Diffusion-time dependence of diffusional kurtosis in the mouse brain

Abstract

Purpose:

To investigate diffusion-time dependency of diffusional kurtosis in the mouse brain using pulsed-gradient spin-echo (PGSE) and oscillating-gradient spin-echo (OGSE) sequences.

Methods:

3D PGSE and OGSE kurtosis tensor data were acquired from ex vivo brains of adult, cuprizone-treated, and age-matched control mice with diffusion-time (t_D) ~20 ms and frequency (f) =70 Hz, respectively. Further, 2D acquisitions were performed at multiple times/frequencies ranging from f =140 Hz to t_D = 30 ms with b-values up to 4000 s/mm². Monte Carlo simulations were used to investigate the coupled effects of varying restriction size and permeability on time/frequency-dependence of kurtosis with both diffusion-encoding schemes. Simulations and experiments were further performed to investigate the effect of varying number of cycles in OGSE waveforms.

Results:

Kurtosis and diffusivity maps exhibited significant region-specific changes with diffusion time/frequency across both gray and white matter areas. PGSE- and OGSE-based kurtosis maps showed reversed contrast between gray matter regions in the cerebellar and cerebral cortex. Localized time/frequency-dependent changes in kurtosis tensor metrics were found in the splenium of the corpus callosum in cuprizone-treated mouse brains, corresponding to regional demyelination seen with histological assessment. Monte Carlo simulations showed that kurtosis estimates with pulsed- and oscillating-gradient waveforms differ in their sensitivity to exchange. Both simulations and experiments showed dependence of kurtosis on number of cycles in OGSE waveforms for non-zero permeability.

Conclusion:

The results show significant time/frequency-dependency of diffusional kurtosis in the mouse brain, which can provide sensitivity to probe intrinsic cellular heterogeneity and pathological alterations in gray and white matter.

INTRODUCTION

Diffusion MRI (dMRI) is a powerful technique to probe various aspects of tissue microstructure. In the presence of restrictions posed by local tissue microenvironment, e.g., cell membranes, organelles, myelinated axons and extracellular structures, the displacement probability distribution of diffusing water molecules in the brain deviates from Gaussianity. The second-order tensor formalism used in diffusion tensor imaging (DTI) fits the diffusion-weighted signal attenuation with increasing b-value to a monoexponential function [1]. As DTI builds on an approximation of Gaussian distribution of molecular displacements, it does not capture the full extent of information about water diffusion dynamics contained in the diffusion-encoded signal acquired from heterogeneous brain tissue microenvironments. dMRI approaches that examine higher-order statistics of random molecular motion, such as diffusion kurtosis imaging (DKI) [2,3] and q-space imaging methods [4], thereby have the potential to lead to additional insights into tissue microstructure. DKI fits the logarithm of the diffusion-weighted signal to the fourth-order cumulant expansion in powers of q [5], to quantify the excess kurtosis of the molecular displacement probability distribution [2]. Recent studies have shown promising applications of DKI to examine changes in conditions such as traumatic brain injury [6,7], ischemic stroke [8-10], and white matter demyelination or hypomyelination [11-13]. While the structural correlates of diffusional kurtosis in gray matter and white matter are not yet fully understood, non-zero kurtosis is a quantitative measure of the non-Gaussianity of the diffusion process which relates to the degree of restriction or tissue heterogeneity [3,14].

A second important consequence of restricted diffusion in the brain is the dependence of dMRI measurements on the effective diffusion time, i.e., the time interval over which random spin displacements are sampled in dMRI experiments [15,16]. By varying the diffusion time (t_D), the length scales probed by diffusing water molecules can be adjusted, which can provide sensitivity to unique properties of tissue microstructure. At relatively long t_Ds attained with pulsed-gradient spin echo (PGSE) sequences, the diffusion-encoded signal reflects the integrated effect of restrictions to free water diffusion over multiple spatial scales. However, the effects of restrictions at short length scales (e.g., those arising from sub-cellular structures) are effectively homogenized and are difficult to experimentally distinguish, as only the asymptotic signal behavior may be observable [17]. Furthermore, in neural tissue environments, the effects of restricted diffusion are entangled with the effects of water exchange across permeable plasma membranes. With increasing t_D, these two processes manifest competing effects on the diffusion signal intensity at high b-values [18]. In other words, t_D is expected to govern the relative influence of diffusion restriction versus compartmental water exchange on the observed signal.

Diffusion-encoding schemes using oscillating gradient waveforms have been increasingly used to probe diffusion dynamics over short length scales, which are not accessible with standard PGSE implementations. Oscillating-gradient spin echo (OGSE) sequences allow encoding sensitivity to spin displacements on the time-scale of the period of oscillation. By increasing the gradient modulation frequency, it is thus possible to reach the regime where changes in the signal behavior due to contributions from small-scale restrictions in the brain may become experimentally observable [19,20]. Several recent studies have reported unequivocal time/frequency-dependence of diffusivity measurements in the brain using OGSE sequences with relatively low diffusion-weighting and monoexponential signal fitting (i.e., using the lowest-order approximation of the cumulant expansion) [21-28]. For typical cellular dimensions in neural tissues, OGSE-based measurements of the diffusion coefficient have been shown to be less sensitive to the effects of exchange than PGSE-based measurements at longer t_Ds [29].

However, the relation between diffusion time and diffusional kurtosis in the brain remains relatively less explored. Monte Carlo simulations of restricted diffusion in a geometry of parallel permeable cylinders indicate a non-monotonic time-varying behavior of kurtosis [30], yet literature concerning the time dependence of kurtosis in neural tissues is sparse. An earlier study by Portnoy et al. using PGSE and OGSE sequences reported time/frequency-dependence of apparent kurtosis in fixed rat hippocampal slices [31]. Pyatigorskaya et al. [32] used PGSE and OGSE sequences to examine the effect of diffusion time on parameters estimated from the bi-exponential and cumulant expansion representations and demonstrated time-dependence of kurtosis in the rat cortex. Recently, Jespersen et al. [33] have reported t_D-dependence of microstructural parameters estimated from DKI in white matter of the fixed porcine spinal cord. Probing the time-dependence of diffusional kurtosis could potentially reveal novel contrasts in the brain that are sensitive to the intrinsic heterogeneity of cellular environments across different regions, and to changes in these microenvironments caused by pathological disruptions. Further, while pulsed- and oscillating-gradient schemes are often used to probe distinct or non-overlapping time regimes within gradient hardware constraints, both the time/frequency and gradient-waveform dependence of non-Gaussian signal behavior in the presence of exchange and varying restriction scales remains less well understood.

In this study, we sought to investigate the time-dependence of diffusional kurtosis in the fixed mouse brain using pulsed- and oscillating-gradient dMRI at 11.7 T. Changes in kurtosis tensor metrics and tissue contrasts derived from PGSE and OGSE acquisitions in both normal and cuprizone-treated mouse brains were compared to further examine the effect of toxic demyelination on t_D-dependence of kurtosis in white matter. Monte Carlo simulations were performed to investigate the coupled effects of permeability and restriction size on PGSE and OGSE acquisitions using the same gradient waveforms and parameters as the mouse brain experiments. Finally, simulations and experiments were used to probe the signal behavior over an extended time/frequency range and under varying number of diffusion-encoding cycles in OGSE waveforms.

METHODS

Animals and specimen preparation

All experimental procedures were approved by the Animal Use and Care Committee at the Johns Hopkins University School of Medicine. Adult C57BL/6 mice (n=4, 12 weeks old, male, Jackson laboratory, Bar Harbor, ME, USA) were used for the dMRI experiments. A separate set of C57BL/6 mice (10 weeks old, male) was divided into a cuprizone-treated (n=5) and an age-matched control (n=5) cohort. Mice in the cuprizone-treated group were placed on a diet supplemented with 0.2% w/w cuprizone (bis(cyclohexanone) oxaldihydrazone) (Sigma-Aldrich, St. Louis, MO, USA), and were sacrificed at 4 weeks following the start of the diet. Mice in the control group were maintained on a normal diet and were sacrificed along with the 4-week cuprizone-treated mice. All animals were fixed by transcardial perfusion with 4% paraformaldehyde (PFA) in phosphate buffered saline (PBS). After perfusion, the brains were removed and immersed in 4% PFA in PBS overnight at 4°C. Prior to MRI, the brains were transferred to PBS for 48 h to wash out the fixative, and were placed in 15-mm-diameter glass tubes which were filled with Fomblin® (Solvay Inc., Princeton, NJ, USA) for susceptibility matching and to prevent tissue dehydration.

Data acquisition

MRI experiments were performed on a vertical-bore 11.7 T scanner (Bruker Biospin, Billerica, MA, USA) equipped with an actively-shielded Micro2.5 gradient system (1000 mT/m maximum gradient strength). A 15-mm-diameter birdcage coil was used as the RF transmitter and receiver. The temperature of the specimens was maintained constant at 37°C during imaging via the spectrometer’s temperature control system.

Three-dimensional (3D) dMRI data from all mouse brains were acquired using a gradient-and-spin-echo (3D-GRASE) readout with adiabatic BIR-4 pulses and twin navigator echoes for phase correction [34]. PGSE data were acquired with diffusion gradient duration/separation (δ/Δ) = 5/22 ms, turbo factor = 4, EPI factor = 3, echo time (TE)/repetition time (TR) = 40/700 ms, 2 signal averages, and receiver bandwidth = 100 kHz. For OGSE experiments, trapezoid-cosine oscillating gradient waveforms were implemented to maximize the b-value within gradient limits [23,35]. The first and last half lobes of the trapezoid-cosine waveform were replaced by full lobes at twice the base frequency, similar to the apodized cosine waveform described in Ref. [36]. The b-value for the trapezoid-cosine gradient waveform including the effect of apodization with the first and last lobes was calculated as $b={\int }_0^{\tau }dt{\left[\gamma {\int }_0^{t}g(t^{\prime })d{t}^{\prime }\right]}^2$ (Eq. [1]). Co-registered OGSE data of the mouse brains were acquired at an oscillation frequency (f) of 70 Hz with one cycle on either side of the 180° pulse, using 3D-GRASE readout with identical TE and imaging parameters as the PGSE experiments. Both PGSE and OGSE data were acquired along 18 non-collinear directions uniformly distributed over a sphere [37] with two b-shells (b = 2000 s/mm² and 4000 s/mm²) and two non-diffusion-weighted images. The b-values selected were empirically optimized by first measuring both PGSE and OGSE signal attenuation curves from one adult mouse brain. Typical field-of-view (FOV) and imaging matrix size were 16 mm x 11.5 mm x 7.5 mm and 128 × 92 × 60, respectively, for a spatial resolution of 125 μm x 125 μm x 125 μm. The total scan time for 3D PGSE and OGSE datasets with two b-shells was ~14 h.

To further investigate the change in the diffusion signal behavior over a range of effective diffusion times, 2D data from one brain in each cohort (total 3 mice) were acquired at multiple t_Ds using tetrahedral diffusion-encoding with 9 b-values (equally spaced between 800 and 4000 s/mm²) and two non-diffusion-weighted images. For PGSE experiments, δ was fixed at 1.9 ms and effective diffusion times (= Δ-δ/3) were set to 7, 10, 15, 20, 25, and 30 ms. Co-registered OGSE data were acquired with f = 70 Hz and 140 Hz (N = 1 and 2 cycles, respectively). A 2D spin-echo readout was used with the following acquisition parameters: three sagittal slices with no gap, slice thickness = 1 mm, in-plane resolution 112 μm x 112 μm, TR = 1000 ms, 6 signal averages, and bandwidth = 70 kHz. The TE was kept constant at 35 ms for all 2D OGSE and PGSE acquisitions, except for the PGSE data at t_D = 30 ms which were acquired with a slightly longer TE of 38 ms to accommodate the Δ. The slice locations were selected carefully to ensure that the midsagittal plane of the brain was imaged. Tetrahedral diffusion encoding [38] was used with gradients of equal amplitude applied simultaneously along all three axes, to achieve a b-value of up to 4000 s/mm² for the highest oscillation frequency (140 Hz). The acquired gradient vectors [1,1,1], [−1,1,1], [1,−1,1], and [1,1,−1] were distributed at the same angle (cos⁻¹(1/√3)) with respect to the long axis of the corpus callosum in the midsagittal plane.

Additionally, to investigate the effect of number of oscillation periods (N) in OGSE acquisitions, data from one adult mouse brain were acquired with f = 105 Hz and N = 1 and 2 cycles, using the same b-values and acquisition parameters as above, with TE of 44 ms and 10 signal averages.

For reference, the same sequences used for the mouse brain experiments were also used to acquire data from a phantom prepared with polyvinylpyrrolidone (PVP) (K value 30, Millipore Sigma, St. Louis, MO, USA) solution in distilled water (40% w/v) in a 15-mm-diameter tube. PVP lowers the water diffusion coefficient, but is not expected to affect the diffusional kurtosis [39]. PGSE and OGSE datasets along one direction ([1,1,1]) were acquired with the same b-values and diffusion times/frequencies as the mouse brain experiments, with the phantom temperature maintained at 25°C.

Data processing and analysis

Images were reconstructed from k-space data using in-house written code in IDL (ITT Visual Information Solutions, Boulder, CO, USA). k-space data were apodized by a symmetric 10% trapezoidal function and zero-filled by a factor of two prior to Fourier transformation. DTI reconstruction for all mouse brains was performed using 3D PGSE and OGSE data acquired with b = 2000 s/mm² [34]. Kurtosis tensors and maps of mean, axial, and radial kurtosis (MK, AK, and RK) were derived from the two-shell (b = 2000 and 4000 s/mm²) PGSE and OGSE data using constrained linear-weighted DKI fitting in Diffusion Kurtosis Estimator [40], with no spatial smoothing. For 2D PGSE and OGSE datasets acquired with tetrahedral encoding and 9 b-values, the raw images were smoothed using a Gaussian kernel with full-width and half-maximum of 1.25. Log-signal intensities were fit voxel-wise to $ln\left(\frac{S}{S_0}\right)=-bD+\frac{1}{6}{b}^2{D}^{2}K$ (Eq. [2]) using the Levenberg-Marquardt algorithm in Matlab (Mathworks, Natick, MA, USA) to derive the average diffusivity and kurtosis estimates over all directions at each diffusion time/frequency. Typical signal-to-noise ratio (SNR) values in the cortex ranged from 176 to 190 in non-diffusion-weighted images and 32 to 57 in b = 4000 s/mm² images across all diffusion times/frequencies.

For group analysis, 3D data from all cuprizone-treated and control mice were registered to one representative control brain chosen as the anatomical reference, using linear rigid alignment followed by dual-channel large deformation diffeomorphic metric mapping [41] driven by fractional anisotropy and non-diffusion-weighted contrasts. The transformation matrices were then applied to co-register the PGSE- and OGSE-based diffusivity and kurtosis maps from each mouse brain. Regions of interest (ROIs) in select gray and white matter structures were manually defined using RoiEditor (www.mristudio.org) based on a reference atlas [42]. Statistical analyses were performed using Matlab. Statistical comparisons of kurtosis and diffusivity values across different diffusion times/frequencies and between OGSE acquisitions with varying N were performed over voxels in the ROIs using nonparametric Wilcoxon signed-rank tests for paired data. Differences in OGSE and PGSE measurements between the control and cuprizone-treated groups were evaluated using Mann-Whitney U tests.

Monte Carlo simulations

Monte Carlo (MC) simulations of OGSE and PGSE signals were performed using random walkers implemented in Camino [43]. Particles were initially uniformly distributed inside and between parallel cylinders with permeable membranes and hexagonal packing. To investigate the coupled effects of varying restriction size and permeability on non-Gaussian diffusion signal behavior and its time/frequency-dependence, substrates with different cylinder radii (from 1.5 μm to 2.5 μm) and different membrane permeabilities were simulated. The intracellular volume fraction was kept fixed at 63%. The intra-cellular and extra-cellular diffusivity of walkers (D₀) was set to 1 μm²/ms. Permeability, specified as the probability (p) of a walker stepping through the membrane, was varied up to 0.01 [44] to span the physiologically relevant range of membrane permeabilities reported in literature [45]. The gradients were applied perpendicular to the cylinder long-axes.

First, OGSE and PGSE signals were simulated for the same diffusion-encoding gradient waveforms and parameters (6 Δs, 2 frequencies, 9 b-values) as the mouse brain experiments, with substrates of varying cylinder radii and permeabilities. The simulations were also repeated for substrates consisting of randomly-packed cylinders with gamma-distributed radii.

Second, to evaluate the effect of the diffusion-encoding gradient waveform on the measured signal behavior and kurtosis, we performed simulations over an extended range of effective diffusion times (t_D = 3.6 to 20 ms) for pulsed-gradients and waveform frequencies (f = 12.5 to 70 Hz) for oscillating-gradients, with the same b-values of 0-4000 s/mm². The number of oscillation periods on either side of the 180° pulse (N) for OGSE was set to 1, and pulse duration (δ) for PGSE was kept the same as the experimental data (1.9 ms). While practical gradient constraints limit the acquisition of OGSE and PGSE data over an overlapping range, MC simulations were performed to probe how changes in permeability affect the time/frequency-dependence of kurtosis with either encoding scheme.

Third, to probe the potential effect of number of oscillation periods in OGSE acquisitions, simulations were performed for the same gradient waveforms as the mouse brain experiments above (f = 105 Hz, N = 1 and 2), with the membrane permeability varied from 0 to 0.01.

All simulations were performed with 10⁵ walkers and a time step (Δt) of 2.5 μs. For each set of simulations, both OGSE and PGSE signals were computed for the same spin trajectories. The simulated signal intensities over all b-values were fit to Eq. 2 using Levenberg-Marquardt nonlinear fitting in Matlab to obtain the diffusivity and kurtosis estimates at each diffusion time/frequency.

Histopathology

Following MRI experiments, the cuprizone-treated and control mouse brains were processed for histology and sectioned into 30-μm thick coronal slices. Select sections were mounted on slides and stained using Black Gold II for assessment of myelin in the genu (gcc) and splenium (scc) of the corpus callosum. Briefly, slides were incubated in 0.3% Black Gold at 60°C for 12–20 min until the thinnest fibers were stained. The slides were then fixed in 1% sodium thiosulfate for 3 min, counterstained with 0.1% Cresyl Violet (Millipore, Burlington, MA, USA) for 3 min, dehydrated using a series of gradated alcohols, cleared in xylene, and coverslipped with VectaMount Permanent Mounting Media (Vector Laboratories, Burlingame, CA, USA). Slides were imaged on a BX41 Olympus microscope using ImageProPlus5.1 software.

RESULTS

Fig. 1 shows comparison of PGSE- and OGSE-based mean diffusivity (MD) and MK maps of an adult mouse brain from 3D tensor data acquired with Δ = 22 ms and f = 70 Hz, respectively. MD and MK exhibited significant region-specific differences between PGSE and OGSE maps, which were reproducible across all adult mouse brains. While MD showed a clear time-dependent decrease in regions corresponding to the cerebellar granule cell layer (Cbgr) and dentate gyrus (Fig. 1a), MK was found to exhibit a dramatic increase in the Cbgr and white matter tracts including the corpus callosum (Fig. 1b). In PGSE-based maps, the Cbgr was marked by significantly (P < 0.005) higher MK (1.09 ± 0.12) compared with other gray matter regions, whereas in OGSE-based maps, MK reduced drastically and approached zero selectively in the Cbgr, falling below MK in other gray matter regions such as the cortex (Fig. 1b, arrowheads). Plots of PGSE and OGSE signal attenuation in the Cbgr and cortex as a function of b-value are shown in Fig. 2. The resulting contrasts highlighting specific gray and white matter regions can be clearly seen in the ΔMD and ΔMK maps, which represent voxel-wise differences between the PGSE and OGSE maps (Fig. 1a-c).

Fig. 1:

Comparison of PGSE- and OGSE-based mean diffusivity (MD) and mean kurtosis (MK) maps of an adult mouse brain. a, b) A horizontal slice from 3D PGSE and OGSE tensor datasets acquired with Δ = 22 ms and f = 70 Hz, respectively, is shown. Maps representing the difference between PGSE- and OGSE-based MD and MK (ΔMD and ΔMK, respectively) reveal selective enhancement of distinct gray and white matter regions in the brain, as seen by comparison with the Nissl-stained section (c). Arrowheads indicate the dramatic reduction in MK observed in the granule cell layer of the cerebellum (Cbgr) in OGSE-based maps. The Nissl-stained section is from [62]. Abbreviations are: Cx: cortex, cc: corpus callosum, DG: dentate gyrus, Cbgr: cerebellar granule cell layer.

Fig. 2:

Plots showing signal attenuation as a function of b-value for PGSE and OGSE in gray matter regions of the mouse brain. a) Sagittal MK sections show sharply reduced kurtosis in the Cbgr compared to the cortex (Cx) in OGSE maps. b, c) Signal attenuation (ln S/S₀) along one representative direction plotted as a function of b-value. The ln(S/S₀) plot for OGSE in the Cbgr is approximately linear, indicating that kurtosis approaches zero at f = 70 Hz. Plots for the cortex exhibit a distinct curvature for both PGSE and OGSE data. Data points in (b, c) denote mean (± standard deviation) measurements over the regions of interest. Solid curves represent fits of the experimental data to Eq. [2].

Plots in Fig. 3 show the time/frequency-dependent changes in kurtosis and diffusivity in the Cbgr and cortex over the range f = 140 Hz to t_D = 30 ms measured with 2D PGSE and OGSE experiments. For both gray matter regions, kurtosis showed a characteristic biphasic behavior with diffusion time. In the Cbgr, kurtosis was close to zero at 140 Hz, continued to increase till it reached a peak value (1.02 ± 0.09) at t_D of 20 ms, and started to decline thereafter (Fig. 3a). In comparison, kurtosis in the cortex showed a more rapid rate of initial increase to reach a relatively lower peak value (0.79 ± 0.03) at t_D of 10 ms, and continued to decline steadily at longer t_Ds (Fig. 3b). Diffusivity in both structures decreased monotonically as t_D increased, with a larger percentage decrease observed in the Cbgr (56.8 ± 3.4%) than the cortex (16.9 ± 3.1%) across the time/frequency range examined (Fig. 3c-d). Corresponding maps of kurtosis and diffusivity showing the t_D-dependent contrast evolution can be found in Supporting Information Figure S1. Diffusivity values for the PVP phantom measured with the same PGSE and OGSE sequences are also plotted in Fig. 3. Signal decay in the PVP phantom was monoexponential for all times/frequencies, with an average diffusivity value of 0.74 ± 0.02 μm²/ms obtained at 25°C, which is in close agreement with previous studies [46].

Fig. 3:

Time-dependence of kurtosis and diffusivity in gray matter regions of the mouse brain. Kurtosis (a, b) and diffusivity (c, d) values fitted from OGSE and PGSE data over 9 b-values at each diffusion time/frequency are plotted. For visualization, OGSE data (squares) are plotted against 1/4f and PGSE data (circles) are plotted against t_D (= Δ-δ/3). Data points represent the mean (± standard deviation) values over regions of interest in the Cbgr and cortex. Asterisks in (a) and (b) indicate the peak kurtosis positions for each ROI. Diffusivity values for the PVP phantom are plotted in (c) for reference. Corresponding maps of diffusivity and kurtosis for the mouse brain at all sampled diffusion times/frequencies can be found in Supporting Information Fig. S1.

Group-averaged ΔMK maps of the control and cuprizone-treated mouse brains (n=5 each), along with corresponding PGSE (Δ = 22 ms) and OGSE (f = 70 Hz) based MK, RK, and AK maps at the level of the scc derived from 3D kurtosis tensor data are shown in Fig. 4. A significant (P < 0.05) localized decrease in ΔMK was observed in the scc of cuprizone-treated mice as compared to the control group (Fig. 4a, arrowheads). This localized decrease was reproducible across all cuprizone-treated mice, and was associated with a larger decrease in PGSE-based MK (33.8 ± 9.2%) than OGSE-based MK (19.4 ± 8.6%) with reference to the control brains. For both PGSE and OGSE data, RK was found to exhibit a larger decrease than AK in the scc of cuprizone-treated brains (Fig. 4b).

Fig. 4:

Group-averaged maps of control and cuprizone-treated mouse brains (n=5 each) showing changes in PGSE (Δ = 22 ms) and OGSE (f = 70 Hz) based diffusion kurtosis contrasts in the corpus callosum. a) Sagittal sections from ΔMK (MK_PGSE-MK_OGSE) maps reveal a selective loss of contrast in the splenium (scc) of cuprizone-treated mice (red arrowheads) as compared to the genu (gcc). b) Group-averaged PGSE- and OGSE-based MK, RK, and AK maps of control and cuprizone-treated mouse brains at the level of the scc. The anatomical location of the coronal slices is indicated by the dashed white line in (a).

Plots of kurtosis and diffusivity in the gcc and scc of control and cuprizone-treated brains measured from 2D OGSE and PGSE experiments over the range f = 140 Hz to t_D = 30 ms are shown in Fig. 5. For the control brain, kurtosis in both ROIs increased significantly up to t_D of 10 ms, with a steeper time-dependent decrease observed in the scc for longer t_Ds up to 30 ms (Fig. 5a,b). Fitted kurtosis and diffusivity values in the gcc of the cuprizone-treated brain did not differ significantly from the control brain (Fig. 5a,c). However, in the scc, the kurtosis curves diverged after f =140 Hz, with significantly (P < 0.005) lower kurtosis in the cuprizone-treated brain observed for t_Ds above 7 ms (Fig. 5b). Diffusivity in the scc was significantly (P < 0.005) elevated for the cuprizone-treated brain at all f and t_D points sampled (Fig. 5d). Histological Black Gold II stained sections from the same brains (Fig. 6) revealed extensive demyelination in the scc of the cuprizone-treated mice after 4 weeks, with relative sparing of the gcc.

Fig. 5:

Time-dependence of kurtosis and diffusivity in the corpus callosum of control and cuprizone-treated mouse brains. Kurtosis (a, b) and diffusivity (c, d) values from OGSE and PGSE data acquired along tetrahedral directions distributed at ~54.7° relative to the long-axis of the corpus callosum in the midsagittal plane are shown. For visualization, OGSE data (squares) are plotted against 1/4f while PGSE data (circles) are plotted against t_D (= Δ-δ/3). Data points represent the mean (± standard deviation) values over regions of interest in the genu (gcc) and splenium (scc) of the corpus callosum.

Fig. 6:

Black Gold II-stained sections of the corpus callosum from control and cuprizone-treated mouse brains. Coronal sections at the level of the genu (gcc) and splenium (scc) of the corpus callosum from a control mouse (left) and two mice after 4-weeks of cuprizone-treatment (right) are shown. Extensive demyelination is seen in the scc of the cuprizone-treated mice, with relative sparing of the gcc. Scale bar = 200 μm.

Fig. 7 shows results of fitted kurtosis and diffusivity from MC simulations for the same diffusion-encoding gradient waveforms and parameters as the 2D mouse brain experiments, demonstrating the coupled effects of varying restriction size and permeability. Values shown in Figs. 7-9 were fit from simulated PGSE and OGSE signals using the Levenberg-Marquardt algorithm with all correlation coefficients ≥ 0.998. For non-zero permeability values, the kurtosis curves exhibited a non-monotonic or biphasic response with diffusion time (Fig. 7, top panel). As permeability increased for fixed cylinder radii, kurtosis showed a lower peak value and the peak position shifted progressively to shorter t_Ds, with a steeper rate of time-dependent decrease observed after the peak. As cylinder radii increased, the kurtosis peak also decreased in magnitude, but the peak occurred at progressively longer t_Ds. Diffusivity decayed monotonically with diffusion time in all cases (Fig. 7, bottom panel).

Fig. 7:

MC simulation results for the same diffusion-encoding gradient waveforms and parameters as those used in the mouse brain experiments, showing the coupled effects of varying restriction size and permeability on the time-dependent kurtosis curves. Fits from simulated signals for OGSE and PGSE waveforms are plotted against 1/4f and t_D, respectively. Shaded area demarcates the approximate effective diffusion-time range (≤ 5 ms) probed with OGSE sequences. Plots of kurtosis (K) and diffusivity (D) for different cylinder radii (R) and membrane permeabilities (p) are shown in the top and bottom rows, respectively. Plots for randomly-packed cylinders are included in Supporting Information Figure S2.

Fig. 9:

Effect of varying number of cycles (N) in OGSE waveforms on diffusivity and kurtosis estimates. a) Results from MC simulations for f = 105 Hz and N = 1 and 2, showing plots of fitted diffusivity (D) and kurtosis (K) as a function of permeability. b) Experimental OGSE data from an adult mouse brain with f = 105 Hz and N = 1 and 2, showing diffusivity and kurtosis measurements (mean ± standard deviation) for two regions corresponding to the cortex (Cx) and deep cerebellar nuclei (CbN). Examples of ln(S) versus b-value curves for N = 1 and 2 in both regions are shown in the plot on the right. Solid lines represent fits of the experimental data to Eq [2].

Results of MC simulations for PGSE and OGSE waveforms over an extended t_D and f range are shown in Fig. 8. For both PGSE and OGSE waveforms, kurtosis showed a non-monotonic behavior with diffusion time/frequency and a monotonic decrease with increasing permeability. However, OGSE-based kurtosis measurements showed a more rapid rate of decay as permeability increased than PGSE-based measurements at similar effective t_Ds (Fig. 8a-b). Fitted diffusivity values for simulated PGSE and OGSE signals were comparable, and showed similar rates of increase with increasing permeability (Fig. 8c-d).

Fig. 8:

Surface plots showing comparison of fitted kurtosis (a, b) and diffusivity (c, d) from MC simulations for pulsed and oscillating encoding waveforms over an extended time/frequency range. The x- and y- axes are normalized to t_D/t_R and t_R/τ_ex, respectively, with t_R= R²/2D₀ and τ_ex=αR/2κ, where α is the extracellular volume fraction and κ is the permeability corresponding to p [63]. Kurtosis for both PGSE and OGSE waveforms showed a biphasic response with diffusion time/frequency, and a monotonic decay with permeability. OGSE-based measurements of kurtosis showed a more rapid rate of decrease with increasing permeability than PGSE-based measurements for similar values of t_D and 1/4f.

Fig. 9 shows results of MC simulations along with experimental OGSE data from a mouse brain for f = 105 Hz and N = 1 and 2. For impermeable membranes (p = 0), varying N had no significant effect on simulated kurtosis (Fig. 9a). For p > 0, the kurtosis curves diverged and kurtosis for N = 2 showed a more rapid rate of decrease with increasing permeability. No significant differences were observed in fitted diffusivities for the two gradient waveforms (Fig. 9a). Experimental data from the mouse brain with N = 1 and 2 (Fig. 9b) were consistent with the simulation results for p > 0. Fig. 9b shows the average diffusivity and kurtosis for two regions corresponding to the cortex and deep cerebellar nuclei. The plots reveal significantly (P < 0.005) lower kurtosis for N = 2, with no significant differences in corresponding diffusivity values in both regions. Examples of signal decay curves for N = 1 and 2, demonstrating the divergence in signal intensity at higher b-values, are shown in Fig. 9b.

DISCUSSION

This results of this study show unique tissue contrasts based on the time-dependence of diffusional kurtosis in both gray and white matter regions of the mouse brain. 3D kurtosis tensor data acquired with b-values of up to 4000 s/mm² for both OGSE and PGSE sequences showed region-specific contrasts in highly reproducible areas of the brain. Further, 2D data over a range of diffusion times/frequencies revealed a biphasic or non-monotonic time-varying behavior of kurtosis, with peak locations that varied across brain structures. In the cuprizone mouse model, a selective decrease in ΔMK was observed in the scc, corresponding to localized demyelination seen with histopathological evaluation. MC simulations were used to probe the effect of varying restriction size and permeability on the non-Gaussian signal behavior, and to characterize differences in PGSE and OGSE based kurtosis measurements in the presence of exchange. Finally, simulations and mouse brain experiments were used to demonstrate the effect of varying number of cycles in OGSE sequences.

ΔMD and ΔMK maps derived from 3D PGSE and OGSE data exhibited significant differences across brain regions. In particular, the Cbgr showed selectively vanishing kurtosis in OGSE maps, which increased to become significantly higher than other gray matter regions (e.g., the cortex) in PGSE maps. Moreover, 2D experiments showed that the time-dependent behavior of kurtosis in both regions was non-monotonic, with the Cbgr consistently showing a higher kurtosis peak at a longer t_D of 20 ms (Fig. 3). The Cbgr consists of densely-packed somata of cerebellar granule cells, which constitute more than half of the neurons in the CNS and have a spherical cell body (5-10 μm in diameter) that is almost entirely occupied by the nucleus [47]. The cellular architecture of the cortex is more heterogeneous, and consists mostly of neuropil (axons, dendrites and glial cell processes, with average diameters < 1 μm) with relatively few neuronal cell bodies [48]. Given the dimensions and morphology of cerebellar granule cells and the plots in Fig. 3, the selectively vanishing kurtosis in the Cbgr observed in OGSE maps likely reflects that diffusion is approaching the Gaussian limit wherein water molecules encounter minimal restrictive effects from membranes or barriers at corresponding length scales. Interestingly, this limit was approached in the Cbgr, but not the dentate gyrus (DG in Fig. 1) which consists of densely-packed granule cells with slightly larger (~10-18 μm) somas [49], even though both regions exhibited a strong change in MD over the time/frequency range examined.

In this short-time limit or Mitra regime, the surface-to-volume ratio (S/V) of restrictions can be further estimated from the time/frequency-dependent signal behavior [50]. OGSE-based diffusivity measurements at 70, 105, and 140 Hz in the Cbgr showed a closely linear dependence on the square-root of the inverse of frequency (Supporting Information Fig. S3). Fitting the data to $D(\omega )\cong D_0\left(1-c(\mathrm{N})\frac{1}{d\sqrt{2}}\frac{S}{V}\sqrt{\frac{D_0}{\omega }}\right)$ (Eq. [3]) derived for cosine-modulated gradients with finite N [51,52], where ω = 2πf, D₀ is the free diffusivity, d = 3 for isotropic diffusion in gray matter, and c(N) is the correction factor for finite number of oscillation cycles [51], yields estimates of mean D₀ = 1.15 μm²/ms and S/V = 1.23 μm⁻¹ with coefficient of determination = 0.998 (Figure S3). This translates to an estimated radius or restriction length of ~2.4 μm for spherical pores, which is in close agreement with the sizes of cerebellar granule cells reported in literature [47].

The non-monotonic behavior of kurtosis observed in our experiments is consistent with simulation studies in a geometric model of packed cylinders [30] and recent reports in the rat cortex [32] and fixed spinal cord white matter [33]. Using MC simulations, Fieremans et al. [30] showed that in the long-time limit, the behavior of kurtosis under sufficiently low permeability (barrier-limited exchange) can be approximated by the Kärger model [53], wherein kurtosis decreases with t_D on a scale set by the exchange time. However, the Kärger model assumes Gaussian diffusion in all compartments, and does not account for the effects of intra-compartmental kurtoses or time-varying diffusivity, which are observed at short to intermediate diffusion times. In our study, for the sampled range of f = 140 Hz to t_D = 30 ms, a biphasic behavior was observed for most brain regions, resulting in kurtosis contrasts that evolved with diffusion time (Figs. 1, S1). The initial increase in kurtosis likely reflects increasing interactions of water molecules with membranes or barriers as t_D increases [3]. Comparison of time-dependent kurtosis curves in Fig. 3 further indicates that for t_Ds in the range of 10 - 20 ms, progressive loss of diffusional heterogeneity is already observable in cortical gray matter, but not in the Cbgr, which accounts for the dramatically elevated MK in the Cbgr seen in 3D PGSE maps acquired with Δ = 22 ms (Fig. 1). These results show that the time-dependence of kurtosis can provide unique sensitivity to probe the intrinsic heterogeneity of cellular microenvironments across different gray matter regions in the brain.

In the corpus callosum, MK, RK, and AK were significantly higher compared with gray matter regions in both PGSE- and OGSE-based maps. Moreover, a sharp region-specific decrease was observed in ΔMK maps following localized demyelination in the cuprizone-treated group (Fig. 4). Axons in the mouse corpus callosum have typical diameters in the range 0.1–3 μm, with a mean of ~0.56 μm reported in electron microscopy studies [54]. Potential microstructural changes contributing to the observed differences in ΔMK maps and t_D-dependent kurtosis curves include decrease in intra-axonal volume fraction and increase in membrane permeability associated with demyelination. The rate of water exchange across axonal membranes is likely to be strongly influenced by the myelin sheath [18]. While estimates of membrane permeability in white matter are limited in literature, a previous study by Stanisz et al. [55] estimated an axonal membrane permeability of 0.009 μm/ms in the ex vivo bovine optic nerve at 20°C, which corresponded to an exchange time of ~60 ms for axons averaging 2.6 μm in diameter. Given these estimates, it is reasonable to postulate that the exchange times for the small and relatively thinly myelinated axons in the fixed mouse scc at 37°C are shorter, and more comparable to the diffusion time range probed in our study. Kurtosis and diffusivity curves from MC simulations of OGSE and PGSE signals with increasing permeability (Fig. 7) showed similar behavior as the experimental data for the scc of control and cuprizone-treated brains. Furthermore, a higher density of unmyelinated axons has been reported in the scc of rodents as compared to the gcc [56, 57], which could account for observed differences in the rate of t_D-dependent kurtosis decay between the two ROIs (Fig. 5a,b). A recent study using numerical modeling also reported reduced intra-axonal residence times in the scc of multiple sclerosis patients, which was attributed to increased permeability due to myelin damage [58]. However, the pathophysiology of cuprizone-induced demyelination is complex, and other factors such as the presence of glial cells, myelin debris, or inflammation associated with myelin breakdown [13,22] could also contribute to the observed changes.

OGSE-based measurements of kurtosis have been reported in recent studies [31,32,59]. Our simulation results show that there are significant differences between PGSE and OGSE based estimates of kurtosis in the presence of exchange (Fig. 8a-b). Previously, Portnoy et al. [31] reported similar findings in fixed rat hippocampal slices, with opposite trends in PGSE and OGSE based kurtosis observed with increasing diffusion time. Our experimental and simulation results show a time/frequency-dependent biphasic behavior for both encoding schemes, however they differ in their sensitivity to exchange. While OGSE sequences at higher frequencies are sensitive to shorter length scales, the notion of an equivalent t_D is not well-defined for oscillating-gradient waveforms. Therefore, OGSE acquisitions are best interpreted using the frequency domain approach [15,60]. Heuristically, an effective t_D for cosine-modulated gradients has been calculated to be 1/4f [21] (or 9/64f [52]), and following the same approach it is ~0.9 times this value for the trapezoid-cosine gradients used in our study. Diffusivities for both schemes fit from MC simulations for similar values of t_D and 1/4f showed close agreement over a wide range of permeability values (Fig. 8). Diffusion-weighting with oscillating gradients accumulates over successive (2N) cycles of dephasing and rephasing lobes, and intra/extra-cellular exchange that occurs on a time-scale comparable to or shorter than the timing of multiple encoding periods can affect the signal differently as compared to PGSE acquisitions.

Results of both MC simulations and mouse brain data acquired with the same gradient waveforms (f=105 Hz, N=1 and 2) showed that kurtosis reduces with increasing N, when all other parameters are kept the same (Fig. 9). Notably, the divergence of kurtosis curves with N in simulations was observed only for non-zero permeability values (p>0) (Fig. 9a), which further shows that the exchange regime (fast vs. slow) determined by both the number and duration of encoding periods affects the kurtosis behavior in OGSE acquisitions. As the gradient waveform duration is lengthened with increasing N, it is more likely for spins to undergo exchange between different diffusion-encoding or oscillation periods. Interestingly, the diffusivity values were relatively independent of N in both simulations and experimental data (Fig. 9a,b), which is reasonable, given that increased sensitivity to exchange is expected at higher b-values [45]. These findings suggest that PGSE and OGSE acquisitions may probe different exchange regimes. It is clear that both schemes reveal highly reproducible kurtosis estimates and signal behavior across brain regions. These results also indicate that in addition to information about varying restriction scales, ΔMK maps reflect sensitivity to differences in exchange regimes across brain regions, and thus carry information that is distinct from that probed with ΔMD maps for the same time/frequency range.

Probing the behavior of kurtosis and diffusivity estimates over a wide time/frequency range can allow novel insights into different microstructural features across brain regions that govern the transition from Gaussian to non-Gaussian diffusion, similar to our observations in the cerebellum. However, this becomes rapidly challenging for higher frequencies, as the b-value scales inversely with f³ [21], and the waveform duration is in practice constrained by T₂-decay and SNR. In our study, kurtosis tensor data with a b-value up to 4000 s/mm² could be acquired for f = 70 Hz, while acquisitions at 140 Hz necessitated the use of tetrahedral encoding. Truncation of the waveforms to one or two cycles also affects the width of frequency peaks in the dephasing spectra, as shown previously [22,23]. Acquisitions over multiple times/frequencies could further benefit from accelerated kurtosis imaging techniques [61]. It is worth noting that the time/frequency range to observe the biphasic kurtosis behavior and tissue contrasts in vivo may differ. Qualitatively, our observations in the cortex are similar to those reported by Pyatigorskaya et al. [32] for the in vivo rat cortex. Similar to their findings, a time-dependent decrease in kurtosis was observed after t_D ~10 ms, while OGSE measurements showed a decrease with increasing frequency. Assigning the t_D-dependent changes to specific tissue compartments, such as intra- or extra-cellular spaces, is not straightforward and requires further investigation. Understanding the signal behavior over a range of diffusion times/frequencies could potentially allow development of models that incorporate the effects of both exchange and geometrical features of tissue [45]. While the focus of MC simulations in our study was to evaluate the effects of varying restriction size and exchange on differences between OGSE and PGSE signals, future work will allow probing the effects of other features such as disorder in axonal packing, intracellular volume fraction, and ratio of intra-to-extracellular diffusivities, which are also likely to influence the time/frequency-dependent signal behavior.

In conclusion, this study demonstrated significant diffusion time/frequency-dependence of kurtosis contrasts in the brain that are sensitive to intrinsic microstructural heterogeneity across both gray and white matter regions, and to changes induced by pathological disruptions such as demyelination. Changes in both cell size and permeability are important during brain development and in many neurological disorders, e.g., cell swelling in edema, traumatic brain injury, and white matter disorders of myelin such as multiple sclerosis. Probing the non-Gaussian signal behavior over a range of diffusion times/frequencies could thus provide valuable insights into regional changes in brain tissue microstructure and pathological alterations. Our simulation results also demonstrate that exchange has a major influence on OGSE and PGSE based estimates of kurtosis, which should be considered when comparing ΔMK and ΔMD maps for the same time/frequency range.

Supplementary Material

supp info

Supporting Information Figure S1: Diffusivity (D) and kurtosis (K) maps of an adult mouse brain from 2D OGSE and PGSE experiments showing diffusion-time/frequency dependent evolution of contrasts over the range f = 140 Hz to t_D = 30 ms. Diffusivity and kurtosis maps were fit using the Levenberg-Marquardt algorithm from OGSE and PGSE data acquired with 9 b-values and tetrahedral encoding at each diffusion time/frequency. OGSE data at 70 Hz and 140 Hz were acquired with N = 1 and 2 cycles, respectively. A clear time-dependent monotonic decrease in D and biphasic behavior of K can be seen for both the Cbgr (top panel) and the cortex (bottom panel). Black arrows indicate the different effective t_Ds corresponding to kurtosis peaks in the Cbgr and cortex.

Supporting Information Figure S2: Simulated kurtosis and diffusivity curves for randomly-packed cylinders with gamma-distributed radii and varying permeability (p). Histograms of two example radii distributions for substrates with fixed intracellular volume fraction = 0.63 are shown on the left, with mean radius (μ_R) = 1.5 μm (top panel) and 2 μm (bottom panel), and standard deviation = 0.3μ_R. Simulations were performed for oscillating- and pulsed-gradient waveforms at the same diffusion times/frequencies of f = 140 Hz to t_D = 30 ms as the plots shown in Figure 7.

Supporting Information Figure S3: OGSE- and PGSE-based diffusivity values in the Cbgr from a representative mouse brain plotted against square-root of the inverse of oscillation frequency ω (OGSE, filled squares) or square-root of diffusion time t_D (PGSE, circles). Data points represent the mean ± standard deviation of diffusivity values over the region of interest at each time-point. OGSE data acquired at 70, 105, and 140 Hz show a closely linear dependence on ω^−1/2. The dashed black line represents fitting of the OGSE data points to Eq. [3].