Computational Neuroanatomy

The following sections are included:

• Preface

• Contents

In the usual structural image analysis, images are segmented and registered to a template. Then various individual imaging characteristics relative to the template are feed into general linear models. The resulting statistical parametric maps are used to quantify and localize signal while accounting for multiple comparisons across different voxels. In this chapter, the basics on general linear models and random field theory based multiple comparisons correction are presented.

The advancement of magnetic resonance imaging (MRI) gives us a new computational tool for the characterization of temporally varying brain morphology and this is emerging as the new field of computational neuroanatomy. MRI depends on the response of magnetic fields to produce digital images that provide structural information about brain tissue nonrinvasively. Compared to the computed tomography (CT), MRI has been mainly used for in vivo imaging of brain due to due to much greater image contrast in soft tissues. The T1-weighted MRI use a gradient echo sequence and it is more often used in anatomical studies compared to the T2-weighted MRI. This noninvasive but somewhat expensive procedure has become a standard brain imaging modality in examining the structure of the brain since it provides a good image contrast between soft tissues…

Another very promising technique for non-ROI based morphometry is tensor-based morphometry (TBM), which uses the spatial derivatives of deformation fields (Ashburner and Friston, 2000; Thompson et al., 2000; Chung et al., 2001a). Similar to DBM, TBM does not require segmentation of a priori regions of interest (Ashburner and Friston, 2000; Davatzikos, 1999). The morphological tensor maps are computed at each voxel level and used to quantify the variations in length, area, volume, the cortical thickness and surface curvature (Thompson et al., 2000; Chung et al., 2003c). From these tensor maps, 3D statistical parametric maps (SPM) are created for a group of subjects to quantify the variations in length,area, volume and surface curvature and to visualize these variations in the 3D is whole brain volume (Chung et al., 2001a), on the 2D cortical surface (Thompson et al., 2001; Andrade et al., 2001; Chung et al., 2003c) and on the surface of the brain substructures such as the hippocampus and amygdala (Wang et al., 2003). In TBM, the Jacobian determinant of the deformation field, which is required to register one brain to another, is mainly used to quantify volumetric changes at voxel level. By definition, the Jacobian determinant of the deformation is the volume of the unit-cube after the deformation. Assuming that one can find the deformation field at each voxel, tissue growth or loss can be quantified at a voxel level…

The following sections are included:

• Image Segmentation
  • Mumford-Shah Model
  • Level Sets
  • Active Contours
  • Deformable Surface Models
  • Thin-Plate Spline Thresholding

• Mixture Models
  • Bayesian Segmentation
  • Mixture Models
  • Expectation Maximization Algorithm
  • Two Components Gaussian Mixtures

• Voxel-Based Morphometry
  • ROI Volume Estimation in VBM
  • Limitations of Witelson Partition
  • General Linear Models on Tissue Densities
  • 2D VBM Applied to Corpus Callosum

Due to the geometric nature of convoluted cortical surfaces, differential geometric approaches have been often used in quantifying shape changes in the cortical surfaces. For an overview of differential geometry, see Boothby (1986), do Carmo (1992) and Kreyszig (1959). Given a cortical surface which we assume to be a smooth twice-differentiable 2-dimensional manifold embedded in ℝ³ (Joshi et al., 1995), we have a parameterization of the surface

In brain imaging, cortical and subcortical surfaces obtained from MRI are often represented as triangular meshes. Image acquisition, segmentation and surface extraction process themselves are likely to introduce noise. Further processing such as thickness computation, surface registration and parameterization on the meshes are expected to introduce more noise in cortical measures. It is thus imperative to reduce noise measurements on the meshes. For instance, in order to increase the signal-to-noise ratio (SNR) and smoothness of data required for the subsequent random field theory based inference, some type of surface-based smoothing is necessary (Kiebel et al., 1999). The SNR is usually defined as the ratio:

The following sections are included:

• Surface Flattening

• Cortical Thickness
  • Cortical Thickness via Laplace Equation
  • Cortical Thickness vs. Gray Matter Density
  • Distance Map

• Partial Correlation Mapping
  • Partial Correlations
  • Statistical Inference on Correlations
  • Brain-Behavior Correlations
  • Facial Emotion Discrimination Tasks

• Tensor-Based Surface Morphometry
  • Surface Deformation
  • Metric Tensor Computation on Surfaces
  • Statistical Inference on Surfaces
  • Quantifying Brain Growth
  • Tensor Computation via SPHARM

• Multivariate General Linear Models
  • Roy’s Maximum Root
  • SurfStat

• Mixed Effect Models on Surface Shape Change
  • Longitudinal Imaging Data
  • Mixed Effect Models
  • Restricted Maximum Likelihood Estimation
  • Longitudinal Hippocampus Shape Model
  • Functional Mixed Effect Models

• Sparse Surface Shape Recovery
  • Sparse Regression on Surface Data
  • Effect of Aging on Hippocampus Shape

There are extensive literature on local cortical shape modeling and analysis (Chung et al., 2005a; Fischl and Dale, 2000; Joshi et al., 1997; Taylor and Worsley, 2008; Thompson and Toga, 1996; Lerch and Evans, 2005; Luders et al., 2006b; Miller et al., 2000)). The medial representation (Pizer et al., 1999) has been also successfully used in modeling various subcortical structures including the cross sectional images of the corpus callosum (Joshi et al., 2002), hippocampus and amygdala complex (Styner et al., 2003), ventricle and brain stem (Pizer et al., 1999). In the medial representation, the binary object is represented using the finite number of atoms and links that connect the atoms together to form a skeletal representation of the object. The medial representation is mainly used with the principal component analysis type of approach for shape classification and group comparison…

The cerebral cortex is composed of many functionally and structurally distinct areas, each receiving and sending various neuronal projections from and to each other. Thus the brain forms a very complex network of connections (Young, 1992). Neuroanatomical connectivity places structural constraints on the functional connectivity of the cerebral cortex (Sporns et al., 2000). Therefore, it is crucial to understand functional connectivity in relation to the underlying structural connectivity…

In this chapter, we present a new framework for characterizing signals in images using techniques borrowed from persistent homology, a branch of computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. This field is usually referred to as topological data analysis. The main aims of the topological data analysis are to (1) infer high dimensional structure from low dimensional representation and (2) assemble discrete points into global structure (Ghrist, 2008)…

The following sections are included:

• Bibliography

• Index