To quantify microstructural parameters in a fiber geometry using PGSE with a finite pulse width. The finite pulse duration $\delta$ is typically a source of bias for quantifying microstructure[1]. Here, we use the diffusivity dependence $D(\Delta,\delta)$ on $\delta$ in order to reveal the correlation length $l_c^\perp$ of the fiber packing, an essential μm–level characteristic of microstructure, thereby turning the finite pulse duration to our advantage. We validate our method in a fiber phantom[2] that mimics an axonal packing geometry. Specifically, we estimate phantom parameters from measuring $D(\Delta,\delta)$ by varying $\delta$ at fixed $\Delta$, and predict the measurement outcome with varying $\Delta$ at fixed $\delta$ without adjustable parameters.

Theory. The instantaneous diffusion coefficient $D_{\rm{inst}}(\Delta)=D_∞+A/\Delta^ϑ$ approaches its bulk value $D_∞$ according to a power law with dynamical exponent $ϑ$, related to the disorder class and the spatial dimensionality $d$ [2,3]; both $D_{\rm{inst}}(\Delta)$ and the corresponding $D(\Delta)$ decrease with $\Delta$ for $\Delta\gg{t_c}$, where $t_c$ is the time to diffuse across the packing correlation length $l_c=\sqrt{2dD_∞t_c}$ [2]. How can one measure $t_c$ if it does not explicitly enter the known power-law tails in $D_{\rm{inst}}(\Delta)$? Here we introduce a soft cut-off at $\Delta\sim{t_c}$ for the tail in $D_{\rm{inst}}(\Delta)$, thereby incorporating the dependence on $t_c$. This enables the interpretation of our measurements at $\Delta,\,\delta\sim{t_c}$, and thus quantifying $l_c$.

The effect of $t_c$ and $\delta$ can be evaluated via a combined low-pass filter $$F(ω)=F_\delta(ω)\cdot\,F_{t_c}(ω),$$ where $$F_\delta(ω)=\left(\frac{sin(ω\Delta/2)}{ω/2}\right)^2\left(\frac{sin(ωδ/2)}{ω/2}\right)^2,\,\,F_{t_c}(ω)=\left(\frac{sin(ωt_c/2)}{ωt_c/2}\right)^2,$$ applied to the velocity autocorrelation function $D(ω)=D_∞-Aϑ\cdot\,Γ(ϑ)\cdot(-iω)^ϑ$ in the frequency domain. The filter $F_\delta$ accounts for finite pulse width[4,5], while the newly introduced filter $F_{t_c}$ describes the behavior of $D_{\rm{inst}}(\Delta)$ for all $\delta\sim{t_c}$. The integration of $D(ω)$ with $F(ω)$ gives Eqs.(1,2) in Fig.1. We apply our general result to $D_\perp(\Delta,\delta)$ measured perpendicular to fibers, for which the exponent $ϑ$=1 reflects short-range disorder of restrictions in a two-dimensional geometry, and the finite-$\delta$ measurement yields Eqs.(1,3). For $\Delta\gg\tilde{t_c}\equiv\max(\delta,t_c)$, Eqs.(1,3) have the asymptotic solution of Eq.(4), which is compatible with previous studies[2,4]. In Eq.(4), the symmetry between $\delta$ and $t_c$ implies that the dependence of the measured $D(\Delta,\delta)$ on $t_c$ is most pronounced when $\delta\sim\,t_c$, which is key to using this dependence to quantify $t_c$, $A$, and $D_\infty$.

MRI. Diffusion measurements were performed on a fiber phantom[2,6,7], composed of aligned Dyneema® fibers (radius 8.5±1.3μm) submersed in water, on a 3T Siemens Prisma with a 64 channel head coil. The water in between the fibers mimics the extra-axonal space, while the fiber radius exceeds the axonal radius by about tenfold, enabling a more convenient range of times to probe experimentally. We used two PGSE variants– STEAM DTI and monopolar DTI (Siemens WIP 511E) on two different days (Table 1)– to show that the resulting $l_c^\perp$ does not depend on the sequence (Table 2). For either STEAM or monopolar PGSE, in scan 1, we measured $D_\perp(\Delta,\delta)$ and $D_{||}(\Delta,\delta)$ with fixed $\Delta$=120ms and varied $\delta$=4-100ms; in scan 2, we fixed $\delta$=10ms and varied $\Delta$=50-600ms. Mean diffusion eigenvalues were extracted over a region of interest in the center of the fiber bundle.

Using scan 1 data, we observed that the $D_\perp(\Delta,\delta)$ decrease with $\delta$ is captured well by Eqs.(1,3) (solid lines), and asymptotically consistent with the limits Eq.(4) and Eq.(5), dashed and dotted lines in Fig.2a and Fig.3. Corresponding parameters from fitting Eqs.(1,3) are shown in Table 2, where the estimated correlation length $l_c^\perp$ ≈ 6μm, compatible to the fiber radius ≈ 8.5μm. $D_{||}(\Delta,\delta)$ does not change appreciably with $\delta$ in the range of 4~100ms. To validate our method, we predict the $\Delta$-dependent scan 2 results according to the parameters from scan 1(Table 2), and show that we capture the systemic decrease of $D_\perp(\Delta,\delta)$ with $\Delta$ (solid lines in Fig.2b). The prediction based on Eqs.(1,3) was done without any adjustable parameters, but based on the phantom properties evaluated in scan 1.Varying $\delta$ allows for quantifying the fiber-packing geometry from $D_\perp(\Delta,\delta)$ and estimating the correlation length $l_c^\perp$, which is the same order as the fiber radius. STEAM and monopolar DTI experiments were performed on different days, and resulted in slightly different fit parameters, except for the estimated $l_c^\perp$, which reflects the phantom’s packing geometry and was very similar between experiments. The consistency between scan 1 and scan 2 validates our framework and the assumption of short-range disorder (random fiber-packing) in the radial direction of this phantom. Future work will focus on the validation of our model, optimization of the acquisition protocol for in vivo brain scans and evaluation of its potential as biomarkers for brain pathology, e.g., by probing axonal density and outer axonal diameters.