Effect of improved TLD dosimetry on the determination of dose rate constants for 125 I and 103 Pd brachytherapy seeds

PURPOSE
To more accurately account for the relative intrinsic energy dependence and relative absorbed-dose energy dependence of TLDs when used to measure dose rate constants (DRCs) for (125)I and (103)Pd brachytherapy seeds, to thereby establish revised "measured values" for all seeds and compare the revised values with Monte Carlo and consensus values.


METHODS
The relative absorbed-dose energy dependence, f(rel), for TLDs and the phantom correction, Pphant, are calculated for (125)I and (103)Pd seeds using the EGSnrc BrachyDose and DOSXYZnrc codes. The original energy dependence and phantom corrections applied to DRC measurements are replaced by calculated (f(rel))(-1) and Pphant values for 24 different seed models. By comparing the modified measured DRCs to the MC values, an appropriate relative intrinsic energy dependence, kbq (rel), is determined. The new Pphant values and relative absorbed-dose sensitivities, SAD (rel), calculated as the product of (f(rel))(-1) and (kbq (rel))(-1), are used to individually revise the measured DRCs for comparison with Monte Carlo calculated values and TG-43U1 or TG-43U1S1 consensus values.


RESULTS
In general, f(rel) is sensitive to the energy spectra and models of the brachytherapy seeds. Values may vary up to 8.4% among (125)I and (103)Pd seed models and common TLD shapes. Pphant values depend primarily on the isotope used. Deduced (kbq (rel))(-1) values are 1.074 ± 0.015 and 1.084 ± 0.026 for (125)I and (103)Pd seeds, respectively. For (1 mm)(3) chips, this implies an overall absorbed-dose sensitivity relative to (60)Co or 6 MV calibrations of 1.51 ± 1% and 1.47 ± 2% for (125)I and (103)Pd seeds, respectively, as opposed to the widely used value of 1.41. Values of Pphant calculated here have much lower statistical uncertainties than literature values, but systematic uncertainties from density and composition uncertainties are significant. Using these revised values with the literature's DRC measurements, the average discrepancies between revised measured values and Monte Carlo values are 1.2% and 0.2% for (125)I and (103)Pd seeds, respectively, compared to average discrepancies for the original measured values of 4.8%. On average, the revised measured values are 4.3% and 5.9% lower than the original measured values for (103)Pd and (125)I seeds, respectively. The average of revised DRCs and Monte Carlo values is 3.8% and 2.8% lower for (125)I and (103)Pd seeds, respectively, than the consensus values in TG-43U1 or TG-43U1S1.


CONCLUSIONS
This work shows that f(rel) is TLD shape and seed model dependent suggesting a need to update the generalized energy response dependence, i.e., relative absorbed-dose sensitivity, measured 25 years ago and applied often to DRC measurements of (125)I and (103)Pd brachytherapy seeds. The intrinsic energy dependence for LiF TLDs deduced here is consistent with previous dosimetry studies and emphasizes the need to revise the DRC consensus values reported by TG-43U1 or TG-43U1S1.


INTRODUCTION
Within the TG-43 formalism for brachytherapy dosimetry, 1,2 Λ, the dose rate constant (DRC) plays a central role since it relates the air-kerma strength of a seed, S K to the dose rate 1 cm from the seed on its transverse axis, D(1 cm, 90 • ) via All other dose rates around the seed are proportional to Λ.
In general, as we show below, DRC measurements for 125 I ( 103 Pd) are systematically higher than Monte Carlo calculated Λ values by an average of 4.9% (4.1%) which led the AAPM TG-43 to define the DRC consensus value as the average of these two values.The measured values are almost universally dependent on measurements with LiF TLDs.The TLDs are irradiated in some sort of phantom and calibrated in terms of dose to water per unit reading in a 60 Co or 6 MV beam.The relative absorbed-dose sensitivity (S rel AD,med , formally defined below) of the TLD is then used to establish the equivalent dose to water per unit reading in the 125 I or 103 Pd field.This dose is the dose to water in the phantom material, and a phantom correction factor (P phant , formally defined below) is used to derive the corresponding dose to water in water from the seed being investigated.
There are many uncertainties associated with the procedure.4][5][6][7][8] The value of 1.4 was widely accepted as this is just the ratio of ratios of LiF to water mass energy absorption coefficients which is a simple theoretical expectation for the ratio in 60 Co and 20-30 keV (specifics below).There are several issues related to using these early values.For one, the shape of the TLD plays a significant role and, as we will show below, the S rel AD,med values vary by 3%-4% from this issue alone and up to 8.4% when all issues are considered.Similarly, we will show that the simple model leading to a value of 1.40 is off by several percent due to absorption effects in the TLDs.Moreover, this simple model is actually only a model for the relative absorbed-dose energy dependence of the LiF, f rel , i.e., the change in the ratio of the dose to the medium per unit dose to the LiF and not the required S rel AD,med .Finally, the measured values are all based on the state-of-the-art methods of dosimetry for the time, but since then, the primary standards and the dosimetry protocols used have all changed considerably.
0][11] This further changes the expected value of S rel AD,med although, as will be shown below, this tends to cancel some of the attenuation effects mentioned above.
The goal of this paper is to reanalyze the published values of measured DRCs making use of state-of-the-art Monte Carlo calculations of f rel , the relative absorbed-dose energy dependence of the LiF detectors, and then determining the value of k rel bq , the intrinsic energy dependence of LiF, by determining the value which makes the measured values most closely agree with our Monte Carlo calculated values of the DRC.It will be shown that the value determined this way is consistent with the directly measured values.The paper then provides a revised set of DRCs for 24 different seed models and compares them to the previously recommended values from TG-43U1 (Ref.2) or TG-43U1S1. 12onsistent with standard medical physics practice to date, we are ignoring the possible variation in the values of the intrinsic energy dependence of LiF detectors depending on the annealing and/or reading protocols followed.While not denying the potential impact of such variations which can be significant (see Refs. 13 and 14 and references therein), it is beyond the scope of this work to include this variable.Also, it is not explicitly corrected for in any of the experimental papers we have reanalyzed nor is there currently adequate knowledge to do so.It should be noted that the up to 3% difference between the measured results of k rel bq in the recent literature 9,10 in the energy range of interest here has been attributed to being most likely due to different protocols although it could be due to different energy spectra or other reasons as well.
While the purpose of this paper is to reanalyze a large number of prior publication's values, we want to be clear that this is not meant as a criticism of these previous papers.In the majority of cases, these papers applied the state-of-the-art procedures and values that were available at that time.However, the field's knowledge and abilities to do calculations with much higher accuracy and statistical precision today make this reanalysis possible.

1.A. Formalism and notation
There is considerable confusion in the literature caused by varying terminologies, so it is essential to define a rigorous terminology, and we follow that used in Chap. 4 of the AAPM's 2009 Summer School book. 15The absorbed-dose sensitivity of a detector is given by where M det is the detector's reading in the beam quality of interest with corrections made for effects such as for recombination, polarity, leakage, and dose rate dependencies, D med is the dose to the phantom material (usually water) in the absence of the detector at the point of measurement (usually the midpoint of the detector) and N D, w is the absorbed-dose calibration coefficient for the detector in the beam quality of interest.The absorbed-dose sensitivity has two components.The first is f , the absorbed-dose energy dependence where D det is the average dose to the detector's sensitive material.For low-energy beams, f is often approximated by the ratio of mass energy absorption coefficients The second component of S AD,med is the intrinsic energy dependence of the detector, k bq given by These definitions lead to a simple relationship From these definitions we have Since TLDs are often calibrated in a high-energy photon beam, Q o , for which the absorbed dose is known (usually 60 Co or 6 MV), and one needs the absorbed-dose sensitivity in a beam 114301-3 of quality Q, hence one needs the relative absorbed-dose sensitivity where the relative intrinsic energy dependence and relative absorbed-dose energy dependence are given by For relative absorbed-dose calibration coefficients one has (k Q in TG-51 terminology 16 ) The quantity f rel can be calculated using Monte Carlo techniques and for photon detectors in low-energy beams is often approximated, following Eq.( 4), as For TLDs in 125 I fields, the value of f rel relative to a 60 Co beam is roughly 0.7 although the literature often deals with its inverse, viz., 1.4.In Chap.14 of the AAPM's 2009 Summer School book, 17 f rel and k rel bq were defined as the inverse of what is given in Eq. ( 9) above which is based on the definitions in Chap. 4 of the same book.Chap.14 would be internally consistent except that it gives the same expression for calculating f rel as in Eq. ( 11) above (i.e., Q over Q o , see 2 lines above Eq.(14.20) in Chap.14) and this is inconsistent with the definitions of Chap.14.The equation for k rel bq in that chapter (14.20) is also inconsistent with the expression for S rel AD,med of Chap.14.The remainder of this paper will use the quantities which are consistent with Chap. 4 definitions and as used elsewhere in the 2009 AAPM Summer School book.
Based on the Chap. 4 definitions and the relations derived above for the relative quantities, the final equations needed when measuring the value of the dose rate constant, Λ, are (ignoring linearity and phantom material effects, i.e., assuming a water phantom):

1.B.1. k rel bq
Based on the information available at that time that k rel bq was unity (albeit with large uncertainties), 18 some papers made the now known to be incorrect assumption that S rel AD is numerically equal to 1/ f rel , but this ignores the relative intrinsic energy dependence of the detector, k rel bq .Many authors have demonstrated that at low energies k rel bq for TLDs is not unity [9][10][11]13,[19][20][21] with a value between 0.90 and 0.94 in the 125 I and 103 Pd energy range [9][10][11] (note that these papers present data for 1/k rel bq since they refer to an increase in the relative energy response which is basically S rel AD,med ∝ 1/k rel bq ). A value  k rel bq less than unity means that the detector's reading per unit dose to the detector material is higher in the relevant beam quality than in the calibration beam.Using x-ray beam energies ranging from 20 to 250 kV(peak), Davis et al. 9 and Nunn et al. 10 60 Co photons.Weaver 4 found a TLD reading per unit dose to water of 1.39 ± 0.03 in the brachytherapy energy range relative to 60 Co.Later, Weaver et al. 5 found values of 1.47 ± 5% and 1.42 ± 5% for 6702 125 I and 6711 125 I seeds, respectively. Meiooni et al. 6,7 also reported a relative absorbed-dose sensitivity of 1.41 ± 3% for low-energy photon beams relative to a high-energy beam (4 MV).Muench et al. 8 also reported an S rel AD value of 1.41 for 60 kV x-rays relative to 4 MV (3%-5% uncertainty estimated based on similar measurements 5,6,22,23 ). The atio of the mass energy absorption coefficient of LiF to water in the brachytherapy energy range relative to that at 60 Co is 1.41.Ignoring the need for the intrinsic energy dependence, the measured values are in good agreement with this simple theoretical expectation.As a result, a value of 1.40 or 1.41 has been generalized and used as the relative absorbed-dose sensitivity in many measurements of the DRC for the brachytherapy seeds currently in the market (see Tables IV and V below for individual values).However, it must be realized that all of these earlier reports on the value of S rel AD were based on the TG-21 24 or similar protocols for the high-energy dosimetry, and the x-ray dosimetry was based on old protocols or procedures, often not specified.Furthermore, the high-energy measurements were usually based on the air kerma (formerly exposure) standards of the National Institute of Standards and Technology (NIST) for 60 Co and these have been revised.25 In short, the measured value of 1.41 is based on many dosimetric quantities and procedures which have experienced changes, and the effects of these changes have not been tracked for their effect on this measured value, thereby making the proper value quite uncertain.In contrast, currently used measurements of dose rate constants are based on NIST's post-1999 S K standard 26 and the TG-51 16 protocol for high-energy beam measurements.However, the values of the relative absorbed-dose sensitivity published in the articles mentioned above were often used to correct the TLD readings.
An exception to this trend is the recent work of Kennedy et al. 27 who explicitly accounted for a k rel bq value of 0.916 ± 0.023 in their measurements of the DRC for the THINSeed 9011 and GE 6711 seed models.These measurements are not included in our analysis but compared to our results below.

1.B.3. f rel
At low energies, TLD material attenuates photons more than water and hence, taking f to be the simple ratio of mass energy absorption coefficients as f is not applicable since the photon energy fluence is not the same in both materials.Mobit and Badragan 28 reported that for 125 I fields, Monte Carlo calculated values of f rel relative to 60 Co vary between 1.32 and 1.41 for 5 and 1 mm diameter microrods.Calculation of the ratio of average attenuation and hence photon fluence between TLD and water for a detector x cm thick (i.e., e −µ TLD x/2 /e −µ w x/2 ) shows that for 2, 1, 0.4, and 0.1 mm thicknesses, the ratios are 0.930, 0.964, 0.986, and 0.996, respectively.Therefore, the finite thickness of the detector significantly affects the absorbed-dose energy dependence, f rel , and consequently the measurement of the DRC of brachytherapy seeds.

1.C. Phantom corrections
Up to this point, the discussion has been about measurements and quantities defined in a water phantom, but for 125 I and 103 Pd seeds, almost all measurements are done in a plastic phantom of some sort.It is usually assumed that the relative absorbed-dose sensitivity of the TLDs, S rel AD,med , is unchanged when the measurement is in a phantom, despite the fact that the different phantom materials could cause the photon spectrum to be different at the location of the detector.Then, for a TLD calibrated at high energies in terms of absorbed dose to water, the TLD reading is thought of as reporting an absorbed dose to water, even when in a plastic phantom, i.e., the TLD is considered to be reporting D med w , the dose to water in the phantom medium.Hence, one measures Λ med ≡ D med w /S K .To extract the DRC in water, Λ w , the phantom correction factor, P phant , is defined such that where D w w is the dose to water at the reference point in a water phantom.Many papers have calculated Λ med using Monte Carlo, shown it agreed with their measured value and then calculated Λ w to allow calculation of P phant .In other words, many measured values of Λ w are actually directly proportional to a Monte Carlo calculated value of the same quantity, albeit in a ratio to Λ med .
The statistical uncertainties on previous P phant calculations are usually much higher than here, and hence we have systematically replaced them with our calculated values.However, as Patel et al. 29 have shown, variations in the actual composition of the phantom material, especially of the high-Z components, represent a significant potential uncertainty which is discussed below.
It is worth noting that TG-43U1 (Ref.2) explicitly defined E(r) to include the phantom correction factor, i.e., in our notation: In a few papers it is actually used this way 29,30 but the much more common use has been E(r) = S rel AD,med or E(r) = ( f rel ) −1 along with a separate assessment of P phant .To further confuse the situation, in at least one paper 29 the calculated value of E(r), based on Eq. ( 17), is said to "agree with previously published energy response measurements, 4,6,31 " whereas each of the papers cited measured S rel AD,med (i.e., with no P phant but including k rel bq ).

2.A. Relative absorbed-dose energy dependence and phantom correction
The relative absorbed-dose energy dependence is calculated as where D TLD is the average absorbed dose in the TLD and D w is the absorbed dose to water in a small voxel (0.1 × 0.1 × 0.05 mm 3 ) at the midpoint of the detector in the absence of the detector.The numerator refers to values determined in the radiation field of interest and the denominator to those in the calibration beam.It is important not to use just the dose to water averaged over the detector volume as this averaging would decrease the value of f rel by up to 2.4% due to 1/r 2 effects, being more of an effect for larger detectors.At brachytherapy energies the absorbed doses are calculated using BrachyDose, a fast EGSnrc-based 32,33 Monte Carlo code developed by Yegin et al. [34][35][36] to perform brachytherapy dose calculations.The voxel-based BrachyDose Monte Carlo calculations of TG-43U1 dosimetry parameters have been benchmarked by Taylor et al. 35 The absorbed dose to a TLD (D TLD ) is calculated in a LiF:MgTi 3 × 3 × 1 mm 3 voxel centered at 1 cm from the axis of the seed on the transverse plane in a 30 × 30 × 30 cm 3 water phantom.TLDs made of TLD-100 (LiF:MgTi) are the most commonly used detectors to measure the DRC.Consequently, the TLD material is simulated as LiF:MgTi material which has a fractional composition by weight of 0.26700 of lithium, 0.73279 of fluorine, 0.00020 of magnesium, and 0.00001 of titanium. 9Since TLDs come in different sizes and forms, simulations are also performed using 6 mm long × 1 mm diameter rods or 1 × 1 × 1 mm 3 cubes.These are the most common TLD sizes used in brachytherapy dose measurements.The rod's longitudinal axis is placed at 1 cm parallel to the longitudinal axis of the seed.The average dose to the detector is also calculated in TLDs with frontal areas of 3 × 3 mm 2 and 1 × 1 mm 2 and thicknesses of 0.8, 0.6, 0.4, and 0.1 mm.Several researchers 26,[37][38][39][40] have demonstrated that 125 I seed models containing silver and those that are silver-free have notable differences in their generated spectra.Therefore, calculations were initially performed using three different sets of seeds: (a) 125 I seed models that contain silver components (GE Oncura 6711, Imagyn IS-12051, and MBI SL-125), (b) 125 I seed models that are silver-free (STM 1251, IBt 1251L, and Best 2301), and (c) 103 Pd seed models (Theragenics 200, MED3633, and Best 2335).
The phantom correction, Eq. ( 15), is calculated as the ratio of the dose to water in water [in a 1 mm 3 voxel] to the dose to water in phantom material 17 at the reference point (1 cm from the center of the seed on the transverse axis).In addition, it is also calculated as the ratio of the dose to TLD in water to the dose to TLD in phantom material at the reference point.Calculations were performed for the different phantom materials used in measurements, solid water 41 (SW), RW-1, also called plastic water, 42 or plastic water PW2030, 43 virtual water 44 (VW), PMMA and some reported variations on these materials.Table I gives the compositions used with the emphasis on the widely used RMI solid water for which several densities and compositions have been reported. 29,45hen determining the k rel bq in this work, the phantom correction (P phant ) used in TLD measurements is replaced by the new phantom correction (P new phant ) calculated here.
Rayleigh scatter, bound Compton scatter, photoelectric absorption, and fluorescent emission of characteristic x-rays are included in the simulations.The photon energy cutoff is set to 1 keV with no electron transport.Photon cross-sections from the XCOM (Ref.46) database are used in all calculations.One standard deviation statistical uncertainties are kept lower than 0.1%.
At high energy, D w /D TLD is calculated using the EGSnrc DOSXYZnrc user code. 47Dose is calculated in a LiF:MgTi 3 × 3 × 1 mm 3 voxel on the central axis with the front face at depths of 2, 5, and 8 cm in a 30 × 30 × 30 cm 3 water phantom.The dose ratio was also calculated for large and small chips and rods at 5 cm depth.The photon sources are a 10 × 10 cm 2 parallel 60 Co beam 48 and a 10 × 10 cm 2 parallel 6 MV Varian spectrum. 49Simulation parameters are the same as for the low-energy simulations except that electron transport is simulated down to 10 keV (ECUT = 521 keV).One standard deviation statistical uncertainties are kept less than 0.1% in all high-energy simulations.

2.B. Relative intrinsic energy dependence
The inverse of the absorbed-dose energy dependence, ( f rel ) −1 , found in this work is used to replace the ≈1.41 value of the relative absorbed-dose sensitivity used in many DRC measurements.As discussed above in Sec.2.A, the finite size of the detector means it experiences a decreased dose compared to a point at the detector's midpoint on the axis.This is accounted for in our calculations and some authors have corrected for this effect [50][51][52]  dose to the point on the detector's axis vs the dose averaged over the detector's volume.Thus, when the value of this correction was reported, we have taken it into account by dividing the reported S rel AD,med value.Similarly, the value of P phant determined in this work is used to replace the values used in the original work.This gives which deliberately ignores the relative intrinsic energy dependence, k rel bq .A global value of k rel bq is determined by scaling Λ nok bq by k rel bq as required by Eq. ( 13) or ( 14) and then minimizing the difference between the calculated and adjusted measured values doing a least squares fit varying k rel bq .Although this case is linear, the value of k rel bq is determined using a graphical method for assigning uncertainties to parameters in nonlinear least squares fittings. 53The χ 2 is given by where s m,i and s c,i are the absolute uncertainties in the ith of l measurements and calculations, respectively, and ∆ i is The calculated values of Λ i are those from column RR (Ref.40) in Table VI discussed below.
In both previous calculations of Λ i , 40,54 the statistical component of uncertainty was 0.3% or better.However, when calculating k rel bq in Eqs.(20) and (21), s c,i also includes Type B uncertainties as recommended by AAPM TG-43U1/U1S1 (Refs. 2 and 12) for Monte Carlo simulations.Type B uncertainties include those generated by uncertainties from initial photon cross-section libraries, seed geometries, and photon energy spectra, whereas Type A uncertainties come from estimates of the statistical uncertainty inherent in the Monte Carlo technique.Type B uncertainty values are assigned based on various studies.Williamson 55 found that shifting of internal components of the 125 I Draximage LS-1 caused a DRC uncertainty of less than 2%.Dolan et al. 30 reported a total uncertainty of 0.75% due to geometry uncertainties of the Oncura 6711 seed model.As a compromise between the two previous values, the current work assigns an uncertainty of 1.2% to the DRCs due to geometric uncertainties.In a recent article, 40 we showed there is a negligible effect on the calculation of air kerma, dose and DRC for 125 I and 103 Pd full seed models when using any of four different initial spectra.AAPM TG-43U1 recommends an uncertainty of 0.1% due to uncertainty in the initial energy spectrum and that is conservatively adopted here.Uncertainty in the TG-43U1 parameters due to uncertainties in the photon cross-section libraries has been investigated by others (Bohm et al., 56 DeMarco et al., 57 Hedtjärn et al., 58 Nunn et al. 10 ).EGSnrc BrachyDose uses the NIST XCOM database, a current state-of-the-art photon cross-section library.Nunn et al. 10 reported an uncertainty of 0.86% on the calculated value of ( f rel ) −1 in this energy region due to cross-section uncertainties.We have adopted this value.Overall the total uncertainty assigned to the Monte Carlo calculated DRCs (s c,i ) is 1.5%.This value is lower than the generic uncertainty value of 2.5% suggested by AAPM TG-43U1 for DRCs calculated by Monte Carlo mainly because of the lower value assigned to the photoionization cross-section and seed geometry uncertainties (0.9% and 1.2% compared to the 1.5% and 2.0%, respectively, in TG-43U1).These lower values are assigned based on recent studies 10,30 not available at the time AAPM TG-43U1 were published.
The uncertainties on the measured DRC values, s m,i , are adjusted by subtracting in quadrature the reported uncertainties of the original relative absorbed-dose sensitivity and phantom correction from the reported total uncertainty due to the fact that the original values are replaced by the relative absorbed-dose energy dependence and phantom correction found in this work so that k rel bq can be determined.The uncertainties on our calculated values of P phant and ( f rel ) −1 are likewise added back in quadrature.As cited explicitly below in the tables, the experimental dose rate constant values have been taken from 10 articles for 7 different 103 Pd seed models and from 25 articles for 17 different 125 I seed models.When appropriate, the corrected values reported in the TG-43 updates are used. 2,12 RESULTS

3.A. Relative absorbed-dose energy dependence
Table II shows ( f rel ) −1 for 125 I and 103 Pd seeds for various TLD shapes.In general, data show that the relative absorbeddose energy dependence varies with the detector thickness in the brachytherapy energy range.The shape of the frontal face (toward the seed) of the detectors is also important when measuring dose delivered by brachytherapy seeds because there is a detector volume effect that needs to be considered when using TLDs to measure dose in the low energy brachytherapy range.Because the linear attenuation coefficient increases quickly at low energies (20-30 keV), the effect is even more notable for 103 Pd seeds compared to 125  For small detectors, as the thickness of the detector reduces down to a thin detector, the attenuation effect is minimal and the value of ( f rel ) −1 approaches the ratio of the mass energy absorption coefficient of LiF to water relative to that at 60 Co.TLDs with thickness of 0.1 mm do not exist in the market but are used here to show the thickness dependency of the ratio.The detector volume effect with rods is more significant, 114301-7 T II.Inverse of the relative absorbed-dose energy dependence, ( f rel ) −1 , for 125 I (GE 6711, IS-12051 and MBI SL-125 with silver and STM 1251, IBt 1251L, and Best 2301 without silver) and 103 Pd (Theragenics 200, MED3633, and Best 2335) seeds relative to 60 Co (D w /D TLD values are 1.202, 1.214, and 1.208 for large chips, small chips, and rods, respectively).( f rel ) −1 values are the average of the three seeds for each group and the value in parenthesis is the absolute difference between the maximum and minimum ( f rel ) −1 over all seeds of that group.Statistical component of uncertainty on ( f rel ) −1 is ≤0.2%.

With silver
Silver-free 103 Pd Large chips (0.009) (0.003) (0.023) 3 × 3 × 1 mm 3 1.365 because 1/r 2 effects lead to a lower dose to the TLD material at the ends of the detector and consequently brings down the average dose in the detector divided by the dose to a small voxel of water at the central point of measurement compared with the same ratio for a 1 × 1 × 1 mm 3 microcube detector.Generally, for small chips, only small differences were observed in f rel values for 125 I seeds within the silver group or within the silver-free group, but a difference of up to 0.5% exists between the two groups for small chips.Large chips and rods, on the other hand, show considerable variation between seeds in each group.The effect is more evident in 103 Pd seed models with seed to seed variations of up to 1% for even the small chips.The values in parenthesis in Table II represent the absolute difference between the maximum and minimum ( f rel ) −1 values for all seeds studied, not just the three presented in more detail.Palladium seeds have differences up to 0.023 which represents approximately 1.8% of ( f rel ) −1 for large chips.Due to these differences, ( f rel ) −1 values were calculated specifically for each 103 Pd seed model and for 125 I seed models that used rods for DRC measurements.
For both high-energy beams ( 60 Co and 6 MV), the calculated dose ratio (D w /D TLD ) does not change as a function of depth.However, different values were observed for each TLD shape.D w /D TLD values for 60 Co are 1.202, 1.214, and 1.208 for large chips, microcubes and rods, respectively.The same ratios for 6 MV are 1.205, 1.219, and 1.211, respectively.Statistical uncertainty for each value is 0.14%.
The ( f rel ) −1 values in column 7 of Tables IV and V presented below correspond to the specific seed, TLD shape and calibration source ( 60 Co or 6 MV) used in each measurement.
In the analysis used here and used at least indirectly in all measurements in plastic phantoms, is the assumption that the value of f rel is the same whether the TLD is in water or in a plastic phantom.We verified this directly for a solid water phantom, and although the dose to a small water volume at 1 cm from the seed decreased by 3.3% near an 125 I seed and by 3.8% near a 103 Pd seed, the ratio of D SW w /D SW TLD vs the ratio of D w w /D w TLD was constant within the 0.1% statistics of the calculation for both seeds, and similarly the ratios at 5 cm depth in a 60 Co beam stayed constant at better than the 0.1% statistics.Hence, the values of f rel are basically identical in water or solid water and we assume the same for all plastic phantoms.

3.B. Phantom correction
Table III presents calculated results for the phantom correction factors, P phant , for the various phantom materials used for measurements of DRCs.P phant values are very similar for different 125 I seed models although values for 103 Pd are generally further from unity and show a greater spread in values for different models.For SW, a 1.5% change in density has about a 0.2% effect on P phant for 125 I seeds but a 1% effect for 103 Pd seeds.More importantly, the SW with a measured reduction in the calcium content has a 4.7% lower value of P phant for 125 I seeds and an 8.1% lower value for 103 Pd seeds, consistent with the 125 I results of Patel et al. 29 Average values of P phant in solid water (1.052 ± 0.003 for 103 Pd and 1.035 ± 0.001 for 125 I) are used in Tables IV and V.This narrow range of values for a given radioisotope compares to the range in the literature of 1.030-1.054for 103 Pd and 1.031-1.05for 125 I seeds, which is mostly due to less statistical precision in many of the prior values and justifies using the values calculated here.In the present analysis, we adopt the average values of P phant from Table I and assign an uncertainty of 3% in recognition of the impact of the uncertainties in the composition.
At high energy, the ratios D w w /D SW w or D w TLD /D SW TLD are 1.002 ± 0.002, which implies that at high energies, both water and solid water provide similar scatter to the detector.Also, explicit calculations for several cases show that the phantom correction does not change with TLD shape.

3.C. Relative intrinsic energy dependence
Tables IV and V present the measured DRCs and their respective uncertainties as reported by the authors [or as updated by TG-43 (Refs. 2 and 12)] along with the phantom correction and the relative absorbed-dose sensitivity used with those measurements.Experimental data are based on the post-1999 S K standard at NIST (Refs.26 and 59) and TG-51 (Ref.16) protocol for high-energy beam calibrations.The tables show the measured dose rate constants of 7 103 Pd seeds models and 17 125 I seed models.Columns 6 and 7 in Tables IV and V show the P phant and ( f rel ) −1 values for each seed model used in the 114301-8 T III.Monte Carlo calculated P phant values to correct doses measured at 1 cm from the seed in a given material to the corresponding dose at 1 cm in water.P phant values are given for four seeds of each radioisotope to show typical variations with seed model.The statistical uncertainty is about 0.2%.The calculations for SW with different densities and compositions show that the systematic uncertainties dominate the P phant values.Values from the literature are shown for comparison, but these are often not for specific seed models.IV for the OptiSeed IBt 1032P seed are anomalous and revised DRCs are not included in the analysis.c 1.048 ± 3%, 68 1.043 ± 3%, 65 1.0315, 69 1.043 ± 3%, 50 1.044 ± 0.2%, 70 1.033 ± 4.6%, 71  i TLD large chips used in measurements.The others used TLD microcubes (1 × 1 × 1 mm 3 ).determination of k rel bq in this work.Columns 2 and 3 in Table VI show the MC calculated dose rate constants for the same seed models.Both sets of Monte Carlo DRCs were calculated using the EGSnrc BrachyDose code and have similar values (but not always within the statistical component of uncertainty) except for two 125 I seed models (NASI MED3631 and STM1251) and three 103 Pd seed models (Theragenics, NASI MED3633, and IsoAid Advantage IAPd-103).Two of these seeds have recently had a revision in the geometry description of the seed in our database.The previous IAPd-103 seed model had a minor mistake in the geometry description (a small region near the seed had water instead of air in it) which was affecting the air-kerma calculation (but not the in-phantom calculations), and the NASI MED3633 geometry was modified and updated as described by Rivard. 60Due to the loss of previous seed input files, it has not been possible to identify the reasons for DRC discrepancies for the other seeds.However, the revised mea-sured values presented below are on average slightly closer to the calculated values in Ref. 40, which are used in the rest of the present analysis.Column 5 presents the Λ nok bq value calculated using Eq. ( 18) and data in Tables IV and V.For seed models with two entries in Tables IV and V, Λ nok bq ± s ′′ m represents the average of the two values.
Figures 1 and 2 show plots of χ 2 as a function of the fitting parameter k rel bq [in Eq. ( 20)] for 125 I and 103 Pd seed models, respectively.The fitted k rel bq value is given at χ 2 min and the uncertainty is determined by taking the corresponding values at χ 2 min + 1 (Ref.53).The k rel bq ((k rel bq ) −1 ) values are 0.931 ± 0.013 (1.074 ± 0.015) and 0.922 ± 0.022 (1.084 ± 0.026) for 125 I and 103 Pd seed models, respectively.
The relative intrinsic energy dependence deduced from the revised measured DRCs is reasonably consistent with values found by some other authors for x-ray beams [9][10][11] as shown in Fig. 3.The intrinsic energy dependence found in this work applies directly to 125 I and 103 Pd seeds rather than to x-ray spectra as in most previous studies.Das et al. 18 reported a relative intrinsic energy dependence value of 1.0, but the reported uncertainties on those data were large.Values of Das et al. at ≈20 keV are not as consistent with ours as those at 28 keV, and their 20 keV values are even farther from the other previous results. 10,11

3.D. Calculated vs revised measured dose rate constants
For each seed model, Table VII shows the Monte Carlo calculated DRC, Λ MC (from column RR, Table VI), and the revised measured DRC, Λ rev meas , which is determined by replacing the phantom correction and relative absorbed-dose sensitivity used originally to determine the experimental values, by the new phantom correction and relative absorbeddose sensitivity values, S rel AD,med = ( f rel • k rel bq ) −1 .The P phant and ( f rel ) −1 values are taken from Tables IV and V, and the (k rel bq ) −1 values are as determined above.After revising the measured DRCs, the average MC calculated values compared to measured values are 1.2% higher and 0.2% lower for 125 I and 103 Pd seeds, respectively.This compares with average calculated DRCs being 4.8% lower than the "as published" DRCs.For each seed model, the table also presents Λ rev avg , the average value of the revised measured DRC and the Monte Carlo calculated DRC, and the percentage difference between that average and the consensus value reported in TG-43U1 (Ref.2) or TG-43U1S1 (Ref.12).The consensus values are on average 3.8% and 2.8% higher than the average of the calculated and revised measured DRCs for 125 I and 103 Pd brachytherapy seeds, respectively.DRCs agree well with their new measured values for both seeds, and our revision of the previous measured DRC for the 6711 seed 30 now agrees better with their new measurement and with the calculated values.

CONCLUSIONS AND DISCUSSION
Important parameters in dosimetry at low energy (e.g., f rel , k rel bq , P phant ) are now measured or calculated with higher accuracy and precision than in the past.This work has focused on calculating the phantom correction and the relative absorbed-dose energy dependence for the energy spectra generated by 125 I and 103 Pd brachytherapy seeds using a stateof-the-art tool for such calculations.Results show that the generalized value of 1.41 used as S rel AD for 125 I and 103 Pd seed dose measurements needs to be updated.Small differences (≈±0.5%)have been detected in S rel AD,med values for 125 I seed models containing silver vs those without silver (Table II) when using small chips, but differences are larger when using F. 2. Same as Fig. 1  large chips and rods.Although these differences are small compared to the typical uncertainty in dose measurements at low energies, they have been included in this work.
Based on the individual values on which values of Table II are based, variations in f rel values of up to 3.6% for 125 I seeds and up to 5.6% for 103 Pd have been calculated amongst the most common TLD sizes used in brachytherapy dosimetry (3 × 3 × 1 and 1 × 1 × 1 mm 3 chips and 6 mm long by 1 mm diameter rods).Furthermore, for a given chip size, the f rel values for 125 I and 103 Pd seed models differ from each other by up to 5.4%, 3.6%, and 6.5% for large chips, small chips, and rods, respectively, and it may be up to 8.4% among any TLD shape and seed model, i.e., one must use f rel values specific to the seed and TLD shape involved.If we further take into account the difference in the relative intrinsic energy dependence, the overall relative absorbed-dose sensitivity values are as much as 4.2%, 2.8%, and 5.0% less for 103 Pd than for 125 I for the large chips, small chips, and rods, respectively.This finding is significant because to date the same value of the relative absorbed-dose sensitivity has been used to correct dose measurements for both 125 I and 103 Pd seed models and any shape of TLD.It was also shown that f rel [as defined in Eqs.(3) and ( 9)] is similar for measurements in water or solid water, and it is assumed it does not change for other plastic phantoms either.
Phantom corrections for a given material are nearly the same for different seed models (range up to 0.7% in the worst case) but do depend on the isotope involved.Uncertainties in the phantom corrections from uncertainties in the composition and density dominate and are likely ≈3%.
Analysis of the calculated and measured data also shows that values of (k rel bq ) −1 of TLDs for brachytherapy dose measurements are 1.074 and 1.084 for 125 I and 103  energies [9][10][11] and with the value reported in Ref. 27 which is based on an unpublished result for 125 I seeds.Thus, significant changes are needed for that subset of previous experimental data that used calculated relative absorbed-dose energy dependence values to correct the measurements based on a relative intrinsic energy dependence of 1.0 . 18n general, the relative absorbed-dose sensitivity needs to be updated taking into account not only the application of the recently and more accurately measured intrinsic energy dependence value but also considering the f rel dependence on TLD shape and seed model.The S rel AD,med value can be determined using Eq. ( 8), the tabulated values of f rel −1 and the values of k rel bq determined in Figs. 1 and 2. Compared to the oft-used value of 1.41 for S rel AD,med , our results imply typical values of 1.47 for 103 Pd seeds with (1 mm) 3 LiF TLDs and 1.46 and 1.51 for 125 I seeds using rods and (1 mm) 3 TLDs, respectively.Our calculated P phant values also modify the DRCs by an average of 1.6% in the same direction for 103 Pd seeds and 0.7% in the opposite direction for 125 I seeds  which leads to the overall decreases in the DRCs of 4% and 6.4%, respectively for 103 Pd and 125 I seeds.The discrepancy between the previous Monte Carlo calculated and measured DRC values caused the AAPM TG-43 to define a consensus value of the DRC of brachytherapy seed models as the average of the two data sets.However, this work finds that by applying the updated and appropriate relative absorbed-dose sensitivity to correct measurements at low energies, on average such differences decrease, and all individual cases agree within the uncertainties of the calculations and measurements.At present, the TG-43U1 (Ref.2) and TG-43U1S1 (Ref.12) consensus values are, on average, 2.8% and 3.8% higher than the average values of the revised and calculated values presented here for 103 Pd and 125 I seeds, respectively.The DRC scales the 3D dose distribution in a brachytherapy treatment plan, and the higher TG-43U1 and TG-43U1S1 values are overestimating the dose delivered to the patient by 3%-4% compared to the dose that would be given using the present results.Perhaps more importantly, in a worst case the dose delivered for two different model seeds would differ by up to 6.1% based on the use of the current consensus values vs the average revised values proposed here.
This work suggests that the dose rate constant consensus value reported by TG-43U1 or TG-43U1S1 for each brachytherapy seed model used clinically should be revised and updated to those determined by the average of values calculated by Monte Carlo simulation and values measured with the appropriate relative absorbed-dose sensitivity.Going one step further, given that, • there is overall agreement between the measured and calculated DRCs, • the overall uncertainty on the calculated DRCs, i.e., 1.5%, is considerably lower than that on the measured values, • most measured DRCs are directly proportional to the MC calculated DRC through the phantom correction factor, and • making accurate TLD measurements is almost impossible without relative absorbed-dose sensitivities which are specifically applicable to the annealing and readout protocols used, one might suggest that the relevant committee consider adopting the Monte Carlo calculated values for clinical use rather than the averaged consensus values.This is already done for g(r) and F(r,θ) values in LDR brachytherapy and for DRCs of HDR 192 Ir sources.In order to avoid possibly significant mistakes in modeling new seeds, a measured verification that the calculated DRCs are reasonable and/or a verification that the spectra from any new seed model agrees with measured spectra 40 would still be necessary before adopting the calculated DRC as the clinical value.
T VIII.Comparison of present results to those of Kennedy et al. (Ref.27) for the THINSeed 9011 and GE 6711 125 I seeds.Kennedy et al. used a value of (k rel bq ) −1 = 1.092 ± 0.027 compared to the value 1.074 ± 0.015 presented here.Λ WAFAC (calc) (cGy h −1 U −1 ) Λ meas (cGy h −1 determined that 1/k rel bq relative to 60 C, ranges from 1.08 to 1.11.Similarly, Tedgren et al. 11 reported values of 1.06-1.07 in this energy range. by increasing the TLD reading by a 1%-2% geometry correction where k geom is the T I. Compositions and densities of phantom materials used in the calculations of P phant .Values shown have considerable uncertainty and many values reported in the literature are just the manufactures stated values.Tello et al. (Ref.61) reported spreads in density of 1.1%-4% for commercial phantom materials.Values are shown as originally reported but for the present calculations are renormalized to give 100%.SW called RMI-451 in Ref. 65 but with lower density.Several papers cite this paper by Williamson for P phant values.
bAs reported by manufacturer.c d P 0.02, K 0.02.e Virtual water: Refs.41 and 62 give a slightly different composition.f Ref. 67 measured 1.0467 g/cm 3 .g Often referred to as plastic water in the literature.h Mg 0.91.i Plastic water used in high-energy beams.j 0.03% Br. k Values from Ref. 104 are within 0.01%.l 8.44% Al.
103 ) of103Pd seed models measured with TLDs.Columns 3, 4, and 5 represent the phantom correction (P phant ± s p ), the phantom material, and the relative dose sensitivity (S rel AD ± s m ) originally used in the measurements.Thelast two columns represent the new phantom correction (P new phant ) and the absorbed-dose energy dependence [( f rel ) −1 ] calculated in this work.Λ values with two references show the original report, but the value is as updated in TG-43U1 (Ref.2).The statistical component of uncertainty on our calculated TLD corrections is ≤0.2%.T V. Same asTable IV except for 125 I seed models. Vale measured back in the 1980s or early 1990s.The other values were determined by Monte Carlo calculation and are actually ( f rel ) −1 values since k rel bq was not accounted for or taken as unity.d Authors cited Weaver et al. (Ref.5) who reported two values.Average is used.e TLD rods used in measurements.f 60 Co used at high energy.The others used 6 MV.g P phant included in published S rel AD .h Energy response for model 6702 measured by Weaver et al. (Ref.5).
1.05 Ref. 72 quoting Ref. 65 (cf.1.043and 1.041 above), 1.036 Ref. 73 citing Ref. 65 (cf 1.043 above).d1.038 (Ref.66),1.041 (Ref.65)explicitly for 6711.eWallace reported a value of 0.995 74 for the phantom correction for PW2030, and this was used in a series of related papers.In a private communication, it became clear that what they correctly used was, in our notation, P phant = (1/0.995)(1/0.996)=1.009,where the 0.996 factor is the required F med factor to convert from PW2030 to water.T IV.Dose rate constant values (Λ± s a References are for seed descriptions.bValue measured back in the 1980s or early 1990s.cTLD rods used in measurements.The others used TLD microcubes (1 × 1 × 1 mm 3 ).d60 Co used at high energy.The others used 6 MV. e The OptiSeed result is presented to show the discrepancy in P phant values, which is the result of the original paper using [in notation of Eq. (15)] P phant = D w w /D med TLD rather than the needed D w w /D med w (Ref.105), and this throws in doubt their measured DRC which is excluded from further analysis.114301-9 a Divided by geometry correction as shown, if reported, as per Eq.(18).b Value deduced from E(1 cm, 90) water PMMA .c 103 VI.Dose rate constant values of103Pd seed models calculated using Monte Carlo simulation.MC calculations from Refs.40and 54 have statistical component of uncertainties of 0.3% or less.The TLD measurements column shows Λ nok bq ± s ′′ m which is the modified measured DRC value with the original P phant and S rel AD values (columns 3 and 5 in Tables IV and V) replaced by the P new phant and ( f rel ) −1 values calculated in this work (columns 6 and 7 in Tables IV and V), and the s ′′ m is the adjusted percentage uncertainty s ′′ m = nok bq deliberately ignores the intrinsic energy dependence, k rel bq .The last column represents the % difference between the Λ MC (column RR) and Λ nok bq .Values differ from Ref. 40 based on further seed model changes identified by Ref. 106.
a Seed models changed.See text.b c Dose rate constants calculated in this work.
Table VIII presents a comparison of our results for the THINSeed 9011 and GE 6711 to those of Kennedy et al. 27 who also properly took into account k rel bq .Our calculated 114301-11 F. 1.Comparison between 25 measured DRCs for 125 I seed models using TLD and the respective Monte Carlo calculated value.(k rel bq ) −1 , consequently, has a value of 1.074 ± 0.015.
103 for six measured DRCs for103Pd seed models.(k rel bq ) −1 is 1.084 ± 0.026.F. 3. Comparison of the inverse of the intrinsic energy dependence relative to high-energy beams, (k rel bq ) −1 , deduced here and as measured by other authors.The values found in this study agree reasonably with those values measured for x-ray beams by Davis et al. (Ref.9), Nunn et al. (Ref.10), and Tedgren et al. (Ref.11).Das et al. (Ref.18) reported an average value of 1.00, but the uncertainty in measurements is large compared to the others.
Pd seeds, respectively.Our values are in agreement with values measured by other authors for x-ray beams with similar mean 114301-12 T VII.Revised measured DRC values of 125 I and 103 Pd seed models.Λ MC is the Monte Carlo calculated dose rate constant (column RR, Table VI).Λ rev meas is the revised measured DRC determined by replacing original S rel AD by ( f rel • k rel bq ) −1 , found in this work (Tables IV and V and Figs. 1 and 2).Percentage uncertainty, s rev m , is determined as s rev m = where s k bq is the percentage uncertainty of the relative intrinsic energy dependence, k rel bq , found in this work.Λ rev avg is the average of Λ MC and Λ rev meas values, and Λ CON is the DRC consensus value reported in TG-43U1 and TG-43U1S1.Last column shows difference between Monte Carlo and revised measured average values and the consensus values of DRCs of the 125 I and 103 Pd brachytherapy seed models currently in the market.
, a Uncertainty of Λ MC is 1.5%.