Cortical thickness analysis in autism with heat kernel smoothing
Introduction
The cerebral cortex is a highly convoluted sheet of gray matter with varying thickness. The pattern of cortical thickness across the cortex varies with different clinical populations (Chung et al., 2003, Kabani et al., 2000). Cortical thickness can be used as an anatomical index to quantify local cortical shape differences. We measure cortical thickness using the inner and outer boundaries of the gray matter. The first step is to obtain T1-weighted magnetic resonance images (MRI) and classify each voxel as one of three tissue types: gray matter, white matter and cerebrospinal fluid (CSF). This classification can be done automatically using, for example, a neural network classifier (Kollakian, 1996) or mixture modeling (Ashburner and Friston, 2000). The boundary between gray and white matter voxels gives a crude approximation of the inner cortical boundary which is then refined using a deformable surface algorithm (Dale and Fischl, 1999, Davatzikos and Bryan, 1995, MacDonald et al., 2000). The outer cortical boundary is obtained similarly. In this study we use the method presented in MacDonald et al. (2000) which results in triangular meshes of 40,962 vertices and 81,920 triangles with the average inter-vertex distance of 3 mm (Fig. 1).
There are many techniques proposed for measuring the cortical thickness from two surface meshes (Fig. 2), for example, the minimum Euclidean distance method of Fischl and Dale (2000), the Laplace equation method of Jones et al. (2000), Bayesian construction of Miller et al. (2000) and the automatic linkage method of MacDonald et al. (2000). In this study, we use the automatic linkage method which has been validated in Kabani et al. (2000) and has been successfully used in Chung et al. (2003) for quantifying normal cortical development.
Before performing cross-subject comparison of spatially-varying data, a spatial normalization step is necessary. Spatial normalization consists of spatially transforming each subject anatomy towards a template anatomy. When analyzing 3D datasets such as fMRI, a 3D spatial transformation is used. In this study, however, the data lies on the relatively-thin cortical sheet so a 3D spatial normalization is not applicable: the population of cortices will not generally overlap after normalization in 3D. Instead, we treat each cortex as a surface (using, say, the inner cortical surface) and perform a 2D normalization along the surface. A number of surface registration methods have been published that differ in the measure of discrepancy between the input and template surface, for example, (Dale and Fischl, 1999, Davatzikos, 1997, Fischl et al., 1999, Liu et al., 2004, Robbins, 2003, Thompson and Toga, 1996, Thompson et al., in press, Van Essen et al., 1998). In particular, Davatzikos (1997) uses curvature, Fischl et al. (1999) and Robbins (2003) use some measure of gyrification, while Liu et al. (2004) use attribute vectors in trying to minimize the discrepancy. Alternately, Van Essen et al. (1998) and Thompson et al. (in press) solve a partial differential equation that models warping as elastic deformation with landmarks.
Each of the segmentation, thickness computation, and surface registration procedures are expected to introduce noise in the thickness measure. To counteract this, data smoothing is used to increase the signal-to-noise ratio (SNR) and the sensitivity of statistical analysis. For analyzing data in 3D whole brain images Gaussian kernel smoothing is widely used, which weights neighboring observations according to their 3D Euclidean distance. In this study, however, the data lie on a 2D surface so the smoothing must be weighted according to distance along the surface (Andrade et al., 2001, Chung et al., 2003, Lerch and Evans, 2005, Thompson et al., in press). One such approach, the “anatomically informed basis function” method (Kiebel and Friston, 2002), constructs a non-stationary anisotropic 3D filter kernel in such a way that effectively smoothes functional MRI data along the cortical sheet rather than normal to the sheet. An alternative approach has been developed, known as diffusion smoothing, that smooths data on an explicit 2D cortical surface representation (Andrade et al., 2001, Cachia et al., 2003a, Chung et al., 2003). Diffusion smoothing is based on the observation that, in Euclidean space, Gaussian kernel smoothing is equivalent to solving an isotropic diffusion equation (Koenderink, 1984). This diffusion equation can also be used on the surface manifold (Andrade et al., 2001, Cachia et al., 2003a, Chung et al., 2003), generalizing Gaussian kernel smoothing. The drawback of previous diffusion smoothing methods is the complexity of setting up a finite element method (FEM) for solving the diffusion equation numerically and making the numerical scheme stable. To address this shortcoming, we have developed a simpler method based on heat kernel convolution.
As an illustration, we apply our methods to detect regions of cortical thickness difference between a group of 16 high functioning autistic children and a group of 12 normal children. Regions of statistically significant thickness difference are detected as rejection regions for group comparison tests based on random fields theory.
Section snippets
Subjects and image processing
Gender and handedness affect brain anatomy (Luders et al., 2003) so all the 16 autistic and 12 control subjects used in this study were screened to be right-handed males except one subject who is ambidextrous. Sixteen autistic subjects were diagnosed with high functioning autism (HFA) via the Autism Diagnostic Interview-Revised (ADI-R) by a trained and certified psychologist at the Waisman center at the University of Wisconsin-Madison. Twelve healthy, typically developing males with no current
Heat kernel smoothing
All the preceding image accession and processing steps introduce unwanted noise into the cortical thickness measurements. Consider the following stochastic model for thickness measurement on cortical manifold ∂Ω:where θ is the mean thickness and ɛ is a zero mean Gaussian random field. The cortical surface ∂Ω can be assumed to be a smooth 2-dimensional Riemannian manifolds (Dale and Fischl, 1999, Joshi et al., 1995). We define the heat kernel smoothing estimator of data θ to
Statistical analysis on cortical manifolds
The first group consists of autistic subjects and the second group consists of normal control. For the ith group, let ni denote the number of subjects and θi denote the population mean thickness. Under stochastic model (1), we are interested in testing if the thickness for the two groups are identical, that is,vs.
The above null hypothesis is the intersection of the collection of hypotheseswhere H0(p) : θ1(p) = θ2(p). Assuming
Results and discussion
Image acquisition and processing were performed as described in Subjects and image processing, resulting in a cortical thickness map and the total gray matter volume (see Table 1) for each subject. The thickness measurements were then smoothed with the heat kernel of size 30 mm FWHM as described in Heat kernel smoothing and used to compute the corrected P value maps for t and F statistics as discussed in Statistical analysis on cortical manifolds. First, we performed analysis without removing
Acknowledgments
Authors with to thank Arnaud Cachia at the Service Hospitalier Frédéric Joliot, CEA, Orsay, France for providing the traces of the central and superior temporal sulcal fundi used in producing Fig. 3. Authors also wish to thank Keith Worsley of the McGill University, Kam Tsui and Shijie Tang of the Department of Statistics, University of Wisconsin-Madison and Tulaya Limpiti of the Department of Electrical Engineering, University of Wisconsin-Madison for valuable discussions on heat kernel
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