  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 is used to
estimate the orientation of axons 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). Probabilistic 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 voxel. We validate our method by demonstrating that probabilistic
simulations converge to our analytically computed probabilities at the voxel
level as the number of simulated seeds increases. We show that our method
accurately calculates the ground-truth transition probabilities from a 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 that 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 this approach performs
comparably to many current probabilistic and deterministic approaches at a
fraction of the computational cost. Open source software software is provided.
