A Computational Geometry Approach for Modeling Neuronal Fiber Pathways

Abstract

We propose a novel and efficient algorithm to model high-level topological structures of neuronal fibers. Tractography constructs complex neuronal fibers in three dimensions 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 aims to simplify the connectivity of white matter fibers. Given the trajectories of neuronal fiber pathways, we model the evolution of trajectories that encodes geometrically significant events and calculate their point correspondence in the 3D brain space. Trajectory inter-distance is used as a parameter to control the granularity of the model that allows local or global representation of the tractogram. Using diffusion MRI data from Alzheimer's patient study, we extract tractography features from our model for distinguishing the Alzheimer's subject from the normal control. Software implementation of our algorithm is available on GitHub (https:// github.com/UCSB-VRL/ReebGraph.

A robust Reeb graph model of white matter fibers with application to Alzheimer’s disease progression⋆

“…Two trajectories (or subtrajectories) T i and T j are ϵ -connected if ∀ p i ∈ T i , ∃ p j ∈ T j such that p i , p j are ϵ -connected. Given ℐ and a distance parameter ϵ , bundling structure of a set of trajectories is modeled using a Reeb graph of the ϵ -connected trajectories [21,22]. We define:…”

“…Max-width ϵ-bundle partition: Given a set of max-width ϵ-bundles, a max-width ϵ-bundle partition is defined as P = {B 1 , B 2 , ..., B l } such that every point in I is assigned to a bundle. To compute P, we solve a subtrajectory clustering problem as discussed in [3,21] by maintaining a spatially dynamic graph G representing the ϵ-connected relation. The graph G changes with increasing spatial position: when trajectories are ϵ-connected we insert new edges into G, and when ϵ-disconnected we remove edges.…”

“…Hence, to computationally model effects such as the increase in sparsity of bundle specific white matter fibers with the progression of Alzheimer’s disease (AD) [20] or the disruptions caused due to stroke [7], a hypothesis-driven computational geometry approach is necessary. Towards that end, a Reeb graph-based approach in [21,22] discovers the branch and merge structure of the streamlines for a topological understanding of white matter fibers. However, basic Reeb graph computation fails to account for false positives and is not robust to the noise that the tractography methods introduce.…”

“…We quantify the white matter degeneration and increase in sparsity in progressive AD using the ADNI dataset. Given I and a distance parameter ϵ, bundling structure of a set of trajectories is modeled using a Reeb graph of the ϵ-connected trajectories [21,22]. We define: Bundle: A Bundle denoted by B is a set of subtrajectories that consists of at most one subtrajectory from each trajectory T .…”

ReeBundle: A Method for Topological Modeling of White Matter Pathways Using Diffusion MRI

“…In contrast, ReeBundle helps to visualize, interact with, and interpret brain connectivity. Preliminary work on the use of a Reeb graph to model white matter fibers appeared in [18], [19]. In this paper, we extend this approach with a novel robust algorithm that is validated and provide practical use cases of the proposed method.…”