Maëliss Jallais^{1} and Demian Wassermann^{1}

^{1}Université Paris-Saclay, Inria, CEA, Palaiseau, France

Non-invasive imaging at the cellular level could lead us to quantify grey matter tissue cytoarchitecture, which has so far been accessible only through histology, or by indeterminate-prone approaches.

We propose a new dMRI-based index to study modulated by the size of the somas. This index can be extracted without indeterminations from any acquisition including three b-values superior to 3 ms/μm^{2}. Simulations were experimentally confirmed by tests on the HCP MGH data set.

Such non-invasive measurement of cellular characteristics has the potential to quantify tissue cytoarchitecture, in a unique-solution system, which has so far been accessible only through histology.

Our contribution focuses on the human brain gray matter, which can be decomposed into three compartments, somas and processes that are modeled by spheres and 0-radius tubes, and extra-cellular water. This work shows the feasibility to extract factor that reflects the averaged diameter of the somas in the voxel with a unique solution. Our three-compartment model, under the hypothesis that there is no exchange between the three compartments

\begin{equation} \frac{S(q)}{S(0)} = f_{t}S_{tubes}(q, D_a) + f_{s}S_{spheres}(q, D_{s}, r_{s}) + (1 - f_{t} - f_{s}) S_{ecs}(q, D_{ecs})\end{equation}

f

The tube signal is modeled by a power-law scaling

Others

\begin{equation} RTOP(q_{max}) = \frac{1}{(2\pi)^{3}} \int_{0}^{q_{max}}{\frac{S(q)}{S(0)}} dq_{max}\end{equation}

For q

\begin{equation} RTOP(q_{max}) = \frac{(1 - f_{t})}{(2\pi)^2 \cdot 8 \sqrt{\pi} \cdot 4 C^{3/2}} + \frac{f_{t}}{4(2\pi)^3\sqrt{\pi \tau D_a^{||}}} \cdot q_{max}^2 + \frac{\gamma}{3(2 \pi)^{3}} \cdot q_{max}^3\end{equation}

By solving an ordinary least square regression we can find the coefficients a

Spiked LEMONADE, low b :

\begin{equation} \begin{cases} M^{(2),0} = f_t D_a + 3 \boldsymbol{f_s C} + 3 (1 - f_t - f_s) D_e \\ \frac{M^{(2),2}}{p_2} = f_t D_a \\ M^{(4),0} = f_t D_a^2 + 5 \boldsymbol{f_s C}^2 + 5 (1 - f_t - f_s) D_e^2 \\ \frac{M^{(4),2}}{p_2} = f_t D_a^2 \end{cases} \end{equation}

RTOP, high b:

\begin{equation} \begin{cases} a_{fit} = \frac{(1 - f_{t}) \sqrt{\pi}}{(2\pi)^3 \cdot 4 \boldsymbol{C}^{3/2}} \\ b_{fit} = \frac{f_{t} \sqrt{\pi}}{2(2\pi)^4\sqrt{\tau D_a^{||}}} \end{cases} \end{equation}

1. Partha P. Mitra and Pabitra N. Sen. Effects of microgeometry and surface relaxation onNMR pulsed-field-gradient experiments: Simple pore geometries.Physical Review B, 453(1):143–156, January 1992. ISSN 0163-1829, 1095-3795. doi: 10.1103/PhysRevB.45.143.URLhttps://link.aps.org/doi/10.1103/PhysRevB.45.143.

2. Palombo M, Shemesh N, Ianus A, et al., Abundance of cell bodies can explain the stickmodel’s failure to describe high b-value diffusion signal in grey matter. Proc. Int. Soc.Magn. Reson. Med. 2018, 1096

3. Jelle Veraart, Els Fieremans, and Dmitry S. Novikov. On the scaling behav-ior of water diffusion in human brain white matter.NeuroImage, 185:379–387,January 2019. ISSN 10538119. doi: 10.1016/j.neuroimage.2018.09.075. URLhttps://linkinghub.elsevier.com/retrieve/pii/S1053811918319475

4. Balinov et al. The NMR Self-Diffusion Method Applied to Restricted Diffusion. Simu-lation of Echo Attenuation form Molecules in Spheres and between Planes. 1993

5. Marco Palombo, Clemence Ligneul, and Julien Valette. Modeling diffusion of intracel-lular metabolites in the mouse brain up to very high diffusion-weighting: Diffusion inlong fibers (almost) accounts for non-monoexponential attenuation: Modeling Diffusionof Brain Metabolites in Vivo up to Very High Diffusion Weighting.Magnetic Resonancein Medicine, 77(1):343–350, January 2017. ISSN 07403194. doi: 10.1002/mrm.26548.URLhttp://doi.wiley.com/10.1002/mrm.26548

6. Rutger H.J. Fick, Demian Wassermann, Emmanuel Caruyer, and Rachid De-riche.MAPL: Tissue microstructure estimation using Laplacian-regularizedMAP-MRI and its application to HCP data.NeuroImage, 134:365–385,July 2016. ISSN 10538119. doi: 10.1016/j.neuroimage.2016.03.046. URLhttps://linkinghub.elsevier.com/retrieve/pii/S1053811916002512

7. Mitra et al. Pulsed-field-gradient NMR measurements of restricted diffusion and thereturn-to-origin probability.Journal of Magnetic Resonance, 114:47–58, 1995

8. Dmitry S. Novikov, Jelle Veraart, Ileana O. Jelescu, and Els Fieremans. Rotationally-invariant mapping of scalar and orientational metrics of neu-ronal microstructure with diffusion MRI.NeuroImage, 174:518–538, July2018.ISSN 10538119.doi: 10.1016/j.neuroimage.2018.03.006.URLhttps://linkinghub.elsevier.com/retrieve/pii/S1053811918301915

9. Evrard, H. C., Forro, T. & Logothetis, N. K. Von Economo Neurons in the Anterior Insula of the Macaque Monkey. Neuron 74, 482–489 (2012)

10. Katrin Amunts and Karl Zilles.Architectonic Mapping of the Hu-man Brain beyond Brodmann.Neuron, 88(6):1086–1107, Decem-ber 2015.ISSN 08966273.doi: 10.1016/j.neuron.2015.12.001.URLhttps://linkinghub.elsevier.com/retrieve/pii/S0896627315010727