Kévin GINSBURGER^{1}, Felix MATUSCHKE^{2}, Fabrice POUPON^{3}, Jean-François MANGIN^{3}, Markus AXER^{2}, and Cyril POUPON^{1}

In this work, we investigate the potential of machine learning techniques to make one step forward by quantitatively estimating beading amplitude, a specific marker of pathological beading using frequency-dependent changes in diffusion measurements. Classification and regression are performed using Extremely Randomized Trees from OGSE signals corresponding to 6 distinct frequencies and synthesized from numerical simulations in realistic white matter phantoms depicting beaded axons.

Material & Methods

2000
virtual white matter tissues each containing 2000 axonal fibers with
randomly sampled mean axonal diameters, packing densities and mean BA
(fig.1) were created, including 500 samples corresponding to healthy
tissues ($$$BA < 0.2$$$)^{2,3} and 1500 samples to pathological
tissues ($$$BA \in [0.2,0.9]$$$). A novel GPU-based phantom simulator
modeling axonal fibers from sets of overlapping spheres and enabling
to model global angular dispersion, axonal tortuosity and beading at
high packing densities was employed to generate the tissues (fig.2).

Monte-Carlo
simulations were subsequently performed on each virtual tissue using 10^{5} spins^{6,7} with a random walk step equal to 2% of the mean
axonal diameter for each tissue. Equal-step-length random leap^{8} was
employed for particle-membrane interaction with impermeable
membranes. OGSE diffusion-weighted signals at 6 distinct frequencies
from 75Hz to 200Hz and constant $$$b$$$-value of $$$450 s/mm^2$$$
were synthesized according to the protocol given in fig.1, adding
noise to reach SNR=30 clinically achievable with a Connectome
gradient set.

A feature vector was computed from each synthesized OGSE signal using a DTI analysis, comprising the mean diffusivity, fractional anisotropy, axial and radial diffusivities for the 6 OGSE frequencies resulting in 24 features per tissue.

An
ExtraTrees classifier from Scikit-Learn^{9} was trained on a subsample
of the dictionary of generated signals containing 500 healthy and 500
pathological samples to discriminate between healthy and pathological
tissues (fig.3). An ExtraTrees regressor was trained on the whole
dictionary (500 healthy and 1500 pathological samples) to learn the
mapping between the extracted features and the underlying
microstructure parameters of interest : intra-axonal volume fraction,
axon diameter and BA. Both classifier and regressor used 100 trees
and a maximum depth of 20 (other parameters were set to default
values). All performance metrics were computed using a 10-fold
cross-validation strategy to reduce bias due to the choice of
training data.

Fig.2 shows example tissues demonstrating the ability of the employed phantom generator to control the intra-axonal fraction and BA of each constructed tissue. The impact of BA on the diffusion-weighted signal at each OGSE frequency is illustrated in fig.4, which indicates that differences in BA are more prominent at higher OGSE frequencies (150-200Hz).

However, using the full frequency spectrum instead of only higher frequencies improves the discrimination between healthy and pathological tissues. Classification accuracies of 75.2/70.4% are obtained with the full spectrum/higher frequencies only, with precision and recall scores of 77.3/74.3% and 91.4/87.8% respectively, showing that the classifier trained with the full spectrum has both a low rate of false positive and false negatives. Regressions of BA and other microstructure parameters from the OGSE signals are also improved using the full OGSE frequency spectrum: $$$R^2$$$ scores of 0.69/0.66, 0.65/0.52 and 0.53/0.44 are thus obtained for intra-axonal fraction, axonal diameter and BA respectively for the global and partial frequency spectra, showing substantial correlations between predictions and ground truth at the employed noise level (SNR=30), which could be further improved using a bigger training dataset. Mean absolute errors (MAE) of 0.039/0.04, 0.26/0.32 $$$\mu m$$$ and 0.13/0.15 were obtained for intra-axonal fraction, axon diameter and BA respectively. The diameter MAE is notably low for a SNR of 30 considering the range of estimated diameters (1-3 $$$\mu m$$$).

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