Corresponding author: Sajad Keshavarz ( sajadkeshavarz7@gmail.com ) Academic editor: Boris Balakin
© 2019 Sajad Keshavarz, Dariush Sardari.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Keshavarz S, Sardari D (2019) Different distributions of gold nanoparticles on the tumor and calculation of dose enhancement factor by Monte Carlo simulation. Nuclear Energy and Technology 5(4): 361371. https://doi.org/10.3897/nucet.5.39096

Gold nanoparticles can be used to increase the dose of the tumor due to its high atomic number as well as being free from apparent toxicity. The aim of this study is to evaluate the effect of distribution of gold nanoparticles models, as well as changes in nanoparticle sizes and spectrum of radiation energy along with the effects of nanoparticle penetration into surrounding tissues in dose enhancement factor DEF. Three mathematical models were considered for distribution of gold nanoparticles in the tumor, such as 1uniform, 2 nonuniform distribution with no penetration margin and 3 nonuniform distribution with penetration margin of 2.7 mm of gold nanoparticles. For this purpose, a cubeshaped water phantom of 50 cm size in each side and a cube with 1 cm side placed at depth of 2 cm below the upper surface of the cubic phantom as the tumor was defined, and then 3 models of nanoparticle distribution were modeled. MCNPX code was used to simulate 3 distribution models. DEF was evaluated for sizes of 20, 25, 30, 50, 70, 90 and 100 nm of gold nanoparticles, and 50, 95, 250 keV and 4 MeV photon energies. In uniform distribution model the maximum DEF was observed at 100 nm and 50 keV being equal to 2.90, in nonuniform distribution with no penetration margin, the maximum DEF was measured at 100 nm and 50 keV being 1.69, and in nonuniform distribution with penetration margin of 2.7 mm, the maximum DEF was measured at 100 nm and 50 keV as 1.38, and the results have been showed that the dose was increased by injecting nanoparticles into the tumor. It is concluded that the highest DEF could be achieved in low energy photons and larger sizes of nanoparticles. Nonuniform distribution of gold nanoparticles can increase the dose and also decrease the DEF in comparison with the uniform distribution. The nonuniform distribution of nanoparticles with penetration margin showed a lower DEF than the nonuniform distribution without any margin and uniform distribution. Meanwhile, utilization of the real Xray spectrum brought about a smaller DEF in comparison to monoenergetic Xray photons.
dose enhancement, gold nanoparticles, distribution, nanoparticle size, Monte Carlo
Cancer is one of the main reasons of fatality and one of the treatment methods for curing this ailment can be radiotherapy. The main purpose in radiotherapy is to give lethal dose to the tumor and protecting the normal tissues. One of the methods we can implement here is the use of nanoparticle in treatment and diagnosis. Nanoparticles can enhance the efficiency of diagnosis and treatment of cancer. The purpose of new radiation therapy techniques is reducing the absorbed dose in healthy tissues (Mousavie Anijdan et al. 2008). Since the major interaction of photons inside the tumor depends on the photon energy and target atomic number, gold with Z=79 is a proper candidate to be formed as nanoparticles and can enter into a tumor. Gold nanoparticles (GNPs) have longer retention time in a tumor. Therefore, they can be attached to the tumor (
The effect of iodine concentration and radiation quality on growing lymphocytes as well as the effect of the dose after injection of iodine in the brain of rabbits was reviewed by
Energy optimization in gold nanoparticle enhanced radiation therapy by Monte Carlo simulation in the brain and breast tumors in kilovoltage and orthovoltage beam energy has been compared by Sung et al. (2018). they showed for breast using a single photon beam, kilovoltage with Gold NanoParticle (kV+GNP) was found to yield up to 2.73 times higher mean relative biological effectiveness (RBE)weighted dose to the tumor than two tangential MegaVoltage (MV) beams while delivering the same dose to healthy tissue and for irradiation of brain tumors using multiple photon beams, the GNP dose enhancement was found to be effective for energies above 50 keV.
Local Effect Model (LEM)based predictions of radioenhancement were suggested based on the ideas of heterogeneous dose distributions inside the cell in GNPenhanced xray therapy (
Sizedependent tissue kinetics of Polyethylene Glycol (PEG)coated gold nanoparticles was studied by
In previous studies, commonly iodine and gadolinium were used, but in this study, instead of iodine and gadolinium, gold nanoparticles due to their high atomic number, inaction, and better biocompatibility are used. The formation of more surface bonds between biomolecules and gold nanoparticles and ~3times more photon absorption than iodine at 20 and 100 keV energy (
As it is illustrated in Fig.
Location of tumor in phantom and the mesh size for computation. (A): uniform distribution of GNPs, (B): Nonuniform distribution of GNPs while the central layer is in the necrosis state. Each layer of thickness 1 mm is divided into sublayers, (C): Nonuniform distribution of GNPs with penetration margin of 2.7 mm for the tumor.
Three kinds of spatial distributions were considered for gold nanoparticles inside the tumor as below:
This is the most ideal case for distribution of nanoparticles. In this model, it was assumed that nanoparticles were uniformly distributed all over the tumor volume and there were no nanoparticles outside the tumor and therefore:
$C\left(r\right)=\{\begin{array}{cc}\mathrm{C}\hfill & \text{if}\phantom{\rule{1.00em}{0ex}}0\le r\le A\hfill \\ 0\hfill & \text{if}\phantom{\rule{1.00em}{0ex}}r>A\hfill \end{array}$ (1)
In this equation, A is the side of the cube and r is the distance of an arbitrary point from the tumor center. C is the concentration of gold nanoparticles.
As it was mentioned before, in Fig.
$DEF=\genfrac{}{}{0.1ex}{}{\text{Averge dose with GNPs}}{\text{Aveage dose without GNPs}}$ (2)
In this study, such computations were carried out for xray spectra generated by 50, 95, 250 keV and 4 MV linear accelerators (
As a matter of fact, the blood circulation in the tumor surface is much more than its internal layers and the tumor core is normally a necrosis volume. Therefore, it is expected to exist less concentration of nanoparticles as it is deeper into the tumor. The cubical tumor was divided into 11 layers for showing the change of absorption gold nanoparticle form surface of tumor to the center of the tumor that it was assumed 1 layer was in the center of the tumor (necrosis) and 5 layers were upper than the necrosis layer and 5 layers were below the necrosis layer. Such distribution of nanoparticles is described by C_{i} = a.exp(z)k. Which C_{i} is the amount of concentration in each layer and a is a constant number that depends on the total concentration, and k is a constant number and z is the distance from the center of the tumor. In the present work, the cubical tumor shape was divided into 11 shells (layers) with different concentrations that reduced as the shell was closer to the center of the tumor. Till in the central layer the concentration becomes zero (C_{i} = 0).With considering C_{i} = 0 in C_{i} = a.exp(z)k, it obtained:
$\{\begin{array}{c}{C}_{i}=0\hfill \\ z=0\hfill \end{array}\to {C}_{i}=a{e}^{z}k\to 0=a{e}^{0}k\to a=k$
${C}_{i}=a({e}^{z}1)$ (3)
Considering the above relation and that the integral of this equation (3) is equal to the total concentration of nanoparticles on the tumor from the center (z = 0) to the surface of the tumor (z = 0.5 cm), and total concentration of nanoparticle in this study is C_{T} = 18 mg/ml and then the value of a and k are calculated:
$\underset{0}{\overset{0.5}{\int}}}{C}_{i}dz={C}_{T$
$\underset{0}{\overset{0.5}{\int}}}{C}_{i}dz={\displaystyle \underset{0}{\overset{0.5}{\int}}}a({e}^{z}1)dz={C}_{T$
$a({e}^{0.5}1.5)=18$
$a=k=0.121$
Therefore, the distribution of nanoparticle in each layer in the tumor is obtained:
${\mathrm{C}}_{\mathrm{i}}=0.121\mathrm{exp}\left(\mathrm{z}\right)0.121$ (4)
As it can be observed in Fig.
In clinical practice, it was observed that deposition of nanoparticles is not restricted to only tumor volume (
${C}_{i}=a({e}^{z}1)$
$\underset{0}{\overset{0.5}{\int}}}{C}_{i}dz={C}_{T$
$\underset{0}{\overset{0.5}{\int}}}{C}_{i}dz={\displaystyle \underset{0}{\overset{0.5}{\int}}}a({e}^{z}1)dz={C}_{T$
$a({e}^{0.5}1.5)=13$
$a=k=0.0087$
Therefore, the distribution of nanoparticle in each layer in the tumor is obtained:
${C}_{i}=0.0087{e}^{z}0.0087$ (5)
In this equation, C_{i} is the amount of concentration in each layer and z is the distance from the center of the tumor.
The calculation an uncertainty was less than 3%. DEF has been computed for Xrays of 50, 95, 250 keV and for Xray generated with 4MV with Varian linac with the same size of nanoparticles. It is noteworthy that the DEF was also obtained for some points outside the tumor in the zdirection.
The results of DEF with different sizes of gold nanoparticles (20, 25, 50, 70, 90 and 100 nm) in 50, 95, 250 keV and 4 MeV energies with 18 mg/cm^{3} concentration in the uniform distribution are revealed in Fig.
Average dose enhancement factor (DEF) in the whole volume for different photon energies and GNP sizes for the uniform model.
Diameter of GNPs  
20 nm  25 nm  30 nm  50 nm  70 nm  90 nm  100 nm  
50 keV  2.46  2.47  2.48  2.55  2.76  2.85  2.90 
95 keV  2.01  2.02  2.03  2.06  2.18  2.25  2.28 
250 keV  1.13  1.13  1.14  1.15  1.18  1.19  1.20 
4 MV  1.04  1.05  1.05  1.05  1.06  1.06  1.07 
Variation of dose enhancement factor based on the size of nanoparticles and energy of Xrays in the uniform model is demonstrated in Figs
The results of DEF with variation of the size of gold nanoparticles (20, 25, 50, 70, 90 and 100 nm) in 50, 95, 250 keV and 4 MV energies in the 18 mg/cm^{3} of concentration in the uniform distribution without penetration margin are displayed in Fig.
Average dose enhancement factor (DEF) in the whole volume for different photon energies and GNP sizes for the nonuniform model.
Diameter of GNPs  
20 nm  25 nm  30 nm  50 nm  70 nm  90 nm  100 nm  
50 keV  1.49  1.51  1.52  1.54  1.60  1.65  1.69 
95 keV  1.23  1.23  1.26  1.27  1.29  1.31  1.33 
250 keV  1.06  1.07  1.08  1.08  1.08  1.09  1.09 
4 MV  1.04  1.05  1.05  1.06  1.07  1.07  1.08 
Variation of dose enhancement factor versus the size of nanoparticles and energy of Xrays in the uniform model without penetration margin is presented in Fig.
The results of DEF with variation of the size of gold nanoparticles (20, 25, 50, 70, 90 and 100 nm) in 50, 95, 250 keV and 4 MV energies with 18 mg/cm^{3} concentration in the uniform distribution with 2.7 mm penetration margin are depicted in Fig.
Average dose enhancement factor (DEF) in the whole volume for various photon energies and GNP sizes for the nonuniform model with a 2.7 mm penetration margin.
Diameter of GNPs  
20 nm  25 nm  30 nm  50 nm  70 nm  90 nm  100 nm  
50 keV  1.25  1.26  1.27  1.30  1.31  1.35  1.38 
95 keV  1.08  1.10  1.12  1.13  1.15  1.16  1.19 
250 keV  1.02  1.03  1.03  1.04  1.04  1.05  1.06 
4 MV  1.01  1.01  1.01  1.02  1.02  1.02  1.02 
Variation of dose enhancement factor with the size of nanoparticles and energy of Xraysin the uniform model with a 2.7 mm penetration margin is shown in Figs
In this study, the effect of uniform and nonuniform distribution of gold nanoparticles in the tumor, the effect of nanoparticle sizes, the effect of energy on the tumor dose enhancement and the effect of the margin of penetration of gold nanoparticles into healthy tissues around the tumor were taken into consideration.
Dose enhancement factor was calculated in three models of uniform distribution, nonuniform distribution with no penetration margin and nonuniform distribution with a margin of 2.7 mm penetration of gold nanoparticles to surrounding healthy tissue. Figrue 2A–D indicates that the absorbed dose is increased in the presence of nanoparticles. In this figure, dose enhancement was drawn versus the depth of the tumor in the direction of the radiation beam. In Table
The results of DEF computations for the nonuniform distribution are shown in Fig.
According to Fig.
In the nonuniform model with a 2.7 mm penetration margin, 5 mg/cm^{3} of 18 mg/cm^{3} of gold concentration penetrates to the healthy tissue. This leads to lower DEF in comparison with two cases of uniform and nonuniform distributions in the tumor. This phenomenon is shown in Fig.
In other words, In the uniform distribution, the maximum DEF of 2.90 occurs for 100 nm diameter of gold nanoparticles and 50 keV Xray energy. In the nonuniform model which lacks the margin case, DEF of 1.69 happens for 100 nm nanoparticles and 50 keV Xray. When a penetration margin is considered for nanoparticles, DEF of 1.38 happens for 100 nm nanoparticles and 50 keV Xrays. Additionally, in the nonuniform distribution of gold nanoparticles, has been used for two distribution models: 1. Nonuniform distribution without penetration margin. 2. Nonuniform distribution with a margin of 2.7 mm penetration of gold nanoparticles to surrounding tissues. It has been clarified that the DEF in the nonuniform model with the penetration margin is less than DEF in the nonuniform model without margin.
In a study by
In another study, in Computational Fluid Daynamics (CFD) model, the effect of the size of magnetic nanoparticles on the energy absorption in solar collector has been investigated. By increasing the nanoparticle size to about 100 nm, thermal efficiency has been increased, but larger than 100 nm, the thermal efficiency has been reduced (
Based on the current study with Monte Carlo simulation code, it is shown that in a certain concentration, GNPs with higher dimensions contribute more dose to the tumor volume; while in a uniform distribution of GNP causes a remarkable increase in the absorbed dose. Also, it was observed that having a penetration margin for gold nanoparticles to healthy tissues can reduce the concentration of gold nanoparticles on the surface of tumor, resulting in reduction in the dose given to the tumor and larger sizes of gold nanoparticles showed an increase in the DEF, it also increases the DEF in lower energies. The use of gold nanoparticles in this study increased the optimal dose in tumor tissue.
When the GNP is large relative to the range of electrons generated therein, many lowenergy electrons are trapped inside the GNP. Decreasing the size of the GNP, on the other hand, decreases the total number of photon interactions, leading to a reduced number of secondary electrons (
We encountered a number of uncertainties in this study: They are listed below.
In this investigation, the source has been assumed as a point which is different from the clinical case. And it could be considered as one of the limitations of this work.
Given the fact that therapeutic applications of GNPs in acquiring the proper dose enhancement have demanded much attention in recent years, defining the proper size and plan of GNP distribution on the tumor area would be considered extremely vital for pretreatment plans.
Also, macroscale and nanoscale simulation is not enough for the correct consideration of the radiosensitizing effect at the cellular level. But this study can be one of the reasons for increasing the dose and the sensitivity.
Gold nanoparticles due to their high atomic number (Z) provide remarkable photoelectric crosssection and lead to enhancement of absorbed radiation dose in a tumor. This is due to a wide ranges of electrons emanated from photoelectric reactions inside and in the vicinity of the tumor. Since the crosssection of the photoelectric reaction falls rapidly at higher photon energies, the effect of dose enhancement vanishes at megavoltage energy range.
The collision of the photons with gold nanoparticles causes significant photoelectric interaction on the tumor, which produces secondary particles such as characteristic Xray, photoelectrons and Auger electrons and they increase the dose as well (
In this work, with 18 mgAu/ml homogeneous GNPs in 50, 95 and 250 keV photon energy for the case of a tumor located at 2 cm depth, DEF is 3.08, 2.41 and 1.22 respectively, In other words, the observations and results confirmed that gold nanoparticles can increase the dose of the tumor during radiotherapy.
The authors would like to thank Mr. Mahdi Ghorbani, Professor of Medical Physics, Shahid Beheshti University of Tehran, Iran for his helful consultation in the preparation of this manuscript.