Estimation of the axon diameter from diffusion MRI (dMRI) may provide a valuable tool for non-invasive studies of brain connectivity [1]. However, accurate quantification of axon diameters in the human brain is challenging, since the MR signal curves are indistinguishable for axon with diameters below the resolution limit (Fig. 1) [2,3]. The purpose of this study was to develop a framework to estimate the resolution limit. To address the growing interest in non-conventional diffusion encodings that go beyond single diffusion encoding (SDE) [4], we generalised the framework to permit prediction of the resolution limit regardless of the gradient waveform.We consider a simple model where the MR signal ($$$S$$$) originates only from water within parallel cylinders, representing axons. We define the resolution limit as the diameter ($$$d_{\rm{min}}$$$) for which $$$S$$$ becomes indistinguishable from that of an infinitesimally thin axon. Hence, the difference in signal between these cases ($$$\Delta{S}$$$) approaches the detection limit ($$$\sigma$$$) for $$$d_{\rm{min}}$$$: $$\Delta{S}=S(b\,|\,d_{\rm{min}})-S(b\,|\,d\,\rightarrow\,0)=\sigma\Rightarrow\Delta{S}=1-\exp\big(-bD_\bot(d_{\rm{min}})\big)\approx\,bD_\bot(d_{\rm{min}}),~~~(1)$$where $$$b$$$ is the b-value and $$$D_\bot$$$ is the apparent diffusivity perpendicular to the axons. The approximation is valid close to the resolution limit, where the attenuation is low. A novel contribution of this work is the notion that $$$D_\bot$$$ can be calculated for arbitrary gradient waveforms by utilising the low-frequency approximation of the diffusion spectrum, i.e., $$$D_\bot(\omega)\propto\omega^2$$$ [5], according to$$D_\bot(d)=k\,D_0^{-1}V_{\omega}\,d^4,~~~(2)$$where $$$k\,=\,(2^73^15^3)^{-1/2}$$$, $$$D_0$$$ is the bulk diffusivity, and $$$V_\omega$$$ is the variance of the encoding spectrum. Omitting the derivation, we found$$bV_\omega\,=\,\gamma^2\,\int\,g^2(t){\rm\,dt},~~~(3)$$where $$$g(t)$$$ is the gradient waveform. The resolution limit for parallel axons $$$d_{\rm{min}}^{\rm (para)}$$$ is now given from Eqs. 1-3 as$$d_{\rm{min}}^{\rm(para)}=\left(\frac{\sigma\,D_0}{k}\frac{1}{bV_\omega}\right)^{1/4}.$$In the case of full orientation dispersion [6], $$S(b|d)=\exp(-bD_\bot)\,h(A)$$where $$$h(A)=\sqrt{\pi/4}~{\rm{erf}}(A)/A$$$, $$$A^2=b(D_\parallel-D_\bot)$$$, and $$$D_\parallel$$$ is the axial diffusivity. In dispersed axons, $$$d_{\rm{min}}^{\rm(disp)}$$$ is thus given by $$d_{\rm{min}}^{\rm(disp)}=d_{\rm{min}}^{\rm(par)}\,h(A)^{-1/4},$$where we assume $$$D_\bot(d_{\rm{min}})\,\approx\,0$$$ and thus $$$h(A)$$$ becomes independent of $$$d$$$. Since $$$h(A)<1$$$, the resolution is worse in the dispersed case.We first compared predictions of $$$D_\bot$$$ from Eq. 2 to results from Monte Carlo simulations [7]. By investigating a multitude of gradient waveforms, we concluded that multiple diffusion encoding (MDE) by square wave gradients yielded the lowest possible value of $$$d_{\rm{min}}$$$. We then searched for the number of diffusion encodings that minimised $$$d_{\rm{min}}$$$. The analysis was performed for parallel axons and for full orientation dispersion (assuming $$$D_\parallel=2$$$ μm²/ms), for various gradient amplitudes and field strengths (3T and 7T). We assumed $$$\sigma=1\%$$$, which is a reasonable detection limit on a clinical MRI system. The waveform duration was 80 ms for 3T and 50 ms for 7T, which accounts for shorter T2 relaxation times at 7T.The low-frequency approximation is demonstrated in Fig. 2. Figure 3 shows $$$\Delta S$$$ for $$$d=3.5$$$ μm, versus the gradient oscillation frequency. Parallel axons show highest value of $$$\Delta S$$$ for the lowest frequency (i.e. SDE). For dispersed axons, $$$\Delta S$$$ is at its highest for a frequency of approximately 100 Hz. Figure 4 shows gradient waveforms that were optimised to minimise $$$d_{\rm{min}}$$$, with corresponding values of $$$d_{\rm{min}}$$$ shown in Figure 5. Waveforms with higher oscillation frequencies were preferred for dispersed axons. Such waveforms featured lower values of $$$b$$$. Orientation dispersion worsened the resolution limit by up to 50\%, and reduced the relative benefit of strong gradients (Table 1). 7T provides higher SNR than 3T, which translates to better resolution only if the gradient amplitude is the same as at 3T.

We assessed the lowest axon diameter that can reliably be recovered from the signal attenuation of intra-axonal water. We label this diameter ’the resolution limit’. Our analysis allows the resolution limit to be assessed for arbitrary gradient waveforms, or conversely, as an approach to minimising the resolution limit by optimising the gradient waveform.

For parallel axons, we found that the SDE experiment yields a better resolution than more complex waveforms. However, axons are rarely parallel, not even in the corpus callosum [8], but rather run in tortuous configurations. In the presence of full orientation dispersion, the effective SNR decrease with higher $$$b$$$, since the intra-axonal water signal is attenuated proportional to the relative alignment of the axon to the direction of the diffusion encoding. Accordgingly, the resolution is at its optimum for lower b-values for dispersed compared to parallel axons. This finding is in agreement with [9]. Our framework provides valuable knowledge for the design of dMRI experiments, although we acknowledge that the model is an oversimplification [10,11].

In conclusion, we recommend MDE by oscillating square waves rather than SDE where there is axonal orientation dispersion, in order to obtain the best possible diameter resolution.