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\section{Discussions}  \subsection{Practical application}  This study used an ultrasound device to image the sample and track shear waves, due to its high temporal resolution, availability and ease of use. However, for a clinical implementation such as brain elasticity imaging, MRI is more suited for tracking shear waves, as acoustic waves used in ultrasound imaging for shear wave tracking are attenuated by the skull. In a practical MRI implementation, no magnet would be necessary, and MRI-compatible coils should be used. As most clinical MRI scanners use 1.5 T magnetic fields, which is more than  ten times higher than the one used in this study, displacement amplitude could be increased by a similar factor. Magnetic Resonance Elastography is usually employing continuous shear waves; but induction of a continuous electrical current by the coil could affect MRI measurements, so "repetitive transient" excitations, which would lead to a continuous wave, could be used.  \subsection{Displacement amplitude}  In our numerical study, Lorentz force magnitude reached about 60 10  N.m$^{-3}$ for a 0.15 T 50 mT  permanent magnetic field and a 5 S.m$^{-1}$ medium. Numerous measurements of grey and white matter electrical conductivity have been performed, and results vary from 0.02 to 2 S.m$^{-1}$ \cite{19636081}. By assuming an average value of 0.1 to 0.3 0.2  S.m$^{-1}$, in a 1.5 T MRI system, the Lorentz force would reach a magnitude of about 18 to 54 12  N.m$^{-3}$. This is comparable to the magnitude of the acoustic radiation force used for shear wave elastography: this force, calculated with the equation $f = 2 \alpha I \Delta t / c$, with $\alpha$ attenuation of the medium, $I$ ultrasound intensity, $\Delta t$ duration of force application and $c$ speed of sound, is about 80 N.m$^{-3}$ (using Nightingale's parameters \cite{Nightingale_2002}: $\alpha$ = 0.4 Np.cm$^{-1}$, $I$ = 1000 W.cm$^{-2}$, $\Delta t$ = 0.7 ms, $c$ = 1540 m.s$^{-1}$). We found in our numerical study a displacement slightly lower than the experimental value in the phantom. Various factors like viscosity and border effects, which were not included in our model, could explain this difference. Moreover, there are uncertainties about electrical current amplitude and shape in the coil, as constructor values were used.