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Water exchange time between gray matter compartments in vivo

Ileana Ozana Jelescu¹ and Dmitry S Novikov²
¹Center for Biomedical Imaging, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, ²Dept. of Radiology, New York University School of Medicine, New York, NY, United States

Synopsis

In the absence of a myelin sheath, exchange generally is non-negligible over the typical diffusion times of MRI experiments (10 – 100 ms) and should be accounted for in gray matter modeling. Here we use time-dependent kurtosis and the Kärger model (KM) of two slowly exchanging compartments to evaluate water exchange time between intra-neurite and extra-cellular compartments in rat GM in vivo. We report exchange times on the order of 10 – 30 ms. Future work will focus on exploring a broader range of diffusion times to test the asymptotic decay of kurtosis toward zero.

Introduction

If anything the many years of white matter modeling taught us, it is that elucidating and validating model assumptions should occur before bringing model fitting into the clinic. For gray matter (GM), the very basic relevant “compartments” are not established yet: Should we account for soma^[1]? Is there a “stick” compartment^[2,
3]? How fast is the exchange? Here we attempt to make first steps into this by quantifying water exchange.
In the absence of a myelin sheath, exchange generally is non-negligible over the typical diffusion times of MRI experiments (10 – 100 ms). The estimation of water exchange time between intra-neurite/axon and extra-cellular spaces has so far yielded very diverse results, from 60 ms in freshly excised bovine optic nerve^[4] to over 500 ms in major human white matter tracts using FEXI^[5] and 100 – 150 ms in astrocyte and neuron cultures, respectively^[6].
Here we use time-dependent kurtosis and the Kärger model (KM) of two slowly exchanging compartments^[7] to evaluate water exchange time in rat GM in vivo. One substantial advantage of rat brain vs. human is the large GM volume which removes confounding effects of partial volume with white matter or CSF.

Methods

All experiments were approved by the local Service for Veterinary Affairs. Three Wistar rats (250 - 300g) were scanned on a 14T Bruker system using a home-built surface quadrature transceiver. Diffusion MRI data were acquired using a PGSE EPI sequence (TE/TR = 50/2500 ms; matrix: 128x96; FOV=25.6x19.2 mm²; 16 0.5-mm thick coronal slices; Partial Fourier = 0.55; b = 0 (4 rep); 7 b-shells, 24 dirs each: b = 1:1.5:10 ms/μm²) at four diffusion times Δ = 12/20/30/40 ms, keeping δ = 4.5 ms constant.
Images were denoised and corrected for Rician bias, Gibbs ringing and motion^[8]. Diffusion and kurtosis tensors were estimated from b=0, 1 and 2.5 shells using a weighted linear least-squares algorithm, from which mean diffusivity and kurtosis were derived. Powder-average signal was also computed for each shell.
Regions of interest (ROI) covering the corpus callosum (CC), hippocampus (HPC) and cortex (CTX) were manually drawn.
KM assumes time-independent diffusivity $$$D_{KM} = f·D_i + (1-f)D_e$$$ in which case the only source of kurtosis is the inter-compartment heterogeneity which decays to zero as 1/t at long t^{[7, 9]}:
$$K_{KM}(t)=K_{0}\frac{2t_{ex}}{t}\left[1-\frac{t_{ex}}{t}\left(1-\exp(-t/t_{ex})\right)\right]\quad (1)$$
where t_ex is the exchange time. Mean kurtosis MK(t) was fit to Equation (1) to estimate K₀ and t_ex, both on the average ROI signal and individually in each voxel.
The isotropic average of the full KM signal $$$S_\mathrm{anisoKM}$$$ for two anisotropic compartments^[10] was also fit to the powder-averaged ROI signal over the whole b-value range (0 – 10 ms/μm²):
$$\overline{S} = \int\!{d{\bf\hat{q}}\over4\pi}\,S_\mathrm{anisoKM}({\bf q},t;\;t_{ex},f,D_i,D_e)\quad (2)$$
where D_i was fixed to 2.4 μm²/ms and the extra-neurite compartment was modeled as Gaussian isotropic with D_e = 0.8 μm²/ms, leaving t_ex and f to be estimated.

Results

Over the range t = 12 – 40 ms explored here, diffusivities displayed little to no time-dependence in GM (Figure 1), consistent with previous literature^[11-13], which supported the applicability of Eq. (1) to estimate the exchange time from MK(t).
Parametric maps of t_ex derived from Equation (1) were homogeneous across GM regions and consistent between rats (Figure 2). Overall, exchange times averaged around 15 – 30 ms in the cortex and around 10 – 15 ms in the hippocampus.
In CC, asymptotic signal decay followed the expected power law $$$b^{-1/2}$$$ characteristic of a stick compartment^[2,
3], while in GM a deviation from this power law, attributable to exchange^[10], was evident (Figure 3).
Estimates of exchange time and compartment fractions from fitting the two-compartment anisotropic Kärger model to powder-average data for each diffusion time were overall consistent with results from fitting K(t) in the cortex (Figure 4). Estimates were less reliable in the hippocampus (data not shown).

Discussion and Conclusions

Time-dependent kurtosis from the KM yielded rather short exchange times, similar to those found on human Connectome scanner^[10], suggesting exchange cannot be neglected in gray matter modeling. The validity of Equation (1) hinges on reaching the Gaussian diffusion limit in all compartments if exchange is switched off. The near-absence of diffusivity t-dependence supports this assumption.
Fitting the full KM was more prone to noise and estimates were unphysical for certain datasets. Given the limited number of data points (i.e. b-shells) available, diffusivities also had to be fixed. More data points at high b-values will be acquired in the forthcoming months.
The low-b part of the acquisition, yielding K(t), is advantageous because it does not require high b-values or special sequences and data can be acquired on any clinical scanner. Future work will focus also on exploring a broader range of diffusion times to test the asymptotic decay of kurtosis toward zero, and estimating the competing role of intra-compartment kurtosis (structural disorder)^[14].

Acknowledgements

The authors thank Analina da Silva and Stefan Mitrea for assistance with animal setup and monitoring. This work was supported by Centre d'Imagerie Biomédicale (CIBM) of the EPFL, Unil, CHUV, UniGe and HUG.

References

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[8] B. Ades-Aron, J. Veraart, P. Kochunov, S. McGuire, P. Sherman, E. Kellner, D.S. Novikov, E. Fieremans, Evaluation of the accuracy and precision of the diffusion parameter EStImation with Gibbs and NoisE removal pipeline, NeuroImage, 183 (2018) 532-543.

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[12] M.D. Does, E.C. Parsons, J.C. Gore, Oscillating gradient measurements of water diffusion in normal and globally ischemic rat brain, Magnetic resonance in medicine : official journal of the Society of Magnetic Resonance in Medicine / Society of Magnetic Resonance in Medicine, 49 (2003) 206-215.

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Figures

Figure 1. Mean diffusivity (MD) and kurtosis (MK) in each of the three ROIs, as a function of diffusion time Δ for one rat. Unlike in the corpus callosum, MD is relatively time-independent in cortex and hippocampus and MK decreases steadily with diffusion time. The fits to Equation (1) for MK in cortex and hippocampus are shown in solid blue and red lines, respectively. Estimated parameters t_ex and K₀ are provided in the table for each rat. Estimates are overall consistent across rats.

Figure 2. Top: Parametric maps of t_ex in a coronal slice covering hippocampus and cortex, matched in each of the three rats (1-3). An example FA map is shown on the right for anatomical reference. Bottom: Histograms of t_ex estimates in cortex (blue) and hippocampus (red) ROIs, for each rat. Histograms are very consistent between rats; the third rat showed longer exchange time in the cortex but was also heavier and potentially at a different developmental stage. The median of each distribution matches the estimate from the ROI-based fits (Fig.1).

Figure 3. Powder-averaged experimental signal decay for each diffusion time. For b≥5.5 ms/μm², the signal decays linearly as 1/√b in corpus callosum, consistent with the impermeable stick model. In cortex and hippocampus, this model does not hold. The noise floor was calculated based on the noise level estimation from [15] and scaled to S₀ without accounting for the increased SNR in denoised data, and is likely overestimated. While some contamination from Rician bias cannot be excluded in GM signal, the curvature is still attributable in part to finite membrane permeability.

Figure 4. Estimates for exchange time t_ex and intra-neurite fraction f from full KM in the cortex ROI, averaged over three rats. Exchange times broadly agree with estimates from fitting K(t). A bias towards longer exchange times for longer diffusion times is present though.

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)

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