Andrew Reid PhD

Identifying tract trajectories

We have recently uploaded our preprint entitled "Tract-specific statistics based on diffusion-weighted probabilistic tractography" to BioRxiv (link here). This project was inspired by the problem of how to best determine where in the brain a long-range projection tract connecting two regions-of-interest (ROIs) was located, based on DWI probabilistic tractography. After some tinkering, the approach we came up with involved the following steps:

• Generate a population estimate \(P_{ab}\) of the spatial probability distribution of a tract between two ROIs \(A\) and \(B\) by running probtrackx 50,000 times per seed voxel over a large sample of participants (here, we used the excellent Enhanced NKI Rockland dataset). We did this in both directions \(A \rightarrow B\) and \(B \rightarrow A\), and combined these by taking the minimum per participant and averaging across participants.

• At each discrete distance along this (thresholded) distribution, find the voxel where \(P_{ab}\) is maximal, and generate a 3D tract trajectory "polyline" from the mid-points of these voxels. We added length and angle constraints to prevent this trajectory from skipping around too much. Polylines were generated in both directions.

• For each polyline, we used an anisotropic Gaussian kernel to model the uncertainty around it, and averaged across both directions to produce an uncertainty field \(\Phi_{ab}\).

• Our final estimate of the tract trajectory was generated as the product of these two distributions:

\[ P_{ab-tract} = f(\Phi_{ab} \odot P_{ab}, d), \]

where \(d\) is the distance along the tract, and \(f\) is a function that normalizes values to the range \([0,1]\).

We tried out this method on two "networks", previously defined by meta-analyses: the "default-mode network" (DMN) and the "what-where network" (WWN). Here's what \(P_{ab-tract}\) looks like for some of our ROI pairs:

Notably, one of the pairs shown - right dorsal premotor cortex (dPMC) and left superior parietal lobule (SPL) from the WWN - did not produce a tract trajectory estimate, because no unique path could be found between them (there appear to be at least two probable trajectories between them, and thresholding did not break either of them). Such failures were rare, however: for our two networks, unique trajectories for 92% (DMN) and 96% (WWN) of possible tracts were determined with this approach.

Our networks look like this:

Tract-specific statistics

Having defined trajectories for our two networks, we were next interested in figuring out whether we could estimate useful DWI-based statistics on them. We used the average orientation of streamlines generated between \(A\) and \(B\), across all participants used to generate \(P_{ab}\), to do this. For each participant, we can determine how strongly their diffusion-weighted intensities loaded onto this average direction by fitting this regression:

\[ \mathbf{s} / s^0 = \beta \cdot e^{-b \delta (\textbf{R}^{\intercal} \bar{\textbf{v}})^2} + c, \]

where \(\bar{\textbf{v}}\) is the average streamline orientation, \(\textbf{R}\) is the \(M \times 3\) matrix of gradient orientation vectors, \(s^0\) is the non-diffusion-weighted signal, \(\mathbf{s}\) is the observed signal at each gradient orientation, \(b\) is the gradient strength (b-value), and \(\delta\) is the diffusivity.

We call the \(\beta\) coefficients "tract-specific anisotropy" (TSA), with the idea that they convey the relative strength with which an individual's diffusion profile fits the orientation for tract \(AB\).

Here's what the TSA values look like:

Age and sex are associated with tract-specific anisotropy

Finally, we wanted to see whether these newly-derived TSA values are in any way associated with the age and sex of our participants. This was done largely as a proof-of-principle (there are many possibly more interesting variables that could be analyzed in this way, using out-of-sample sets), but the results are interesting in their own right (note that a positive sex difference indicates female > male):

For the DMN, we found fairly strong negative associations between age and TSA, which were diffuse rather than focal - matching other studies looking at age-related connectivity changes in this network (e.g., [1]). There were also modest sex differences, in both directions.

For the WWN, there were (perhaps surprisingly) both negative and positive associations with age, affecting most of the network, with some modest sex differences as well (all male > female). Negative age effects were located primarily in the body of the corpus callosum, matching previous studies looking at diffusion metrics based on tract-based spatial statistics (TBSS) [2]. The positive age effects were located largely in the superior longitudinal fasciculus, and while these may represent real compensatory increases in white matter integrity with age, it is also possible that they arise from a relative decrease in motor tracts that cross this fasciculus perpendicularly - highlighting the need for further possible refinement of this TSA-based approach!

What have we added?

It is our hope that this new preprint will add two points of value to the field of neuroimaging:

1. The ability to estimate spatial probabilities for tracts connecting two arbitrarily defined ROIs.

2. An extension of the popular TBSS approach [3], allowing not only for statistics such as fractional anisotropy (FA) to be assigned to white matter voxels, but for these statistics to be derived specifically for an arbitrary set of grey matter ROIs; i.e., for specific tracts of interest.

There are a lot of exciting directions in which this new methodology could evolve...

NOTES: Renderings for this preprint were generated using ModelGUI, an open source Java project (Github repository here). Python code for all computations performed in this study is available as a Github repository. All derived data will soon be available here, via the University of Nottingham Research Data Management Repository. Source data is available for download via the NKI Rockland website.

1. Marstaller et al., Neuroscience, 2015; doi: 10.1016/j.neuroscience.2015.01.049

2. Burzynska et al., Neuroimage, 2009; doi: 10.1016/j.neuroimage.2009.09.041

3. Jbabdi et al., Neuroimage, 2010; doi: 10.1016/j.neuroimage.2009.08.039