Monte Carlo investigation of energy response of various detector materials in I125 and Yb169 brachytherapy dosimetry

Relative absorbed‐dose energy response correction R for different detector materials in water, PMMA and polystyrene phantoms are calculated using Monte Carlo‐based EGSnrc code system for I125 and Yb169 brachytherapy sources. The values of R obtained for I125 source are 1.41, 0.92, 3.97, 0.47, 8.32 and 1.10, respectively, for detector materials LiF, Li2B4O7,Al2O3, diamond, silicon diode and air. These values are insensitive to source‐to‐detector distance and phantom material. For Yb169 source, R is sensitive to source‐to‐detector distance for detector materials other than air and Li2B4O7. For silicon, R increases from 3 to 4.23 when depth in water is increased from 0.5 cm to 15 cm. For Yb169 source, the values of R obtained for air and Li2B4O7 in PMMA and polystyrene phantoms are comparable to that obtained in water. However, LiF, Si and Al2O3 show enhanced response and diamond shows decreased response in PMMA and polystyrene phantoms than in water. PACS number: 87.53.Jw

used MCNP (version 4) Monte Carlo code to calculate absorbed-dose energy response correction for 169 Yb source. Mobit et al. (22) used the EGSnrc Monte Carlo system (24) to calculate the corrections for LiF TLD rods for 125 I sources.
According to TG43U1 update (1) for brachytherapy dose measurements a well-characterized energy response function should be quantitatively accounted for. The objective of the present study is to calculate the relative absorbed-dose energy response correction as a function of depth in water for different solid-state detector materials for 125 I (model selectSeed) and high-dose rate (HDR) 169 Yb (model 4140) brachytherapy sources. The study also includes air as detector material representing an ionization chamber. Measurement of dose distribution is usually performed in 'water-equivalent' solid phantoms. The solid phantoms have the advantage that they can be precisely machined to accommodate sources and detectors, and distances can be accurately determined. Therefore, the study also includes calculation of relative absorbed-dose energy response correction for the investigated detector materials in solid phantoms polystyrene and polymethyl methacrylate (PMMA). We have employed the Monte Carlo-based DOSRZnrc and FLURZnrc user-codes (24) of the EGSnrc code system (25) in the present work.

A. radioactive sources and detectors
The geometric details and composition of the 125 I selectSeed and 169 Yb model 4140 are taken from the published studies. (26,27) The photon energy spectra of the 125 I and 169 Yb sources needed for the Monte Carlo calculations are taken from literature. (1,27) The detector materials investigated in the present study are LiF, Li 2 B 4 O 7 , A1 2 O 3 , diamond, silicon and air. Table 1 presents the values of Z eff (effective atomic number), 〈Z/A〉 (electron density) and ρ (mass density) of the investigated detector materials.

B. Energy dependence of the detector
The energy dependence of the detector may be separated into two components. (28,29,30) One function, called the intrinsic energy-dependence, k bq (Q), relates the detector output, M det (Q), to the average dose to the material of the sensitive detector element, D det (Q), as a function of beam quality, Q. (1) The other function, denoted the absorbed-dose energy dependence, f(Q), relates D det (Q) to the dose to another medium, D med (Q), in the absence of the detector, as a function of Q.
(2) For a cavity (detector) that is large in comparison to range of electrons, where is ratio of mean mass-energy absorption coefficient of medium-to-detector at Q. The above equation is applicable when there is charged particle equilibrium and the energy fluence spectrum of photon is not perturbed by the detector. For more details on f(Q), see DeWerd et al. (28) In brachytherapy, quantity of interest is dose to water. The detectors are generally calibrated against a reference beam, which is usually 60 Co. The relative absorbed-dose energy response correction R is the largest single source of Type B (systematic) uncertainty for TLD and other secondary dosimeters used in brachytherapy dosimetry. For a given detector material and a beam quality Q, R is defined as: where the numerator represents detector-to-water dose ratio at Q ( 125 I or 169 Yb), and the denominator represents the same dose ratio at 60 Co.
In the presence of charged particle equilibrium and when the detector material does not alter the photon energy fluence spectrum (see Eq. (3)), the above equation can be written as: Here, the numerator represents ratio of mean mass-energy absorption coefficient of detectorto-water at Q, and the denominator represents the same ratio at 60 Co. Figure 1 presents the values of R for the investigated detector materials shown as a function of photon energy in the range 10 keV to 1.5 MeV. The values are based on the mass energy absorption coefficients data by Hubbell and Selzter. (31)

C. Monte Carlo calculations C.1 DOSRZnrc simulations of dose ratios for 60 Co beam
Calculation of dose ratios at 60 Co is important to derive R (see denominator of Eq. (4)). Dose ratios in water phantom for the investigated detector materials for the 60 Co beam, , are calculated using the DOSRZnrc user code (24) of EGSnrc code system. (25) Here, D det and D wat represent dose to detector and dose to water, respectively. In the Monte Carlo calculations, a parallel 60 Co beam is incident on a 20 cm radius by 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 (25) is used in the calculations. Cylindrical detector materials of 0.5 cm diameter and varying thicknesses are positioned at a depth 0.5 cm along the central axis of the water phantom. The thicknesses of the detector material are varied from 0.1 cm to 0.5 cm to study the influence of the thickness of the detector on .

C.2 FLURZnrc simulations of collision kerma and mean energies for 125 I and 169 Yb sources
For the calculation of dose ratio of detector-to-water for the 125 I and 169 Yb sources (numerator of Eq. (4)), we used the FLURZnrc user-code. (24) In the calculations, the photon fluence spectrum is scored in 0.5 mm thick and 0.5 mm high cylindrical shells, along the transverse axis of the sources (distances, 0.5 cm-15 cm) in the 20 cm radius by 40 cm high cylindrical phantoms. 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. (28) Using the values of collision kerma to water and collision kerma to detector materials, the denominator of Eq. (1) is obtained for the 125 I and 169 Yb sources. In the calculation of collision kerma to detector materials, no detector material is present. We have assumed that the presence of the detector materials does not affect the photon fluence spectrum. At 125 I and 169 Yb photon energies, charged particle equilibrium exists and the collision kerma may be approximated to absorbed dose. The fluence weighted mean energy and the detector-kerma weighted mean energy (suffix m represents detector material) are calculated as a function of distance from the source along the transverse axis using the following expressions: where E is the kinetic energy of photon in keV, Φ(E) is the differential photon fluence spectrum at E about dE and (μ en (E)/ρ) m is the mass energy absorption coefficient of the detector material m at the photon energy E. The values of and are calculated for the 125 I (selectSeed) and 169 Yb (model 4140) as well as for the bare sources.

C.3 Monte Carlo parameters and statistical uncertainties
The PEGS4 dataset needed for Monte Carlo calculations described above is based on XCOM (32) compilations. We set AE = 0.521 MeV (kinetic energy of the electron is 0.01 MeV) and AP = 0.001 MeV while generating the PEGS4 dataset, where the parameters AE and AP are the low-energy thresholds for the production of knock-on electrons and secondary bremsstrahlung photons, respectively. All the calculations utilized the PRESTA-II step length and EXACT boundary crossing algorithms. In all calculations, electron range rejection technique is used to save computation time. We set ESAVE = 2 MeV for this purpose.
The photon transport cut off energy PCUT is chosen at 1 keV in all calculations. In DOSRZnrc calculations, we set AE = ECUT = 0.521 MeV (10 keV kinetic energy). In the FLURZnrc calculations, electrons are not transported by setting electron transport cutoff parameter ECUT = 2 MeV (kinetic energy). Up to 10 9 photon histories are simulated. The 1 σ statistical uncertainties on the calculated DOSRZnrc-based dose values are generally within 0.3%. The 1 σ statistical uncertainties on the calculated FLURZnrc-based collision kerma values are usually 0.1% and never exceeded 0.2%. The statistical uncertainties on the calculated values of R are less than 0.6%. Throughout the text, the number shown in parentheses following a value represents the absolute uncertainty on the last digit of the value with a coverage factor k = 1.

A. Mean energies
An analysis of XCOM data shows that the interaction mechanisms at 27 keV photons in water are 46.4% photo electric absorption, 41% Compton scattering and 12.6% coherent scattering. At this energy, even after multiple Compton scattering in water, the energy of the scattered photons does not change significantly. Hence, the mean energies of the 125 I source do not change with the depth in water. For example, for the selectSeed 125 I source and the bare 125 I line source is about 28 keV in water, independent of distance.
For the 169 Yb source, mean energies decrease with distance in water. This is due to substantial degradation in the photon energy after scattering. Figure 2 presents the values of and for the 169 Yb (model 4140) source at various transverse axis distances in water. The energy degradation is significant in PMMA and polystyrene phantoms when compared to water phantom because scattering is high in PMMA and polystyrene. Table 2 presents the values of for different detector materials at 0.5 cm depth in water phantom for various detector thicknesses. Also shown in this Table are the values  of  calculated at 1.25 MeV, and , for comparison purposes. For a given detector material, the dose ratio is independent of detector thickness. It is interesting to see that the values of and agree well with the values of (a maximum difference of 1.8% is observed for the air material). This suggests that at the 60 Co energies, the investigated detectors (thickness from 0.1 cm to 0.5 cm) behave like a photon detector, as Compton scattering is the predominant interaction in all the detector materials. This implies that dose to detector is related to dose to water by the relation . As the difference between the values of and is small (see Table 2), we have used values for calculating R.  Table 2. Monte Carlo-calculated ratio of dose to detector and dose to water for different detector materials for 60 Co beam presented for different detector thickness. The number shown in parentheses following a value represents the absolute uncertainty on the last digit of the value with a coverage factor k=1. The radius of the detector is 5 mm. These detectors are at a depth of 0.5 cm in a 20 cm radius by 40 cm height unit density water phantom. The 60 Co beam has a radius of 5.64 cm at the phantom surface. Also shown in this table are the values of ratio of mass-energy-absorption coefficients of detector to water calculated at the 60 Co energy (1.25 MeV) and the values of ratio of <Z/A> of detector to water.  The value of R = 1.41 for LiF-TLD calculated in the present work is consistent with the published value of 1.41 by Meigooni et al. (8) Mobit and Badragan (22) have also studied the energy response correction for LiF-TLD micro rods of different diameters. Their study showed that R is sensitive to diameter of LiF rod (i.e. the values of R are 1.406 ± 0.2% and 1.323 ± 0.2% for rods of 0.1 cm and 0.5 cm diameter (0.6 cm length), respectively.) (22) The authors also studied the angular and radial distance dependence of R. For a LiF-TLD of diameter 1 mm calibrated at 1 cm on the transverse axis of the 125 I source in water, R decreases by a maximum of 3.5% within the 6 cm × 6 cm × 6 cm calculation region. For the 5 mm diameter LiF-TLD, R decreases by a maximum of 5% in the same region. Note that a 5% uncertainty is assigned to R for the LiF TLD-100 based dosimetry. (33) Figure 3 presents the Monte Carlo-calculated values of R for the 169 Yb (model 4140) source as a function of distance along the transverse axis of the source for different detector materials. For a given detector material, the 169 Yb source (model 4140) shows R is distance-dependent (change in R is not significant for distances 5 cm and above). For example, R increases from 1.11 to 1.18, 1.80 to 2.28, 2.97 to 4.17 and 1.036 to 1.052, respectively for the LiF, A1 2 O 3 , silicon and air detector materials when the distance is varied from 0.5 cm to 15 cm. This is because mean energy decreases with depth in water (see Fig. 2), which results in increase in R. Increase in R is substantial for the Si diode detector (up to 40%) as the atomic number is high (see Table I). For the air material, increase in R is within 6% as its Z eff is comparable to that of water. The detector materials, Li 2 B 4 O 7 and diamond have Z eff values smaller than that of water. Hence, the values of R decrease from 0.972 to 0.954 and 0.832 to 0.781, respectively, for the Li 2 B 4 O 7 and diamond detectors, for the above-mentioned distance range.

C.2 169 Yb source
MacPherson and Battista (15) studied the response for LiF for 169 Yb point source as a function of distance in water using the Monte Carlo-based MCNP code. The values of R reported are 1.16 and 1.2 at 1 cm and 4 cm, respectively. In our study, we obtain the values of 1.12 and 1.15 at 1 cm and 4 cm for the bare 169 Yb line source, respectively.
The Monte Carlo-calculated values of R for the investigated detectors for the bare 169 Yb source are different from the 169 Yb 4140 source model. This is shown in Fig. 4 for the LiF detector. The calculations with the bare 169 Yb line source have resulted in overestimation of R when compared to the encapsulated 169 Yb source (model 4140) for LiF, A1 2 O 3 and Si detectors. For example, depending upon the distance, the overestimation is about 3% for LiF (independent of distance), 15% to 22% for silicon, and 11% to 15% for A1 2 O 3 . For Li 2 B 4 O 7 and air, the variation in the values of R between the bare 169 Yb source and the encapsulated 169 Yb source (model 4140) is only 1%. This difference is comparable to the statistical uncertainty of about 1%. For diamond, the bare 169 Yb source has resulted in underestimation of R by 5%, independent of distance.

C.3 Effect of detector thickness
In general, R includes corrections for volume averaging (influence of dose gradients in the detector volume) and self-absorption by the detector. The energy response of a Scanditronix (IBA Dosimetry GmbH, Schuarzenbruck, Germany) p-type diode detector (active volume is 2.5 mm diameter and 60 μm thick) (12,14,17) for 125 I source has been established to be within ± 3.5% for diode-to-source distance range of 0.5 cm to 10 cm. (12) The published value of average absolute response with respect to dose in water for 125 I source is 6.75. (12) This value includes self-attenuation of the diode, which is 0.911. (12) The present study gives absolute response value of 7.44 (independent of distance), which does not include self-attenuation of the diode, because we have not modeled the full diode. When including the published self-attenuation of 0.911, we obtain average absolute response value of 6.78, which agrees with (within 0.4%) the above-mentioned published value of 6.75.
In the calculations, we have not modeled the detectors due to limitations associated with the DOSRZnrc and FLURZnrc codes. (24) For example, simulation of a cylindrical detector whose axis is parallel to the source axis (cylindrical source) cannot be modeled using the above usercodes. To quantify approximately the influence of finite detector dimensions on the calculated dose values of 125 I and 169 Yb sources, we modeled the detectors as cylindrical shells using the DOSRZnrc user-code. In this study, the Scanditronix p-type diode detector (60 mm thick active volume of the diode embedded in a circular disk of silicon substrate of diameter 3.5 mm and thickness 0.45 mm) (12,14,17) is modeled as cylindrical shells of height 1 mm. The 60 μm thick sensitive silicon diode (cylindrical shell) material is embedded between 0.225 cm thick silicon substrate shells. Both kerma and absorbed dose are scored in the 60 μm thick sensitive diode region. The values of kerma to Si diode and absorbed dose to Si diode obtained from the DOSRZnrc simulation are statistically indistinguishable. A comparison of dose results obtained from the DOSRZnrc simulations with the FLURZnrc simulation (detector is not modeled in collision kerma calculations) gives self-attenuation by the diode detector. The value of self-attenuation by the diode detector obtained for the 125 I source is 0.889(1) (independent of distance), which compares reasonably well with the published value of 0.911. (12) For the 169 Yb source, the values of self-attenuation by the diode detector obtained at depths of 1 cm, 5 cm, 10 cm and 15 cm in water are 0.992(5), 0.983(5), 0.973 (7), and 0.970 (7), respectively. A similar study using the 1 mm thick and 1 mm height LiF and A1 2 O 3 detectors in water gives negligible self-attenuation for the 169 Yb source at all distances. However, for 125 I source, self-attenuation by LiF is 0.975(5) and by A1 2 O 3 is as large as 0.850(1), independent of distance.

C.4 Influence of phantom materials on energy response
The relative absorbed-dose energy response corrections obtained in the solid phantom materials PMMA, polystyrene and water are designated as R PMMA , R Poly , and R Water respectively. The ratios R PMMA /R Water and R Poly /R Water would demonstrate the influence of solid phantoms on the energy response of the detectors compared to the water phantom. Except for the air and Li 2 B 4 O 7 detector materials, the FLURZnrc-based collision kerma ratios (numerator of Eq. (1)) obtained in the solid phantom materials are different from that in water phantom. Figures 5  and 6 present the values of R PMMA /R Water and R Poly /R Water for the 169 Yb source (model 4140). These figures demonstrate that both PMAA and polystyrene materials produce similar energy response corrections for air and Li 2 B 4 O 7 detector materials, at all distances. Whereas, for the rest of the detector materials, the values of R PMMA /R Water and R Poly /R Water deviate from unity (larger than unity implies over-response and smaller than unity implies under-response) as the distance increases. For example, for the Si detector material, the values of R PMMA /R Water and R Poly /R Water are 1.18 and 1.30, respectively, at 15 cm depth. For the diamond detector, the values of R PMMA /R Water and R Poly /R Water are 0.93 and 0.88, respectively, at 15 cm depth.

IV. ConCLuSIonS
The Monte Carlo-based relative absorbed-dose energy response corrections as a function of depth in water, PMMA and polystyrene phantoms for detector materials such as LiF, Li 2 B 4 O 7 , A1 2 O 3 , diamond, silicon and air are calculated for the 125 I and 169 Yb brachytherapy sources. For the 125 I source, the relative absorbed-dose energy response correction for a given detector is independent of distance in the phantom materials, suggesting that all the detector materials are good for relative dose measurements. For the 169 Yb source, the correction is distancedependent. As opposed to 125 I source, detailed modeling of actual 169 Yb is important, as the bare line source-based response is significantly different from the encapsulated 169 Yb source (model 4140). The relative absorbed-dose energy response of a given detector for the 125 I source is insensitive to the phantoms investigated. Whereas, for the 169 Yb source, PMMA and polystyrene phantoms demonstrate over-response for LiF, A1 2 O 3 and Si diode and under-response for diamond when compared to that in water medium. The over-or under-response is significant at large distances. The corrections for the detector materials, air and Li 2 B 4 O 7 are almost identical in all the phantoms.