Synopsis

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.

Introduction

Methods

In this study, we consider the time-dependent dMRI indices⁴. 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 scheme⁴ 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 model⁹ for down-sampling purpose. The data consist of $$$80 \times 160 \times 5$$$ voxels of size $$$0.1 \times 0.1 \times 0.5$$$ mm³ 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.

Results and discussion

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 cuprizone¹⁰. 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 origin⁷.

Conclusion

Acknowledgements

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.

References

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.