# Estimation of Dose Enhancement for Inhomogeneous Distribution of Nanoparticles: A Monte Carlo Study
## 2 Materials and Methods
### Nanoparticles Modeling  
Nanoparticles can be modeled in the simulation either as physical volumes embedded in the phantom or by defining them as a mixture material. In this study, the physical volumes model was initially considered, as it has been reported to be a more accurate model [29]. Initial runs showed the difficulties of using this model, as it would require approximately 10\({}^{12}\) volumes to be created, which would require an impractical amount of memory. Shrinking the phantom size (\(\leq\)1 mm\({}^{3}\)) to reduce the number of NPs to a memory of a manageable amount resolves that issue but causes a limitation on the tumor size. The size of the tumor is essential in this study, as it affects some of the dosimetric quantities to be evaluated. Another noticeable issue with this technique is the significantly longer simulation time, especially when used with the DNA physics models, and the difficulty in distributing the NPs inhomogeneously. The mixture technique used in this study relies on defining the NPs as a mixture with its parent volume (the volume that the NPs are embedded within). This technique was reported to overestimate the dose in some cases [29]. When photons interact with NPs, ejecting secondary electrons in the process, these electrons might be absorbed before leaving the NP volume. As a result, these secondary electrons do not contribute to dose the tissue. This effect, known as self-absorption, is not considered in the mixture technique, as all secondary electrons emitted and absorbed in the mixture contribute to the dose. However, the mixture technique is faster to compute, easier to implement, and is useful when the objective is to compare different types of NPs and provide a general understanding of their effect on dosimetric quantities. The major drawback of this technique is that it does not allow studying the effect of size or shape of the NPs on the dose enhancement, which is out of this study's scope. A mathematical model to mimic the spatial distribution of the NP within the tumor was constructed. The distribution was homogeneous within each of the layers and followed an inhomogeneous, outwardly radial exponential growth, reaching a plateau at the edge of the tumor, \(C\times\left(1.64-5.724\times 10^{-0.294x}\right)\), where C is the total concentration in the tumor and \(x\) is the maximum radial distance from the center of the tumor. For instance, for 30 mg/g, the distribution would be 9.7, 19.8, 27.2, 32.8, 37.04 and 40 mg/g for layers 2 to 7 respectively. The concentration in the necrotic region is constrained to 0. This model, similar to others used by several authors [42; 43; 44], was based on the fact that the vascularization of the tumor tissue follows an exponential decrease from the periphery to the necrotic center [45; 46]. A wide choice of nanomaterial platforms, such as inorganic nanomaterials, are used for dose enhancement because they can accumulate in the tumor tissue via extravasation from leaky vasculature [47]. This phenomenon, known as enhanced permeability and retention (EPR) [48; 49], depends on the specific pathophysiological difference between tumor tissues and healthy tissues. EPR is present in angiogenic tumor vessels, having  
Figure 1: 3D Illustration of the tumor, including all layers, inside the water phantom. gaps larger than 100 nm between adjacent vascular endothelial cells, unlike the blood vessels found in healthy tissues [50]. As a result, these nanoparticles exhibit improved accumulation in the tumors than in normal tissues, thus, selectively enhancing cancer cell killing. Three types of NPs were considered in this work: gold (Z = 79), gadolinium (Z = 64), and silver (Z = 47). The tumor NPs concentrations used in this study are 10, 20, and 30 mg/g of water. The biotoxicity and biocompatibility of the NPs are limiting factors in the choice of NP concentrations [51; 52]. NPs induce human cytotoxicity via mechanisms such as apoptosis, ROS generation, inhibition of mitochondria function, membrane leakage, and membrane lipid peroxidation.