Marco Palombo^{1}, Ioana Hill^{1}, Mathieu David Santin^{2,3}, Francesca Branzoli^{2,3}, Anne-Charlotte Philippe^{2,3}, Demian Wassermann^{4,5}, Marie-Stephane Aigrot^{2}, Bruno Stankoff^{2,6}, Hui Zhang^{1}, Stephane Lehericy^{2,7,8}, Alexandra Petiet^{2,7}, Daniel C. Alexander^{1}, and Ivana Drobnjak^{1}

Estimating axonal permeability reliably is
extremely important, however not yet achieved because mathematical models that
express its relationship to the MR signal accurately are intractable. Recently
introduced machine learning based computational model showed to outperforms
previous approximate mathematical models. Here we apply and validate this novel
method experimentally on a highly controlled *in-vivo* mouse model of axonal demyelination, and demonstrate for
the first time in practice the power of machine learning as a mechanism to
construct complex biophysical models for quantitative MRI.

Quantitative MRI relies on biophysical models
to relate MR signals to tissue properties, but mathematical models rapidly
become intractable beyond very simple descriptions of tissue, and parameters
such as permeability remain elusive. A recent study^{1} demonstrated
that using machine learning we can construct a computational model that
outperforms approximate mathematical models in estimating permeability via the
residence time τ_{i} of water inside axons. τ_{i} is a
potentially important biomarker for white matter pathologies, as myelin damage
is hypothesized to affect axonal permeability, and thus τ_{i}.

Here we for the first time apply and validate
this novel idea on a highly controlled *in-vivo*
mouse model of axonal demyelination. We use Monte Carlo simulations and random
forest (RF) regression^{1} to build a mapping between
diffusion-weighted MR signals and ground-truth microstructure parameters. We then
estimate differences in τ_{i} between *in-vivo* healthy and
cuprizone-treated (CPZ) mice brain, a well-known model of white matter (WM)
demyelination.

*In-vivo data*. We used a diffusion-weighted Pulsed-Gradients-Spin-Echo (DW-PGSE) protocol (**Tab.1**).
We scanned sixteen C57BL6J mice on BrukerBioSpec 11.7T: eight CPZ mice at 6 weeks
post intoxication (0.2% cuprizone) and eight healthy age-matched wild-type (WT)
mice of the same background (data available on https://zenodo.org/record/996889#.WgH5E9vMx24).
Images are corrected for eddy-currents using FSL-eddy. DTI and NODDI^{2}
maps are computed for comparison.

*Simulation data*. We used MC simulations in CAMINO^{3}.
To mimic the *in-vivo* mouse brain, WM
was modelled as a collection of non-abutting parallel cylinders with gamma-distributed
radii with parameters randomly
selected for 11000 substrates in the ranges: mean and standard deviation of
axonal radius μ_{R}∈[0.1,1.0]μm and σ_{R}∈[min(0.1,μ_{R}/5),μ_{R}/2]μm, intra-axonal volume
fraction f_{i}∈[0.40,0.75], τ_{i}∈[0.02,1.00]s, intrinsic diffusivity d∈[0.8,2.2]μm^{2}/ms.
Signal-to-noise ratio 30.

*Data quality test*.
We assessed the sensitivity of our imaging protocol to τ_{i} using synthetic
data as in^{4}, and the quality of the synthetic data by direct comparison
with the *in-vivo* data.

*Machine Learning*. We adapted the well-known RF regressor in
the scikit-learn toolkit^{5} to learn the mapping between the
microstructure parameters defining our simulation substrates and the normalized
DW-MRI signal (**Fig.1**). The RF had
decision trees/maximum depth=100/20. We trained the RF on simulated data using 11000
feature vectors (synthetic diffusion-weighted signals) through a greedy
splitting process that constructs a linear mapping from the features to the
corresponding ground truth parameters. The trained RF estimates f_{i}, τ_{i} and d,
which we tested on simulated data using 1500 previously unseen feature
vectors and then applied on *in-vivo* data.
We manually segmented the corpus callosum (CC) and Fornix and compared WT and CPZ
mice estimates using 2-tailed t-test with equal variance.

*Data quality test*.
**Fig.2A** shows that our imaging protocol has good sensitivity to
τ_{i}, with sufficient
change in DW-MRI signal to provide reliable τ_{i} estimates for τ_{i}≤500 ms. **Fig.2B**
shows an excellent match between the simulated and* in-vivo* diffusion-weighted signals, demonstrating that our training
data set is a good representation of the *in-vivo*
data.

*Simulation data estimates*. Testing shows strong correlations between RF estimates
and the ground truth parameters (R^{2}=0.86/0.94/0.88 for f_{i}/τ_{i}/d
respectively). A comprehensive sensitivity/accuracy analysis of the proposed
method is the subject of another abstract that we submitted to this symposium.

*Machine Learning*. **Fig.3** shows parametric maps from
the *in-vivo* data (WT versus CPZ) of conventional DTI, NODDI
orientation-dispersion index (ODI) and using RF approach. RF estimates in CPZ
mice show a statistical significant reduction in f_{i}, and τ_{i}
compared to WT (**Fig.4**). DTI shows
reduction in FA and increase in RD, in agreement with^{6}. ODI is not statistical different
between the two groups (**Fig.4**). An electron
microscopy image of CC is also reported to show the underpinning
microstructural changes in the tissue: demyelination and partial axonal loss.

Our analysis of DW-MRI from a well-known mouse model of WM
demyelination **(Fig.3,4**) shows that
machine learning approach allows an accurate and robust estimation of the
expected changes in tissue-microstructure. While DTI gave only indirect
information, our model found: 1) a decrease in f_{i} as direct measure of partial axonal loss;
2) a decrease in τ_{i} as direct measure of axonal membrane
permeability increase due to demyelination. Furthermore, our estimates show no
bias due to fiber orientation dispersion, as shown by the negligible changes in ODI. Finally, our results
match the expected tissue-microstructure changes (electron microscopy in
**Fig.3**).

Standard methods for estimating permeability so far have
been using the Karger model^{7}, however in^{1} has been shown
that machine learning approach works much better. Here we provide a clear
validation of the machine learning approach introduced in^{1} affirming
a wider potential of the approach as a mechanism to construct complex biophysical
models for quantitative MRI.

^{1} Nedjati-Gilani, G.L., Schneider, T.,
Hall, M.G., Cawley, N., Hill, I., Ciccarelli, O., Drobnjak, I.,
Wheeler-Kingshott, C.A.M.G., Alexander, D.C. : Machine learning based
compartment models with permeability for white matter microstructure imaging,
NeuroImage, http://dx.doi.org/10.1016/j.neuroimage.2017.02.013 (2017).

^{2} Zhang G, et al. NODDI: practical in vivo neurite orientation dispersion
and density imaging of the human brain. Neuroimage 2012; 61 (4):
1000-1016.

^{3} Cook, P. A., Bai, Y., Nedjati-Gilani,
S., Seunarine, K.K., Hall, M.G., Parker, G.J., Alexander, D.C. : Camino:
Open-Source Diffusion-MRI Reconstruction and Processing, 14th Scientific
Meeting of the International Society for Magnetic Resonance in Medicine,
Seattle, WA, USA, p. 2759, May 2006.

^{4} Drobnjak, I., Zhang, H., Ianuş, A., Kaden, E., & Alexander, D. C.:
PGSE, OGSE, and sensitivity to axon diameter in diffusion MRI: Insight from a
simulation study. Magnetic resonance
in medicine, 75 (2), 688-700 (2015)

^{5} Pedregosa, F., Varoquaux, G.,
Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer,
P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D.,
Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: Machine Learning in
Python. JMLR 12, 2825-30 (2011).

^{6} Zhang, J., Jones, M. V., McMahon, M.
T., Mori, S., Calabresi, P. A. : In vivo and ex vivo diffusion tensor imaging
of cuprizone‐induced demyelination in the mouse corpus callosum. Magnetic resonance in medicine, 67(3), 750-759 (2012).

^{7} Karger, J., Pfeifer, H., Wilfried, H.: Principles and
application of self-diffusion measurements by nuclear magnetic resonance. Adv
Mag Res 12(1), 1-89 (1988)