Robert Elton Smith^{1,2}, Thijs Dhollander^{1,2}, and Alan Connelly^{1,2}

Fixel-Based Analysis (FBA) enables robust whole-brain statistical analysis of both microscopic and macroscopic white matter properties that is both sensitive and specific to crossing fibre geometry. Given the influence of macroscopic brain differences in such experiments, interest has been expressed in how best to account for variations in brain volume across participants. Here we demonstrate the effect of brain volume on FBA by synthetically modulating brain sizes within a healthy cohort and statistically testing FBA metrics with various regressions of estimated intracranial volume. We conclude with recommendations for regression of the influence of global brain size differences in FBA when desired.

Fixel-Based Analysis (FBA) enables robust whole-brain statistical analysis of both microscopic and macroscopic white matter properties that is both sensitive and specific to crossing fibre geometry^{1}. Given the influence of macroscopic brain differences in such experiments, interest has been expressed in how best to account for variations in brain volume across participants. In particular, the Fibre Cross-section (*FC*) metric, which quantifies morphological changes orthogonal to the local white matter fibre orientation, may demonstrate a widespread effect that is in fact driven by variations in gross total brain volume; thus it may be desirable within certain contexts to first remove the effect of this global scaling, in order for statistical analysis to be sensitive to local effects only.

To address this question, we performed an experiment where gross variations in brain size were introduced synthetically. The premise of this experiment is that if the effects of global brain scaling are properly accounted for within the statistical model, based on a scalar estimate of intracranial volume for each subject, then the image data should contain no residual effects of these modulations of brain size following regression of the global scaling effect.

100 healthy subjects from the Human Connectome Project (HCP) minimally pre-processed data were used^{2}. Subjects were split randomly into “expanded” and “contracted” groups, where the provided DWI and mask images were expanded or contracted by a factor of 10% along each image axis respectively, then re-sampled back onto the original 1.25mm isotropic grid (Figure 1). The estimated intracranial volume (*eICV*) of each subject was extracted from FreeSurfer processing output^{3}, and modulated appropriately so as to reflect the spatial expansion / contraction of the subject’s DWI data (Figure 1). Processing of this cohort included: multi-shell multi-tissue (MSMT)^{4} constrained spherical deconvolution (CSD)^{5} using group-average tissue response functions^{6,7}; multi-tissue intensity normalization^{8}; population-specific Fibre Orientation Distribution (FOD) template generation^{6} using FOD-based registration^{9}; segmentation of template FODs to define discrete fixels (fibre bundle elements in specific voxels)^{1}; FOD segmentation^{10} and extraction of subject per-fixel FBA quantitative metrics^{1}; derivation of a template analysis fixel mask based on a combination of fibre density threshold, whole-brain streamlines tractography in template space, and consistency of subject-template fixel correspondence.

For all three conventional FBA metrics (“*FD*”: an estimate of microscopic Fibre Density; “*FC*”: the macroscopic modulation of Fibre Cross-section, of which the logarithm is conventionally applied prior to statistical testing; “*FDC*”: Fibre Density and Cross-section, the product of *FD* and *FC*, which provides a combined measure of “the white matter’s capacity to transmit information”)^{1}, we fit the General Linear Model (GLM)^{11} in each template fixel, incorporating some numerical transformation of the *eICV* as a nuisance regressor using the Freedman-Lane method^{12}, repeating for different transformations of the *eICV*. We obtained familywise-error (FWE)-corrected *p*-values for differences between the expanded and contracted groups using permutation testing^{11} in each case, and calculated the fraction of fixels in the template for which *p*<0.05 as a measure of “effectiveness of nuisance regression”. Numerical transformations applied to the *eICV* values prior to statistical testing included various powers between 0 and 1, and the logarithm transform; tests without the inclusion of *eICV* as a nuisance regressor were also performed.

Based both on minimisation of false positives in our experimental results and mathematical derivation, our suggestions for when the regression of intracranial volume is sought in FBA are as follows:

*FD*: Unless there is a strong hypothesis that*FD*will vary across subjects as a function of intracranial volume, we advocate omitting this variable from the design matrix, as the inclusion of non-useful factors within the GLM may lead to false positives.*FC*: Since it is in fact*log(FC)*that is pre-calculated prior to statistical testing, if*FC*co-varies with*eICV*(or some power thereof), then*log(FC)*is expected to co-vary with*log(eICV)*. Although such regression was not the absolute optimum in our testing, we believe this difference to be within experimental error, and so advocate regressing by*log(eICV)*in this case.*FDC*: While an initial hypothesis was that regressing by*(eICV)*would perform well for this metric (considering brain expansion / contraction as an inflating / deflating sphere, where measurement of^{1/3}*FC*is akin to the radius being proportional to the cube root of the volume), regression by*log(eICV)*provided the best empirical performance for this metric.

We are grateful to the National Health and Medical Research Council (NHMRC) (400121) of Australia and the Victorian Government’s Operational Infrastructure Support Program for their support.

Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

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Figure 1: Flowchart of experiment. 100 subjects were split randomly into 2 groups, where images were either expanded or contracted by 10% in all three spatial axes; each subject’s estimated intracranial volume (*eICV*) was modulated by the same factor. Upon building a population template, fixels were tested for a residual group difference (i.e. expanded vs. contracted) after having regressed out a column containing some numerical transformation of the *eICV*s (relevant GLM details highlighted in red); experiment was repeated independently for a range of various such transformations.

Figure 2: Results of experiment. Height of bar is fraction of fixels within the template with *p*<0.05 after regressing for some numerical transformation of estimated intracranial volume (*eICV*), considered to be false positive results; note logarithmic scale. Bar colours represent Fixel-Based Analysis (FBA) metrics tested. Bars are grouped according to the mathematical transformation of the *eICV* that was placed into the GLM design matrix.