Distance‐dependent distribution thresholding in probabilistic tractography

Abstract Tractography is widely used in human studies of connectivity with respect to every brain region, function, and is explored developmentally, in adulthood, ageing, and in disease. However, the core issue of how to systematically threshold, taking into account the inherent differences in connectivity values for different track lengths, and to do this in a comparable way across studies has not been solved. By utilising 54 healthy individuals' diffusion‐weighted image data taken from HCP, this study adopted Monte Carlo derived distance‐dependent distributions (DDDs) to generate distance‐dependent thresholds with various levels of alpha for connections of varying lengths. As a test case, we applied the DDD approach to generate a language connectome. The resulting connectome showed both short‐ and long‐distance structural connectivity in the close and distant regions as expected for the dorsal and ventral language pathways, consistent with the literature. The finding demonstrates that the DDD approach is feasible to generate data‐driven DDDs for common thresholding and can be used for both individual and group thresholding. Critically, it offers a standard method that can be applied to various probabilistic tracking datasets.


| INTRODUCTION
Recent methodological advances in diffusion neuroimaging and probabilistic tractography have enabled the delineation of white matter fibre pathways in vivo (Conturo et al., 1999;Hagmann et al., 2006;Parker et al., 2002). For example, studies have reconstructed pathways in the human brain that are known to sustain higher-level cognitive functions such as language where non-human analogues are difficult to define and track (Catani et al., 2005;Catani & Thiebaut de Schotten, 2008;Cerliani et al., 2012;Cloutman et al., 2012;Parker et al., 2005;Saur et al., 2008;Saur et al., 2010). Whilst probabilistic tractography allows us to investigate fibre pathways in vivo, there are at least two fundamental challenges with interpreting outputs. First, there is no consensus on how to threshold connectivity matrices, which impacts on the ability to perform statistical analyses that are comparable across studies. Second, brain networks typically comprise of short-and long-range connections but the latter inherently have distance dependant reductions in streamline likelihoods due to the inherent nature of probabilistic tractography. To gain a better understanding of long connections and networks, a more sophisticated treatment for distance effects is required. Thus, the present study set out to tackle these two key issues.
While the probabilistic tracking approach has proved useful in reconstructing complex structural connectivity networks, it has critical challenges especially with respect to false positives (Maier-Hein et al., 2017;Roberts et al., 2017;Sarwar et al., 2019;Yeh et al., 2018).
By definition, probabilistic tractography is designed to explore all possible streamlines, which are generated from fibres orientation distributions with differing degrees of non-random noise. Consequently, this method often produces widely connected brain networks (Roberts et al., 2017); however, thresholds can be applied to allow for more interpretable outcomes and is particularly useful in reconstructing complex crossing fibres (Sarwar et al., 2019). Previous research has introduced several thresholding approaches. At the individual level, one could apply a set threshold to all individuals to remove subthreshold connections or apply different thresholds across individuals to retain the same number of connections (Betzel et al., 2019;Hagmann et al., 2007;Rubinov & Sporns, 2010). Group-level networks are commonly generated based on the consensus of connections, which retains connections present in some fraction of participants (i.e., at least 50% of subjects) (Betzel et al., 2019;de Reus & van den Heuvel, 2013). However, typically, the threshold is arbitrary and as such a range of thresholds are often used to characterise the network (Rubinov & Sporns, 2010). Taking the field as a whole, given that there are inconsistencies in methodologies across datasets, tracking procedures and algorithms, it makes formal comparisons across studies difficult.
Regardless of the specific threshold used, it is common to apply a uniform threshold to all connections regardless of length/distance. This could be problematic, especially in a large-scale distributed brain network, because noise accumulates and streamline likelihoods decrease as a function of path length (Morris et al., 2008). Thus, if a strict threshold is applied in order to minimise false positives near the seed region, then it is likely that long-range connections would not exceed that threshold, resulting in distance false negatives. A recent study by Roberts et al. (2017) developed consistency-based thresholding, which computed variation of unthresholded connectivity strength across individuals. A threshold was applied to connections with high consistency (i.e., low variation) across the group to preserve a desired level of connectivity density. It was assumed that if longrange connections were truly connected, the connections would be consistent across individuals such that consistency could be high even though the connection strengths were low. Given that the individual connectivity matrix was not thresholded, it remains unclear if this approach may overestimate connections as an unthresholded matrix is almost always fully connected (Roberts et al., 2017). Indeed, the authors noted that the consistency-based approach could potentially be biased by specific types of data acquisition and pre-processing.
More recently, Betzel et al. (2019) developed a distance-dependent consensus thresholding approach. This approach is similar to the consistency-based approach in that no threshold was applied at the individual level while group-level consensus was obtained by computing the fraction of individuals that had connection weights greater than zero. The novel aspect is related to applying a correction based on distance (across a range of bins), where for each bin the pair of brain regions that have the highest consensus across individuals is preserved. One potential caveat is that the bins used should be reasonably wide to keep the same number of pairs as in the original matrix. Also, similar to Roberts et al. (2017), this approach was primarily operated at the group level.
Despite the fact that a variety of thresholding approaches have been proposed, critical issues remain. There is still a lack of empirical support for a common thresholding regime that allows for the comparison across studies and one that resolves the distance artefact at both individual and group levels. A previous study by Cloutman et al. (2012) did propose an approach towards standardisation that was inspired by statistical hypothesis testing. In that study, probabilistic tracking was conducted to map connectivity between the sub-regions of the insula and the rest of the brain. For each individual, the distribution of connection scores between the seed region and each region of an atlas was fitted with a Poisson distribution, and that was used to identify a threshold value at α = 5%. At the group level, the consensus approach was used, in which connections were selected only when they were consistently identified across at least 50% or 75% of participants. This distribution-thresholding approach at the individual level was the first attempt to determine individual thresholds based on a statistical metric (alpha) using the connectivity distributions; however, it did not take into account distance effects, as in Betzel et al. (2019) and Roberts et al. (2017). The present study extended this distribution approach by developing a distance-dependent distribution (DDD) method in order to establish a common ground for thresholding based on significance levels of alpha. Specifically, as per standard Monte Carlo type statistical methods, we generated normative distributions of randomly sampled connectivity and then set different levels of alpha to identify the corresponding thresholds. Critically, we generated the DDDs at different distances to derive thresholds for connections of varying length. To illustrate the novel approach, we applied DDD thresholding to generate a language connectivity matrix and evaluated if it could capture a priori language networks reported in the literature (Catani et al., 2005;Catani & Thiebaut de Schotten, 2008;Parker et al., 2005;Saur et al., 2008). Specifically, it was expected that the resulting connectivity matrix could capture long-range connections and reproduce key white matter tracts in the language network including arcuate fasciculus (AF) and/or superior longitudinal fasciculus (SLF), which are generally associated with the dorsal language pathway; and middle longitudinal fasciculus (MdLF), inferior frontal-occipital fasciculus (IFOF) and uncinate fasciculus (UF), which are generally associated with the ventral language pathway.
2 | METHODS 2.1 | Human connectome data, pre-processing and tracking Fifty-four participants' pre-processed structural and diffusion datasets were downloaded from the WU-Minn 1200 Human Connectome Project (HCP) (Fischl, 2012;Glasser et al., 2013;Jenkinson et al., 2002Jenkinson et al., , 2012Sotiropoulos et al., 2013;Van Essen et al., 2013). In this HCP dataset, the participants were scanned on a 3T Siemens Skyra The acquired images were pre-processed by the HCP and the pipeline is briefly described here. The T1w and T2w images were aligned in native space using FSL's FLIRT and FNIRT functions. A reverse-phase distortion correction was used to estimate field map distortions and the resulting corrected data were registered to T1w and T2w images using FSL's FLIRT boundary-based registration (BBR) algorithm. The structural images were subjected to bias field correction . Each participant's native structural volume space was registered to MNI space using FSL's FLIRT and FNIRT functions. As the diffusion data were collected with reversed phase encoded polarities, these pairs of images were utilised to estimate the susceptibility-induced off-resonance field, and they were combined into a single image using FSL's TOPUP and EDDY functions for distortion-correction (Andersson et al., 2003;Andersson & Sotiropoulos, 2016;Smith et al., 2004).
After the pre-processing procedures, we took the distortioncorrected data and submitted it to the MRtrix3 toolbox (Tournier et al., 2019) (https://www.mrtrix.org/) for further processing and whole-brain probabilistic tracking. Specifically, the diffusion data were first subjected to bias correction using the ANTS flag (Avants et al., 2009). Next, a response function was estimated using spherical deconvolution for grey, white matter and cerebrospinal fluid (CSF) compartments using the 'dhollander' algorithm . Subsequently, we averaged the response function across subjects for each tissue type to obtain a group average, which was then used to estimate fibre orientation distributions (FOD) using the multi-shell multi-tissue constrained spherical deconvolution algorithm (Tournier et al., 2019). Finally, intensity normalisation (in the logdomain) was applied to all FOD outputs. Whole-brain tractography was performed using MRtrix3 with anatomically constrained priors (obtained using the five-tissue segmentation function) and the iFOD2 algorithm. We obtained 10 million streamlines per subject with a maximum streamline length of 250 mm and a fractional anisotropy (FA) cut-off value of 0.06. The seed points were determined dynamically using the spherical-deconvolution informed filtering of tractograms (SIFT) model in order to improve the distribution of reconstructed streamlines density. The resultant whole-brain tractogram was further filtered using SIFT2 to improve the quantification and biologically meaningful nature of whole-brain connectivity .

| Distance-dependent sampling distributions of connectivity
To generate sampling distributions of connectivity using the Monte-Carlo simulation at different distances, we created a grid of 230 cortical regions of interest (ROIs) of 3D spheres in volume space with a radius of 8 mm that covered the entire left hemisphere in MNI space.
For each participant, ROIs were inverted to native diffusion space (using inverse FLIRT and FNIRT transforms) and binarised over the grey and white matter interface (GWI). The GWI-ROIs were used as a mask to extract the number of streamlines connecting all other ROIs based on the whole brain tractogram. Thus, no streamlines were extracted for those ROIs that were entirely on white or grey matter.
A connection score between a pair of ROIs was quantified by computing the proportion of streamlines emerging from the seed and ending at the target ROI with a radial search distance of 4 mm (a default setting in MRtrix3). The connection score did not include streamlines passing through an ROI. As there were two directions of connectivity (i.e., from ROI A to ROI B and from ROI B to ROI A), we computed the average of the two connection scores generated from each direction of tracking. It was expected that closer ROI pairs would have higher connection scores while distant ROI pairs would have lower scores, reflecting the fact that the number of tractography streamlines and the fraction of true connections (Markov et al., 2013) would decrease with increasing distance. However, as the ROIs were randomly placed, many ROI pairs would have connection scores close to zero.
The distance between ROI pairs could be measured using different approaches. One simplest way was to compute the Euclidean distance between the centres of the ROI coordinates. Alternatively, one could use streamline length between ROIs pairs as a measure of distance. Compared to the Euclidean distance, the streamline approach could be more realistic in reflecting how the two ROIs were connected; however, the computation is more complex and sometimes there is a null connection between the ROI pairs. Here, we computed the distance between ROI pairs using both approaches and compared the results.
For each ROI pair, the Euclidean distance between the centres of the ROI coordinates was computed and rounded to the nearest integer, resulting in 87 unique distances. As there were varying numbers of connection samples for each distance, the distance values were further grouped into 26 distance bins to ensure that each had at least 1000 samples. The procedure for generating the ROI distance based on streamline length was not straightforward because the streamline length for a given ROI pair for each participant varied widely. Thus, for each participant, we first obtained all the streamlines of each of the ROI pairs in native space and projected them onto MNI space by using warpcovert and tcknormalise functions in the MRtrix3. Then, for each ROI pair, we took the average of the minimum streamline lengths across all the participants as the measure of the distance. If, in any of the participants' data, there was an absence of streamlines between the ROI pairs, it was treated as missing data and discarded from the averaging process. In this way, the streamline distance between ROI pairs was quantified by the average of minimum streamline length across participants. Again, the streamline distance was rounded to the nearest integer, resulting in 237 unique distances, and the distance values were further grouped into 26 distance bins to ensure that each had at least 1500 samples.
To generate a sampling distribution of connectivity, we adopted the Monte-Carlo simulation technique. That was for each distance bin, we randomly drew a sample from all the candidate samples with replacement for 100,000 times, resulting in 26 distance-dependent sampling distributions of connectivity. For each distribution, we set three alpha levels of 0.1, 0.2, and 0.3, which meant that thresholds where 10%, 20%, or 30% of the sampling distribution, respectively, were above the threshold. The same procedures were applied to generate sampling distributions of connectivity for both distance bins generated from the two distance measures.

| Language network ROIs
To generate a language connectivity matrix, we selected 13 left hemisphere ROIs (a radius of 8 mm) associated with language processing and reading (see Supplementary Figure S1). The peak coordinates were taken from the literature with slight modifications to avoid any overlap between ROIs (Binney et al., 2010;Cohen & Dehaene, 2004;Hoffman et al., 2015;Jobard et al., 2003;Noonan et al., 2013;Saur et al., 2008). The ROIs included the middle and posterior fusiform gyrus (mFG and pFG), the anterior and posterior superior temporal gyrus (aSTG and pSTG), the anterior and posterior middle temporal gyrus (aMTG and pMTG), the opercularis, triangularis and orbital parts of the inferior frontal gyrus (IFG Oper, IFG Tri and IFG Orb), the premotor cortex (PMC), posterior supramarginal gyrus (pSMG) and both the lateral and ventral anterior temporal lobe (vATL and lATL). The language connectivity matrix was generated by computing the connection strength between all of the language ROIs pairs. A group-level language connectivity matrix was generated by computing the average connectivity matrix across individuals. We then applied the three levels of the DDD thresholds to the average language connectivity matrix according to the distances between the ROI pairs.    Figure 3 illustrates that distance scores generated from the two approaches for the corresponding ROI pairs were highly correlated, r = 0.804, p < .001, with the streamline approach generating distance scores at a finer scale. Figure 4 shows the sampling distributions for each distance range based on the Euclidean distance and streamline distance. For the distributions based on the Euclidean distance (Figure 4a), the short-range connections tended to have higher connection scores compared to the long-range connections, as noted by the right skewed distributions. For the longer connections, the majority of connection scores were close to zero; however, as expected given the presence of longrange white matter and fasciculi in the brain, there were notable extremes that had strong connectivity differed from the majority of F I G U R E 2 The number of ROI paired samples for each of the 26 distance ranges based on Euclidean distance (a) and streamline distance (b).

F I G U R E 3
The scatter plot of the distance scores based on Euclidean distance and streamline distance for all the ROI pairs. the null connectivity. For the distributions based on the streamline distance (Figure 4b), the patterns of connection strength looked similar to that based on the Euclidean distance except that higher connection scores for long-range connections were observed. Figure 5 shows that the DDD thresholds at three alpha levels of 10%, 20% and 30% across the 26 distance bins based on Euclidean distance and streamline distance. For each alpha level, the connectivity threshold decreased with increased distance. As a result, the shortdistances had higher thresholds than long-distance connections. The 'rate' of decrease was more pronounced for the thresholds based on the Euclidean distance than that based on the streamline distance.
Note that the three alpha levels were selected arbitrary in order to demonstrate their impact on the thresholds. It is likely that a different range of alpha levels might be needed for different distance measures.
Nevertheless, it was clear that the connectivity thresholds were moderated by the alpha level, where the highest thresholds were noted for the smallest alpha and the lowest thresholds for the largest alpha.

| Language network connectivity
As we have seen from previous sections that the distance scores generated from the two distance approaches were highly correlated. The resulting patterns of connectivity thresholds also followed a similar trajectory. Furthermore, the resulting patterns of average language connectivity matrices generated from the two distance approaches were highly similar (r = 0.82, p < .001 based on the alpha level of 20%). For concise presentation, here we reported the language F I G U R E 4 The sampling distribution of connectivity for each of the 26 distance ranges based on the Euclidean distance (a) and streamline distance (b). The x-axis indicates connection strength and the y-axis indicates the number of samples on a log scale. DR: distance range (mm). connectivity matrix generated by using Euclidean distance. However, the result of the average language connectivity matrix generated by using streamline distance was reported in the Supplementary Figure S6. Table 1 shows the distance between 13 language ROIs included in the analyses (See Methods for details). The distance based on Euclidean distance varied widely as expected, where IFG Tri and pFG were furthest apart (101.45 mm) and the mFG and pFG were closest together (16.28 mm). We first converted all distance values to integers, which were used to compute the average language connectivity matrix across individuals. The three levels of the DDD thresholds were applied to the outputs (as in Figure 5a) according to the distance of the ROI pairs. Note that we did not threshold the connectivity matrix at the individual level because the DDD approach did not require sparse individual connectivity matrices which are typically required for group-level consensus thresholding. For completeness, we also demonstrated that the DDD thresholds could also be applied to individual connectivity matrices, where the thresholded individual matrices had high correlations with the thresholded average matrix (see Supplementary Figures S7 and S8).
After thresholding, the average connectivity score was binarised and coded with three different colours according to the alpha levels (red = 10% of alpha, green = 20% and blue = 30%) as in Figure 6.
We projected the connection scores (without binarisation) to the brain for an intuitive way to visualise the results (Figure 7) using the BrainNet Viewer (Xia et al., 2013). The width of the line indicates the strength of connection. With the most stringent threshold (i.e., the alpha level of 10%), the connectivity between the ROIs in both the close and distant cortical regions was reconstructed. Specifically, we observed neighbouring ROIs to be connected in the occipito-temporal regions (mFG and the pFG); temporal lobe (aSTG with pSTG and aMTG with pMTG); anterior temporal lobe (vATL and lATL); and frontal lobe (IFG Orb, IFG Tri and IFG Oper) as well as premotor cortex (PMC). We also observed distant ROIs being connected between the frontal and the parietal lobes, the temporal and the parietal lobes, the frontal and the temporal lobes, and the frontal and the occipitotemporal lobes. Specifically, the IFG Tri and IFG Oper were connected to the pSMG presumably via the AF, which could be part of the dorsal language pathway. The aSTG, pSTG, aMTG and pMTG were also connected to the pSMG presumably via the MdLF, which could be associated with the ventral language pathway. Moreover, there was longrange connectivity between the IFG Tri and IFG Oper and the pMTG via the AF, as well as between the IFG Orb and the pFG via the IFOF and/or part of the ILF, which could be part of the ventral language pathway.
With lenient thresholds (i.e., the alpha levels of 20% and 30%), more cross-region connectivity could be observed, especially for the inferior frontal, temporal and occipital regions. The IFG Orb was connected with the aMTG via the uncinate fasciculus (UF), and the IFG Oper and the IFG Tri were connected to the mFG and pFG via the AF. Collectively, these results demonstrated that the connectivity matrix based on the DDD thresholding approach was able to reconstruct the key white matter tracts that sustain language processing in the language network as previously reported in tractography studies (Catani et al., 2005;Catani & Thiebaut de Schotten, 2008;Parker et al., 2005;Saur et al., 2008) and the three inferior frontal regions had different connectivity profiles with temporal and parietal regions, consistent with the cortico-cortical evoked potential study (Nakae et al., 2020).
For comparison, we also thresholded the average language connectivity matrix using a common arbitrary thresholding approach at 5%, 10%, 20% and 40% of percentiles. As showed in Figure 8, with a threshold of 5% percentile, almost all ROI pairs were connected to each other, showing false positives for short-range connections. With an increase in threshold from 5% to 40% of percentile, false positives F I G U R E 5 The distance-dependent distribution thresholds at three alpha levels of 10%, 20% and 30% varied with the 26 ROI distance bins based on Euclidean distance (a) and streamline distance (b). for short-distance connections could be gradually reduced. Consequently, however, it also led to an increase in false negatives for longrange connections (e.g., the connection between mFG and IFG Oper via the IFOF).

| DISCUSSION
Probabilistic tractography has become an increasingly important tool in neuroimaging studies to delineate white matter fibre pathways F I G U R E 6 The average language connectivity matrix across individuals after thresholding and binarization based on Euclidean distance. The different levels of alpha with 10% on the top, 20% on the middle, and 30% on the bottom were superimposed onto one matrix wherein anything that survives the 10% would also survive 20% and 30%.
F I G U R E 7 The projections of the average language connectivity matrix generated by using Euclidean distance in the brain with three alpha levels of 10% (red), 20% (green) and 30% (blue) and the difference plots between them (gold). The wider the connection line indicates the stronger the connections. To date, most connectivity thresholds are chosen heuristically and, arguably, arbitrarily based on existing literature or exploratory outcome. For a given dataset, one could use a more lenient criterion to explore the probable tracts in contrast with a more stringent criterion, which could be used to identify core bundles. Indeed, it has been suggested that networks might be better characterised with a broad range of thresholds (Rubinov & Sporns, 2010). Additionally, thresholding might also be related to the demographic, psychosocial and medical information of the individuals, such as age, gender and mental illness (Buchanan et al., 2020). Although there may be justifications to apply specific thresholds to a given dataset, the outcomes may not be comparable with or generalise to other datasets. The ability to compare across studies is critically important in validating and testing the reliability of key findings. Thus, while studies have focused on probabilistic tractography and improving tractography algorithms (Behrens et al., 2003(Behrens et al., , 2007Parker et al., 2003;Smith et al., 2013), only a handful of studies have developed thresholding approaches that make the resulting connectivity close to the 'ground true' or comparable across the species (Betzel et al., 2019;de Reus & van den Heuvel, 2013;Roberts et al., 2017). Our data-driven DDD approach was designed to work with any probabilistic tracking dataset and to provide a common ground (i.e., the alpha level) to relate thresholds across datasets regardless of the specific tracking approach and parameters used (provided that random sampling distributions are established per dataset). Obviously, if studies have used similar datasets, tracking tools and parameters, then the same DDDs can be directly applied.
Our DDD approach dealt with the distance artefact by generating higher thresholds for close regions and lower thresholds for distant regions ( Figure 5). As long-range connections tend to have smaller connection strength, some tools such as FSL (Behrens et al., 2003(Behrens et al., , 2007) apply a distance correction by multiplying the distance with the probability of connection strength. This can lead to probabilities greater than one, which makes interpretations difficult. More importantly, it is not clear what mathematical form best characterises the relationship between distance and the probability of connection strength for correction. Two studies have tried to overcome these F I G U R E 8 The average language connectivity matrix generated by using Euclidean distance with a common arbitrary thresholding approach at 5%, 10%, 20% and 40% of percentiles. The different levels of thresholding with 5% on the top, 10% and 20% on the middle, and 40% on the bottom were superimposed onto one matrix wherein anything that survives the 5% would also survive 10%, 20% and 40%.
challenges. Roberts et al. (2017) introduced consistency-based thresholding, which does not directly deal with distance but it could effectively preserve long-range connections using high consistency thresholds (possible if variance across individuals is small). In contrast, Betzel et al. (2019) developed distinct thresholds for the group-level consensus scores of different lengths. Although both approaches have proved their effectiveness for group-level thresholding, the approaches cannot be applied to individual-level thresholding. In contrast, we have demonstrated that our DDD approach can work directly with the average connectivity matrix to identify plausible connections related to language processing ( Figure 6) and can also be applied at the individual level (see Supplementary Figure S7). Thus, our DDD approach extends previous distance-related thresholding approaches, while supporting both individual-level and group-level analyses. Regarding the issue of how to measure the distance between ROIs, the results demonstrated that the DDD approach could work with two different common measures: one based on Euclidean distance and the other based on streamline length. As can be seen in Figure 2, a wider distance range with more samples was observed for streamline distance compared to Euclidean distance, and that was because streamline length could be any value ranging between 0 and 250 mm (a maximum streamline length set during tracking). Consequently, when unique distances were grouped into the same number of distance bins, there tended to have more samples in each distance bin for streamline distance than for Euclidean distance. A direct comparison of the computed ROI-to-ROI Euclidean distances and streamline lengths (see Figure 3) showed that (i) the two measures were strongly correlated (r = 0.8) and (ii) that, as one would expect, some streamline distances through and around the brain's anatomy are longer than the direct Euclidean distance.
Figure 3 also shows that, at the shorter distances, there are a small number of occasions where the reverse is true. This is because the Euclidean distance is always the distance between the centres of the 'sending' and 'receiving' ROIs (not the distance between the individual voxel start and end points). The streamline length is the distance along the shortest connecting streamline (voxel-to-voxel). This can be less than the assigned ROI-to-ROI Euclidean distance for two reasons (i) the start and end voxels are towards or at the edge of the neighbouring ROIs (in the limit they could be neighbouring voxels at the boundaries); and (ii) MRtrix utilizes a 4 mm search zone which 'captures' any streamlines that terminate within 4 mm of the target position. These two factors allow for the streamline length to be occasionally shorter than the assigned Euclidean centre-to-centre ROI distance. It also explains why this is not seen when the ROIs are much further apart. Furthermore, for the DDDs generated from streamline length (Figure 4b), high connection scores were also observed for long-range connections in addition to short-range connections. It is likely because some ROIs in neighbouring gyri could be strongly connected with long and complex U-shape fibres. Thus, using the streamline approach may better characterise this phenom- of the corpus callosum to inferior and lateral brain regions (Chao et al., 2009;Park et al., 2008); and (b) to keep computational costs down. However, we envisage that the appropriate DDDs could also be generated following our methodological descriptions. Further work can be extended to generate DDDs for the right hemisphere and the whole brain. Thirdly, most raw tracking outputs can be biased over certain white matter tracts (e.g., corpus collosum). Filtering methods such as SIFT (2) (Smith et al., , 2015 have been developed to limit this bias that involve selective filtering of streamlines. While these methods can impose some constraints and make the measurements biologically plausible, they may selectively alter short/long connections. The impact of this with the DDD method is unknown and could be the focus of future research. Lastly, our DDD approach is designed to work with the outcome of tractography and to provide a method of comparing thresholding results regardless of tractography algorithm/settings. Thus, we did not seek to identify optimal thresholds for a 'ground truth'. Instead, we considered a range of comparable thresholds that can help characterise the structural connectivity (Rubinov & Sporns, 2010) and bridge studies using different tractography approaches. We acknowledge the key role of thresholding techniques in reducing the probability of spurious connectivity in probabilistic tractography, which can allow us to identify 'real' networks.

| CONCLUSION
To conclude, the DDD approach was developed as a strategy to formally threshold structural connectivity maps. The approach is in principle applicable to various tractography datasets and allow for comparisons across studies with different levels of alpha. The DDD approach also addressed the distance artefact by providing different thresholds for short-and long-range connections.