Mode-based morphometry: A multiscale approach to mapping human neuroanatomy

Abstract

Voxel-based morphometry (VBM) and surface-based morphometry (SBM) are two widely used neuroimaging techniques for investigating brain anatomy. These techniques rely on statistical inferences at individual points (voxels or vertices), clusters of points, or a priori regions-of-interest. They are powerful tools for describing brain anatomy, but offer little insights into the generative processes that shape a particular set of findings. Moreover, they are restricted to a single spatial resolution scale, precluding the opportunity to distinguish anatomical variations that are expressed across multiple scales. Drawing on concepts from classical physics, here we develop an approach, called mode-based morphometry (MBM), that can describe any empirical map of anatomical variations in terms of the fundamental, resonant modes--eigenmodes--of brain anatomy, each tied to a specific spatial scale. Hence, MBM naturally yields a multiscale characterization of the empirical map, affording new opportunities for investigating the spatial frequency content of neuroanatomical variability. Using simulated and empirical data, we show that the validity and reliability of MBM are either comparable or superior to classical vertex-based SBM for capturing differences in cortical thickness maps between two experimental groups. Our approach thus offers a robust, accurate, and informative method for characterizing empirical maps of neuroanatomical variability that can be directly linked to a generative physical process.

Figure 1:

SBM and MBM analysis pipelines. (a) In SBM, a t-statistic is calculated independently at each vertex, quantifying point-wise group differences in CT. A thresholded t-map is derived by comparing the observed t-map and the distribution of null t-maps after permutation testing. (b) In MBM, eigenmodes are derived from a cortical surface mesh (solving Eq. (2)). The modes are ordered in increasing spatial frequency or decreasing spatial wavelength. Values in each mode are arbitrarily defined, with negative–zero–positive values in colored as blue–white–red. (c) An empirical t-map can be decomposed as a weighted sum of eigenmodes and errors using a GLM (Eq. 6), with weights given by β_j. The set of β_j is called the β spectrum. (d) An example β spectrum with large β₄₅ indicating a dominant contribution from mode 45. (e) An example of statistically significant βs derived by comparing the observed β spectrum and the distribution of null β spectra after permutation testing.

Figure 2:

Framework for ground-truth simulations. (a) We generate a CT map using the model in Eq. (7). Here, we show an example with σ_C = 0.4 and σ_S = 0.8. (b) Using the simulated CT maps for groups A and B, we estimate a t-map and its corresponding β spectrum (d). (c) The ground-truth (GT) difference map is given by the subtraction of ${\mathbf{M}}_C^A$ and ${\mathbf{M}}_C^B,$ from which the ground-truth β spectrum (e) is obtained.

Figure 3:

Comparing spatial variograms of empirical and simulated CT maps for different combinations of σ_C and σ_S. (a) Two examples of mean variograms (after subtracting the minimum offset) of empirical and simulated maps for parameter pairs (σ_C = 1,σ_S = 0) and (σ_C = 1,σ_S = 0.8). The error bars (vertical bars) show the variance of the variograms. (b) Norm distances between the empirical and simulated mean variograms (after subtracting the minimum offset) for combinations of σ_C and σ_S. The green boxes highlight the realistic regimes where the generated maps have a similar spatial structure as the empirical data (norm distance ≈ 0).

Figure 4:

Accuracy of SBM and MBM with respect to ground-truth simulations for different combinations of σ_C and σ_S. (a) Mean correlation between the t-map of an experiment and the ground-truth difference map. (b) Mean correlation between the β spectra of the t-map of an experiment and the ground-truth difference map. The green boxes highlight the realistic parameter regimes where the generated maps have a similar spatial structure as the empirical data, as shown in Fig. 3.

Figure 5:

Distributions (in log scale) of correlations between pairs of 100 experiments for different combinations of σ_C and σ_S. The panels show correlations between pairs of experimental t-maps (SBM), correlations between pairs of experimental β spectra (MBM), binary correlations between thresholded t-maps (SBM, thres), and binary correlations between statistically significant β spectra (MBM, thres). The green boxes highlight the realistic parameter regimes where the generated maps have a similar spatial structure as the empirical data, as shown in Fig. 3.

Figure 6:

Distribution (in log scale) of pairwise correlations between experiments in the realistic parameter regimes and for different smoothing kernels. The panels show correlations for experimental t-maps (SBM), correlations for experimental β spectra (MBM), binary correlations for thresholded t-maps (SBM, thres), and binary correlations for statistically significant β spectra (MBM, thres).

Figure 7:

SBM and MBM analyses of CT differences between sexes. (a) Unthresholded tmap. (b) Thresholded t-map at p_unc ⩽ 0.05. Red and blue denote significantly thicker CT in females and males, respectively. (c) β spectrum of the unthresholded t-map. The β’s of the significant modes, obtained via permutation testing at p_unc ⩽ 0.05, are colored green. (d) Significant pattern obtained by combining the significant modes weighted by their β’s. (e) Six most influential modes, i.e., significant modes with largest absolute β values. The signs of modes with negative β’s were flipped to better visualize the similarity between the modes and the significant patterns. The number denotes the order of influence, not the mode index. (f) Smoothed t-map at FWHM = 30 mm.

Figure 8:

Distribution of pairwise correlations between experiments for different smoothing kernels in the empirical analysis of sex differences under resampling. The panels show correlations for experimental t-maps (SBM), correlations for experimental β spectra (MBM), binary correlations for thresholded t-maps (SBM, thres), and binary correlations for statistically significant β spectra (MBM, thres).

Figure 9:

MBM analyses of CT differences between patients with Alzheimer’s disease and healthy controls across three sites. (a) Unthresholded t-maps of the three sites. Negative–zero–positive values are colored as blue–white–red, with positive values indicating reduced thickness in patients. (b) Thresholded t-map at p_unc ⩽ 0.05. Red and blue denote significantly thinner CT in patients and controls, respectively. (c) β spectrum of the unthresholded t-map. The β’s of the significant modes, obtained via permutation testing at p_unc ⩽ 0.05, are colored green. (d) Significant pattern obtained by combining the significant modes weighted by their βs. (e) Six most influential modes, i.e., significant modes with largest absolute β values. The signs of modes with negative β’s were flipped to better visualize the similarity between the modes and the significant patterns. The number denotes the order of influence, not the mode index.

Figure 10:

Comparing the performance of SBM and MBM analyses of CT differences between patients with Alzheimer’s disease and healthy controls across three sites. (a) Unthresholded t-maps of the three sites in the Alzheimer’s study with FWHM = 0, 10, 20, 30 mm. Negative–zero–positive values are colored as blue–white–red, with positive values indicating reduced thickness in patients. (b) Thresholded t-maps (p_unc ⩽ 0.05) of the three sites with FWHM = 0, 10, 20, 30 mm. Red and blue denote significantly thinner CT in patients and controls, respectively. (c) Absolute values of β spectra of the three sites without smoothing in log scale. Green and gray bars show significant and not significant βs, respectively. (d) The proportion of significant modes in each approximate eigengroup. (e) Correlation between the empirical t-map and its mode-derived reconstruction obtained using the full β spectrum, after removing modes in order of decreasing or increasing spatial wavelength. (f) Correlation between the empirical t-map and its mode-derived reconstruction obtained using only significant modes from the full β spectrum, after removing modes in order of decreasing or increasing spatial wavelength.

Figure 11:

Consistency of SBM and MBM results in explaining multi-site CT differences between patients with Alzheimer’s disease and healthy controls at different smoothing kernels. (a) Pairwise correlations between sites for unthresholded results. The light and dark blue horizontal lines represent the mean correlation of the unthresholded t-maps (for SBM) and of β spectra (for MBM), respectively. (b) Pairwise binary correlations between sites for thresholded results. The light and dark blue horizontal lines represent the mean binary correlation of the thresholded t-maps (for SBM) and of β spectra (for MBM), respectively.