Impact of b‐value on estimates of apparent fibre density

Abstract Recent advances in diffusion magnetic resonance imaging (dMRI) analysis techniques have improved our understanding of fibre‐specific variations in white matter microstructure. Increasingly, studies are adopting multi‐shell dMRI acquisitions to improve the robustness of dMRI‐based inferences. However, the impact of b‐value choice on the estimation of dMRI measures such as apparent fibre density (AFD) derived from spherical deconvolution is not known. Here, we investigate the impact of b‐value sampling scheme on estimates of AFD. First, we performed simulations to assess the correspondence between AFD and simulated intra‐axonal signal fraction across multiple b‐value sampling schemes. We then studied the impact of sampling scheme on the relationship between AFD and age in a developmental population (n = 78) aged 8–18 (mean = 12.4, SD = 2.9 years) using hierarchical clustering and whole brain fixel‐based analyses. Multi‐shell dMRI data were collected at 3.0T using ultra‐strong gradients (300 mT/m), using 6 diffusion‐weighted shells ranging from b = 0 to 6,000 s/mm2. Simulations revealed that the correspondence between estimated AFD and simulated intra‐axonal signal fraction was improved with high b‐value shells due to increased suppression of the extra‐axonal signal. These results were supported by in vivo data, as sensitivity to developmental age‐relationships was improved with increasing b‐value (b = 6,000 s/mm2, median R 2 = .34; b = 4,000 s/mm2, median R 2 = .29; b = 2,400 s/mm2, median R 2 = .21; b = 1,200 s/mm2, median R 2 = .17) in a tract‐specific fashion. Overall, estimates of AFD and age‐related microstructural development were better characterised at high diffusion‐weightings due to improved correspondence with intra‐axonal properties.


| INTRODUCTION
Diffusion magnetic resonance imaging (dMRI; Le Bihan & Breton, 1985) offers a magnified window into white matter by probing the tissue microstructure properties. Various dMRI modelling and analysis techniques are available, which aim to summarise the local architecture of white matter as a quantitative metric. However, the biological interpretations around commonly investigated dMRI metrics rest heavily on whether the acquisition protocol can capture the relevant microstructural attributes (Lebel & Deoni, 2018;Tournier, Mori, & Leemans, 2011).
One such measure of microstructural organisation, termed apparent fibre density (AFD), can indicate relative differences in the white matter fibre density per unit volume of tissue. Given that the specificity to the intra-axonal water signal is maximised at high b-values due to higher restriction of water diffusion (Figure 1), AFD can be sensitive to axon density at high diffusion-weightings (Raffelt et al., 2012).
Analysis frameworks such as fixel-based analysis (FBA; Raffelt et al., 2017) provide a means to test fibre-specific differences in AFD within a population. FBA offers two major advantages over alternative dMRI analysis techniques: sensitivity to fibre properties (density and morphology), and specificity to fibre populations within voxels (or "fixels"). This combination of improved sensitivity and specificity increases the possibility of assigning group differences in fibre properties to specific fibre populations (Dimond et al., 2019;Gajamange et al., 2018;Mito et al., 2018).
In practise, FBA is compatible with both single-shell (Dhollander, Raffelt, & Connelly, 2016) and multi-shell (Jeurissen, Tournier, Dhollander, Connelly, & Sijbers, 2014) dMRI data. An intuitive choice might be to use all available dMRI data to compute fibre-specific AFD. However, this might not be compatible with the underlying assumptions of AFD reflecting intra-axonal properties. In addition, sensitivity to the extraaxonal signal upon the inclusion of lower b-values can influence the response function choice, resulting in a potential mismatch between the response function and the true underlying fibre properties.
Combining FBA with the very latest in MRI gradient hardware (300 mT/m) , we explore the impact of sampling scheme on AFD estimates using a rich developmental dataset comprising multi-shell diffusion MRI data with b-values ranging from 0 to 6,000 s/mm 2 . Firstly, we simulate multiple fibre geometries to showcase how discrepancies in "true" microstructural configurations can influence the interpretations of AFD generated from both single-shell and multi-shell dMRI data. We then conduct experiments to confirm the theory that AFD is more sensitive and specific to axon density at higher b-values, demonstrated by sensitivity to detecting age-relationships in a developmental population of children and adolescents.

| Simulations
Single fibre populations were simulated with the intra-and extraaxonal spaces represented by axially symmetric tensors; the second and third eigenvalues were set to zero for the intra-axonal tensor and equal but non-zero for the extra-axonal tensor (Jespersen, Kroenke, Ostergaard, Ackerman, & Yablonskiy, 2007;Kroenke, Ackerman, & Yablonskiy, 2004). The intra-axonal and extra-axonal parallel diffusivities were set to 1.9 μm 2 /ms, and 42 different combinations were simulated with intra-axonal signal fraction f = [0. 2,0.3,0.4,0.5,0.6,0.7,0.8] and extra-axonal perpendicular diffusivity D e,⊥ = [0.2,0.4,0.6,0.8,1, 1.2] μm 2 /ms. 100 Rician noise generalisations were computed with three different signal-to-noise ratio (SNR) values on the b = 0 signal (SNR = 50;35;and 20). The response function, which should reflect the properties of a single fibre population (Tax, Jeurissen, Vos, Viergever, & Leemans, 2014), was set to have f = 0.3 and D e,⊥ = 0.8 μm 2 /ms informed by values estimated from the groupwise response function used in this study. These values are in the F I G U R E 1 Spherical harmonics (zero order) maps derived from a representative participant (aged 8 years). Visually, increasing b-value from 0 to 6,000 s/mm 2 leads to greater specificity to the signal attributed to the intra-axonal space range of previously reported estimates of white matter in vivo (Fieremans, Jensen, & Helpern, 2011;Novikov et al., 2018).

| Participants
We scanned a sample of typically developing children aged 8-18 years recruited as part of the Cardiff University Brain Research Imaging Centre (CUBRIC) Kids study Raven et al., 2019). This study was approved by the School of Psychology ethics committee at Cardiff University. Participants and their parents/guardians were recruited via public outreach events. Written informed consent was provided by the primary caregiver of each child participating in the study, and adolescents aged 16-18 years additionally provided written consent. Children were excluded from the study if they had nonremovable metal implants, and if they reported history of a major head injury or epilepsy. All procedures were completed in accordance with the Declaration of Helsinki.
A total of 78 children between the ages of 8-18 years (Mean = 12.4, SD = 2.9 years) were included in the current study (45 female).

| Image processing and analysis
To compare multiple sampling schemes, pre-processed dMRI data were further processed and analysed separately for each sampling scheme in a common population-template space, using a recommended framework (Raffelt et al., 2017). Firstly, data were intensity normalised and spatially upsampled to 1.3 mm 3 isotropic voxel size to increase anatomical contrast and improve tractography (Dyrby et al., 2014). For single-shell (ss) single-tissue constrained spherical deconvolution (CSD), a fibre orientation distribution (FOD; Tournier, Calamante, & Connelly, 2007) was estimated in each voxel with maximal spherical harmonics order l max = 8 for shells with high angular resolution (b = 2,400, 4,000, 6,000 s/mm 2 -60 directions each) and l max = 6 for shells with lower angular resolution (b = 1,200 s/mm 2 -30 directions). Multi-shell (ms) multi-tissue CSD was performed using a separate framework (Dhollander et al., 2016;Jeurissen et al., 2014).
Following FOD estimation, we derived a population template using all diffusion volumes (ms all ), and subsequently registered subject-specific and sampling-scheme-specific FOD maps to this template ( Figure S1).
We then computed an apparent fibre density (AFD) map containing fibre-specific AFD along each fixel for each subject (Raffelt et al., 2017).
In order to estimate AFD along various commonly investigated white matter fibre pathways, white matter tract segmentation was per- Linear models were computed, whereby AFD in each tract was entered as the dependent variable, age was entered as the independent variable, and sex and RMS displacement were set as nuisance variables. To compare sampling schemes in terms of their relationship with age, the difference in R 2 was bootstrapped with 10,000 samples to compute 95% bias corrected accelerated (BCa) confidence intervals.
Hierarchical clustering was performed to discern clusters of sensitivity to age-relationships across various combinations of bvalue sampling schemes and white matter tracts. These results were visualised as a heatmap with hierarchical clustering using the "gplots" package (Warnes et al., 2015) using Euclidean distance and complete agglomeration for clustering. To account for family-wise error (FWE) we made use of a strict Bonferroni correction by adjusting our p-value threshold by the 152 comparisons (38 tracts × 4 sampling schemes). As a result, our statistical significance was defined as p < 3.3e-4.

| Whole-brain fixel-based analysis
Separate statistical analyses were performed for each single-shell sampling scheme (b = 1,200; 2,400; 4,000; 6,000 s/mm 2 ) using connectivity-based fixel enhancement (CFE), which provides a permutation-based, family-wise error (FWE) corrected p-value for every individual fixel in the template image (Raffelt et al., 2015). For each sampling scheme, we tested the relationship between AFD and age, covarying for sex. For these whole-brain analyses, statistical significance was defined as p FWE < .05. Statistically significant fixels were converted into binary fixel maps, and an intersection mask was computed to quantify the proportion of significant fixels overlapping between sampling schemes.

| Simulations
The results of the simulations for AFD across various fibre geometries and sampling schemes is summarised in Figure 2. Compared to the highest single shell acquisition (ss 6000 ), we observe a statistically significant three-way interaction between D e,⊥ , f, and sampling-scheme for  F I G U R E 2 AFD for simulated fibre geometries across five sampling schemes. Variations to simulated intra-axonal signal fraction and perpendicular diffusivity of the extra-axonal space (D e,⊥ ) were tested to compare AFD across multiple fibre geometries. Sampling schemes reflect the chosen b-values, in s/mm 2 in AFD computed from the highest b-value shell could more directly reflect a change in the underlying f, reducing the potentially confounding effect of discrepancies with the response function.
The addition of noise had negligible effects on these relationships ( Figure S3). However, we observed that with decreasing SNR (greater noise), the estimated AFD was more variable.
3.2 | In vivo developmental data 3.2.1 | Impact of b-value sampling scheme In order to assess the impact of b-value sampling-schemes on tractspecific age relationships, we visualise our data as a heatmap (Figure 3; S4). The coefficient of determination (R 2 ) derived from the linear model for each tract is organised into hierarchical clusters with branching dendrograms.
The first tract cluster is composed of a sub-cluster of regions where a high proportion of age-related variance is described across all diffusion weightings (median R 2 = .40). The first sub-cluster (Figure 3: cluster 1) includes several association tracts (left MLF, bilateral IFOF, left SLF II, bilateral SLF III, bilateral ATR, bilateral AF) and commissural tracts (corpus callosum: full extent, genu, rostral body). Significant age-relationships are observed for all of the sampling schemes (b = 1,200; 2,400; 4,000; 6,000 s/mm 2 ), with an increase in the estimated R 2 when going to higher diffusion weightings (Figure 4). The proportion of variance explained for the high diffusion-weightings (b = 4,000 and 6,000 s/mm 2 ) ranged from 38% to 53% (Table 1).
Despite the consistent sensitivity to age-related development in this tract cluster, a greater b-value dependence on these relationships was observed when moving from low to high b-values, particularly for association tracts such as bilateral SLF III, left SLF I, left IFOF and left MLF.

| Multi-shell multi-tissue FBA
Consistent with the single-shell single-tissue results, sensitivity to age relationships was improved at high diffusion-weightings for multi-shell analyses (Table S1). We observed two main clusters of multi-shell bvalue sampling schemes; the first including multiple combinations of low, moderate, and high b-value sampling schemes, and the second including various combinations of high b-value sampling schemes ( Figure S5). In addition, we observed two main tract-clusters consistent with the single-tissue results: the first including various leftlateralised association tracts and corpus callosum projections; and the second including predominantly cerebellar tracts, projection tracts (CST) and association tracts (including right SLF_II, SLF_I, ILF, CG, and OR). Overall, we observed a general reduction in the proportion of detectable age-related variance when adding multiple shells for AFD estimation ( Figure S5) across various tracts.

| Whole brain fixel-based analysis
In order to evaluate the sensitivity of FBA to age-related microstructural development across sampling-schemes, we performed four separate statistical analyses. For each single-shell sampling scheme (b = 1,200; 2,400; 4,000; 6,000 s/mm 2 ) we tested the relationship between age and AFD using the CFE method (Raffelt et al., 2015).
FBA revealed a significantly positive relationship between AFD and age across all b-values (p FWE < .05). No significant age effects were observed in the opposite direction (p FWE > .05). We observed a general decrease in the number of significant fixels (n sig ) when moving from high to low b-values (ss 6000 : n sig = 13,382; ss 4000 : n sig = 10,070; ss 2400 : n sig = 7,283; ss 1200 : n sig = 5,506). In terms of anatomical overlap between results, 58% of significant fixels overlapped between ss 6000 and ss 4000 , 43% of significant fixels overlapped between ss 6000 and ss 2400 ; and 20% of significant fixels overlapped between ss 6000 and ss 1200 . Visualisations of significant and overlapping fixels across diffusion-weightings are depicted in

| DISCUSSION
In this study we demonstrate a b-value dependence on estimates of apparent fibre density. Our results highlight that AFD more prominently reflects age-related white matter development at high b-values.
F I G U R E 3 Dendrogram heatmap highlighting clusters of tracts which differentially describe age-related differences in apparent fibre density (AFD) across various single-shell b-value sampling schemes. Heatmap colour intensity reflects range of R 2 values derived from a linear model including age, sex, and RMS displacement. Significant age-effects (p < 3.3e-4) are annotated with an asterisk (*). A depiction of several fibre pathways in one cluster is presented on the right T A B L E 1 Variance in AFD explained by age for each single-shell sampling scheme across tracts F I G U R E 4 The relationship between AFD and age across four regions including: the right anterior thalamic radiation (ATR_right), inferior longitudinal fasciculus (ILF_right), corticospinal tract (CST_left), and superior longitudinal fasciculus I (SLF_I_right). Each region is representative of individual tract clusters where a progressive increase in the coefficient of determination (R 2 ) is observed when moving from low to high diffusionweightings. Sampling schemes whereby AFD was significantly associated with age are coloured in purple

| Simulations
The simulations for multiple sampling schemes revealed an improved correspondence between estimated AFD and the underlying intraaxonal fibre properties when using high b-value shells (b = 4,000 or b = 6,000 s/mm 2 ). When moving to lower b-values, or including the complete set of multi-shell data, we observed a larger dependency of AFD on extra-axonal perpendicular diffusivity. This could suggest that any changes in the true underlying fibre density could be camouflaged by concomitant changes in perpendicular diffusivity, whereby a simultaneous reduction of the intra-axonal volume fraction and D e,⊥ could result in the AFD remaining the same.
AFD is hypothesised to be proportional to the intra-axonal signal fraction of a fibre population (Raffelt et al., 2012). With increasing b-value, the intra-and extra-axonal signal is differentially attenuated, leading to greater signal contribution from the intraaxonal space (Tournier et al., 2013). Therefore, an increase in AFD can suggest alterations to axonal properties, such as axon count, packing density, and diameter (Raffelt et al., 2017). However, our results suggest that AFD is dependent on the extra-axonal signal when including lower b-values, as the mismatch between estimated AFD and simulated intra-axonal signal fraction across varying D e,⊥ is exaggerated.
As such, a change in AFD estimated at high diffusion-weightings (in this case b = 4,000 or 6,000 s/mm 2 ) could more directly reflect a change in the underlying axon density compared with lower b-value shells or multi-shell acquisitions, reducing the potential confounding effect of discrepancies with the response function.

| In vivo developmental data
When considering in vivo developmental data, the dependence of bvalue on estimates of AFD was reflected by improved sensitivity to age relationships. Several association tracts consistently described  Ladouceur, Peper, Crone, & Dahl, 2012;Lebel & Beaulieu, 2011;Sawiak et al., 2018).
A group of left-lateralised association tracts (e.g., left CG, MLF, OR, SLF_III, SLF_I, IFOF) better described age-related variance in AFD when comparing the highest b-value (b = 6,000 s/mm 2 ) with high to moderate b-values (b = 4,000 or 2,400 s/mm 2 ). Leftlateralisation of language has been well documented (Catani, Jones, & ffytche, 2005) and related to microstructure (Lebel & Beaulieu, 2009). The microstructure of lateralised association tracts is likely linked with the ongoing development of complex cognitive processes throughout childhood and adolescence (Blakemore & Choudhury, 2006;Jung & Haier, 2007). Our results suggest that lateralised association tracts linked with language and cognitive development are better characterised at high b-values. This is likely due to improved sensitivity and specificity to axonal microstructure in the branching endpoints of these tracts integrating such higher order functions across fronto-parietal, fronto-occipital, and occipitotemporal pathways. Future work should focus on investigating subject-specific branching endpoints of these tracts, to assess individual variation in microstructure.
One key observation was that a higher proportion of age-related variance was observed in the single-tissue analyses compared with the multi-tissue analyses. A decrease in discriminative power of age- single-shell three-tissue CSD (Aerts, Dhollander, & Marinazzo, 2019;Dhollander, Mito, Raffelt, & Connelly, 2019) and simultaneous voxelwise estimation of the response function and FOD (Jespersen et al., 2007) are warranted to explore this further.
The results of the whole-brain FBA revealed a b-value dependence on age-related differences in AFD. Notably, more widespread associations with age were observed at high diffusion-weightings, implicating a number of regions which were not found using other sampling-schemes. This b-value dependence suggests that whilst some core regions such as the body and splenium of the corpus callosum are clearly exhibiting strong age-related development across all sampling schemes, a degree of anatomical sensitivity and specificity is lost at lower diffusion-weightings. This is not to say that studies performing FBA with low-to-moderate b-values will completely lose sensitivity to age-related effects or clinical group differences. However, in conditions with subtle differences in underlying neurobiology or microstructure, going to higher b-values may improve the characterisation of AFD and thus improve the detectability of clinically significant group differences.
Overall, AFD derived from high b-values (b = 4,000 or 6,000 s/mm 2 ) best modelled age-relationships for the majority of white matter tracts tested. These results, combined with the simulations, suggest that axonal properties (such as axon density) dominate age-related variance in AFD at high b-values, whereas extra-axonal signal contamination at decreasing diffusion-weightings incrementally suppress this effect.

| Implications
Our results bear implications for fixel-based analysis applications using retrospectively collected dMRI data which may not be optimal for the estimation of AFD. The biological interpretation of group differences in AFD should be tailored to the acquisition scheme used. Promisingly, our simulation results suggest that the effect of b-value and discrepancy with the response function dominates the effect of noise ( Figure S3), even at a lower SNR which closely matched our in vivo data (SNR = 50). Therefore, we expect that our observations at high b-values may be reproducible on a standard 3.0T system. As strong gradient systems become increasingly available, the practicalities of acquiring such high quality dMRI data at higher b-values is becoming less cumbersome (Chamberland, Tax Whilst in this study we have used a developmental population of children and adolescents as an exemplar of a b-value dependence on estimates of AFD, these findings can be applied more broadly and bear implications for a range of group studies (e.g., clinical groups or ageing adults).

| Limitations and future directions
One limitation of the current study is that we have no ground truth on the development of axonal density over childhood and adolescence. Therefore, our interpretations of improved intra-axonal signal sensitivity rests on age-relationships investigated here, which has also been used previously (Maximov, Alnaes, & Westlye, 2019;Pines et al., 2019). Whilst we have attempted to understand how AFD can vary across multiple simulated fibre geometries, we do not know how the underlying fibre properties (such as axon diameter) vary with age.
Despite this consideration, a recent study of histological validation suggests that AFD is a reliable marker of axonal density in the presence of axonal degeneration (Rojas-Vitea et al., 2019). This is a promising indicator of the neurobiological properties proportional to AFD.
Future work should adopt multi-dimensional approaches to extract meaningful components , enhance data quality (Alexander et al., 2017) and harmonise existing data (Maximov et al., 2019;Tax et al., 2019).

| CONCLUSION
We summarise our findings with three main conclusions: (a) the correspondence between apparent fibre density and simulated intra-axonal signal fraction is improved with high b-value shells; and (b) AFD better reflects age-related differences in axonal microstructure with increasing b-value (b = 4,000 or 6,000 s/mm 2 ) over childhood and adolescence; and (c) these relationships differ across the brain, with a greater b-value dependence in association tracts and posterior projections of the corpus callosum. Together, our results suggest that axonal properties dominate the variance in AFD at high b-values.

ACKNOWLEDGMENTS
We are grateful to the participants and their families for their par-

CONFLICT OF INTEREST
All authors disclose no real or potential conflicts of interest.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.