doi:10.1016/j.neuroimage.2004.12.052 
Copyright © 2004 Elsevier Inc. All rights reserved.
Cortical thickness analysis in autism with heat kernel smoothing
Moo K. Chunga, b, c, 
, 
, Steven M. Robbinsf, Kim M. Daltonc, Richard J. Davidsonc, d, Andrew L. Alexanderc, e and Alan C. Evansf
aDepartment of Statistics, University of Wisconsin, Madison, 1210 West Dayton Street, WI 53706, USA
bBiostatistics and Medical Informatics, University of Wisconsin, Madison, 1210 West Dayton Street, WI 53706, USA
cW.M. Keck Laboratory for Functional Brain Imaging and Behavior, University of Wisconsin, Madison, WI 53706, USA
dDepartment of Psychology and Psychiatry, University of Wisconsin, Madison, WI 53706, USA
eDepartment of Medical Physics, University of Wisconsin, Madison, WI 53706, USA
fMontreal Neurological Institute, McGill University, Canada
Received 5 October 2004;
revised 22 November 2004;
accepted 3 December 2004.
Available online 10 March 2005.
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Abstract
We present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold. Gaussian kernel smoothing, which weights neighboring observations according to their 3D Euclidean distance, has been widely used in 3D brain images to increase the signal-to-noise ratio. When the observations lie on a convoluted brain surface, however, it is more natural to assign the weights based on the geodesic distance along the surface. We therefore develop a framework for geodesic distance-based kernel smoothing and statistical analysis on the cortical manifolds. As an illustration, we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique.
Keywords: Cortical thickness; Autism; Brain; Heat kernel; Diffusion smoothing
Fig. 1. Left: a typical triangular surface representation of the brain cortical manifold. Right: typical triangular surface with m = 6 neighboring vertices around p = q0.
Fig. 2. Anatomy of brain cortex. Left: part of the cortical surface showing both outer (yellow) and inner surface (blue) that bound gray matter. Right: enlargement of the boxed region. The cortical thickness measures the distance between outer and inner surfaces. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Automatically generated traces of the central and superior temporal sulcal fundi (Cachia et al., 2003b). The first column shows the traces generated for the template surface. The second column shows the probability of sulcal matching based on 149 normal subjects before any surface normalization. The third column shows the probabilities after surface normalization. The first row is the left hemisphere and the second row is the right hemisphere. Note that the distribution is much more spatially concentrated and the probabilities are much greater after normalization.
Fig. 4. Probability of sulcal matching, after normalization, for 39 manually identified central sulci defined as the surface region surrounded by gyri, not just the fundus. The views are illustrated on a slightly-opened version of the template cortical surface in order to better view inside the sulcus. The warping in 2D localizes the central sulcus nearly completely inside the template central sulcus. Left (right) figure is the left (right) central sulci.
Fig. 5. Top: average surface of the normal subjects, on which statistical maps are projected, constructed via the surface registration method in Chung et al. (2003). First two images show the outer cortical surface and the next two images show the inner cortical surface. Bottom: average template constructed via the improved surface warping method providing more detailed anatomy.
Fig. 6. Heat kernel smoothing on real and simulated data. Top: the sample mean and the sample variance of 12 normal subjects. These are used in generating simulated data. Middle: iterated heat kernel smoothing of real data with σ = 1 and k = 20, 100, 200. Bottom: iterated heat kernel smoothing of simulated data with σ = 1 and k = 20, 200, 5000. At k = 5000 iterations, it shows the increasing convergence to the within-subject mean thickness (Property 3).
Fig. 7. Left: within-subject variance plotted over the number of iterations of heat kernel smoothing. Decreasing variance implies the convergence of the heat kernel smoothing to the within-subject mean (Property 3). Right: between-subject variance plotted over the number of iterations illustrating Property 2.
Fig. 8. Corrected P value maps projected onto the average outer (1st and 3rd rows) and inner surfaces (2nd and 4th rows). First two rows: two-sample t test results. Red is the regions of thicker gray matter while blue is thinner gray matter in the autistic subjects. Last two rows: F test results removing the effect of age and relative gray matter volume difference. F test results shows relatively asymmetric thickness difference between two groups. Comparing two P value maps, it can be seen that the thicker gray matter region is largely due to the effect of age and gray matter volume difference.
Table 1.
Age and relative total gray matter volume distribution (×105 mm3 for volume measurements)
The mean gray matter volumes are 7.08 ± 0.46 mm3 for the autistic group and 6.95 ± 0.33 mm3 for the control group. The method for estimating the total gray matter volume is given in Chung et al. (2003).
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