Optimising microstructural characterisation of white-matter phantoms: impact of gradient waveform modulation on Non-uniform Oscillating Gradient Spin-Echo sequences

We focus on extracting information regarding the microstructure sizes within tissues or porous media by monitoring the diffusion of water molecules in confined compartments. The nuclear magnetic resonance (NMR) signal is affected by the dephasing effects due to the Brownian motion of the molecules within these compartments.

The NMR signal evolution is determined from the interaction of the ensemble of 1H nuclear $1/2$-spins with an external static magnetic field $B_{0}$ and an applied magnetic field gradient $G(t)$ along a spatial direction $\hat{r}$ grebenkov_nmr_2007 ; callaghan_translational_2011 . The gradient strength $G(t)$ is modulated as a function of time as $G(t)$ = $Gf(t)$, varying its sign and strength with the function $\left|f(t)\right|\leq 1$ stepisnik_time-dependent_1993 ; callaghan_frequency-domain_1995 ; gore_characterization_2010 . Here, $G$ is the gradient strength. The static field $B_{0}$ defines the Larmor frequency $\gamma B_{0}$, with $\gamma$ the gyromagnetic ratio of the nucleus. In the rotating frame of reference at such frequency, the spin’s precession frequency $\omega(t)$ = $\gamma G(t)r(t)$ is determined by the instantaneous gradient strength and position of the spin, thus fluctuating due to diffusion-induced displacements of the water molecules grebenkov_nmr_2007 ; callaghan_translational_2011 . The spin position $r(t)$ describes the diffusing trajectory of the nuclear spin along the field gradient direction.

The spins acquire a phase $\phi(t_{D})$ according to their random displacements as a function of the total diffusion time $t_{D}$. The random phase $\phi(t_{D})$ typically follows a Gaussian phase distribution klauder_spectral_1962 ; stepisnik_analysis_1981 ; stepisnik_validity_1999 . Hence, the diffusion-weighted NMR signal becomes

which is determined from the spin ensemble average. It decays as a function of time, depending on the random phase’s variance $\langle\phi^{2}(t_{D})\rangle$ over the spin ensemble.

The attenuation factor $\beta(t_{D})$ can be written in a Fourier representation as stepisnik_time-dependent_1993 ; lasic_displacement_2006 ; shemesh_measuring_2013 ; alvarez_coherent_2013 ; zwick_maximizing_2016

where $F(\omega,t_{D})$ is a filter function defined by the Fourier Transform (FT) of the gradient modulation $G(t)$ and $S(\omega)$ is the displacement spectral density given by the FT of the spin’s displacement correlation function $\langle\Delta r(0)\Delta r(t)\rangle$. The spin’s displacement $\Delta r(t)$ is given by $r(t)-\langle r(t)\rangle$. Within this FT representation, the signal decay is determined by the integral of the overlap between the filter function $F(\omega,t_{D})$ and the spectral density $S(\omega)$. The filter $F(\omega,t_{D})$ depends exclusively on the setup parameters of the chosen MGSE sequence, while the spectral density $S(\omega)$ depends on the characteristics of the diffusion-driven fluctuations. We consider that in typical restricted diffusion conditions, it is well approximated by a Lorentzian function stepisnik_time-dependent_1993 ; shemesh_measuring_2013 ; alvarez_coherent_2013 ; zwick_precision_2020

The free diffusion coefficient is $D_{0}$ and $\tau_{c}$ the time required for most molecules to fully probe the compartment boundaries. Considering the Einstein’s expression $l_{c}^{2}$ = $2D_{0}\tau_{c}$, the characteristic time $\tau_{c}$ is related to the diffusion restriction length $l_{c}$ imposed by the microstructure compartment in which diffusion is taking place callaghan_translational_2011 .

Therefore, by suitably choosing the oscillating gradient modulation parameters to steer the NMR signal decay, one can probe the water diffusion process and, consequently, extract microstructural restriction sizes from its time dependence gore_characterization_2010 ; shemesh_measuring_2013 ; alvarez_coherent_2013 ; xu_mapping_2014 ; shemesh_size_2015 ; capiglioni_noninvasive_2021 .

Morphological heterogeneities in most biological tissues and porous media lead to multiple compartment sizes. To describe these complex systems, a probability distribution for the compartment sizes is typically used assaf_axcaliber_2008 ; barazany_vivo_2009 ; nilsson_role_2013 . In the particular case of white-matter tissue, a lognormal probability distribution $P(l)$ is often observed for the restriction sizes $l$ pajevic_optimum_2013 ; liewald_distribution_2014 ; shemesh_size_2015 ; capiglioni_noninvasive_2021 . Hence, we consider such distribution to describe the interstitial compartment sizes on the white-matter phantoms used in this work, which is given by

with the mean size $l_{c}$ = $e^{\mu+\sigma^{2}/2}$, the standard deviation $\sigma_{c}$ = $\sqrt{e^{\sigma^{2}+2\mu}\left(e^{\sigma^{2}}-1\right)}$, the median $e^{\mu}$ and the mode ${e^{\mu-\sigma^{2}}}$.

The total signal originated from the spins diffusing within this compartment size distribution model is therefore determined by the weighted average

where the magnetisation signal $M_{l}(t_{D})$ corresponds to the signal given by spins diffusing within a compartment size $l$. In the implemented fitting model we consider a discrete version of the lognormal distribution. Including the factor $1/\Sigma_{l}P(l)$ we ensure the normalisation of the distribution.

The NOGSE sequence shemesh_measuring_2013 ; alvarez_coherent_2013 is depicted in Figure 1a, schematized with a sharp modulation, i.e. with instantaneous switching of the gradient’s sign. It comprises two concatenated sequences that produce a signal decay similar to that of a CPMG (OGSE) modulation carr_effects_1954 ; meiboom_modified_1958 followed by a Hahn (Gradient Echo) modulation hahn_spin_1950 . The total diffusion time is $t_{D}$ and the total number of gradient lobes is $N$. For the sharp gradient modulation, refocusing lobes of duration $t_{C}$ are repeated $N-1$ times, corresponding to the CPMG modulation. Notice that, as the modulation shape is based on a cosine symmetry, the first and last gradient pulses are of duration $t_{C}/2$. The last refocusing period corresponds to the Hahn counterpart with a total duration $t_{D}-(N-1)t_{C}$. As $t_{D}$, $N$ and the Echo Time between the acquired images are kept fixed, gradient switching imperfections and additional relaxation effects are factored out shemesh_measuring_2013 ; alvarez_coherent_2013 .

As instantaneous switching of the modulating gradients is not experimentally achievable, we performed the experiments with trapezoidal waveforms with the fastest slew rate possible. This sets a paradigmatic sharp gradient modulation sequence, which one can approximate as an instantaneous switch of the gradient sign, if gradient ramp times are much shorter than the lobes duration $t_{C}$. If not, they are assumed as gradient switching imperfections that are factored out by the NOGSE approach shemesh_measuring_2013 ; alvarez_coherent_2013 .

The NOGSE sequence selectively probes microstructure compartment sizes from a signal decay-shift, rather than from a signal decay-rate shemesh_measuring_2013 ; alvarez_coherent_2013 ; shemesh_size_2015 ; capiglioni_noninvasive_2021 . Figure 1b shows the signal decay of the sharp-modulated NOGSE at the limiting values of the lobe duration $t_{C}$, $M_{NOGSE}(t_{D},t_{C}\sim 0,N)$ = $M_{Hahn}(t_{D})$ for $t_{C}$ = 0 and $M_{NOGSE}(t_{D},t_{C}=t_{D}/N,N)$ = $M_{CPMG}(t_{D},N)$ for $t_{C}$ = $t_{D}/N$. Analysing the magnetisation signal as a function of the diffusion time $t_{D}$, the NOGSE contrast $\Delta M$ is determined from the signal difference between the magnetisation decay at $t_{D}$ at the limiting values of $t_{C}$, described above. The difference between these two limiting values is the NOGSE modulation contrast $\Delta M$ = $M_{CPMG}(t_{D},N)-M_{Hahn}(t_{D})$.

In the restricted diffusion regime, the diffusing spins experience a spatial restriction imposed by the confining cavity. To sense these restrictions, the lobe duration $t_{C}$ needs to be longer than the restriction correlation time $\tau_{c}$. At $t_{C}$ = 0 the NOGSE signal decays with the attenuation factor

while at $t_{C}=t_{D}/N$, with

Therefore, the NOGSE contrast with a sharp gradient modulation at the diffusion restricted regime is (see Appendix A.1) shemesh_measuring_2013 ; alvarez_coherent_2013 ; capiglioni_noninvasive_2021

This signal evolution evidences a decay-rate $\gamma^{2}G^{2}D_{0}\tau_{c}^{2}$ that is independent of the number of lobes and it is proportional to $\tau_{c}^{2}$, as in OGSE-type experiments klauder_spectral_1962 ; stepisnik_time-dependent_1993 ; callaghan_frequency-domain_1995 ; baron_oscillating_2014 ; drobnjak_pgse_2016 . The signal behaviour manifests a decay-shift, which is a constant term $\gamma^{2}G^{2}D_{0}\tau_{c}^{3}(1+2N)$ independent of the total diffusion time $t_{D}$ and varies as $\tau_{c}^{3}$ shemesh_measuring_2013 ; alvarez_coherent_2013 . This NOGSE decay-shift arises if the gradient modulations are sharp, with a parametric sensitivity to microstructure compartment sizes proportional to $l_{c}^{6}$, enabling selective probing of specific restriction sizes capiglioni_noninvasive_2021 .

The NOGSE contrast $\Delta M^{restr}_{sharp}$ of Equation 8 results from considering ideal gradient pulses, with instantaneous ramps for the switching of the applied gradient amplitude (Figure 1a). We define this case as the sharp gradient modulation that gives a decay-shift independent of the diffusion time $t_{D}$. However, we observe that if we deviate from the ideal case scenario, i.e. if ramps are smooth (Figure 1c), the decay-shift decreases as a function of $t_{D}$ (see Figure 1d). In this study we consider for smooth gradient modulations the paradigmatic case of a sinusoidal functional dependence. Here, the first $N-2$ lobes, with $N\geq 4$, correspond to the CPMG modulation and the last two to the Hahn modulation, with a duration $t_{D}-(N-2)t_{C}$.

In order to make both implementations of NOGSE modulations comparable in the scanner, $t_{D}$, $N$, $t_{C}$ and $G$ take the same acquisition parameters for both types of gradient modulations.

The attenuation factor for this smooth NOGSE version at the restricted diffusion regime is (see Appendix A.2)

for the Hahn modulation at the limiting value $t_{C}\rightarrow 0$ and

for the CPMG modulation at the limiting value $t_{C}=t_{D}/N$. Therefore, for the smooth gradient modulation, the NOGSE contrast is

Here, the smooth modulation evidences a parametric sensitivity to probe compartment sizes proportional to $l_{c}^{8}$. However, the shift decays as a function of the diffusion time as a power law proportional to $1/t_{D}$. This indicates that at very long diffusion times $t_{D}\gg\tau_{c}$, the decay-shift might become non-observable.

We mimic axon bundles with phantoms comprising aramid fibres of $\sim$12 $\mu$m mean diameter. A Scanning electron microscope (SEM) image of the fibres is shown in Figure 2a and a histogram to determine their diameter distribution is shown in Figure 2b. The samples for the SEM were placed on a sample holder and glued on a carbon tape. Subsequently, the sample holder was placed in a gold bath for one minute, generating a gold layer of approximately 10 nm.

To prepare the phantoms, fibres were aligned together and positioned inside a thermo-shrinkable plastic tube to form a fibre bundle. Then, it was heated to enhance the compaction degree. Several fibre bundles were put into a 15 ml conical centrifuge tube, and then they were slowly filled with demineralized water so as to let the water enter the fibre bundles by capillary effect. Once the phantom was completely immersed in water, it was left in ultrasound to remove the remaining air bubbles.

As the fibres are both solid and impermeable, the water diffuses within the interstices of the extra-fibre region. The presence of fibres provides a tortuous environment with displacement restrictions for the diffusing water molecules, mimicking the extra-axonal space in white matter tracts.

A transverse plane DWI of the phantom is shown in Figure 2c, where parallel aramid fibre bundles of different packing densities are shown. We define several regions-of-interest (ROIs) on the transversal plane to evaluate the microstructure restriction-size distribution of the interstitial compartments. Outside the fibre bundles in the phantom (ROI5), there is freely diffusing water that is used for calibration, control purposes and to determine the free diffusion coefficient.

MRI scans were performed at 9.4T on a Bruker Avance III HD WB NMR scanner with a 1H resonance frequency of $\omega_{z}$ = 400.15 MHz. We used a Bruker Micro 2.5 probe capable of producing gradients of up to 1500 mT/m in the three spatial directions. We programmed and implemented with ParaVision 6 the NOGSE MRI sequences shown in Figure 3 capiglioni_noninvasive_2021 . All experiments have the following sequence parameters: the Repetition time TR = 2000 ms; Field of view (FOV) = 16$\times$16 mm², with a matrix size of 192$\times$192, leading to an in-plane resolution of 83$\times$83 $\mu$m²; and slice thickness of 1 mm.

For experiments involving the analysis of the NOGSE signal vs the duration $t_{C}$ in Sections IV.2 and IV.4, the images were acquired using echo planar imaging (EPI) encoding with 4 segments with a scan time of $\sim$ 4 minutes with 16 averages and an echo time TE = 70.5 ms; a diffusion time $t_{D}$ varying from 12.5 ms to 24.5 ms, although in most experiments $t_{D}$ was fixed at 21.5 ms. We used $2<N<20$ for sharp modulations and $4<N<20$ for smooth modulations. The time $t_{C}$ for the lobes duration varied between 0.3 ms and $t_{D}/N$; the gradient strength $G$ varied from 20 mT/m to 600 mT/m, with ramp times of 50 mT/m/us, and were applied perpendicular to the main axis of the aramid fibre bundles.

For experiments involving the analysis of the NOGSE signal vs the diffusion time $t_{D}$ (Section IV.3), the images were acquired with an EPI encoding with 4 segments, with an image scan time of $\sim$ 2.5 hours with 512 averages and an echo time TE = 171.5 ms; a diffusion time $t_{D}$ varied from 1.5 ms to 80 ms. We used $N$ = 4 for sharp and smooth modulations. The lobe duration $t_{C}$ = 0.3 ms was the shortest time used to approximate the Hahn-limit modulation for NOGSE and $t_{C}$ = $t_{D}/N$ for the CPMG-limit NOGSE modulation. The gradient strength $G$ was $G_{sharp}$ = 100 mT/m and $G_{smooth}$ = 141 mT/m for the sharp and smooth gradient modulations, respectively, with ramp times of 50 mT/m/us. These values were chosen to make the signal decay rate in the restricted diffusion regime equal (see Equations 6 and 9). The gradients were applied perpendicular to the main axis of the aramid fibre bundles.

We determined the relaxation times $T_{1}$ = 1794.5 ms, $T_{2}$ = 70.41 ms, and $T_{2}^{*}$ = 16.294 ms in ROI1 using RARE-VTR, MSME and MGE sequences, respectively. The free water relaxation times were determined from ROI5, giving $T_{1}$ = 3409.4 ms, $T_{2}$ = 154.4 ms, $T_{2}^{*}$ = 55.181 ms.

The NOGSE signal arising from each voxel and ROI of the acquired images was analysed and plotted as a function of the NOGSE’s acquisition parameters $t_{C}$, $t_{D}$, $N$ or $G$. Fittings to the experimental data were done based on the NOGSE theoretical model considering Equation 5 with the microstructure size parameters that describe the distribution, i.e. the mean size $l_{c}$ and standard deviation $\sigma_{c}$. A uniform free diffusion coefficient $D_{0}$ = 2.30 $\mu$m²/ms of water molecules at room temperature 25 °C was used based on its experimental estimation using the signal arising from ROI5 (see Section IV.2).

To evaluate the optimal acquisition setup for estimating microstructure restriction-size distributions, we use an information gain metric $S$ jimenez_optimization_2023 . It is the derivative of the mean square distance of the NOGSE curve with respect to changes in the parameters to be estimated, as these are obtained by fitting a theoretical representation to the experimental data based on reducing the mean square distance between the corresponding points. The highest derivative predicts the optimal NOGSE acquisition parameters for a given size distribution, consistently with quantum information estimation bounds zwick_precision_2020 ; jimenez_optimization_2023 .

More precisely, the measured magnetisation signal $M(\theta,\epsilon)$ is defined by the acquisition parameters $\theta=\{t_{d},G,N\}$ and the parameters that characterise the underlying lognormal restricton-size distribution of the sample under study, $\epsilon=\{\mu,\sigma\}$ (see Equation 4). Thus, the aim is to find an optimal set of parameters $\theta$ to measure $\epsilon$ with the greatest precision possible.

We can define the information gain metric $S$ as the sensitivity of the result of a measurement against perturbations $\epsilon^{\prime}=\{\mu^{\prime},\sigma^{\prime}\}$ in the real parameters to be measured $\epsilon=\{\mu,\sigma\}$.

The optimal acquisition parameters $\theta=\{t_{d},G,N\}$ are those that maximise the $D_{L2}(\theta,\epsilon,\epsilon^{\prime})$ distance for small variations $\epsilon^{\prime}=\{\mu^{\prime},\sigma^{\prime}\}$ in the vicinity of the true values $\epsilon=\{\mu,\sigma\}$. This $D_{L2}$ distance is the canonical distance for Lebesgue square-integrable functions (L2), since we consider the signal measurement as a function of the gradient-lobe duration $t_{c}$, and it is given by

As we aim to optimise two acquisition parameters – the diffusion time $t_{D}$ and gradient strength $G$ – we define the information gain metric

as the mean of the distance $D_{L2}$ over an ellipsoidal curve $C(\epsilon,\Delta)$, which is given by

This curve $C(\epsilon,\Delta)$ parametrises an ellipse with minor and major radius $r_{1}=\mu\Delta$ and $r_{2}=\sigma\Delta$. The parameter $\Delta$ defines the $\epsilon^{\prime}$ over which the $D_{L2}$ distance will be calculated. The minimum value $\Delta=0$ corresponds to taking $\epsilon^{\prime}=\epsilon$ and the maximum value $\Delta=1$, to $\epsilon^{\prime}=0$ and $\epsilon^{\prime}=2\epsilon$.

In this way, we obtain the sensitivity of the result of a measurement to changes in the set of parameters as a function of $\theta$. Therefore, the optimal control parameters will be those that maximise this information gain metric $S$.

$T_{2}$ relaxation was included in the calculation of the information gain metric $S$ assuming ideal modulations of the NOGSE sequences and echo time $TE$, without accounting for the noise introduced on the signal acquisition process. The echo time was calculated as $TE$ = $2t_{D}$ + $t_{p}$, with $t_{p}$ being the time left for the RF pulses and crusher gradients. The maximum diffusion time $t_{D}$ considered was $T_{2}/2$.

We first performed a preliminary estimation of the restriction size distribution in ROI1 of the fibre phantom (Figure 2c) to determine the NOGSE optimal configuration using the method described in Section III.3. We obtained $l_{c}$ = 7.3 $\mu$m and $\sigma_{c}$ = 2.8 $\mu$m from a fitting of the theoretical model of Equation 5 to the experimental data. These values of $l_{c}$ and $\sigma_{c}$ are the averaged parameters that describe the size distributions obtained using the smooth NOGSE modulation, with $N$ = 16, and several measasurements performed within the diffusuon time $t_{D}$ and gradinet strength $G$ ranges 40 - 60 ms and 70 - 300 mT/m, respectively. They were used as reference values to predict the optimal values of $N$, $G$ and the NOGSE diffusion time $t_{D}$ for each type of gradient modulation using the protocol described in Section III.3.

According to previous works, the number of refocusing periods should be kept to a minimum for a better inference of the microstructure-size restriction parameter zwick_precision_2020 . We found consistent results in the numerical evaluation; therefore, we kept the number of refocusing periods at the lowest achievable value. According to the available NOGSE modulation sequence design (Figure 1), the lowest $N$ used are 2 and 4 for the sharp and smooth gradient modulation, respectively.

Then, we performed a systematic exploration of the information gain metric described in Section III.3, as a function of $G$ and $t_{D}$. Figure 4 shows three information gain maps for the NOGSE modulations with $N_{sharp}$ = 2 or $N_{sharp}$ = 4 and $N_{smooth}$ = 4, respectively. They show the optimal values of the acquisition parameters $t_{D}$ and $G$ to estimate the restriction size distribution, corresponding to the highest information gain. For the NOGSE sharp modulation: $t_{D,\,opt}$ = 35.9 ms and $G_{opt}$ = 94.6 mT/m for $N$ = 2, and $t_{D,\,opt}$ = 37 ms and $G_{opt}$ = 99.5 mT/m for $N$ = 4. Whereas for the NOGSE smooth modulation: $t_{D,\,opt}$ = 35.2 ms and $G_{opt}$ = 144.3 mT/m. Although we find the optimal diffusion time $t_{D}$ around 36 ms, we used a lower value to improve signal-to-noise ratio (SNR) in all images. By experimental observation, we found $t_{D}$ = 21.5 ms to have a good compromise between the predicted optimal value and the SNR of the images.

At $t_{D}$ = 21.5 ms, the predicted optimal gradient strengths for the sharp modulations are $G_{sharp}$ = 169 mT/m with $N_{sharp}$ = 2 and $G_{sharp}$ = 184 mT/m if $N_{sharp}$ = 4, and for the smooth modulation $G_{smooth}$ = 259 mT/m with $N_{smooth}$ = 4. As evidenced in Figure 4, there is a plateau on the information gain where NOGSE reaches a comparable information gain (yellow-red region). Hence, we expect a similar estimation accuracy of the underlying microstructure size distribution by choosing $t_{D}$ and $G$ within this region. Furthermore, we do not find a significant difference between using sharp or smooth modulations. However, the required gradient strength increases for the smooth modulations, and gives a slightly better precision.

We estimate the free diffusion coefficient $D_{0}$ based on the free water signal in ROI5 (Figure 2c). When the diffusion time is much shorter than $\tau_{c}$, we achieve the free diffusion limit. The NOGSE magnetisation decays with the attenuation factor

for the sharp gradient modulation and

for the smooth NOGSE modulation. Notice that both expressions are independent of $\tau_{c}$, and consequently, of the restriction size $l_{c}$. We use these expressions to estimate $D_{0}$. Figure 5 shows the estimated $D_{0}$ values as a function of the NOGSE acquisition parameters $G$, $t_{D}$ and $N$, for the smooth and sharp modulations.

Figure 5 demonstrates that there is a range for the values of the acquisition parameters $G$, $t_{D}$ and $N$ where the estimated $D_{0}$ remains stable around $D_{0}$ = (2.30 $\pm$ 0.03) $\mu$m²/ms (averaged value of all the experiments performed in the shaded regions of the figure). These regions are thus more sensitive for accurately estimating $D_{0}$.

Outside of these regions, for gradient strengths $G$ exceeding approximately 300 mT/m, the signal experiences low SNR, resulting in an inefficient estimation. Similarly, excessively long diffusion times reduce SNR, leading to a diminished estimation efficiency. In contrast, for very low gradient strengths, selective RF pulses and image encoding introduce diffusion weighting effects, eclipsing the NOGSE diffusion weighting. The variation of the number of gradient lobes within the explored dynamic range does not yield significant changes. Nonetheless, a slight improvement is observed by increasing $N$.

We use the estimated free diffusion coefficient $D_{0}$ = 2.30 $\mu$m²/ms to obtain the restriction size distributions present in the different aramid fibre bundles of the phantom. Notice that the optimal acquisition parameters to estimate restriction size distributions are not in general the same as those identified for estimating $D_{0}$. For instance, an optimal diffusion time $t_{D}$ to sense restriction size distributions needs to be long enough to allow water molecules to fully explore the compartment restrictions zwick_precision_2020 . On the contrary, the optimal inference of the free diffusion coefficient does not require the water molecules to sense the compartment confinement.

We experimentally assess the manifestation of the NOGSE decay-shift by applying both gradient modulations on the white-matter phantom. Figure 6a displays the experimental measurements of the NOGSE signal decay, depicted in pink for the sharp modulation and light blue for the smooth modulation. The NOGSE decay is shown for the limiting values of $t_{C}\rightarrow 0$ (Hahn modulation, triangles) and $t_{C}$ = $t_{D}/N$ (CPMG modulation, circles) with $N$ = 4 as a function of the total diffusion time $t_{D}$, for both sharp and smooth gradient modulations.

The selected acquisition parameters for the NOGSE sequence setup ensure that both modulations yield the same decay rate within the restricted diffusion regime for a single restriction size. Based on Equations 8 and 11, this condition is achieved when $G_{smooth}$ = $\sqrt{2}G_{sharp}$. Hence, we chose $G_{sharp}$ = 100 mT/m and $G_{smooth}$ = 141 mT/m as in Figures 1b and d.

Comparing the signal decays shown in Figure 6a with the ones predicted by the theoretical model for a single restriction size in Figures 1b and d, we find that the overall behaviour of the predicted shifts is observed. The main difference is due to the restriction size distribution effects on the experimental data regardless of the chosen modulation type. Each restriction size from the probability distribution modelled by Equation 5, contributes with a different decay rate and shift, which are manifested differently as a function of the diffusion time $t_{D}$. Still, in the region of the normalised signal within 0.3 and 0.6 an effective restriction size seems to dominate on the decaying curve manifesting the expected shift of the exponentially decaying curves as in Figures 1b and d. At longer times, the spin signal contribution from smaller restrictions becomes dominant and the decaying laws change. Moreover, Rician noise effects may also become significant.

The NOGSE signal shift reduces as a function of the diffusion time $t_{D}$ significantly faster for the smooth modulation compared with the sharp modulation as expected. For the sharp modulation, it also seems to be reduced. In this latter case, it might be due to the fact that the sharp modulations are performed with trapezoidal pulse gradients, which are not instantaneous as in the theoretical model. It might also be due to the size-distribution effects or Rician noise manifestation.

The fit of the theoretical representations to the experimental data provides comparable size distributions for both types of gradient modulations. The modes of the distributions are shown with black dots which are indistinguishable between the different cases. However, the distribution width evidences some variations (see the caption of Figure 6). We provide further discussions about these differences in the following section.

We also present the NOGSE signal as a function of the lobe duration $t_{C}$ in Figure 6b. As predicted, the NOGSE contrast $\Delta M$ is larger for the sharp modulation. As shown in the inset, the extracted size distributions from the fitting of the theoretical model of Equation 5 to the experimental data are also comparable. Again, the mode is robust, i.e. they are indistinguishable in both cases, yet there are differences on the distribution widths.

We performed a more systematic evaluation of the reconstructed restriction size distributions determined from the different NOGSE sequences using sharp and smooth gradient modulations. We carried out NOGSE experiments for different $t_{D}$, $N$ and $G$ as a function of the lobe duration $t_{C}$, as shown in Figure 6b. Figure 7 shows the extracted parameters that characterise the corresponding estimated size distributions for all the considered cases. We present the mode, median, mean size and standard deviation $\sigma_{c}$ of the restriction size distribution obtained from the fitted curves to the experimental data of the averaged signal of ROI1. The symbols represent the fitted size distribution parameters as a function of different NOGSE modulations and acquisition parameters: $G$ (Figure 7a), $t_{D}$ (Figure 7b) and $N$ (Figure 7c).

For both modulation types, the mean size and standard deviation have more dispersion than the mode and, in second place, than the median. The mode evidences to be a more robust parameter for characterising the restriction-size compared to the mean size. For all the cases, including variations of $G$, $t_{D}$ and $N$, the mode is quite stable near to the optimal setup parameters of the sequence for the restriction-size inference predicted in Section IV.1. It is even robust when comparing the different modulation shapes of the gradient. However, the mean and median sizes do not manifest robustness, as they are dependent on the value of the estimated standard deviation $\sigma_{c}$.

As the sequence is not sensitive to sizes larger than those explored during the diffusion time, the estimation of size distributions appears to incline towards a distribution that encompasses larger sizes beyond the range of exploration due to the lack of sensitivity to them. The estimation of larger sizes is correlated with the inclusion of smaller sizes when increasing $\sigma_{c}$. For this reason, despite the dispersion of the distribution shape, the mode seems always to be a robust parameter to characterise the system.

Both NOGSE modulations estimate similar lognormal distributions, although we find that, in general, smooth gradient modulations give slightly smaller microstructure-size mode and mean values, as well as for the distribution widths. Similar qualitative behaviours are observed for the extracted size distribution parameter from different ROIs in the phantom.

The estimated values have less dispersion when estimated with NOGSE acquisition parameters near to the optimal region predicted in Section IV.1. Notice that higher parametric fluctuations are observed at the lower and higher ends of the gradient strengths and also at the extremes of the explored range for the diffusion time. These extremes are less sensitive for estimating the size distribution parameters as they are far from the optimal setup for the inference. We find, in general, that the regions with less dispersion in the estimated size distribution parameters, are at a bit higher gradient strengths compared with the predicted optimal value. In particular this is more noticeable for the sharp modulation. This seems to be consistent with the fact that the numerical predictions assume ideal squared shapes, while in practice they have trapezoidal shapes. By increasing $N$, we find that the estimation provides slightly lower restriction sizes. This is consistent with the fact that the lower $N$ possible is the most sensitive for estimating restriction sizes zwick_precision_2020 and that increasing $N$ for the same diffusion time reduces the maximum restriction sizes that can be estimated.

In addition to evaluate the consistency of the estimations, we compare the resulting restriction size distributions obtained from the averaged signal of a ROI as shown in Figure 7 with the ones obtained from single-restriction size fits from the voxel-by-voxel data. In this case, we use the NOGSE parameters derived from the region in Figure 7a that exhibits the lowest dispersion of the estimated values. Figure 8a,b are maps of the determined single-restriction sizes for each voxel, for the sharp and smooth NOGSE modulations, respectively. The observed variations in fibre densities, based on the restriction size estimation, are consistent with the level of compaction achieved during the phantom preparation process. Notice that the assumption here is that the size-distribution width at every voxel is negligible.

Single-size histograms from the voxel-by-voxel fits of ROI1 are shown in Figure 8c. Smooth and sharp modulations provide very similar size distributions. We also compare each histogram with the restriction-size distribution that arises from the averaged-signal fit of ROI1, for both types of gradient modulation. The distribution from the voxel-by-voxel differs from the one obtained via the averaged-signal. The distribution size mode is robust as both approaches give indistinguishable values. Nevertheless, there is an inconsistency in the distribution-width determination.