# Estimation of Dose Enhancement for Inhomogeneous Distribution of Nanoparticles: A Monte Carlo Study
## 3 Results  
The DEF along depth for AgNPs is illustrated in Figure 3. The highlighted area in the figure represents the necrotic region. For 50 keV, a significant dose enhancement is observed for each concentration: up to a factor of nearly seven for 30 mg/g concentration, five for 20 mg/g concentration and three for 10 mg/g concentration. Overall, the DEF decreases with increasing the beam energy and decreasing the NPs' concentrations. The enhancement difference between the first tumor measurement point in the beam path and the last point before the necrotic region is higher for higher concentrations and lower energies. For instance, the DEF for the last measurement point is approximately 48% of the DEF of the first point for 10 mg/g Ag. Increasing the concentration reduces the percentage to 41% and 35% for 20 and 30 mg/g Ag, respectively. Increasing the energy for 10 mg/g Ag increases the percentage to 77%, 85% and 92% for 100, 150 and 200 keV, respectively. Furthermore, given that the concentration is constant within each layer, the DEF values for the same layer should be similar. However, by examining the first and last DEF tumor measurements (both represent layer 7) in Figure 3a, it can be noted that the first measurement shows distinctively higher DEF. At low concentrations and higher energies, the beam has higher penetration power. As a result, the reduction in flux along the depth becomes less affected by the presence of NPs. Therefore, the ratio between the flux in the presence of NPs to flux in the absence of NPs becomes approximately constant and independent of depth. In other words, under these conditions, any two measurement points that are located at different depths but have the same NPs concentrations should show a similar DEF. For example, the differences between the DEF of the first and last measurement points at the tumor (both have the same concentration) are 0.35, 0.79 and 1.61 for 10, 20 and 30 mg/g Ag, respectively. Increasing the energy for 30 mg/g Ag reduces the difference to 0.05, 0.01 and 0.01 for 100, 150 and 200 keV, respectively. Figure 2: (**a**) Side and (**b**) top views showing the voxelization of the phantom. The white area represents the central voxels used to estimate the DEF along depth. Each central 4 voxels DEF is averaged for each depth. As the concentration increases or the photon energy decreases, more photons are attenuated as the beam enters the tumor. As a result, flux decreases more rapidly with higher concentrations and lower energy. The decrease in flux means that, under these conditions, any two measurement points exhibiting the same concentration but located at different depths might show different DEFs. This leads to the conclusion that the DEF for a given concentration and energy is affected by depth. However, the effect of tumor depth on DEF is negligible at higher energies and lower concentrations. The GdNPs DEF over the depth of the tumor is illustrated in Figure 4. A significant dose enhancement in the layers containing NPs for each concentration up to a factor of four for 30 mg/g concentration for both 50 keV and 100 keV is observed. For 20 mg/g concentration, the DEF is approximately three and for 10 mg/g, the DEF is approximately two for both 50 and 100 keV. After that, the DEF decreases with increasing beam energy and decreasing NPs concentration. Figure 5 illustrates the DEF variation over the diameter of the tumor for AuNPs. The 50 keV energy beam shows a significant dose enhancement in the layers containing NPs for each concentration up to a factor of 6.4 for 30 mg/g concentration, just under 5 for 20 mg/g concentration and 3 for 10 mg/g. The DEF decreases with increasing the beam energy and decreasing the NPs' concentrations. Table 1 summarizes the maximum DEF for all energies considered in this study. It is evident that at 400 keV and below, the maximum  
Figure 3: DEF along depth for (**a**) 50, (**b**) 100, (**c**) 150 and (**d**) 200 keV for 10, 20, and 30 mg/g concentrations of AgNPs.