Scale-invariant brain morphometry: application to sulcal depth

TL;DR

This work provides the first quantitative analysis of the influence of brain size on sulcal depth measurements, and introduces a novel, scale-invariant method for sulcal depth estimation based on an original formalization of the problem.

Abstract

The geometry of the human cortex is complex and highly variable, with interactions between brain size, cortical folding, and age well-documented in the literature. However, few studies have explored how global brain size influences morphometry features of the cortical surface derived from anatomical MRI. In this work, we focus on sulcal depth, an imaging phenotype that has gained attention in both basic research and clinical applications. We make key contributions to the field by: 1) providing the first quantitative analysis of the influence of brain size on sulcal depth measurements; 2) introducing a novel, scale-invariant method for sulcal depth estimation based on an original formalization of the problem; 3) presenting a validation framework and sharing our code and benchmark data with the community; and 4) demonstrating the biological relevance of our new sulcal depth measure using a large sample of 1,987 subjects spanning the developmental period from 26 weeks post-conception to adulthood.

Figures (11)

• Figure 1: Illustration of the dissociation between the value of the parameter $\alpha$ and the size of the surface. Top: for both DPF and DPF*, larger values for $\alpha$ correspond to smaller scale of interest on a given surface. We also show the spherical representations to better visualise the variations in the scale of interest. Bottom: thanks to the adaptation function $f$ in DPF*, once the scale of interest has been fixed for a given surface through the value of $\alpha$, it is automatically adapted according to the global size of the other surfaces.
• Figure 2: Results from Expe #1. A: From top to bottom, we show the distributions of the three measures $Dev$, $StdCrest$, and $Sep$ across all the brains from the dataset #1 as violin plots, for the different values of $\alpha$. We highlight in gray the range of values for which the distributions are statistically equivalent to the one obtained using the best value for $\alpha$ (Wilcoxon-test with p-value$>$.05). B: For each measure, we show the distributions obtained for the method SULC, the mean curvature and the classical $DPF$ from auzias_deep_2015, for comparison.
• Figure 3: Results from Expe #2. A. The scaled surfaces obtained as described in Sec.\ref{['sec:desc_expe2']}. B. Depth maps generated using the $DPF^*$ method (with $\alpha=500$). The scale-invariance is confirmed empirically: the estimated depth values are not influenced by the scaling. The correlation is thus perfect for all surfaces and the regression lines are super-imposed. C. Left: Depth estimations resulting from the SULC method applied to the scaled surfaces (y-axis) with respect to SULC on the initial mesh (x-axis). Both the slope and the correlation values depend on the scaling factor. Right: Spatial distribution of residuals from the linear regression models derived from the left plot. Consistently with the regression plot, the residuals increase with the scaling factor. The regions corresponding to most concave and convex geometric configurations show higher magnitude of error.
• Figure 4: Analysis of the distribution of SULC, $DPF^*$ and $DPF^*_{abs}$ across the 879 surfaces from the dHCP newborns and 1108 surfaces from HCP young adults. From left to right, the three columns correspond resp. to SULC, $DPF^*$ and $DPF^*_{abs}$. On the top row, we show the distribution of each sulcal depth estimation for all subjects, ordered following their global size. For each method, we show the mean in purple, the median in blue, the 25th and 75th centiles in green, and the 5th and 95th centiles in orange. On the second row, we show the corresponding matrices of the Wasserstein distance between every pair of surfaces. On the third row, we show for each method the distributions of the Wasserstein distances within the four sub-populations corresponding to the blocks illustrated on the matrix for the method SULC. At the bottom, we show for each method the KS statistic comparing the distribution in the block corresponding to the subgroup of largest brains, to the other blocks corresponding to smaller brains. See the text for further description.
• Figure 5: Top: Labeled sulcal regions shown on three time steps of the spatio-temporal atlas. The nomenclature of sulci and associated colors are shown on the right. The colors are indicative of the expected timing of formation: sulci colored in levels of blue are expected to appear before the sulci in red that appear before the sulci in green. Bottom: Curves showing the average sulcal depth estimated in the different sulcal regions for SULC (left) and $DPF_{abs}^*$ (Right).

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof