

ORIGINAL ARTICLE 



Year : 2012  Volume
: 37
 Issue : 3  Page : 129137 

An analytic approach to the dosimetry of a new BEBIG ^{60} Co highdoserate brachytherapy source
Subhalaxmi Bhola, T Palani Selvam, Sahoo Sridhar, Ramkrishna S Vishwakarma
Radiological Physics and Advisory Division, Bhabha Atomic Research Centre, Anushaktinagar, Mumbai, India
Date of Submission  20Sep2011 
Date of Decision  14Apr2012 
Date of Acceptance  14Apr2012 
Date of Web Publication  1Aug2012 
Correspondence Address: Subhalaxmi Bhola Radiological Physics and Advisory Division, Health, Safety and Environment Group, Bhabha Atomic Research Centre, Anushaktinagar, Mumbai  400 094 India
Source of Support: None, Conflict of Interest: None  Check 
PMID: 22973079
Abstract   
We present a simple analytic tool for calculating the dose rate distribution in water for a new BEBIG highdoserate (HDR) ^{60} Co brachytherapy source. In the analytic tool, we consider the active source as a point located at the geometric center of the ^{60} Co material. The influence of the activity distribution in the active volume of the source is taken into account separately by use of the line sourcebased geometric function. The exponential attenuation of primary ^{60} Co photons by the source materials ( ^{60} Co and stainlesssteel) is included in the model. The model utilizes the pointsourcebased function, f(r) that represents the combined effect of the exponential attenuation and scattered photons in water. We derived this function by using the published radial dose function for a point ^{60} Co source in an unbounded water medium of radius 50 cm. The attenuation coefficients for ^{60} Co and the stainlesssteel encapsulation materials are deduced as bestfit parameters that minimize the differen
Keywords: Analytic method, brachytherapy, highdoserate, monte carlo, treatment planning
How to cite this article: Bhola S, Selvam T P, Sridhar S, Vishwakarma RS. An analytic approach to the dosimetry of a new BEBIG ^{60} Co highdoserate brachytherapy source. J Med Phys 2012;37:12937 
How to cite this URL: Bhola S, Selvam T P, Sridhar S, Vishwakarma RS. An analytic approach to the dosimetry of a new BEBIG ^{60} Co highdoserate brachytherapy source. J Med Phys [serial online] 2012 [cited 2023 Mar 21];37:12937. Available from: https://www.jmp.org.in/text.asp?2012/37/3/129/99228 
Introduction   
In clinical practice, highdoserate (HDR) ^{192} Ir and ^{60} Co brachytherapy sources are used. ^{192} Ir sources are most commonly used, but the use of ^{60} Co sources has increased because of its longer halflife (5.25 years) and its availability in miniaturized forms (with dimensions comparable to those of ^{192} Ir HDR sources). The Ralstron remote afterloader which uses type 1, type 2 and type 3 ^{60} Co HDR sources, was introduced for intracavitary treatments because of the longer halflife. ^{[1]} Presently, BEBIG HDR ^{60} Co brachytherapy sources (old and new designs) are in widespread use for intracavitary treatments. ^{[2],[3]} In a recently published study by Richter et al., ^{[4]} the authors have compared the physical properties of ^{60} Co and ^{192} Ir HDR sources. They demonstrated that the integral dose due to radial dose falloff is higher for ^{192} Ir than for ^{60} Co within the first 22 cm from the source. At larger distances, this relationship is reversed. Their study suggests that no advantage or disadvantage exists for ^{60} Co sources compared with ^{192} Ir sources with regard to clinical aspects. However, there are potential logistical advantages of ^{60} Co sources because only 33% of the activity of ^{192} Ir sources is needed to yield an equivalent doserate. Further, because of relatively long half life, ^{60} Co sources can be used for much longer duration resulting in reduced operating costs.
The use of a brachytherapy source for clinical trials requires an extensive dosimetric data set either in the form of an American Association of Physicists in Medicine (AAPM) TG43 parameters or in the form of a 2D doserate lookup table. ^{[5],[6]} According to AAPM TG56, such data are needed for commissioning and verification purposes in radiotherapy treatment planning systems (RTPS). ^{[7]} The dosimetry data are usually generated by use of Monte Carlo methods. For example, Monte Carlobased dosimetry data for HDR ^{60} Co sources are reported in the literature. ^{[1],[2],[3]} and ^{[8]} An EGSnrcbased published study for new and old sources by Selvam and Bhola ^{[8]} demonstrated that the doserate data compare well with the GEANT4based published data ^{[2]} and ^{[3]} for radial distances larger than 0.5 cm. Selvam and Bhola ^{[8]} have shown differences in dose values up to 9% for regions close to BEBIG ^{60} Co sources when compared with GEANT4based published work for these sources. ^{[2]} and ^{[3]} It was also demonstrated that the length of stainlesssteel cable for the new BEBIG ^{60} Co source considered by Granero et al. ^{[3]} in their GEANT simulations was 1 mm, although it was mentioned to be 5 mm. ^{[8]}
The Sievert integral algorithm is generally used in RTPS for dose calculation around brachytherapy sources. ^{[9]} For ^{125} I, ^{169} Yb, ^{137} Cs and ^{192} Ir brachytherapy sources, such analytic methods have already been established. ^{[10],[11],[12],[13],[14],[15],[16],[17],[18]} Our objective in the present study was to develop a simple analytic tool for calculating the 2D doserate distribution in water for the new model (Co0.A86) of a BEBIG ^{60} Co HDR source. Using the analytic model, we calculated AAPM TG43 dose parameters such as the doserate constant, radial dose function and anisotropy function for the above source in a 50cm radius in an unbounded water medium. We also calculated the doserate lookup table in a Cartesian format. A comparison was made with previously published work. ^{[8]} The doserate data calculated with the use of the proposed analytic model could be used for verifying the results of treatmentplanning systems or directly as input data for RTPS.
Materials and Methods   
New BEBIG ^{60} Co source
Analytic calculations were performed for the new model of the BEBIG ^{60} Co HDR (model Co0.A86) brachytherapy source [Figure 1]. ^{[3]} The new BEBIG ^{60} Co HDR brachytherapy source ^{[3]} is very much similar, both in materials and design, to the old BEBIG ^{60} Co HDR brachytherapy source (model GK60M21). ^{[2]} The new source design has a smaller active core of diameter 0.5 mm with a rounded capsule tip, whereas the old design has an active core of diameter 0.6 mm. The new source has a more rounded capsule tip. Both sources consist of a central cylindrical active core of length 3.5 mm, which is made of metallic ^{60} Co. The active core is covered by a cylindrical stainlesssteel capsule with an external diameter of 1 mm.  Figure 1: Schematic diagrams of the new BEBIG ^{60}Co highdoserate brachytherapy source (model Co0.A86), depicting geometric characteristics and materials. The coordinate axes used in this study are also shown with their origin situated in the geometric center of the active volume. All dimensions are in millimeters. Figures not drawn to scale
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TG43 dose calculation formalism
The TG43 report ^{[5],[6]} has recommended the dose calculation algorithm for establishing the 2D doserate distribution in a water medium around cylindrically symmetric photonemitting brachytherapy sources. The doserate at polar coordinates (r,θ) is written as
Here, the airkerma strength, S_{k} , is defined as the product of the airkerma rate measured at a calibration distance r _{c} along the transverse bisector of the source in free space and the square of the distance r _{c}. S_{k} has units of U (1U=1 μGy m ^{2} /h = 1 cGycm ^{2} /h).
The doserate constant, Λ, is defined as the doserate per S_{k} along the transverse source bisector at the reference distance r _{0} = 1 cm. The reference angle, θ_{0} defines the source transverse plane, and is specified to be 90 ^{0} or /2 radian. Λ has the units of cGy/h/U, which reduces to cm.
The radial dose function, g_{L}(r), accounts for the effect of absorption and scatter in the medium along the transverse axis of the source, defined as follows:
The anisotropy function, F(r, θ), accounts for anisotropy of the dose distribution around the source, and is defined as follows:
The geometry function, G_{L}(r,q), accounts for the spatial distribution of radioactivity within the source, and is defined as follows:
where β (in radians) is the angle subtended by the source to the point of interest, (r, θ). ^{[5],[6]}
Analytic approach
Monoenergetic point photon source in water
The absorbed doserate in water at a distance r (cm) away from a point isotropic monoenergetic photon source is given by,
where A denotes the activity of the source (in Bq), E is the energy emitted by the source (in MeV) per photon, k is the constant converting the unit MeV/gm to the mass energy absorption coefficient of water in units of cm ^{2} /gm for photon of energy E, and B is the energyabsorption buildup factor. ^{[19]} B is defined as the absorbed doserate from both the primary and the scattered photons in an infinite water medium divided by the absorbed doserate from only primary photons. The assumption made in equation 7 is that the energy lost by photons in the scattering and absorption events is absorbed locally in the medium. This means that the range of secondary electrons (photo electrons, compton electrons and deltarays) is assumed to be negligible. In the energy range relevant to brachytherapy sources, this assumption has little impact on the calculated doserate.
S _{k} at a distance r away from a point monoenergetic photon source of energy E and activity A can be obtained analytically:
where is the massenergy absorption coefficient of air for the energy E. It should be noted that S _{k} is expressed in terms of the recommended unit U (= 1 cGycm ^{2} /h).
The doserate in water per S _{k} due to a point monoenergetic point photon source of energy E is then given by
where is the ratio of the massenergy absorption coefficient of water to air at photon energy E, which is equal to 1.112 at ^{60} Co energies. ^{[20]} The value of is constant (=1.112) in the photon energy range between 0.15 MeV and 3 MeV. ^{[20]} As compton scattering is the predominant process in water at ^{60} Co energies, one can write is the ratio of electrons/g between water and air. The values of for water and air are 0.555 and 0.499, respectively. ^{[20]}
The functional form of f(r) for ^{60} Co brachytherapy sources has been presented by many authors. Kartha et al. ^{[21]} have given an analytical expression, where E is the photon energy (for ^{60} Co, it is 1.25 MeV), μ is the linear attenuation coefficient in water and r is the distance from the source. Meisberger ^{[22]} approximated f(r) by the ratio of the air kerma in water to air kerma in air. He fitted the f(r) data to a thirdorder polynomial function, f(r) = A+ Br +Cr ^{2} +Dr ^{3} (valid up to 10 cm from the source), with coefficients A = 0.99423, B = 5.318x10 ^{ 3} / cm, C= 2.610x10 ^{3} /cm ^{2} and D = 13.27 x 10 ^{5} /cm ^{3} . Van Kleffens and Starr ^{[23]} provided an expression, f(r) = (1+ ar ^{2} ) / (1+br ^{2} ), with a= 10x10 ^{3} /cm ^{2} and b=14.50x10 ^{3} /cm ^{2} . Kornelson and Young ^{[24]} have given a functional form for the buildup factor, where μ is the linear attenuation coefficient at ^{60} Co energies, k _{a} = 0.896 and k _{b} = 1.063. Angelopoulos et al. ^{[25]} have provided data for f(r) for distances r = 19 cm in a 10cmradius water phantom. The most recent published study for a point ^{60} Co source in water gives a radial dose function, g_{p}(r) in an unbounded water medium by Papagiannis et al. ^{[1]}
Note that equation 10 is based on waterkerma as it was verified in a previously published work. ^{[8]} From equation 10, f(r) can be derived as follows:
According to the TG43 protocol, ^{[5],[6]}
where r_{0}= 1 cm.
By using equation 7 in 11, one obtains
Here, f(r_{0}) = 0.9864 is calculated at r_{0} = 1 cm with the use of Mesiberger's polynomial ^{[22]} for f(r). In our analytical model, we make use of equation 13.
Monoenergetic bare cylindrical source in water
The doserate at a point (r, θ) in water per S _{k} for a bare cylindrical source of photon energy, E, can be written as
In the above formalism, the active cylindrical source is divided into N active segments, and each segment is treated as a point source and r _{i} is the distance between the i^{th} source element to the point (r, θ). The above equation is further simplified as follows:
where, r is the distance between the center of the active length and the point of interest (r, θ). In equation 15, an assumption is made that the entire activity is concentrated at the geometric center of the cylinder. The influence of the activity distribution in the cylindrical volume is taken into account separately by use of the linesourcebased geometry function, G_{L}(r,q). A simple calculation for a cylindrical bare active ^{60} Co source of 3.5 mm length and 0.5 mm diameter (these are typical active dimensions of the new BEBIG ^{60} Co source) by using equations 14 and 15 give a doserate value of 4.23 cGy/h/U at 5 mm along the transverse axis of the source. This suggests that equation 15 can be used for dose calculations.
Papagiannis et al. ^{[26]} observed that close to ^{192} Ir HDR sources, it is the inherent influence of the ''exact'' geometry function ^{[5],[18],[27]} that determines the doserate distribution. In order to verify that the use of G_{L}(r,q) in equation (15) produces reasonably accurate results, we calculated an "exact" geometry function, G_{ex}(r,q) by using the Monte Carlo integration approach as adapted by Karaiskos et al. ^{[18]} ,
where with being the distance between the i ^{th} Monte Carlo generated point and the calculation point (r, θ). The Monte Carlo values of G_{ex}(r,θ) at distances 1, 2 and 5 mm from the source center along the transverse axis (q = 90°) are larger only by 1.23%, 0.5% and 0.1%, respectively, when compared with the corresponding values of G_{L}(r,θ).
Real cylindrical source in water
Cassell proposed the quantization method (decomposition of the source into small cells) for brachytherapy dose calculations. ^{[28]} This algorithm is similar to the Sievert integral model described by Williamson. ^{[10]} According to Cassell, ^{[28]} the doserate in water at a point P( r, θ), in units of cGy/h can be obtained from the following equation:
Note that the reference airkerma rate, is equivalent to S _{k} of the source. ^{[6]}
In the quantization method, the active part of the cylindrical source is divided into N source elements, which are treated as point sources. For each elemental source, the doserate is calculated by multiplying and correcting for the inverse square of the distance, tissue attenuation, selfabsorption and filter attenuation by use of an exponential correction over the line between the elemental source and the calculation point. Symbols in equation 17 have the following meaning: μ_{s} , μ_{f} and μ_{w} are the linear attenuation coefficient of the active source, of the filtration material and of water, respectively. t _{1} and t _{2} are the active radii of the source and the encapsulation thickness, respectively. , and are the distances traveled by photons within the source core, filter material and the water medium, respectively, and is the distance between the center of the source and the calculation point P(r, θ). Photon paths in different media are depicted in [Figure 2].
 Figure 2: Simplified geometry used in the present analytical model. A point ^{60}Co source is positioned at the geometric center of the inactive metallic ^{60}Co material
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Motivated by our analytic model described for a bare cylindrical source, we consider that the total activity is concentrated at the center of the ^{60} Co material instead of being distributed throughout its volume. The influence of the distribution of activity in the source on the doserate is taken into account separately by the linesourcebased geometry function G_{L}(r,θ). According to our simplified model, the doserate per S _{k} at a point (r, θ) can be written as follows:
where r _{s} , r _{f} and r _{w} (= r  r _{s}  r _{f} ) are the distances traveled by photons within the source core, filter material and water, respectively, as is shown in [Figure 2]. In the calculations, the function f(r) (equation 13) is evaluated at r _{w} . When θ = θ_{0} , r _{s} = t _{1} , r _{f} = t _{2} . For the new design of the ^{60} Co source, t _{1} = 0.025 mm, t _{2} = 0.015 cm and t _{1} + t _{2} = 0.04 cm. Therefore, r _{w} = r and f(r _{w} ) = f(r) for values of r larger than 2.5 mm. For θ = θ_{0} , equation 19 can be written as:
Equation 20 is same as equation 15 when we set θ = θ_{0} in equation 15. For the calculation of the transverse axis doserate distribution, equation 20 is good enough. When we set r = r _{0} = 1 cm in equation 20,
Equation 21 is a general expression for the doserate constant of a monoenergetic photon source of active length L.
Doserate calculation for BEBIG source by use of the analytic tool
We have adapted the analytical tool described above for calculating doserate distributions in water around the new BEBIG ^{60} Co HDR source. We used equation 19 for this purpose. The lengths of stainless steel cable considered in the analytical calculations are 1 mm and 5 mm. Doserate calculations are carried out as functions of polar coordinates (r, θ) and Cartesian coordinates (y,z). In the calculations, the radial distance r is varied from 1 mm to 14 cm (in 2.5 mm intervals up to 3 cm and 1cm intervals from 3 cm to 14 cm), and the polar angle θ is varied from 0° to 179° (in 2° intervals) for each r, with the 180° angle referring to the source cable side.
The analytically calculated doserate values for the new BEBIG ^{60} Co source with 1 mm and 5 mm cable lengths by use of equation 19 compare well with the published values for regions other than those close to the source axis. ^{[3],[8]} For example, for the regions close to the source axis, the analytically calculated doserates are higher by up to 10% when compared with the published Monte Carlobased values. ^{[3],[8]} In order to verify whether this disagreement is due to simplifications made in the analytic model, we carried out a test calculation by using equation 17. Yet, the same disagreement was observed. For the analytical calculations, we used μ_{s} = 0.47/cm, μ_{f} = 0.43/cm and μ_{w} = 0.063/cm, all obtained at ^{60} Co energies. ^{[13]}
Most of the currently available RTPS make use of the Sievert algorithm ^{[9]} to generate dose distributions for filtered line sources. Frequently, the RTPS, based on this algorithm, does not produce accurate calculations ^{[5]} for the regions close to the source axis. TG43 ^{[5]} recommends treating the attenuation coefficients as parameters of the best fit for minimizing the deviations between the Sievert model predictions and the other calculated results. Selfabsorption by the source core (μ_{s} ) and attenuation by the filtration material (μ_{f} ) to be used in such algorithms are generally derived by comparing the dose results with the Monte Carlo results. For example, Ballester et al. ^{[13]} and Casal et al. ^{[14]} adapted this approach in their Sievert integralbased ^{137} Cs dosimetry and derived bestfit parameters for μ_{s} and μ_{f} . Similarly, Pérez Calatayud et al. ^{[16]} derived bestfit parameters for μ_{s} and μ_{f} in their quantization methodbased dosimetry study on CDCtype miniaturized ^{137} Cs sources ^{[15]} and the bestfit parameter for μ for ^{192} Ir wires.
Guided by the abovementioned published work, we treated the parameters μ_{s} and μ_{f} as freefit parameters. Instead of using the actual values of μ_{s} (= 0.47/cm) and μ_{f} (= 0.43/cm) at ^{60} Co energies, ^{[13]} we used the fitted values μ_{s} = 0.25/cm and μ_{f} = 0.25/cm for the new BEBIG ^{60} Co HDR source [Table 1]. For the abovedescribed analytic method, computer software has been developed in C _{++} computerprogramming language. The software generates a complete dosimetry dataset around the source. The data include TG43 parameters and a 2D lookup table. For the calculation of g_{L}(r) and F(r,θ) we used G_{L}(r,θ). This is consistent with the updated TG43 formalism. ^{[6]}  Table 1: Actual and fitted values of linear attenuation coefficients for source and filtration materials μ_{s} and μ_{f}, respectively, for the new BEBIG ^{60}Co source
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Results and Discussion   
[Table 2] compares the values of the doserate constants Λ of BEBIG sources to published values. ^{[2],[3],[8]} In the present study, this is 1.088 cGy/h/U (for both new and old designs). The GEANT4based published values of Λ are 1.084 cGy/h/U and 1.087 cGy/h/U for the old and new source designs, respectively. ^{[2],[3]} When r _{0} = 1 cm, equation 9 represents the doserate constant for a point source, Λ_{p} . The calculated value of Λ_{p} for a ^{60} Co point source is 1.097 cGy/h/U, which is consistent with the value of 1.094 cGy/h/U reported by Papagiannis et al. ^{[1]} and the EDKnrcbased value of 1.107 ± 0.01 cGy/h/U reported by Selvam and Bhola. ^{[8]}  Table 2: Comparison of dose rate constants of old and new BEBIG ^{60}Co sources
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[Table 3] compares the values of g_{L}(r) for BEBIG ^{60} C sources calculated in the present study and the EGSnrcbased published work. ^{[8]} At r = 2.5 mm, the EGSnrcbased published value ^{[8]} is higher by about 6% when compared with the value obtained in the present work. This is because the analytic calculation considers that there is charged particle equilibrium for all calculation points, including for regions close to the source. [Table 4] presents values of the anisotropy function for various radial distances from the source. The source cable length considered in this calculation was 1 mm for comparison with values published by Granero et al. ^{[3]} The analytically calculated data compares with the published data within 3%. [Table 5] and [Table 6] present the 2D doserate distribution in water (in cGy/h/U) for the new source model with 5mm and 1mmlength stainlesssteel cable, respectively. For the 1mm cable length, the analytical data agree with the data published by Granero et al. ^{[3]} within 1%, and for the 5mm cable length, the agreement is within 3%. For regions where chargedparticle equilibrium exists, a comparison of these data with the corresponding EGSnrcbased published data ^{[8]} suggests that the analytically calculated values are comparable to within 0.5% for most points, and the maximum deviation is about 3%.  Table 3: Comparison of analytically calculated (this work) and Monte Carlobased published data of radial dose function, g_{L}(r), of the new BEBIG ^{60}Co HDR source
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 Table 4: Anisotropy function F(r,θ) values for the new (model Co0.A86) BEBIG ^{60}Co HDR source.
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 Table 5: Dose rate distributions per unit airkerma strength (cGy/h/U) around the new (model Co0.A86) BEBIG ^{60}Co HDR source.
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 Table 6: Dose rate distributions per unit airkerma strength (cGy/h/U) around the new (model Co0.A86) BEBIG ^{60}Co HDR source.
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Conclusions   
We have proposed a pointsourcebased simple analytic method for calculating the doserate distribution in water in units of cGy/h/U for a BEBIG ^{60} Co HDR source. Using this method, we calculated TG43 parameters such as the doserate constant, radial dose function and anisotropy function. We also calculated a 2D doserate table in Cartesian format. The proposed analytic method needed bestfit parameters for linear attenuation coefficients of source and filtration materials for regions close to the source axis. The analytic model proposed is easy to implement in radiotherapy treatmentplanning dose calculations. For regions where electronic equilibrium exists, a comparison between the analytically calculated and published Monte Carlobased data shows good agreement (for most calculation points, agreement was within 0.5%, and the maximum deviation was about 3%). The doserate data calculated with this method could be used for verifying the results of RTPS or directly as input data for radiotherapy treatmentplanning dose calculations.
Acknowledgment   
The authors wish to thank Dr. Y. S. Mayya, Head, Radiological Physics and Advisory Division, Bhabha Atomic Research Centre (BARC), and Dr. G. Chourasiya, BARC, for their encouragement and support throughout this project.
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[Figure 1], [Figure 2]
[Table 1], [Table 2], [Table 3], [Table 4], [Table 5], [Table 6]
