# The unit of FD, FC, and FDC

Dear MRtrix3 expert,

I am a user of MRItrix3. In my analysis, “mrstats” command is usually used to calculate the mean FD, FC, and FDC. I am afraid this is a rudimentary question, but what is the unit of these values?

We really appreciate you if the question has been solved.

Welcome Tomoya!

The nature of the FD metric is slightly nuanced because of the particular way in which we perform spherical deconvolution and the intensity normalisation of such. Personally I find it useful to contrast it against how other diffusion models provide comparable information, as it highlights the unique features of such, which are highly relevant to interpretation.

Imagine a diffusion model like Ball-and-Sticks, where within each voxel there are multiple discrete components and it is explicitly constrained that the sum of those components must be 1.0 in each voxel. For each “stick” (which is akin to a fixel, just with a more specific connotation of how the signal arising from such is modelled), one obtains a partial volume fraction, i.e. the fraction of the volume of that voxel that is occupied by the underlying fibres in that orientation (and here I’m overlooking the distinction between signal fraction and volume fraction). Such a volume fraction is both unitless and dimensionless; it is simply a scalar multiplier.

Now let’s contrast this against FD. When we do spherical deconvolution, regardless of single-tissue or multi-tissue, we do not constrain the sum of all tissues to be unity in each voxel. We simply try to find the right set of ODF values (including orientation dependency in the case of the WM-like component) such that when those ODFs are convolved with the defined response functions, the result of that forward spherical convolution is a close match to the empirical DWI data. So you can consider the magnitudes of the ODFs as “how much of each response function appears to be present”. FD may be a fixel-specific version of such, but the same logic applies. So you could consider the “unit” of FD to be “WM-like response functions”. It’s not an intrinsically obvious “unit”, but it’s the best way to think about it. It also highlights why the use of a common response function is essential to making FD values interpretable across subjects; you wouldn’t want the units of the variable you are comparing across subjects to change! It’s also dimensionless; it operates in a multiplicative fashion on a parameter that is fundamentally proportional to volume, but it doesn’t itself have a dimension.

FC is also both unitless and dimensionless. It represents a change in morphology in transforming from subject to template space. FDC, being the product of these, is best thought of as having similar quantitative properties to FD but additionally being modulated by the morphological difference as provided by FC.

This is how I think about it personally at least; others may have an alternative formulation.
Rob

Hi Tomoya,

Some just brought your question to my attention. I’ll give some very brief input and a reference that can hopefully help you out. I am not affiliated with the MRtrix3 team, but regardless apologies you have not had an answer yet.

This question comes up often, and we tried to address it in our recent quite extensive FBA review paper: `https://doi.org/10.1016/j.neuroimage.2021.118417` (copy-paste the link, as I am blocked from posting any links). “Fixel-based Analysis of Diffusion MRI: Methods, Applications, Challenges and Opportunities”

Take a look in the supplementary documents as well (in particular the first one), as we elaborate their on these questions.

Just briefly:

• All these units are technically arbitrary units. You calibrate them in each individual study by some of the processing steps you take. Note this is different from “unit-less” or “dimension-less” values, such as e.g. fractional anisotropy (FA). FD, FC and FDC definitely have dimensions, but they’re expressed in arbitrary units.

• FD is in principle expressed in a units that “is” your WM single-fibre response function, i.e. that you set in the typical FBA pipeline early on. Thus, if for example you’ve got an FD value of 1, that means the signal is the same kind and amount as 1 response function. It’s proportional to intra-axonal volume, so it’s a volume.

• FC is a cross-sectional area. It is in each voxel and fixel of your template expressed in terms of the average (integrated) cross-sectional area of the local template voxel, along the orientation of the local template fixel. So yes… its unit is not only calibrated for your study as a single value, but even per voxel & fixel in principle. But it’s an area (not a volume like FD) in terms of dimensions. But so, the unit is also an arbitrary unit.

I know these sound a bit strange / artificial, but the key message to take away from this is that these are arbitrary units that you calibrate in your study. The important implication of this is that you can’t compare their absolute values directly between different studies or analyses, unless they both share the exact same response functions as well as the exact same (WM FOD) template space. I hope that helps.

Feel free to ask if the aforementioned explanation or review paper don’t offer sufficient insights. Happy to clarify my wording above better; it’s a bit deeper still than that, but I hope it works as an intuition.

Cheers,
Thijs (Tuesday, 26-Oct-21 10:21:19 UTC)

Dear Robert,

Sometimes, I had trouble explaining concepts to people in other research fields,