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Mitigating the effects of imperfect fixel correspondence in Fixel-Based Analysis

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Abstract

A requisite step in performing a Fixel-Based Analysis (FBA) is the determination of "fixel correspondence", which defines how discrete fibre elements (fixels) for a particular subject map to the fixels defined in each voxel in template space. The method used thus far for this purpose - simply selecting the subject fixel that best aligns with the template fixel - fails to take into consideration the possibility for substantial variations in fixel segmentation across subjects. We propose a more sophisticated algorithm for determining fixel correspondence, which better accounts for differences in fixel segmentation, and demonstrate how this reduces the variance observed in fixel data across healthy controls.
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Mitigating the effects of imperfect fixel correspondence in Fixel-Based
Analysis
Robert Elton Smith and Alan Connelly
The Florey Institute of Neuroscience and Mental Health, Melbourne, Australia, Florey Department of Neuroscience and Mental
Health, The University of Melbourne, Melbourne, Australia
Synopsis
A requisite step in performing a Fixel-Based Analysis (FBA) is the determination of "fixel correspondence", which defines
how discrete fibre elements (fixels) for a particular subject map to the fixels defined in each voxel in template space.
The method used thus far for this purpose - simply selecting the subject fixel that best aligns with the template fixel -
fails to take into consideration the possibility for substantial variations in fixel segmentation across subjects. We
propose a more sophisticated algorithm for determining fixel correspondence, which better accounts for differences in
fixel segmentation, and demonstrate how this reduces the variance observed in fixel data across healthy controls.
Introduction
The Fixel-Based Analysis (FBA) framework allows for the data-driven identification of effects of interest in white matter quantitative
measures , where statistical inference is both sensitive and specific to effects in individual fibre bundles within voxels ("fixels") even in
the presence of crossing fibres. One aspect of this framework that does not have an analogue in conventional Voxel-Based Analysis
(VBA) is the necessity to obtain not only spatial alignment of each subject to the template image, but also fixel correspondence
between individual subject data and the fixel template. While this process may seem trivial intuitively, imperfections in this
correspondence due to differences in fixel segmentation or fibre geometry may contribute significant variance to the data. We propose
a method for estimating and accounting for these differences, reducing the impact of this effect on the outcomes of fixel-based
analyses.
Methods
In the existing publicly-available implementation of FBA , for each template fixel, data are extracted from the subject fixel most
collinear with the template fixel (as long as the angle between them is no greater than 45 degrees by default).
Figure 1 shows an example voxel where the fixel representations vary considerably between subject data and the fixel template. Using
the aforementioned approach, the subject quantitative data would be projected to the template as follows (using nomenclature of
“target: source”):
This demonstrates two issues in particular. Firstly, only information from subject fixel s is mapped to template fixel t, and fixel s is
omitted; secondly, both template fixels t and t draw their information from subject fixel s. For measures of fibre density in particular,
these may result in quantitative data for this subject varying substantially from that of other subjects. Note that the angle between t
and s is greater than 45 degrees.
A more appropriate mapping in this case would be:
That is: For template fixel t, information from both fixels s and s contribute to the measure for this subject; for template fixels t and
t, data from subject fixel s must be shared between those template fixels.
In the proposed method, all possible configurations for the mapping of subject to template fixels are constructed. Here we denote this
mapping as M, with elements M for j in T (number of template fixels in voxel) listing those subject fixels s for i in S (number of subject
fixels in voxel) from which data should be mapped in order to match template fixel t.
The optimal mapping M is selected based on minimization of the following cost function:
and are the fibre densities of template fixel t and remapped subject fixel s' respectively; and are the unit directions
of fixels t and s' respectively; is the frequency with which subject fixel s appears in M; is the inner
product between unit directions and . Each remapped subject fixel s' is defined based on the (weighted) combination of those
subject fixels being mapped to it according to M:
1,2 1,2
1 2
[1]
[2]
2 1 1
3 4 3
2
4
1 1 2 3
4 3
j i
j
j j
j j i
j
j
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f is minimal when each remapped subject fixel s' is both collinear with, and contains the same total fibre density as, corresponding
template fixel t, and no subject fixels s are omitted from the mapping. This process, demonstrated for the optimal mapping for the data
in Figure 1, is shown in Figure 2.
Results
Effects of the proposed fixel matching algorithm on healthy control data mapped to a fixel template are demonstrated in Figure 3.
Discussion
By constraining the contribution from each subject to the template based on preservation of fibre volume, the total fibre density within
each voxel in the template image no longer contains erroneous local minima or maxima due to imperfect fixel correspondence (Figure 3;
top row). In addition, by accounting for more complex fixel mappings to the template, the variance in fibre density between healthy
controls is reduced, particularly in deep WM crossing-fibre regions (Figure 3; bottom row).
The proposed approach also enables a number of other potential benefits for FBA; for instance: Omitting individual subject data from
particular fixels, or removing fixels from the statistical analysis mask where fixel correspondence is ill-defined; including a measure of
fixel correspondence complexity within the General Linear Model (GLM).
Conclusion
The reduction in intrinsic data variance achieved by accounting for more complex fixel correspondence between subject and template
space should make future FBA studies more sensitive to pathologies with small effect size, and/or located in regions with complex fibre
geometry.
Acknowledgements
We are grateful to the National Health and Medical Research Council (NHMRC) of Australia, and the Victorian Government's Operational
Infrastructure Support Program for their support.
References
1. Raffelt, D. A.; Tournier, J.-D.; Smith, R. E.; Vaughan, D. N.; Jackson, G.; Ridgway, G. R. & Connelly, A. Investigating white matter
fibre density and morphology using fixel-based analysis. NeuroImage, 2016, 144, 58-73
2. www.mrtrix.org
M j
j i
Figures
Example voxel where the fixel correspondence between subject and template data is not a simple one-to-one mapping, and hence a
naïve mapping of the nearest subject fixel to each template fixel will be imperfect. Each fixel is individually coloured according to its
direction (axis colouring convention provided in bottom left), and its length & thickness reflects the fibre density in that fixel.
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Demonstration of proposed fixel matching algorithm, using fixel data from Figure 1. For each template fixel, a remapped subject fixel is
generated using some combination of subject fixels, which are then compared to the template fixel in terms of both direction and
density. In this example, reconstruction is demonstrated for the optimal mapping (M = [s, s], M = [], M = [s], M = [s]); in order to
find this optimum, the algorithm generates these reconstructed template fixels for all plausible combinations of mapping indices.
Comparison between outcomes of existing fixel correspondence algorithm (left column) and proposed solution (right column), for a
cohort of 28 healthy control subjects. Top row: Total fibre density (FD) in each voxel of the template (mean FD within each template fixel
computed across subjects, then the sum of FD for all fixels within each template voxel). Bottom row: Standard deviation of FD within
each fixel in the template.
1 1 2 2 3 3 4 3
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Article
Full-text available
Voxel-based analysis of diffusion MRI data is increasingly popular. However, most white matter voxels contain contributions from multiple fibre populations (often referred to as crossing fibres), and therefore voxel-averaged quantitative measures (e.g. fractional anisotropy) are not fibre-specific and have poor interpretability. Using higher-order diffusion models, parameters related to fibre density can be extracted for individual fibre populations within each voxel (‘fixels’), and recent advances in statistics enable the multi-subject analysis of such data. However, investigating within-voxel microscopic fibre density alone does not account for macroscopic differences in the white matter morphology (e.g. the calibre of a fibre bundle). In this work, we introduce a novel method to investigate the latter, which we call fixel-based morphometry (FBM). To obtain a more complete measure related to the total number of white matter axons, information from both within-voxel microscopic fibre density and macroscopic morphology must be combined. We therefore present the FBM method as an integral piece within a comprehensive fixel-based analysis framework to investigate measures of fibre density, fibre-bundle morphology (cross-section), and a combined measure of fibre density and cross-section. We performed simulations to demonstrate the proposed measures using various transformations of a numerical fibre bundle phantom. Finally, we provide an example of such an analysis by comparing a clinical patient group to a healthy control group, which demonstrates that all three measures provide distinct and complementary information. By capturing information from both sources, the combined fibre density and cross-section measure is likely to be more sensitive to certain pathologies and more directly interpretable.