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.

Appendices

Appendix A: The myelin thickness upper bound used for the segmentation could influence g-ratio estimations

The estimated g-ratio values could be influenced by the myelin thickness upper bound used for dilating the segmented IAS to further segment the myelin sheath (“Myelin sheath and g-ratio”, “Materials and methods”). In Fig. 7, six different upper bound values, varying from 0.1 to 0.6 µm, are used to segment the myelin sheath and calculate the mean g-ratio, ranging from 0.77 to 0.55, indicating that the upper bound has to be carefully chosen for an accurate g-ratio estimation. A small upper bound (e.g., Fig. 7, upper left) could lead to a underestimated myelin thickness and an overestimated g-ratio; in contrast, a large upper bound (e.g., Fig. 7, upper right) could lead to an overestimated myelin thickness and a underestimated g-ratio.

Fig. 7

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)

Appendix B: Axonal diameter estimates per various definitions

In this section, distributions of axonal diameters are shown based on different definitions, such as equivalent circle diameter (Fig. 8a), short and long axis length of the fitted ellipse (Fig. 8b, c), and inscribed circle diameter (Fig. 8d). To compare with a previous study (Sepehrband et al. 2016a), we fitted the axonal diameter histogram to Gamma distribution and generalized extreme value (GEV) distribution in Fig. 8, which shows that GEV distribution fits better to the experimental diameter distribution (of all four definitions) than Gamma distribution does, consistent with the conclusion in (Sepehrband et al. 2016a). Also, GEV distribution has a longer tail than Gamma distribution does for thick axons in diameters > 3–5 µm, manifested by semi-logarithmic plots of diameter distributions in the bottom row of Fig. 8.

Fig. 8

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)

Appendix C: Relations between 2d and 3d dispersion angles

The purpose of this Appendix is to relate the 2d dispersion angle ${\theta _{2d}}$ derived from 2d histology (using, e.g., structure tensor) to the 3d dispersion angle ${\theta _{3d}}$ (defined, e.g., as ${\theta _{{\text{eff}}}}$ or ${\theta _{{p_2}}}$ in the main text). It is quite obvious that, generally, ${\theta _{2d}} \leq {\theta _{3d}}$, since the projection onto a plane removes part of the orientational variance (in the direction transverse to that plane). Here we address this relation quantitatively, and also estimate the 2d dispersion angle in terms of the 3-dimensional FOD’s SH coefficients and rotational invariants.

Aligning the z-axis with the main direction of a fiber bundle, the $i$-th fiber segment is defined by the polar and azimuthal angles $\left( {{\theta _i},{\phi _i}} \right)$. Its 2d projection angle $\theta _{i}^{'}=\theta _{i}^{'}({\theta _i},{\phi _i})$ within a plane (e.g., x–z plane in Fig. 9a) parallel to the main direction (z-axis), can be determined as

$$\cos \theta _{i}^{'}=\frac{z}{{\sqrt {{x^2}+{z^2}} }}=\frac{{\cos {\theta _i}}}{{\sqrt {{{\sin }^2}{\theta _i}{{\cos }^2}{\phi _i}+{{\cos }^2}{\theta _i}} }}~.$$

We now define the 2d dispersion angle. One can average the above $\theta _{i}^{'}$ over the FOD,

$$\theta _{{2d}}^{2} \equiv \left\langle {\theta _{i}^{{'2}} } \right\rangle ,~$$

such that

$$\cos ^{2} \theta _{{2d}} = \cos ^{2} \sqrt {\left\langle {\theta _{i}^{{'2}} } \right\rangle } \simeq 1 - \left\langle {\theta _{i}^{{'2}} } \right\rangle ~ \simeq \left\langle {\cos ^{2} \theta _{i}^{'} ~} \right\rangle ,$$

or, alternatively, adopt the above Taylor approximation as a definition, since it is actually more natural to average $\cos ^{2} \theta _{i}^{'}$ rather than the angle itself, as $\cos ^{2} \theta _{i}^{'}$ corresponds to the structure tensor component.

Fig. 9

a Considering a fiber bundle with its main direction aligned to the z-axis, the 3d dispersion angle ${\theta _{3d}}$ is defined by the fiber segment (black) orienting into $\left( {{\theta _i},{\phi _i}} \right)$ in 3d space, and the 2d dispersion angle ${\theta _{2d}}$ is defined by the fiber segment projection (red) on a 2d plane (e.g., x–z plane) with a 2d projection angle $\theta _{i}^{'}$. b The 3d dispersion angle (e.g., ${\theta _{{p_2}}}$ in Eqs. 4a, 4b) 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. (9) (red solid line), has a 3% error. Similarly, the prediction based on the 3d dispersion angle (e.g., ${\theta _{{p_2}}}$), 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

The FOD average will be performed in two steps. First, we average ${\cos ^2}\theta _{i}^{'}$ over the azimuthal angle ${\phi _i}$. This can be explicitly done if the FOD is axially symmetric. We also note that random histological sampling performed on a sufficiently large scale effectively performs such azimuthal averaging. The azimuthal averaging can be performed exactly for any fixed ${\theta _i}$:

$$\langle{\cos ^2}\theta^{'}_i\rangle_{{{\phi _i}}} \equiv \mathop \int \limits_{0}^{{2\pi }} {\cos ^2}\theta _{i}^{'}~\frac{{d{\phi _i}}}{{2\pi }}~=\left| {\cos {\theta _i}} \right|~.~$$

(7)

Hence, we explicitly see that the azimuthal 2d dispersion variance is given by the first power of $\cos {\theta _i}$, which is greater than the 3d variance ${\cos ^2}{\theta _i}$, corresponding to a narrower 2d dispersion, ${\theta _{2d}} \leq {\theta _{3d}}$, cf. Equation (6) of the main text.

At the second step, we average Eq. (7) over the remaining polar angle ${\theta _i}$ to obtain ${\cos ^2}{\theta _{2d}}=\left\langle {|\cos {\theta _i}|} \right\rangle$. We can already see that for narrow FODs, Taylor-expanding up to $\left\langle {\theta _{i}^{2}} \right\rangle \simeq \theta _{{3d}}^{2}$, we find ${\theta _{2d}} \simeq {\theta _{3d}}/\sqrt 2$, which is just a statement that the variance of the axial radius $\langle{x^2}+{y^2}\rangle=2\langle{x^2}\rangle$ is given by twice the variance of its x- or y-coordinate. Note, however, that this approximation ceases to be correct for the higher orders of $\theta$, essentially because of the non-trivial denominator $\sqrt {{x^2}+{z^2}}$ in the definition of $\cos \theta _{i}^{'}$, as opposed to $\sqrt {{x^2}+{y^2}+{z^2}} =1$.

To estimate the 2d dispersion angle based on FOD’s SH coefficients ${p_{lm}}$ in Eq. (2), we average the right-hand side of Eq. (7), where only the $m$ = 0 SH contribute due to the axial symmetry. Using Eq. (2) and ${Y_{l0}}\left( \theta \right)=\sqrt {\frac{{2l+1}}{{4\pi }}} {P_l}\left( {\cos \theta } \right)$, where ${P_l}\left( {\cos \theta } \right)$ are the Legendre polynomials, we obtain

$$\begin{aligned} {\cos ^2}{\theta _{2d}} & \simeq \langle{\cos ^2}\theta _{i}^{'}\rangle=\mathop \int \limits_{{ - 1}}^{1} \left| {\cos {\theta _i}} \right|\mathcal{P}\left( {\hat {{\varvec{n}}}} \right) \cdot \frac{1}{2}d\left( {\cos {\theta _i}} \right) \\ & =\frac{1}{2}+\mathop \sum \limits_{{l=2,4, \ldots }}^{\infty } {p_{l0}} \cdot \sqrt {\frac{{2l+1}}{{4\pi }}} \cdot \mathop \int \limits_{0}^{1} z~{P_l}\left( z \right)~{\text{d}}z~,~~ \\ \end{aligned}$$

(8)

where the integral

$$\mathop \int \limits_{0}^{1} z~{P_l}\left( z \right)~{\text{d}}z=\frac{{{{\left( { - 1} \right)}^{l/2+1}}~}}{{\left( {l - 1} \right)\left( {l+2} \right)}} \cdot \frac{{\left( {l - 1} \right)!!}}{{l!!}}~,~~~~~l=2,4, \ldots ~$$

can be evaluated using the generating function of Legendre polynomials $\frac{1}{{\sqrt {1 - 2tz+{t^2}} }}=\mathop \sum \nolimits_{l} {P_l}\left( z \right){t^l}$, such that $\mathop \int \nolimits_{0}^{1} \frac{z}{{\sqrt {1 - 2tz+{t^2}} }}~{{\rm d}}z=~\mathop \int \nolimits_{{ - \infty }}^{\infty } \frac{{{{\rm d}}\lambda }}{{\sqrt \pi }}~\mathop \int \nolimits_{0}^{1} z{{\rm d}}z~{{{\rm e}}^{ - {\lambda ^2}(1 - 2tz+{t^2})}}$ and the subsequent integral is reduced to a few Euler’s Gamma functions.

Finally, we use the definition in Eq. (3) to express ${p_{l0}}$ via the rotational invariants ${p_l}$, when the other $m$ ≠ 0 FOD harmonics are either zero (axial symmetry) or negligible. As a result, we find

$${\cos ^2}{\theta _{2d}} \simeq \frac{1}{2}+\mathop \sum \limits_{{l=2,4, \ldots }}^{\infty } {p_l} \cdot \frac{{{{\left( { - 1} \right)}^{l/2+1}}~(2l+1)}}{{\left( {l - 1} \right)\left( {l+2} \right)}} \cdot \frac{{\left( {l - 1} \right)!!}}{{l!!}}~.~$$

(9)

It is important to note that the 2d dispersion angle ${\theta _{2d}}$ appears to depend on the SH and rotational invariants ${p_l}$ with all $l$, in contrast to the 3d dispersion angle ${\cos ^2}{\theta _{3d}}=(2{p_2}+1)/3$ in Eqs. (4a, 4b), which only involves the irreducible representation of the SO(3) group of rotations with the weight $l$ = 2. It is quite obvious that the 3d definition of the dispersion angle is more natural (after all, the FOD is a 3-dimensional object), and mathematically, it is a better quantity, since it only depends on the $l$ = 2 invariant, and does not mix the irreducible representations of SO(3). The above equation gives the precise way to compare 2d and 3d FOD estimates.

Figure 9b shows that the predicted 2d dispersion angle based on rotational invariants and Eq. (9) is close to the value calculated by projecting fiber segments on 2d planes, with only 3% error due to a lack of perfect axial symmetry of our FOD.

We note that Eq. (7) derived above has an interesting geometric meaning. Its right-hand side, averaged over the FOD, provides the cosine transform of the FOD [cf. Eq.(8) above], with respect to the z-axis. We now recall that the cosine transform of the FOD, multiplied by the length-to-volume ratio of all fiber segments, equals to the number density of intersections of the fibers by the x–y plane. This statement is intuitively obvious (fiber length projected onto the z-axis contributes to the intersection); it is rigorously discussed for the point processes, e.g., in Eq. (4.40) of the book (Schneider and Weil 2008). This statement is valid in any basis, and serves as a foundation for constructing the Steiner compact, which is an equivalent FOD representation. In other words, the FOD, whose direct estimation relies on calculating the gradients to determine the local directions, through the cosine transform is related to a simpler, counting problem, of calculating the numbers of intersections by all possible planes. Going back from the Steiner compact to the FOD by inverting the cosine transform is essentially rederived above, cf. the right-hand side of Eq. (9). Remarkably, we can see that the left-hand side of Eq. (7) relates the Steiner compact to the structure tensor calculation in the x–z plane. This means that the angular-averaged 2d structure tensor calculation (in any plane parallel to z) can also be performed as a counting problem, by counting the density of fiber intersections by the plane orthogonal to z.