deletions | additions
diff --git a/Introduction.tex b/Introduction.tex
index 6922b6e..c88d49a 100644
--- a/Introduction.tex
+++ b/Introduction.tex
...
In this study, we show that shear waves can be induced in soft tissues by a laser beam. We also propose a model for the underlying physics. We finally applied the technique in a biological tissue to evaluate its potential application in shear wave elastography.
In the first experiment, illustrated by Figure \ref{Figure1}, we used a Q-switch Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a pulse of energy $E$ = 200 mJ at a central wavelength of 532 nm during
20 10 ns in a beam of section $S$=20 mm$^2$. We defined Z as the laser beam axis, and the laser beam impact on the medium is the origin of coordinates (0,0,0). The laser beam was absorbed in a 4x8x8 cm$^3$ tissue-mimicking phantom made of water and of 5\% polyvinyl alcohol, 1 \% black graphite powder and 1\% salt. A freezing/thawing cycle was applied to stiffen the material to a value of 15$\pm$5 kPa \cite{17375819}.
The laser is absorbed in the medium with an exponential decay of the optical intensity $I(z)=I_0 \exp(- \gamma z)$ along medium depth $z$, where $I_0=\frac{1}{S}\frac{d E}{dt}$ is the incident intensity distribution at the surface (the reflection on the black mat material being neglected) and $\gamma$ the absorption coefficient of the medium.
!!!We We measured the fraction of light which go through different thicknesses of the medium with a laser beam power measurement device (): it indicated that $\gamma \approx$
??? m$^{-1}$ 20 mm$^{-1}$ in our
sample,!!! sample, meaning that
most 63\% of the radiation is absorbed in the first
hundred of micrometers.
%Even if the sample is mainlyeFor low concentration medium, $\gamma$ is hard to calculate in our case, as the sample is composed of different materials, but the graphite, even in low concentration, absorbate much more than other components, so we can approximate $\gamma \approx \gamma_{graphite}$. For graphite particles of 1.85 $\mu$m at a concentration of 10 g.L$^{-1}$, the order of magnitude of $\gamma$ is 10$^4$ m$^{-1}$, meaning that most of the radiation is absorbed in the first hundred of 50 micrometers of
the sample. This is quite higher than metals where the radiation is absorbed within a few
nanometres. nanometers. The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $q = \gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ in absence of convection and of phase transition can be computed using the heat equation:
\begin{equation}
\frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}
\label{eq:eqChaleur}
\end{equation}
where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ =
20 10 ns the laser emission duration and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately equal here to
0.1 $\mu$m. 80 nm. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so that equation \ref{eq:eqChaleur} can be simplified as $\frac{\partial T}{\partial t} = \frac{q}{\rho C}$.
The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium.
In the thermoelastic regime, a local dilatation of the medium occurs. We suppose later on that the medium is
homogenous homogeneous and isotropic, and
we adopt a 1D model, as the depth of absorption is small compared to the beam
diameter. diameter, we adopt a 1D model. The stress $\sigma_{zz}$ is the sum between the axial strain component and the thermal expansion component \cite{scruby1990laser}:
\begin{equation}
\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \zeta}
\label{eq:stressThermo}
\end{equation}
where $\lambda + 2 \mu$ is the P-wave modulus and $\lambda + \frac{2}{3}\mu$ the bulk
modulus, modulus with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\zeta$ the average depth of laser beam absorption. In the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressUnidim} can be integrated, giving a displacement $u_z$ from the surface:
\begin{equation}
u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}
\label{eq:deplThermo}
\end{equation}
As in a biological soft tissues, $\mu \gg \lambda$, the displacement $u_z$ can be approximated as $\frac{3 \alpha E}{\rho C S}$. Taking as an order of magnitude $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 200 mJ, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.5 $\mu$m. This value is slightly smaller than the typical ultrasound resolution of displacement, typically of a few micrometers. In a
tridimensional 3D model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface.
In the ablative regime, the local increase of temperature is so high that the surface of the medium is vaporized. This phenomenon creates a stress $\sigma_{zz}$ in the medium, as the sum of the P-wave modulus and a term given by the second law of motion \cite{scruby1990laser}:
\begin{equation}
...
u_z = \frac{\zeta}{\rho (\lambda + 2 \mu)}\frac{I^2}{(L+C(T_V-T_0))^2} \approx \frac{\zeta}{\rho \lambda}\frac{E^2/S^2 \tau^2}{(L+C(T_V-T_0))^2}
\label{eq:deplAbla}
\end{equation}
Estimating $\zeta$ equal to
100 50 $\mu$m (average depth of absorption), $\lambda$ = 2 GPa (first Lamé's coefficient of water), $L$ = 2.2 MJ.kg$^{-1}$ (vaporization latent heat of water), $C$ = 4180 J.kg$^{-1}$.K$^{-1}$ (water heat capacity), $T_V-T_0$ = 373-293 = 80 K (water vaporization temperature minus laboratory temperature), $\rho$ = 1000 kg.m$^{-3}$ (water density), $E$ = 200 mJ, $S$ = 20
mm$^2$, mm$^2$ and $\tau$ =
20 10 ns, we obtain a displacement $u_z$ approximately equal to
2.0 3.9 $\mu$m. This value is
four eight times higher than the one obtained with thermoelastic expansion.
In both cases, absorption of the laser by the phantom leads to a local displacement which can propagate as elastic waves in the medium. To observe the shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements linked to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was used in ultrafast mode \cite{bercoff2004supersonic}, acquiring 1500 ultrasound images per second. Due to the presence of graphite particles, the medium presented a speckle pattern on the ultrasound image. Tracking the speckle spots with an optical flow technique (Lucas-Kanade method) allowed to compute one component of the displacement in the medium (Z-displacement or Y-displacement, depending on the position of the probe on the medium). The laser beam was triggered 10 ms after the first ultrasound acquisition, $t$ = 0 ms being defined as the laser emission.
...
