Investigating exchange, structural disorder, and restriction in gray matter via water and metabolites diffusivity and kurtosis time-dependence

Diffusion-weighted MRI (dMRI) is a valuable radiological tool to non-invasively quantify the brain structure at the cellular scale, the so-called microstructure (Alexander et al., 2019; Jelescu et al., 2020; Novikov et al., 2019). Among the different dMRI approaches investigated in the past decades, measurements of the diffusion-time (t_d) dependent dMRI signal have been shown to provide information on the brain tissue inter-compartmental exchange (Hill et al., 2021; Jelescu et al., 2022; Nedjati-Gilani et al., 2017; Nilsson et al., 2009; Olesen et al., 2022; Stanisz et al., 1997; Zhang et al., 2021), structural disorder (Novikov et al., 2014; Palombo et al., 2013; Palombo, Ligneul, et al., 2018), and the restricted environment where the molecules diffuse over ~1-100 µm (Assaf & Cohen, 2000; Does et al., 2003; Drobnjak et al., 2010; Jespersen et al., 2018; Palombo et al., 2016; Panagiotaki et al., 2012).

In particular, it has been shown that measurements of the t_d-dependence of water diffusivity (Aggarwal et al., 2012; Arbabi et al., 2020; Baron & Beaulieu, 2014; Does et al., 2003; Wu et al., 2014), D_W(t_d), and kurtosis (Aggarwal et al., 2020; Lee et al., 2020; Pyatigorskaya et al., 2014; Wu et al., 2018), K_W(t_d), provide unique quantitative insight into these tissue properties. Varying the t_d allows probing different length scales and thus assessing the spatial heterogeneity of the cellular microenvironment. For instance, at short t_d, obtained with the oscillating gradient spin echo (OGSE) sequence, a large D_W variation as a function of the oscillating-gradient frequency (inversely proportional to t_d) was measured in cell-dense regions compared to other regions of the mouse brain, which is thought to reflect, among other things, restricted diffusion on a length scale of a few micrometers, thus in the order of the cell nucleus size (Aggarwal et al., 2012). At longer t_d, which can be probed using, for example, pulsed gradient spin echo (PGSE), the length scales probed are more on the order of tens of micrometers. For such t_d, D_W and K_W are sensitive to demyelination in white matter (Aggarwal et al., 2020) or to 1D structural disorders associated with neurites in gray matter (GM) (Lee et al., 2020; Novikov et al., 2014), but it has also been shown to be influenced by exchange (Jelescu et al., 2022; H. Li et al., 2017; Olesen et al., 2022; Yang et al., 2018), and disentangling the different contributions remains complex. Most importantly, some exchange regimes can also influence K_W(t_d), as shown by numerical simulations (Aggarwal et al., 2020): increasing the diffusion time will result in increased K_W until t_d approaches a characteristic time corresponding to a diffusion length comparable to the typical (permeable) restriction size; for longer t_d, K_W will then decrease due to exchange, with a slope which is steeper as exchange gets faster (i.e., higher permeability). However, as valuable as these simulations are, they do not fully mirror the true structural complexity of the GM tissue, in particular the structural disorder of the intra- and extracellular space. Disentangling compartment-specific structural disorder from inter-compartment exchange, either experimentally or in simulations, remains very challenging.

In contrast and complementary to water, diffusion measurements of purely intracellular metabolites by diffusion-weighted MRS (dMRS) offer information about the brain microstructure that is specific to the intracellular space and is not affected by inter-compartmental exchange (Assaf & Cohen, 1999; Palombo, Shemesh, et al., 2018; Ronen & Valette, 2015). Some brain metabolites are also preferentially compartmentalized within neurons, such as N-acetyl aspartate (NAA), and glial cells, such as myo-inositol (Ins), providing a probe of neuronal or glial structure, and can, for example, be used to estimate neurite radii and intracellular diffusivity (Ligneul et al., 2017; Palombo, Shemesh, et al., 2018). Interestingly, t_d-dependent dMRS is an efficient tool for probing complex GM cell microstructure, as it was shown that dMRS is sensitive, for example, to spines and leaflets density (Palombo, Ligneul, et al., 2018) and cellular processes branching (Palombo et al., 2016). Specifically, t_d-dependence of metabolite diffusivity D_M(t_d) and kurtosis K_M(t_d) is expected to be influenced by features of the cellular structure at different length scales, but without influence of extracellular space or exchange with extracellular space (Ianus et al., 2021). Micrometric restriction (soma and projection size) or exchanges between soma and projection (Ianus et al., 2021) could be all specific information of the intracellular environment. D_M(t_d) and K_M(t_d) would thus provide valuable and complementary information to water to investigate GM. By comparing the t_d-dependent diffusivity, D_M(t_d), and kurtosis, K_M(t_d), of intracellular metabolites with D_W(t_d)-K_W(t_d) one can potentially separate and quantify the relevant mechanism(s) driving each t_d-dependency in GM.

The aim of the present work is to shed some light on the role of exchange, structural disorder, and restriction in GM, by exploiting the complementary information of D_W(t_d)-K_W(t_d) and D_M(t_d)-K_M(t_d) measured by dMRS. More specifically, it aims at:

• (I)

  revisiting water diffusion time-dependence D_W(t_d)-K_W(t_d) with excellent SNR conditions as provided by dMRS (i.e., measured in a large volume) and up to very long t_d (500 ms)

• (II)

  reporting for the first time, to our knowledge, intracellular metabolite diffusion time-dependence D_M(t_d)-K_M(t_d) up to very long t_d (500 ms) to determine the signature of intracellular diffusion

• (III)

  combining results from (I) and (II) to try to disentangle the effect of exchange and the signature of the extracellular diffusion.

All experimental protocols were reviewed and approved by the local ethics committee (CETEA N°44), and authorized by the French Ministry of Education and Research. They were performed in an approved facility (authorization #B92-032-02), in strict accordance with recommendations of the European Union (2010-63/EEC). All efforts were made to minimize animal suffering, and animal care was supervised by veterinarians and animal technicians. Mice were housed under standard environmental conditions (12-hour light-dark cycle, temperature: 22 ± 1°C and humidity: 50%) with ad libitum access to food and water.

Experiments were performed on an 11.7 T BioSpec Bruker scanner interfaced to PV6.0.1 (Bruker, Ettlingen, Germany). A quadrature surface cryoprobe (Bruker, Ettlingen, Germany) was used for transmission and reception. Wild-type C57BL/6 mice were anesthetized with ~1.6% isoflurane and maintained on a stereotaxic bed with one bite and two ear bars. Throughout the experiment, body temperature was monitored and maintained at ~36°C by warm water circulation. Breathing frequency was monitored using PC – SAM software (Small Animal Instruments, Inc., Stony Brook, NY). The sequence used for all acquisition was a diffusion-weighted stimulated-echo sequence (Ligneul et al., 2017) followed by a LASER localization (STE-LASER). Echo time was set to TE = 33.4 ms (including the 25-ms LASER echo time) and was held constant for each acquisition. Repetition time was set to TR = 2500 ms. The duration of the pulsed diffusion gradient was set to δ = 3 ms. Diffusion gradients were separated by the delay ∆, which was varied according to the protocol described in the next two sections. In addition, a VAPOR water-suppression module supplemented by an additional 21-ms hermite inversion pulse inserted during the mixing time was used for metabolite acquisition, respectively.

Water diffusion was measured in four mice. A spectroscopic volume of interest (1.5 × 0.8 × 2 mm³ = 2.4 µL) was placed in the hippocampus, in such a way that it consisted almost exclusively of gray matter (no visible white matter, and CSF contamination <<1%). For each delay = 21.8, 31, 43.5, 101, 251, 501 ms, that is, resulting respectively in diffusion time t_d = 20.8, 30, 42.5, 100, 250, 500 ms, attenuation was measured for b = 0.2, 0.7, 1.2, 2.0, 2.5 ms/µm² along one fixed direction, with 16 repetitions.

Metabolite diffusion was measured in seven mice. A spectroscopic voxel (7 × 1.5 × 3 mm³ = 31.5 µL) covering the two hemispheres was centered in the hippocampus so that CSF contamination was minimal. Voxel composition (gray matter~94%, white matter~5%, and CSF~1%) was estimated a posteriori on anatomical images acquired with a RARE sequence with TE/TR=30/2500 ms, 78.1-µm isotropic resolution, and 0.5-mm slice thickness, using manual segmentation with the Fiji distribution of the ImageJ software, in the whole volume. For each delay = 43.5, 101, 251, 501 ms, that is, resulting respectively in diffusion time t_d = 42.5, 100, 250, 500 ms, attenuations of N-acetylaspartate (NAA), choline compounds (tCho), creatine + phosphocreatine (tCr), inositol (Ins), and taurine (Tau) were measured for b = 0.2, 1.0, 2.0, 3.2, 4.5, 6.0, 8.0 ms/µm² along one direction. Two blocks of 32 repetitions interleaved between each t_d and b value were acquired for t_d = 42.5, 100, 250 ms and three blocks of 32 repetitions were acquired to improve the SNR for t_d = 500 ms.

Individual scans were frequency- and phased-corrected before averaging. For water acquisitions, the integral of water peak was computed on a 0.5-ppm window centered on the peak maximum at each b and t_d. For metabolites, each spectrum was analyzed with LCModel (Provencher, 2001). Experimental macromolecule (MM) spectra, acquired for each t_d using a double inversion recovery module (TI₁ = 2200 ms and TI₂ = 770 ms), at b = 10 ms/µm², were included in LCModel basis-sets containing simulated metabolite spectra.

Remaining within these predefined b-value ranges (according to the convergence radius of the cumulant expansion (Frøhlich et al., 2006) for one hand water and other hand metabolites, which exhibit about four times slower diffusivity than water), for each t_d, the diffusion-weighted signal S as a function of b was fitted with a non-linear least-square method implemented in Matlab’s “lsqcurvefit” function to estimate the t_d-dependent apparent diffusivity of water or metabolite (D_W/M(t_d)) and kurtosis (K_W/M(t_d)), using equation 1 (Jensen & Helpern, 2003), for each t_d.

The STE-LASER sequence provides a good water signal even at a long diffusion time and a high b value, allowing D_W and K_W to be evaluated over a wide range of diffusion times [20.8; 500] ms and reaching a fairly long diffusion time (t_d~500 ms), longer than in previous studies using PGSE sequence (Jelescu et al., 2022; Lee et al., 2020). With respect to attenuation (Fig. 1B), the curvature of signal attenuation clearly attenuates with increasing diffusion time, approaching but not reaching near mono-exponential diffusion at long diffusion time (K~0.3). The model given by equation [1] fits the attenuation of water well (as shown for one mouse on Fig. 1B), and this protocol provides a small inter-animal distribution of diffusion parameters (D_W and K_W), with a standard error of the mean between mice of 2% at shortest t_d and ~15% at longest t_d for D_W and ~2% for K_W.

Regardless of the precise time-dependence, average D_W estimated in this study are in the same order of magnitude as ADC measured in rat (Jelescu et al., 2022) and mouse GM (Aggarwal et al., 2020; Zhang et al., 2021), and slightly lower than the ADC in human brain (Lee et al., 2020). K_W is also in the same order of magnitude as in previous studies (Jelescu et al., 2022; Lee et al., 2020). This spectroscopic acquisition using the STE-LASER sequence is therefore in good agreement with the literature.

Both D_W and K_W are time-dependent (Fig. 1.C-D), over the whole t_d range. D_W decreases significantly (~12% difference with an F-test (F(5, 15) = 31.102, p < 0.001) for a Repeated-Measure ANOVA), reaches a plateau ~0.5 µm²/ms, when D_W tends to the tortuosity asymptote (Fig. 1C). K_W also decreases (~50% with an F-test (F(5, 15) = 13.321, p < 0.001) for a Repeated-Measure ANOVA) and approaches ~0.3 (Fig. 1D), as the diffusion time increases, consistent with the diffusion not being Gaussian even at the longest diffusion time probed in this experiment. Given the composition of the water spectroscopic voxel used (nearly totally GM), we are also confident that the presence of multiple T₁ pools, or exchange between multiple T₁ pools during TM, is very unlikely to bias our diffusion measurements (Fig. S1). Furthermore, by plotting log S(b = 0) = f(TM) (not reported), TM being the mixing time, we did not observe multiple T₁ pools that could bias the signal, or diffusivity estimates as a function of t_d.

The main difference from previous results (Jelescu et al., 2022; Lee et al., 2020) is in the D_W time-dependence. On the one hand, we measured a significant decrease in D_W (from 0.62 to 0.57 µm²/ms) over the t_d range 20-42.5 ms, while on the other hand Jelescu et al. (2022) and Lee et al. (2020) report an almost constant D_W over this same t_d range in rat cortex and hippocampus and over a t_d range 20–100 ms in human cortical GM, respectively. On closer inspection, the ~0.05–µm²/ms difference measured in our study is within the range of inter-animal variability reported in these other studies, which may explain the lack of significant difference in these other studies.

A decrease in D_W and K_W over the entire t_d range suggests a nontrivial influence of microstructure, such as structural irregularities, and therefore the diffusion cannot be considered Gaussian, even at the longest diffusion time measured in this experiment. Moreover, these observations may also indicate an influence of the exchange between the intracellular and extracellular spaces, or a contribution of the two phenomena, structural disorder and exchange.

According to the literature (Lee et al., 2020; Novikov et al., 2014), intra-compartmental diffusion can be influenced by structural disorder, and thus the assumption of Gaussian diffusion is broken. In particular, works based on the coarse-graining approach have shown that D_W(t_d) and K_W(t_d) can reflect different classes of structural disorder in 1D, 2D, or 3D (Lee et al., 2020; Novikov et al., 2014). Therefore, we compared different functional forms to fit the data and identify which structural disorder might dominate the water behavior.

First, three models were applied to determine whether 1D or 2D/3D structural disorder could dominate. Based on Lee et al. (2020) and Novikov et al. (2014), we consider models given by

with the constants C_D, C_K, $D∞=limtd→∞D$ and $K∞=limtd→∞K$ left as free parameters to be determined by fitting the experimental data and setting the “universal” dynamical exponent θ = (p+d)/2 to 0.5 (representing 1D structural disorder with a dimensionality d = 1 and a structural exponent p = 0, which determines the structural universality class) and 1 (representing either 2D/3D structural disorder, with either p = 0 and d = 2 corresponding, for example, to random disks or p = -1 or d = 3 corresponding, for example, to random rods), respectively (Fig. 2A-B). In gray matter, neuronal and glial cell soma, represented as spherical objects with ~5–8-µm radii, can also induce a 1/t power law decay of D_W(t_d)/K_W(t_d) for long diffusion times t_d >> R²/D₀. If water probes an extracellular space mostly comprising randomly oriented fibers, D_W(t_d) and K_W(t_d) may also exhibit a logarithmic singularity of the form ln(t_d/tc)/t_d at long t_d (Burcaw et al., 2015). Hence, we also consider the alternative model

with A a coefficient proportional to the universal dimensionless tail ratio, and B a coefficient corresponding to the long time limit (Fig. 2C).

Visually, it is not clear whether D_W and K_W follow a power law with exponent θ = 0.5, θ = 1 or equation [4]. The results of the fit with the specific 1D structural disorder form are not too different from the results of the fit with ln(t_d/tc)/t_d, according to cAIC that is low for both models. In addition, it is not clear that D_W and K_W follow exactly the same law. At this stage, it is therefore difficult to determine whether one or the other form of structural disorder dominates.

To go further, a model with exponent θ in equation [2] and equation [3] left as an additional free parameter to be determined from the data was used to fit D_W and K_W (Fig. 3). The fit gives θ = 0.2 ± 0.2 (which is closer to 0.5 than 1), and the universal dimensionless tail ratio $ξ=CK/(CD/D∞)=1.3±0.2$ (which is close to 2). Values of θ ~0.5 and ~2 are expected for 1D short-range structural disorder (Lee et al., 2020; Novikov et al., 2014). We also found a tail ratio of ~2 regardless of the model used for fitting, with a best precision. Fit with free exponent is less accurate because of the small number of t_d.

Spectra acquired in the hippocampal voxel have a good signal-to-noise ratio (>20 for NAA for the longest t_d and highest b value) even at the longest diffusion times and highest b values (Fig. 4) and the LCModel analysis (not shown here) is robust (CRLB ~2% for all metabolites even at the longest diffusion time and highest b value) and we find a correct estimate of S/S0 with low inter-animal variability. The model given by equation 1 also fits the data well for each metabolite and animal (Fig. 4).

For each metabolite, with the exception of Ins, the inter-animal estimate of D_M and K_M is relatively consistent between animals with a standard error of the mean of less than 9% for D_M and 30% for K_M (Fig. 5). D_M decreases to a plateau of ~0.08 µm²/ms for NAA, Glu, tCr, and Tau, and ~0.06 µm²/ms for tCho and Ins with increasing t_d for all metabolites, and the D_M estimate is clearly in agreement with previous results (Palombo et al., 2016). Despite the low number of t_d, imposed by an in vivo experimental setup, statistical analysis with a paired-sample t-test shows a significant increase in K_M and decrease in D_M for each metabolite, with at least a power of 0.7 for our sample size N = 7. The diffusivity of predominantly intra-glial metabolites (Fisher et al., 2002) is lower than that of other metabolites. These differences are also noticeable in the estimate of K_M, which increases with t_d for NAA, Glu, tCr, and Tau and tends almost to a plateau ~2.3–2.5 while Ins and tCho remain almost constant ~1.5.

Such an increase in K is expected for diffusion in a restricted environment and thus strongly affected by the microstructure, in agreement with the numerical simulations of Ianus et al. (2021), which account for restriction in soma, exchange between soma, and cellular processes and cellular processes branching. The differences on the diffusion parameters observed between glial and neuronal cells are therefore not surprising because of the rather different morphology of these two cell types (Palombo, Shemesh, et al., 2018). Besides the high sensitivity of these dMRS methods for probing microstructure, already reported in previous studies (Ligneul et al., 2019), this work shows that even at lower b values (≲10 ms/µm²), suitable for the validity range of the kurtosis representation, different cell microstructures can be distinguished. dMRS thus appears a powerful tool to probe the cell morphology with potential feasible translation to in vivo human measurements, using also clinical scanners with relatively low maximum gradient strength (~60-80 mT/m).

To go further, we therefore proposed to evaluate the influence of the restricted environment on diffusion by fitting to D_M(t_d) and K_M(t_d) a simple model representing the intracellular compartment through the soma and projections formed by 20% spheres and 80% randomly oriented cylinders. Due to the low number of measurements, this ratio is imposed to avoid overfitting the model. Using an in-house iterative algorithm, the model was fitted to the D_M and K_M of mostly intra-neuronal (NAA and Glu) and predominantly intra-glial (tCho and Ins) metabolites (Fisher et al., 2002; Gill et al., 1989; Griffin et al., 2002; Urenjak et al., 1993), to extract radii from the soma and projections of (hypothetical) neuronal and glial cells, respectively (Fig. 6). First, this restriction model appears to be suitable for describing the behavior of D_M(t_d) and K_M(t_d). For neuronal cells, this model gives estimates with low standard error of the radii of spheres and D_M(t_d = 0), and the estimates are consistent with previous results, whereas the radii of cylinders are rather poorly estimated. For intra-glial metabolites, the uncertainty in the sphere and cylinder radii is high and the estimate is overestimated, but the D_M(t_d = 0) is consistent with the literature in normal mouse GM (Ligneul et al., 2019; Palombo et al., 2020). Of course, the aim of this work is not to propose a new model for estimating biophysical parameters, but these results suggest that even such a simple model, like SANDI (Palombo et al., 2020), describes metabolites data well.

In conclusion, the t_d-dependent diffusion of intracellular metabolites is dominated by restriction in the intracellular space, in line with a previous study with various dMRS methods (Assaf & Cohen, 1999; Kroenke et al., 2004; Ligneul et al., 2017; Shemesh et al., 2014, 2017; Valette et al., 2018; Vincent et al., 2020) and it is not surprising that it is clearly influenced by cell morphology (cell projection size and soma size at least). Importantly, as K_M does not decrease at increasing t_d, no signature of 1D structural disorder along the neurites is observed for the metabolites, in contrast with what has been speculated for intracellular water. The t_d dependence of metabolite diffusion seems thus clearly more specific to cell morphology than water.

It is insightful to examine water and metabolites results in the light of the work of Aggarwal et al. (2020). Our experimental results about intracellular metabolite kurtosis time-dependence, that is, kurtosis increasing with t_d, are in line with the behavior simulated for long exchange times in Aggarwal et al. (2020). In contrast, water kurtosis time-dependence exhibits the opposite behavior, which is in line with the behavior expected for a system with significant exchange in Aggarwal et al. (2020). This strongly suggests fast water exchange relative to the diffusion times used in the present study, which is in agreement with previous works suggesting short exchange time (Jelescu et al., 2022; Lee et al., 2020). The influence of such fast exchanges is also suggested in our estimate of the free exponent, θ = 0.2, found in Figure 3, which is close to that found by Jelescu et al. when they fitted their simulated exchange-driven K for a short exchange time ~25 ms, with the same power-law decay with a free theta exponent.

To provide a quantitative estimate of exchange time for our experimental conditions, we fit an increasingly used (Fieremans et al., 2010; Jelescu et al., 2022; Jensen, 2023; C. Li et al., 2023; Moutal et al., 2018; Olesen et al., 2022; Zhang et al., 2021) model of exchange: the Kärger Model

to the data for the longest diffusion time [42.5; 500] ms, to satisfy one of the assumptions of the Kärger model, that is, constant D_W(t_d) (Fig. 7). The estimated exchange time is very imprecise and spans hundreds of ms: from 46 (±412) ms (using equation [6]) to 196 (±238) ms (using equation [5]). Regardless of the poor precision despite high data quality, the contradiction with the previous estimate of exchange time shorter than ~25 ms (as discussed in the previous paragraph) might be due to the fact that the assumptions behind the Kärger Model are not satisfied. According to Fieremans et al. (2010), the non-applicability of the Kärger Model suggests that the membrane between the intracellular and extracellular spaces is too permeable to allow for an accurate description using the Kärger Model. Other recently proposed models of exchange are NEXI (Jelescu et al., 2022) and SMEX (Olesen et al., 2022); both based on the same modified Kärger Model to account for potential exchange between neurites and extracellular space. Fitting NEXI/SMEX to the water data as a function of b value (for extra b values up to 6 ms/µm²) and diffusion time (Fig. 8) yields once more very imprecise estimates of exchange time: 1.7 (±4.9) ms (other model parameters estimates are: intraneurite diffusivity 1.56 ± 1.28 µm²/ms; extraneurite diffusivity 0.75 ± 1.10 µm²/ms and neurite signal fraction 0.64 ± 0.52). Although imprecise, the NEXI/SMEX estimate of exchange time is more in agreement with the evidence of exchange time shorter than 10 ms discussed in the previous paragraph and supported by recent works (Cai et al., 2022; Olesen et al., 2022; Williamson et al., 2019, 2020, 2023).

In conclusion, these spectroscopic measurements with lower variability and over a wider range of t_d than the previous studies (Jelescu et al., 2022; Olesen et al., 2022) corroborate the influence of exchange on water diffusion as a function of time (in vivo in the mouse GM). The comparison of the very different functional forms of the measured D_W(t_d)-K_W(t_d) and purely intracellular D_M(t_d)-K_M(t_d) supports the hypothesis that the signature of 1D or 2D/3D short range structural disorder observed for D_W(t_d)-K_W(t_d) might mostly come from the water diffusion in the extracellular space, which “contaminates” the whole water pool signature due to fast exchange and hence hides intracellular signature. Our measurements reinforce the need for a model describing the t_d-dependence of diffusion in GM, with specific signatures in intra- and extracellular compartments and inter-compartmental exchange, to precisely evaluate how exchange can mask the signatures of intracellular space.

The range of t_d values over which we were able to carry out our measurements remains rather restricted, even if it is the widest ever probed in this type of in vivo measurement. On the one hand, a constraint linked to the duration of the spoiling gradient applied during the mixing time in this sequence (Ligneul et al., 2017) prevents us from having access to smaller t_d values. With a shorter t_d, we could observe the inflection point of the time-dependent kurtosis of water, which could help determine the average permeability. Future work could try to combine OGSE and PGSE measurements to try to characterize the time-dependence at shorter t_d. On the other hand, the increase in t_d is limited by the SNR, which becomes too low beyond 500 ms. Among other things, this increase could allow us to better discriminate which class of structural disorder dominates, which would be a first step towards disentangling the different contributions of exchange and structural disorder.

Our aim here is not to propose a new modeling method, but to present experimental data that could support theoretical models to better describe gray matter. We have chosen to place ourselves in a diffusion regime suited to a representation with the cumulant exponent, which seems quite sensitive to exchange. Nevertheless, with our experiments we were not able to precisely determine the exchange time in gray matter (using currently available biophysical models, that is, Kärger model and NEXI/SMEX), and to determine the influence of structural disorder on the water diffusion properties. The complexity of both exchange mechanisms (e.g., multiple exchange times) (Nilsson et al., 2013) and microstructure (e.g., glial and neuronal cells and complex extracellular space) (Palombo et al., 2020) will need to be taken into account in future work to disentangle the contributions of exchanges and structural disorder. Towards this goal, future work could harness advanced computational modeling tools for gray matter microstructure simulations (Callaghan et al., 2020; Palombo et al., 2016, 2020).

We also chose a smaller voxel size for water, in order to focus only on gray matter without multiple compartment effects that can complicate interpretation of the t_d dependence of the diffusion and to be in conditions similar to those encountered in the literature for segmented images of rodent brain (Jelescu et al., 2022). In particular, in this voxel, we minimize the contribution of CSF and white matter. For metabolites, we have no choice to take this large voxel to have sufficient signal at long diffusion time; however, the contribution of CSF and WM remains small (respectively ~5% and 1%), which can certainly have a negligible impact on our results. Indeed, we did not notice a strong signal attenuation at low b which would have suggested a high CSF contribution whatever the metabolite. This is in line with measurements carried out by Lundell et al. (2021) in the human brain, where they noted that for a CSF contribution of ~2% there was no visible effect on diffusion, whereas for a contribution of ~10% there was. Finally, by fitting a bi-exponential model to log(S0) as a function of mixing time, only a small fraction of the short T₁ component was estimated, suggesting that signal is not affected by multi-T₁ pool, and certainly not influenced by short T₁ of WM (see Supplementary Materials, Figs. S1 and S2).

In this work, we focused on healthy mice; future work will explore the ability of combined D_W/M(t_d)-K_W/M(t_d) to characterize pathological microstructural changes. Moreover, it could be informative to extend in future work the investigation of D_M(t_d)-K_M(t_d) to also metabolites that are in both intra- and extracellular space and potentially in exchange. Further extension of this work will include measurements in humans to assess the feasibility and translatability of this approach.

This experimental study proposes for the first time a comparison of the t_d dependence of the diffusion of two different endogenous probes, water and intracellular metabolites. The information provided by these two sets of t_d-dependent acquisitions is complementary, especially since they probe different compartments of the GM. In particular, the t_d dependence of water diffusion is mainly dominated by short-range structural disorder in the extracellular space and rapid exchanges between intra- and extracellular spaces, whereas the t_d dependence of intracellular metabolite diffusion is more specific to restriction within the intracellular space. Other underlying features could be further investigated to fully decipher the sources contributing to total kurtosis (beading, undulation,…), as already proposed by Shemesh Lab (Alves et al., 2022; Henriques et al., 2020). Nevertheless, these new results help interpret D(t_d)-K(t_d) measurements in terms of the underlying microstructure of GM and suggest that D(t_d)-K(t_d) measurements of metabolites may offer a new way to quantify microstructural restrictions in GM.

All the data and analysis codes underpinning the results presented here can be found in the Cardiff University data catalogue at the http://doi.org/10.17035/d.2024.0314230392.

E.M.: Conceptualization, Data acquisition, Formal analysis, Investigation, Methodology, Validation, Visualization, Writing—original draft, and Writing—review & editing. J.V.: Conceptualization, Methodology, Writing—original draft, Writing—review & editing, Funding acquisition, and Original idea. M.P.: Conceptualization, Formal analysis, Methodology, Writing—original draft, Writing—review & editing, Funding acquisition, and Original idea.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programs [grant agreement No 818266] and the UKRI Future Leaders Fellowship MR/T020296/2. The 11.7 T MRI scanner was funded by a grant from “Investissements d'Avenir - ANR-11-INBS-0011 - NeurATRIS: A Translational Research Infrastructure for Biotherapies in Neurosciences.” M.P. is supported by the UKRI Future Leaders Fellowship MR/T020296/2.

The authors declare no competing financial interests.

Supplementary material for this article is available with the online version here: https://doi.org/10.1162/imag_a_00123.