Monte Carlo‐based beam quality and phantom scatter corrections for solid‐state detectors in 60Co and 192Ir brachytherapy dosimetry

Beam quality correction, kQQ0(r), for solid‐state detectors diamond, LiF, Li2B4O7,Al2O3, and plastic scintillator are calculated as a function of distance, r, along the transverse axis of the 60Co and 192Ir brachytherapy sources using the Monte Carlo‐based EGSnrc code system. This study also includes calculation of detector‐specific phantom scatter correction, kphan(r), for solid phantoms such as PMMA, polystyrene, solid water, virtual water, plastic water, RW1, RW3, A150, and WE210. For 60Co source, kQQ0(r) is about unity and distance‐independent for diamond, plastic scintillator, Li2B4O7 and LiF detectors. For this source, kQQ0(r) decreases gradually with r for Al2O3 detector (about 6% smaller than unity at 15 cm). For 192Ir source, kQQ0(r) is about unity and distance‐independent for Li2B4O7 detector (overall variation is about 1% in the distance range of 1–15 cm). For this source, kQQ0(r) increases with r for diamond and plastic scintillator (about 6% and 8% larger than unity at 15 cm, respectively). Whereas kQQ0(r) decreases with r gradually for LiF (about 4% smaller than unity at 15 cm) and steeply for Al2O3 (about 25% smaller than unity at 15 cm). For 60Co source, solid water, virtual water, RW1, RW3, and WE210 phantoms are water‐equivalent for all the investigated solid‐state detectors. Whereas polystyrene and plastic water phantoms are water‐equivalent for diamond, plastic scintillator, Li2B4O7 and LiF detectors, but show distance‐dependent kphan(r) values for Al2O3 detector. PMMA phantom is water‐equivalent at all distances for Al2O3 detector, but shows distance‐dependent kphan(r) values for remaining detectors. A150 phantom shows distance‐dependent kphan(r) values for all the investigated detector materials. For 192Ir source, solid water, virtual water, RW3, and WE210 phantoms are water‐equivalent for diamond, plastic scintillator, Li2B4O7 and LiF detectors, but show distance‐dependent kphan(r) values for Al2O3 detector. All other phantoms show distance‐dependent kphan(r) values for all the detector materials. PACS numbers: 87.10.Rt, 87.53.Bn, 87.53.Jw

brachytherapy sources, (5,6) different solid phantoms are also used to overcome practical problems such as waterproofing and precise positioning of detectors. In a previously published article, (7) relative absorbed-dose energy response corrections, R, were reported for different solid-state detectors for 169 Yb and 125 I brachytherapy sources. The study also included the influence of solid phantom materials such as PMMA (polymethylmethacrylate) and polystyrene on R. In another study, (8) the values of R were reported for different radiochromic films for high-energy brachytherapy sources such as 60 Co, 137 Cs,192 Ir, and 169 Yb in liquid water, PMMA, and polystyrene phantom materials. In a recently published article by Selvam et al., (9) beam quality corrections (which is the inverse of R) were reported for different solid-state detectors at 137 Cs energy. In addition, detector-specific phantom scatter corrections for different solid phantoms were also reported in their study. The purpose of this study is to calculate the beam quality corrections for solid-state detectors for 60 Co and 192 Ir brachytherapy sources. This study also includes detector-specific phantom scatter corrections for different solid phantoms for these sources. The investigation of phantom scatter also includes water as a detector material. The EGSnrc-based (10) user-codes DOSRZnrc and FLURZnrc (11) are used in the study.

A. Radioactive sources
The brachytherapy sources investigated in this study are BEBIG HDR 60 Co (model Co0.A86; Eckert & Zielger BEBIG BmbH, Berlin, Germany) (3) and HDR 192 Ir (model MicroSelectron; Elekta, Stockholm, Sweden). (2) In the Monte Carlo calculations, two gamma lines 1.17 MeV and 1.33 MeV are considered for the 60 Co source. For 192 Ir source, the spectrum is taken from the literature. (12)

B. Detector and phantom materials
The detector materials investigated in this study are diamond, LiF, Li 2 B 4 O 7 , plastic scintillator, and Al 2 O 3 . The solid phantom materials investigated are PMMA, polystyrene, solid water, virtual water, plastic water, RW1, RW3, A150, and WE210. The atomic composition and density details of RW1 and virtual water phantoms are taken from the published studies. (13,14) Remaining phantom data are taken from the study by Seco and Evans (15) C. Beam quality and phantom scatter corrections As described in the published study by Selvam et al., (9) beam quality correction, k QQ 0 (r), and phantom scatter correction, k phan (r), can be calculated at a brachytherapy beam quality Q for solid-state detectors by using the following relations: Here, D w,Q (r) and D det,Q (r) are the absorbed dose to water and absorbed dose to detector in liquid water at a distance, r, along the transverse axis of the photon emitting brachytherapy source of beam quality Q (in the present study, it is 192 Ir or 60 Co), respectively. D w,Q (r) and D det,Q (r) are the absorbed dose to water and absorbed dose to detector in water at the reference beam quality Q 0 (realistic 60 Co teletherapy beam), respectively. D det,phan,Q (r) is the absorbed dose to detector in the solid phantom at r from the brachytherapy source of beam quality Q.
For the calculation of k QQ 0 (r), the values of water-to-detector dose ratio at Q 0 (denominator of Eq. (1)) are taken from the published article. (7) Note that k phan (r) converts absorbed dose to detector at r for the brachytherapy source (of beam quality Q) in a solid phantom to absorbed dose to detector in liquid water phantom at the same r. Numerator of k QQ 0 (r) corrects for the difference in the energy absorption properties of water and detector at brachytherapy beam quality Q at r, and the denominator of k QQ 0 (r) corrects for the same, but at reference beam quality Q 0 .

D. Monte Carlo calculations
Dose ratios of water-to-detector at beam quality Q (numerator of Eq. (1)) are based on the FLURZnrc user-code (11) as described in the published study. (7)(8)(9) In the Monte Carlo calculations, the source is positioned at the centre of a 40 cm diameter × 40 cm height cylindrical phantoms (liquid water and solid phantoms). The photon fluence spectrum is scored along the transverse axis of the source (r = 1-15 cm) in 0.5 mm high and 0.5 mm thick cylindrical shells. The fluence spectrum is converted to collision kerma-to-water and collision kerma-to-detector materials by using the mass-energy absorption coefficients of water and detector materials, respectively. (16) Note that the FLURZnrc (11) simulations also provide fluence-weighted mean energy of photons, E fl . Up to 10 9 photon histories are simulated. The 1 σ statistical uncertainty on the calculated absorbed dose and collision kerma values are about 0.2%. The statistical uncertainties on the calculated values of k QQ 0 (r) and k phan (r) are less than 0.5%. The Monte Carlo parameters used in the calculations are same as that used in the earlier work. (8,9)

III. RESULTS & DISCUSSION
A. Fluence-weighted mean energy, E fl Tables 1 and 2 present the values of E fl as a function of r for 60 Co and 192 Ir sources in various phantoms, respectively. E fl decreases with distance due to degradation in the photon energy after scattering. The degree of decrease depends on the type of the phantom as well as the type of source. For the 60 Co source, the decrease in E fl is higher in PMMA, polystyrene, and A150 phantom as compared to other phantoms. For example, E fl decreases from 1.134 MeV to 455 keV in PMMA, from 1.146 MeV to 486 keV in polystyrene, and from 1.140 MeV to 481 keV in A150 phantom when the distance is increased from 1 cm to 15 cm. For phantoms such as water, RW1, RW3, and solid water, E fl decreases from about 1.15 MeV to 520 keV in the above distance range. For the virtual water and WE210 phantoms, E fl decreases from For 192 Ir source, decrease in E fl is higher for PMMA, A150, and polystyrene phantoms as compared to other phantoms. E fl decreases from about 320 keV to 140 keV when the distance is increased from 1 cm to 15 cm. For phantoms such as water, WE210, virtual water, and solid water, E fl decreases from about 325 keV to 160 keV in the above distance range. For RW1 and RW3 phantoms, E fl decreases from about 325 keV to 156 keV in the above distance range. In the case of plastic water phantom, E fl decreases from 327 keV to 178 keV when the distance is increased from 1 cm to 15 cm.
B. Beam quality correction, k QQ 0 (r) Table 3 presents the values of k QQ 0 (r) for the 60 Co and 192 Ir sources, respectively. For Li 2 B 4 O 7 detector, k QQ 0 (r) is about unity, and is independent of r for both the sources. For the 60 Co source, k QQ 0 (r) is about unity and distance independent for diamond, plastic scintillator, and LiF detectors. Whereas for the 192 Ir source, k QQ 0 (r) increases gradually about 6% and 8% larger than unity for diamond and plastic scintillator, but decreases about 4% smaller than unity for LiF detector with r over the distance range of 1-15 cm. For Al 2 O 3 detector, k QQ 0 (r) decreases with r gradually about 6% and steeply about 25% smaller than unity for 60 Co and 192 Ir sources respectively, in the above distance range.
C. Phantom scatter correction, k phan (r) Table 4 presents the summary of k phan (r) results for diamond, Al 2 O 3 , Li 2 B 4 O 7 , LiF, and plastic scintillator detectors in the investigated phantom materials for the 60 Co and 192 Ir sources, respectively. In this table, phantoms which are water-equivalent (i.e., k phan (r) is unity) at all distances (1-15 cm) are designated as "Yes". "No" implies that the phantoms show distance-dependent k phan (r) values. For such phantoms results are discussed below.

C.1 60 Co source
Phantoms such as solid water, virtual water, RW1, RW3, and WE210 are water-equivalent (i.e., k phan (r) is unity) at all distances (1-15 cm) for all the solid-state detectors (maximum deviation from unity is about 1% at 15 cm for Al 2 O 3 detector in solid water, RW1, and RW3). Polystyrene and plastic water phantoms are water-equivalent at all distances for all the detectors (with a maximum deviation of about 1% from unity for LiF), other than Al 2 O 3 . Figure 1 presents the distance-dependent k phan (r) values for the Al 2 O 3 detector in plastic water and polystyrene phantoms. PMMA is water-equivalent at all distances for Al 2 O 3 detector (larger than unity by about 1% at 15 cm), whereas k phan (r) increases with r for remaining detector materials, including water (see Fig. 2). In this phantom, k phan (r) values are comparable for diamond, plastic scintillator, Li 2 B 4 O 7 , LiF, and water detectors at all distances. For A150 phantom, k phan (r) increases with r for all the detectors, including water (see Fig. 3). For this phantom, k phan (r) values are comparable for the detectors diamond, plastic scintillator, Li 2 B 4 O 7 , LiF. and water at

C.2 192 Ir source
Phantoms such as solid water, virtual water, RW3, and WE210 are water-equivalent in the distance range of 1-15 cm for all the detectors other than Al 2 O 3 (with a maximum deviation of about 2% at 15 cm for solid water and RW3 phantoms). Figure 4 presents the distance-dependent k phan (r) values of Al 2 O 3 detector for the above four phantom materials. For this detector, k phan (r) increases with r for solid water, virtual water, and WE210 phantoms and decreases with r for RW3 phantom. k phan (r) is comparable for solid water, virtual water, and WE210 phantoms. PMMA is water-equivalent for LiF detector. Figure 5 presents k phan (r) values for all the detector materials other than LiF. For this phantom, k phan (r) decreases with r for Al 2 O 3 detector (about 10% at 15 cm), whereas k phan (r) increases with r for all the other detectors. The degree of increase is higher for diamond detector and plastic scintillator (maximum deviation from unity at 15 cm is about 5% and 6%, respectively).
The phantoms polystyrene, RW1, plastic water, and A150 show distance-dependent k phan (r) values which are presented in Figs. 6 to 9. k phan (r) decreases with r for all the detector materials in polystyrene and RW1 phantoms (Figs. 6 and 7). However, the degree of decrease is higher for Al 2 O 3 detector compared to all other detectors. For example, the value decreases to 0.821 and 0.960 at 15 cm for polystyrene and RW1 phantoms, respectively. For plastic water phantom, k phan (r) values increase with r for all the detector materials, including water (Fig. 8). The degree  of increase is higher for Al 2 O 3 detector (about 20% larger than unity at 15 cm) compared to all other detectors (minimum deviation of about 5% from unity at 15 cm for diamond and plastic scintillator detector).   In the case of A150 phantom, k phan (r) value increases with r for all the detector materials (maximum deviation of about 6% from unity at 15 cm for diamond detector) other than Al 2 O 3 detector (Fig. 9). For Al 2 O 3 detector, k phan (r) decreases from 0.997 (at 1 cm) to 0.978 (at 7 cm) and thereafter increases to unity at a distance of 15 cm. In order to verify this trend beyond 15 cm, auxiliary simulations are carried out using the FLURZnrc user-code (11) with larger dimensions (50 cm diameter × 50 cm height) of A150 and water phantoms, to calculate k phan (r) for r = 1-20 cm. Figure 10 compares k phan (r) values obtained in 50 cm diameter × 50 cm height and 40 cm diameter × 40 cm height phantoms for Al 2 O 3 detector. Up to 15 cm k phan (r) values are comparable in both the phantom dimensions. For 50 cm diameter × 50 cm height phantom, k phan (r) reaches the value of 1.032 at r = 20 cm. To verify any possible influence of the detector dimensions on k phan (r), separate auxiliary simulations are also carried out with 50 cm diameter × 50 cm height phantom by using the DOSRZnrc user-code, (10) in which Al 2 O 3 detector is modeled as a 1 mm thick × 2 mm high cylinder. The values of k phan (r) are calculated along the transverse axis of the 192 Ir source for r = 1, 5, 10, 15, and 20 cm. The study shows that DOSRZnrc-based k phan (r) values are statistically identical to the corresponding FLURZnrc-based k phan (r) values.

D. Influence of detector dimensions on detector response
The above-described FLURZnrc-based calculated values of k phan (r) and k phan (r) are based on the assumption that charged particle equilibrium exists and the presence of detector does not  affect the above corrections. In order to quantify the influence of detector thicknesses on the calculated response, auxiliary simulations are carried out in water phantom using the DOSRZnrc user-code. (10)

IV. CONCLUSIONS
Beam quality correction, k QQ 0 (r), for solid-state detector materials such as diamond, plastic scintillator, LiF, Li 2 B 4 O 7 , and Al 2 O 3 are calculated as a function of distance along the transverse axis of the 60 Co and 192 Ir brachytherapy sources using the Monte Carlo-based EGSnrc code system. For 60 Co source, k QQ 0 (r) is about unity and distance independent for diamond, plastic scintillator, Li 2 B 4 O 7 , and LiF detector, and decreases gradually with r for Al 2 O 3 (about 6% lesser than unity at 15 cm). For 192 Ir source, k QQ 0 (r) is about unity and independent of distance for Li 2 B 4 O 7 detector. k QQ 0 (r) increases with distance for diamond and plastic scintillator (about 6% and 8% larger than unity at 15 cm, respectively). k QQ 0 (r) decreases gradually with r for LiF and steeply for Al 2 O 3 .
Phantom scatter correction, k phan (r), for various solid phantoms are calculated for the above detectors along the transverse axis of 60 Co and 192 Ir sources. For 60 Co source, phantoms such as solid water, virtual water, RW1, RW3, and WE210 are water-equivalent for all the investigated detectors. Polystyrene and plastic water phantoms are water-equivalent for diamond, plastic scintillator, Li 2 B 4 O 7 , and LiF detectors, but shows distance-dependent k phan (r) values for Al 2 O 3 detector. PMMA is water-equivalent at all distances for Al 2 O 3 , but shows distancedependent k phan (r) values for remaining detectors. A150 phantom shows distance-dependent k phan (r) values for all the detector materials. For 192 Ir source, solid water, virtual water, RW3, and WE210 phantoms are water-equivalent for diamond, plastic scintillator, Li 2 B 4 O 7 , and LiF detectors, whereas these phantoms show distance-dependent k phan (r) values for Al 2 O 3 detector. Remaining phantom materials demonstrated distance-dependent k phan (r) values, but the degree of dependence depends on the type of solid phantom and the detector. Li 2 B 4 O 7 detector shows k phan (r) values identical to that of water detector, and diamond detector shows k phan (r) values identical to that of plastic scintillator detector for all the investigated phantoms for 192 Ir and 60 Co sources.