### Synopsis

**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 model

^{1} 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} mm

^{2}/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} mm

^{2}/s,
r

_{ax} = [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 system

^{2}, 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.

### 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 b

_{max,Δ,δ}
was in the low (≤ 2000 s/mm

^{2}), medium (4000 s/mm

^{2} ≤ 8000
s/mm

^{2}) and high b-value range (≥ 10000 s/mm

^{2}),
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.

### Results

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} mm

^{2}/s.
The previously reported “turned pipe”-shape

^{3} 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.

### Discussion

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}.

### Acknowledgements

No acknowledgement found.### References

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