Evaluation of Scatter Contribution and Distance Error by Iterative Methods for Strength Determination of HDR 192Ir Brachytherapy Source

Medical Dosimetry, Vol. 35, No. 3, pp. 230-237, 2010 Copyright © 2010 American Association of Medical Dosimetrists Printed in the USA. All rights reserved 0958-3947/10/$–see front matter

doi:10.1016/j.meddos.2009.06.008

EVALUATION OF SCATTER CONTRIBUTION AND DISTANCE ERROR BY ITERATIVE METHODS FOR STRENGTH DETERMINATION OF HDR 192 IR BRACHYTHERAPY SOURCE SUDHIR KUMAR, M.SC., PANCHAPAKESAN SRINIVASAN, M.SC., SUNIL D. SHARMA, PH.D., KAMATAM V. SUBBAIAH, PH.D., and YELIA S. MAYYA, PH.D. Radiological Physics and Advisory Division, Bhabha Atomic Research Centre; Radiation Safety Systems Division, Bhabha Atomic Research Centre, Mumbai, India; and Safety Research Institute, Atomic Energy Regulatory Board, Kalpakkam, India (Received 16 February 2009; accepted 18 June 2009)

Abstract—High-dose rate (HDR) 192Ir brachytherapy sources are commonly used for management of malignancies by brachytherapy applications. Measurement of source strength at the hospital is an important dosimetry requirement. The use of 0.6-cm3 cylindrical ionization chamber is one of the methods of measuring the source strength at the hospitals because this chamber is readily available for beam calibration and dosimetry. While using the cylindrical chamber for this purpose, it is also required to determine the positioning error of the ionization chamber, with respect to the source, commonly called a distance error (c). The contribution of scatter radiation (Ms) from floor, walls, ceiling, and other materials available in the treatment room also need to be determined accurately so that appropriate correction can be applied while calculating the source strength from the meter reading. Iterative methods of Newton-Raphson and least-squares were used in this work to determine scatter contribution in the experimentally observed meter reading (pC/s) of a cylindrical ionization chamber. Monte Carlo simulation was also used to cross verify the results of the least-squares method. The experimentally observed, least-squares calculated and Monte Carlo estimated values of meter readings from HDR 192Ir brachytherapy source were in good agreement. Considering procedural simplicity, the method of least-squares is recommended for use at the hospitals to estimate values of f (constant of proportionality), c, and Ms required to determine the strength of HDR 192Ir brachytherapy sources. © 2010 American Association of Medical Dosimetrists. Key words: Brachytherapy, Source strength, Iterative method, Scatter factor, Monte Carlo.

terms of reference air kerma rate (RAKR), whereas NIST (USA) recommends the quantity air kerma strength (AKS).1– 6 Recommended experimental methods for source strength measurement at the hospital5– 8 are: (i) cylindrical ionization chamber (typical volume 0.6 cm3) method; and (ii) well-type ionization chamber method. IAEA recommends the use of cylindrical ionization chamber (0.6 cm3) in addition to suitable well-type chamber for the RAKR measurement of HDR sources.5 While using the cylindrical chamber method, it is also required to determine the positioning error of the ionization chamber, with respect to the source, commonly called distance error. The contribution of scatter radiation from floor, walls, ceiling, and other materials available in the treatment room also need to be determined accurately so that appropriate correction can be applied while calculating the source strength from the measured data. Both experimental and theoretical methods have been described in literature to determine the distance error and scatter correction factor.5–10 The analytical methods suggested in the literature require the use of dedicated software such as MATLAB (The MathWorks, Natick, MA, USA) for evaluating these parameters. In this paper we describe simple iterative methods to determine distance error and scatter contribution, which can

INTRODUCTION 192

Ir sources are being used increasingly in high dose rate (HDR) brachytherapy treatment of malignant diseases. At present, there are more than 105 192Ir HDR brachytherapy machines in India and the number is increasing steadily. It is important to determine the absorbed dose accurately because the dose is the basis for reliable evaluations and comparisons of treatment techniques and radiation modalities. The overall uncertainty in dose delivered to the patient is attributed to uncertainties resulting from both physical and clinical parameters of treatment. A major contribution to the physical part of the uncertainty in brachytherapy stems from the source strength measurement to which the absorbed dose is directly proportional. The accuracy of dose delivery to the tumor directly depends on the accuracy of the source strength. It is therefore essential to determine accurately the strength of HDR 192Ir brachytherapy sources so that the prescribed dose can be delivered to the tumor within a stipulated limit. The BCRU (UK), ICRU, and IAEA have recommended specifying brachytherapy sources in Reprint requests to: Sunil D. Sharma, Ph.D., RP&AD, BARC, Department of Atomic Energy, CTCRS, Anushaktinagar, Mumbai, Maharashtra 400094, India. E-mail: [email protected] 230

Strength determination of HDR

be used by a hospital physicist/dosimetrist without the aid of dedicated mathematical software. Monte Carlo simulation was also carried out in this work. The results of iterative methods were compared with the results of Monte Carlo calculations to validate the iterative methods.

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231

Steel encapsulation (1.1 mm) diameter)

Active Ir-192 source core (0.6 mm diameter)

MATERIALS AND METHODS Ionization chamber The ionization chamber simulated and experimentally used in this work is a 0.6-cm3 Farmer chamber type 30001 (PTW Freiburg, Germany), which is a typical reference dosimetry ionization chamber.11 The chamber cavity is made up of polymethylmethacrylate (PMMA) ([C5H8O2]n) wall (thickness ⫽ 0.275 mm; physical density, ␳ ⫽ 1.19 g cm⫺3), the inner surface of which is covered by a graphite layer (thickness ⫽ 0.15 mm, ␳ ⫽ 0.82 g cm⫺3). The inner wall of this cavity serves as an outer electrode. The chamber also contains a 1-mmdiameter and 21.2-mm-long aluminium central electrode, which serves as an inner electrode. The chamber is also equipped with an acrylic (PMMA) 60Co build-up cap. The chamber is fully guarded up to the measuring volume and has guard ring potentials equal to collection electrode potential. The actual geometry of the simulated chamber is presented in Fig. 1. HDR brachytherapy source Calculation and measurements were performed using old design microSelectron HDR 192Ir brachytherapy

Steel cable

Fig. 2. Schematic cross-sectional view of HDR 192Ir brachytherapy source encapsulated in a stainless steel capsule.

source (Nucletron B.V., Veenendaal, The Netherlands). The internal construction and dimensions of the old HDR source used in the simulation was taken from the value reported in the literature.12,13 The old microSelectron HDR source is cylindrical in geometry, with a 0.6-mm diameter and 3.5-mm length. It is assumed that the radioactive 192Ir is uniformly distributed within this geometry of the source core. This source is encapsulated in an AISI 316L stainless steel capsule, with an outer diameter of 1.1 mm, and a spherical distal end, with a radius of curvature of 0.55 mm. The distance from the distal face of the active source core to the physical source tip is 0.35 mm. The cable of old source design was approximated by AISI 316L material. Schematic diagram of the source is shown in Fig. 2.

Build-up cap (PMMA) Graphite layer (0.15mm)

Central electrode (Al)

PMMA wall (C5H8O2) n Air gap Sensitive volume (0.6cc)

Fig. 1. Schematic drawing of 0.6 cm3 PTW 30001 cylindrical ionization chamber.

Experimental procedure A specially designed Perspex (PMMA) jig was fabricated with provisions of holding the HDR 192Ir brachytherapy source and the 0.6-cm3 ionization chamber in a perfectly aligned geometry. The jig has the provision to hold the chamber at 10, 15, 20, 25, 30, 35, and 40-cm distances from the source. The air kerma calibration factor of this ionization chamber for HDR 192Ir brachytherapy source was approximated by interpolating between 2 known air kerma calibration factors as suggested in the literature.7,14 During the measurement, the 60Co acrylic (PMMA) build-up cap was placed over the cavity of the chamber. The alignment of the center of the source and the center of the chamber was verified before recording the electrometer readings. The chamber was moved from 10 to 40 cm, whereas the source was stationary during the measurement. The average reading at each distance was corrected for variation in environmental conditions, ion-recombination, and air attenuation.5 Nonuniformity correction was also applied.15–17

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Seven distance method As stated earlier, it is required to know the distance error and room scatter contribution precisely to determine RAKR or AKS while using the cylindrical ionization chamber. The multiple-distance (typically 7 distances) measurement technique has been recommended to determine these 2 parameters.5– 8 The output of the source in air, Md, which is the sum of primary and scattered radiation, can be written as Md ⫽ M p ⫹ Ms Md ⫺ Ms ⫽

f (d ⫹ c)2

(1) (2)

where d is the apparent distance between source and chamber centers, c is the error in the distance measurement, Ms is the scattered dose contribution assumed to be independent of distance, and f is the constant of proportionality. We used 2 different iterative methods to estimate c, Ms and f. Iterative method 1. Before applying the Newton Raphson Method18 to determine the value of c, we eliminated f and Ms from the above equations as described below. Eq. 2 can be rewritten as: f ⫽ (M d ⫺ M s) * (d ⫹ c)2

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Ms ⫽

f ⫽ 共M d1 ⫺ M s兲 * (d1 ⫹ c)2

(4)

f ⫽ 共M d2 ⫺ M s兲 * (d2 ⫹ c)2

(5)

f ⫽ 共M d3 ⫺ M s兲 * (d3 ⫹ c)2

(6)

Subtracting Eq. 4 from Eq. 5, we get:

RAKR ⫽

⫽ M s * (d1 ⫺ d2) * (d1 ⫹ d2 ⫹ 2c) (7)

Subtracting Eq. 5 from Eq. 6, we get:

L⫽

L⫽

7

兺 [f ⫺ (Mi ⫽ Ms)(di ⫹ c)2]2 i⫽1

(12)

7

兺 关 f ⫺ 2 Midic ⫹ Msd2i ⫺ Mid2i ⫺ Mic2 i⫽1

⫹ 2dicM s ⫹ M sc2兴2 (13)

where L is the sum of the square of the errors. To minimize L, the partial derivatives with respect to c, f, and Ms (i.e. ⭸L ⭸c,⭸L ⭸f &⭸L ⭸M s ) were set equal to zero. The first iteration was carried out under the assumption that c and Ms in the nonlinear terms are zero. Resulting equations in terms of f, c, and Ms are: ⭸L ⭸f

⫽ M s * (d2 ⫺ d3) * (d2 ⫹ d3 ⫹ 2c) (8)

Eq. 7 can be re-arranged as follows:

(11)

1002

where Mi is the meter reading at distance di

M d2 * (d2 ⫹ c) ⫺ M d3 * (d3 ⫹ c)

共Md1 ⫺ Md2兲c2 ⫹ 2共M d1d1 ⫺ M d2d2兲c ⫹ 共M d1d21 ⫺ M d2d22兲 (d1 ⫺ d2)(d1 ⫹ d2 ⫹ 2c) ⫽ (d2 ⫺ d3)(d2 ⫹ d3 ⫹ 2c) 共M d2 ⫺ M d3兲c2 ⫹ 2共M d2d2 ⫺ M d3d3兲c ⫹ 共M d2d22 ⫺ M d3d23兲

Nk f

Iterative Method 2. In this method, least square technique18 was used to determine f, Ms, and c. This involves minimizing the following expression:

2

Dividing Eq. 7 by Eq. 8 and re-arranging the terms, we get the following cubic equation in c:

(10)

where Nk,RAKR is the air kerma calibration factor of the chamber for 192Ir brachytherapy source.

Ⲑ

M d1 * (d1 ⫹ c)2 ⫺ M d2 * (d2 ⫹ c)2

2

(d1 ⫺ d2)(d1 ⫹ d2 ⫹ 2c)

Equation 9 was solved using the Newton Raphson iterative technique. Three sets of data points were used to determine the value of c. The value of c so evaluated was used in Eq. 10 to find the value of Ms. The values of c and Ms thus obtained were used to determine the value of f by using either of Eqs. 4 – 6. Corresponding to each triplet (e.g., d1, d2, and d3; d1, d2, and d4; etc.), we get values of c, Ms, and f. The value of f averaged over 35 (7C3) datasets was then used to determine the RAKR (Gy/hr at 1 meter distance) using the following formula:

(3)

For distances d1, d2, and d3, Eq. 3 can be modified as:

共Md1 ⫺ Md2兲c2 ⫹ 2共Md1d1 ⫺ Md2d2兲c ⫹ 共M d1d21 ⫺ M d2d22兲

⫽ 0 ) 7f ⫺ 2c

7

兺

i⫽1

Ⲑ

M idi ⫹ M s

Ⲑ

7

兺

i⫽1

d2i ⫽

7

兺 Mid2i

i⫽1

(14) ⭸L ⭸c

⫽0) f

7

兺

i⫽1

M idi ⫺ 3c

7

兺

i⫽1

M 2i d2i ⫹ 2M s

7

兺 Mid3i

i⫽1

⫽

7

兺 M2i d3i

(15)

i⫽1

⭸L (9)

⭸M s

⫽0) f

7

7

7

7

兺 d2i ⫺ 4ci⫽1 兺 Mid3i ⫹ Msi⫽1 兺 di4 ⫽ i⫽1 兺 Midi4 i⫽1 (16)

The values of c and Ms obtained in solving Eqs. 14 –16 were substituted in the nonlinear terms of Eq. 13 and new values of f, c, and Ms were determined. Four iterations were carried out, which gave 3 sets of the

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Ceiling

Ir-192 Source

100 Left Wall

300 10 - 40

Ionisation chamber

100 300 Floor

Fig. 3. MCNP plot of the entire problem geometry showing the locations of the source and the ionization chamber.

values of f, c, and Ms. Further iterations were not required because the values of f, c, and Ms were converging. The converging value of f was used to determine the RAKR of the source by using Eq. 11. More details about the iteration procedure used for determination of f, c, and Ms are described in the Appendix. Monte Carlo simulation The simulation of physics and geometry of the problem was realized using the Monte Carlo N-Particle transport code System (MCNP) version 4B.19 This is a general purpose continuous-energy transport code that deals with transport of neutrons, photons, and coupled electron-photons, i.e., transport of secondary electrons resulting from gamma interactions. Gamma photon cross sections are available for elements with naturally occurring atomic abundances. A cross-section library containing data from the ENDF/B-VI (evaluated nuclear data file version B-VI) is included with the MCNP code. The code solves the Boltzmann transport equation using random sampling techniques to obtain information about the energy, flux, and angular distribution of photons at any given point in the phase space. The simulation of a photon history begins in the MCNP code with random sampling its initial position, energy and direction from the source distribution. The photon cross section for the sampled energy E of the source medium is computed and the next collision distance is randomly sampled to determine the distance to be travelled by the photon. From the probability density function for the outcome of the next collision event (photoelectric/Compton/pair production) the photon’s fate is decided. For photons, the code takes account of incoherent and coherent scattering, the possibility of fluorescent emission after photoelectric absorption, absorption in pair production with local emission of anni-

hilation radiation and bremsstrahlung. Photoelectric event is considered as absorption and the photon history is terminated in such an interaction. In the case of a Compton interaction, the final energy and position of the photon are random sampled from the Klien-Nishina relation– based angular probability and energy distribution functions. After ascertaining the phase space coordinates of the photon, its history is continued as described before. In a collision resulting in pair production, either electron-positron pairs are created for further transport and the photon disappears, or it is assumed that the kinetic energy of the electron-positron pair produced is deposited as thermal energy at the time and point of collision, with isotropic production of one photon of energy (0.511 MeV) headed in one direction and another photon of energy (0.511 MeV) headed in the opposite direction. The rare single 1.022-MeV annihilation photon is ignored. Geometry simulation in MCNP. An MCNP model, including the geometrical representation of the locations of the PTW 0.6 cm3 cylindrical ionization chamber with the PMMA build-up cap and the 192Ir HDR source, which are housed inside a concrete walled room to include the effect of Compton scattered photons, was realized using a variety of card options available in the code. In the Monte Carlo simulation of the experimental conditions, the HDR source was located 100 cm above the ground at a distance of 100 cm from the left wall of the brachytherapy treatment room (dimensions 3 ⫻ 3 ⫻ 3 m3) as indicated in Fig. 3. The distance between the chamber center and source was varied from 10 to 40 cm in steps of 5 cm. The top curved portion of the stainless steel encapsulation around the source was simulated by considering a hemispherical cell intersecting with the lateral section of the cylinder, as shown in graphic rep-

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H(E) ⫽ ␴(E)

3

兺 pi(E) * (E ⫺ Eesc) i⫽1

(18)

with ␴ (E) being the total microscopic cross section as a function of energy E, pi(E) ⫽ probability of interaction of type I, Eesc⫽ average exiting gamma energy, i ⫽ 1 incoherent (Compton) scattering with form factors, i ⫽ 2 pair production (Eesc ⫽ 1.022 MeV ⫽ 2 m0c2), and i ⫽ 3 photoelectric interaction (Eesc ⫽ 0)

Fig. 4. Photon emission line spectrum of 192Ir HDR source as a function of energy.

resentation. Fig. 1 shows the MCNP geometry plot (expanded side view) of the PTW 0.6 cm3 Farmer cylindrical ionization chamber, and Fig. 2 shows the 192Ir source encapsulated in stainless steel. The entire problem geometry depicting the source and the 0.6 cm3 ionization chamber, which are housed inside a concrete walled room is shown in Fig. 3. The Monte Carlo computations have been performed by taking care of self attenuation in the source. The room was filled with dry air (density 1.2905 ⫻ 10⫺3 g/cm3), and a concrete wall thickness of 50 cm was assumed along the floor, ceiling, and walls. The experimentally determined source strength (316.72 GBq ⬇ 8.56 Ci) of 192Ir was considered in computing the initial particle weight in the MCNP model. Source energy distribution. The gamma ray photon energies and corresponding emission probabilities from the 192Ir isotope was taken from literature20 –22 for simulation of the HDR brachytherapy source. The gamma photon emission spectrum of 192Ir is shown as a histogram representation in Fig. 4. The energy distribution of the photon source used for this study is realized using the SDEF card options of the MCNP viz. SI, SP cards. Tally specification MCNP was used to predict the total fluence rate, scattered fluence rate, and the RAKR of the chamber. The MCNP tally options used for this problem are photon fluence tally (F5) and photon energy deposition tally (F6). The fluence tally was scored in such a way that contributions from each cell are accounted for separately to enable computation of scattering correction factor as a result of the surrounding concrete structure. Energy deposition rates in the detector sensitive volume is estimated by F6 tally that uses the following equation: F6 ⫽ ␳'

兰兰兰 V t E

冋 册

H(E)⌽(r, E, t)dEdt

dV V

(17)

where ␳= is the ratio of the atom density to the gram density and H (E) is the heating response function given by:

All the energy transferred to electrons is assumed to be deposited locally. The energy deposition rate thus obtained from F6 tally in units of MeVg⫺1s⫺1 from MCNP was converted to Pico-Coulombs/sec by using the appropriate conversion factor as the tally multiplier. The computation of the conversion factor is outlined in following section. RESULTS AND DISCUSSION Iterative methods Average values of f, c, and Ms calculated using Newton-Raphson method are 1970.62 pC/s cm2, 0.1354 cm, and 0.0574 pC/s, respectively. Table 1 shows the values of f, c, and Ms derived using successive iterations by the least-squares method. As per this method, converging values of f, c, and Ms, are 1975.43 pC/s cm2, 0.1435 cm, and 0.0101 pC/s, respectively. The 2 values of f obtained by the 2 iterative methods are in good agreement within 0.23%. However, the standard deviation of f in the Newton-Raphson method was about 4%. In addition, the Newton-Raphson method is quite cumbersome because a large number of values of a parameter needs to be computed (e.g., 35 values of f was computed in this work). In the case of the least-squares method, we could obtain the converging values of these 3 parameters in 3 to 4 successive iterations. The least-squares method is preferred when the number of data points exceeds the number of unknowns and the data points are subjected to experimental errors. Further, the least-squares method takes into account the entire experimental data obtained, whereas in the Newton-Raphson method only 3 data points are used. Monte Carlo method The primary aim of the Monte Carlo simulation was to compute the contribution of scattered radiation to the observed meter reading. To achieve this, we used the

Table 1. Values of c, Ms, and f determined using the leastsquares method Iteration

c

Ms

f

1 2 3 4

0 0.1936 0.1412 0.1435

0 0.0762 0.0098 0.0101

— 1936.73 1974.34 1975.14

Strength determination of HDR

192

Table 2. Comparison of room scatter correction factors calculated using the Monte Carlo method with that published by Selvam et al.9 Room Scatter Correction Factor, Ksc Source to Chamber Distance, d (cm)

This Work (Monte Carlo)

Selvam et al.

% Difference

13 16 19 22 25

0.9782 0.9769 0.9742 0.9723 0.9708

0.995 0.994 0.992 0.989 0.988

1.68 1.71 1.79 1.67 1.73

Room size 3 ⫻ 3 ⫻ 3 m3.

photon fluence tally, scoring individual contributions from each cell. Table 2 presents the Monte Carlo calculated scatter correction factor, Ksc [Ksc ⫽ (Md ⫺ Ms)/Md] and its comparison with published values of Selvam et al.9 This exercise was undertaken to gain confidence in our simulation approach. Values of Ksc calculated in this work show good agreement with the corresponding values of Selvam et al. The maximum difference between our data and published data is about 2%. This difference in Ksc can be attributed to the fact that MCNP version 3.1 (1980) was used by Selvam et al., whereas MCNP version 4B (1997)19 was used in the present study. The later version of MCNP used by us incorporates updated photon interaction cross-section data and electron transport methods in contrast to the earlier version of the MCNP code used by Selvam et al. The F6 tally of MCNP code estimates the mean energy deposited per unit mass for one starting particle per second. The following equation was used to convert the photon energy deposition rate per unit mass (MeV/ g⫺1s⫺1) to meter reading (pC/s) so that computed meter reading can be compared with the experimentally observed meter reading Meter reading (pC ⁄ s) A1(MeVg⫺1s⫺1) * B(JMeV⫺1) * M(g) (19) ⫽ C1(JCoulomb⫺1)

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235

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 numerical value equal to 1.602 ⫻ 10⫺13 J MeV⫺1, C1 ⫽ 33.97 (J/Coulomb) and M is the mass of air in the sensitive volume of the chamber in grams (M ⫽ 7.23048e ⫻ 10⫺4 g). For the F6 tally, the relative error R of the mean value was less than ⫾0.5% for 2 ⫻ 108 particle histories considered in the simulation. Experimentally observed meter readings (pC/s) and meter readings (pC/s) calculated by the least-squares method and Monte Carlo simulation at different source to chamber distance are listed in Table 3. Percentage difference between experimental and corresponding theoretical values is also listed in Table 3. Meter readings calculated using the least-squares method are in excellent agreement with the experimental values at all source-tochamber distances. However, meter readings estimated using the Monte Carlo simulation show a percentage difference up to 1.84% from the respective experimental values. In addition, contributions to the meter reading because of primary, scattered and total photons were estimated from each of the problem cells simulated. The primary radiation emitted from source includes the inscattering and self-absorption phenomena in the source medium. Figure 5 presents the variation of various components of meter reading (primary, scatter and total), with source-to-chamber distance as predicted by Monte Carlo simulation. It may be observed from the graph that the primary contribution decreases according to inverse square law as one moves away from the source, whereas the scattered contribution does not. CONCLUSION Iterative methods of Newton-Raphson and leastsquares were presented in this work as a simple mathematical tool to determine scatter contribution in the experimentally observed meter reading of a cylindrical ionization chamber. Monte Carlo simulation was also used to cross verify the results of the least-squares

Table 3. Comparison of experimentally observed meter readings and meter readings calculated by the least-squares method and Monte Carlo simulation Meter Reading (pC/s) % Deviation Between Experimental and Source to Chamber Distance, d (cm)

Experimental

Least-squares

Monte Carlo

Least-squares

Monte Carlo

10 15 20 25 30 35 40

19.2008 8.6185 4.8939 3.1305 2.1774 1.6115 1.2362

19.207 8.623 4.878 3.134 2.184 1.609 1.236

19.220 8.589 4.986 3.173 2.272 1.736 1.341

⫺0.032 ⫺0.052 0.325 ⫺0.112 ⫺0.303 0.155 0.016

⫺0.099 0.342 ⫺1.841 ⫺1.349 1.369 ⫺1.516 1.237

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Fig. 5. Variation of various components of meter reading (primary, scatter, and total) with source-to-chamber distance as predicted by the Monte Carlo simulation.

method. The experimentally observed least-squares calculated and Monte Carlo estimated values of meter readings from HDR 192Ir brachytherapy source are in good agreement. Considering procedural simplicity, the leastsquares method is recommended for use at the hospitals to estimate values of f, c, and Ms required to determine the strength of HDR 192Ir brachytherapy sources.

Acknowledgements—The authors wish to express their gratitude to Shri H. S. Kushwaha, Director, Health Safety and Environment Group; Dr. D. N. Sharma, Head, Radiation Safety Systems Division; Dr. J. S. Bisht, Ex-Scientist and Dr. G. Chourasiya, Head, Medical Physics & Training Section, RP&AD, Bhabha Atomic Research Centre, Mumbai, for their encouragement and support during this work.

REFERENCES 1. Nath, R.; Anderson, L.L.; Meli, J.A.; et al. Code of practice for brachytherapy physics: Report of the AAPM Radiation Therapy Committee Task Group No. 56. Med. Phys. 24:1557–98; 1997. 2. Kubo, H.D.; Glasgow, G.P.; Pethel, T.D.; et al. High dose rate brachytherapy treatment delivery: Report of the AAPM Radiation Therapy Committee Task Group No. 59. Med. Phys. 25:375– 403; 1998. 3. British Commission on Radiation Units and Measurements (BCRU). Specification of brachytherapy sources. Br. J. Rdiol. 57:941–2; 1984. 4. International Commission on Radiation units and Measurements (ICRU). Radiation quantities and units. Report No. 33; 1980. 5. International Atomic Energy Agency (IAEA); Calibration of photon and beta ray sources used in brachytherapy. TECDOC-1274; IAEA; Vienna; 2002. 6. International Atomic Energy Agency (IAEA); Calibration of brachytherapy sources. Guidelines of standardized procedures for the calibration of brachytherapy sources at secondary standard dosimetry laboratories (SSDLs) and hospitals. TECDOC-1079; IAEA; Vienna; 1999. 7. Goetsch, S.J.; Attix, F.H.; Pearson, D.W.; et al. Calibration of 192Ir high dose rate afterloading system. Med. Phys. 18:462–7; 1991. 8. Stump, K.E.; DeWerd, L.A.; Micka, J.A.; et al. Calibration of new high dose rate 192Ir sources. Med. Phys. 29:1483– 8; 2002. 9. Selvam, T.P.; Govinda Rajan, K.N.; Nagarajan, P.S.; et al. Monte Carlo aided room scatter studies in the primary air kerma strength standardization of remote afterloading 192Ir HDR source. Phys. Med. Biol. 46:2299 –315; 2001.

10. Chang, L.; Ho, S.Y.; Chui, C.S.; et al. Room scatter factor modelling and measurement error analysis of 192Ir HDR calibration by a Farmer chamber. Phys. Med. Biol. 52:871–7; 2008. 11. Pena, J.; Sanchez-Doblado, F.; Capote, R.; et al. Monte Carlo correction factors for a Farmer 0.6 cm3 ion chamber dose measurement in the build-up region of the 6 MV clinical beam. Phys. Med. Biol. 51:1523–32; 2006. 12. Devan, K.; Aruna, P.; Manigandan, D.; et al. Evaluation of dosimetric parameters for various 192Ir brachytherapy sources under unbounded phantom geometry by Monte Carlo Simulation. Med. Dosim. 32:305–15; 2007. 13. Daskalov, G.M.; Baker, R.S.; Rogers, D.W.O.; et al. Dosimetric modelling of the microSelectron high dose rate 192Ir source by the multigroup discrete ordinates method. Med. Phys. 27:2307–19; 2000. 14. Mainegra-Hing, E.; Rogers, D.W.O. On the accuracy of techniques for obtaining the calibration coefficient Nk of 192Ir HDR brachytherapy sources. Med. Phys. 33:3340 –7; 2006. 15. Bielajew, A.F. Correction factors for thick-walled ionization chambers in point-source photon beams. Phys. Med. Biol. 35:501– 16; 1990. 16. Bielajew, A.F. An analytical theory of the point-source nonuniformity correction factor for thick-walled ionization chambers in photon beams. Phys. Med. Biol. 35:517–38; 1990. 17. Kondo, S.; Randolph, M.L. Effect of finite size of ionization chambers on measurements of small photon sources. Radiat. Res. 13:37– 60; 1960. 18. Sastry, S.S. Introductory Methods of Numerical Analysis. 2nd ed. Prentice-Hall Press; New Delhi, India, 1995. 19. Briesmeister, J.F. MCNP—A General Monte Carlo N-Particle Transport Code, Version 4B, Transport Methods Group. Los Alamos National Laboratory Report; 1997. 20. Sureka, C.S.; Sunny, S.C.; Subbaiah, K.V.; et al. Dose distribution for endovascular brachytherapy using Ir-192 sources: Comparison of Monte Carlo calculations with radiochromic film measurements. Phys. Med. Biol. 52:525–37; 2007. 21. Sureka, C.S.; Aruna, P.; Ganesan, S.; et al. Computation of relative dose distribution and effective transmission around a shielded vaginal cylinder with 192Ir HDR source using MCNP4B. Med. Phys. 33:1552– 61; 2006. 22. Williamson, J.F., Li, Z. Monte Carlo aided dosimetry of microSelectron pulsed and high dose rate 192Ir sources. Med. Phys. 22:809 –19; 1995. APPENDIX Basic equations used in the least-squares method are: L⫽

7

[f ⫺ (M i ⫺ M s)(di ⫹ c)2]2 兺 i⫽1

(1)

Strength determination of HDR

L⫽

7

关 f ⫺ 2Midic ⫹ Msd2i ⫺ Mid2i ⫺ Mic2 ⫹ 2dicMs ⫹ Msc2兴2 兺 i⫽1

(2)

192

Ir brachytherapy ● S. KUMAR et al.

Taking partial derivatives of L, with respect to f, c, and Ms, and equating to zero, we get the following equations:

On putting c ⫽ 0 and Ms ⫽ 0 in the nonlinear terms during first iteration and equating partial derivatives to zero, we get the following equations: ⭸L ⭸f ⭸L ⭸c

⫽ 0 ) 7f ⫺ 2c

⫽0) f ⭸L

⭸M s

7

7

7

M idi ⫹ M s兺 兺 i⫽1 i⫽1 7

d2i ⫽

7

兺 i⫽1

M id2i

7

⫽0) f

7

7

7

(3)

(4)

7f ⫺ 13337.197 * C ⫹ 5075M s ⫽ 13685.0

(6)

668.5985f ⫺ 237534.5288 * C ⫹ 686706.4M s ⫽ 1301254

(7)

5075f ⫺ 1373413 * C ⫹ 5481875 * M s ⫽ 9980818

(8)

[f 兺 i⫽1

f ⫺ 135.31159 * C ⫹ 1080.1724M s ⫽ 1965.8205

(12)

Ms ⫽ 0.00977 (pC/s)

f ⫽ 1974.37 (pC/s cm2)

f ⫽ 1936.73 (pC/s cm )

L⫽

7

[f ⫺ M id2i ⫺ 0.01993532M i ⫺ 2M idiC ⫹ M sd2i ⫹ 1.948145E 兺 i⫽1

⫺ 04 ⫹ 2.759552E ⫺ 03di]2 (13)

Taking partial derivatives of L, with respect to f, c, and Ms, and equating to zero, we get the following equations: f ⫺ 191.0281429 * C ⫹ 725M s ⫽ 1955.05192

(14)

f ⫺ 236.848202 * C ⫹ 513.541629M s ⫽ 1946.344449

(15)

f ⫺ 135.31159 * C ⫹ 1080.1724M s ⫽ 1966.629246

(16)

2

Substituting these values in nonlinear terms of Eq. 2, we get: L⫽

(11)

Substituting these values in non linear terms of the Eq. 2, we get

On solving Eqs. 6 – 8, we get the following values of c, Ms, and f:

7

f ⫺ 236.848202 * C ⫹ 513.541629M S ⫽ 1945.94689

(5)

On putting the experimental values of M and d in Eqs. 3–5, we get the following equations:

c ⫽ 0.1936 cm Ms ⫽ 0.07622 (pC/s)

(10)

c ⫽ 0.1412 cm

7

d2i ⫺ 4c兺 M id3i ⫹ M s兺 di4 ⫽ 兺 M idi4 兺 i⫽1 i⫽1 i⫽1 i⫽1

f ⫺ 191.0281429 * C ⫹ 725M s ⫽ 1954.482876

On solving Eqs. 10 –12, we get following values of c, Ms, and f:

7

M idi ⫺ 3c兺 M 2i d2i ⫹ 2M s兺 M id3i ⫽ 兺 M 2i d3i 兺 i⫽1 i⫽1 i⫽1 i⫽1

237

⫹ 0.0295138di]2

On solving Eqs. 14 –16, we get following values of c, Ms, and f: c ⫽ 0.1434 cm

⫺ M id2i ⫺ 0.03748754M i ⫺ 2M idiC ⫹ M sd2i ⫹ 0.00285718 (9)

Ms ⫽ 0.0101 (pC/s)

f ⫽ 1975.14 (pC/s cm2)

Successive iterations can also be repeated in similar fashion.