Synopsis

Diffusion inside axons is restricted and thus non-Gaussian, with diffusion MRI (dMRI) signal strongly sensitive to the shape of the confining axon. This sensitivity is confounded by the coarse-graining of the diameter/shape variation along the fiber during the diffusion time. Here, we analytically relate dMRI metrics to the axonal shape, and validate our theory using 3d Monte-Carlo simulations in beaded cylinders and realistic axons reconstructed from electron microscopy images of the mouse brain white matter. Our simulation results show that the intra-axonal space has a non-trivial kurtosis transverse to axons. Its value is different from that in a perfectly straight cylinder, and needs to be considered in axonal diameter measurements (e.g., spinal cord, strong gradients, intra-axonal metabolites).

Purpose

Methods

$$$\rm\bf{Theory}$$$

Our starting point is the diffusion-diffraction long-time limit of the propagator for a 2d confined pore[13,14],

$$G(q_\perp,t\to\infty)\simeq|V(q_\perp)|^2\,,\qquad(1)$$

where $$$V(q_\perp)$$$ is the Fourier transform of the pore shape. When the diffusion can occur in the 3rd dimension, this equation must be modified:

$$G(q_\perp,t\to\infty)\simeq|\langle{V(q_\perp)}\rangle|^2\,,\qquad(2)$$

where $$$\langle…\rangle$$$ is the coherent averaging over cross-sections within a coarse-graining window $$$L(t)\sim\sqrt{D_\parallel t}$$$ along the axon.

For a cylinder of varying radius, Fig.1, the result of this average is

$$\langle{V(q_\perp)}\rangle=1-\tfrac{1}{8}q_\perp^2\langle{r^2}\rangle_{\rm{v}}+\tfrac{1}{192}q_\perp^4\langle{r^4}\rangle_{\rm{v}}+…\,,\qquad(3)$$

where $$$V(q_\perp)$$$ is the Fourier transform of disc shape, and $$$\langle\cdot\rangle_{\rm{v}}$$$ denotes volume-weighted quantity for the disc volume $$$\propto{r^2}$$$. In Eq.(3), $$$V(q_\perp)$$$ is averaged first along the fiber, before calculating the propagator. This coherent averaging/coarse-graining effect within a fiber should be followed by the incoherent averaging over an ensemble of non-communicating fibers with different shapes.

Eqs.(2-3) yield for the radial diffusivity

$$ D_\perp(t)={r_{\rm eff}^2}/4t\,,\quad r_{\rm{eff}}^2\equiv\langle{r^4}\rangle/\langle{r^2}\rangle\,,\quad\quad(4)$$

generalizing the concept of effective radius $$$r_{\rm eff}$$$[15,16] onto variable axon shape.

For the radial kurtosis (RK) in the $$$t\to\infty$$$ limit, we obtain

$$K_\infty=\frac{\langle{r^4}\rangle_{\rm{v}}}{\langle{r^2}\rangle_{\rm{v}}^2}-\frac{3}{2}=\frac{\langle{r^6}\rangle\langle{r^2}\rangle}{\langle{r^4}\rangle^2}-\frac{3}{2}\,.\quad\quad(5)$$

$$$\rm\bf{Cylinders\,with\,periodic\,beads}$$$

To test the applicability of Eqs.(4,5), we designed cylinders with periodic beads, resulted by the radius variation along the z-axis,

$$r_b(z)=r_0+r_1\cos\left(\tfrac{2\pi{z}}{a}\right)\,,$$

where $$$a$$$ is the distance between beads, and $$$r_0$$$ and $$$r_1$$$ are parameters to tune the mean cross-section-area, $$$V/L$$$, and coefficient of variation of radii, CV($$$r$$$)$$$\,\equiv\frac{\sigma_r}{\bar{r}}$$$:

$$\tfrac{V}{L}=r_0^2+\tfrac{1}{2}r_1^2\,,\quad\quad{\rm{CV}}(r_b)=\frac{r_1}{\sqrt{2}r_0}\,.$$

Based on histological observations in the previous study[7], we fixed $$$a=5.4\,$$$µm and $$$V/L=\pi$$$×(1µm)$$$^2$$$, and varied CV($$$r$$$) from 0 (no beads) to 0.5 (big beads, Fig.1).

$$$\rm\bf{Realistic\,microstructure}$$$

The brain tissue from a female 8-week-old C57BL/6 mouse’s genu of CC was processed and analyzed with an SEM (Zeiss Gemini 300), in a volume of 36×48×20μm$$$^3$$$. We employed a simplified seeded-region-growing algorithm[7,17] to segment IAS, leading to 227 long axons (≥20μm in length). The IAS mask was down-sampled into a resolution of (0.1μm)$$$^3$$$ and aligned to the z-axis, as in Fig.2.

$$$\rm\bf{Numerical\,simulation}$$$

MC simulations of random walkers were implemented in MATLAB in continuous space within the 3d micro-geometry. For simulations in cylinders with periodic beads, 1×10$$$^5$$$ random walkers per cylinder diffuse over 2.5×10$$$^5$$$ steps with a duration 2×10$$$^{-4}$$$ms and a length 0.049µm. For simulations in realistic IAS, 1.2×10$$$^7$$$ random walkers in total diffuse over 5×10$$$^5$$$ steps with a duration 2×10$$$^{-4}$$$ms and a length 0.049µm. Intrinsic diffusivity $$$D_0$$$=2µm$$$^2$$$/ms. Total calculation time ~4.5days on 200 CPU cores at the NYU high-performance-computing cluster.

Diffusivity and kurtosis were estimated based on cumulants ($$$\langle{x^2}\rangle$$$,$$$\langle{x^4}\rangle$$$).

Results

For simulations in cylinders with periodic beads, simulated time-dependent RD, $$$D_\perp(t)$$$ in Fig.3a, scales as $$$1/t$$$ as discussed in Theory, and simulated RK, $$$K_\perp(t)$$$ in Fig.3b, is constant over diffusion times>5ms. The effective radius $$$r_{\rm{eff}}$$$ fitted based on simulations in Fig.3a is consistent with the theoretical predictions given by Eq.(4), as shown in Fig.3c. Similarly, RK in $$$t\to\infty$$$ limit, $$$K_\infty$$$, estimated based on simulations in Fig.3b agrees with the theoretical predictions given by Eq.(5), as shown in Fig.3d. In Fig.3e-f, $$$r_{\rm{eff}}$$$ and $$$K_\infty$$$ grows with the strength of beading, CV($$$r$$$), except for $$$K_\infty$$$ when CV($$$r$$$)>0.4.

For simulations in realistic IAS, simulated RD, $$$D_\perp(t)$$$ in Fig.4a, scales as $$$1/t$$$, and simulated RK, $$$K_\perp(t)$$$ in Fig.4b, gradually approaches to a constant at long times. However, the effective radius $$$r_{\rm{eff}}$$$ fitted based on simulations in Fig.4a is slightly larger than the theoretical predictions in Eq.(4), as shown in Fig.4c, probably due to the fiber undulations[18]. The RK in $$$t\to\infty$$$ limit, $$$K_\infty$$$, is centered around -0.2 (Fig.4d), different from the case of the perfect cylinder = -0.5.

Discussion and Conclusions

Acknowledgements

References

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