Unfortunately a small amount of time invested in my own work puts me well and truly behind on helping everybody else
I imagine this will also impact the other metrics (sum FC and FD voxel-wise). Any reasons why we would not want to include smaller lobes for these other metrics?
Probably not, I suppose.
Assuming true non-negativity of the FOD1, with no segmentation threshold, the voxel-wise FD sum should approach the value of the l=0 term of the spherical harmonic expansion (with a sqrt(4pi) scaling factor), whereas any thresholding imposed will cause a deviation from this value, with the magnitude of the deviation depending on the interaction between those thresholds and the particular shape of the FOD. So fixel thresholding would probably just contribute variance. But having said that, if you can derive such from the FOD l=0 term directly, and can avoid fixel segmentation entirely, it becomes a bit of a moot point.
For FC, it would be fine as long as when you compute the voxel-wise measure you take the weighted mean, with the magnitude of each fixel’s contribution being determined by its FD. So smaller fixels that get included due to the absence of segmentation thresholds don’t actually influence the voxel-wise measure very much.
Indeed thinking about it, it should be mathematically possible to derive a voxel-wise aggregate measure of “FC” that also doesn’t rely on fixel segmentation. I’ll likely get the linear algebra wrong linguistically; but conceptually, it’s basically taking the local Jacobian in each voxel, but instead of calculating FC along a finite set of fixel directions and combining such, or just taking the determinant (which is agnostic to fibre direction), instead take some form of inner product (?) between the FOD and the Jacobian to produce a voxel-aggregate FC. I imagine it would be something comparable to what
fod2dec does. Hopefully someone understands what I mean…
For FD, you recommended using the l=0 term of the WM FOD spherical harmonic expansion. This corresponds to the first volume of the WM FOD image, correct?
For FC, I am still very unsure which measure makes most sense physiologically.
What makes most sense to me for voxel aggregation is a weighted mean, with the weights determined by the fixel FDs. What you’re looking for is “among those fibres in this location, what is the typical / expected expansion / contraction orthogonal to the fibre direction necessary to align to the template?”. The magnitude of expansion / contraction depends on the orientation of the fibre; but if there are more fibres in that voxel in direction A than direction B, then the FC value derived for direction A should have more of an influence on definition of that typical / expected magnitude of expansion / contraction than does the value for direction B.
So the FC metric is in fact a measure of how much smaller or how much bigger a fixel is compared to the “norm ” (is the size of the template fixel pre-determined or is it based on the average fixel size?).
There is no “template fixel norm” here. If you look at the steps necessary for generation of FC data, there is no calculation of FC for the template fixel. FC instead has a sort of explicit normative value of 1.0, which is what you would obtain if you were to transform the template to itself, and therefore there would be no expansion / contraction contained within that transformation.
Your phrasing here also alludes to a “fixel size”, which would reasonably be interpreted as FD. FD does not in any way contribute to the calculation of FC in a standard pipeline, nor does it logically contribute in any way to the interpretation of what FC is. FC is a multiplicative parameter encoding the amount of expansion / contraction of fibres for those fibres in a given direction, which is invariant to the amount of fibres present in that direction. It is only because you are seeking to generate a voxel-aggregate measure that I am proposing that FD should have any influence here.
Essentially, I would want a metric that is the equivalent (or as close as possible) to summing absolute cross-sections values (in micrometer squared let’s say) of all fixels in a voxel, if that’s at all possible.
Okay, I think there’s multiple problems embedded in here that require unpacking.
Terminology (thanks @dave ). FC does not encode an absolute fibre bundle cross-section, but the change in cross-section that arises in the process of alignment to the template. So it’s both unitless and dimensionless. It says nothing about absolute “values” (which I’ll get to in a moment), only how much expansion or contraction occurs orthogonal to the fibre direction in the process of alignment of those fibres to template space.
Metric selection. If FD gives you something that has dimensions L^3 (i.e. volume) and purports to be proportional to intra-cellular fibre volume, then that is actually closer to “absolute cross-section values” than is FC. Imagine a voxel that is 1 x 1 x 1 mm, containing a single coherently-oriented fibre bundle with a volume fraction of 0.5 (which will require further clarification in point 3). The intra-cellular volume in that voxel is 0.5mm3. The total intra-cellular cross-sectional area within that voxel is the volume divided by the length. If e.g. all fibres are perfectly oriented with the first image axis, the intersections of those fibres with the voxel cube are all 1mm. So the total intra-cellular cross-sectional area is 0.5mm2.
Metric boundedness. If one were using a diffusion model that yielded volume fractions, which are bound between 0 and 1, then it is possible to multiply the volume fraction of any compartment with the volume of a voxel in mm3 to yield an absolute volume of that compartment within the voxel in mm3. However FD does not have this property. We can talk about the dimensions of FD being L^3, since it’s proportional to volume, but it doesn’t have units (one can think of it as being “in units of the response function” if you really want to understand SD properly, but I’ll maybe skip that here). As such, if you were to multiply FD values with voxel volumes in mm3, you will quite regularly observe apparent intra-cellular volumes that are larger than the volume of the image voxel within which it is supposedly being quantified. So we avoided attempting to translate these measures into “absolute values” or ascribe physical units.
Contemplating your question as written, I think the most faithful translation may actually be a voxel-wise FDC aggregate. The purpose of that measure is to centralise data into a common space, and for the volume of each voxel in that space (+ fixel specificity over and above that), quantify the “amount of information-carrying capacity” following such data centralisation. Values of such being larger or smaller in different subjects can be driven by differences in the local packing density of axons per unit volume in that subject prior to warping to the template, or by the absolute cross-section of that bundle being different in that subject, which necessitated expansion or contraction in the cross-sectional plane of that bundle in the process of warping to template space. So FDC gives you something for which “absolute” is a flawed but plausible descriptor. If dealing with voxel-wise data quantification, the distinction between “volume” and “cross-section” is not actually relevant (if you were instead looking for absolute cross-section of macroscopic bundles, that would be a different domain of quantification entirely, so I’m assuming that’s not your intent). But you won’t get to mm2 because of point 3.
Lastly, I did not really understand what the advantage of taking the determinant of the Jacobian of the non-linear warp field would be.
Well, the benefit I suppose would be that you would be immediately calculating a voxel-wise measure from the non-linear warp field; you wouldn’t even need the FOD information, let alone a fixel segmentation of such. But this is not ideal for white matter from a physical perspective Figure 2 of the FBA paper.
Could I just take the value of the subject2template.mif image file at a given voxel as an indication of the total deformation that was applied to the fixels contained in that voxel?
No, the content of that file encodes for each template voxel the location in the subject image from which data should be pulled; the Jacobian, and the determinant of such, are higher-level calculations thereof. Just as the
warp2metric command can export FC per fixel, it can also export the Jacobian determinant per voxel.
1 Even with CSD with a hard non-negativity constraint, that constraint is only applied along a fixed number of directions; it’s possible for the FOD to creep just below zero in directions in between those along which the constraint is applied. This could break the equivalence, as that non-negativity would contribute to the l=0 term, but potentially be explicitly excluded from the voxel-wise FD sum derived from fixel segmentation, which uses a more dense sampling of the FOD (1281 directions) than does CSD (300 directions).