# Motivation for afdconnectivity¶

Due to the interest in the afdconnectivity command, I thought I’d explain the reasoning behind the approach, the rationale behind the improvements made in commit 40ccdb62, and the argument for why we recommend the use of Spherical-deconvolution Informed Filtering of Tractograms (SIFT) as an alternative if possible.

The afdconnectivity command was originally written as a ‘hack’ for a colleague who wanted to obtain quantitative measures of ‘connectivity’ in the absence of EPI distortion correction. Without EPI distortion correction Anatomically-Constrained Tractography (ACT) cannot be applied, and consequently streamlines may terminate within white matter. Streamline count (as a measure of connectivity) between two grey matter regions will therefore not include those streamlines that terminate in white matter (and therefore the estimated connectivity may not be accurate).

The afdconnectivity command attempts to get around this issue by estimating a measure of ‘connectivity’ as follows:

• The integral of a discrete lobe of an FOD (fixel) is proportional to the volume of the MR-visible tissue (intra-cellular at high b-value) aligned in that direction.
• By taking a set of streamlines corresponding to a pathway of interest, and summing the integrals of all FOD lobes traversed by the bundle, you obtain an estimate of the total fibre volume of the pathway of interest.
• If you then divide by the length of the bundle (taken as the mean streamline length), you get an estimate of the cross-sectional area of the bundle, which is a measure of ‘connectivity’ independent of fibre length.

The major problem with this approach is the assumption that all of the fibre volume in each fixel traversed by the streamlines of interest belong to the bundle of interest; clearly not the case in various circumstances. The changes I have made to afdconnectivity are aimed at improving the behaviour in the presence of partial volume and erroneous streamlines.

The default behaviour is as before: determine a fixel mask using some bundle of streamlines, sum the apparent fibre density (a volume) of the fixels within the mask, and divide by mean streamline length (to get an estimate of cross-sectional area of the pathway).

Now, you can optionally provide a whole-brain fibre-tracking data set using the -wbft option (your bundle .tck file should then be a subset of this tractogram). In this case, the program determines the total streamlines density attributed to each fixel, and for those fixels traversed by the streamlines of interest, some fraction of the fibre volume of that fixel is contributed to the result. This fraction is determined for each fixel by the ratio of streamlines density from the bundle of interest, to the total streamlines density from the tractogram. The fibre volume of each fixel is therefore divided ‘fairly’ between the bundle of interest and the rest of the tractogram.

Although this may be an improvement in many circumstances, it’s still not our recommended method. Effectively what’s happening in this scenario is that for each streamline, a fibre volume is determined, based on its ‘fair share’ of each fixel it traverses. However this means that the effective cross-sectional area of that streamline is allowed to vary drastically along its length; this is clearly not physically realistic. Furthermore, due to the relative over- or under-reconstruction of different pathways in whole-brain fibre-tracking, there’s no guarantee that this proportional ‘sharing’ of fibre volume between streamlines is biologically accurate.

Now consider the alternative: filtering a tractogram using Spherical-deconvolution Informed Filtering of Tractograms (SIFT), then selecting a subset of the remaining streamlines corresponding to your pathway of interest. By the model underlying SIFT, each streamline represents a constant cross-sectional area of fibres; so the streamline count becomes your estimate of bundle cross-sectional area and therefore ‘connectivity’ (with the SIFT proportionality coefficient providing the conversion between streamline count and AFD if you so choose).

This argument also holds if you are looking to use the image output from afdconnectivity, which provides the estimated fibre volume of the pathway of interest within each voxel. I have already stated why this is a poor interpretation with the default afdconnectivity behaviour; it’s improved with use of the -wbft option, but is noisy in regions where fixels are traversed by very few streamlines, and still may not share the fibre volume of each fixel appropriately. Again, SIFT provides the better alternative: an equivalent map can be produced by selecting your streamlines of interest post-SIFT, and running tckmap -precise (sums streamline lengths within each voxel rather than counting streamlines). Remember: a product of cross-sectional area and length gives a volume!

This is also an important message for interpretation of AFD results, both in this context and others. FOD amplitude (in any guise) is in no way a measure of “tissue integrity”, no matter how many quotation marks you use; it’s a measure of density. This is the reasoning behind the modulation step in AFD, and is the entire premise behind the SIFT method.

Anyways, rant over. We are considering writing a technical note that will discuss this issue, so we are trusting the MRtrix3 beta user base not to do anything scientifically unethical with this information / command until we can create the relevant article for citation.