Topological characterizations of neuronal fibers and its implications in comparing brain connectomes

Our brain consists of approximately 100 billion neurons that form functional networks across different brain regions. Brain functions involve complex interactions between these regions that are poorly understood and lack quantitative characterizations. Such complex data analysis requires the integration of multiple imaging modalities and methods. In this work, we apply topological data analysis to diffusion magnetic resonance imaging (dMRI) and show that it has great potential in comparing brains. This is motivated by our previous research work that combines features from neuronal fibers along with the brain lesion segmentation mask, thereby, improving the performance of a patch-based neural network. The novel use of tractographic features appears to be promising in the overall survival prediction of brain tumor patients. However, previous studies ignore the geometry and topology of the white matter fibers. Towards this, we propose a novel and efficient algorithm to model high-level topological structures of neuronal fibers using the construct of the Reeb graph. Tractography generates complex neuronal fibers in 3D that exhibit the geometry of white matter pathways in the brain. However, most tractography analysis methods are time consuming and intractable. We develop a computational geometry-based tractography representation that simplifies the connectivity of white matter fibers into a graph-based mathematical model. We present an application of the Reeb graph model integrated with a machine learning model to the classifications tasks from Alzheimer's studies. Experimental results are reported on the ADNI (\url{http://adni.loni.usc.edu/}) dataset.