Khieu Van NGUYEN^{1}, Edwin Hernandez Garzon^{1}, and Julien Valette^{1}

Here we implement a GPU-based Monte-Carlo simulation of diffusion to efficiently simulate diffusion-weighting in realistic, complex cellular structures such as astrocytes as directly derived from confocal microscopy. This opens new possibilities to better understand intracellular diffusion, validate diffusion models, and create dictionaries of intracellular diffusion signatures.

**Methods **

*Triangular surface mesh of astrocyte*: Mouse brain slices were acquired using a Leica SP8 confocal system and confocal image resolution was 0.07443×0.07443×0.2985 µm^{3}. The volume image of mouse's astrocytes was extracted from the confocal microscopy images, and was then used for generating the triangular surface mesh by using an open-source meshing toolbox "*Iso2Mesh*"^{1} adapted with the *cgalmesh*^{2} library. Typical astrocytic surfaces consist of 5×10^{5} - 1×10^{6} faces.

*GPU-based Monte-Carlo simulation*: Random walk of particles (N = 2^{17}) was simulated inside the structure with free intracellular diffusivity *D _{intra}* = 0.5 µm

All codes were implemented in C++ using the CUDA library to interface with NVIDIA GPUs (Tesla K40c) and performed on an HP workstation (Intel(R) Xeon(R) CPU E5-2630 v4 @ 2× 2.20GHz) on Windows 7 professional.

First of all, we validated the numerical simulation of the diffusion signal attenuation, as well as the ADC (calculated as $$$\frac{-ln(S(b=3))}{3}$$$) as a function of ∆, in the simple case of restricted diffusion in the box of size Lx=4.02, Ly=9.02 and Lz=14.02 µm (fig. 1A). The good agreement between the simulated signal attenuation and analytical signal attenuation^{6} in the case of infinite short gradient duration (δ=0 ms), diffusion time ∆=50 ms and very high b-values (b from 0 to 60 ms/µm^{2}) is shown on fig. 1B. Fig. 1C showing the superimposed between the simulated and analytic ADC behavior when increasing the diffusion time from 50 ms to 500 ms.

Moreover, we ran the simulation for diffusion in astrocyte reconstructed from confocal microscopy image (fig. 2A) for three different diffusion times (∆=50, 250 and 500 ms) and high b-values (up to 60 ms/µm^{2}). The simulated signal attenuation is shown in fig. 2B. The corresponding ADC (calculated as $$$\frac{-ln(S(b=3))}{3}$$$ ) as a function of ∆ is shown in fig. 2C.

Simulated data can then be analyzed using existing models, which might be interesting to investigate the validity of these models. For example, it is possible to fit the high-b data simulated at ∆=50 ms using an analytical model of diffusion in randomly oriented cylinders of infinite length but finite diameter, as we introduced for experimental intracellular metabolite diffusion measured by diffusion-weighted MRS^{7}. If we do so, we extract *D _{intra}*=0.31 µm

**Discussion and conclusion**

- Qianqian Fang and David Boas, Tetrahedral mesh generation from volumetric binary and gray-scale images,
*Proceedings of IEEE International Symposium on Biomedical Imaging*,**2009**, pp. 1142-1145 - CGAL, Computational Geometry Algorithms Library,
*http://www.cgal.org* - Donald Meagher, Geometric modeling using octree encoding
*, In Computer Graphics and Image Processing, Volume 19, Issue 2, 1982, Pages 129-147, ISSN 0146-664X, https://doi.org/10.1016/0146-664X(82)90104-6.* - Hall, M. G. and Alexander, D. C., Convergence and Parameter Choice for Monte-Carlo Simulations of Diffusion MRI,
*IEEE Transactions on Medical Imaging*,**2009**, 28, 1354-1364 - Waudby, C. A. and Christodoulou, J., GPU accelerated Monte Carlo simulation of pulsed-field gradient NMR experiments,
*Journal of Magnetic Resonance*,**2011**, 211, 67 - 73 - Tanner, J. E. and Stejskal, E. O. Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method,
*The Journal of Chemical Physics, AIP*,**1968**, 49, 1768-1777 - Palombo, M.; Ligneul, C. and Valette, J., Modeling diffusion of intracellular metabolites in the mouse brain up to very high diffusion-weighting: Diffusion in long fibers (almost) accounts for non-monoexponential attenuation,
*Magnetic Resonance in Medicine*,**2017**, 77, 343-350 - Palombo, M.; Ligneul, C.; Hernandez-Garzon, E. and Valette, J., Can we detect the effect of spines and leaflets on the diffusion of brain intracellular metabolites?
*NeuroImage*,**2017** - VAA3D, 3D Visualization-Assisted Analysis,
*www.vaa3d.org*

A) The
box meshed by triangles (438 vertices and 872 triangle faces) used for validation.
B) The simulated signal attenuation (dot) in x, y and z directions is in good agreement with the theoretical signal attenuation (dash) in case
of δ=0
ms, ∆=50 ms and b-values from 0 to 60 ms/µm^{2}. C) The simulated ADCs
for differ diffusion time from 50 to 500 ms are superimposed with the analytic
ADC values. The simulation parameter is 2^{20} particles, the number of time-step is 5000, the simulation time approximates
~19.5 hours for 3 diffusion directions, 60 b-values per diffusion time.

The simulated signal attenuation (B) and the simulated ADC values (C) in one astrocyte (A) for three different diffusion times 50, 250 and 500 ms. The black square box represents 5×5 µm^{2}. The simulation time was ~21 hours 30 minutes for 60 b-values in the case of ∆=50 ms.