# Numerical Study of Photoacoustic Pressure for Cancer Therapy
## 3 Numerical Results and Discussion
Here, with the used materials, the shear stress components are: \(\sigma_{zz}=\Delta P\approx 300\) Pa, \(\sigma_{xx}=\sigma_{yy}=\Delta PR/(2e)\approx 0.05\) MPa with \(R\) being the radius of the cell, \(e\) being the width of the cell's membrane and the associated shear modulus being \(\tau\approx 0.01\) to 0.1 GPa [23]. Moreover, the maximum rate of strain is about \(S_{\max}\approx 10^{4}\)-\(10^{5}\) s\({}^{-1}\) (i.e., \(S_{\max}\approx\delta R/R/\Delta t\)[24] where \(\Delta t\) is the evolution time). That leads to a maximum tension in the membrane \(\tau_{\max}\approx\mu S_{\max}R\) varying in the range \([10^{-4};10^{-1}]\) N\(\cdot\)m\({}^{-1}\), where \(\mu\approx 0.1\) to 0.001 kg\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\) is the dynamic viscosity of the aqueous solution in the cell (i.e., depending on the cell composition). Such a tension is sufficient to stabilize and neglect the thermal fluctuation of the membrane area, but will only expand the actual membrane area by a fraction \(\Delta S/S\approx 2.0\times 10^{-3}\) (with an expansion modulus of about 0.24 N\(\cdot\)m\({}^{-1}\)[24]). To achieve the shear of the membrane (or at least opening of pores), a ratio of area \(\Delta S/S\geq 0.03\) is necessary [24]. To extend the application in cancer therapy, the illumination must use wavelengths in the biological window (800 to 1000 nm). Such an illumination of a spherical gold nanoparticle at \(\lambda=800\) nm is shown in Figure 5. In such a window, the imaginary part of the permittivity of the gold nanoparticle is smaller and requires us to design the particle shape or to increase the energy laser [25]. Moreover, the use of larger cell (typically a cell size is varying from 1 to 100 \(\upmu\)m) or a mix of larger nanoparticles, greater pulse energy or a higher repetition rate of ultra-short illumination shocks (which will induce an increase in the local pressure variation at the membrane boundary and therefore the maximum rate of strain) can open the way to a process inducing a local stress (shear stress) at the membrane in order to produce a disruption of the cell's membrane. Here, we focus on the study of one nanoparticle, nevertheless, metal nanoparticles can also be taken up by cells in 1-15 nanoparticles/vesicles [26,27]. The increase of the number of nanoparticles, if aggregated, has a similar effect of increasing the size and shape of the nanoparticle. Indeed, each nanoparticle plays the role of local temperature source inducing a photoacoustic pressure wave. Therefore, multiple pressure waves can be induced at the same time and the pressure shock at the membrane can also be increased due to the aggregated nanoparticles. Such a disruption could be produced by cytolysis (or osmotic lysis) that occurs when a cell bursts due to an osmotic imbalance that has caused excess water being released into the cell. Figure 1: Geometry (**a**) and associated mesh (**b**) of the 3D water domain including the cell; Zoomed-in view in the xy plane on the cell (**c**) including the gold nanoparticle and (**d**) the associated mesh. Figure 2: Time evolution of the variations of the acoustic pressure (in Pa) computed at **(a)** 0.5 ns; **(b)** 1.5 ns; **(c)** 2.5 ns; **(d)** 4.5 ns; **(e)** 14.5 ns and **(f)** 40.0 ns after the initial shock in the whole computationnal domain. The size of the cell’s membrane is also illustrated at the beginning and the end of the wave propagation **(a–f)**. The radii of the nanoparticle, the cell and the computationnal domain are: 50 nm, 1 μm and 22 μm, respectively. Figure 4: **(a)** Time evolution of the variation of the pressure (in Pa) in the cell (i.e., \(|y|\leq 1000\) nm) and in the water (i.e., \(|y|\geq 1000\) nm); **(b)** The spatial distribution of the pressure variation is computed from the center of the nanoparticle (i.e., at \(|y|\approx 0\) nm) and propagated to the cell membrane (i.e., at \(|y|\approx 1\) μm).