Data was acquired from a C57Bl6 wild-type mouse on a 11.7T Biospec 117/16 (Bruker, Germany) equipped with a mouse cryoprobe. We performed the dMRI acquisition using 2D-EPI with TR/TE=2500/40ms, δ=5ms, and Δ=(11, 20, 30)ms. For each Δ we acquired 10 different shells at gradient strengths equally distributed between 40 and 450mT/m at a total of 182 non-collinear directions per shell³. Half of these were acquired with blip from top to bottom of the head and half with the inverse blip. Each DW image consisted of six 0.40mm thick sagittal slices with a 24×14.4mm FOV at a 0.15×0.15mm resolution. We corrected eddy current distortions using FSL.

We fitted the attenuation $$$E^*$$$ of all DW images with our novel 4D continuous representation²

$$E_{\textbf{c}}(\textbf{q},\tau)=\sum_{\{jlm\}}^{N_{\textrm{max}}}\sum_{o=0}^{O_{\textrm{max}}}c_{jlmo}\,S_{jlm}(\textbf{q})T_{o}(\tau),\quad \tau=\Delta-\frac\delta 3,\quad\textbf{c}=\{c\}_{jlmo}\qquad(1)$$

generalising the MAP representation of multi-shell images⁴ to represent the DW attenuation at different diffusion times. For this, we solved the problem

$$\arg\min_{\textbf{c}}\left\|E^*(\textbf{q},\tau)-E_{\textbf{c}}(\textbf{q},\tau)\right\|^2+Reg(\textbf{c}),$$

regularised as in Fick et al² with $$$E^*$$$ the MRI-measured attenuation.

In Eq. (1), $$$T_{o}(\tau)$$$ is our temporal basis with order $$$o$$$ and $$$S_{jlm}(\textbf{q})$$$ is the 3D-SHORE basis with basis orders $$$jlm$$$. Here $$$N_{\textrm{max}}$$$ and $$$O_{\textrm{max}}$$$ are the maximum spatial and temporal order of the bases, which can be chosen independently. We formulate the bases themselves as

$$S_{jlm}(\textbf{q},u_s)=\sqrt{4\pi}i^{-l}(2\pi^2u_s^2q^2)^{l/2}e^{-2\pi^2u_s^2q^2}L_{j-1}^{l+1/2}(4\pi^2u_s^2q^2)Y_l^m(\textbf{u})$$

$$T_o(\tau, u_t)=\exp(-u_t\tau/2)L_o(u_t\tau)$$

where $$$u_s$$$ and $$$u_t$$$ are the spatial and temporal scaling factors. Here $$$\textbf{q}=q\textbf{u}$$$, $$$L_n^{(\alpha)}$$$ is a generalized Laguerre polynomial and $$$Y_l^m$$$ is the real spherical harmonics basis⁵. Here $$$j$$$, $$$l$$$ and $$$m$$$ are the radial order, angular order and angular moment of the MAP basis⁴ which are related as $$$2j+l=N+2$$$ with $$$N\in\{0,2,4\ldots N_{\textrm{max}}\}$$$.

A main methodological contribution is the estimation of the apparent axonal diameter distributions. Rather than choosing arbitrary direction perpendicular to the predominant axonal orientation $$$d$$$ to fit the AxCaliber model¹, we exploit the assumed axial symmetry in the signal and our model in Eq. 1. At each q‑value magnitude and diffusion time, we compute the perpendicular average as the circle integral at radius $$$|q|$$$ on the plane perpendicular to $$$\textbf{q}_\parallel$$$ which is aligned with $$$d$$$. For this, we used the Legendre polynomial $$$P_l$$$ to compute the analytical circle integral generalizing the q-ball estimation method of Descoteaux⁵:

$$E(\hat{\textbf{q}}_\perp,\tau)\triangleq\frac{1}{2\pi |\hat{\textbf{q}}_\perp|}\int_{\textbf{q}: |\textbf{q}|=|\hat{\textbf{q}}_\perp|,\,\textbf{q}\cdot\textbf{q}_\parallel=0} E(\textbf{q},\tau)d\textbf{q}=\sum_{\{jlm\}}^{N_{\textrm{max}}}\sum_{o=0}^{O_{\textrm{max}}}c_{jlmo}P_l(0)S_{jlm}(\textbf{q}_\parallel,u_s)T_o(\tau, u_t),\quad|\textbf{q}_\parallel|=|\hat{\textbf{q}}_\perp|$$

Once the averaged perpendicular attenuation for the CC voxels of the mouse brain was computed, we fitted the AxCaliber Model to $$$E(\hat{\textbf{q}}_\perp,\tau)$$$ extracting the parameters of the Γ-distributed apparent axonal diameter and the apparent axonal volume fraction.

Our main result is the estimation of average apparent axonal diameters in the Genu of the mouse CC, which we illustrate in Fig 1. To reach this result we first analysed the quality of fitting of our 3D+t model to the distortion-corrected dMRI signals. For this, we computed the $$$r^2$$$ at each voxel. As shown in Figure 2, the majority of the voxels had $$$r^2\geq 0.8 $$$ with few below $$$0.4$$$. This denotes a general good fit of our model to dMRI signal.

Then we computed for each voxel the AxCaliber model, which resulted in the parameters of the diameter-$$$\Gamma$$$ distribution: shape $$$\alpha=0.66\pm 0.09\mu m$$$, dimensionless scale $$$\beta=1.67\pm 0.30$$$, and the dimensionless volume fraction corresponding to the intra-axonal compartment $$$f=0.83\pm 0.08$$$, shown in Figure 3. Finally we computed the histogram of mean apparent axonal diameters: $$$1.09\pm 0.04\mu m$$$, shown in Figure 4.

In this work we used, for the first time, our 3D+t dMRI model² to fit in-vivo dMRI data acquired at different gradient strengths, directions and diffusion times. Then, we exploited the analytical formulation of our model and the symmetry of the diffusion signal to average the attenuation perpendicular to the predominant axonal direction at each voxel of the genu of a mouse's CC and fitted the AxCaliber model to this signal. Although there was previous attempt using CHARMED⁷, we have shown that our method fits the attenuation better² and our approach can compute the averaged perpendicular attenuation analytically. Overall, our model allowed us to fit the AxCaliber model without the requirement of prior knowledge of predominant axonal directions and taking advantage of the dMRI signal symmetry.

In exploiting our novel 3D+t model to fit AxCaliber, we obtained parameters for the apparent axonal diameter Γ distribution, average apparent axonal diameter and intra-axonal volume fraction similar to those previously reported⁶. Although further validation is needed, our first experiment shows the potential of our novel 3D+t model. Summarising, our novel model shows potential to be a cornerstone in reducing the requirements of dMRI acquisitions in angular, q-space and diffusion time coordinates to perform dMRI-based white matter microscopy.