Determination of water equivalent ratio for some dosimetric materials in proton therapy using MNCPX simulation tool

Reza Bagheri·Alireza Khorrami Moghaddam·Bakhtiar Azadbakht·Mahmoud Reza Akbari·Seyed Pezhman Shirmardi

Abstract The water equivalent ratio(WER)was calculated for polypropylene(PP),paraf fin,polyethylene(PE),polystyrene(PS),polymethyl methacrylate(PMMA),and polycarbonate materials with potential applications in dosimetry and medical physics.This was performed using the Monte Carlo simulation code,MCNPX,at different proton energies.The calculated WER values were compared with National Institute of Standards and Technology(NIST)data,available experimental and analytical results,as well as the FLUKA,SRIM,and SEICS codes.PP and PMMA were associated with the minimum and maximum WER values,respectively.Good agreement was observed between the MCNPX and NIST data.The biggest difference was 0.71%for PS at 150 MeV proton energy.In addition,a relatively large positive correlation between the WER values and the electron density of the dosimetric materials was observed.Finally,it was noted that PE presented the most analogous Depth Dose Characteristics to liquid water.

Keywords Water equivalent ratio·Proton therapy·Dosimetric materials·MCNPX code

1 Introduction

Proton radiotherapy is developing into an established approach for the treatment of localized tumors adjacent to critical organs,due to the superior conformity of the dose within the target volume associated with this technique[1,2].Protons interact with the electrons and the nuclei of the medium through Coulomb forces.Inelastic collisions with the former result in the absorbed dose.Rare collisions with the nuclei cause nuclear reactions.Bremsstrahlung interactions with the nuclei are also possible,but are rare,and hence,negligible[3].

For a monoenergetic proton beam,there is an initial slow increase in dose with depth,followed by a sharp increase near the end of the range,called the Bragg peak[3,4].The depth of the Bragg peak can be adjusted according to the depth and extent of the tumor region by selecting the appropriate proton beam energies[5,6].

The application of proton radiation therapy to medical practice is predicated on appropriate calibration to simulate human organs and tissues.It is rarely possible to directly measure the dose distribution in patients treated with any type of radiation.

The data on the dose distribution are almost entirely derived from measurements in phantoms made from tissue equivalent materials[7-9].In clinical proton dosimetry,the measured dose distribution data are usually acquired in a liquid water phantom,which closely approximates the properties of soft tissue.However,water phantom has several disadvantages when used in conjunction with detectors.Numerous water equivalent solid phantoms have been developed as substitutes for water in the measurement of the absorbed dose[10-12].For a given material to be tissue or water equivalent,it must have the same mass stopping power and secondary particle production,effective atomic number,electron density,and mass density as water[13,14].

A proton range,which was measured for the selected material and liquid water,is used as an equivalence estimator.For this purpose,the water equivalent ratio(WER)parameter was considered for comparing the dosimetric equivalency of the materials with water.Various experimental and theoretical works have been performed to facilitate the calculation of this quantity in a wide range of materials.Zhang and Newhauser[15]calculated WER values for some materials using the numerical method[NM]presented by Newhauser et al.[16].Also,in another work,Zhang et al.[17]calculated analytical values of the WER for some selected materials using the Bragg-Kleeman[BK]rule.This method has also been used by Zhang and Newhauser[15].The BK rule determines the range of the proton beam in a given material using two material and energy-dependent constants. Extensive energy-range tables were prepared by Janni[18]for protons with energy in the range of 1 keV to 10 GeV for all the elements and many compounds and mixtures,using the Bethe equation with all the necessary corrections.WER values are easily calculated thanks to these tables.

Furthermore,computerized codes were utilized to obtain WER values.Akbari et al.[19]calculated WER values for liquid water,PMMA,PS,and Al using FLUKA code and the SRIM program.The WER values for these materials and graphite,Cu,Ti,and Au,were calculated according to the SEICS Monte Carlo code as well as by de Vera et al.[20].Konefał et al.[21]studied the in fluence of the energy spectrum and the spatial spread of 50-70 MeV proton beams applied to the treatment of eye tumors,on the Depth Dose Characteristics in water using the GEANT4 Monte Carlo simulation tool kit.

In addition,Moyers et al.[22]measured the relative linear stopping powers of 21 different tissue substitutes for proton beams accelerated to the energies of 155,200,and 250 MeV with the LLUPTF synchrotron and compared their results with the calculated and literature values.Typically,the difference between the measured and the literature or calculated values was generally less than 1.5%and largerthan the derived 2σ uncertainty in the measurement.

In the present work,a Monte Carlo evaluation of the water equivalent ratio of six phantom materials,originally developed for conventional radiation therapy,was studied using the MCNPX code in the range of 100-225 MeV proton energies with 25 MeV energy steps(usually applied in proton therapy).The Depth Dose Pro files,range,penumbra width,electron density(ρe),effective atomic number(Zeff),and the relative electron density(RED)values of the phantom materials were also calculated.

In order to verify and validate the results from the simulation,the acquired values were compared with theoretical values from the National Institute of Standards and Technology(NIST).Furthermore,the simulated and theoretical values of WER were compared with available experimental and analytical data as well as the FLUKA,SRIM,and SEICS codes values for some investigated materials.

2 Materials and methods

2.1 Simulation

Polypropylene(PP),paraf fin(PA),polyethylene(PE),polystyrene(PS),polymethyl methacrylate(PMMA),and polycarbonate(PC)were considered as the phantom materials.The MCNPX code(version 2.6.0)was used for Monte Carlo evaluation of the water equivalent of these materials.This code is a general-purpose Monte Carlo radiation transport algorithm for modeling the interaction of radiation with matter[23,24].Since this code considers all interactions that affect the dose distribution within the material,the obtained results were found to be accurate and reliable[20,25].

For this purpose,cylindrical geometries with a height of 100 cm and a 15 cm radius were employed for modeling the phantom materials.A collimated monoenergetic proton beam with a 1 cm diameter that emitted protons perpendicular to the front face of the phantom materials was considered.A disk source was de fined in the MCNP data card with ERG,PAR,POS,and DIR commands for energy,type of particle,position,and direction of the proton beam,respectively.In order to obtain the WER values,cylindrical detectors with a 2 cm height and 1 mm diameter were simulated along the beam's trajectory in the phantom materials.To pave the way for this process,a cylindrical mesh tally was utilized to obtain the MCNPX simulation data.A third type of mesh tally was also included to record the energy deposition data per unit volume of material for all the tracked particles.The elemental composition and densities of the studied dosimetric materials were adopted from the National Institute of Standards and Technology(NIST)data using ICRU results.Simulations were performed with 5 to 50 million histories depending on the phantom material and the proton beam energy.All the results generated by the MCNPX code were reported with less than 0.02%statistical uncertainty.In order to reduce the relative error of a tally,the model geometry and the physics of the problem were simpli fied,and the technique of geometry splitting(using imp card)was used for the transport of particles in the cells of interest.

In addition,the NIST extracted data were processed using the PSTAR program,which calculates the stopping power and the range of protons in various materials according to the methods described in the ICRU reports 37 and 49[12,26].

The comparison between the calculated and experimental data of WER values was evaluated according to relative deviation(RD)values.

2.2 Theory

The following formula was applied for calculating WER values[20]:

where d80,waterand d80,materialare the ranges of the proton in water and the phantom material,respectively[9,27,28].A range is de fined as the distance between the entrance surface of the beam and the distal point of the 80%maximum dose.Likewise,the penumbra width is de fined as the distance between the points of 80%and 20%absorbed dose in the distal region of the Bragg curve.The curve fitting program,OriginPro 9.1,was used to fit an asymmetric Gaussian curve on the Bragg curves and to extract speci fic parameters.

The electron density(ρe,electron/cm3)of the phantom material was calculated from its mass density(ρm)and its atomic composition according to the formula[3]:

where

where NAis the Avogadro constant and aiis the fraction by weight of the ith element with Ziand Aiatomic number and weight,respectively.Also,the effective atomic numbers(Zeff)of the phantom materials were derived by the following equation[3]:

where aiis the fractional contribution of the ith element to the total number of the electrons in the mixture.The relative electron density(RED)values were calculated as the ratio of the electron density of the phantom material to the water phantom.

We determined the rate of the energy loss or the linear stopping power for the charged particles(such as protons)in a given material using the Bethe formula[29]:

where v and z are the velocity and charge of the incident particle,respectively;N and Z are the atom number density and atomic number of the material,respectively;m0is the electron rest mass;e is the electronic charge;and I represents the average excitation and ionization potential of the material.The first expression is related to the characteristics of the incident particle,which was identical for all the materials used in this research(z=1 for proton)and the second expression represents the features of the absorbing material.

Table 1 presents the properties of the phantom materials.The effective atomic numbers and electron densities of the investigated dosimetric materials were calculated according to Eqs.(2)-(4).Moreover,the weights which were used in the material card of the MCNP code are given in percentage of each element in the materials and are presented in Table 2.

3 Results and discussion

The calculated Depth Dose Pro files for the materials are shown in Fig.1.It was found that the relative dose ratio of the plateau region to the Bragg peak increases with an increase in the proton energy,and likewise,the Bragg peaks broaden with an increase in energy.

In order to compare the Bragg peak position of the investigated materials for identical proton energy,the Depth Dose Pro files are shown in Fig.2 for 225 MeV proton energy.

It can be clearly seen from this figure that for a given proton energy,the largest differences in the Bragg peak position with respect to liquid water are observed in polymethyl methacrylate(PMMA)and polypropylene(PP)in two sides of the water Bragg peak.In addition,the results for polyethylene(PE)and paraf fin(PA)present Depth Dose Characteristics comparable to that of liquid water.These solid plastics exhibit the most analogous Depth Dose Characteristics to liquid water as an accessible phantom which closely approximates the radiation absorption and scattering properties of muscle and other soft tissue.

As shown in Figs.1 and 2,unexpected wavelets are observed in the region before the Bragg peak.This is due to the 1 mm thick layers in the mesh tally.As previously indicated,cylindrical detectors with a 2 cm height and 1 mm diameter were utilized for calculation of the WER values.The MCNPX code calculates the average deposited energy at each cell volume with a 1 mm thickness.By reducing the thickness of the cells,the observedfl uctuations(wavelets)will be removed in the plateau regions of the Depth Dose Pro files.Since the Bragg peak region is required to calculate the quantities of interest in this research,an attempt was made to choose an optimum thicknessvaluesto minimize the waveletswithout increasing relative errors and computational time.

Table 1 Chemical formula,density(ρm,g cm-3),electron density(ρe× 1023electron cm-3),effective atomic number(Zeff),and relative electron density(RED)for materials discussed in this work

Table 2 Percentage of atomic composition for studied materials[12]

The range values of the studied materials obtained using the MCNPX code and NIST data(using PSTAR program)are presented in Table 3 for the selected proton energies.

Figure 3 shows the difference between the ranges of proton in water and the investigated dosimetric materials versus the proton energy using the results of Table 3.This difference becomes larger as the energy of the proton increases.Generally,polyethylene(PE)and polymethyl methacrylate(PMMA)have the smallest and largest range difference relative to water(45.00 and 0.10 mm for PMMA and PE,respectively,at 225 MeV proton energy).

Calculated WER values for the investigated materials using the MCNPX code and the PSTAR program(NIST data)for different proton energies as well as the acquired relative deviation(RD)values are shown in Figs.4 and 5,respectively.The WER values were calculated using Eq.(4)and Table 3 data.

The RD difference between the MCNPX and NIST data of WER values at selected proton energies was calculated according to Eq.(6):

From Figs.4 and 5,the MCNPX and the NIST data are in good agreement with each other.Polypropylene(PP)and polymethyl methacrylate(PMMA)have the minimum and maximum WER values,respectively.WER values were approximately constant for all proton energies for all the investigated materials.

The RD values which ranged from-0.06 to 0.72%were found to be less than 0.8%for all the phantom materials.An average difference of 0.27%was obtained for the MCNPX and NIST data of WER values.The biggest observed difference was 0.71% forpolystyrene at 150 MeV.In the case of polycarbonate,good agreement with the NIST data was observed(RD value less than 0.05%).In addition,it was determined that the RD values in Fig.5 are energy independence.

The correlation of the WER values to the effective electron densities and the RED values of the dosimetric materials are shown in Fig.6.The average WER value for the investigated range of proton energies(100 to 225 MeV)was considered for each material.

A strong correlation of the WER values to the RED values and the electron densities of the dosimetric materials was observed(the correlation coef ficient=+0.999).As previously indicated,for a given incident particle(proton in this research),the second expression in Eq.(5)describes the characteristics of the absorbing material.A comparison of different materials revealed that the rate of proton energy loss and therefore its range,(integrating the reciprocal of the energy loss)and in turn the WER value,depends mainly on the product NZ.This product represents the electron density of the material.High electron density materials(or RED value)will therefore result in a high linear stopping power,short range and high WER values(water range per material range).As shown in Table 1 and Fig.6,PMMA has the greatest electron density value(or RED value)and WER value.Figure 6 indicates that the WER value is directly proportional to the electron density magnitude(or RED value)of the studied materials with high accuracy(the correlation coef ficient=+0.999).

Fig.1(Color online)Calculated Depth Dose Pro files for the studied materials for selected proton energies.Data are normalized to the dose of the Bragg peak

The penumbra width was calculated using MCNPX which is given as the distance between points 80%and 20%of the absorbed dose in the distal region of the Bragg curves as shown in Table 4 for selected proton energies.

It can be clearly seen from Table 4 that the penumbra width increases as the proton energy increases.In addition,the results indicate that for a given proton beams energy,a higher material density generally results in a wider penumbra.

Finally,the MCNPX and NIST results of WER values are compared with available experimental and analytical data as well as with the FLUKA,SRIM,and SEICS codegenerated values.The WER values of PMMA,PS,PE,and PC solid plastics forthe proton energy range of 100-225 MeV were considered using the available literature data.

In addition,Fig.7 indicates the difference between the calculated(simulated and theoretical)and experimental data for WER values at 135,175,and 225 MeV proton energies.The relative deviation(RD)values were calculated using the following equation:

As seen in Fig.7 and Table 5,in general,a good agreement is observed between the calculated and the experimental values.Moreover,the difference between the two datasets is in the range of the reported experimental error,which was less than 3%[21].

Fig.2(Color online)Depth Dose Pro files of the studied materials for 225 MeV proton energy as a function of depth

Table 3 MCNPX and NIST data of proton range(mm)for studied materials at different proton energies

Fig.3(Color online)Differences between proton range in water(Rw)and studied dosimetric materials(Rm)versus proton energy

Fig.4(Color online)WER values for different proton energies in the studied materials

Fig.5(Color online)Difference(%)between the MCNPX and NIST data of WER values in the studied materials

The RD values,which ranged from-3.77 to 5.29%,were found to be less than±6%for all the phantom materialsand methods.Polyethylene (PE)exhibited the greatestrelative deviation values.High-density

Fig.6 A strong correlation of WER values to the RED values and the electron densities of the dosimetric materials(the correlation coef ficient=+0.999)

Table 4 MCNPX results of penumbra width(mm)for studied materials at different proton energies

The result of the application of Monte Carlo codes(MCNPX,FLUKA,SEICS,and SRIM)and the analyzed NIST data produced the lowest RD values.Likewise,the analytical data suggested by Zhang et al.[17]produced acceptable RD values less than±1.35%.

The observed differences in the WER values may be due to differences in the selected values of the variables used in the calculations such as,the density and elemental compositions of the studied materials,the energy of the protons which enter the phantoms or slabs in the experiment,differences in the proton beam characteristics,errors in the different techniques and methods applied in the analytical procedures and simulations of the physical and mathematical models,uncertainties in the nuclear/atomic data,improper modeling of the source energy and actual geometry,etc.It should be noted that the model used in this simulation was ideal and simple.The suggested model estimates the Depth Dose Characteristics of the studied materials to an acceptable extent and compares them with each other.

The experimental results indicate that the energy of the proton beam and the electron density(or RED value)of the irradiated material strongly affect the Depth Dose Pro files polyethylene(HDPE)with a density of 0.964 g cm-3was investigated by Zhang et al.[17],however,the application of computational methods resulted in a lower value being assigned to this material.In addition,the RD values are the greatest when the difference between the density values is the largest,as indicated by the results and demonstrated by Janni[18].including the intensity and depth of the Bragg peak,range,and penumbra width.

Fig.7(Color online)Difference(%)between calculated and experimental results of WER values at 135,175,and 225 MeV proton energies

Table 5 Simulated,theoretical,and experimental values of water equivalent ratio for various incident proton energies for investigated materials.MCNPX,NIST,and SRIM data were analyzed in this research

4 Conclusion

In the present work,the Bragg curves and their related parameters(range,WER,and penumbra width)were calculated using MCNPX code for proton beams with energies from 100 to 225 MeV for several materials of interest(PP,PA,PE,liquid water,PS,PMMA,and PC).The results were compared with reported theoretical and experimental results.We determined that the results from NIST,MCNPX code,our experiment,and other Monte Carlo simulations are in good agreement with one another.

The calculated WER values using the MCNPX code suggest that the simulation geometry method can be utilized to determine these values and Bragg curves characteristics.This approach is particularly useful for materials for which experimental values are unavailable.It is concluded that polyethylene and paraf fin have the most similar dose characteristics to liquid water.

AcknowledgmentsThe authors thank Dr.M.A.Roshanzamir(Nuclear Science and Technology Research Institute,Iran)for modifying and editing the manuscript.