Elsevier

NeuroImage

Volume 18, Issue 2, February 2003, Pages 198-213
NeuroImage

Regular article
Deformation-based surface morphometry applied to gray matter deformation

https://doi.org/10.1016/S1053-8119(02)00017-4Get rights and content

Abstract

We present a unified statistical approach to deformation-based morphometry applied to the cortical surface. The cerebral cortex has the topology of a 2D highly convoluted sheet. As the brain develops over time, the cortical surface area, thickness, curvature, and total gray matter volume change. It is highly likely that such age-related surface changes are not uniform. By measuring how such surface metrics change over time, the regions of the most rapid structural changes can be localized. We avoided using surface flattening, which distorts the inherent geometry of the cortex in our analysis and it is only used in visualization. To increase the signal to noise ratio, diffusion smoothing, which generalizes Gaussian kernel smoothing to an arbitrary curved cortical surface, has been developed and applied to surface data. Afterward, statistical inference on the cortical surface will be performed via random fields theory. As an illustration, we demonstrate how this new surface-based morphometry can be applied in localizing the cortical regions of the gray matter tissue growth and loss in the brain images longitudinally collected in the group of children and adolescents.

Introduction

The cerebral cortex has the topology of a 2-dimensional convoluted sheet. Most of the features that distinguish these cortical regions can only be measured relative to that local orientation of the cortical surface (Dale and Fischl, 1999). As brain develops over time, cortical surface area as well as cortical thickness and the curvature of the cortical surface change. As shown in the previous normal brain development studies, the growth pattern in developing normal children is nonuniform over whole brain volume Chung et al 2001a, Giedd et al 1999, Paus et al 1999, Thompson et al 2000. Between ages 12 and 16, the corpus callosum and the temporal and parietal lobes show the most rapid brain tissue growth and some tissue degeneration in the subcortical regions of the left hemisphere Chung et al 2001a, Thompson et al 2000. It is equally likely that such age-related changes with respect to the cortical surface are not uniform as well. By measuring how geometric metrics such as the cortical thickness, curvature, and local surface area change over time, any statistically significant brain tissue growth or loss in the cortex can be detected.

The first obstacle in developing surfaced-based morphometry is the automatic segmentation or extraction of the cortical surfaces from MRI. It requires first correcting RF inhomogeneity artifacts. We have used the nonparametric nonuniform intensity normalization method (N3), which eliminates the dependence of the field estimate on anatomy (Sled et al., 1998). The next step is the tissue classification into three types: gray matter, white matter, and cerebrospinal fluid (CSF). An artificial neural network classifier Ozkan et al 1993, Worsley et al 1999 or a mixture model cluster analysis (Good et al., 2001) can be used to segment the tissue types automatically. After the tissue classification, the cortical surface is usually generated as a smooth triangular mesh. The most widely used method for triangulating the surface is the marching cubes algorithm (Lorensen and Cline, 1987). Level set method (Sethian, 1996) or deformable surfaces method (Davatzikos, 1995) are also available. In our study, we have used the anatomic segmentation using the proximities (ASP) method (MacDonald et al., 2000), which is a variant of the deformable surfaces method, to generate cortical triangular mesh that has the topology of a sphere. Brain substructures such as the brain stem and the cerebellum were removed. Then an ellipsoidal mesh that already had the topology of a sphere was deformed to fit the shape of the cortex guaranteeing the same topology. The resulting triangular mesh will consist of 40,962 vertices and 81,920 triangles with the average internodal distance of 3 mm. Partial voluming is a problem with the tissue classifier but topology constraints used in the ASP method were shown to provide some correction by incorporating many neuroanatomical a priori information MacDonald 1997, MacDonald et al 2000. The triangular meshes are not constrained to lie on voxel boundaries. Instead the triangular meshes can cut through a voxel, which can be considered as correcting where the true boundary ought to be. Once we have a triangular mesh as the realization of the cortical surface, we can model how the cortical surface deforms over time.

In modeling the surface deformation, a proper mathematical framework might be found in both differential geometry and fluid dynamics. The concept of the evolution of phase-boundary in fluid dynamics (Drew, 1991; Gurtin and McFadden, 1991), which describes the geometric properties of the evolution of boundary layer between two different materials due to internal growth or external force, can be used to derive the mathematical formula for how the surface changes. It is natural to assume the cortical surfaces to be a smooth 2-dimensional Riemannian manifold parameterized by u1 and u2: X(u1, u2) = {x1(u1, u2), x2(u1, u2), x3(u1, u2) : (u1, u2) ∈ D ⊂ R2}. A more precise definition of à Riemannian manifold and a parameterized surface can be found in Boothby 1986, Do Carmo 1992, and Kreyszig (1959). If D is a unit square in R2 and a surface is topologically equivalent to a sphere then at least two different global parameterizations are required. Surface parameterization of the cortical surface has been done previously by Thompson and Toga (1996) and Joshi et al. (1995). From the surface parameterization, Gaussian and mean curvatures of the brain surface can be computed and used to characterize its shape Dale and Fischl 1999, Griffin 1994, Hamilton 1992, Dale and Fischl 1999. In particular, Joshi et al. (1995) used the quadratic surface in estimating the Gaussian and mean curvature of the cortical surfaces.

By combining the mathematical framework of the evolution of phase boundary with the statistical framework developed for 3D whole brain volume deformation (Chung et al., 2001), anatomical variations associated with the deformation of the cortical surface can be statistically quantified. Using the same stochastic assumption on the deformation field used in Chung et al. (2001), we will localize the region of brain shape changes based on three surface metrics: area dilatation rate, cortical thickness, and curvature changes and show how these metrics can be used to characterize the brain surface shape changes over time.

As an illustration of our unified approach to deformation-based surface morphometry, we will demonstrate how the surface-based statistical analysis can be applied in localizing the cortical regions of tissue growth and loss in brain images longitudinally collected in a group of children and adolescents.

Section snippets

Modeling surface deformation

Let U(x, t) = (U1, U2, U3)t be the 3D displacement vector required to deform the structure at x = (x1, x2, x3) in gray matter Ω0 to the homologous structure after time t. Whole gray matter volume Ω0 will deform continuously and smoothly to Ωt via the deformation xx + U while the cortical boundary ∂Ω0 will deform to ∂Ωt. The cortical surface ∂Ωt may be considered as consisting of two parts: the outer cortical surface ∂Ωtout between the gray matter and CSF and the inner cortical surface ∂Ωtin

Surface parameterization

The ASP method generates triangular meshes consisting of 81,920 triangles evenly distributed in size. In order to quantify the shape change of the cortical surface, surface parameterization is an essential part. We model the cortical surface as a smooth 2D. Riemannian manifold parameterized by two parameters u1 and u2 such that any point x ∈ ∂Ω0 can be uniquely represented as x=X(u1, u2) for some parameter space u = (u1, u2) ∈ DR2. We will try to avoid global parameterization such as tensor

Surface-based morphological measures

As in the case of local volume change in the whole brain volume (Chung et al., 2001), the rate of cortical surface area expansion or reduction may not be uniform across the cortical surface. Extending the idea of volume dilatation, we introduce a new concept of surface area, curvature, cortical thickness dilatation, and their rate of change over time.

Suppose that the cortical surface ∂Ωt at time t can be parameterized by the parameters u = (u1, u2) such that any point x ∈ ∂Ωt can be written as x

Surface-based diffusion smoothing

In order to increase the signal-to-noise ratio (SNR) as defined in Dougherty 1999, Rosenfeld and Kak. 1982, and Worsley et al. (1996b), Gaussian kernel smoothing is desirable in many statistical analyses. For example, Fig. 5 shows fairly large variations in cortical thickness of a single subject displayed on the average brain atlas Ωatlas. By smoothing the data on the cortical surface, the SNR will increase if the signal itself is smooth and in turn, it will be easier to localize the

Statistical inference on the cortical surface

All of our morphological measures such as surface area, cortical thickness, curvature dilatation rates are modeled as Gaussian random fields on the cortical surface; i.e., Λ(x)=λ(x)+ϵ(x), x ∈ ∂Ωatlas, where the deterministic part λ is the mean of the metric Λ and ϵ is a mean zero Gaussian random field. This theoretical model assumption has been checked using both Lilliefors test (Conover, 1980) and quantile-quantile plots (qqplots) (Hamilton, 1992). The qqplot displays quantiles from an

Results

Twenty-eight normal subjects were selected based on the same physical, neurological, and psychological criteria described in Giedd et al. (1996). This is the same data set reported in Chung et al. (2001), where the Jacobian of the 3D deformation was used to detect statistically significant brain tissue growth or loss in 3D whole brain via deformation-based morphometry. 3D Gaussian kernel smoothing used in this study is not sensitive to the interfaces among the gray, white matter, and CSF.

Conclusions

The surface-based morphometry presented here can statistically localize the regions of cortical thickness, area, and curvature change at a local level without specifying the regions of interest (ROI). This ROI-free approach might be best-suited for exploratory whole brain morphometric studies. Our analysis successfully avoids artificial surface flattening Andrade et al 2001, Angenent et al 1999, which can destroy the inherent geometrical structure of the cortical surface. It seems that any

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