## 2 Materials and Methods
### Numerical Investigation  
The study subject is a skin cancer tumor located near the skin surface as shown in Figure 5. The simulation in this study is a 3D model that contains a cylindrical tumor tissue within the cubic shape of the normal tissue. It was assumed that the laser was irradiated in a circle, and the size of normal tissue was set large enough to maintain a radial directional symmetry of the tumor. Figure 5 shows the result of one x-z plane of the numerical model for better understanding. The skin is composed of a total of four layers. The diameter of the normal tissue is 30 mm, and the diameter of the tumor is 10 mm. The wavelength of the light source was 1064 nm, and the thickness of each skin layer and thermal/optical properties of the tumor and skin layers are shown in Table 3.  
The thermal damage of tissue depends on the duration of temperature impulse. To validate this, after fixing the laser intensity to 0.02 W, the temperature variation for various depths at the center was checked, as shown in Figure 6. It is evident that the temperature of the medium constantly rises and then converges. Accordingly, in this study,  
Figure 4: Numerical validation result: (**a**) temperature variation at center position of tumor (point A); (**b**) temperature distribution at different depths. Figure 3: Schematic figure of validation numerical analysis model. the laser irradiation time was fixed at 1000 s, and the effect of photothermal treatment was investigated through the results in the steady state. In addition, thermal behavior was confirmed for the depth (depth of the tumor) and length, laser intensity, and volume fraction of the injected GNPs from the surface of the tumor, and the conditions are shown in Table 4.  
For the depth and length of the tumor, the penetration depth of the tumor that satisfies all stages of Breslow's depth was considered [40]. The profile of the irradiated laser was assumed to be a top-hat distribution, and the laser diameter was set to be 10 mm, which  
Figure 5: Schematic figure of numerical analysis. Figure 6: Temperature variation for various depth of tumor at center. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(t\) [mm] & \(\rho\) [kg/m\({}^{3}\)] & \(c_{p}\) [J/kgK] & \(k\) [W/mK] & \(\mu_{a}\) [1/mm] & \(\mu_{s}\) [1/mm] & \(g\) \\ \hline Epidermis & 0.08 & 1200 & 3589 & 0.235 & 0.4 & 45 & 0.8 \\ Papillary dermis & 0.5 & 1200 & 3300 & 0.445 & 0.38 & 30 & 0.9 \\ Reticular dermis & 0.6 & 1200 & 3300 & 0.445 & 0.48 & 25 & 0.8 \\ Subcutaneous & & & & & & \\ fat & 7.82 & 1000 & 2500 & 0.19 & 0.43 & 5 & 0.75 \\ Tumor & – & 1030 & 3852 & 0.558 & 0.08 & 1.28 & 0.925 \\ \hline \end{tabular}
\end{table} 
Table 3: Thermal and optical properties of skin layer and tumor [18; 33; 34; 35; 36; 37; 38; 39]. is the same as the tumor diameter. The laser intensity (P) was set from 0 W to 0.2 W at 0.01 W intervals, and the volume fraction of GNP in the tumor to be checked in this study was divided into seven steps from \(10^{-3}\) to \(10^{-6}\), and the temperature distribution was accordingly confirmed. The optical properties of GNPs are determined through the volume fraction (\(f_{v}\)) of GNPs in the tumor, the effective light absorption area (\(r_{eff}\)), and the absorption efficiency (\(Q\)) as shown in Equation (14) [41]. In this study, gold nanorods (GNRs), which have a high efficiency compared to other shapes, such as the sphere, cage, and shell, with a diameter of 10 nm and a length of 67 nm were used [42], and the light absorption efficiency of GNRs was calculated using the discrete-dipole approximation (DDA) technique [43]. It was assumed that the input GNRs were uniformly distributed in the medium.