# Numerical Study of Photoacoustic Pressure for Cancer Therapy
## 3 Numerical Results and Discussion  
Here, we consider a 3D spherical gold nanoparticle of radius \(R_{p}=50\) nm embedded in a 3D spherical cell of radius \(R_{c}=1.0\) um illuminated by an electromagnetic wave at \(\lambda=532\) nm and immersed in a water domain \(\Omega\) of radius \(R_{w}=22\) um at initial temperature \(T_{0}=37.5\)\({}^{\circ}\)C (i.e., 310.5 K). This wavelength is chosen to be near the plasmon resonance for such a gold nanoparticle where light absorption is maximum and was already used as a contrast agent for in vivo tumor imaging [20]. The illumination pulse of duration \(\tau_{pulse}=5\) ns is achieved with a laser with energy \(E_{laser}=10\) mJ and a radius of the spotsize \(R_{beam}=15\) mm. The parameters of Equations (4)-(5) for the thermic and acoustic problems are given in Table 1.  
Figure 1a-d show the geometries of the problem with the meshes' associated materials for the computational domain and the cell, respectively. The relative permittivities of materials (which describe the electromagnetic problem of Equations (1) and (2)) are given in Table 1. Here, the materials of the problem are considered to be isotropic and homogeneous. The results of the computation of the pressure maps are illustrated in Figure 2. Figure 2a-f shows the associated variations of the pressure at 0.5, 1.5, 2.5, 4.5, 14.5 and 40 ns after the illumination. The variations and the propagation of the pressure wave is shown in the hole computational domain. The numerical scheme not only takes into account the shape and size of the nanoparticle but also the variations induced by the electromagnetism, the temperature and the pressure. The pressure is initiated at the beginning of the temperature shock that is induced by the modulation of the illumination. Such an acoustic pressure wave propagates in the whole media (cell and water, corresponding to a spatial extension of some micrometers the cell membrane has spatial extension \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(\rho\) (\(\mathbf{kg}\cdot\mathbf{m}^{-3}\)) & \(C_{p}\) (\(\mathbf{m}^{2}\cdot\mathbf{s}^{-1}\cdot\mathbf{K}^{-1}\)) & \(\kappa\) (\(\mathbf{kg}\cdot\mathbf{m}\cdot\mathbf{s}^{-3}\cdot\mathbf{K}^{-1}\)) & \(v\) (\(\mathbf{m}\cdot\mathbf{s}^{-1}\)) & \(\beta\) (\(\mathbf{K}^{-1}\)) & \(e_{r}\) \\ \hline water & 1000 & 4185 & 0.60 & 1493 & 0.000215 & 1.768 \\ cell & 1090 & 2185 & 1.20 & 1535 & 0.000515 & 1.768 \\ Au & 19,300 & 129 & 250 & 3240 & 0.000014 & \(-5.58+\mathrm{j}2.25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the parameters for the photothermal model for the three media (water, cell, gold) and \(\beta^{2}=-1\). 1 \(\upmu\)m) before vanishing. That contrasts with the temperature propagation (see Figure 3a,b) that is mainly located in the near vicinity of the gold nanoparticle (i.e., only a few nanometers from the gold nanoparticle, corresponding to the spatial expansion of the nanoparticle along the \(y\)-axis: \(-50\) to 50 nm). Therefore, only the nearest zone around the nanoparticle sees the temperature elevation. Figure 4 illustrates that the cell's membrane is receiving a pressure shock wave (i.e., the variation of pressure is non vanishing at the position of the cell membrane). Figure 5 shows the maximum of the pressure at the position of the cell membrane (i.e., at 1 \(\upmu\)m) at the initiation of the temperature and pressure shock and the evolution of the pressure at the limit of the membrane. For an illumination at \(\lambda=532\) nm, the two curves are obtained for two nanoparticle shapes. The first one consists of a spherical gold nanoparticle of radius \(R=50\) nm, the second one is an ellipsoidal particle of semi-axis a = 100 nm and b = 50 nm. Such a pressure variation can induce a deformation of the cell. The deformation is determined by the shear rate of the streaming flow field (computed from the strain tensor) and the cell elasticity [21,22].