Paul Thompson's Research Publications

Mathematical/Computational Strategies for Creating a
Probabilistic Atlas of the Human Brain

Workshop on Statistics of Brain Mapping,
McGill University, June 13-14 1998

Paul Thompson and Arthur W. Toga

Laboratory of Neuro Imaging, Dept. Neurology, Division of Brain Mapping,
UCLA School of Medicine, Los Angeles CA 90095, USA

Analyzing the Cortex. Key questions in brain mapping focus on how cortical features (e.g., gray matter thickness, panel 1) differ across subjects. To relate measures obtained in one subject to another, the anatomy of one cortex is smoothly overlaid onto the other (covariant flow, panel 2). Statistical inferences can then be made about cross-subject and cross-group differences using a Beltrami flow to develop a Riemannian metric space (panel 3) in which the normalized residuals of the statistical fields are stationary.


In this talk, I review our recent progress as part of a collective effort to construct a probabilistic atlas of the human brain. Extreme variations in brain structure, especially in the gyral patterns of the human cortex, present two major challenges in brain mapping studies. First, anatomic variations make it especially difficult to design computerized strategies to detect abnormal brain structure. Second, integrating and comparing brain data from multiple subjects and groups is hampered by the extreme complexity of anatomic variations.

To illustrate these challenges, we describe several hybrid approaches for high-dimensional brain image registration, data integration, and pathology detection. In these approaches, computer vision algorithms and statistical pattern recognition measures are integrated with anatomically-driven elastic transformations which encode complex shape differences between systems of anatomic surfaces. These algorithms help to integrate brain data from many subjects, and detect structural anomalies and abnormal asymmetries in Alzheimer's Disease, as well as gyral and sulcal anomalies in schizophrenia and neurodevelopmental disorders. Exciting developments are occurring in pathology detection algorithms which encode patterns of brain variation using random vector fields (Thompson and Toga, 1998; Cao and Worsley, 1998; Thirion et al., 1998), shape-theoretic approaches (Bookstein et al., 1997), and pattern-theoretic approaches (Grenander and Miller, 1998).

Two new brain mapping algorithms are introduced: (1) a tensor-based approach for mapping dynamic (4D) growth patterns in the developing human brain, and (2) an approach termed covariant deformable templates (Thompson and Toga, 1998), which has a variety of applications in brain image registration, automated structure extraction, and in developing probabilistic models of the human cortex. Quantitative comparison of cortical models can be based on the mapping which drives one cortex onto another (Van Essen et al., 1997; Worsley et al., 1997; Thompson et al., 1997); elastic matching of cortical regions also factors out a substantial component of confounding variance in functional imaging studies (Collins et al., 1996; Grenander and Miller, 1998). In a covariant template approach, we first establish a cortical parameterization, in each subject in an image database, as the solution of a time-dependent partial differential equation (PDE) with a spherical computational mesh (MacDonald et al., 1993; cf. Davatzikos, 1996; Sereno et al., 1996). This procedure sets up an invertible parameterization of each surface in deformable spherical coordinates, and defines a Riemannian manifold (Bookstein, 1995). This Riemannian manifold is then not flattened (as in Drury et al., 1996; Van Essen et al., 1997), but serves as a computational mesh on which a second covariant PDE is defined which matches sulcal networks from subject to subject, encoding variations in gyral topography from one subject to another. Dependencies between the metric tensors of the underlying surface parameterizations and the matching field itself are eliminated through the use of generalized coordinates and Christoffel symbols (Thompson and Toga, 1998). This mathematical strategy was introduced by Einstein (1914) to allow the solution of physical field equations defined by elliptic operators on manifolds with intrinsic curvature. Similarly, the problem of deforming one cortex onto another involves solving a similar system of elliptic partial differential equations (Drury et al., 1996; Davatzikos, 1996; Thompson and Toga, 1998), defined on an intrinsically curved computational mesh (in the shape of the cortex). Using this approach, patterns of abnormal structure, 3D anatomic variation and asymmetry are mapped out in patient groups with Alzheimer's disease and schizophrenia.

Covariant deformable templates also show promise for automated structure extraction, in that they invoke an auxiliary tensor field which can be used to bias the surface dynamics of the deforming model in favor of certain expected target geometries. This behavior offers advantages in extracting models of anatomic surfaces with extremely complex geometry, such as the ventricles and caudate. Such algorithms for rapid anatomical model extraction and matching are a vital part of the efforts which focus on the structural and functional mapping of the human brain. Finally, progress in mapping dynamic (4D) patterns of brain growth in serial pediatric MR scans will be discussed, with a focus on integrating dynamic maps of growth into population-based atlases of the human brain.

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    Paul Thompson
    73-360 Brain Research Institute
    UCLA Medical Center
    10833 Le Conte Avenue
    Westwood, Los Angeles CA 90095-1761, USA.

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