Analytic tractography: A closed-form solution for estimating local white matter connectivity with diffusion MRI

Abstract

White matter structures composed of myelinated axons in the living human brain are primarily studied by diffusion-weighted MRI (dMRI). These long-range projections are typically characterized in a two-step process: dMRI signal is used to estimate the orientation of axon segments within each voxel, then these local orientations are linked together to estimate the spatial extent of putative white matter bundles. Tractography, the process of tracing bundles across voxels, either requires computationally expensive (probabilistic) simulations to model uncertainty in fiber orientation or ignores it completely (deterministic). Furthermore, simulation necessarily generates a finite number of trajectories, introducing "simulation error" to trajectory estimates. Here we introduce a method to analytically (via a closed-form solution) take an orientation distribution function (ODF) from each voxel and calculate the probabilities that a trajectory projects from a voxel into each directly adjacent voxels. We validate our method by demonstrating experimentally that probabilistic simulations converge to our analytically computed transition probabilities at the voxel level as the number of simulated seeds increases. We then show that our method accurately calculates the ground-truth transition probabilities from a publicly available phantom dataset. As a demonstration, we incoroporate our analytic method for voxel transition probabilities into the Voxel Graph framework, creating a quantitative framework for assessing white matter structure, which we call "analytic tractography". The long-range connectivity problem is reduced to finding paths in a graph whose adjacency structure reflects voxel-to-voxel analytic transition probabilities. We demonstrate that this approach performs comparably to the current most widely-used probabilistic and deterministic approaches at a fraction of the computational cost. We also demonstrate that analytic tractography works on multiple diffusion sampling schemes, reconstruction method or parameters used to define paths. Open source software compatible with popular dMRI reconstruction software is provided.

Finding maximally disconnected subnetworks with shortest path tractography

“…White matter voxel graphs were constructed for each HCP subject using the double-ODF method in MITTENS (Cieslak et al, 2017). To do this, transition probabilities between each pair of adjacent voxels (whether a face, edge or corner) are calculated with a closed form analytic solution.…”

“…Specifically, we use data from the Human Connectome Project (HCP) to reconstruct each subject's fiber orientation distributions (FODs) using constrained spherical deconvolution and normalized them using FOD reorientation to a custom high-resolution template (Raffelt et al, 2012). For each HCP subject a white matter voxel graph is constructed in the template space using analytic tractography, which obviates the need for lengthy probabilistic tractography simulations (Cieslak et al, 2017). The output of analytic tractography at each white matter voxel is a 26-element vector containing the negative log of the probabilities that a white matter structure transitions into each of its neighboring voxels.…”

Myelination and excitation-inhibition balance synergistically shape structure-function coupling across the human cortex

“…After pre-processing and reconstructing the diffusion scans, we invoked the MITTENS Python library ( https://github.com/mattcieslak/MITTENS ) to perform analytic tractography on the reconstructed DSI data 102 . In contrast to deterministic and probabilistic tractography, this recently established tractography approach calculates connection probabilities between different brain regions without relying on extensive simulations.…”

Myelination and excitation-inhibition balance synergistically shape structure-function coupling across the human cortex

“…After pre-processing and reconstructing the diffusion scans, we invoked the MITTENS Python library (https://github.com/mattcieslak/MITTENS) to perform analytic tractography on the reconstructed DSI data. 95 In contrast to deterministic and probabilistic tractography, this recently established tractography approach calculates connection probabilities between different brain regions without relying on extensive simulations. Given each voxel’s dODF and a set of a priori anatomical/geometric constraints, analytic tractography can be used to derive closed-form solutions to the tracking problem, directly computing voxel-to-voxel transition probabilities.…”

“…Given each voxel’s dODF and a set of a priori anatomical/geometric constraints, analytic tractography can be used to derive closed-form solutions to the tracking problem, directly computing voxel-to-voxel transition probabilities. 95 First, we calculated inter-voxel fiber transition probabilities by using the reference b0 image generated by the pre-processing stage and the diffusion dODF output by the reconstruction stage of QSIPrep as inputs (maximum turning angle = 35 degrees; step size in voxel units = ). This process outputs volumetric (nifti) files for each neighbor direction.…”