The dosimetry protocol TRS-398 developed by the International Atomic Energy Agency (IAEA) presents a protocol to measure absorbed doses of external radiation [1]. Absorbed doses are measured by using an ionization chamber placed inside a water phantom, with the ionization chamber being periodically calibrated to determine the correct absorbed dose. Most ionization chambers are calibrated based on ⁶⁰Co gamma rays, but since patient treatments actually use radiation of various particles and energies, a correction compensating for the difference in beam quality is required. TRS-398 proposes a correction factor k_Q,Q0 to correct for this difference in beam quality.

Among the radiation used for patient treatment, the physical characteristic of a single-energy proton beam is an energy peak that is generated by transferring a large amount of energy at the end of the range, which is called a Bragg peak. In addition, the Spread Out Bragg Peak (SOBP), which is made by the superposition of proton beams of various energies, can irradiate radiation uniformly to the target volume. Due to these physical characteristics, the proton beam can focus radiation on the target and minimize damage to surrounding normal tissues compared to treatments using gamma and electron beams.

The determination of the beam quality correction factor through experimental measurements presents many difficulties. Monte Carlo simulations can be used for an efficient and accurate calculation of the beam quality correction factor, as shown in previous studies [2,3]. The Monte Carlo simulation toolkit used in this study is TOPAS [4], which is developed based on Geant4. It was shown that TOPAS is able to calculate a sufficiently accurate beam quality correction factor in comparison with previous Monte Carlo simulations [5].

The k_Q,Q0 correction factor for the beam quality of the SOBP presented in TRS-398 is calculated based on the residual range between the center of the SOBP and the practical range (R_p). Proton beam therapy requires accurate dosimetry in all areas of the SOBP. However, since TRS-398 currently only presents one single k_Q,Q0 factor calculated at the center of the SOBP, it is necessary to verify the accuracy of this correction factor in other areas of the SOBP. In this study, Monte Carlo simulations were used to calculate the beam quality correction factor for the PTW30013 farmer chamber at the center of the SOBP, which is the reference depth presented in TRS-398, and also at other positions on the SOBP to determine the position dependence.

The beam quality correction factor k_Q,Q0 is a factor correcting for the difference in effect between the reference beam quality Q₀ used to calibrate the chamber and the user beam quality Q used for actual treatment. Under the conditions in which the Bragg-Gray capacity theory is applied, the calculation of the k_Q,Q0 factor is as follows [1].

Here, s_w,air is the stopping-power ratio of water and air, p_Q is the overall perturbation factor for the ionization chamber, and W_air is the average energy that generates ion pairs in air for each beam quality. Instead of calculating separately the factors s_w,air and p_Q, it has been proposed to calculate the k_Q,Q0 factor in a more direct way by using Monte Carlo simulations as follows [6,7].

The f_ch factor, which depends on the chamber and the beam quality, is the ratio between the absorbed dose D_w (P) for water at the reference point in the absence of the chamber and the average absorbed dose ${\overline{D}}_{ch-air}$ for air in an air-filled cavity. In this study, in order to calculate the beam quality correction factor k_Q,Q0 , the dose values D_w (P) and ${\overline{D}}_{ch-air}$ were calculated for both the reference beam quality Q₀ (⁶⁰Co) and the user beam quality Q (protons) by using Monte Carlo simulations. For W_air, we simply used the known values of 33.97 eV and 34.44 eV for ⁶⁰Co and for protons, respectively [8].

The simulation toolkit used in this study is TOPAS [4] (TOol for PArticle Simulation) version 3.2.p2 based on the Monte Carlo code Geant4 version geant4-10-05- patch-01 [9].

1. Geometries and materials

In order to calculate the beam quality correction factor, the geometries of the ionization chamber and of the point volume are required to determine ${\overline{D}}_{ch-air}$ and D_w (P), respectively. The geometry of the PTW30013 farmer chamber was implemented in the simulation. The point volume used to calculate D_w (P) is a disk with a diameter of 10 mm and a thickness of 2 mm. Fig. 1 shows the geometries of the point volume and ionization chamber. For the densities of the PMMA and graphite inside the ionization chamber, one should refer to the manufacturer’s blueprint [10]. For the densities of water, air, and aluminum, and the I-value of all materials, the information described in ICRU-90 was used [8]. I-values and density values of the materials are given in Table 1.

Table 1 . The density and average excitation energy of the materials used in the simulation.

Material	I-value	ρ (g/cm3)
Water	78 eV	1
Air	85.7 eV	1.2048 × 10-3
PMMA	74 eV	1.19
Graphite	81.1 eV	1.85
Aluminum	166 eV	2.7

Figure 1. (Color online) (a) The point volume used for the calculation of D_w (P) is a disk filled with water with a thickness of 2 mm and a diameter of 10 mm. (b) The geometry of the PTW30013 farmer chamber used for the calculation of ${\overline{D}}_{ch-air}$. From outermost to innermost, the different parts composing the PTW30013 farmer chamber are PMMA (in green), graphite (in red), air-filled cavity (in black), and aluminum (in yellow).

The beam quality correction factor of the proton beam used in this study was calculated for three depth values. The correction factor was first calculated at the depth of 7.5 g/cm², which is the central depth of the SOBP and also the reference depth suggested by TRS-398. Then, the correction factors were calculated at the depths of 4 g/cm² and 13 g/cm² to observe any position dependence.

2. Beam conditions

Fig. 2 shows the proton beam nozzle of the National Cancer Center (NCC) as reconstructed in TOPAS. The proton beam is modulated with a range of 15 cm and a width of SOBP of 15 cm, and the field size of the proton beam on the water phantom surface is 10×10 cm². The proton beam passes through the nitrogen gas between the nozzle and the water phantom and enters the water phantom which has a size of 30 × 30 × 30 cm³.

Figure 2. (Color online) Proton beam nozzle of the National Cancer Center (NCC) reconstructed in TOPAS [11].

3. Simulation parameters

When performing the simulations with TOPAS, the physics lists of radioactive decay process g4decay, inelastic nuclear interaction process g4hphy_QGSP_BIC_HP, ion physics process g4ionbinarycascade, elastic hadron processes g4helastic_HP, stopping power process g4stopping, electromagnetic process g4emstandard_opt4 were used [5].

The parameter dRoverRange that controls the step size corresponds to the maximum ratio of the step size to the particle range. The parameter finalRange sets the minimum step size at which particles are absorbed. Thus, as the particle progresses and energy decreases, the step size decreases until the particle range becomes smaller than the finalRange. In this study, dRoverRange is set to 0.05 and finalRange is set to 100 nm [12]. In the Monte Carlo simulations, secondary particles with a continuous slowing down approximation range (R_CSDA) lower than the production cut are absorbed in place. The production cut for the region of interest (ROI) including the reference volume and ionization chamber was set to 1 µm for all particles, and to 500 µm elsewhere [13].

1. Percent depth dose

Fig. 3 presents a graph comparing the simulation results with experimental measurement results of percent depth dose (PDD) to check whether the proton beam is generated well as intended. In the case of the simulation, the absorbed dose was measured by placing a voxel of 0.5 × 0.5 × 0.5 mm³ on the water phantom. For the experimental measurement, IBA Zebra was used to measure the proton beam at NCC. IBA Zebra measured the absorbed dose using vented ionization chambers spaced 2 mm apart [14]. The results from the simulations and the experiment show a slight difference in the absorbed dose on the surface of the water phantom, but the width of SOBP is about 1% different, with TOPAS and IBA Zebra presenting values of 14.9 g/cm² and 15.0 g/cm², respectively.

Figure 3. (Color online) Comparison of measurement results of percent depth dose between Monte Carlo simulations and experiment with an actual beam, for a proton beam modulated with an SOBP width of 15 cm and a range of 15 cm. The experiment measurements were made using IBA Zebra. Both results were normalized based on the absorbed dose at 7.5 g/cm².

2. Position dependence of the beam quality correction factor

Fig. 4 shows the beam quality correction factor calculated by using Monte Carlo simulations at several depths to determine the position dependence. The calculated values at the depths of 4 g/cm², 7.5 g/cm², and 13 g/cm² are 1.041, 1.045, and 1.048, respectively. Thus the beam quality correction factors calculated by using TOPAS are within the error range of the value presented by TRS-398. However, the results show that the beam quality correction factor gradually increase with the depth. Compared to the value at 7.5 g/cm², the value at 4 g/cm² was found to be about 0.4% smaller, and the value at 13 g/cm² is about 0.3% larger. These results imply that the beam quality correction factor may not be constant depending on the position of the ionization chamber in the SOBP.

Figure 4. (Color online) Monte Carlo simulation results of beam quality correction factor for the PTW30013 farmer chamber. The error bars represent the standard deviation, which is around 1%. The solid line at 1.030 is the k_Q,Q0 factor at the reference depth provided by TRS-398, and the dotted line shows the corresponding standard uncertainty of 1.7%.

In this study, the proton beam nozzle of NCC was simulated using the Geant4-based Monte Carlo simulation toolkit TOPAS to generate a proton beam and calculate the beam quality correction factor k_Q,Q0 of the ionization chamber for various positions in the SOBP of the proton beam. In order to observe the variation of the beam quality correction factor in the SOBP, it was calculated at the depths of 4 g/cm² and 13 g/cm², along with the reference depth of 7.5 g/cm² suggested by TRS-398. While all the values calculated in this study are within the error range of the value provided by the TRS-398 protocol, the simulation results show that the beam quality correction factor increases gradually with the depth, and thus a position-dependent correction factor may be required for the SOBP. In order to verify the Monte Carlo simulation results of this study, it is necessary to conduct a precise experiment using a calorimeter that can measure the absolute dose. In addition, while this study was conducted for one particular ionization chamber, further studies using other ionization chambers and beam conditions are required.

This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government, Ministry of Science, ICT and Future Planning (MSIP) (NRF-2019R1F1A1060665), and was also supported by the Korea Research Institute of Standards and Science (KRISS), the research program of “Proton beam (double scattered) water absorbed dose precision measurement technology development”.