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Reeb graphs for topological connectomics

Abstract

Our brain consists of approximately 100 billion neurons that form functional neural networks across different brain regions. Brain functions involve complex interactions between these regions that are poorly understood and lack quantitative characterizations. Diffusion MRI of the brain generates millions of complex curvilinear fibers (streamlines) in 3D that exhibits the geometry of white matter pathways in the brain. It has been shown that a critical element in neurological and developmental disorders is the topological deterioration in streamlines. Despite this, most existing methods model the neural connections with connectivity matrices that overlook the topology of connections.

Towards addressing this shortcoming, we model neuronal fibers as context-aware geometrical objects in three-dimensional space. We introduce a novel Reeb graph-based method that efficiently encodes the topology and geometry of white matter fibers. Given the trajectories of neuronal fiber pathways of a neuroanatomical bundle, we re-bundle the streamlines by modeling their spatial evolution to capture geometrically significant events (akin to a fingerprint). Reeb graph parameters control the granularity of the model and handle the presence of improbable streamlines commonly produced by tractography. Our method meaningfully describes the bundle tracts without oversimplifying them into skeletons.

To quantify the difference between the connections, we introduce a new Reeb graph-based distance metric that quantifies the topological differences in bundle comparison. For the longitudinal repeated measures in the Cognitive Resilience and Sleep History (CRASH) dataset, repeated scans of a given subject acquired weeks apart lead to provably similar Reeb graphs that differ significantly from other subjects, thus highlighting our method's potential for clinical fingerprinting of brain regions. Reeb graph and topological distance metric are sensitive to variations in brain structure, such as those observed in the developing brain and in the presence of tumors. This sensitivity is beneficial in the context of longitudinal studies of brain development and the topological evolution of the brain. This thesis highlights the potential utility of our topology-based distance metric in tracking Alzheimer's disease progression using the ADNI dataset, evaluating the effects of surgical interventions for brain tumor using the OpenNeuro brain tumor datasets and quantifying the effect of shunt surgery for NPH brain condition in collaboration with UCI Medical Center.

Our method is not limited to neuroscience and is general-purpose in its applicability. This is demonstrated in our application of Reeb graphs for structure discovery in spatio-temporal trajectories where we use the model for detecting anomalous trajectories. Human behavior typically follows a pattern of normalcy in day-to-day activities. This is marked by recurring activities within specific time periods. Our method models this behavior using Reeb graphs where any deviation from usual day-to-day activities is encoded as nodes in the Reeb graph.

In summary, this thesis build up the theoretical framework for modeling spatio-temporal trajectories in 2D and 3D space, with applications to brain connectome analysis. A promising direction for future research is to link the Reeb graph-based structural model presented here with the behavioral model derived from functional MRI. This multi-modal strategy is expected to yield novel insights into the structural and functional connectivity of the human brain.

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