Building Large-Scale Brain Atlases for Disease and Genetic Applications: Covariant PDEs and Probability Distributions on Manifolds

Paul Thompson
Assistant Professor of Neurology
UCLA School of Medicine and Laboratory of Neuro Imaging

[Powerpoint Slides: .ppt (32MB)]

Tutorial as part of the:
Conference on Discrete Geometry with Applications to Science and Medicine, May 16-19, 2002
Wakulla Springs State Park and Lodge, Wakulla Springs, FL

(Hosted by the Focused Research Group on Computational Conformal Mapping and Scientific Visualization:
De Witt Sumners, Monica Hurdal, Phil Bowers: Mathematics, Florida State U., Tallahassee;
Ken Stephenson, Chuck Collins: Mathematics, U. of Tennessee, Knoxville;
David Rottenberg, Neurology & Radiology, U. of Minnesota, Minneapolis;
with funding support provided by NSF).

ABSTRACT

Neuroscience and medicine are increasingly empowered by new mathematics that derives information from brain imaging databases. Algorithms now uncover disease-specific patterns of brain structure and function in whole populations. These tools chart how the brain grows in childhood, detect abnormalities in disease, and visualize how genes, medication, and demographic factors affect the brain.

We review recent developments in brain image analysis, focusing on mathematical challenges whose solution will advance the field. We describe our construction of statistical brain atlases that store detailed information on how the brain varies across age and gender, in health and disease, and over time. Specifically, we introduce a mathematical framework to analyze variations in brain organization, cortical patterning, asymmetry and tissue distribution in several collaborative studies of brain development and disease (N>1000 scans). Mathematics based on Grenander's pattern theory, covariant partial differential equations (PDEs), pull-backs of mappings under harmonic flows, and high-dimensional random tensor fields are employed to encode anatomic variations in population-based brain image database. We use this reference information to detect disease-specific abnormalities in Alzheimer's disease and schizophrenia, including dynamic changes and medication response over time. We will show illustrative examples resolving disease-specific patterns that are not apparent in individual brain images. We also discuss four-dimensional (4D) maps that store probabilistic information on the dynamics of development and disease. Digital atlases can identify general patterns of structural and functional change in diseased populations, and link them to therapeutic and genetic parameters. Finally, we introduce a framework to map how genes affect brain structure. The resulting genetic brain maps can be used in data mining applications, to help investigate inheritance patterns and search for susceptibility loci in diseases with known genetic risks.

For more information please visit: http://www.loni.ucla.edu/~thompson/thompson.html where PDFs of tutorial papers and chapters are available.

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