# Numerical study of a macroscopic finite pulse model of the diffusion MRI signal

1 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. The dMRI signal from a heterogeneous sample includes the contribution of the water proton magnetization from all spatial positions in a voxel. If the voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). \soutnew{Recently, a new macroscopic ODE model of the dMRI signal, the Finite Pulse ODE (FP-ODE) model, was derived from the Bloch-Torrey partial differential equation (PDE), without the narrow pulse restriction, using periodic homogenization techniques.}{Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch-Torrey partial differential equation (PDE) using periodic homogenization techniques.} \soutnew{When restricted to narrow pulses, the FP-ODE model has the same form as the Karger model.}{When restricted to narrow pulses, this new homogenized model has the same form as the Karger model.} We conduct a numerical study of the \soutnew{FP-ODE}{new homogenized} model for voxels that are made up of periodic copies of a representative volume that contains spherical and cylindrical cells of various sizes and orientations and show that the signal predicted by the \soutnew{FP-ODE}{new} model approaches the reference signal obtained by solving the full Bloch-Torrey PDE in $O(\veps^2)$, where $\veps$ is the ratio between the size of the representative volume and \soutnew{the diffusion displacement}{a measure of the diffusion length}. When the narrow gradient pulse assumption is not satisfied, the \soutnew{FP-ODE}{new homogenized} model offers a much better approximation of the full PDE signal than the Karger model. Finally, preliminary results of applying the \soutnew{FP-ODE}{new} model to a voxel that is not made up of periodic copies of a representative volume are shown and discussed.
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Journal articles

Cited literature [51 references]

https://hal.archives-ouvertes.fr/hal-01111058
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Jing-Rebecca Li, Hang Tuan Nguyen, Dang Van Nguyen, Houssem Haddar, Julien Coatléven, et al.. Numerical study of a macroscopic finite pulse model of the diffusion MRI signal. Journal of Magnetic Resonance, Elsevier, 2014, pp.54-65. ⟨10.1016/j.jmr.2014.09.004⟩. ⟨hal-01111058⟩

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