Abib O. Y. Alimi^{1}, Alexandra Petiet^{2}, Mathieu Santin^{2}, Anne-Charlotte Philippe^{2}, Stephane Lehericy^{2,3}, Rachid Deriche^{1}, and Demian Wassermann^{1,4}

We study the
sensitivity of time-dependent diffusion MRI indices or qτ-indices to demyelination
in the mouse brain. For this, we acquire *in vivo* four-dimentional
diffusion-weighted images -varying over gradient strength, direction
and diffusion time- and estimate the qτ-indices from the
corpus callosum. First order Taylor approximation of each index gives fitting coefficients α and β whose variance we investigate. Results indicate that, cuprizone intoxication
affects mainly index coefficient β by introducing inequality of variances
between the two mice groups, most significantly in the splenium and that MSD increases and RTOP decreases over diffusion
time τ.

In this study, we consider the time-dependent dMRI indices^{4}. These microstructure indices correspond to the three-dimensional q-space scalar indices^{7,8}
evaluated over diffusion time $$$\tau$$$. That is, the time-dependent Return-To-Origin Probability
(RTOP), Return-To-Axis
Probability (RTAP), Return-To-Plane Probability (RTPP) and Mean
Squared Displacement (MSD). These are referred to as q$$$\tau$$$-indices and
are estimated using the sparse four-dimensional diffusion signal
representation.
For illustration, we focus on MSD and RTOP in this preliminary study.

*Index
coefficients approximation*:
To compare the q$$$\tau$$$-indices from the corpus callosum regions of the two mice groups,
each index is formulated as a function of time $$$\tau$$$: $$$index(\tau)$$$.
Then, in the logarithmic scale illustrated in Fig. 2, the first
order Taylor approximation of $$$\log{}(index(\tau))$$$
fits a linear function of $$$\log{}\tau$$$, $$$\alpha\log{}\tau +\log{}\beta$$$
to $$$\log{}(index(\tau))$$$. For
each q$$$\tau$$$-index, the coefficients $$$\alpha$$$ and $$$\log{}\beta$$$ are estimated by a least
square approach then the exponent of the linear line is taken $$index(\tau) \simeq \exp(\alpha\log{}\tau+\log{}\beta+\mathcal{\Omega}(\log{}\tau)) = \beta \tau^\alpha\mathcal{\Omega}(\log{}\tau) \quad (1) $$

Given an index, $$$\alpha$$$ and $$$\log{}\beta$$$ respectively correspond to the slope and intercept of the linear system that best fits the index in the log scale.

*In
vivo acquisitions*:
We acquire *in
vivo*
diffusion images of the brains of controls and cuprizone-treated mice
on a $$$11.7$$$ Tesla Bruker scanner. We use the q$$$\tau$$$-dMRI acquisition
scheme^{4}
defined in q$$$\tau$$$ space so as to account for both three-dimensional q-space and
diffusion time $$$\tau$$$, but with drastically less q$$$\tau$$$-samples using the
relaxed probabilistic model^{9}
for down-sampling purpose. The data consist of $$$80 \times 160 \times 5$$$ voxels of
size $$$0.1 \times 0.1 \times 0.5$$$ mm^{3}
for a total of $$$515$$$ diffusion-weighted images from each mouse.

*Data
preparation*:
We use FSL’s eddy to correct the data from eddy currents and motion
artifacts. We then manually create a mask from the fractional anisotropy (FA) map, as pictured in Fig. 1, to segment the corpus callosum and separate three regions of interest:
genu, body and splenium.

Fig.
2 indicates that, at short diffusion $$$\tau$$$, time-dependent q$$$\tau$$$-indices are approximated by the first order Taylor in the
logarithmic scale. To assess cuprizone effects on q$$$\tau$$$-indices, our analysis focuses on the derived $$$\beta$$$ coefficients. Results concerning the slope $$$\alpha$$$ are discarded since $$$\alpha$$$ does not vary very notably between mice groups. As cuprizone induces myeline changes mainly in the corpus callosum, we test the homogeneity of variances of $$$\beta$$$ estimated from MSD and RTOP. In Fig. 3, for both q$$$\tau$$$-indices, difference of variances is significant in the splenium of the corpus callosum. It is the same in the body for MSD but no significant inhomogeneity is observed elsewhere. Fig. 4 supports this trend by showing large variances of $$$\beta$$$ between mice groups, more specifically in the splenium, considering both q$$$\tau$$$-indices. Overall, $$$\beta$$$ coefficients of indices determined from the splenium are the most impacted as expected because this region is the most readily affected by cuprizone^{10}. Moreover, MSD increases and RTOP
decreases over time, see Fig. 2. Indeed as diffusion $$$\tau$$$ increases, spins have more time to diffuse, traveling longer distances and reducing their chance to return at the origin^{7}.

This work was partly supported by ANR “MOSIFAH” under ANR-13-MONU-0 0 09-01, the ERC under the European Union’s Horizon2020 research and innovation program (ERC Advanced Grant agreement no 694665: CoBCoM), MAXIMS grant funded by ICM’s The BigBrain Theory Program and ANR-10-IAIHU-06.

1. Bihan, Denis Le. "Molecular diffusion, tissue microdynamics and microstructure." NMR in Biomedicine 8.7 (1995): 375-386.

2. Novikov, Dmitry S., et al. "Revealing mesoscopic structural universality with diffusion." Proceedings of the National Academy of Sciences 111.14 (2014): 5088-5093.

3. Fieremans, Els, et al. "In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter." NeuroImage 129 (2016): 414-427.

4. Fick, Rutger HJ, et al. "Non-parametric graphnet-regularized representation of dMRI in space and time." Medical Image Analysis 43 (2018): 37-53.

5. Praet, Jelle, et al. "Cellular and molecular neuropathology of the cuprizone mouse model: clinical relevance for multiple sclerosis." Neuroscience & Biobehavioral Reviews 47 (2014): 485-505.

6. Jelescu, Ileana O., et al. "In vivo quantification of demyelination and recovery using compartment-specific diffusion MRI metrics validated by electron microscopy." Neuroimage 132 (2016): 104-114.

7. Özarslan, Evren, et al. "Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure." NeuroImage 78 (2013): 16-32.

8. Fick, Rutger HJ, et al. "MAPL: Tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data." NeuroImage 134 (2016): 365-385.

9. Filipiak, Patryk, et al. "Spatio-Temporal dMRI Acquisition Design: Reducing the Number of q$$$\tau$$$ Samples Through a Relaxed Probabilistic Model." MICCAI 2017 Workshop on Computational Diffusion MRI (CDMRI 2017). 2017.

10. Steelman, Andrew J., Jeffrey P. Thompson, and Jianrong Li.
"Demyelination and remyelination in anatomically distinct regions of the
corpus callosum following cuprizone intoxication." Neuroscience research 72.1 (2012): 32-42.

Figure 1. Illustrations of
segmentation of the corpus callosum, colored in red from the FA map, of mice brains selected from the control group (top) and the cuprizone-treated group (bottom), respectively.

Figure 2. Mean and standard deviation of MSD and RTOP estimated from the genu, the body and the splenium of the corpus callosum. In each region, the qτ-index is fitted in the logarithmic scale by a linear function (dashed line) of log(τ) derived from the first order Taylor approximation.

Figure 3. Levene's test for equality of variances of estimated β between controls and cuprizone-fed mice. After Bonferroni correction, significant results are highlighted in cyan (p<0.05).

Figure 4. Histogram comparison of variances of approximated β coefficients: inhomogeneities of variances are observed between mice groups. The vertical lines indicate the mean of each distribution.