We first acquired an ex vivo diffusion MRI image of the whole monkey brain, which was reconstructed with isotropic 0.5×0.5×0.5 mm³ voxel sizes (Figure 1A) (see Methods section for details). We then obtained a thin needle biopsy sample of 1 mm diameter from the mid-body of the corpus callosum (CC) (Figure 1B), which we imaged twice with phase contrast tomography; at DESY with 550 nm voxel size, providing fascicular resolution, and at ESRF with 75 nm voxel size, providing axonal resolution. At ESRF, we stacked four consecutive image volumes to create a large effective FOV (Figure 1C). The relative sizes of an MRI voxel and the various synchrotron volumes are shown in Figure 1D. The dark image intensities in the synchrotron x-ray images (Figure 1B and C) are due to osmium staining of the myelin.

As seen in the image acquired at DESY in Figure 1B, the staining only penetrated the outer rim of the biopsy sample. At this image resolution, resolvable structures include the largest axons (with diameters roughly >2.5 μm), blood vessels, clusters of cell bodies, and vacuoles. Cell bodies and vacuoles appear bright in contrast to their outlining myelinated axons, which appeared to be oriented mainly in the left-right (L-R) axis, as expected for a CC sample.

The high-resolution ESRF data has a field-of-view (FOV) of 0.21×0.21×0.21 mm. Figure 1B depicts the placement of the FOV in the osmium-stained rim of the CC sample. The four stacked scans provide an extended FOV of 0.21×0.21×0.78 mm to match better that of the DESY volume and an MRI voxel (Figure 1C). In this data, it is possible to observe and quantify fine microstructural details, i.e., axon morphology, myelin thickness, nodes of Ranvier, clusters of cell bodies, vacuoles, and blood vessels (for a detailed description see Andersson et al., 2020).

The CC, the largest white matter tract in the brain, has high tissue anisotropy, being dominated by densely packed axon fasciculi that all run in the right-left direction between the two hemispheres (Andersson et al., 2020). The directional colour-coding of the three eigenvectors from the diffusion MRI tensor model suggests that the axonal directions are locally consistent and symmetric around the midsagittal plane (Figure 2A). The MRI voxel corresponding to the location of the synchrotron samples is located in the mid-body, approximately 2 mm lateral to the midsagittal plane (the white box in Figure 2A). In this voxel, the estimated orientations of the diffusion tensor and CSD models (Figure 2B and C) both have a main R-L component (red) with a strong I-S inclination (blue), and very little response in the A-P direction (green). The multi-fibre (CSD) model shows a single fibre peak, with a relatively narrow and isotropic spread of the fibre orientation distribution (FOD) when mapped onto a spherical polar histogram (Figure 2B). This is consistent with a homogeneous microstructural environment dominated by densely packed parallel axons. Indeed, the diffusion tensor model predicted a high fractional anisotropy (FA) value of 0.861 (Basser and Pierpaoli, 1996) and even higher FA anisotropy in the micro-tensor domain of 0.997, i.e., the micro (μ)FA calculated as (Kaden et al., 2016).

Our analysis of the high-resolution DESY data of the CC sample is shown in Figure 3A, C and E. The structure tensor was estimated based on a single set of scale-space parameters, corresponding to an integration patch size of 12 microns (see methods for further details), which is sufficient to detect all tissue-relevant orientation features. Interestingly, the structure tensor directional colour map in Figure 3A (yellow-green bands) indicates axonal fasciculi disposed with a mainly planar organisation, thus forming laminae with a thickness of up to 40–45 μm projecting throughout the whole cross-section of the CC sample. The laminar thickness was determined by manual measurements of laminae visually identified in the 3D volume. These laminae have inclination angles up to ~35 degrees from the right-left (R-L) axis. Figure 3C shows a subset of the structure tensors visualised as 3D glyphs overlaid onto an image from the synchrotron volume. At this image resolution, the structure tensor directions follow either blood vessels or axonal structures.

Upon mapping the structure tensor principal directions within the image volume onto a spherical polar histogram (Figure 3E), we observe a FOD having a single fibre peak. The I-S inclination angle is smaller than that of CSD from diffusion MRI (Figure 2B and C). Figure 3F illustrates the elongated shape of the structure tensor FOD towards the I-S axis, reflecting a degree of fibre dispersion originating mainly from the axonal laminar organisation visible in Figure 3A (regions marked in white square).

From the ultra-high image resolution ESRF data shown in Figure 3(B and D), we estimated the structure tensor at different scales, corresponding to patch sizes ranging from 9.2 to 2 μm. This allowed for the detection of tissue anisotropy on different anatomical length scales, i.e., corresponding to blood vessels, large axons, and small axons. The structure tensor directional colouring (Figure 3B) does not clearly capture the layered laminar organisation seen in the DESY data. This may be explained by the smaller FOV (210 μm) not covering a volume in which the laminar organisation appears.

The direction vectors of the structure tensor analysis from all four stacked ESRF volumes are converted to spherical polar coordinates in Figure 3H (see Appendix 1—figure 3 for each volume individually). The histogram resembles that representing the DESY data (Figure 3E), except that the main direction of the ESRF data has a lower inclination angle in the I-S axis. An exact co-registration of the two synchrotron volumes was not feasible, and some discrepancies between the FODs are to be expected. The spread and shape of the FODs are quantified using the Orientation Dispersion Index (ODI) and Dispersion Anisotropy (DA) (see Methods section). For the DESY and ESRF samples, the ODI values are 0.038 and 0.069, respectively, while the DA values are 0.55 and 0.59. Despite the different FOVs, these values are similar. The low ODI indicates a high degree of axon alignment, as expected in the CC. The mid-level DA values suggest some anisotropic spread of the directions, reflecting the angled laminar organisation observed in the DESY sample. Interestingly, the DA value for the ESRF sample is almost identical, despite the laminar bands being less visually apparent.

In Figure 3G, we present the calculated distribution of FA values of the structure tensor for the four stacked ESRF volumes, intended to match approximately a single MRI voxel. The mean FA value of the ESRF data was higher than in the DESY data (0.67±0.05 vs 0.54±0.09, respectively) and the ESRF data distribution was narrower than in the DESY data, i.e., (the ESRF median/Inter quartile range (IQR): 0.68/0.06, DESY median/IQR: 0.55/0.12). The high anisotropy at the micrometre-scale image resolutions is in accord with the FA value of 0.86 and μFA of 0.99 in the dMRI voxel, as shown in Figure 2C. Although the two synchrotron volumes originated from the same biopsy, the FA distributions are not identical. Visual inspection reveals that smaller axons and cell bodies are hardly distinguishable in the DESY data (Figure 3A) but are clearly evident in the higher-resolution ESRF data (Figure 3B).

We next investigated the axonal organisation in the centrum semiovale (CS) in deep white matter. The region contains up to three major pathways that are potentially crossing, i.e., the corticospinal tract, the CC, and the superior-longitudinal fasciculus (Schmahmann and Pandya, 2009). We imaged this complex tissue sample with dMRI and ultra-high resolution synchrotron imaging at the ESRF, and then applied a structure tensor analysis using the same range of scale-space parameters as for the CC sample, i.e., focusing on contrasted tissue features of size in the range of 9.2–2 μm.

Figure 4A shows the CSD FOD multi-fibre reconstructions from dMRI of a region encompassing the sample puncture. Due to the image resolution of dMRI and the known regional complexity of fibre organisation, most voxels contained more than one fibre peak, which indicates the expected presence of crossing fibres. (Figure 4A) (white circle) highlights the dMRI voxel best matching the FOD in the synchrotron imaging volume; it contains three fibre components, as shown by the spherical polar histogram (Figure 4B) and the corresponding glyph (Figure 4C). The two largest peaks (‘parallel fibres’) are S-I directed (blue) and the smaller peaks point mostly in the R-L direction (red).

The synchrotron data from centrum semiovale shows axons that visually tend to organise as laminae similarly to the CC. The laminar thickness ranges from as little as 5–10 μm to as much as 35–40 μm. Some laminae are predominantly populated by axons of small diameter, too small to segment individually at this image resolution. Other laminae include a mixture of populations of different diameters. Some axons intermingle with a neighbouring lamina as shown in Figure 4E. The laminar findings are consistent across the four stacked sub-volumes (see the supplementary animation, Figure 4—video 1).

The structure tensor FOD within the four stacked ESRF volumes (Figure 4F and G) (see Appendix 1—figure 3 for each volum for each volume individually) has one strong S-I and one weak R-L directed peak, much as seen in the diffusion MRI (Figure 4B and C). The largest discrepancy between the FODs of the two modalities lies in the presence of a secondary S-I-directed peak in the dMRI data (Figure 4B and C). Interestingly, visual inspection of the colour-coded structure tensor directions in Figure 4E shows the existence of voxels whose primary direction is along the A-P axis. However, this represents a small enough portion of the volume that it does not appear as a distinct peak on the FOD.

Estimation of ODI and DA was not performed in this case, as these indices are uninterpretable when the FOD has contributions from multiple WM pathways. It was not possible to robustly isolate the different pathways at the voxel level in this sample.

Finally, the estimated FA of the diffusion tensor within the single dMRI voxel (0.35) was much lower than that in the micro-tensor (0.99). The low FA value of the diffusion tensor model is expected as it cannot handle the complex fibre crossings (Kaden et al., 2016; Jespersen et al., 2013; Lasič et al., 2014), whereas the micro-tensor is modelled to be independent of the axonal organisation.

The structure tensor FA histogram from the ESRF data (Figure 4H) had a median value of 0.66 and of IQR 0.07. Interestingly, the structure tensor FA histograms of the ‘complex region’ and the ‘straight fibre’ regions (Figure 3G vs. Figure 4H, respectively) are almost identical, suggesting that the given scale-space structure tensor is independent of fibre organisation, similar to the diffusion micro-tensor model (μFA).

To explore the macroscopic axonal organisation and trajectory variations of fasciculi at the MRI sub-voxel level, we applied streamlined tractography (Tournier et al., 2019) to the main direction of the structure tensor analysis of the synchrotron data. We manually drew seeding regions to delineate where we expected axons to project towards within the sampled volume (see Methods section). To ease visualisation and quantification, we used QuickBundle clustering (Garyfallidis et al., 2012) to group neighbouring streamlines with similar trajectories into a centroid streamline. This centroid streamline serves as an approximation of the actual trajectory of a fasciculus. For simplicity, we designate such a centroid streamline as streamline.

Tractography in the DESY corpus callosum data revealed evenly distributed streamlines throughout the sample (Figure 5A). The streamlines all follow a primary L-R direction, and a large portion of the trajectories follow a small S-I bending (coloured purple), which is in good agreement with the local FODs both from structure tensor and CSD dMRI data (Figure 2 vs. Figure 3). Depicting the portion of streamlines without a strong S-I direction component with a separate colour (Figure 5A, cyan streamlines, and the supplementary animation, Figure 5—video 1) conveys the laminar organisation also observed in the structure tensor analysis (Figure 3A). The inclination angle between the trajectory of the two laminae is about 35 degrees.

For the ESRF data, the streamlines typically represent the trajectory of a small number of axons or, in some cases, of single large-diameter axons. As such, the streamlines in ESRF data (Figure 5C) better reflect detailed axonal trajectory variations as compared to the DESY data. For example, streamlines in the CC show non-straight trajectories that circumvent neighbourhood obstacles such as clusters of cell bodies, other axons, and blood vessels (orange structures). Interestingly, we observed a few axon fasciculi crossing the FOV orthogonally to the otherwise dominant R-L organisation. This phenomenon was particularly evident along the blood vessels, as illustrated in Figure 5C (red streamlines).

As expected, streamlines in the ESRF centrum semiovale data, followed the main direction of the laminae identified in the structure tensor directional map seen in Figure 4F. Figure 5D shows the trajectory of streamlines coloured according to the local direction, i.e., R-L (red), A-P (green) and S-I (blue). Independent of the main direction, we observe streamlines skirting around local obstacles and other axons, resulting in more complex and less straight trajectories than in the CC region.

To quantify the shape of fasciculi trajectories through the image volume, we characterised each streamline with two scalar values: (1) the streamline tortuosity index, which is a unitless score in [1, ∞] describing the deviation from linearity, and (2) the calculated maximum physical deviation from a straight line of each streamline, which depicts the maximal axonal ‘amplitude’ or dispersion. Both scalar metrics are independent of the general fasciculus orientation, but reflect micro-dispersion that can impact dMRI estimations (Nilsson et al., 2012; Andersson et al., 2020). We estimated the distributions of tortuosity and maximum deviation for all streamlines in both the DESY and ESRF samples (Figure 5B).

In the CC, the median tortuosity index of the distribution was almost identical in the DESY (1.01, IQR: 0.01) and ESRF sample (1.01, IQR: 0.01). A tortuosity index close to unity is equivalent to highly straight fibres, consistent with both the visual inspection of the streamlines (Figure 5A and C), the estimated ODI, and likewise with our expectation for the CC. In contrast, the median observed maximum deviation within the samples differed distinctly between DESY (17.1 μm, IQR: 10.7 μm) and ESRF (2.8 μm, IQR: 3.1 μm). In ESRF data, the small FOV limits the analysis to capture only microscopic effects. Therefore, the maximum deviation is driven by the interaction of the fasciculus with neighbourhood obstacles, i.e., micro-dispersion. We observed maximum deviations as high as 12 μm, which were due to obstacles like blood vessels, thus in good agreement with observations reported in Andersson et al., 2020. In DESY data, deviations were as high as 40 μm, which we attribute to more macroscopic effects such as the bending of pathways, as observed in Figure 5A. We suppose that this may arise from the U-shaped macroscopic trajectories of CC fibres towards the cortex.

In the ESRF centrum semiovale sample, the median tortuosity index was significantly higher and more variable (1.04, IQR: 0.07) than in the CC. This agrees with the more complex crossing fibre trajectories observed in Figure 5D (and the supplementary animation, Figure 5—video 2). Interestingly, the distribution of maximum deviations had a peak of around 5.3 μm and a median of 7.1 μm (IQR: 9.6 μm). While it is an increase compared to the CC, the magnitude of streamline dispersion is on a comparable scale.

We conducted three non-parametric statistical tests of significance (⍺=0.05) between the two CC samples (DESY vs. ESRF) and between the two ESRF samples (CC vs. CS). These tests included: a Kolmogorov-Smirnov (two-sample) test for equality of distributions, a Wilcoxon rank sum test for equality of medians, and a Brown-Forsythe test for equality of variance (Hollander et al., 2015; Brown and Forsythe, 1974; Massey, 1951). For streamline tortuosity (Figure 5B, top), the null hypothesis was accepted for all three tests between CC samples (p=0.2, p=0.7, p=0.2), but rejected for all tests comparing CC and CS (p<0.001). For streamline maximum deviation (Figure 5B, bottom), all six null hypotheses were rejected (p<<0.001).

Finally, we investigated the organisation of fasciculi in both healthy mouse brains and a murine model of focal demyelination induced by 5 wk of cuprizone (CPZ) treatment. This allowed for the exploration of the disease-related influence on axonal organisation, particularly under inflammation-like conditions with high glial cell density at the demyelination site (He et al., 2021). The experimental setup for DESY and ESRF is similar to that described for the monkey, with the exception that we did not perform dMRI and synchrotron imaging on the same brains, and only collected MRI data for healthy mouse brains. This approach allowed us to apply the same structure tensor and tractography streamline analysis used previously, but in a healthy versus disease comparison, demonstrating the methodology’s ability to provide insights into pathological conditions.

The dMRI is a whole-brain FOV, whereas the FOVs of DESY and ESRF are similar to those for the monkey (see Table 1). The image voxels of dMRI, DESY and ESRF have isotropic side lengths of 125 μm, 550 nm and 75–100 nm, respectively. Although the mouse brain image resolution of the DESY and ESRF data is similar to that acquired on the monkey, the white matter structures in the mouse brain are considerably smaller in proportion, and the resolvable anatomical length scale differs. In the mouse, the FOV of the DESY data included both the splenium region of the corpus callosum and parts of the neighbouring Cingulum pathway (Figure 6A). Furthermore, in the ESRF data, the image resolution did not suffice to outline the diameters of even the largest axons (Figure 6B), in contrast to the monkey CC. However, blood vessels and glial cells are outlined as the stained myelinated axons skirt around them. Unlike in the monkey samples, we did not observe vacuoles in the mouse samples.

DESY and ESRF data both revealed demyelinated regions in the CC of the cuprizone-treated mouse. The lesion area is demarcated by an intensity gradient increasing from dark (intact myelination) toward a brighter signal (demyelination) (Figure 6C and D). In the demyelinated area, the ESRF data revealed a higher density of cell bodies and small accumulations of heavily stained material. Cellular structures were manually segmented in a small representative region to reveal their 3D shapes (Figure 6E).

In dMRI of the healthy mouse brain, the colour-coded diffusion tensor directions of the first (major) principal direction have the same R-L direction (red) as seen in the monkey (Figure 7A vs. Figure 2A). However, Figure 7A shows a shift of the second and third principal directions in the mouse CC, as compared to the monkey.

In the healthy and demyelination mouse brains, we use only a single scale-space level for the structure tensor patch size (see method section) both in the DESY and ESRF data sets. The integration patch sizes correspond to 18.7 μm for the DESY and 9.4 μm for the ESRF samples (see Table 2).

In the DESY data from healthy mice, the directional colour coding of the structure tensor was in accord with the diffusion tensor MRI (Figure 7A vs. Figure 7B). We found sporadic directional deviations in the coronal view, represented as local regions with different directional colours in Figure 7B, which indicate local crossing axon fasciculi and/or blood vessels. Nevertheless, the visualisation of orientations did not reveal any axonal organisation in the mouse CC due to the lack of local angular contrast, unlike the clear laminar structures seen in the monkey sample (Figure 3A). Any parallel organisation in tissue remains undetectable because our visual contrast relies on angular differences.

The two major pathways, the CC and cingulum, appear in the mouse as two sharply separated structures, facilitating the generation of image masks for regionally independent quantifications of the structure tensor analysis. These structures cross almost orthogonally, as shown in the 3D glyphs of the structure tensor FODs from both the ESRF and DESY datasets (Figure 7E). In all cases, the ODI was estimated to be below 0.06, indicating highly directed pathways, similar to those values measured in the monkey CC. The DA indices show more variation: In the CC, DA was 0.49 and 0.32 for the DESY and ESRF samples, respectively, and in the cingulum, 0.78 and 0.29. These values are on the same scale as those for the monkey and are non-zero, suggesting some axonal organisation is present, even though it was not visually confirmed.

The structure tensor FA histograms of the mouse CC and cingulum, are both shifted toward lower values in the DESY data as compared to the ESRF data (Figure 7G). The two histograms overlap in the DESY data, whereas the ESRF data shows a shift towards higher FA values in the cingulum as compared to CC, indicating a dependency of the estimated FA on image resolution relative to the size of structures. From the ESRF data, we were able to roughly segment cell clusters and blood vessels. Since the volume consists of about 13.5% cells/blood vessels in the cingulum, versus only 8.5% in CC, the higher density of extra-axonal structures may factor into the differences in estimated FA values.

In the CPZ-treated mouse brain, the directional colour-coded structure tensors in the DESY data match that in the healthy mouse, even in the demyelination region, which shows reduced image contrast (Figure 7C vs. D). The fact that there even is directional contrast in the demyelinated region, depicts the advantage of phase-contrast imaging, where the obtained contrast is not solely dependent on absorption. Note that the demyelination region is confined to axons within the CC, with sparing of the cingulum. The lower image contrast (lack of strong edges) in the demyelinated region contributes to the blurrier tensor shapes and lower FA values as compared to the normal-appearing white matter (mean FA: 0.44 vs 0.52), as shown in Figure 7H. The FODs are similar (Figure 7F) with ODI = 0.03 and 0.04 for the normal-appearing and lesioned regions respectively as expected for the highly aligned axons in CC. In the same order DA = 0.18 and 0.40.

For better comparisons, we registered the ESRF data of mouse CC onto the DESY data. In healthy and cuprizone-lesioned mice, Figure 8A and B shows that the CC and cingulum are both mappable using streamlines. Irrespective of image resolution, the cingulum did have a lower tortuosity and maximal deviation than the CC (Figure 8C and E), which could be due to differences in densities of extra-axonal structures. This is suggested by the streamlines projecting through the demyelinated region (Figure 8B), which contains more cells (Figure 6D and E). Indeed, both indices broadened with lower image resolution in combination with a larger FOV that covers macroscopic shape changes. An example is the bending of the corpus callosum detected by the longer streamlines.

Like before, we conducted three non-parametric statistical tests of significance (⍺=0.05): a Kolmogorov-Smirnov test, a Wilcoxon rank-sum test, and a Brown-Forsythe test. These tests compared measurements between the two pathways from each dataset (CC vs. Cg) and within the same pathway across different modalities (DESY vs. ESRF). For tortuosity (Figure 8C), the null hypotheses for distribution and median equality were rejected in all cases (p<<0.001). The null hypotheses for variance equality were accepted when comparing the CC across samples (p=0.35, red vs. blue) and comparing the CC and cingulum in the DESY dataset (p=0.66, red vs. yellow), but were otherwise rejected (p<<0.001). For maximum deviation (Figure 8E), all 12 null hypotheses were rejected (p<<0.001).

Interestingly, both in healthy and demyelination samples, some streamlines from the CC corpus deviate to project into the cortex, as illustrated in Figure 6 and Figure 8A and B (pink extensions of streamlines). Those that project into the cortex also tend to originate from a more superiorly placed seed point. As such, the streamlines visually present a vertically layered organisation within the CC.

Our white matter samples from monkey and mouse brains were geometrically organised as stacked laminae across anatomical length scales extending from individual axons to axon fasciculi and the major WM pathways.

In the monkey CC DESY data, which has a field of view (FOV) comparable to a dMRI voxel, a columnar laminar organisation at a macroscopic level was visually revealed from the structure tensor (ST) direction colouring. However, this laminar organisation was not visible in the higher-resolution ESRF data for the same tissue sample. Although the two samples were not co-registered, the size of a single ESRF FOV within the DESY sample is illustrated in Figure 3A. This demonstrates the possibility of placing the ESRF sample where the observed laminar structure is absent. Consequently, knowledge of the tissue structural organisation and its orientation is important to fully benefit from the stacked FOV of the ESRF sample and when choosing appropriate minimal FOV sizes in future experiments.

Interestingly, when characterising FODs with measures like ODI and DA as indicators of fibre organisation, rather than relying on visualisation, results from large- and small-FOV data show no discrepancies. This statistical approach discards the spatial context (visually perceived as laminae), highlighting the need to combine both methods.

Invasive tracer studies and histology in monkeys (Caminiti et al., 2009; Howard et al., 2023) and histology of humans (Mollink et al., 2017) previously hinted at a columnar organisation in the ventral part of midsagittal CC. However, the earlier approaches failed to show the finer 3D columnar organisation of 10 μm thick laminae as revealed by the present structure tensor analysis of the synchrotron data. The occurrence of these thin layers with slightly different angles may be confirmed by visual inspection of polarised light imaging (PLI) data collected in a human brain coronal CC slice (in Figure 6 of Mollink et al., 2017). In a coronal view of the mid-sagittal region, PLI showed fibres arising from the right and left hemispheres reaching the midsagittal CC as a layered structure, based on its orientational colour coding. The greater slice thickness (in PLI 100 μm) in a coronal view may have resulted in a partial volume effect arising from the thinner non-parallel columnar laminar sheets that we observed in the sagittal view of our synchrotron data (Figure 3A).

Structure tensor analysis of the present ESRF data in the complex fibre region clearly shows the crossing of three pathways composed of laminar sheets of fasciculi (Figure 4E). Compared to the parallel within-tract axonal fasciculi in the monkey CC, the boundaries between fasciculi in the complex fibre region are thin (tens of micrometres), and, therefore, likely to be less well defined when visualised as streamlines. However, we did not see such an intermingling of streamlines at the interface between the crossing of the larger well-defined tracts such as the CC and cingulum in the mouse brain (Figure 7B–D).

The structure tensor method performed very well in analysing the fasciculi without segmentation, through its detection of tracts and fasciculi based on their angular differences and highlighting of their sheet-like and laminar organisation. The observed tendencies regarding fasciculi organisation can be grouped into two: Within-WM pathways where inclination angles are low, including 0 degree parallel alignment, and between-WM pathways where the angles are high.

Within major WM pathways, neighbouring axonal fasciculi appeared to follow the same principal direction but could have inclination angles up to the order of 35 degrees in both species.

In the monkey CC (mid-body), we observed laminar organisation indicated by clear spatial angular differences in the ST directions in the sample (Figure 3A). Quantifications of the FOD shape showed DA indices of 0.55 and 0.59 for the DESY and ESRF samples, respectively. In contrast, the mouse CC (splenium) did not visually reveal a similar angled laminar organisation (Figure 7C), and the DA indices were lower, at 0.49 and 0.32, respectively. Two possible explanations exist. First, the within-pathway laminar organisation may not be identical across the entire CC. Consequently, more scans from other CC regions would be required to confirm. Second, the different species might account for the differences. Larger brains like the monkeys might foster a different level of within-pathway axon organisation compared to the smaller mouse. Although we could not visually detect laminar organisation from the colour coding of the ST direction in the mouse, the non-zero DA values suggest some level of organisation. This is supported by our streamline tractography, which indicates a vertical layered organisation (Figure 8A and B). It further aligns with studies using histological tracer mapping that shows a stacked parallel organisation of callosal projections in mice, between cortex regions M1 and S1 (Zhou et al., 2013). Nevertheless, we cannot rely solely on voxel-wise ST directions to fully describe the axonal organisation, as this method does not contrast almost parallel fasciculi (inclination angles approaching 0 degrees). Analysing patterns in tractography streamlines would be an interesting future direction for this purpose.

Interestingly, the default multi-fibre dMRI CSD model for the monkey CC showed only a single fibre direction with an isotropic FOD, indicating a coherent fibre pathway (Figure 2C). This approach did not reveal any of the layering and organisation evident from structure tensor analysis of the synchrotron data (Figure 3). Nevertheless, when investigating the directions of the diffusion tensor model we see signs that the diffusion signal is indeed sensitive to the complexity of fasciculi organisation; the principal direction agrees with the CSD and is locally consistent across the entire CC. However, the second and third eigenvectors are also informative, in that they behave symmetrically across the midline, and are locally consistent, although their order can shift at specific locations. This is likely an indication of the local organisation, similar to PLI-based observations (Mollink et al., 2017). Additionally, Tax et al., 2016 have demonstrated the calculation of a grid-crossing sheet probability index from diffusion MRI data, which suggested the presence of sheet-like features in a crossing fibre region, which is in line with our findings in the synchrotron data. Note that the method by Tax et al. only detects sheet-like structures crossing on a grid and does not reveal laminar structures with lower inclination angles, as we observed in the monkey CC.

By further exploring the diffusion tensor modelling of the CC in the Human Connectome Data set (N=4, Appendix 1—figure 1) we found an order of the second and third eigenvectors resembling that in the dMRI of the monkey. This observation supports the future exploration of the diffusion-weighted signal to reveal insights about axonal organisation within a WM pathway. For the present, we have demonstrated axonal organisation only in selected regions of the WM, but further research may establish more generalised rules governing fasciculi organisation in WM pathways, and potential species-related differences.

Between-WM pathways, the fasciculi cross with high inclination angles approaching 90 degrees. This is readily detectable in diffusion MRI multi-fibre CSD modelling (Tournier et al., 2004) and in the structure tensor analysis. We observed this right angle crossing at the centrum semiovale, a complex fibre region with the crossing of three different pathways (Figure 4). Using tracer labelling of axonal projections, Mortazavi et al., 2018 revealed grid-like right-angle crossing of axons in a similar region in the monkey brain. Tractography and micro-dissection findings of Wedeen et al., 2012 of the same region similarly showed grid-like crossing of laminar from different pathways.

Interestingly, we observed in the ESRF data a single axonal fasciculus composed of a few axons that crossed the main callosum pathway at an almost 90 degree angle running parallel to a blood vessel (Figure 5C). Such phenomena may be overlooked in lower resolution data and have such a small contribution to the diffusion MRI signal as to be dismissed as noise/artefact. Without prior knowledge of their existence, such crossings are apt to be disregarded by modelling choices, such as orders of spherical harmonics, model regularisation, and number of fibre populations.

Our findings of laminar organisation and the distinction between high and low inclination angles seem to support a simple topological rule across anatomical length scales. This holds especially for the low inclination cases, which retain their laminar organisation with minimal intermingling in the neighbourhood, meaning that we expect that topological organisation should be kept at a distance. If so, the (inclination angle) information might serve to form rules for low-resolution diffusion MRI-based tractography about how best to project through crossing fibre and bottleneck regions, which is currently a source of false-positives trajectories (Maier-Hein et al., 2017). The reason is that standard tractography methods do not ‘remember’ or follow anatomical organisation rules as they trace through complex regions. Our findings on pathway lamination and inclination angles—low for parallel-like trajectories and high for crossing-like trajectories—can help incorporate trajectory memory into these methods, reducing the risk of false trajectories.

We believe our observed topological rule of white matter laminar organisation can be explained by a biological principle known from studies of nervous tissue development. The first axons to reach their destination, guided by their growth cones, are known as ‘pioneering’ axons. ‘Follower’ axons use the shaft of the pioneering axon for guidance to efficiently reach the target region (Breau and Trembleau, 2023). Axons can form a fasciculus by fasciculating or defasciculating along their trajectory through a zippering or unzipping mechanism, controlled by chemical, mechanical, and geometrical parameters. Zippering ‘glues’ the axons together, while unzipping allows them to defasciculate at a low angle (Šmít et al., 2017). Although speculative, the zippering mechanism may be responsible for forming the laminar topology observed across length scales. The defasciculation effect can explain our results in the corpus callosum (CC) of monkeys, with laminar structures at low angles (~35 degrees) also observed by Innocenti et al., 2019; Caminiti et al., 2009, as well as in other major pathways (Sarubbo et al., 2019). In contrast, a fasciculation mechanism may be observed in the mouse CC (0 degrees). If the geometrical angle between two axons is high, i.e., toward 90 degrees, the zippering mechanism will not occur, and the two axons (fasciculi) will cross (Šmít et al., 2017). This supports our and other findings that crossing fasciculi or pathways occur at high angles toward 90 degrees in the fully matured brain (Wedeen et al., 2012). Once myelination begins, the zippering mechanism is lost (Šmít et al., 2017), suggesting that laminar topology is established at the earliest stages of brain maturation.

Applying streamlined tractography to the main fibre orientations in the structure tensor analysis enabled a geometric quantification of the non-linearity of axon fasciculi trajectories as a tortuosity metric (Pingel et al., 2022). Surprisingly, we observed that the maximal amplitude of the axon fasciculi was comparable when measured in the crossing and straight fibre regions. This suggests that the geometrical variation in an axon fasciculus trajectory may hold similarly throughout the white matter.

We also detected differences in minor pathways: In the mouse brain ESRF data, axon fasciculi were straighter (i.e. having lower tortuosity) in the cingulum than those in the corpus callosum. Similarly in monkey ESRF data, fasciculi were straighter in the CC than in the crossing fibre region.

For the monkey and mouse brain samples, the distributions of maximal deviation in the ESRF and DESY data showed main peaks around 5–7 μm. This range of maximal deviation aligns with the typical radius of the cell bodies. Indeed, axons and fasciculi have been observed to skirt around oligodendrocytes in the ESRF monkey data set (Andersson et al., 2020). In addition, the maximal deviation is in accord with the size of a single axonal fasciculi (Figure 5C) or blood vessels.

Even in the case of regional demyelination in the cuprizone-treated mice, we saw minor changes in tortuosity compared to normal axon fasciculi (Figure 8D). We suppose that microglia and macrophages invading the site of demyelination are sufficiently similar in size to oligodendrocytes that they have little net effect on the distribution of tortuosity. The measured max deviation was higher in the cuprizone-treated mouse (Figure 8F), but given the magnitude of approx. 30 μm, this effect is more likely a description of the macroscopic bending of the CC, rather than a result of the demyelination. As our streamlines went through both demyelinated and normal-appearing regions of the CC, we experienced a similar dilemma found in MRI-based tractography applied in humans. The streamlines are based on only the direction of the tensor and are not guaranteed to be sensitive to local pathology. Additionally, demyelination is a dynamic process. We only sampled a single time point, and more must be included to explore if the tractography-based analysis has the sensitivity to correlate with the temporal dimension of demyelination.

Diffusion MRI is sensitive to the micro-dispersion effects (Lee et al., 2019; Andersson et al., 2020), but our findings suggest that the amplitude of micro-dispersion is primarily related to the sizes of extra-axonal structures independent of the complexity of the major white matter pathway. We also observe that the tortuosity can change between different white matter pathways. Assuming that the local distributions of cell bodies, vessels, and axon fasciculi can generalise within a pathway, then we can expect the micro-dispersion effects on the dMRI signal to be homogeneous.

Notably, high-resolution light-microscopy-techniques such as Polarised Light imaging with in-plane 5 μm and 30–100 μm thick slices (Axer et al., 2011) as well as light-sheet imaging are expected to be too low in resolution to detect the micro-dispersion effects.

We found a general agreement in the FODs between the different image modalities and across image resolutions. This is in line with other studies comparing dMRI with structure tensor analysis using both single and multi-fibre models (Khan et al., 2015; Budde and Frank, 2012; Leuze et al., 2021; Schilling et al., 2016).

The FOD discrepancies that we did observe could be attributed to the differences in the FOVs across the imaging modalities. As shown in Figure 1, we prepared biopsy samples for synchrotron imaging that was on a sufficiently large scale to cover several MRI voxels of isotropic 500 μm size. Indeed, the DESY sample covered multiple MRI voxels, except that a relatively large unstained region in the middle of the sample was not included in the generated FOD. Similarly, the four stacked image volumes of the ESRF measuring 780 μm in length by 210 μm in diameter only covered a fraction of the DESY volume and MRI image voxel. Since the FOD is dependent upon fibre organisation, the observed FOD differences reflect actual anatomical differences, for example with more volumetric weighing in one fibre direction. Therefore, although FODs are independent of image resolution, care must be taken when comparing two modalities without covering the same 3D volume, as observed in the CC (Figure 3E and F vs. Figure 3H, I). Examples are validation studies comparing the FOD from diffusion MRI fibre models with that derived from 2D or 3D histology which only rarely cover the same identical volume (Khan et al., 2015; Budde and Frank, 2012). In cases of differing FOV, there may be a risk of misattributing an actual difference in anatomical information across length scales as a methodological difference.

We show that tissue anisotropy metrics are comparable across modalities but vary with the anatomical length scale. The simple FA metric in the diffusion tensor MRI model was, as expected, sensitive to the average tissue organisation/anisotropy in the voxel. Thus, we found higher anisotropy in the CC (Figure 2C) as compared to the centrum semiovale, a more complex fibre region, much as observed by others (Andersen et al., 2020). In contrast, the μFA dMRI metric (Kaden et al., 2016; Jespersen et al., 2013; Jespersen et al., 2013; Lasič et al., 2014) is sensitive to tissue anisotropy only in micro-domains of 10 μm length scale (molecular displacement). The μFA generates a mean anisotropy value within a voxel, returning similarly high FA values in both the CC and centrum semiovale, thereby confirming the expected independence of FA on the structural organisation (Andersen et al., 2020).

In the DESY and ESRF data, anisotropy measured by the structure tensor model is also determined within a micro-domain controlled by the patch-size and produces a distribution of values from the full volume. We introduced a scale-space parameter that automatically adjusts the patch-size to ensure optimal sensitivity to anisotropic features. Patch-sizes of 18 μm for DESY and 5 μm for ESRF data depict similarly sized micro-domains as in the μFA diffusion MR model and were shown to be largely independent of an axonal organisation (Figure 4H and Figure 7G).

The same anatomical features, namely cell membranes and myelin, dominate the contrast of diffusion vs. structure tensor techniques. In dMRI, these structures are the main sources for restricted and hindered water diffusion, and their osmium staining for the synchrotron scans provides strong image gradients modelled by the structure tensor. However, tissue preparation differs substantially for the two image modalities. In the case of ex vivo MRI, Sun et al., 2005 showed that anisotropy of perfusion fixated, hydrated tissue should closely match that in vivo. However, synchrotron imaging calls for an extra dehydration step before tissue embedding in EPON, which changes the intra- and extracellular volume fractions due to tissue shrinkage (Korogod et al., 2015). Similarly, there are shrinkage effects between dehydrated and hydrated synchrotron image samples (Töpperwien et al., 2019). Given the large differences in tissue processing, the obtained anisotropy measures naturally differ across the modalities. Nevertheless, as both the diffusion- and structure tensor models were sensitive to the same anatomical features, the observed strong correlation in FA values was expected.

Across image resolutions, the distributions of structure tensor anisotropy also changed, being generally lower and broader in DESY data compared to ESRF data. This could reflect greater partial volume effects in the lower-resolution DESY data, thus decreasing the apparent separation of cells and axons. Such image blurring can change the image gradient information used by the structure tensor model, thereby making the anisotropy metric dependent on the image resolution. Therefore, structure tensor quantification of anisotropy metrics calls for invariant image resolution.

In our comparison of structure tensor anisotropy in healthy mice versus cuprizone-induced demyelination, the data clearly shows a lowered anisotropy distribution in focally demyelinated regions compared with healthy WM regions (Figure 7H). We observed an increased extra-axonal content in the lesion area, thus in agreement with the observed higher density of cells in demyelinated tissue (He et al., 2021). By outlining the various cellular structures in 3D, we could visualise what we believe to be cell bodies, macrophages, and the myelin debris engulfed inside macrophages (Figure 6E).

Interestingly, despite reduced anisotropy in demyelinated regions, the structure tensor still detects a clear directionally dependent anisotropy as in the healthy brain, consistent with the persistence of axons in this animal model despite demyelination (He et al., 2021). We did not collect diffusion MRI for the cuprizone-lesioned brains. However, we recently reported relatively preserved μFA values in demyelinated regions of the rat CC (He et al., 2021). Similar findings were made in a group of multiple sclerosis patients compared to normal (Andersen et al., 2020) supporting our synchrotron anisotropy results.

The tissue came from a 32-mo-old female perfusion-fixated vervet (Chlorocebus aethiops) monkey brain, obtained from the Montreal Monkey Brain Bank. The monkey, cared for on the island of St. Kitts, had been treated in line with a protocol approved by The Caribbean Primate Center of St. Kitts.

C57BL/6 female mice were obtained from Taconic Ltd. (Ry, Denmark). Mice were bred at the Biomedical Laboratory, University of Southern Denmark according to protocols and guidelines approved by the Danish Animal Health Care Committee (2014-15-00369). All animal experiments complied with the EU Directive 2010/63/EU for animal experiments.

Repeated oral administration of the copper chelator bis-cyclohexanone-oxalyldihydrazone (cuprizone) leads to demyelination and oligodendrocyte loss notably in the CC, and thus serves as a model for the demyelination lesions in patients with multiple sclerosis (MS) (Torkildsen et al., 2008). Cuprizone (Sigma Aldrich, MO, USA) was administered as 0.4% cuprizone in powdered standard chow to female mice aged 8–9 wk for 5 wk. Control mice were kept on a normal diet. During experiments, mice were weighed every second day to monitor the characteristic weight loss due to cuprizone exposure, with euthanasia of mice losing more than 20% of their baseline body weight. After 5 wk on the cuprizone diet, the mice were euthanized with an overdose of pentobarbital (Glostrup Apotek, Glostrup, Denmark) followed by perfusion with DPB and 4% paraformaldehyde (PFA). The brains were stored in 4% PFA at 4 °C.

Two acquisition setups were used to collect the ex vivo diffusion MRI data sets on whole brains from the monkey (N=1) and normal mouse (N=1). The protocol for the monkey brain was from Andersson et al., 2020 and includes three shells: b-values of [2011, 2957, 9259] s/mm^2, (gradient strength (G) = [300, 219, 300] mT/m, gradient duration (δ) = [5.6, 7.0, 10.5] ms, gradient separation (Δ) = [12.1, 20.4, 16.9] ms), [84, 87, 68] non-collinear diffusion encoding directions and the number of b=0 s/mm² was [15, 16, 13]. The higher b-value was adjusted to the lowered ex vivo diffusivity compared to in vivo (Dyrby et al., 2011). Correspondingly, the mouse brain protocol is from Perens et al., 2023 and includes a single shell: b-value of 4000 s/mm^2 (G=456 mT/m, δ=5 ms, Δ=13 ms), 60 non-collinear diffusion encoding directions and five b=0 s/mm². Indeed, whole-brain MRI scanning was performed on both the normal and cuprizone mice prior to synchrotron sample preparation. However, the MRI image quality was poor and could not be improved, as this was only realised after the tissue had been processed for synchrotron imaging. Therefore, the collected MRI mice data from Perens et al., 2023 was included in the study.

Tissue preparation in both species followed a standard pipeline for diffusion MRI ex vivo (Dyrby et al., 2011). To reduce susceptibility artefacts and avoid air bubbles, we scanned the monkey brain in a double-sealed plastic bag filled with PBS, whereas the mouse brain was kept in the skull and placed in a sealed plastic bag with PBS. The monkey brain data was collected on an experimental 4.7 Tesla Agilent MRI scanner, whereas the mouse brain data were collected on an experimental 7 Tesla Bruker Biospec MRI scanner. We used a quadrature radio frequency volume coil for the monkey, and a 2-parallel cryo-coil probe for the mouse brains. The isotropic image voxels were 0.55 mm for the monkey brain and 0.125 mm for the mouse. The monkey brain acquisition used an optimised three-shell ActiveAx MRI protocol based on a maximal gradient strength of 300 mT/m for ex vivo tissue as in Dyrby et al., 2013. The mouse acquisition used a single-shell diffusion MRI protocol for diffusion tensor imaging (Perens et al., 2023). All whole-brain diffusion MRI data sets are available at https://www.drcmr.map/.

Before local fibre modelling, the diffusion MRI datasets were denoised (Veraart et al., 2016) and processed in the MRTrix3 software toolbox (RRID:SCR_ 006971)(Andersson et al., 2020; Perens et al., 2023) to remove Gibbs ringing artefacts (Kellner et al., 2016). Then, we fitted the single-fibre diffusion tensor and the multi-fibre constrained-spherical deconvolution models in the MRtrix3 software toolbox (Tournier et al., 2019). For the monkey data, we fitted the constrained-spherical deconvolution using only a single-shell b-value i.e., 9686 s/mm², from which we estimated the fibre orientation distribution (Tournier et al., 2004). The diffusion tensors were fitted with a single-shell b-value, i.e., 2957 s/mm² for the monkey (Andersson et al., 2020) and 4000 s/mm² for the mouse (Perens et al., 2023). From the diffusion tensor, we estimated the three Eigenvectors representing fibre orientations as well as the fractional anisotropy (FA) metric (Basser et al., 1994). Since the FA value of microstructure anisotropy is biased by fibre architecture (Andersson et al., 2020), we also fitted a micro-tensor model to estimate μFA from the monkey three-shell diffusion MRI data set of monkey (Kaden et al., 2016). The diffusion tensor model assumes a single tensor per voxel, whereas the micro-tensor model assumes a micro-tensor regime with many micro-tensors on the diffusion length scale. Hence, the micro-tensor model is not sensitive to fibre organisation (Kaden et al., 2016; Lasič et al., 2014; Jespersen et al., 2013).

We used data from four subjects of the Human Connectome Project (HCP) Adult Diffusion database to compute the eigenvalues and eigenvectors of the diffusion tensor (Basser et al., 1994) via a two-step weighted and iterated least-squares method (Veraart et al., 2013) as implemented in MRtrix3 (Tournier et al., 2019). Data were denoised as indicated in Pizzolato et al., 2023 using a Rician variance stabilisation transform (Foi, 2011) in combination with PCA optimal shrinkage (Gavish and Donoho, 2017), with subsequent application of Gibbs ringing removal (Kellner et al., 2016) and eddy current distortion correction (Andersson and Sotiropoulos, 2016). For the estimation of the tensor, we selected only the b=0 and the 64 volume directions corresponding to b=1000 s/mm². The resulting eigenvector colour-coded maps (Pajevic and Pierpaoli, 2000) are shown in Appendix 1—figure 1, which is organised similarly to Figure 2A.

To perform the SRI experiments, small tissue samples extracted from the monkey and normal (N=1) and cuprizone (N=1) mouse brains were processed and embedded in EPON.

The monkey brain was sliced in the sagittal plane with a monkey brain matrix. Cylindrical samples of 1 mm diameter were extracted from the mid sagittal CC and the centrum semiovale (CS) with a biopsy punch. After post-fixation in 2.5% glutaraldehyde for 24 hr, they were stained by immersion in 0.5% osmium tetroxide (OsO₄), followed by an embedding in EPON resin and shaped into blocks measuring 1×1×4 mm. For details of sample preparation, see Andersson et al., 2020.

The brains of the normal and cuprizone mice were cut into 1 mm thick coronal slices using a mouse brain matrix. For the cuprizone mouse, we selected a slice where a demyelination lesion in the white matter was visible. The same slice position was selected in the normal mouse brain for comparison. After selecting the slice containing the splenium of the CC, we carefully excised the part of the splenium traversing the mid-sagittal plane and extending approximately 2 mm into the left hemisphere, using a scalpel under a microscope. The mouse brain samples were then processed as described above. Once the EPON resin had polymerized, we used a metallographic grinder to polish the blocks to have a smooth surface, a thickness of approximately 700 µm, and a length of a few millimetres.

The specimens were imaged at beamline ID16A of the European Synchrotron Radiation Facility (ESRF) with x-ray nano-holotomography, as described in Andersson et al., 2020. In short, the samples were illuminated with a nano-focused cone (Cesar da Silva et al., 2017) 17 keV x-ray beam. The samples were rotated over 180 degrees, and tomographic scans were acquired at four different propagation distances (Hubert et al., 2018). Each scan consisted of 1800 projections with exposure times of 0.22 s, and a pixel size of 100 nm or 75 nm, taking approximately 4 hr to acquire. Upon performing the phase retrieval and tomographic reconstruction, the resulting volumes had dimensions 2048×2048 ×2048 voxels. In the case of the healthy mouse sample, the reconstruction was performed in an extended FOV providing a volume of 3200³ voxels as presented in Table 1. For the monkey CC and CS samples, we collected four consecutive image volumes with a small overlap to extend the total FOV.

The specimens were also imaged at Deutsches Elektronen-Synchrotron (DESY), at the synchrotron radiation nano-CT end station (GINIX) of the P10/PETRA III beamline (Salditt et al., 2015). Here, a 13.8 keV x-ray beam illuminated the EPON samples, which were rotated through 180 degrees to acquire tomographic scans at the lens-coupled detector (XSight Micron, Rigaku). Each tomographic scan consisted of 1000 projections with 20 ms of exposure. The detector was placed in the direct-contrast regime of the sample, and a phase retrieval was performed with an in-house Bronnikov-aided correction-based algorithm (Lohse et al., 2020; De Witte et al., 2009), prior to tomographic reconstruction, which produced volumes of voxel size 550 nm and variable dimensions, as presented in Table 1. Only one volume was collected per sample.

Note that several separate beamline experiments were conducted to collect the volumes listed in Table 1. In the first two experiments, samples from the monkey brain were scanned at ESRF and DESY, respectively. The samples from the mouse brain were imaged in two subsequent experiments. Consequently, the location of the identified demyelinating lesion in the cuprizone mice, which cannot be precisely controlled, did not match the location of the CC biopsies in the monkey.

Due to the size of the data selected, processed volumes, masks and results are available at https://zenodo.org/records/10458911. Other datasets can be shared on request.

Finding spatial correspondences within and across image modalities was largely a manual process. For aligning DESY/ESRF images to MRI, photo documentation of tissue extraction sites was visually compared with the dMRI scan, and the best matching dMRI voxel or region was selected manually. For registering ESRF images to DESY scans, manual initialisation in ITK-SNAP was followed by rigid registration refinement, employing rigid transformation and mutual information-based image similarity. Stitching individual ESRF fields of view involved translation-only registration, where the approximate translation shift - known from the scan setup- was manually refined using the ITK-SNAP (RRID:SCR_002010).

For each synchrotron volume, we computed the 3D structure tensor for each voxel (Jeppesen et al., 2021; Khan et al., 2015). The processing involves three steps: (1) Computation of the image gradients, by filtering with the derivative of a Gaussian kernel, where the parameter σ is the standard deviation of the Gaussian; (2) Calculating the outer product of the gradient with itself, yielding a tensor in each image voxel; (3) Aggregation of tensor information within a local neighbourhood, by filtering with a Gaussian kernel i.e., also known as the patch size, where the parameter ρ is the standard deviation of the Gaussian.

Similar to the Diffusion Tensor in MRI (Basser et al., 1994), the eigendecomposition of the structure tensor defines a 3D ellipsoid, whose axes are scaled according to the eigenvalues λ1*, λ2* and λ3* (where λ1* > λ2* > λ3* and may be normalised to ensure that ∑λi∗=1), and their orientations are defined by the three orthogonal eigenvectors v1, v2, and v3. Throughout this paper, the estimated structure tensor decomposition is by default converted to a diffusion-like tensor, as the structure tensor and the diffusion tensor are ‘inverted’ to one another (Khan et al., 2015). We use the following single parameter (γ) model for converting the eigenvalues of the structure tensor:λi=e−λi∗/γ

Afterwards, the values are normalised to have ∑λi=1.

From the eigenvalues, the anisotropy of the structure tensor i.e., its shape is characterised by the same Fractional Anisotropy (FA) metrics as defined for the diffusion tensor (Basser et al., 1994) i.e.,FA=32((λ1−λ^)2+(λ2−λ^)2+(λ3−λ^)2λ12+λ22+λ32)

where λ^= (λ1+λ2+λ3)/3 is the mean of the three eigenvalues.

We introduce a scale-space structure tensor approach to ease parameter tuning and provide a scale-invariant analysis. The variation of axon diameters within some of the samples makes it challenging to capture all relevant structural orientation information with a single set of (σ,ρ)-parameters, i.e., for a single patch size. Therefore, we employed a ‘scale space structure tensor’ approach (Lindeberg, 1998). Here, the structure tensor is computed multiple times using a suite of varying (σ,ρ)-parameters called scales. Finally, for each voxel, we search for the dominating scale from which we retain the tensor. The criterion for selecting the dominant scale is based on the scale-wise relative maximum fractional anisotropy (FA) value. Let FAji be the calculated FA-value in voxel j at scale i using the parameter set (σ,ρ)ⁱ. The dominant scale index, d_j, for voxel j is then selected asdj=argmaxi(FAjimaxj(FAji)) ,j=1,2,...,N

In other words, at a given scale, the maximum observed FA value across all voxels, maxj(FAji), is used as a normalisation factor. With this approach, we select the structure tensor that provides the most anisotropic response across the set of manually defined scales. This ensures that the tensor-derived quantities—eigenvectors, eigenvalues, and FA—become independent of the axon diameter. Using the FA value for scale selection is reasonable, as we focus on the fibre-like axons. Appendix 1—figure 2 in the Appendix illustrates the scale-space structure tensor concept applied to the monkey CC ESRF data, where the FA of three selected scales of gradually decreasing patch sizes (scales 2, 6, and 8), are shown to emphasise distinctly different microstructural features. The dominant scale illustration shows how scales with large patch sizes are selected where the axons are large.Structure tensor-based quantifications.

The ST analysis, whether using the scale space or standard variant with a single scale, provides one tensor per voxel. Each sample is then quantified by the statistical distribution of FA values and the FOD based on the principal directions. The distributions are estimated from the combination of the following inclusion and exclusion regions of interest (ROIs).

The ROI segmentation masks were generated either manually using ITK-SNAP RRID:SCR_002010 or through intensity thresholding followed by morphological operations.

The statistical FA distributions were generated based on the selected voxels by making a kernel density estimation of the probability density function over the FA values.

The Fibre Orientation Distribution (FOD) was generated by binning the principal (unit) vectors of all selected voxels into a spherical histogram parametrized by azimuth and elevation angles. The poles for this representation are poorly defined and hard to visualise. Therefore, the polar direction is aligned along the anatomical axis where we expect the least directional contribution for the particular sample. Additionally, since the bins in the parameterisation do not have equal areas, the actual area is estimated and used to normalise the histogram, obtaining a fibre orientation based probability density function.

To characterise the FOD, we fitted a spherical Bingham distribution to the FOD using the mtex toolbox (https://mtex-toolbox.github.io/BinghamODFs.html). From the Bingham parameters, we derive two indices: the Orientation Dispersion Index (ODI), which describes the dispersion of the fibres on the surface of the unit sphere, and the Dispersion Anisotropy Index (DA), which expresses the anisotropic shape of the fibre dispersion (Tariq et al., 2016). Both metrics range from 0 to 1. A low ODI indicates a population of fibres with a narrow spread (a focused main direction), while a low DA suggests that the shape of the fibre dispersion is close to isotropic. It should be noted that the use of ODI and DA is only meaningful for a single pathway FOD.

The principle and the process of structure tensor-based tractography are much the same as in diffusion MRI-based tractography, but applied to the synchrotron volumes using the direction vectors of the structure tensor and the deterministic FACT algorithm (Tournier et al., 2019). The process is controlled by defining seeding point regions, masks for rejection, inclusion/termination, and various streamlined filtering parameters, such as minimum length. We utilise an a priori anatomical understanding of the sample and it’s the axonal organisation to manually generate the seeding region(s). For example, in the CC samples, the axons should primarily run in the L-R direction. We therefore create a mask at the left and right ends of the sample (with some margin from the sample edge). When seeding points in the left region of interest (ROI), streamlines must reach the right ROI to be included, and vice versa.

In several instances, we employed inclusion or rejection masks, which defined the permitted boundaries of travel for the streamlines. In essence, these masks should roughly represent axonal tissue segmentations. Depending on the sample, such a mask can be generated by thresholding either FA - or image intensity values, followed by morphological operations (opening, closing) to close holes and remove small spurious regions.