1. Tensor-based Surface Modeling and Analysis
Computer Vision and Pattern Recognition 2003
Medical Image Analysis
Tensor-based Surface Modeling and Analysis
Moo K. Chung123, Keith J. Worsley4, Steve Robbins4, Alan C. Evans4
1Department of Statistics, 2Department of Biostatistics and Medical Informatics, 3W.M. Keck Laboratory for functional Brain Imaging and Behavior
University of Wisconsin, Madison, USA
4Montreal Neurological Institute, McGill University, Montreal, Canada
1. Motivation
We present a unified tensor-based surface morphometry in characterizing the gray matter anatomy change in the brain development longitudinally collected in the group of children and adolescents. 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.
5. Random fields theory
Statistical analysis is based on the random field theory (Worsley et al., 1996). The Gaussianess of the surface metrics is checked with the Lilliefors test. The isotropic diffusion smoothing is found to increase both the smoothness as well as the isotropicity of the surface data. For a paired t-test for detecting the surface metric difference, we used the corrected P-value of t random field defined on the manifold which is approximately
where is the 2-dimensional EC-density given by
and is the total surface area of the template
brain estimated to be 275,800mm2. The validity of our
modeling and analysis was checked by generating
null data. The null data were created by reversing
time for the half of subjects chosen randomly. In the
null data, most t values were well below the threshold
indicating that our image processing and statistical
analysis do not produce false positives.
2. Magnetic Resonance Images
Two T1-weighted magnetic resonance images (MRI) were acquired for each of 28 normal subjects at different times on the GE Sigma 1.5-T superconducting magnet system. The first scan was obtained at the age 11.5 years and the second scan was obtained at the age 16.1 years in average. MRI were spatially normalized and tissue types were classified based a supervised artificial neural network classifier (Kollakian, 1996). Afterwards, a triangular mesh for each cortical surface was generated by deforming a mesh to fit the proper boundary in a segmented volume using a deformable surface algorithm (MacDonald et al., 2000). This algorithm is further used in surface registration and surface template construction (Chung et al., 2003).
Top: Thin-plate spline energy functional computed on the inner cortical surface of a 14-year-old subject. It measures the amount of folding of the cortical surface. Bottom: t-statistic map showing statistically significant region of curvature increase between ages 12 and 16. Most of the curvature increase occurs on gyri while there is no significant change of curvature on most of sulci. Also there is no statistically significant curvature decrease detected, indicating that the complexity of the surface convolution increase.
6. Morphometric changes between ages 12-16
Gray matter volume: total gray matter volume shrinks. Local growth in the parts of temporal, occipital, somatosensory, and motor regions.
Cortical Surface area: total area shrinks. highly localized area growth along the left inferior frontal gyrus and shrinkage in the left superior frontal sulcus.
Cortical thickness: no statistically significant local cortical thinning on the whole cortex. Predominant thickness increase in the left superior frontal sulcus.
Cortical curvature: no statistically significant curvature decrease. Most curvature increase occurs on gyri. No curvature change on most sulci. Curvature increase in the superior frontal and middle frontal gyri.
4. Surface data smoothing: Beltrami flow
To increase the signal-to-noise ratio and to generate smooth Gaussian random fields for statistical analysis, surface-based data smoothing is essential. Isotropic diffusion smoothing or Beltrami-flow is developed for this purpose. It is not the surface fairing of Taubin (1995), where the surface geometry is smoothed. We solve an isotropic heat equation on a manifold with an initial condition.
where the Laplacian is the Laplace-Beltrami operator defined in terms of the Riemannian metric tensor g:
We estimate the Laplace-Beltrami operator on a triangulated cortical surface directly via finite element method (Chung, 2001). Let F(pi) be the signal on the i-th node pi in the triangulation. If p1,...,pm are m-neighboring nodes around p=p0, the Laplace-Beltrami operator at p is estimated by
with the weights where  and are the two angles opposite to the edge pi - p in triangles and is the sum of the areas of m-incident triangles at p. Then the diffusion equation is solved via the finite difference scheme:
Left: The Gyri are extracted by thresholding the thin-plate energy functional on the inner surface. Middle & Right: Individual gyral patterns mapped onto the template surface. The gyri of a subject match the gyri of the template surface illustrating a close homology between the surface of the individual subject and the template.
References
Chung, M.K., Statistical Morphometry in Neuroanatomy, PhD Thesis, McGill University, Canada
Chung, M.K. et al., Deformation-based Surface Morphometry applied to Gray Matter Deformation, NeuroImage. 18:198–213, 2003.
Kollakian, K., Performance analysis of automatic techniques for tissue classification in magnetic resonance images of the human brain. Master’s thesis, Concordia Univ., Canada
MacDonald, J.D. et al., Automated 3D Extraction of Inner and Outer Surfaces of Cerebral Cortex from MRI, NeuroImage. 12: , 2000.
Taubin, G., Curve and surface smoothing without shrinkage. The Proceedings of the Fifth International Conference on Computer Vision, , 1995.
Worsley, K.J., et al., A unified statistical approach for determining significant signals in images of cerebral activation, Human Brain Mapping. 4:58-73, 1996.
Left: Individual cortical surfaces (blue: interface between the gray and white matter, yellow: outer cortical surface). Right: The surface template is constructed by averaging the coordinates of homologous vertices.
3. Tensor geometry
Based on the local quadratic surface parameterization, Riemannian metric tensors were computed and used to characterize the cortical shape variations. Then based on the metric tensors, the cortical thickness, local surface area, local gray matter volume, curvatures were computed.