We have organized the modeling research described into two specific Aims. Each Aim tackles a fundamental problem in the computational modeling of biological structures. The proposed solutions and techniques build on our recent advances in image analysis, differential equations, numerical analysis, and differential geometry, as well as our many yearsâ€™ work in mathematics and neuroscience.

**Surface Modeling and Conformal Mapping**

We are developing a general framework for modeling, mapping, and analyzing biological structures represented computationally as 3D surfaces. Using novel results in conformal mapping theory and variational methods, we are developing mathematics to impose parametric surface coordinate systems on biological structures.

**Image Segmentation Using Level Sets, and Geometrical PDEs**

We are creating, testing, and validating a set of methods to find, identify, and isolate specific biological structures in biomedical images. Known collectively as segmentation methods, these approaches will allow the automatic identification and measurement of structures in brain imaging data, although they apply to all types of medical images.

**Motivating Problems**

One way to analyze and compare brain data is to map them into a canonical space while retaining as much geometric information on the original structures as possible. Fischl demonstrated that surface-based brain mapping may offer advantages over volume based brain mapping, especially when localizing cortical deficits and functional activations. We have introduced a mathematical framework based on covariant partial differential equations and pull-backs of mappings under harmonic flows to analyze signals localized on brain surfaces. We have applied these methods to questions in Alzheimer’s disease, schizophrenia, development, and functional neuroimaging of tasks that activate frontal and hippocampal cortices.

Many of our anatomical segmentation methods automatically produce models of the white matter, gray matter, and other elements in the brain as triangulated meshes. Through careful automated topology correction, these surfaces can in fact be guaranteed to be homeomorphic to a sphere, i.e. they have no holes or handles and they have an underlying spherical topology. However, unless a consistent coordinate system is imposed on these meshes, they cannot easily be compared, matched, or averaged together. Conformal mapping provides one approach to produce regular grids on surfaces, which can then serve as the basis for a broad range of computations including the solution of PDEs, computation of 3D flows, statistical maps, and vector or tensor fields that describe morphometric differences within and between population. The overarching goal of these conformal parameterization methods is to provide a local coordinate structure to represent anatomy and perform grid based computations on it. Because the conformal maps to a sphere induce a spherical coordinate system on the brain surface, we are pursuing work that analyzes surface shape using spherical harmonics. These harmonic shape coefficients are provided for analysis of brain shape differences and the domain of normal shape variation in a population, using statistical methods of principal component analysis (PCA), or singular value decomposition (SVD), to extract low-frequency shape information from the anatomical models.

Many processing tasks that use the geometric surface of the brain can be accomplished in the frequency domain more efficiently, such as geometric compression, matching, surface denoising, feature detection, and shape analysis. Similar to image compression using Fourier analysis, geometric brain data can be compressed – and its shape coefficients computed – using spherical harmonic analysis. Global geometric information is concentrated in the low frequency components, whereas local detail and noise is concentrated in the high frequency part. By using low pass filtering, we can keep the major geometric features and compress the brain surface without losing too much information.

**Image Segmentation**

An important application of image segmentation methods is the separation of the brain from non-brain tissue, also known as skull-stripping. This is an important and difficult image processing problem in brain mapping research. The problem requires that extra-cortical voxels in MR brain images, which represent the skull, skin, fat and dura, are removed in order to facilitate accurate analysis of cortical structures (Dale et al., 1999). Removing extra-cortical tissue is also performed to improve the registration of brain surfaces in the creation of an atlas or in warping brain images to an existing atlas.

Many skull-stripping methods have been proposed. Of these, the Brain Surface Extractor (BSE) uses a Marr-Hildreth edge detector to identify anatomical boundaries, and a sequence of morphological and connected component operations to separate connected regions. BSE requires three parameters to perform skull-stripping, and these parameters are sensitive to minor differences in the data sets. Moreover, methods based upon edge detection and threshold classification, are generally not as robust and accurate as deformable models. There is a deformable model for skull-stripping called the Brain Extraction Tool (BET), which utilizes a set of forces to explicitly propagate a front interface.

We are developing new deformable models, based upon Level Set methods, for skull-stripping brain MR images. The front interface is implicitly represented by the level set function and is potentially more stable than explicit methods. We are interested in Level Set methods because they have the advantage of allowing complex topological changes, which improves robustness and stability, and there exist efficient numerical techniques that enable faster curve evolution.