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 calculated WER values for some materials using the numerical method [NM] presented by Newhauser et al. [15,16]. Also, in another work, Zhang et al. calculated analytical values of the WER for some selected materials using the Bragg-Kleeman [BK] rule [17]. 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 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 [18]. WER values are easily calculated thanks to these tables.

Furthermore, computerized codes were utilized to for obtain WER values. Akbari et al. calculated WER values for liquid water, PMMA, PS, and Al using FLUKA code and the SRIM program [19]. 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. studied the influence 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 [21].

In addition, Moyers et al. 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 calculated and literature values [22]. Typically, the difference between the measured and the literature or calculated values was generally less than 1.5% and larger than the derived 2σuncertainty 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, were studied using the MCNPX code in the range of 100 to 225 MeV proton energies with 25 MeV energy steps (usually applied in proton therapy). The depth dose profiles, range, penumbra width, electron density (ρ_e), effective atomic number (Z_eff) 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

3 Results and discussion

The calculated depth dose profiles 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.

Fig. 1Full-screen View

(Color online) Calculated depth dose profiles for the studied materials for selected proton energies. Data are normalized to the dose of the Bragg peak.

In order to compare the Bragg peak position of the investigated materials for identical proton energy, the depth dose profiles are shown in Fig. 2 for 225 MeV proton energy.

Fig. 2Full-screen View

(Color online) Depth dose profiles of the studied materials for 225 MeV proton energy as a function of depth.

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 paraffin (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 observed fluctuations (wavelets) will be removed in the plateau regions of the depth dose profiles. Since the Bragg peak region is required to calculate the quantities of interest in this research, an attempt was made to choose an optimum thickness values to minimize the wavelets without increasing relative errors and computational time.

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 become 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).

Fig. 3Full-screen View

(Color online) Differences between proton range in water (R_w) and studied dosimetric materials (R_m) versus 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 equation 4 and Table 3 data.

Fig. 4Full-screen View

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

Fig. 5Full-screen View

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

The RD difference between the MCNPX and NIST data of WER values at selected proton energies were 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 Poly-methyl 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% for polystyrene 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.

Fig. 6Full-screen View

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

A strong correlation of the WER values to the RED values and the electron densities of the dosimetric materials was observed (the correlation coefficient = +0.999). As previously indicated, for a given incident particle (proton in this research), the second expression in Equation 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. Fig. 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 coefficient = +0.999).

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 was calculated using MCNPX is 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 code generated values. The WER values of PMMA, PS, PE, and PC solid plastics for the proton energy range of 100 to 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:

Fig. 7Full-screen View

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

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 are in the range of the reported experimental error, which was less than 3% [21].

The RD values, which ranged from –3.77% to 5.29%, were found to be less than ±6% for all the phantom materials and methods. Polyethylene (PE) exhibited the greatest relative deviation values. High-density polyethylene (HDPE) with a density of 0.964 g cm^-3 was investigated by Moyers 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 are the largest, as indicated by the results and demonstrated by Janni [18].

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. produced acceptable RD values less than ±1.35% [17].

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 profiles including the intensity and depth of the Bragg peak, range and penumbra width.

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 paraffin have the most similar dose characteristics to liquid water.