Mark Drakesmith^{1} and Derek K Jones^{1}

Microstructural MRI provides non-invasive measures of the microstructure of white-matter axons, including diameter and g-ratio. These can theoretically be used to infer axonal conduction velocity (CV). However, several other morphological and physiological parameters also contribute to CV, which are not accessible through MRI. Using sensitivity analysis on a comprehensive model of axon electrophysiology, we test the feasibility of modelling CV and associated uncertainty using only MRI-accessible parameters. Results show 88.5% of the variance in CV is accounted for by axon diameter and g-ratio and uncertainty due to non-imageable parameters can be easily modelled by a simple linear function. When these measurements can be made reliably it is feasible to obtain estimates of axonal CVs from micro-structural parameters obtained from MRI.

Axonal Modelling: Richardson et al.’s axon ‘Model C’ [9] was used to analyse the sensitivity of CV to variance in 14 different parameters (listed in Table 1). Model parameters derived from optic nerve [10] were used as a proxy for CNS axons. Some parameters were assumed to be well constrained across individuals and fibre populations and thus not tested (Table 1). Others, including number of myelin wraps and myelin thickness, are dependent on g-ratio, axon diameter and myelin periodicity, and so were not directly manipulated. Each model axon was stimulated with 3nA for 10μs applied to the first node. The resultant CV was then measured between the 6th and 13th node. For each simulation, we ensured that action potentials were propagated consistently, with membrane potential peaks of at least -40mV.

Sensitivity Analysis: An exhaustive analysis of 2^{14}=16,384 comparisons was
made by sampling the corners of a 14-dimensional hypercube in the parameter
space, i.e., for every possible combination of positive and negatives changes
in each parameter. The dimensions of the hypercube were set to ±1 s.d., where
s.d. was determined from experimental observations in optic nerve [10,11], or ±20% where no such data were available.

A Model To Predict CV From
MRI-Accessible Parameters:
We aimed to derive an expression for CV (and
associated variance) from the two MRI-accessible parameters of g-ratio (*g*) and internal axon diameter (*d*). We tested the model across a grid comprising
10 values of d (0.25 to 8 μm) and 12
values of g (0.4 to 0.95). For each grid-point,
we repeated the sensitivity hypercube analysis by running ‘Model C’ [9] across
all possible combinations of the remaining non-MRI accessible parameters, to
generate a distribution of CVs
for each point on the grid. This resulted in 10×12×2^{12}=491,520
model runs. The mean and standard deviation of CV at each point was derived. We
then fitted the original Rushton formula [8] for CV, and an alternative
model that better fits the CV estimates.

Figure 1(a) show the distributions of relative changes in CV, due
to change in each model parameter. Figure 1(b) shows
the total variances in CV due to variance in each parameter. The majority of
the variance is explained by *d*,
followed by internode length (non-MRI-accessible) and then *g*. Combined, *d* and *g* explain 88.5% of the model variance in
CV.
The distribution of CVs across *d* and *g* are shown in Figure 2. The mapping of CV to *d* appears to follow a square-root function, while *g* follows an square-root
log function, similar to that given by Rushton [8]:

$$v=pd\sqrt{-\log(g)}$$

where *p*
is some constant, (fitted to *p*=6.25 s^{-1}). The original Ruston model yielded a good fit (SSE=434.5, R^{2}= 0.9615), but the fit was poor for large *d* and *g*. A better fit was derived using a 3rd-order bivariate polynomial
fit in *d* and $$$\sqrt{-\log(g)}$$$ (of which the Rushton model is one term). This yielded a much more accurate fit (SSE=4.439,R^{2}=0.9996)
but required fitting of 11 coefficients. A good fit was also achieved when
considering only cross-terms in the polynomial, (SSE=36.39,R^{2}=0.9968)
which only requires 3 coefficients. However, the AIC and BIC were lowest for
the Rushton model, which therefore remains the preferred model (Figure 3). The s.d.
of the modelled CVs scaled linearly with the mean CV (coefficient=0.046, SSE=0.15,R^{2}=0.9933)
(Figure 4).

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