Reykjavik Sample

The AGES-Reykjavik sample had a mean age of 75.7 (standard deviation = 5.2) years, of which 59.8% were women. ICV ranged from 1,116 to 2,162 cm³; 1,116 to 1,868 cm³, in women and 1,232 to 2,162 cm³ in men. Women had on average lower educational level (P < 0.0001), higher BMI (P = 0.003), were diagnosed less often with diabetes (P < 0.0001), and had smoked (P < 0.0001) and drank alcohol (P < 0.0001) more sparingly compared with men. Women had lower means of ICV and all VOI compared with men (P < 0.0001) (Table 1).

Allometric Scaling Coefficients of All VOI

All VOI scaled non-isometrically to ICV (Fig. 1). After correction for age and sex, a positive allometric coefficient of 1.14 (95% confident interval = 1.11–1.17)) was estimated for WM volume and negative allometric coefficients were found for frontal GM [0.76 (0.73–0.79)], temporal GM [0.75 (0.72–0.78)], parieto-occipital GM [0.79 (0.76–0.83)], MTL [0.60 (0.56–0.64)], thalamus [0.59 (0.56–0.62)], striatum [0.41 (0.37–0.45)], and cerebellum [0.55 (0.52–0.59)]. All were found significantly differently from 1 [1:1 scaling law to ICV (P < 0.0001)].

Significant Scaling Differences between VOI

Results from the marginal model showed that the α-coefficient of WM volume to ICV was significantly different from the α-coefficients of all GM VOI (Table 2) in the entire sample, and in women and men separately. The α-coefficients of the different neo-cortical GM areas to ICV were not significantly different from each other in women and men separately. Also, the α-coefficients of MTL, thalamus, and cerebellum were not significantly different from each other in women and men separately. However, in the entire sample the α-coefficient of the MTL was not significantly different from the α-coefficient of the parieto-occipital GM, but was significantly different from the thalamus and the cerebellum. The α-coefficient of the striatum was significantly different from all other α-coefficients except for the α-coefficient of the thalamus and cerebellum in the entire sample, and the α-coefficient of the cerebellum in men only.

Allometric Scaling in Different Age Groups

The α-coefficients of VOI to ICV for each quartile of age are shown in Table 3. The α-coefficients of the both cortical and deep GM structures and cerebellum in the older quartiles appeared somewhat lower compared with the younger quartiles and the α-coefficient of WM appeared higher in the older quartiles. However, these differences were non-significant, except for temporal GM which was significantly lower in the older quartiles compared with the younger (P-value of 0.004).

Little Allometry Introduced by Automated Segmentation Pipeline

Figure 2 displays log of global tissue volumes plotted against log ICV obtained by the automated segmentation pipeline based on the artificially linearly scaled data set of a relatively large and small brain. The α-coefficients were 0.99 for GM, 1.02 for WM, and 1.00 for CSF for dataset based on the relatively large brain and 0.98 for GM, 1.01 for WM, and 10.2 for CSF for the dataset based on the relatively small brain. Because of the almost perfect fit of the points and the regression line these α-coefficients were significantly different from the isometric scaling law of 1.0 (all P-values < 0.0001), except for the CSF in the large brain (Table 4).

Comparable Fit of Allometric Model and Linear Regression Model

Figure 3 superimposes the line of prediction of the allometric model (and associated α-coefficient and R²) with line of prediction of the linear model (and associated β-coefficient and R²). Compared with the R² of the linear model, the R² of the allometric model was a few per mille smaller for cerebellar, cortical and deep GM structures and a few per mile larger for WM. The models have a comparable fit and can substitute each other.

Allometric Scaling in CARDIA and ADNI

The CARDIA sample consisted of individuals with a mean age of 50 (3.5) years, of which 52.9% were women. ICV in the CARDIA sample, including only supratentorial areas, varied from 999 to 1,643 cm³. The ADNI sample consisted of individuals with a mean age of 76 (5.0) years, of which 49.4% were women. ICV in the ADNI sample varied from 1,116 to 1,985 cm³. We found the highest allometric coefficients for WM volume in both CARDIA (α = 1.05) and ADNI (α = 1.00). All GM areas had negative allometric coefficients with the lowest coefficients in the deep GM areas (Table 3). Roughly, results suggest similar trends compared with those found in the AGES Reykjavik data. However, important differences were (1) allometric coefficients of the neo-cortical GM areas to ICV in CARDIA with values between 0.90 and 0.94 were higher, compared with those in the AGES and ADNI samples, and (2) WM volume in the ADNI data set seems to increase isometrically with ICV.

Different Allometric Scaling of Neo-Cortical WM, Neo-Cortical GM, and Deep GM

One goal of the present study was to assess and compare scaling coefficients of different VOIs to ICV in the AGES-Reykjavik dataset. We found all VOI to scale allometrically with ICV. One could roughly discern three patterns of scaling, that is, WM scaling, neocortical GM scaling and deep GM scaling. First, neo-cortical WM was the only structure to proportionally increase in larger ICV with a positive allometric coefficient of 1.14. Scaling coefficients of WM were found significantly different from all GM structures and cerebellum. Second, negative allometric coefficients were found for frontal (0.76), temporal (0.74), and parieto-occipital (0.79) cortical GM structures. Scaling coefficients of neo-cortical GM structures (frontal, temporal, and parieto-occipital) were not significantly different from each other, but were significantly larger than scaling coefficients of the deep GM structures when women and men were separately assessed. Also scaling coefficients of MTL (0.60), thalamus (0.59), and the cerebellum (0.55) were not significantly different from each other in women and men separately. The scaling coefficient for striatal volume (0.41) was relatively invariant over the range of ICV and was significantly different from all other structures, except the cerebellum.

Allometric Scaling Cannot Solely Be Explained by Age, Sex, Ethnicity, or a Systemic Bias from Segmentation Pipeline

One important limitation of our study was that the sample consisted of older individuals, who have experienced various amounts of brain atrophy. Therefore, the observed scaling coefficients cannot be extrapolated to younger samples. After stratifying the AGES Reykjavik sample into quartiles of age, we found most structures to have similar scaling coefficients except for temporal GM (not including the MTL), which had lower α-coefficients in older individuals. We do not have an explanation for the significant difference in scaling found for temporal GM, but it prompted us not to rule out the possibility that allometric scaling of sub structures of the brain may vary with age in a way that we could not detect in the age span of our sample. Nonetheless, the findings of this article show that allometric scaling is a feature of the brain in the older population, which cannot be accounted for by adjusting for age when performing brain comparative studies.

A second important limitation was that all participants were Icelandic and the sample was genetically relatively homogeneous. Ancillary analyses in ADNI and CARDIA, with participants of younger age and different ethnicities, also showed that WM increased proportionally steepest with increasing ICV, followed by proportionally decreases in GM, with greatest decreases in the deep GM structures, similar to our observations in the AGES Reykjavik Study. Still, there were also differences in the results between the studies. The allometric coefficients of the cortical GM areas to ICV in CARDIA seemed higher compared those in the AGES and ADNI samples. A potential explanation could be that “ICV” in CARDIA was constructed from supratentorial structures only, and as a result allometric coefficients were higher. Another explanation could be that in the relatively young sample of the CARDIA allometric scaling is less pronounced. Further studies are needed to specifically examine this hypothesis. A second difference between the results of the additional analyses and our primary analyses in AGES was that WM volume in the ADNI data set seemed to increase isometrically with ICV. This may be explained by differences in tissue segmentation between GM and WM, as suggested by the higher mean volume of WM and lower mean volume of GM in ADNI compared with AGES Reykjavik. Depending on how border voxels are assigned to the GM and WM tissue classes, the difference between allometric coefficients may differ. Because of these differences in scaling coefficients among the study samples, it is at the moment not possible to establish fixed reproducible allometric coefficients for the human brain and more studies are needed.

A third potential limitation of the study was the use of an automated MR segmentation technique. Systematic errors, such as improper skull stripping, incorrect intensity thresholds, difficulty in segmenting sulcal CSF or imprecise template warping could all be possible sources of finding allometric correlations between VOI to ICV. However, when we fed artificially linearly scaled scans in the segmentation pipeline, the scaling coefficients of the output only showed small deviations from the isometric scaling law of 1, at maximum in the order of 2%. This could not explain the much larger deviations from 1 of the different scaling laws of VOI in the study sample. Therefore, we did not find evidence for a possible systematic error in the segmentation pipeline that could explain the allometry.

Allometric Scaling as True Feature of Brain Geometry

Differences in geometric or cyto-architectural properties of different brain structures may underlie differences and similarities in scaling to ICV. We observed similar scaling coefficients of different neo-cortical GM areas, which suggest they preserve proportionality to one another regardless of ICV. However, cortical GM and WM had significantly different scaling coefficients, indicating they do not preserve proportionality with varying brain size. This can be explained by differences in topology, where GM can be regarded as a surface of neural tissue covering an associated volume of WM [Dale et al., 1999]. The different lobes of the neo-cortex are similarly organized in repetitive cortical columns [Mountcastle, 1997]. Assuming a stable thickness of the neo-cortical GM “surface” across various brain sizes, as suggested by several studies [Hofman 1985; Hofman 1988; Mountcastle 1997], neo-cortical GM to WM should scale by an exponent of two-third (square-cube). If we focus on the results based on the AGES-Reykjavik study sample, we can observe that scaling coefficients of neo-cortical white to gray matter range from 0.65 to 0.70 (0.76/1.14 for frontal GM, 0.74/1.14 for temporal GM, and 0.79/1.14 for parieto-occipital GM), which approximates the geometric square-cube scaling law. Nevertheless, we did not establish the same results in the younger sample of CARDIA or the smaller sample of the ADNI and caution should be taken to apply a purely square-cube scaling law to the architecture of neo cortex. In a previous study slight increases of neo-cortical thickness (scaling of 0.2) with increase in ICV were observed [Im et al., 2008]. Another recent study showed the neo-cortical GM to have a more extensive gyrification, that is, to be “twistier,” in larger brains compared with smaller ones [Germanaud et al., 2012]. Also, for other parameters, such as cell soma size or amount of supporting glial cells, the extent to which they vary with increasing brain size is unclear. Some studies have also pointed to possible constraints in WM expansion, which should lead to scaling factors of white to gray that are higher than the square-cube law of two-third. It has been proposed that hemispheric specialization increases with increasing brain volume, which would lead to a decrease in inter-hemispheric connections and thus a decrease in WM volume [Ringo et al., 1994]. However, the coefficients reported in the present study for the AGES-Reykjavik study provide no evidence for such a limitation on WM expansion

The disproportionally lower scaling coefficients of deep GM structures to ICV compared with the cortex are not readily explained. The cortex gives rise to connections with striatum and thalamus, thus these structures could be expected to expand with neo-cortical GM volume. However, we found no evidence for preserved proportionality of the striatum and thalamus with cortical GM with scaling. Possibly, the deep GM structures are also influenced by other factors during brain development than neo-cortical growth. Brain structures grow in asynchronous patterns from birth through early adulthood and the striatum has been shown, together with frontal brain areas, to undergo more extensive developmental changes relatively late in early adulthood compared with other brain areas [Sowell et al., 1999]. Also, genetic factors could influence variation in regional brain volumes and lead to disproportional neo-cortical and deep GM volume increases with ICV, especially for the striatum. One twin study showed that the volume of the striatum, thalamus, and cerebellum were significantly more influenced by genetic factors compared with neo-cortical structures that were influenced more by environmental factors [Yoon et al., 2011]. And another twin-study concluded the phenotypic covariance of the striatal structures, hippocampus, and thalamus was primarily due to patterns of genetic covariance [Eyler et al., 2010].

Implications of Allometric Scaling for Methods of Head/Brain Size Adjustment

Knowledge on the allometric scaling of regional brain volumes is important for the discussion of adjustment methods for normal variation in comparative brain studies to volumetric and morphological changes. Allometric scaling implies both non-proportionality and non-linearity of scaling. Our results contribute to the understanding why certain methods should not be used. Ratios of brain structure volume over ICV or stereotaxic normalization by means of linear affine transformation assume isometric scaling of the brain, that is, proportionate scaling, which may lead to over- or underestimation of results. Therefore the use of these methods should be avoided, except when studying disproportionality is the purpose. The erroneous effect of linear spatial normalization on groups with differences in ICV was illustrated in a study comparing neo-cortical thickness differences in stereotaxic and native space between men and women [Luders et al., 2006]. The normalized data showed a disproportionately increased neo-cortical thickness in women compared with men, which was considerably attenuated in the unscaled data. Another important finding of our study was that allometric scaling was most apparent in deep GM structures. Unwanted effects of spatial registration therefore may be expected to be especially problematic in deep gray matter structures. Previously, a study reported that spatial-transformation based methods indeed produce significantly different proportions in smaller structures such as the hippocampus [Allen et al., 2008]. Lastly, we compared the fit of the allometric model to a linear model in predicting the relationship of VOI to ICV. We found very small differences in R², which implies the allometric model and linear model could substitute each other in the range of total brain size variation among humans. Therefore, we conclude that it is important in brain comparative studies to adjust for non-proportionality, but not for non-linearity.