Along-axon diameter variation and axonal orientation dispersion revealed with 3D electron microscopy: implications for quantifying brain white matter microstructure with histology and diffusion MRI

Abstract

Tissue microstructure modeling of diffusion MRI signal is an active research area striving to bridge the gap between macroscopic MRI resolution and cellular-level tissue architecture. Such modeling in neuronal tissue relies on a number of assumptions about the microstructural features of axonal fiber bundles, such as the axonal shape (e.g., perfect cylinders) and the fiber orientation dispersion. However, these assumptions have not yet been validated by sufficiently high-resolution 3-dimensional histology. Here, we reconstructed sequential scanning electron microscopy images in mouse brain corpus callosum, and introduced a random-walker (RaW)-based algorithm to rapidly segment individual intra-axonal spaces and myelin sheaths of myelinated axons. Confirmed by a segmentation based on human annotations initiated with conventional machine-learning-based carving, our semi-automatic algorithm is reliable and less time-consuming. Based on the segmentation, we calculated MRI-relevant estimates of size-related parameters (inner axonal diameter, its distribution, along-axon variation, and myelin g-ratio), and orientation-related parameters (fiber orientation distribution and its rotational invariants; dispersion angle). The reported dispersion angle is consistent with previous 2-dimensional histology studies and diffusion MRI measurements, while the reported diameter exceeds those in other mouse brain studies. Furthermore, we calculated how these quantities would evolve in actual diffusion MRI experiments as a function of diffusion time, thereby providing a coarse-graining window on the microstructure, and showed that the orientation-related metrics have negligible diffusion time-dependence over clinical and pre-clinical diffusion time ranges. However, the MRI-measured inner axonal diameters, dominated by the widest cross sections, effectively decrease with diffusion time by ~ 17% due to the coarse-graining over axonal caliber variations. Furthermore, our 3d measurement showed that there is significant variation of the diameter along the axon. Hence, fiber orientation dispersion estimated from MRI should be relatively stable, while the "apparent" inner axonal diameters are sensitive to experimental settings, and cannot be modeled by perfectly cylindrical axons.

Keywords: 3d axon segmentation; 3d electron microscopy; Axonal diameter distribution; Axonal diameter variation; Corpus callosum; Diffusion coarse-graining; Diffusion time-dependence; Fiber orientation distribution; g-Ratio.

Fig. 1.

The semi-automatic IAS segmentation pipeline. (a) A tissue sample of genu in CC, in a volume of 36 × 48 × 20 μm³, was acquired by sequential SEM. (b) The myelin mask (red) was obtained by using a pixel-wise classifier for further segmentation of the intra-axonal space (IAS). (c) Seeds (red dots) for random diffusion grid-hopping process were assigned manually over one central slice (451 seeds). The random-hopping trajectory was bounded by the myelin mask in (b). (d) IAS (colors) was filled by all random-hopping trajectories (321 segmented IAS). The IAS from axons with leaky myelin mask has been excluded by proofreading. (e) The individual myelin sheath (colors) is the overlap of the myelin mask and the expanded IAS dilated by ≤ 0.4 μm. Touching myelin sheaths of adjacent axons are separated based on a non-weighted watershed algorithm. (f-g) By transforming each segmented IAS and individual myelin sheath into polyhedrons, it is feasible to perform numerical simulations in such 3d realistic microstructure

Fig. 2.

IAS segmented (a) manually with the initialization facilitated by the ilastik Carving function (blue pixels), (b) by using RaW (red pixels), and (c) the intersection of both methods (yellow pixels) in a representing slice. The histogram of the (d) Jaccard index and the (e) Sørensen-Dice index for the comparison of IAS segmentations from the two methods. The scale bar in (a-c) is 4 μm

Fig. 3.

The inner axonal diameter (2r) variation, estimated by the cross-sectional area perpendicular to the skeleton and displayed along the main direction of each axon (z_axon), was smoothed by a Gaussian kernel mimicking the diffusion process with an effective diffusion time t = (a) 1 ms. (b) 10 ms, and (c) 100 ms. (d) The diameter histogram becomes narrower with longer diffusion time. (e) The average diameter 2⟨r⟩ has no significant time-dependence, whereas the dMRI-sensitive effective diameter 2r_eff, where $r_{\text{eff}}^4=\langle r^6\rangle ∕\langle r^2\rangle$ (Burcaw et al. 2015), has a non-trivial time-dependence for diffusion time t < 50 ms

Fig. 4.

The histogram of (a) outer axonal diameter, (b) inner axonal diameter, and (c) genuine g-ratio. The relation of g-ratio and inner diameter is shown in (d) as a 2d histogram, fitted by the log-linear equation (red curve) proposed by (Berthold et al. 1983). The histogram of (e) coefficient of variation (CV) of outer axonal diameter and (f) CV of inner axonal diameter show that axons are not perfect cylinders of CV(diameter) = 0.

Fig. 5.

(a) The skeleton of each segmented axon is smoothed to mimic the diffusion time-dependent coarse-grained microstructure along each axon’s main direction with diffusion time t = [1, 10, 100] ms. (b) The skeleton of each segmented axon in (a) was viewed from another view angle. Each axon becomes effectively straighter for longer diffusion times. (c) The FOD of tangent vectors of all axon segments, starting at the center of a unit sphere, shows the intrinsic axonal dispersion. The unit of the colorbar is steradian⁻¹. (d) The 3d FOD glyph was generated by fitting the FOD in (c) to spherical harmonics up to the order of l = 10. Arrows in (c) indicate the view angle for FOD glyphs in (d)

Fig. 6.

(a) Projected dispersion angle θ_2d(ϕ) and (b) dMRI-sensitive dispersion angle θ_eff(ϕ) (Eq. (1)), calculated within a bin width Δϕ = 12°, with respect to the azimuthal angle ϕ at diffusion time t = [1, 10, 100] ms. (c) The rotational invariants (P₂, P₄, P₆ in Eq. (3)) show a small time-dependence for diffusion time t = 20-100 ms. (d) The dispersion angle averaged over all ϕ shows a time-dependence of ≈ 1.7° for diffusion time t = 1-100 ms. (e) Rotational invariants p_l with respect to the even orders l = 2, 4, …, 10 at diffusion time t = [1, 10, 100] ms. (f) Dispersion parameters of the modified power-law relation (λ, C in Eq. (5)) obtained by using a linear fit of log p_l with respect to l = 2-10 for diffusion time t = 1-100 ms

Fig. 7.

The artificial upper bound applied for myelin sheath segmentation influences the estimated mean g-ratio. A small upper bound for the myelin thickness leads to under-segmented individual myelin sheaths (top left, upper bound = 0.1 μm). In contrast, a large upper bound causes over-expanded individual myelin sheaths (top right, upper bound = 0.6 μm). In this study, an upper bound of 0.3-0.4 μm results in appropriate individual myelin sheaths (top middle, upper bound = 0.4 μm).

Fig. 8.

The distribution of axonal diameters, defined by (a) equivalent circle diameter calculated from the cross-sectional area, (b) short axis length and (c) long axis length of the fitted ellipse, and (d) inscribed circle diameter. The upper row shows an exemplified axon cross-section (gray area) and the corresponding diameter estimates (double-arrowed lines). The middle row shows experimental diameter distributions (gray bars) and the fits based on the Gamma distribution (red) and the generalized extreme value distribution (GEV) (blue). The bottom row is the middle row displayed in a semi-logarithmic scale for experimental data (data points) and the fits (solid lines)

Fig. 9.

(a) Considering a fiber bundle with its main direction aligned to the z-axis, the 3d dispersion angle θ_3d is defined by the fiber segment (black) orienting into (θ_i, ϕ_i) in 3d space, and the 2d dispersion angle θ_2d is defined by the fiber segment projection (red) on a 2d plane (e.g., x-z plane) with a 2d projection angle ${\theta }_i^{\prime }$. (b) The 3d dispersion angle (e.g., θ_ρ2 in Eq. (4)) is larger than the 2d dispersion angle as in Fig. 6d. The prediction of 2d dispersion angle based on FOD’s rotational invariants up to the order l = 20, Eq. (C3) (red solid line), has a 3% error. Similarly, the prediction based on the 3d dispersion angle (e.g., θ_p2), Eq. (6) (blue dash-dotted line), has a 7% error. These errors are potentially caused by the axial asymmetry in our FOD, as shown in Fig. 5.