## 4 Discussion  
### Dependence of DNA Damege on Number of GNPs  
In Figure 4, it is evident that the most extensive DNA damage and the greatest number of strand breaks occurred when four GNPs were present. Increasing the number of GNPs from 0 to 4 resulted in an increase in the total number of strand breaks. When the simulations held the volume of gold constant, adding more nanoparticles led to a reduction in nanoparticle size or an increase in the surface-to-volume ratio of the GNPs. This change intensified interactions between the nanoparticles and the radiation beam, thereby increasing the secondary electron yield. These secondary electrons could then be transported to the DNA to cause strand breaks. Another important factor to consider is the self-absorption of the GNP. It was observed that the self-absorption effect was more significant when the nanoparticle size increased. This effect resulted in a reduction in the secondary electrons generated by the nanoparticle, which subsequently reduced DNA damage. However, in the simulation geometry of the four GNPs, each nanoparticle was smaller than in the single GNP geometry. Therefore, the self-absorption effect was less significant, resulting in the generation of more secondary electrons from the four GNPs geometry. This caused more strand breaks and DNA damage.  
Figure 4 also demonstrates that the total number of strand breaks for zero and one GNP was almost identical, while the differences between the number of strand breaks for zero and multiple GNPs were more significant. This discrepancy was primarily due to the single GNP geometry, in which all the gold volume contributed to a single nanoparticle, resulting in a larger nanoparticle size than in other GNP geometries. This caused an enhanced self-absorption effect, which offset the dose enhancement. Additionally, the presence of a single GNP acted as a \"shield\", attenuating the electron beams that irradiated the DNA, thereby further reducing the enhancement of DNA damage.  
Table 1 shows that the number of DSBs for a single GNP was slightly lower than that for the water NP or no GNP. This is due to the self-absorption effect of the single GNP, which reduced the number of secondary electrons that moved to the DNA. This effect was less pronounced when the GNP size was smaller. Since the total gold volume in each simulation with different nanoparticle quantities was constant, the amount of gold used in the clinic would remain the same if the number of gold nanoparticles increased. Even with smaller sizes, the same amount of gold would lead to enhanced DNA damage. Regarding the average number of DSBs per GNP, larger GNPs still had more DSBs than the smaller ones (e.g., 79 vs. 61 for one vs. four GNPs). On the other hand, increasing the number of GNPs and decreasing their size led to a higher total number of DSBs, as shown by the simulations with the four GNPs. This suggests that to enhance DNA damage at the same nanoparticle concentration, increasing the number of GNPs and decreasing their size is recommended. However, when comparing GNPs of different sizes while keeping the nanoparticle number constant, larger GNPs produce more DSBs than smaller ones do.