Great progress has been made in developing mathematical algorithms to analyze 3D
brain images. Drawing upon large image databases, powerful computer algorithms can
now detect disease-specific patterns of brain structure and function. In one approach, we
and others have created statistical atlases to measure how the brain varies across age
and gender, across time, in health and disease, and in large human populations. We
model brain structures as 3D curves and surfaces. Flows, metrics, and statistical fields
are defined on these manifolds, and used to detect anatomical differences across
subjects or groups. A mathematical framework based on covariant partial differential
equations (PDEs), pull-backs of mappings under harmonic flows, and high-dimensional
random fields is used encode anatomical variations in a brain image database (N>1000
scans). We use this reference information to detect brain abnormalities in Alzheimer's
disease and schizophrenia, including how the brain changes over time, and responds to
medication. This has revealed surprising patterns that were not apparent in individual
brain images, visualizing in detail how disease and development impact the brain.
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