We aim to increase the scanner’s sensitivity to small axon diameters by generalizing signal attenuation inside the cylinder from the transverse direction to the geocentric direction in the longitudinal plane of the axons.

Given Neuman’s signal attenuation model for water trapped inside cylinder and ActiveAx orientationally invariant model, we generalize the signal attenuation (S_c) to account for geocentric rotation in the longitudinal plane to axons:$$-\ln S_{c}(\theta,\mathbf{g},\Delta,\delta)=\begin{cases}(\Delta-\frac{\delta}{3})\gamma^2 \delta^2 \mathbf{|g|}^2 D & \theta = \mathbf{g}_{||}\\ \\ 2\gamma^2\mathbf{|g|}^2\sum_{m=1}^\infty\frac{2D\alpha_m^2\delta-2+2e^{-D\alpha_m^2\delta}+2e^{-D\alpha_m^2\Delta}-e^{-D\alpha_m^2(\Delta-\delta)}-e^{-D\alpha_m^2(\Delta+\delta)}}{D^2\alpha_m^6(\frac{1}{4}\alpha_m^2A\csc(\theta)-1)} & \mathbf{g_{||}}<\theta<\mathbf{g_{\perp}}\end{cases}$$For a given θ, apparent axon diameter defines as:

$$A_{app}=A\csc(\theta)$$

Prior to the model fitting we searched for a θ, in which the apparent axon diameter falls above minimum resolvable axon diameter of the scanner (A_s):

$$ \theta':A_s<A_{app}<\sqrt{2D(\Delta-\delta/3)}\\ A_s:S_c(\theta=\pi/2,\mathbf{g},\Delta,\delta)=1-\frac{1}{SNR}$$

Then, by fitting diffusion-weighted measurement to the generalized model above, the apparent geocentric axon diameter can be obtained and axon diameter derived (Figure 1).

First, we simulated transverse Sc for a range of axon diameters as a function of maximum gradient strength to obtain minimum resolvable axon diameter (A_s). Four types of scanners previously used for axon diameter mapping^4-6 were considered: two human scanners with maximum gradient strengths of 60 and 300mT/m and two animal scanners with maximum gradient strengths of 300 and 1350mT/m.

Second, for the human scanner with 300mT/m, we simulated S_c as a function of θ for three different axon diameter values (1, 3 and 5μm) to identify the minimum θ for which the axon diameter can be resolved.

Third, we tested the proposed method on a post-mortem sample of human corpus callosum, scanned with maximum gradient strength of 300mT/m. We acquired 5 non-diffusion-weighted images and 60 diffusion-weighted directions for each of five shells with the following b-values: 1K, 3K, 4K, 8K, and 12Ks/mm², spatial resolution of 150μm isotropic. To avoid the possible bias of non-Gaussian diffusion in extra-axonal space⁷, only the two largest b-values were used in this study. To ameliorate the bias from axonal dispersion, only voxels with high anisotropy (FA>0.7) were used. Signal values were fitted to a three-compartment model (intra-axonal, extra-axonal and CSF) using dictionary-based approach as explained in⁶. Axonal diffusion was fixed to 1.6μm²/ms ²; extra-axonal diffusion was fixed to 0.6 μm²/ms ⁷, given the high b-value range employed.

None of the simulated transverse S_c models were able to resolve axon diameter of <2μm (Figure 2). With a liberal SNR of 5, the A_s of each scanner were around: 10μm, 5μm and 3μm for gradient strengths of 60mT/m, 300mT/m and 1350mT/m, respectively.

Figure 3 shows that, with the proposed model, an axon diameter of 1.5μm can be resolved with θ=20° using a scanner with maximum gradient strength of 300mT/m. A range of θs in which apparent axon diameter is higher than scanner’s minimum resolvable value are plotted in Figure 4.

Axon diameter of 1.92μm was obtained from the human corpus callosum tissue from the acquisition with θ of ≈20°. This value is close to the expected range based on previous histology studies⁸. Apparent axon diameter was 5.61μm, higher than scanner’s minimum resolvable axon diameter of at least 5μm.

Here we generalized Neuman’s signal attenuation model of transverse water displacement inside cylinders to the geocentric direction of the longitudinal plane of the axons (θ). The proposed model increases sensitivity to small axons for most scanners. Model assumptions remains the main limitation of a study of this kind, particularly, as we do not consider axon dispersion, which is expected to have extra influence apparent axon diameters for θ close to the axon direction. It should be noted that in the geocentric directions of the longitudinal plane data-driven axon diameter mapping might become possible, demanding further investigations.