Mark Drakesmith^{1}, Elena Kleban^{1}, Fabrizio Fabrizio^{1,2}, and Derek K Jones^{1}

g-ratio is an important parameter of axon physiology and there is great interest in estimating it non-invasively in MRI. Existing approaches rely on fitting to a multi-compartment model and calculating g-ratio from the estimated volume fractions (Stikov et al, 2015). Here, we show that we can get improved estimates of the g-ratio by modelling its contribution to frequency offsets in GRE data using a hollow cylinder fibre model. Through simulations and model fitting to in vivo human GRE data we show g-ratio estimates are improved and closer to values obtained from histology compared with the existing approach.

The g-ratio, (the ratio of the inner to outer axon diameter), is of considerable
importance to axon physiology and as such, considerable efforts to image this
property non-invasively, in vivo have been made.
A simple approach^{1} computes the g-ratio from estimated axonal and myelin
volume fractions (AVF / MVF, denoted Va / Vm):

$$g = \frac{1}{\sqrt{1+\frac{V_m}{V_a}}}$$

AVF and MVF estimates, can be obtained by modelling the contribution of myelin, axonal and extracellular compartments to MRI signals, in
diffusion MRI^{2,3} or relaxometry.^{4} More recently, attempts to estimate volume fractions (and hence g-ratio) have been made by fitting to the complex
gradient-recalled echo (GRE) signal evolution ^{5, 6}:

^{ }

$$S(t)=\rho V_me^{-t\left(\frac{1}{{T^*_2}_m}-i\omega_m\right)}+V_ae^{-t\left(\frac{1}{{T^*_2}_a}-i \omega_a\right)}+V_ee^{-t\left(\frac{1}{{T^*_2}_e}-i\omega_e\right)}$$

where *V* is the volume fraction of each compartment, *T*_{2}^{∗} is the transverse
relaxation time, *ω* is the angular frequency offset and *t* is echo time. The subscripts *m,a,e* denote the myelin, axonal and extracellular compartments, respectively. *ρ* is
a myelin concentration parameter used to convert signal fractions to volume
fractions.
While this simple multi-compartment model (SMCM) with the Stikov approach
has been used to estimate g-ratios, we can potentially obtain better results by
fitting the signal to the hollow cylinder fibre model (HCFM), that explicitly includes the g-ratio as a parameter in the expressions for the frequency offsets.^{7,8}

$$\frac{\omega_m}{\gamma B_0}=\frac{\chi_i}{2}\left(\frac{2}{3}-\sin^2{\theta}\right)+\frac{\chi_a}{2}\left( \frac{1}{4}-\left( \frac{3}{2} \left( \frac{g^2}{1-g^2} \right)\ln\left(\frac{1}{g}\right)\right)\sin^2{\theta}-\frac{1}{3}\right)+E$$

$$\label{omega_a}\frac{\omega_a}{\gamma B_0}=\frac{3 \chi_a}{4}\sin^2{\theta}\ln\left(\frac{1}{g}\right)$$

$$\label{omega_e}\frac{\omega_e}{\gamma B_0}=0$$

where χ_{i} and χ_{a} are the isotropic and anisotropic tissue susceptibilities, *θ*
is the orientation of the fibers relative to the **B**_{0} field and *E* is an exchange
parameter.
In this model, the frequency offsets are no longer treated as independent
parameters. With the geometric relationships between MVF, AVF, FVF and *g*
(see Table 1), there are additional interdependencies between these parameters,
making it a better conditioned model. The HFCM can be regarded as equivalent
to the SMCM with constraints on the form of the frequency offsets. Here we
compared using simulations and in vivo GRE data, g-ratios estimated
using the SMCM and HCFM^{7,8}.

Complex-valued multi-echo GRE data were simulated in MATLAB using the
HCFM with the parameters in Table 1 . FVF, g-ratio and susceptibility values were varied to test fitting across ranges given in Table 1, resulting in 875 parameter configurations. For each configuration, data with
acquisition parameters matching those given in Table 2 were simulated. Both
models (SMCM and HCFM) were fitted using particle swarm optimisation^{9} (to
avoid local minima) using bounds given in Table 1. For HCFM, *θ*, *ρ* and *E* were assumed to be
constant. Relative errors of all parameter fits were then computed.
Human data acquisition was performed on a single participant (F, 29y)
on a Siemens 7T research-only scanner equipped with a 32-channel head receive/volume transmit coil (Siemens Healthcare, Erlangen, Germany) on a single mid-saggital slice. Image phase data were reconstructed from the raw k-space data using ASPIRE^{10} and combined with the amplitude data to generate complex-valued images. Background field effects were removed by normalising each complex-valued image as follows:^{11}

$$S'(t_i)=\frac{S(t_i)}{S(t_1)} \quad S''_i(t)=\frac{S'(t_i)}{S'(t_2)^{i-1}}$$

The normalised data were then fitted to the HCFM and SMCM. g-ratios
were computed for the SMCM using eq. 1. For
the HCFM, the fibre orientation was assumed to be perpendicular to **B**_{0}
(*θ* = π/2). The corpus callosum. (CC) was manually segmented according to the Witelson scheme^{12} and fitted parameters averaged within each segment.

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Table 1:
biophysical parameters used for the MRI simulations. Table 2:
MRI acquisition parameters used in simulations and in vivo human
data.

Figure 1: Distribution of fitted parameters for (a) HCFM and (b) SMCM and
(c) SSEs for g-ratio and volume fractions for each model.

Figure 2: Parameter fits to a human GRE data set, masked to corpus callosum
with HCFM and SMCM.

Figure 3: g-ratio and volume fraction estimates for segments of corpus callosum
for HFCM and SMCM.