Anisotropic kernels for coordinate-based meta-analyses of neuroimaging studies

Abstract

Peak-based meta-analyses of neuroimaging studies create, for each study, a brain map of effect size or peak likelihood by convolving a kernel with each reported peak. A kernel is a small matrix applied in order that voxels surrounding the peak have a value similar to, but slightly lower than that of the peak. Current kernels are isotropic, i.e., the value of a voxel close to a peak only depends on the Euclidean distance between the voxel and the peak. However, such perfect spheres of effect size or likelihood around the peak are rather implausible: a voxel that correlates with the peak across individuals is more likely to be part of the cluster of significant activation or difference than voxels uncorrelated with the peak. This paper introduces anisotropic kernels, which assign different values to the different neighboring voxels based on the spatial correlation between them. They are specifically developed for effect-size signed differential mapping (ES-SDM), though might be easily implemented in other meta-analysis packages such as activation likelihood estimation (ALE). The paper also describes the creation of the required correlation templates for gray matter/BOLD response, white matter, cerebrospinal fluid, and fractional anisotropy. Finally, the new method is validated by quantifying the accuracy of the recreation of effect size maps from peak information. This empirical validation showed that the optimal degree of anisotropy and full-width at half-maximum (FWHM) might vary largely depending on the specific data meta-analyzed. However, it also showed that the recreation substantially improved and did not depend on the FWHM when full anisotropy was used. Based on these results, we recommend the use of fully anisotropic kernels in ES-SDM and ALE, unless optimal meta-analysis-specific parameters can be estimated based on the recreation of available statistical maps. The new method and templates are freely available at http://www.sdmproject.com/.

Keywords: activation likelihood estimation; anisotropic kernel; coordinate-based meta-analysis; effect size; magnetic resonance imaging; neuroimaging; signed differential mapping.

Figure 1

Main steps of activation likelihood estimation (ALE) and effect-size signed differential mapping (ES-SDM). ALE (left approach) aims to estimate the likelihood that a peak lies in any given voxel. To this end, it first applies a Gaussian kernel so that the likelihood is high in the voxel where the peak is reported and similar but slightly lower in the close voxels. Afterward, it calculates the probability of the union of the likelihoods estimated from the different peaks and studies. ES-SDM algorithms (middle and right approach) are different, as this method aims to estimate the effect size rather than the peak likelihood. However, the first step also consists in applying an (un-normalized) Gaussian kernel, this time to achieve that voxels around a reported peak have an estimated effect size which is similar but slightly smaller to that of the peak. Afterward, effect-sizes recreated from the different peaks of a study are combined using a weighted average, i.e., when a voxel is close to two peaks, it has an effect size that depends on both peaks. Finally, the effect size maps as well as their variance maps are introduced in a meta-analytic random-effects general linear model.

Figure 2

Recreation of clusters using isotropic kernels in previous versions of effect-size signed differential mapping (ES-SDM) and activation likelihood estimation (ALE). Note that the recreation of the effect size (or the estimation of the activation likelihood) does not depend on the strength of the spatial correlations, but only on the Euclidean distance between each voxel and the peak.

Figure 3

Recreation of clusters using the anisotropic kernel in the updated version of effect-size signed differential mapping (ES-SDM). Note that the recreation of the effect size does depend on the strength of the spatial correlations, with the cluster being stretched toward voxels highly correlated with the peak.

Figure 4

Deformed distance between two adjacent voxels depending on the correlation between them. Deformed distances in this example have been calculated for σ = 8.5 mm (FWHM = 20 mm). Note, however, that the recreation of effect size map does not indeed depend on FWHM when full anisotropy is used [see text, Eq. 2 and Figure 6].

Figure 5

Main correlation maps for white matter volume. For illustrative purposes, this Figure only shows correlations along the three main directions (left-right, back-front and bottom-up). The templates created in this study include the correlations with all 26 voxels surrounding each voxel.

Figure 6

Relative mean square error (MSE) of the recreation of the statistical maps used in this study depending on the degree of anisotropy and the full-width at half-maximum (FWHM). Relative MSE was defined as the MSE obtained with the current set of parameters divided by the MSE obtained after applying effect-size signed differential mapping (ES-SDM) standard isotropic kernel (FWHM = 20 mm). Please note that optimal degree of anisotropy and FWHM were different when using other datasets (not reported here), but use of full anisotropy was still associated to a substantial decrease of MSE.