Assuming a single fiber population per voxel a two-compartment CylinderZeppelin model¹ was used in this study. The parallel diffusivity in the hindered compartment and the diffusivity parameter of the restricted compartment were fixed to 1.7x10^-3 mm²/s and the direction of the fiber was assumed to be known. The remaining three free parameters of the model were the restricted volume fraction f_res, the orthogonal diffusivity D_orth and the axon radius r_ax. These parameters were each set to three different values (f_res = [0.3, 0.5, 0.7], D_orth = [0.3, 0.6, 0.9] x10^-3 mm²/s, r_ax = [2, 5, 8] µm) and all 27 possible combinations were used as input parameters for the simulations.Targeting the hardware specifications of a dedicated high-performance gradient system², the following boundary conditions of the acquisition protocol were assumed: G_max = 80 mT/m, SR = 700 mT/m/s, TE = 85 ms, 0.1 ≤ δ/Δ ≤ 0.9, 10 ms ≤ Δ ≤ TE - 2τ, τ = 4ms, δ ≥ 2ms. These assumptions constrain the 2D space of feasible (Δ, δ)-combinations and the corresponding maximum b-values b_max,Δ,δ shown in Fig. 1. To narrow the search space a discrete grid of (Δ, δ)-points within the design space was defined and used in the following simulations. For a given (Δ, δ)-pair, the acquisition protocol was designed such that the diffusion gradient along a single direction orthogonal to the known fiber orientation was increased linearly in N steps up to G_max. In this study we compared the performance of acquisition schemes comprising K (Δ, δ)-pairs and N gradient steps such that KxN is constant, yielding identical acquisition times. More precisely, three types of schemes were investigated: KxN = 5x8, 8x5 and 10x4, with each type comprising 40 individual measurements. In addition, five b=0 acquisitions were assumed.The chosen K (Δ, δ)-pairs covered the feasible range of acquisition protocols. They were considered valid for the simulation if they had sufficient b-value coverage, i.e. if at least one of the K b_max,Δ,δ was in the low (≤ 2000 s/mm²), medium (4000 s/mm² ≤ 8000 s/mm²) and high b-value range (≥ 10000 s/mm²), respectively. This yielded +3000 acquisition protocols for each type (e.g., 3620 protocols for the 5x8 scheme). For all protocols, all 27 tissue configurations were used to compute synthetic data sets. Fifty instances of Rician noise (SNR=30) were added to the data. The fairly mild noise level was chosen to create some variability in the data but to ensure that the results of the study were not dominated by SNR-effects. In the latter case variable TEs would need to be taken into account. The free model parameters (f_res, D_orth, r_ax) were fitted to the data in a least-squares sense using a two-stage solver (grid-search followed by interior-point optimization). Relative errors for the three fitted parameters were computed as well as their mean (bias) and standard deviation (precision). The corresponding average values over all 27 tissue configuration were used as a quality metric.Fig. 2 displays the cost function of the least-squares fitting routine for three different values of r_ax and f_res, for a standard and an optimized protocol. The green x indicates the true solution, when f_res = 0.5, and D_orth = 0.6x10^-3 mm²/s. The previously reported “turned pipe”-shape³ of the valley of the cost function (i.e. the blue area in the plots) can be confirmed. This causes the high noise sensitivity of axon diameter fittings. Although the optimized protocol is not capable of changing this principle structure, it yields a reduced size of the valley (indicated by red arrows), and hence improves the stability of the fit. Fig. 3 compares the results of the simulation study for all acquisition protocols of 5x8 type. The acquisitions have been sorted by the bias of the r_ax estimation, in order to visualize the consistent trend of all quality metrics. This illustrates the importance of choosing a “good” acquisition protocol, but also highlights that even for such a protocol the errors remain quite high. Lastly, Fig. 4 displays three of the best acquisition protocols for each type found in this study. Unfortunately, no protocol could be found with superior performance for all six quality metrics.In this study we have presented a framework for designing acquisition protocols for axon diameter measurements. The key idea is the representation of (Δ, δ)-pairs as points in a 2D design space. We performed extensive simulations trying to find a best-performing protocol, keeping the total number of acquisitions constant. Notably all of the identified “good” protocols employ a broad coverage of the Δ-space which is in agreement with the protocols used in most studies^4,5. However, the widely used minimum-δ-assumption^4,6 could not be confirmed, as it seems to be important to also use larger δ-values in order to achieve ultra-high b-values^5,7,8. Lastly, we can confirm the ill-posed nature of the fitting problem even for a simple two-compartment model with only three free parameters. This motivates the development of even more powerful gradient systems as well as the investigation of more advanced acquisition strategies such as oscillating gradients^9,10.