## 2 Results
The latter suggests that the heat transfer was predominantly conductive. For the case of the predicted thermal damage shown in Figure 4b,c, it can be seen that the entire tumor area, plus margins of up to 1 mm around it, was completely destroyed (\(\Omega=100\%\)). A comparison of temporal response curves for temperatures (Figure 4d) at different control points (Figure 3) within the tumor (P1)and at the tumor-gland boundaries (P2-P4) revealed that the temperature rise as well as the final value was higher at (P1) relative to the boundaries: P2 (top), P3 (bottom) and P4 (side). This phenomenon can be attributed to factors such as relatively low blood perfusion and high metabolic heat of the tumor leading to high retention of heat within the tumor [37]. However, at all the locations, the temperature plateaued after about 2-3 min. The consequence of the high temperature within the tumor is revealed in corresponding  
Figure 3: FEM geometry. Schematic of the photothermal therapy consisting of a normal multi-tissue breast domain with an embedded spherical tumor (blue sphere) and NIR (810 nm) laser source. Inset: Fragment of geometry showing controls point P1\(-\)P4, where temperature were recorded. predicted temporal curves for the thermal damage (Figure 4e), which shows that 100% thermal damage occurs faster in the innermost part of tumor (P1)--\(\approx\)3 min--compared to the peripherals, which take up to about \(\approx\)10 min (P4). Consistent with the literature [33; 38], the model predictions showed the dependence of thermal damage spatial profile on the temperature distribution, which decreased with distance away from center of the tumor (see Figure 4f). Ablative temperatures between 60 and 100 \({}^{\circ}\)C cause irreversible damages to key cytosolic and mitochondrial enzymes [39; 40]. For any tumor ablation therapy to be considered successful and thus reduce the chance of recurrence, it is critical to ensure that the entire volume of the tumor reaches therapeutic temperatures that ensures complete thermal damage (\(\Omega=100\%\)). Such a goal can be achieved through the use of an appropriate maximum temperature, which takes into consideration the tumor dimensions. For NP assisted photothermal therapies such as the one being proposed in this study, maximum ablative tumor temperatures, \(T_{\text{max}}\), can be controlled by varying parameters such as NP number density, \(N\) (or volume fraction, \(\phi_{\text{V}}\)), the laser power, and treatment duration, \(t\). To demonstrate this, a parametric study was used to determine \(N\) required to achieve a given \(T_{\text{max}}\) (70, 85, 100 \({}^{\circ}\)C) and the corresponding volume of the lesion \(V_{\text{L}}\) for different tumor sizes, \(R\) (1, 2.5, 5 mm). \(V_{\text{L}}\), was assumed to be spherical [41; 42]; its radius, \(R_{\text{L}}\), was calculated as half the axial length of the predicted cross-sectional area where \(\Omega=100\%\) (see Figure 4c). A summary of the results is presented in Table 1. The simulations were run with \(P_{0}=1\) W and \(t=15\) min. Generally, it can be observed that \(T_{\text{max}}\) required to achieve complete thermal damage increased with size of the tumor. For instance, \(T_{\text{max}}=70\)\({}^{\circ}\)C produced a lesion with \(V_{\text{L}}=2.95\) mm\({}^{3}\), which was insufficient  
Figure 4: Simulation results. Cross-sectional view of the (**a**) temperature distribution, (**b**) thermal damage, (**c**) thermal damage showing the lesion parameter. Temporal response curves for (**d**) temperature and (**e**) thermal damage at the control points (P1–P4, cf. Figure 3). (**f**) Temperature and thermal damage as a function distance from P1. Simulation settings: \(P_{0}=1\) W, \(t=15\) min and \(T_{\text{max}}=85\)\({}^{\circ}\)C.  
to completely ablate the entire volume of tumor with \(R_{\rm T}=1\) mm (\(V_{\rm L}=4.19\) mm\({}^{3}\)). On the other hand, \(T_{\rm max}=85\) produced a lesion with \(V_{\rm L}=2.95\) mm\({}^{3}\), which was big ensure to ensure complete thermal damage.