Acquisition Protocol Optimization for Axon Diameter Mapping at High-Performance Gradient Systems – A Simulation Study
Jonathan I Sperl1, Ek Tsoon Tan2, Miguel Molina Romero1,3, Marion I Menzel1, Chris J Hardy2, Luca Marinelli2, and Thomas K.F. Foo2

1GE Global Research, GARCHING, Germany, 2GE Global Research, NISKAYUNA, NY, United States, 3Institute of Medical Engineering, Technische Universität München, GARCHING, Germany


The measurement of axonal diameter by diffusion MRI techniques has assumed major interest in the research community. While most work has focused on developing and comparing various multi-compartment models, only minor efforts have been undertaken to optimize corresponding acquisition protocols. In this work we perform simulations using a rather simple two-compartment model, but study the effect of various choices of acquisition parameters on the precision and the bias of the fitted parameters. More precisely, we analyze potential sampling strategies in the 2D design space spanned by the two timing parameters (Δ, δ) of the diffusion encoding.

Methods – two-compartment model

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 fres, the orthogonal diffusivity Dorth and the axon radius rax. These parameters were each set to three different values (fres = [0.3, 0.5, 0.7], Dorth = [0.3, 0.6, 0.9] x10-3 mm2/s, rax = [2, 5, 8] µm) and all 27 possible combinations were used as input parameters for the simulations.

Methods – acquisition protocols

Targeting the hardware specifications of a dedicated high-performance gradient system2, the following boundary conditions of the acquisition protocol were assumed: Gmax = 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 bmax,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 Gmax. 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.

Methods – simulations & fitting

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 bmax,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 (fres, Dorth, rax) 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 rax and fres, for a standard and an optimized protocol. The green x indicates the true solution, when fres = 0.5, and Dorth = 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 rax 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 studies4,5. However, the widely used minimum-δ-assumption4,6 could not be confirmed, as it seems to be important to also use larger δ-values in order to achieve ultra-high b-values5,7,8. 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 gradients9,10.


No acknowledgement found.


1) Panagiotaki et al., NeuroImage 59, 2012

2) Huston et al., ISMRM 2015, 971

3) Jelescu et al., ISMRM 2015, 1024

4) Assaf et al., MRM 59, 2008

5) Ferizi et al., NeuroImage 118, 2015

6) Huang et al., ISMRM 2015, 470

7) Alexander, MRM 60, 2008

8) Schneider et al., ISMRM 2012, 350

9) Gore et al., NRM BioMed 23(7), 2010

10) Drobnjak et al., ISMRM 2015, 347


(Δ, δ)-design space and corresponding bmax,Δ,δ-values.

Contour of the least-squares cost function in Dorth vs. rax space, for a standard (5x8, constant δ) and an optimized protocol (see Fig 4. 5x8, scheme #3), for three different values of rax and fres. Green x’s mark true solution for fres = 0.5, Dorth = 0.6x10-3 mm2/s.

Simulation results for all 5x8 scheme types sorted by bias of rax.

Optimized acquisition schemes. The color coding represents a classification into good (green), average (yellow) and poor (red) results (relative to the other eight values).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)