First we test whether the microcapillary diameter and the intrinsic diffusivity can be estimated based on the entire trapezoidal OGSE imaging protocol in Fig. 2. We then test which of the OGSE sequences out of those in Fig. 2 provide the most accurate parameter estimates by analysing each shell separately, and we compare the results with the parameters obtained from the standard SDE with long diffusion time.

Figure 3a, b display the parameter maps (diameter and diffusivity, respectively) for the ROIs of our plates. Both pairs of 10 and 20 |xm plates have accurate and precise (indicated by the homogeneous maps) estimates. The parameter maps for the 5 |xm plates are partially inhomogeneous and they significantly underestimate the diameter. Figure 3c, d reflect the accuracy and precision of the parameters, displayed in Fig. 3a, b, as the mean and standard deviation of the estimated a and Dj calculated across the ROI. The figure also shows very similar parameter estimates within each pair of plates suggesting that the results are reproducible. For the first set of 5,10 and 20 |xm plates, the estimates of mean±standard deviation for

Fig. 3 The (a) diameter, a, and (b) diffusivity, D_{i}, maps, respectively, across the ROIs of the 5 p,m (plates 1 & 2), 10 p,m (plates 1 & 2) and 20 p,m (plates 1 & 2) plates. All images have been cropped and magnified by the same amount for visual clarity. The graphs show the mean and standard deviation of the (c) diameters of the microcapillaries (p,m) and (d) intrinsic diffusivities (p,m^{2}/ms), which are calculated over the ROIs. The diamond and triangle data points represent the first and second set of plates, respectively. The dashed line represent the line of equality for (c), and for (d) it represents the theoretical water diffusivity calculated using [25] for water at 20 °C

[a,Di] are [1.5 ± 2.4 |xm, 2.0 ± 0.1 |xm^{2}/ms], [10.1 ± 0.5 |rm, 2.0 ± 0.1 |xm^{2}/ms] and [19.8 ± 0.4 |xm, 2.0 ± 0.1 |xm^{2}/ms], respectively. For the second set of 5, 10 and 20 |xm plates, the values of [a,D,] are: [0.7 ± 1.9 |rm, 1.9 ± 0.1 |xm^{2}/ms], [10.3 ± 0.2 |xm, 2.1 ± 0.1 |xm^{2}/ms] and [19.8 ± 0.6 |rm, 2.0 ± 0.1 |xm^{2}/ms], respectively. We observe the highest accuracy and precision for 10 |xm plate pairs, and the worst for 5 |xm plate pairs.

Figure 4 shows the quality of fit by comparing measurements with predictions from the fitted model (dashed line) and the ground truth (solid line) in the central voxel of each plate ROI. The ground truth curve was generated using the manufacturer provided diameters and a diffusivity constant (2.0 |xm^{2}/ms) calculated for the free water compartment at 20 °C [25]. The representative voxels chosen here are typical for the ROIs. A good agreement can be observed between the measurements and the fitted curve and the ground truth curve for 10 |xm and 20 |xm plates. However, slight differences between the fitted curve and the ground truth

Fig. 4 Plots of normalised signal from central voxel of each ROI in Fig. 3a against absolute dot product between the gradient directions and the estimated direction of the microcapillaries; signals from perpendicular gradient direction are towards 0 on the x-axis, and from parallel directions towards 1. The measurements are represented by markers, while the solid (-) and dashed (- -) lines show the predicted signal from the ground truth and estimated parameters, respectively. The colours indicate the different N of the imaging protocol. The black dotted lines show the b=0 measurements. All measurements are normalised by the averaged b0 signal per voxel. The parameter estimates for the representative voxels here are: [a,D_{i}] = [0.0^,m, 2.0 ^,m^{2}/ms],[10.2 ^.m, 2.0 ^m^{2}/ms] and [20.1 ^.m, 2.0 ^m^{2}/ms] for the first pair of 5, 10 and 20 ^m plates, respectively. For the second pair, the respective [a,D_{i}] are [0.0 ^m, 1.8 ^m^{2}/ms], [10.4 ^m, 2.1 ^m^{2}/ms] and [20.5 ^m, 2.1 ^m^{2}/ms]. (a) 5 ^m, plate 1; (b) 5 ^m, plate 2; (c) 10 ^m, plate 1; (d) 10 ^m, plate 2; (e) 20 ^m, plate 1 (f) 20 ^m, plate 2

curve can be observed in the second plates of 10 and 20 p,m. This can be due to the overestimated diffusion constant caused potentially by partial volume effects. For this central voxel, in the case of 5 p,m plates (Fig. 4a, b), some differences can be seen between the measurements, the fitted curve and the ground truth curve. In this case, due to the low signal attenuation for gradients almost perpendicular to the fibre, measurement noise results in normalized signal values S/S_{0} > 1 which are not well captured by the model. Moreover, the differences between signals predicted using the known parameters [a,Df] = [5.0 p,m, 2.0 p,m^{2}/ms] and model estimates [a, D_{i}] = [0.0 p,m, 2.0 p,m^{2}/ms] for the first 5 p,m plate are small, despite the model estimates of diameter being so different. The difference is slightly larger in the second 5 p,m plate ([a,D,] = [0.0|i,m, 1.8 p,m^{2}/ms]) but this is most likely due to an underestimation in the diffusion constant. These results suggest the change in measured signal is negligible for microcapillaries with diameters at or below 5 p,m, i.e the measured signal is not very sensitive to diameters at or below 5 p,m.

Figure 5 shows the mean and standard deviation of the estimated diameter and diffusivity obtained by separately analysing each individual shell with N lobes (from Fig. 2). Here, results from a standard SDE sequence (N=1, 1 = 10 ms) are also included for comparison. 10 and 20 p,m plate diameter estimates are close to the ground truth values for the majority of N, whereas 5 p,m estimates are largely underestimated for all N. Focusing on 10 and 20 p,m plates, N e {2,3,4} perform very well, while for N > 5, the estimates are progressively less accurate and precise as N increases. This is due to insufficient diffusion weighting as N increases. At

Fig. 5 Mean diameter (a) and diffusivity (b) estimates calculated for each N from Fig. 2 (labelled as 1-9 (39 ms), where 1 = 39 ms) and also from the standard SDE sequence (labelled as 1 (10 ms), where 1 = 10 ms), for all plates. The same central row of voxels, as in Fig. 3, is used to calculate the mean and the standard deviation. The dashed lines represents the real nominal diameters in (a), and the calculated diffusivity from [25] in (b). N = 3 produces the best diameter and diffusivity for both pairs of 10 ^m and 20 ^m plates low N (N = 1 (1 = 39 ms)), the fitting fails to correctly estimate the parameters for 20 p,m plates because of the strong diffusion attenuation, forcing the model to fit to the noise floor. As a sanity check we compare the results to (N=1,1 = 10 ms) and find that diameter and diffusivity of microcapillaries with diameter of 20 p,m are estimated accurately for this SDE sequence, however, 10 p,m plates are poorly estimated. Hence, for this particular TE and diffusion gradient duration, we find that N>1 gives better results overall.

N = 3 gives the best estimates for both 10 and 20 p,m plates. N = 3 outputs [а,Д] of [9.7 ± 0.5 p,m, 2.0 ± 0.0 p,m^{2}/ms] and [20.1 ± 0.5 p,m, 1.9 ± 0.1 p,m^{2}/ms] for the first pairs of 10 and 20 p,m plates, respectively. We also see consistency in our estimates because the estimates ([a,A]) for the second pair are [9.9 ± 0.3 p,m, 2.1 ± 0.0 |xm^{2}/ms] and [20.1 ± 0.8 p,m, 1.9 ±0.1 p,m^{2}/ms]. The diameter estimates from N = 3 are close to the ground truth and are also within the confidence limits of the estimates from the combined OGSE protocol shown in Fig. 3. The diffusivity estimates have slightly higher accuracy and slightly lower precision for both pairs of 10 and 20 p,m plates in comparison to the combined OGSE protocol. The diffusivity estimates are also very close to the estimates from the combined OGSE protocol. These results suggest that, for the case of idealised systems, one OGSE shell can perform similarly compared to a combination of OGSE shells.