## 3 Numerical Results & Discussion
### Thermal Hazard Value of Normal Tissue  
When a laser is irradiated to the tumor tissue for treatment, the temperature rises not only in the tumor tissue but also in the surrounding normal tissue by heat conduction. Accordingly, there is a possibility of thermal damage to surrounding normal tissues. The thermal damage to the tumor tissue can be inferred by checking the apoptosis ratio, but the thermal damage to the surrounding normal tissues cannot be confirmed. Thus, it is necessary to minimize thermal damage by quantitatively evaluating the thermal damage occurring in the surrounding normal tissue through new parameters. Accordingly, in this study, the thermal hazard value was used to derive the degree of thermal damage to the surrounding normal tissue.  
Since the thermal hazard value is defined as the weighted sum of the volume ratios in each temperature range, the value varies depending on the range that serves as the standard. Therefore, the results were derived after selecting the distance from the end of the tumor tissue to 50% of the length of the tumor tissue as the standard range for normal tissue.  
Figure 10 illustrates graphs showing the thermal hazard value of the surrounding normal tissue with respect to the intensity of the irradiated laser according to the tumor length and the volume fraction of the injected GNRs when the tumor depth is 0.1 mm. As shown in Figure 10, it is demonstrated that, as the volume fraction of GNRs in the tumor increases, the intensity of the laser should be lowered to a certain level or lower for reducing the thermal risk. In addition, in the case where the intensity of the irradiated laser is the same, the thermal risk of the normal tissue decreases as the length of the tumor increases; therefore, the degree of the irradiating laser intensity can be increased. This is because as the length of the tumor increases, the amount of heat delivered to the bottom of the tumor decreases, resulting in less temperature rise in the region corresponding to the normal tissue at the bottom of the tumor. However, it can also decrease the apoptosis ratio; therefore, the condition should be optimized after considering all the factors such as thermal risk and apoptosis ratio. This will be further discussed later in the results of the effective apoptosis ratio.  
Figure 10: Thermal hazard value of normal tissue (\(\theta_{H,n}\)) for various volume fractions of GNRs (\(f_{v}\)) (\(D_{t}=0.1\) mm) (**a**) \(L_{t}=0.5\) mm, (**b**) \(L_{t}=1.5\) mm, (**c**) \(L_{t}=2.5\) mm, (**d**) \(L_{t}=3.5\) mm.  
Figure 11 illustrates graphs showing the thermal hazard value of the surrounding normal tissue with respect to the intensity of the irradiated laser according to the tumor depth and the volume fraction of the injected GNRs when the length of the tumor is 2.5 mm. It is shown that the thermal hazard value of the normal tissue decreases as the depth of the tumor increases in the case of the same tumor length. This is because, in the same way as the apoptosis ratio varies according to depth, as the depth increases, the amount of heat from the laser reaching the tumor tissue decreases; in turn, the effect on the surrounding normal tissue decreases. However, since the decrease in the amount of laser power reaching the tumor tissue is accompanied by a decrease in the apoptosis ratio, the effective apoptosis ratio must also be considered.