Mathematical/Computational Challenges in Population-Based Brain Mapping

Invited Talk,
Annual Conference of the Society for Industrial and Applied Mathematics:
1st SIAM Conference on the Life Sciences, Boston, Sept. 22-26 2001;
Minisymposium: Mapping the Human Brain, organized by Monica Hurdal, Florida State University
Speakers: Bruce Fischl, MGH; Paul Thompson, UCLA School of Medicine; Eric Schwartz, Boston University; and Monica Hurdal, Florida State University

We describe some exciting mathematical advances in our work as part of a collective effort to create a population-based atlas of the human brain, based on a database of over 1000 3-dimensional brain images. One goal of the atlas is to resolve disease-specific patterns of brain structure and function in several patient populations. We are also investigating how these patterns vary by age, gender, genotype, over time, and in response to therapy. Nonetheless, brain structure is so complex and variable across subjects that it is extremely challenging to (1) automatically identify structures in brain images; (2) detect abnormal brain structure; (3) relate abnormalities to genetic risk or demographic factors; and (4) compare the dynamics of 4-dimensional growth or degenerative processes in the human brain. Probabilistic brain atlases address these challenges by encoding dynamic variations in brain structure and function. We describe our development of atlases that synthesize data across age, gender, time, and multiple brain imaging modalities, to represent normal populations and diseased subpopulations with Alzheimer's Disease and schizophrenia. To identify disease-specific patterns of brain structure and function, we combine mathematical approaches from Riemannian geometry, the theory of Gaussian random fields, variational calculus, and covariant partial differential equations, with supercomputing algorithms for image warping and analysis. These approaches are beginning to uncover intriguing waves of brain growth in childhood, patterns of genetic influences on brain structure and function, and new aspects of how diseases evolve dynamically in the brain. Current challenges will be described that will be of interest to applied mathematicians and statisticians in general, as well as researchers in image analysis and biomedical fields.

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