Monte Carlo calculation of beam quality correction for solid‐state detectors and phantom scatter correction at 137Cs energy

Beam quality correction kQQ0 (r), which reflects the absorbed energy dependence of the detector, is calculated for solid‐state detector materials diamond, LiF, Li2B4O7, and Al2O3 for the 137Cs RTR brachytherapy source using the Monte Carlo‐based EGSnrc code system. The study also includes calculation of detector‐specific phantom scatter corrections kphan(r) for solid phantoms such as PMMA, polystyrene, RW1, solid water, virtual water, and plastic water. Above corrections are calculated as a function of distance r along the transverse axis of the source. kQQ0 (r) is about unity for the Li2B4O7 detector. LiF detector shows a gradual decrease in kQQ0 (r) with r (decrease is about 2% over the distance range of 1‐15 cm). Diamond detector shows a gradual increase in kQQ0 (r) with r (about 3% larger than unity at 15 cm). In the case of Al2O3 detector, kQQ0 (r) decreases with r steeply (about 14% over the distance range of 1‐15 cm). The study shows that some solid‐state detectors demonstrate distance‐dependent kphan(r) values, but the degree of deviation from unity depends on the type of solid phantom and the detector. PACS number: 87.10.Rt, 87.53.Bn, 87.53.Jw, 87.56.Bg


I. IntroduCtIon
American Association of Physicists in Medicine (AAPM) Task Group reports AAPM TG43 (1) and TG43U1 (2) recommend water as a reference medium for dosimetry of interstitial brachytherapy sources. Due to high-dose gradients near brachytherapy sources and specification of the dose parameters within few centimeters of the source, source-detector distance should be specified very accurately for dosimetric measurements. Precise positioning of detectors, reproducibility of source and detectors in reference liquid water medium, and water proofing of detectors posses a practical problem. Solid phantom materials can be easily machined to accommodate the source and detectors in a precise geometrical configuration, facilitating an accurate measurement and reproducibility in source-detector geometry.
In a previously published article, relative absorbed-dose energy response corrections R for detector materials such as air, LiF, Li 2 B 4 O 7 , Si diode, diamond, and Al 2 O 3 were presented for 169 Yb and 125 I brachytherapy sources. (3) The corrections were calculated using the EGSnrcbased (4) Monte Carlo code system for liquid water, PMMA, and polystyrene phantom materials. The present study is aimed at investigating absorbed-dose energy dependence of solid-state detector materials such as diamond, LiF, Li 2 B 4 O 7 , and Al 2 O 3 at the 137 Cs energy. This investigation also includes calculation of detector-specific phantom scatter correction for different solid phantoms such as PMMA, polystyrene, RW1, solid water, virtual water, and plastic water. The EGSnrc-based (4) user-codes DOSRZnrc and FLURZnrc (5) are used in the study.

A. rtr 137 Cs source
The geometric details and material data of the RTR 137 Cs are from the published work. (6) The active length and active radius (active material is gold) of the source are 1.5 cm and 0.04 cm, respectively. The outer radius of the source is 1.5 mm. For Monte Carlo calculations, we have considered only the 662 keV gamma energy of 137 Cs emission, as in a previously published study by Selvam et al., (7) it was demonstrated that 137 Ba X-rays were not important.

B. Phantom materials
Elemental composition, mass fraction, mass density, < Z/A >, and effective atomic number (Z eff ) of water and solid phantom materials are presented in Table 1. The atomic composition and density details of the phantoms are taken from the literature. (8)(9)(10)(11) Z eff values are calculated at 662 keV using the Auto-Z eff software by Taylor et al. (12) C. theoretical background of measurement of absorbed dose to water at brachytherapy energies

C.1 Dose measurements in water phantom
Following discussion is based on the published study by Adolfsson et al. (13) Primary standards of absolute measurements of absorbed dose to water D w are based on water calorimetry. (14) 60 Co or megavoltage (MV) photon beam serves as a reference beam quality Q 0 for this purpose. A dosimeter, for example, ionization chamber calibrated to measure D w at the primary or secondary standards can be used in other beam quality Q (example, other clinical MV photon beams) by using the beam quality correction factor . (15)(16) The other dosimeters, such as solid-state dosimeters, can therefore be calibrated to measure D w at Q traceable to the primary standard. Note that may be calculated at a brachytherapy beam quality, Q, involving a solid-state detector. Consider a solid-state detector is used for measuring D w at Q 0 . This quantity is denoted by . The output measured by the solid-state detector is denoted by . An absorbed doseto-water calibration coefficient can be obtained by using the following the relation: The absorbed dose to the material of the sensitive detector element at Q 0 , , and are related as follows: (17)(18)(19) (2) where the function (called intrinsic energy-dependence (17)(18) ) relates and as below: (3) Let us now consider a cylindrical photon emitting brachytherapy source (beam quality Q, in this study it is 137 Cs) is immersed in a liquid water phantom. The absorbed dose to water in the liquid water phantom at r along the transverse axis of the source is denoted by . The output measured by the detector at r is . Likewise Eq. (3), absorbed dose to the detector at Q, and are related by is obtained by using the following relation: where is the beam quality correction and is given by Using Eqs. (3) and (4) in Eq. (7) gives is relative absorbed dose energy response correction. (3,(17)(18) As described in the previously published work, (3,(17)(18)(19) absorbed-dose dependence at Q, f(Q) relates absorbed dose to medium of interest (usually water), D w,Q and absorbed dose to detector, D det,Q , as below: (13) Similarly at Q 0 : (14) Equation (12) is therefore written as: (15) Equation (10) has two components: (a) , relative intrinsic energy dependence of the detector which can only be determined experimentally, and (b) , inverse of relative absorbed-dose energy response correction. Investigations on photon energy dependence of LiF:Mg,Ti TLDs were published in the 1960s and 1970s, with a summary of the results presented by Budd et al. (20) Most of the studies measured an intrinsic energy dependence that was greater than unity for photon energies below about 150 keV, relative to TLDs that had been calibrated using 60 Co photons. On average, the measured light output was about 10% higher than would be expected based solely on the absorbed-dose energy dependence. For detailed discussion on intrinsic energy dependence of TLDs, readers may consult the literature. (17) As mentioned by Adolfsson et al., (13) when an ion chamber is used, where W is the mean energy imparted to air to form an ion pair in air at Q, and W 0 is the corresponding quantity at Q 0 . The value of W is usually considered to be independent of the beam quality in MV photon and electron beams, but may take other values in beams of protons and heavier charged particles due to the increased ion density along the tracks of the heavy charged particles compared to that along electron tracks. (21) Note that if the yield of radiation-induced products in the detector is independent of the radiation beam quality (i.e., yield is constant), then . Therefore Eq. (9) becomes (16)

C.2 Brachytherapy dose measurements in a solid phantom
Generally, in brachytherapy, absorbed dose measurements involving solid-state detectors are carried out in solid phantoms. The absorbed dose to detector at r in the solid phantom at Q is denoted by . It is recalled that is absorbed dose to detector at Q at r in the liquid water phantom.
and are related as follows: (17) where accounts for influence of solid phantom on the response of the detector, which is known as phantom scatter correction at beam quality Q. Therefore, when measurements are carried out in solid phantoms at Q, in addition to the application of (Eq. (16)), the detector response is required to be corrected for to account of phantom scatter. The final expression for obtaining absorbed dose to water in the liquid water phantom is given by (18) where is output measured by the solid detector at Q in a solid phantom at r.

D.1 FLURZnrc simulations of collision kerma and mean energies for 137 Cs RTR source
The approach adapted for the Monte Carlo calculations of dose ratio of detector to water is as described in the published study. (3) The source is positioned at the centre of a 40 cm diameter by 40 cm height cylindrical phantoms (liquid water and solid phantoms). The photon fluence spectrum in 10 keV energy intervals is scored along the transverse axis of the source (r = 1-15 cm) in 2 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 energyabsorption coefficients of water and detector materials, respectively. (11)

D.2 Calculations of dose ratios at Q 0
In the published study, (3) it was demonstrated that the detector-to-water dose ratio calculated at the reference beam quality Q 0 ( 60 Co beam) at 0.5 mm depth in water phantom was independent of the detector thickness (0.1 mm-5 mm). In the present study, we calculated the above dose ratio for depths 5 cm and 10 cm along the central axis of the water phantom. We used detector dimensions of 5 mm radius × 1 mm thickness. In the Monte Carlo calculations, a parallel 60 Co beam is incident on a 20 cm radius × 40 cm height cylindrical water phantom. The beam has a radius of 5.64 cm at the front face of the phantom (field size is 100 cm 2 ). A realistic 60 Co spectrum from a telecobalt unit distributed along with the EGSnrc code system (4) is used in the calculations. This investigation produced similar dose ratios as obtained at 5 mm depth. This suggests that is independent of depth in the water phantom. We also calculated the dose ratio at depths 5 mm, 5 cm, and 10 cm in the PMMA phantom using the detector dimensions 5 mm radius × 1 mm thickness. The results obtained from the PMMA phantom compare well with the results of water phantom. We have therefore used the values of published in the previous work (3) for deriving . The parameters ECUT and PCUT electron and photon transport cutoff, respectively. ESAVE is a parameter related to range rejection technique. Table 2 presents the values of E fl as a function of r for the 137 Cs RTR source in various phantoms. As r increases E fl decreases, but the degree of decrease depends on the type of phantom. For the phantoms such as water, virtual water, RW1, and solid water, E fl decreases from about 565 keV to 260 keV when the distance is increased from 1 cm to 15 cm. In the case of plastic water phantom, E fl decreases from 570 keV to 285 keV in the above distance range. The values of E fl at 15 cm are 228 keV and 239 keV, respectively, for PMMA and polystyrene phantoms.

B. Phantom scatter correction
The investigation of phantom scatter also included water as a detector material. Values of calculated for the phantoms polystyrene, PMMA, virtual water, RW1, solid water, and plastic water are presented in Figs. 1 to 6. A solid phantom may be termed as water-equivalent when the value of is unity. The investigation suggests that some solid-state detectors demonstrate distance-dependent values, but the degree of dependence depends on the type of solid phantom and the type of detector. For example, the phantoms such as RW1, virtual water, and solid water almost behave like water-equivalent at all distances (1-15 cm) for all the investigated detectors (with a maximum deviation of about 2% from unity for the Al 2 O 3 detector in RW1 phantom). Polystyrene, virtual water, RW1, and solid water phantoms are water-equivalent for the diamond detector as is about unity, independent of distance (maximum deviation is about 1% in the distance range of 1-15 cm for polystyrene phantom). Whereas, for the phantoms PMMA and plastic water, increases with r for the diamond detector. The value increases to 1.0607 in PMMA and 1.0212 in plastic water at 15 cm for the diamond detector. For the LiF, Li 2 B 4 O 7 detectors, virtual water, RW1, and solid water are waterequivalent (within 1%). Note that Li 2 B 4 O 7 detector behaves like water detector at all distances for all the solid phantom materials investigated. For the Al 2 O 3 detector, the phantoms such as Polystyrene, PMMA, and RW1 show decrease in with r and the degree of decrease is higher for polystyrene phantom. For example, the value decreases to 0.9075, 0.9697, and 0.9794 at 15 cm for the phantoms polystyrene, PMMA, and RW1, respectively. The degree of decrease is higher in polystyrene phantom.     Figure 7 presents the values of for the 137 Cs RTR source obtained using Eq. (16). The numerical values of this figure are given in Table 3. For the Li 2 B 4 O 7 detector, is about unity, and is independent of r. The LiF detector shows a gradual decrease in with r. The decrease is 2% over the distance range of 1-15 cm. Diamond detector shows a gradual increase in with r (about 3% larger than unity at 15 cm). For the Al 2 O 3 detector, decreases with r steeply (about 14% over the distance range of 1-15 cm).

D. Influence of detector dimensions on detector response
Dimensions of TLD-100 (LiF:Mg,Ti) chips reported in the literature (22)(23) are 3 × 3 × 0.9 mm 3 , and 1 × 1 × 1 mm 3 , and 3.2 × 3.2 × 0.38 mm 3 . Carbon-doped cylindrical discs of Al 2 O 3 detectors (4 mm diameter × 1 mm height) are used in radiotherapy photon beams. (24) Al 2 O 3 :C chips (2 mm long and 0.5 × 0.5 mm 2 in cross-sectional area) are used in 192 Ir high-dose-rate dosimetry. (25) The sensitive volume of the PTW/diamond detector is a disk made from natural diamond (density 3.51 g/cm 3 ) with a radius ranging from 1.0 to 2.2 mm and a thickness ranging from   (27) In order to quantify the influence of detector thicknesses on the calculated response, we adapted an approach as applied in a previously published work (3) due to limitations associated with the DOSRZnrc user-code. LiF, Li 2 B 4 O 7 , and Al 2 O 3 detectors are modeled as cylindrical shells of thickness 1 mm and height 2 mm along the transverse axis of the source. The phantoms considered are water, polystyrene, and plastic water. Absorbed dose and collision kerma to these detectors are calculated at r = 1 and 15 cm. The DOSRZnrcbased collision kerma values are statistically identical to the FLURZnrc-based collision kerma values. This suggests that detector dimensions do not affect the calculated values. In the case of diamond detector, the calculations are carried out for 0.2 mm and 0.4 mm thicknesses separately (height is 2 mm). DOSRZnrc calculations using these thicknesses show collision kerma values comparable to those obtained using the FLURZnrc user-code. Whereas the absorbed dose calculated for the 0.2 mm thick diamond detector is smaller by about 1% when compared to the collision kerma. In the case of 0.4 mm thick diamond detector, both collision kerma and absorbed dose are statistically identical.

IV. ConCLuSIonS
Absorbed-dose energy dependence of solid-state detector materials such as diamond, LiF, Li 2 B 4 O 7 , and Al 2 O 3 for the 137 Cs RTR brachytherapy source is studied using the Monte Carlo-based EGSnrc code system. Beam quality correction , which reflects absorbeddose energy dependence of the detector, shows a gradual decrease with r for the LiF detector (decrease is about 2% over the distance range of 1-15 cm). Diamond detector shows a gradual increase in with r (about 3% larger than unity at 15 cm). For Al 2 O 3 detector, decreases with r steeply (about 14% over the distance range of 1-15 cm). Li 2 B 4 O 7 does not show energy dependence. The study shows that some solid-state detectors demonstrate distancedependent values, but the degree of dependence depends on the type of solid phantom and the detector.