# Estimation of Dose Enhancement for Inhomogeneous Distribution of Nanoparticles: A Monte Carlo Study
## 3 Results
The enhancement to healthy tissues in the primary beam path ranged from 5% to 19% for beams below 40 keV energy and from 1% to 3% for above 400 keV beams. In all cases, the maximum DEF in healthy tissues occurred within a 3 mm distance from the tumor boundaries. The average enhancement was not considered in this study, as it is highly dependent on the tumor size [11]. Changing the tumor size will affect the dose enhancement distribution and, consequently, the average DEF. On the other hand, the maximum DEF depends on the source location relative to the tumor. For external photon beam therapy, the maximum usually occurs at the first layer of the tumor that interacts with the beam, as it holds the highest concentration in the case of inhomogeneous NPs distribution. As for homogenous NPs distribution, the maximum occurs in the first layer as long as the tumor is located at depths within or higher than the maximum dose depth. In that case, the photon fluence will be the highest at the tumor first layer. As photons are attenuated along the depth, less of the dose is deposited, resulting in lower DEF along the depth. For a tumor located at a depth less than the maximum dose depth, the maximum DEF might be affected by the shift in the maximum dose depth caused by the introduction of NPs. To emphasize more on the characteristic X-ray emission, Figure 7 depicts the number of characteristic X-rays produced in the tumor and the percentage of X-rays that escape the tumor for AgNPs, GdNPs, and AuNPs respectively. It can be seen that for AuNPs and GdNPs, the maximum number of characteristic X-rays produced occurs at the first incident energy that exceeds the K-edge energy. Increasing the effective atomic number by using a higher \(Z\) NP or a higher concentration increases the number of characteristic X-rays produced, given that the incident energy exceeds the K-edge. This also leads to more characteristic X-rays absorption within the tumor, as will be discussed later. Since lower concentrations have a lower effective \(Z\) and produce fewer X-rays, \(\varepsilon\) is higher for lower concentrations. Considering that most X-rays are produced at the outermost tumor layer and that AuNPs and GdNPs K-shell X-rays are more energetic than AgNPs X-rays, AuNPs and GdNPs X-rays have a higher probability of escaping the tumor and thus have higher \(\varepsilon\). The average differences between \(\varepsilon\) at 10 and 30 mg/g for AuNPs, GdNPs and AgNPs are 4%, 5% and 8% respectively. The larger difference with AgNPs is probably caused by the larger increase in the probability of absorbing the 25.5 keV X-rays with higher concentrations in comparison with the 80 keV and 50.2 keV emitted by AuNPs and GdNPs, respectively. The maximum X-ray energy produced in this study is 80 keV. It would be difficult, even for 80 keV photons, to escape both the tumor and healthy tissue without interaction. Thus, most X-rays escaping the tumor will most likely end up depositing their energy in the healthy tissue. Characteristic X-rays showed to play a significant role in enhancing the dose. Figure 8 shows the total absorbed dose in the tumor and the contribution of X-rays to that dose. Generally, the dose increases with energy and becomes independent of the NPs type and concentration beyond 400 keV. The highest contribution of X-rays results from AgNPs and 50 keV photons. At 50 keV, the probability of absorption is the highest and with the incident photon energy exceeding the K-edge energy, more X-rays are produced. Two factors contribute to the decrease in the characteristic X-ray dose contribution as  
Figure 7: (**a**,\(c\),**e**) The number of characteristic X-rays produced in the tumor as a function of incident photon energy for AgNPs, GdNPs, and AuNPs respectively. (**b**,**d**,**f**) The percentage of X-rays that escaped the tumor for AgNPs, GdNPs, and AuNPs respectively. energy increases. Firstly, the increase in energy decreases the occurrence of photoelectric absorption and consequently, fewer characteristic X-rays are produced.