To determine which compartment, intra- or extra-axonal, is predominantly responsible for the change in diffusion signal at low $$$b$$$ with varying diffusion time. Quantifying microstructure with diffusion entails determination of a length scale. This requires modeling time dependence [1] of diffusion metrics. Models providing length scales for white matter, such as axonal diameters [2], need validation. In particular, it is still under debate [3], whether it is the intra-, or extra-axonal water, that contribute most to the observed changes of diffusion signal with diffusion time $$$\Delta$$$. Here we show that the intra- and extra-axonal contributions to the diffusion coefficient dependence on gradient pulse duration $$$\delta$$$ and on $$$\Delta$$$ perpendicular to axons are qualitatively different. Human measurements reveal the predominance of extra-axonal contribution.

Theory. Intra- and extra-axonal spaces both contribute to the diffusivity dependence on $$$\Delta$$$ and $$$\delta$$$ [3-5]; fitting just intra-axonal dependence [2,6,7] (Eqs.(1-2) in Fig.1) yields strongly over-estimated axonal diameters, which prompts asking whether extra-axonal water provides a nontrivial contribution. Here we use the distinct functional form of $$$D_\perp(\Delta,\delta)$$$ in order to select which contribution dominates.

Fig 1 provides our extra-axonal results, Eqs.(3-4), together with conventional intra-axonal model, Eqs.(1-2). In Eqs.(1-2), vanGelderen's/Neuman's model [4,5] is shown for axonal water fraction $$$f_{\rm{int}}$$$, axonal radius $$$r$$$, and free diffusivity $$$D_0$$$. The extra-axonal model involves exrtra-axonal water fraction $$$f_{\rm{ext}}$$$, bulk diffusivity $$$D_\infty^{\rm{ext}}$$$, and the strength $$$A_{\rm{ext}}$$$ of restrictions [3,9]. Empirically, $$$ A_{\rm{ext}}\sim{0.2l_c^\perp}^2$$$, where $$$l_c^\perp$$$ is the packing correlation length[3].

It is crucial that intra-axonal contribution, from Neuman's limit to van Gelderen's formula, scales as $$$\sim1/(\Delta\delta)$$$ when $$$\delta\gg{r^2/D_0}$$$, Eq.(2) [2,7]. In contrast, the extra-axonal contribution obeys Eq.(3), with an asymptotic behavior of $$$\sim\ln(\Delta/\delta)/\Delta$$$ when $$$\Delta\gg\delta$$$, described by Eq.(4). The behaviors of $$$1/(\Delta\delta)$$$ and $$$\ln(\Delta/\delta)/\Delta$$$ for intra-axonal and extra-axonal space have very different trends with respect to $$$1/\delta$$$, which we use here to determine which compartment’s signal dominates at low $$$b$$$.

MRI. Diffusion MRI was performed on five healthy subjects (4 males/1 female, 25-41 years old) using a 3T Siemens Tim Trio scanner with a 32-channel head coil [8,9]. We also demonstrate a similar experiment on two healthy females, 26 and 28 years old, using a 3T Siemens Prisma scanner with a 64-channel head coil. The STEAM DTI sequence provided by the vendor (Siemens WIP 511E) was used to perform two different scans for each subject. The acquisition setup is shown in Table 1. For each scan, we obtained 3-5 $$$b_0$$$ images and $$$b$$$=500$$$s/mm^2$$$ images along 20 directions with isotropic resolution of $$$(2.7 mm)^3$$$ and a FOV of $$$(221mm)^2$$$.

In scan 1, we varied $$$\Delta$$$ and a fixed $$$\delta$$$; in scan 2, we fixed $$$\Delta$$$ and varied $$$\delta$$$. A series of white matter ROIs were created by thresholding the fractional anisotropy map at 0.3, 0.4, and 0.5. The mean values of $$$D_\perp(\Delta,\delta)$$$ were computed within each ROI.

Using data in scan 1, $$$D_\perp^{\rm{int}}(\Delta,\delta)$$$ in Eq.(2) and $$$D_\perp^{\rm{ext}}$$$ in Eq.(4) both fit measurements well (see fitting curves in Fig.2a and Fig.3a/3c). The acquired parameters are shown in Table 2. To illustrate the $$$\delta$$$-dependence, we predicted scan 2 results based on fit parameters in Table 2; the predictions based on Eq.(2) and Eq.(4) are very different asymptotically, shown in Fig.2b and Fig.3b/3d, where $$$D_\perp^{\rm{ext}}(\Delta,\delta)$$$ (Eq.(4)) captures the systematic bend in the curves, and $$$D_\perp^{\rm{int}}(\Delta,\delta)$$$ (Eq.(2)) increases with $$$1/\delta$$$ almost linearly and deviates from experimental results. The prediction of scan 2 was done without any adjustable parameters, since tissue properties are found in scan 1, and $$$\Delta$$$- and $$$\delta$$$-dependence is shown in Eqs.(2,4). Hence, this prediction provides a parameter-free test of the models involved. Both Fig.2 (average over 5 subjects) and Fig.3 (individual subjects) show that it is $$$D_\perp^{\rm{ext}}(\Delta,\delta)$$$, rather than $$$D_\perp^{\rm{int}}(\Delta,\delta)$$$, that demonstrates good consistency between scans 1 and 2, indicating that water in extra-axonal space dominates the diffusion signal decay at low $$$b$$$-value. Additionally, using fitting parameters based on $$$D_\perp^{\rm{int}}(\Delta,\delta)$$$ (Eq.(2)) and physiological values of $$$f_{\rm{int}}\approx 0.5$$$ and $$$D_0\approx 1.7μm^2/ms$$$, the estimated fiber diameter $$$2r\approx$$$12 μm, which is much larger than reported histology radius values ~1 μm for most axons [12]. On the other hand, the values of the correlation length ~ 3-5 μm are reasonable for extra-axonal space packing length scale. Future work will focus on optimization of acquisition protocol for in vivo human brain scans, particularly by exploring combinations of $$$\Delta$$$ and $$$\delta$$$ to achieve better diffusion sensitivity, and combining data from oscillating gradients methods [10,11].