Surface-to-Volume ratio (S/V) quantification from the asymptotic short time (Mitra) limit¹ is essential for disentangling free diffusivity $$$D_0$$$ from restrictions to diffusion. Conventional Pulsed Gradient Spin Echo (PGSE) measurements fail to probe the Mitra limit on clinical MRI scanners due to limited gradient strength. Oscillating Gradient Spin Echo (OGSE) remedies this problem by increasing diffusion weighting and therefore shortening the diffusion time. The success of OGSE led to multiple in-vivo studies for tissue characterization.^2,3,4 While the Mitra limit was recently demonstrated in a mouse glioma model², none of the human measurements have been shown to be in the Mitra limit. Here we validate the S/V quantification using $$$D(\omega)$$$ in four fiber bundles with a range of known S/V values, and demonstrate the first instance of achieving the Mitra limit⁵ on a clinical 3T scanner.

Theory: The analytic form of the Mitra limit in the frequency domain, $$$D(\omega)$$$, was recently derived²

$$D(t)=D_0\left(1-\frac4{3d\sqrt{\pi}}\frac{S}{V}\sqrt{D_0t}\right)\leftrightarrow\text{}D(\omega)=D_0\left(1-\frac1{d\sqrt{2}}\frac{S}{V}\sqrt{D_0\over\omega}\right)(1)$$

Although the conventional identification of “effective diffusion time” is $$$t_{\rm eff}=1/4f$$$, where $$$f=2\pi\omega$$$, matching $$$D(\omega)$$$ and $$$D(t)$$$ is model dependent⁶. Understanding this, we can derive the effective diffusion time within the mitra limit: $$$t_{\rm eff}^{\rm Mitra}=9/64f=9/16\cdot t_{\rm eff}$$$. Here we neglect small corrections from finite number N of oscillations⁷ as our N ranges between 4 and 44, which is chosen by maximizing N for each $$$t_{eff}$$$.

Phantom: Four bundles, consisting of Dyneema fibers ($$$2r=17\mu\text{m}$$$ diameter) are held tightly together by a shrinking tube, are created according to the method described in [8]. The total fiber count in each bundle is varied (50, 75, 100, 110 turns, respectively) to create different fiber densities. The contrast in fiber density in each bundle manifests into a difference in water fraction, $$$\phi$$$, and S/V. With a quantitative measurement of proton density (PD) and knowledge of the fiber radius, S/V can be estimated via: $$\frac{S}{V}=\frac{2}{r}\bigg(\frac{1-\phi}{\phi}\bigg)\textrm{ where }\phi=\frac{PD_{Fiber}}{PD_{Water}}\textrm{ }(2)$$Spin Echo (SE) and Plug-and-Play MR Fingerprinting (PnP-MRF)⁹ are used to calculate PD maps [Fig.1(C-D)]. To account for B1 non-uniformity in SE N3 bias field correction¹⁰ from FreeSurfer is applied. Unlike the original MRF methodology¹¹, PnP-MRF⁹ is a radial fingerprinting approach that separates out the B1+ from the, PD, T2, and T1 maps. For each modality, the water fraction in each bundle is calculated based on the PD maps by the ratio of a fiber ROI inside the bundle and water ROI in close proximity to the bundle, in order to account for residual receive sensitivity variations.

OGSE and PG-STEAM: Measurements were performed on a MAGNETOM 3T PRISMA system (Siemens AG, Erlangen, Germany) with a 20-channel head coil. Measurement parameters are summarized in [Table.1]. A stretched cosine gradient¹¹ [Fig.2(A)] with $$$\alpha=2$$$ is implemented for increased b-value while suppressing zero-frequency lobes Fig.2(B)]. Pulsed Gradient diffusion using STEAM EPI (WIP511E) is also acquired to examine diffusion behavior at long t in comparison to OGSE. DTI metrics are generated using in-house software written in MATLAB. S/V is calculated by fitting Eq(1) to the perpendicular diffusivity, derived as the average of $$$\lambda_2$$$ and $$$\lambda_3$$$. The linear regime as a function of $$$1/\sqrt{\omega}$$$ is determined by selecting the range for which the $$$D_0$$$ will converge to the median of the non-dispersive $$$\lambda_1(\omega)$$$ [black dashed line [Fig.3]].

Increasing the number of fibers within the bundle results in denser fiber bundles with a corresponding lower diffusion coefficient, higher FA, and decreased water fraction [Fig.1]. With increasing fiber density, the Mitra limit is achieved in a shorter range of frequencies [Fig.3], as evidenced by the linear dependence of D on $$$1/\sqrt{\omega}$$$. S/V estimates derived from SE and PnP-MRF showed strong correlation ($$$\rho_{PnPMRF}=0.9951$$$ and $$$\rho_{SE}=0.9943$$$) with S/V estimates from OGSE. Particularly, the PnP-MRF derived S/V is in near exact agreement with the OGSE derived S/V [Fig.4]. The better agreement with PNP-MRF is explained by the more precise estimation of PD of this method⁹. Furthermore, the correspondence between PGSE (STEAM) and OGSE [Fig.1] when using $$$t_{\rm eff}^{\rm Mitra}=9/16\cdot\text{}t_{\rm eff}$$$, suggests that for the short-time limit (only), the extra factor of 9/16 is appropriate. Here we introduce a well-characterized fiber phantom that can be used for the calibration of OGSE and diffusion modeling techniques as the S/V ratio can be measured independently using other MR modalities such as PnP-MRF. Furthermore, as the measurement parameters are within a clinically acceptable range, this calibration experiment offers an exciting perspective of mapping S/V in humans. This is of particular interest in body diffusion applications where the sizes of restrictions match or even exceed those of the fibers in our phantom.