A simplified analytical approach to estimate the parameters required for strength determination of HDR 192Ir brachytherapy sources using a Farmer-type ionization chamber

Applied Radiation and Isotopes 70 (2012) 282–289

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A simplified analytical approach to estimate the parameters required for strength determination of HDR 192Ir brachytherapy sources using a Farmer-type ionization chamber Sudhir Kumar a, P. Srinivasan b, S.D. Sharma a,n, Y.S. Mayya a a b

Radiological Physics and Advisory Division, Bhabha Atomic Research Centre, CTCRS, Anushaktinagar, Mumbai 400094, India Radiation Safety Systems Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 March 2011 Received in revised form 29 June 2011 Accepted 6 August 2011 Available online 18 August 2011

Measuring the strength of high dose rate (HDR) 192Ir brachytherapy sources on receipt from the vendor is an important component of a quality assurance program. Owing to their ready availability in radiotherapy departments, the Farmer-type ionization chambers are also used to determine the strength of HDR 192Ir brachytherapy sources. The use of a Farmer-type ionization chamber requires the estimation of the scatter correction factor along with positioning error (c) and the constant of proportionality (f) to determine the strength of HDR 192Ir brachytherapy sources. A simplified approach based on a least squares method was developed for estimating the values of f and Ms. The seven distance method was followed to record the ionization chamber readings for parameterization of f and Ms. Analytically calculated values of Ms were used to determine the room scatter correction factor (Ksc). The Monte Carlo simulations were also carried out to calculate f and Ksc to verify the magnitude of the parameters determined by the proposed analytical approach. The value of f determined using the simplified analytical approach was found to be in excellent agreement with the Monte Carlo simulated value (within 0.7%). Analytically derived values of Ksc were also found to be in good agreement with the Monte Carlo calculated values (within 1.47%). Being far simpler than the presently available methods of evaluating f, the proposed analytical approach can be adopted for routine use by clinical medical physicists to estimate f by hand calculations. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Brachytherapy Analytical approach Farmer-type ionization chamber HDR 192Ir source Monte Carlo

1. Introduction Measuring the strength of HDR brachytherapy sources on receipt from the vendor is an important component of quality assurance (QA) programs (Kubo et al., 1998; Nath et al., 1997). Reference air-kerma rate (RAKR) or air-kerma strength (AKS) are the recommended quantities to specify the strength of gamma emitting brachytherapy sources. RAKR is the air-kerma rate to air, in air, at a reference distance of one meter, corrected for attenuation and scattering and refers to the quantity determined along the transverse bisector of the source. AKS is the air-kerma rate in air at a given distance corrected for attenuation and scattering and multiplied by the square of the given distance (Kubo et al., 1998; Nath et al., 1997; ICRU, 1997, 1985, 1980; BCRU, 1984). Recommended experimental methods for source strength measurement at the hospital (Stump et al., 2002; IAEA, 2002, 1999; Goetsch et al., 1991) are (i) cylindrical ionization

n

Corresponding author. Tel.: þ91 22 25598713; fax: þ91 22 25519209. E-mail address: [email protected] (S.D. Sharma).

0969-8043/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2011.08.009

chamber (typical volume 0.6 cm3) method, and (ii) well-type ionization chamber method. International Atomic Energy Agency (IAEA) recommends the use of a cylindrical ionization chamber (0.6 cm3) in addition to a suitable well-type chamber for the RAKR (or AKS) measurement of HDR sources (IAEA, 2002, 1999). European Society of Therapeutic Radiology and Oncology (ESTRO) recommends the use of a Farmer-type cylindrical ionization chamber for calibration of HDR 192Ir brachytherapy sources at hospitals (ESTRO, 2004). Considering the convenience of use and reproducibility of measurement results, well-type ionization chambers are preferred over cylindrical chambers for strength determination of HDR 192Ir brachytherapy sources. However, the Farmer-type cylindrical ionization chamber is also used for RAKR or AKS measurement of HDR 192Ir brachytherapy sources (Soares et al., 2009; Douysset et al., 2008, 2005; Patel et al., 2006). This is due to the fact that cylindrical ionization chambers are readily available in hospitals and in case of non-availability of a well-type ionization chamber; the use of a cylindrical ionization chamber is an obvious choice. Further, the Farmer-type ionization chamber is used at standards laboratories for measuring the RAKR or AKS of HDR 192Ir sources, which in turn is used for calibrating the well-

S. Kumar et al. / Applied Radiation and Isotopes 70 (2012) 282–289

type ionization chamber of the hospitals. The use of a Farmertype cylindrical ionization chamber requires the estimation of the scatter correction factor along with other parameters to determine the strength of HDR 192Ir brachytherapy sources. A number of experimental and theoretical methods have been described in the literature to determine these parameters (Soares et al., 2009; Douysset et al., 2008, 2005; Chang et al., 2007; Patel et al., 2006; Stump et al., 2002; IAEA, 2002, 1999; Selvam et al., 2001; Goetsch et al., 1991). The Monte Carlo method (Selvam et al., 2001) is generally complex, requiring special expertise in the use of elaborate software and cross section data, demanding long run time and hence cannot be used routinely. Chang et al. (2007) and Patel et al. (2006) presented empirical formulae to determine the room scatter correction factor only. Other analytical methods (Soares et al., 2009; Douysset et al., 2008, 2005; Stump et al., 2002; Goetsch et al., 1991) presented in the literature require lengthy calculations and cumbersome mathematical manipulations (e.g. the calculations of 35 different f values). As of yet, there is no simple analytical formula available for estimating f and Ms by simple hand calculation. To fulfill this requirement of clinical medical physicists, simple formulae in a compact form for calculating f and Ms using a simplified analytical approach were developed and presented in this paper. The approach for estimating f and Ms is obtained by re-working the dose distance relationship and choosing appropriate variables for regression analysis. The value of c can be calculated using the values of f and Ms determined by the proposed analytical approach. Analytically calculated values of Ms were used to determine the room scatter correction factor (Ksc). The Monte Carlo calculations were also carried out to calculate f and Ksc to verify the magnitude of the parameters determined by the proposed simplified analytical approach.

2. Materials and methods 2.1. Ionization chamber and HDR brachytherapy source A Farmer-type 0.6 cm3 ionization chamber (Model 30001, PTW, Freiburg, Germany) was used in this work. The source used for the calculations and measurements was an old design microSelectron HDR 192Ir brachytherapy source (Nucletron B.V., Veenendaal, The Netherlands). The internal construction and dimensions of the ionization chamber and the microSelectron HDR 192Ir source used for measurements and modeling by the Monte Carlo method have been taken from published literature (Pena et al., 2006; Williamson and Li, 1995). 2.2. Experimental procedure To experimentally determine the RAKR of an HDR 192Ir source using a Farmer-type cylindrical ionization chamber, it is necessary to estimate (i) the contribution of scatter radiation (Ms) from the floor, walls, ceiling, and other materials in the treatment room; (ii) positioning error of the ionization chamber with respect to the source which is commonly called distance error (c); and (iii) a constant of proportionality f. The seven distance method (Goetsch et al., 1991) was followed in this work to estimate these parameters. For this purpose, a specially designed perspex (PMMA) jig was fabricated with provisions for holding an HDR 192Ir source and a 0.6 cm3 ionization chamber. As shown in Fig. 1, the jig consists of a rectangular PMMA bar (length 130 cm, width 12 cm, thickness 3.5 cm) placed horizontally over two cylindrical PMMA rods (each of diameter 6 cm, length 150 cm and a metallic base plate) at a height of 120 cm from the floor. The rectangular bar contains eight holes (one for mounting the source

192

Ir source

283

Ionization chamber

10 cm 15 cm 20 cm 25 cm 30 cm

120 cm

35 cm 40 cm

Fig. 1. Schematic cross-sectional view of the experimental arrangement used to measure the signal of the ionization chamber at seven different distances.

and seven for placing the ionization chamber). The centers of the holes for placing the ionization chamber are located at distances of 10 cm, 15 cm, 20 cm, 25 cm, 30 cm, 35 cm, and 40 cm from the center of the source hole. The separation between the center of the source and measuring point is known to within 70.1 cm. During the measurement, the source was fixed in its hole and the chamber, fitted with its 60Co build-up cap, was positioned at 10 cm distance from the source. The source center was aligned with the chamber center mechanically by adjusting the height of the chamber as well as dosimetrically by recording the electrometer (Unidos, PTW Freiburg, Germany) reading and noting the position of maximum response. In this experiment, the 0.6 cm3 chamber was irradiated continuously while electrometer readings were recorded at one minute intervals. The chamber was placed in other holes and the experiment was repeated in a similar manner. The average reading at each distance was corrected for variation in environmental conditions (as air-kerma calibration coefficient of the ionization chamber was known at 20 1C temperature and 101.325 kPa pressure), ion-recombination and air attenuation (IAEA, 2002, 1999). A non-uniformity correction was also applied (Bielajew, 1990a, 1990b; Kondo and Randolph, 1960). The air-kerma calibration coefficient of this ionization chamber for an HDR 192Ir brachytherapy source was approximated by averaging the reciprocal of its air-kerma calibration coefficients at 250 kV [HVL(Cu)¼2.47 mm] x-rays and 60Co gamma rays [i.e. (1/NK,Ir-192)¼1/2{(1/NK,250 kV) þ(1/NK,Co-60)}] as suggested in the literature (Mainegra-Hing and Rogers 2006). Here NK,Ir-192, NK,250 kV, and NK,Co-60 are the air-kerma calibration coefficients of the 0.6 cm3 ionization chamber at photon energies corresponding to HDR 192Ir gamma rays, 250 kV x-rays, and 60Co gamma rays, respectively. The 0.6 cm3 chamber with its 60Co build-up cap was irradiated while determining its air-kerma calibration coefficients at 250 kV x-ray and 60Co gamma ray energies.

2.3. Seven distance method In this method, the output of the source in air is measured at seven different distances (Stump et al., 2002; IAEA, 2002, 1999; Goetsch et al., 1991) and the corresponding meter reading (Md), which is the sum of contributions from primary and scattered

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f 1=2 ¼ a

radiation, can be written as Md ¼ Mp þ Ms

ð1Þ

where Mp is the meter reading due to primary radiation and Ms is the meter reading due to scattered radiation, which is assumed to be independent of distance. As the primary radiation follows the inverse square law with respect to distance, we have Mp ¼ ðMdi Ms Þ ¼

f ðdi þ cÞ

, where i ¼ 0,. . .,6 2

ð2Þ

where di is the apparent distance between source and chamber centers, c is the offset error in the distance measurement, and f is defined as the constant of proportionality, which is independent of distance (Goetsch et al., 1991). The problem has three unknowns, namely f, Ms, and c. In order to determine them from the measured data, we proceed as follows. If Mi is the meter reading at distance di from the source, Eq. (2) may be written as ðMi Ms Þ ¼

The above relationship may be re-expressed as  1=2   f Ms 1=2 1 ) ðdi þ cÞ ¼ Mi Mi

ð3Þ

Here we treat the parameters related to Mi as the independent variable and the distances di as the dependent variable. Eq. (3) is a nonlinear relationship which cannot be easily regressed for practical routine use. However, we may note that invariably in all practical situations, the scatter radiation Ms is only a small percentage of the meter reading (Mi) (i.e. Ms 5Mi) (Verhaegen et al., 1992; Stump et al., 2002; Douysset et al., 2005; Patel et al., 2006; Soares et al., 2009). This fact permits us to develop a linear approximation by expanding the fractional power-term in terms of a Taylor series to a linear order (  !) 1=2 f Ms f 1=2 ðdi þ cÞ ¼ þ , i ¼ 0,1,. . .,6 ð4Þ 3=2 Mi 2Mi The contribution of the neglected terms will be less than 10  4. There are 3 (three) unknowns to be determined by regressing Eq. (4) over 7 data points. The solution will be simpler as well as more robust if one can reduce the number of unknowns to two. This can be achieved by making use of the first distance itself to eliminate the unknown, c i.e. by noting that (  !) 1=2 f Ms f 1=2 ðd0 þ cÞ ¼ þ ð5Þ 3=2 M0 2M 0

Upon subtracting Eq. (5) from Eq. (4), we obtain the following equation for the relative distances (li ¼di  d0) between the successive points (i¼ 1,y,6) ( ! !) ( ! !) 1 1 Ms f 1=2 1 1 li ¼ ðdi d0 Þ ¼ f 1=2  þ  1=2 1=2 3=2 3=2 2 M M M M 0

i

Ms f 1=2 ¼b 2

then Eq. (6) can be re-written as the following linear equation: li ¼ axi þ byi ,

0

Eq. (6) has only 2 (two) unknowns, which have to be determined by regressing over 6 (six) points (instead of 7 points). Introducing the notations ! ! 1 1 ð7Þ  ¼ xi 1=2 1=2 Mi M0

3=2

Mi

! 

1 3=2

M0

! ¼ yi

i ¼ 1,2,. . .,6

ð11Þ

Given the set of values (li, xi, yi) from each of 6 (six) experimental data sets, we may determine the coefficients a and b (hence f and Ms) by bi-variate linear regression analysis. We adopted the standard least squares method for this purpose. The least squares method is the preferred choice whenever the number of data points exceeds the number of unknowns and the data points are subject to experimental errors. Further, the method of least squares takes into account the entire range of experimental data. It gives the best fit to the experimental results by minimizing the sum of the squares of the errors. At the same time the statistics of goodness of fit can also be evaluated.

L¼

6 X

ð8Þ

½li ðaxi þ byi Þ2

ð12Þ

i¼1

where L is the sum of the squares of the errors. To minimize L, the partial derivatives with respect to a and b (i.e. @L=@a and @L=@b) were set equal to zero. The resulting equations in terms of a and b are: 6 6 6 X X X @L ¼0 ) li xi ¼ a x2i þb xi yi @a i¼1 i¼1 i¼1

ð13Þ

6 6 6 X X X @L ¼0 ) li yi ¼ a xi yi þ b y2i @b i¼1 i¼1 i¼1

ð14Þ

Eqs. (13) and (14) are simultaneous equations, which can be easily solved for a and b. Using these values of a and b in Eqs. (9) and (10), respectively, we obtain the values of f and Ms as given below 8  P 92 P6 P6 P6 6 > > 2 < = i ¼ 1 li xi i ¼ 1 yi  i ¼ 1 li yi i ¼ 1 xi yi ð15Þ f¼     2 P6 P P6 > > : ; x2 6 y2  xy i¼1

i

i¼1

i

i¼1

i i

and n P  P o P6 P6 6 6 2 2 i ¼ 1 xi i ¼ 1 li yi  i ¼ 1 xi yi i ¼ 1 li xi  P  Ms ¼ P P6 P6 6 6 2 i ¼ 1 li xi i ¼ 1 yi  i ¼ 1 li yi i ¼ 1 xi yi

ð16Þ

If we assume the room scatter to be spatially homogeneous, Ms so obtained from Eq. (16) can be used to compute the value of the room scatter correction factor, Ksc, as follows:

ð6Þ

1

ð10Þ

2.3.1. Least squares method The least squares technique (Sastry, 1995) was used to determine a and b. This involves minimizing the following expression:

f ðdi þ cÞ2

i

ð9Þ

Ksc ¼

Md Ms Md

ð17Þ

The values of f and Ms thus obtained using Eqs. (15) and (16), respectively, were used to determine the value of c from Eq. (2). Having determined the value of f using Eq. (15), the air-kerma rate (AKR) (Gys  1) can be calculated using the formula (Stump et al., 2002) AKR ¼

NK f ðd þ cÞ2 Dt

ð18Þ

S. Kumar et al. / Applied Radiation and Isotopes 70 (2012) 282–289

where NK is the interpolated air-kerma calibration coefficient of the chamber for an HDR 192Ir brachytherapy source and Dt is the time interval of the measurement. RAKR can then be determined using the following equation:  2 d þc RAKR ¼ AKR ð19Þ dref where dref is the reference distance (1 m) (Williamson and Nath, 1991).

285

0.6 cm3 ionization chamber is shown in Fig. 2. Source self attenuation and the attenuation and scattering in the surrounding materials were considered in the Monte Carlo computations. The room was filled with air of density 1.205  10  3 g cm  3 (20 1C temperature and 101.325 kPa pressure; Rivard et al., 2004) and 50 cm thick concrete of density 2.35 g cm  3 was assumed along the floor, ceiling and side walls. The experimentally determined source strength (34,147 m Gym2 h  1; IAEA, 2002, 1999) of the HDR 192Ir source was considered in computing the starting source particle weight in the MCNP model.

2.4. Monte Carlo method The generalized approach developed in this paper to calculate f and Ms was verified using the Monte Carlo simulation studies. The Monte Carlo N-Particle transport code (MCNP) version 4C developed at the Los Alamos National Laboratory was used for the present computations. MCNP4C is a general purpose continuous energy, generalized geometry, time dependant code, which deals with transport of neutrons, photons, and coupled electron photon transport, i.e., transport of secondary electrons resulting from gamma interactions (MCNP, 2000). The photon physics option of the code utilized in this work includes coherent scattering, incoherent (Compton) scattering, the photoelectric effect (with K and L shell fluorescence) and pair production (Bohm et al., 2003). MCNP was run in the Mode: P with an energy cutoff value of 1 keV to terminate particle transport. 2.4.1. Geometry simulation in MCNP An MCNP model, including the geometrical representation of the locations of the PTW 0.6 cm3 Farmer-type cylindrical ionization chamber with the PMMA build-up cap and the HDR 192Ir source, was realized using the geometry input options available in the code. In the Monte Carlo simulation of the experimental conditions, the measurement setup was housed inside a concrete walled room to include the effect of Compton scattered photons. During the simulations, the HDR 192Ir source was located 100 cm above the ground at a distance of 100 cm from the left wall of a brachytherapy treatment room of dimensions 3.0 m  3.0 m  3.0 m in order to represent the actual experimental setup used during measurements, and is shown in Fig. 2. Keeping the source at a fixed position, the position of the chamber was varied along the axis perpendicular to the long axis of the source. The center to center distance between the chamber and source was varied from 10 cm to 40 cm in steps of 5 cm. The concrete walled room housing the measurement setup including the source and the

Fig. 2. MCNP Plot of the cross-sectional view of the problem geometry showing the locations of the 192Ir source and the ionization chamber (all dimensions are in cm).

2.4.2. Source energy distribution The gamma ray photon energies and corresponding emission probabilities from the 192Ir radionuclide were taken from the literature for simulation of the HDR brachytherapy source. The decay photon energy spectrum of 192Ir used as the input spectrum for the Monte Carlo simulations in this work consisted of 31 lines, with energies ranging from 8.91 keV to 1.601 MeV (Watanabe et al., 1998). The old design microSelectron HDR brachytherapy source was normalized to one Bq of 192Ir radionuclide. This was achieved by using the sum of emission probabilities of all photons per decay ( ¼2.363) of 192Ir as the initial source weight in the simulation. The term ‘‘source weight’’ denotes the total number of particles emitted from the source. This may be normalized with respect to unit decay, unit Becquerel, unit Curie, or any other parameter of interest. 2.4.3. Tally specification MCNP was utilized to predict the total fluence rate, scattered fluence rate, and the RAKR as measured by the chamber. The MCNP tally options utilized for this problem are photon fluence tally (F5) and photon energy deposition tally (F6), including the secondary electron transport. In the F5 tally, the radius of the sphere of exclusion of 1/8 to 1/2 mean free paths was maintained for particles of average energy at the sphere (Sureka et al., 2006). In MCNP, the average energy of the particle tracks in a given problem cell/region is printed in the standard output. Taking this as the average particle energy for those particle tracks reaching the detector region, the mean free path (for this average energy) was chosen for the sphere of exclusion around the F5 tally point. This would be more appropriate than using the source particle energies for the F5 tally points. The ring detector option of the F5 tally was employed at different distances of the detector with respect to the source in order to achieve reduced variance. The individual cell score contribution option of the F5 tally was utilized to arrive at the photon fluence contribution from each cell separately. This was done to enable computation of the scattered contributions due to surrounding materials and concrete structure housing. The values of scattered and primary photon fluence estimated by the F5 tally were used to compute the scatter correction factors, Ksc [Ksc ¼(Md–Ms)/Md]. An MCNP model consisting of the HDR 192Ir brachytherapy source, the ionization chamber and the geometry of the room (used for the experiment) was benchmarked with the published results (Selvam et al., 2001). Thereafter, the model was used to verify the scatter correction factor estimated using the analytical approach presented in this paper. For the purpose of benchmarking the Ksc values with the published values, the source to detector distances chosen for the F5 tally were 13 cm, 16 cm, 19 cm, 22 cm, and 25 cm whereas for the sake of comparison with the experimentally determined scatter correction factors, the source to detector distances chosen were 10 cm, 15 cm, 20 cm, 25 cm, 30 cm, 35 cm, and 40 cm. The energy deposition rates were estimated by using the F6 tally of MCNP. The energy deposition rate thus obtained from the

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F6 tally in units of MeV g  1 s  1 from the MCNP code was converted to pCs  1 using the appropriate conversion factor as the tally multiplier. The F6 tally was scored in the active volume of the ionization chamber (0.6 cm3). The F6 tally estimates the mean energy deposited per unit mass for one starting particle. The energy deposition rate for the given source strength was computed by multiplying the F6 tally result by the source particle emission rate (photons s  1). The following equation was used to convert the photon energy deposition rate per unit mass (MeV g  1 s  1) to meter reading (pCs  1) Meter readingðpCs1 Þ ¼

A1 ðMeVg1 s1 ÞBðJ MeV1 ÞMðgÞDðpC C1 Þ C1 ðJ C1 Þ

F6 tally estimates obtained from the tally statistics and the uncertainty in the input parameters used in the simulations, including the uncertainty in the conversion factors. The uncertainty in the F6 tally estimate is 0.01% (k¼1). The uncertainty in the input parameters include the density of dry air (0.01%, k¼1; Sander and Nutbrowm, 2006), sensitive volume of the ionization chamber (0.01%, k¼ 1), the compiled photon cross-sections (0.2%, k¼1), geometry modeling (0.2%, k¼1), source energy spectrum (0.5%, k¼1; Medich and Munro, 2007) and the conversion factor J C  1(0.05%, k¼1; Sander and Nutbrowm, 2006). The uc in the evaluated value of f is obtained by summing these individual uncertainties in quadrature and is worked out to be 0.57% (k¼1), with a U of 1.14% (k¼2).

ð22Þ where A1 is the energy deposition rate per unit mass in the sensitive volume of the chamber computed using the F6 tally of the MCNP code, B is the conversion coefficient having a numerical value equal to 1.602  10  13 J MeV  1, D ¼1012 (pC C  1), C1 ¼ 33.97 (J C  1), and M is the mass of air in the sensitive volume of the chamber in units of grams (M¼7.23  10  4 g at 20 1C temperature and 101.325 kPa pressure). The meter reading due to the primary radiation (Mp) was determined from the F6 tally using Eq. (22). The value of Mp thus determined was used in Eq. (2) to arrive at the value of f, assuming c to be equal to zero since the distance error does not have any significance in the Monte Carlo simulations. 2.5. Uncertainty analysis The uncertainty analysis for the experimental determination of RAKR of an HDR 192Ir source was carried out to determine the combined standard uncertainty (uc) and expanded uncertainty (U) following the methodologies outlined in IAEA TECDOC 1585 (IAEA, 2008). Accordingly, the Type A uncertainties were statistically evaluated whereas the Type B uncertainties were recorded from the specifications of the dosimeter, fabrication details of the PMMA jig and published data. The uncertainty associated with the measurement of reference air-kerma rate of an HDR 192Ir brachytherapy source includes uncertainty in deriving the calibration coefficient of the ionization chamber by the interpolation method, uncertainty in the experimental measurement, and uncertainty in the analytical calculation of f. The uncertainty in the calibration coefficient of the ionization chamber at 192Ir gamma ray energies derived by the interpolation method is 1.3% (k¼1), which includes the uncertainties in calibration coefficients of the chamber at 60Co gamma ray and 250 kV x-ray beams. Total uncertainty in the experimental measurement is 0.79% (k¼1), which includes charge collection reproducibility (0.04%, k¼1), electrometer calibration coefficient (0.2%, k¼1), charge leakage (0.01%, k¼1), air density correction (0.1%, k¼1), timer reproducibility (0.01%, k ¼1), ion-recombination correction (0.04%, k¼1), non-uniformity correction (0.1%, k¼1), air attenuation correction (0.07%, k¼1), positional accuracy (0.25%, k¼ 1), and uncertainty in the calculation of f by the analytical method (0.7%, k¼1). The uc in the measurement of RAKR is 1.52% at k¼1. The value of U in RAKR determination of HDR 192Ir is 3.04% (k¼2). This value of U corresponds to a 95% confidence level, calculated from the effective number of degrees of freedom, neff ¼18. For the F5 and F6 tallies of the MCNP code, the relative standard error of the mean value is less than 0.01% for 2  108 particle histories considered in the simulations. This error value includes the uncertainty due to the inherent statistical fluctuations in the obtained quantities from the MCNP runs. The uncertainty associated with the meter reading value (pCs  1), simulated using the MCNP code, includes the uncertainty in the

3. Results and discussion 3.1. Analytical approach The values of f, Ms, and c calculated from the above approach are 1972 pCs  1 cm2, 0.013 pCs  1, and 0.13 cm, respectively. Table 1 presents a comparison of experimentally recorded values of li with those obtained by regression analysis. The maximum difference between the two values is 0.4%, while the sum of error and variance (Appendix A) are 0.39% and 0.096%, respectively. The values of the regression coefficients a and b are 44.40 and 0.277, respectively, and the estimates of variance in a and b are 0.013 and 0.017, respectively (Appendix B). The values of a and f with their confidence intervals are 44.4070.31 and 1972 727, respectively, at a 95% confidence level (Appendix C). The coefficient of determination (R2), which is the ratio of explained to the total variation is equal to 0.9999 (Appendix D), which is very close to unity, indicating the goodness of fit. The analytical method presented here for calculation of f and Ms is simple to implement in practice, as calculation of these parameters can be done using a pocket calculator without requiring a computer program. 3.2. Monte Carlo method The values of Ksc computed using the simulated photon fluence (F5 tally) were compared with values reported in the literature. This exercise was carried out to obtain confidence in our simulation approach. The values of Ksc calculated by the MCNP model adopted for this study showed good agreement (within 1.65%) with the corresponding published values, which are presented in Table 2. This difference in Ksc values can be attributed to the fact that MCNP version 3.1 (MCNP, 1983) was used by Selvam et al. (2001), whereas MCNP version 4C (MCNP, 2000) was used in the present study. It may be noted that all of the simulation parameters such as the source location, source to detector distances, the material densities, the room dimensions, and the concrete wall thickness are the same as those used by Selvam Table 1 Comparison of experimentally recorded and analytically calculated values of li [ ¼ (di  d0)] using the proposed simplified approach. li ¼(di–d0) (cm) Experimental

Analytical

% Deviation

5 10 15 20 25 30

4.999 9.961 15.009 20.041 24.977 30.001

0.014 0.396  0.061  0.204 0.094  0.003

% Sum of error

% Variance

0.39

0.096

S. Kumar et al. / Applied Radiation and Isotopes 70 (2012) 282–289

Table 2 Comparison of room scatter correction factors calculated by the Monte Carlo simulation in this work with those published by Selvam et al. (2001) (room size: (3  3  3) m3). Source to chamber distance, d (cm)

13 16 19 22 25

Room scatter correction factor, Ksc This work (Monte Carlo)

Selvam et al. (2001)

% Difference

0.9789 0.9776 0.9757 0.9735 0.9719

0.995 0.994 0.992 0.989 0.988

1.62 1.65 1.64 1.57 1.63

Table 3 Comparison of room scatter correction factors calculated using the Monte Carlo method and the proposed analytical approach (room size: (3  3  3) m3). Source to chamber distance, d (cm)

10 15 20 25 30 35 40

Room scatter correction factor, Ksc Analytical

Monte Carlo

% Difference

0.9994 0.9986 0.9975 0.9960 0.9943 0.9923 0.9899

0.9849 0.9852 0.9836 0.9826 0.9748 0.9777 0.9758

1.45 1.34 1.39 1.35 1.46 1.47 1.42

et al. (2001). The later version of MCNP used in this study is based upon updated photon interaction cross section data and electron transport methods in contrast to the older version of the MCNP code used in the earlier study (Selvam et al., 2001). The MCNP computed values of Ksc are presented in Table 3. It is observed here that the value of Ksc decreases with increasing source to chamber distance. Further, the scattered component of the meter reading (Ms) is not constant but decreases with increasing distance as discussed earlier (Selvam et al., 2001). The values of f simulated using MCNP were found to be independent of the distance between the source and the chamber, which is in conformity with the observation of Goetsch et al. (1991) and is equal to (1958722) pCs  1 cm2. 3.3. Validation of the analytical approach Table 3 presents the Ksc values estimated using the analytical approach and those computed by MCNP at different distances. The maximum difference between the analytically and MCNP calculated values of Ksc was found to be 1.47%, which in turn validates the analytically calculated values of Ksc. The difference between f values calculated by the simplified analytical approach (1972727) and that obtained by the Monte Carlo method (1958722) is 0.7%. This excellent agreement between the f values validates the suitability of the proposed analytical approach.

4. Conclusions A simplified analytical approach based on the least squares method was proposed to evaluate f and Ms, required for estimating the strength of HDR 192Ir brachytherapy sources using a Farmer-type cylindrical ionization chamber. The most crucial parameter f, calculated using the analytical approach proposed in this paper, was verified using the Monte Carlo simulation. The

287

values obtained from both techniques were found to be in good agreement with each other. It is notable that this approach is very compact and user friendly, dispensing with the use of a computing aid to determine the values of f, Ms, and c. Being far simpler than the presently available methods of evaluating f, the proposed analytical approach can be adopted for routine use by clinical medical physicists to estimate f.

Acknowledgments The authors wish to express their gratitude to Shri S.K. Ghosh, Director, Health Safety and Environment Group (HSEG), Dr. D.N. Sharma, Associate Director, HSEG & Head, Radiation Safety Systems Division, Dr. J.S. Bisht, Ex-Scientist, Radiological Physics & Advisory Division (RPAD), and Dr. G. Chourasiya, Head, Medical Physics & Training Section, RPAD, Bhabha Atomic Research Centre, Mumbai, for their encouragement and support during this work. The first author (S. K) is indebted to Dr. J. S. Bisht for his stimulating interest during the development of this work and for his suggestions and contributions in continued fruitful discussion.

Appendix A. Estimation of sum of error and variance of l Let us define a matrix X: 0 1 0:1124 0:0276 B 0:2238 0:0805 C B C B C B 0:3370 0:1687 C B C X¼B C B 0:4495 0:2994 C B C @ 0:5595 0:4769 A 0:6712

0:7157

where the first column consists of xis and the second column of yis, which are defined in Eqs. (7) and (8), respectively.   1:1419 0:9597 X0 X ¼ ðA1Þ 0:9597 0:8650 where X0 refers to the transpose of matrix X.   12:9838 14:4063 ðX 0 XÞ1 ¼ 14:4063 17:1408 0

5:0

ðA2Þ

1

B 10:0 C B C B C B 15:0 C B C Y ¼B C B 20:0 C B C @ 25:0 A 30:0 where the elements of matrix Y are di  d0 for ^l ¼ 1–6.   50:9685 X0 Y ¼ 42:8550

ðA3Þ

The regression coefficients (a and b) can be calculated by defining a matrix called B such that B ¼ ðX 0 XÞ1 X 0 Y

ðA4Þ

Using Eqs. (A2) and (A3) in Eq. (A4) we get     44:40 a B¼ , where B ¼ 0:277 b

ðA5Þ

The sum of squares of the error in matrix form is given by SSE ¼ Y 0 YB0 X 0 Y 0

ðA6Þ 0

where Y is the transpose of matrix Y and B is the transpose of matrix B.

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P SSE corresponds to i ðli ^li Þ2 , where ^li is the expected value of li from regression Eq. (11). By substituting the matrices Y0 , Y, B0 , and X0 Y (Eq. (A3)) we get SSE ¼ 0:0039

ðA7Þ

Variance is defined using the following equation: SSE Varianceðs Þ ¼ Number of degrees of freedom It may be noted that the number of degrees of freedom is equal to (n  k), where n is the number of modified data points and k is the number of estimated regression coefficients (a,b). 0:0039 ¼ 9:63  104 Varianceðs Þ ¼ 4

The sum of the squares due to the regression plane (explained variation) in matrix form is given by SSR ¼ B0 X 0 XBnl

2

2

Appendix D. Estimation of coefficient of determination

2

ðD1Þ

P SSR corresponds to i ð^li lÞ2 , where li is the mean of observed li values (Eq. (11)). Now, B0 X 0 XB ¼ ð 44:40

0:277 Þ



50:97



42:86

¼ 2:275  103

ðA8Þ 2

SSR ¼ B0 X 0 XBnl ¼ 2:275  103 6  309:76 ¼ 416:43 Appendix B. Calculation of regression coefficients (a and b) and estimate of variances

Now the total sum of the squares (SST) of observed points from the mean of li i.e., the total variation is given as

From Eq. (A5), we get the values of a and b i.e.

SST ¼ SSR þSSE

a ¼ 44:40 and b ¼ 0:277

ðD3Þ

where SST corresponds to P i

ðli lÞ2 ¼

X

i

fExplained variation ðSSR Þg

k

P

i ðli lÞ

2

. Eq. (D3) can be written as

X ðli ^l Þ2

ð^li lÞ2 þ

i

fTotal variation ðSST Þg

i

fUnexplained variation ðSSE Þg

k

k

ðTotal sum of squaresÞ ðSum of squares due to regressionÞ ðSum of squares due to errorsÞ

An estimator of the variance–covariance matrix is given by V ¼ CovðBÞ ¼ s2 ðX 0 XÞ1

ðD2Þ

ðB1Þ

Using Eqs. (A8) and (A2) in Eq. (B1), we get   0:0125 0:0139 V¼ 0:0139 0:0165 Let us rewrite the diagonal elements of the above matrix (V11 and V22) as V11 ¼0.013 and V22 ¼0.017. These are estimates of the variances of a and b, respectively.

Appendix C. Estimation of confidence interval in the value of regression coefficient ‘‘a’’ and constant of proportionality ‘‘f’’ ‘‘a’’ is the most important regression coefficient because it is used to compute f (Eq. (9)), which is in turn used to calculate the RAKR (Eqs. (18) and (19)). Hence we found the 95% confidence level for it. The 95% confidence interval for the regression coefficient ‘‘a’’ is given as a  ¼ a 8t , n SEa ðC1Þ 2 where n denotes the degrees of freedom in student’s t distribution, a is the level of significance using the two tailed test (Spiegel and Stephens, 2000), and SEa is the standard error in a (SEa ¼OV11 ¼O0.0125 ¼0.112) ¼ 44:40 8tð0:05,4Þ0:112 ¼ 44:40 80:31 By applying the law of error propagation (in Eq. (9)), the value of the constant of proportionality ‘‘f’’ at a 95% confidence interval is equal to 1972 727.

ðD4Þ

Using Eqs. (D2) and (A7) in Eq. (D3) we get SST ¼ 416:44 The coefficient of determination (R2), which is defined as the ratio of SSR to SSE, is R2 ¼

SSR 416:43 ¼ 0:9999 ¼ 416:44 SST

ðD5Þ

References BCRU, 1984. Specification of brachytherapy sources. Br. J. Radiol. 57, 941–942. Bielajew, F., 1990a. Correction factors for thick-walled ionization chambers in point-source photon beams. Phys. Med. Biol. 35, 501–516. Bielajew, F., 1990b. An analytical theory of the point-source non-uniformity correction factor for thick-walled ionization chambers in photon beams. Phys. Med. Biol. 35, 517–538. Bohm, T.D., DeLuca Jr, P.M., DeWerd, L.A., 2003. Brachytherapy dosimetry of 125I and 103Pd sources using an updated cross section library for the MCNP Monte Carlo transport code. Med. Phys. 30, 701–711. Chang, L., Ho, S.Y., Chui, C.S., Du, Y.C., Chen, T., 2007. Room scatter factor modeling and measurement error analysis of 192Ir HDR calibration by a Farmer-type chamber. Phys. Med. Biol. 52, 871–877. Douysset, G., Gouriou, J., Delaunay, F., DeWerd, L., Stump, K., Micka, J., 2005. Comparison of dosimetric standards of USA and France for HDR brachytherapy. Phys. Med. Biol. 50, 1961–1978. Douysset, G., Sander, T., Gouriou, J., Nutbrown, R., 2008. Comparison of air-kerma standards of LNE-LNHB and NPL for 192Ir HDR brachytherapy sources: EUROMET project no 814. Phys. Med. Biol. 53, N85–N97. ESTRO, 2004. A practical guide to quality control of brachytherapy equipment, ESTRO booklet no. 8. Belgium. Goetsch, S.J., Attix, F.H., Pearson, D.W., Thomadsen, B.R., 1991. Calibration of 192Ir high dose rate afterloading system. Med. Phys. 18, 462–467. IAEA, 1999. Calibration of Brachytherapy Sources: Guidelines of Standardized Procedures for the Calibration of Brachytherapy Sources at Secondary Standard Dosimetry Laboratories (SSDLs) and Hospitals, TECDOC-1079. Vienna, Austria. IAEA, 2002. Calibration of Photon and Beta Ray Sources used in Brachytherapy, TECDOC-1274. Vienna, Austria. IAEA, 2008. Measurement Uncertainty: A Practical Guide for Secondary Standards Dosimetry Laboratories, TECDOC-1585, Vienna, Austria. ICRU, 1980. Radiation quantities and units, ICRU Report No. 33, Washington, D.C. ICRU, 1985. Dose and Volume Specification for Reporting Intracavitary Therapy in Gynecology, ICRU Report No. 38, Washington, D.C.

S. Kumar et al. / Applied Radiation and Isotopes 70 (2012) 282–289

ICRU, 1997. Dose and Volume Specification for Reporting Interstitial Therapy, ICRU Report No. 58. Bethesda, M.D. Kondo, S., Randolph, M.L., 1960. Effect of finite size of ionization chambers on measurements of small photon sources. Radiat. Res. 13, 37–60. Kubo, H.D., Glasgow, G.P., Pethel, T.D., Thomadsen, B.R., Williamson, J.F., 1998. High dose rate brachytherapy treatment delivery: report of the AAPM Radiation Therapy Committee Task Group No. 59. Med. Phys. 25, 375–403. Mainegra-Hing, E., Rogers, D.W.O., 2006. On the accuracy of techniques for obtaining the calibration coefficient NK of 192Ir HDR brachytherapy sources. Med. Phys. 33, 3340–3347. MCNP, 1983. Monte Carlo Neutron and Photon Transport Code, version 3.1. Los Almos National Laboratory. MCNP, 2000. A General Monte Carlo N-Particle Transport Code, Version 4C, Los Alamos National Laboratory report LA-13709-M. Medich, D.C., Munro, J.J., 2007. Monte Carlo characterization of the M-19 high dose rate Iridium-192 brachytherapy source. Med. Phys. 34, 1999–2005. Nath, R., Anderson, L.L., Meli, J.A., Olch, A.J., Stitt, J.A., Williamson, J.F., 1997. Code of practice for brachytherapy physics: report of the AAPM Radiation Therapy Committee Task Group No. 56. Med. Phys. 24, 1557–1598. Patel, N.P., Majumdar, B., Vijayan, V., 2006. Study of scattered radiation for in-air calibration by a multiple distance method using ionization chambers and an HDR 192Ir brachytherapy source. Br. J. Radiol. 79, 347–352. Pena, J., Sanchez-Doblado, F., Capote, R., Terron, J.A., Gomez, F., 2006. Monte Carlo correction factors for a Farmer-type 0.6 cm3 ion chamber dose measurement in the build-up region of the 6 MV clinical beam. Phys. Med. Biol. 51, 1523–1532. Rivard, M.J., Coursey, B.M., DeWerd, L.A., Hanson, W.F., Huq, M.S., Ibbott, G.S., Mitch, M.G., Nath, R., Williamson, J.F., 2004. Update of AAPM Task Group No. 43 Report: a revised AAPM protocol for brachytherapy dose calculations. Med. Phys. 31, 633–674.

289

Sander, T., Nutbrowm, R.F., 2006. The NPL Air Kerma Primary Standard TH100C for High Dose Rate 192Ir Brachytherapy Source. NPL REPORT DQL-RD-004. National Physical Laboratory, United Kingdom. Sastry, S.S., 1995. Introductory Methods of Numerical Analysis, 2nd ed. PrenticeHall Press, New Delhi. Selvam, T.P., Rajan, K.N.G., Nagarajan, P.S., Sethulakshmi, P., Bhatt, B.C., 2001. Monte Carlo aided room scatter studies in the primary air-kerma strength standardization of remote afterloading 192Ir HDR source. Phys. Med. Biol. 46, 2299–2315. Soares, C.G., Douysset, G., Mitch, M.G., 2009. Primary standards and dosimetry protocols for brachytherapy sources. Metrologia 46, S80–S98. Spiegel, M.R., Stephens, L.J., 2000. Theory and Problems of Statistics, 3rd ed. Tata McGraw-Hill Publishing Company Ltd., New Delhi. Stump, K.E., DeWerd, L.A., Micka, J.A., Anderson, D.R., 2002. Calibration of new high dose rate 192Ir sources. Med. Phys. 29, 1483–1488. Sureka, S., Aruna, P., Ganesan, S., Sunny, S.C., Subbaiah, K.V., 2006. Computation of relative dose distribution and effective transmission around a shielded vaginal cylinder with HDR 192Ir source using MCNP4B. Med. Phys. 33, 1552–1561. Verhaegen, F., Van Diik, E., Thierens, H., Aalbers, A., Seuntjens, J., 1992. Calibration of low activity 192Ir brachytherapy sources in terms of reference air-kerma rate with large volume spherical ionization chambers. Phys. Med. Biol. 37, 2071–2082. Watanabe, Y., Roy, J., Harrington, P.J., Anderson, L.L., 1998. Experimental and Monte Carlo dosimetry of the Henschke applicator for high dose-rate 192Ir remote afterloading. Med. Phys. 25, 736–745. Williamson, J.F., Nath, R., 1991. Clinical implementation of AAPM Task Group 32 recommendations on brachytherapy source strength specification. Med. Phys. 18, 439–448. Williamson, J.F., Li, Z., 1995. Monte Carlo aided dosimetry of microSelectron pulsed and high dose rate 192Ir sources. Med. Phys. 22, 809–819.