An analytic approach to the dosimetry of a new BEBIG (60)Co high-dose-rate brachytherapy source.

We present a simple analytic tool for calculating the dose rate distribution in water for a new BEBIG high-dose-rate (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 source-based geometric function. The exponential attenuation of primary (60)Co photons by the source materials ((60)Co and stainless-steel) is included in the model. The model utilizes the point-source-based 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 stainless-steel encapsulation materials are deduced as best-fit parameters that minimize the different.

Introduction

In clinical practice, high-dose-rate (HDR) 192Ir and 60Co brachytherapy sources are used. 192Ir sources are most commonly used, but the use of 60Co sources has increased because of its longer half-life (5.25 years) and its availability in miniaturized forms (with dimensions comparable to those of 192Ir HDR sources). The Ralstron remote after-loader which uses type 1, type 2 and type 3 60Co HDR sources, was introduced for intracavitary treatments because of the longer half-life.[1] Presently, BEBIG HDR 60Co brachytherapy sources (old and new designs) are in widespread use for intra-cavitary treatments.[23] In a recently published study by Richter et al.,[4] the authors have compared the physical properties of 60Co and 192Ir HDR sources. They demonstrated that the integral dose due to radial dose fall-off is higher for 192Ir than for 60Co 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 60Co sources compared with 192Ir sources with regard to clinical aspects. However, there are potential logistical advantages of 60Co sources because only 33% of the activity of 192Ir sources is needed to yield an equivalent dose-rate. Further, because of relatively long half life, 60Co 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) TG-43 parameters or in the form of a 2-D dose-rate lookup table.[56] According to AAPM TG-56, 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 Carlo-based dosimetry data for HDR 60Co sources are reported in the literature.[1-3] and[8] An EGSnrc-based published study for new and old sources by Selvam and Bhola[8] demonstrated that the dose-rate data compare well with the GEANT4-based 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 60Co sources when compared with GEANT4-based published work for these sources.[2] and[3] It was also demonstrated that the length of stainless-steel cable for the new BEBIG 60Co 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 125I, 169Yb, 137Cs and 192Ir brachytherapy sources, such analytic methods have already been established.[1011-18] Our objective in the present study was to develop a simple analytic tool for calculating the 2D dose-rate distribution in water for the new model (Co0.A86) of a BEBIG 60Co HDR source. Using the analytic model, we calculated AAPM TG-43 dose parameters such as the dose-rate constant, radial dose function and anisotropy function for the above source in a 50-cm radius in an unbounded water medium. We also calculated the dose-rate look-up table in a Cartesian format. A comparison was made with previously published work.[8] The dose-rate data calculated with the use of the proposed analytic model could be used for verifying the results of treatment-planning systems or directly as input data for RTPS.

Materials and Methods

New BEBIG 60Co source

Analytic calculations were performed for the new model of the BEBIG 60Co HDR (model Co0.A86) brachytherapy source [Figure 1].[3] The new BEBIG 60Co HDR brachytherapy source[3] is very much similar, both in materials and design, to the old BEBIG 60Co 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 60Co. The active core is covered by a cylindrical stainless-steel capsule with an external diameter of 1 mm.

Figure 1

Schematic diagrams of the new BEBIG 60Co high-dose-rate 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

TG-43 dose calculation formalism

The TG43 report[56] has recommended the dose calculation algorithm for establishing the 2D dose-rate distribution in a water medium around cylindrically symmetric photon-emitting brachytherapy sources. The dose-rate at polar coordinates (r,θ) is written as

Here, the air-kerma strength, Sk, is defined as the product of the air-kerma rate measured at a calibration distance rc along the transverse bisector of the source in free space and the square of the distance r. Sk has units of U (1U=1 μGym2/h = 1 cGycm2/h).

The dose-rate constant, Λ, is defined as the dose-rate per Sk along the transverse source bisector at the reference distance r0 = 1 cm. The reference angle, θ0 defines the source transverse plane, and is specified to be 90° or p/2 radian. Λ has the units of cGy/h/U, which reduces to cm.

Λ = Ḋ(r0, q0)/S.      …..(3)

The radial dose function, g(r), accounts for the effect of absorption and scatter in the medium along the transverse axis of the source, defined as follows:

g(r) = Ḋ(r,q0)G(r0,q0) / Ḋ(r0,q0)G(r,q0)      …..(4)

The anisotropy function, F(r, θ), accounts for anisotropy of the dose distribution around the source, and is defined as follows:

F(r,q) = Ḋ (r,q)G(r,q) / sḊ (r,q)G(r,q).      …..(5)

The geometry function, G(r,q), accounts for the spatial distribution of radioactivity within the source, and is defined as follows:

point source approximation, G(r,q) = 1 / r2,      …..(6.a)

where β (in radians) is the angle subtended by the source to the point of interest, (r, θ).[56]

Analytic approach

Monoenergetic point photon source in water

The absorbed dose-rate 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 Gy,[m(E)/ r] the mass energy absorption coefficient of water in units of cm2/gm for photon of energy E, and B is the energy-absorption build-up factor.[19] B is defined as the absorbed dose-rate from both the primary and the scattered photons in an infinite water medium divided by the absorbed dose-rate 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 delta-rays) is assumed to be negligible. In the energy range relevant to brachytherapy sources, this assumption has little impact on the calculated dose-rate.

S at a distance r away from a point monoenergetic photon source of energy E and activity A can be obtained analytically:

where [m(E)/ r] is the mass–energy absorption coefficient of air for the energy E. It should be noted that S is expressed in terms of the recommended unit U (= 1 cGycm2/h).

The dose-rate Ḋ(r) in water per S due to a point monoenergetic point photon source of energy E is then given by

where f(r)=Be–mwr, [m(E)/ r] is the ratio of the mass–energy absorption coefficient of water to air at photon energy E, which is equal to 1.112 at 60Co energies.[20] The value of [m(E)/ r] 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 60Co energies, one can write is the ratio of electrons/g between water and air. The values of 〈Z/A〉 for water and air are 0.555 and 0.499, respectively.[20]

The functional form of f(r) for 60Co brachytherapy sources has been presented by many authors. Kartha et al.[21] have given an analytical expression, f(r)=exp{[0.73/E0.05s]–1}mr, where E is the photon energy (for 60Co, 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 third-order polynomial function, f(r) = A+ Br +Cr2+Dr3 (valid up to 10 cm from the source), with coefficients A = 0.99423, B = -5.318×10-3/ cm, C=- 2.610×10-3/cm2 and D = 13.27 × 10-5/cm3. Van Kleffens and Starr[23] provided an expression, f(r) = (1+ ar2) / (1+br2), with a= 10×10-3/cm2 and b=14.50×10-3/cm2. Kornelson and Young[24] have given a functional form for the build-up factor, B(r)=1+ka(mr) where μ is the linear attenuation coefficient at 60Co energies, ka = 0.896 and kb = 1.063. Angelopoulos et al.[25] have provided data for f(r) for distances r = 1–9 cm in a 10-cm-radius water phantom. The most recent published study for a point 60Co source in water gives a radial dose function, g(r), in an unbounded water medium by Papagiannis et al.[1]

g(r) = –1.418 ×10–4 r2 –1.470 ×10–2 rs +1.015      …..(10)

Note that equation 10 is based on water-kerma as it was verified in a previously published work.[8] From equation 10, f(r) can be derived as follows:

According to the TG-43 protocol,[56]

where r0 = 1 cm.

By using equation 7 in 11, one obtains

f(r) = f(r0)g(r)      …..(13)

Here, f(r0) = 0.9864 is calculated at r0 = 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 dose-rate Ḋ(r,q) at a point (r, θ) in water per S 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 is the distance between the ith 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 line-source-based geometry function, G(r,q). A simple calculation for a cylindrical bare active 60Co source of 3.5 mm length and 0.5 mm diameter (these are typical active dimensions of the new BEBIG 60Co source) by using equations 14 and 15 give a dose-rate 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 192Ir HDR sources, it is the inherent influence of the “exact” geometry function[51827] that determines the dose-rate distribution. In order to verify that the use of G(r,q) in equation (15) produces reasonably accurate results, we calculated an “exact” geometry function, G(r,q) by using the Monte Carlo integration approach as adapted by Karaiskos et al.[18],

where r2 = |r‘–r|2 with r being the distance between the ith Monte Carlo generated point and the calculation point (r, θ). The Monte Carlo values of G(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(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 dose-rate Ḋ(r, θ) in water at a point P(r, θ), in units of cGy/h can be obtained from the following equation:

Note that the reference air-kerma rate, is equivalent to Sk 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 dose-rate is calculated by multiplying [m / r] and correcting for the inverse square of the distance, tissue attenuation, self-absorption 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: μs, μf and μw are the linear attenuation coefficient of the active source, of the filtration material and of water, respectively. t1 and t2 are the active radii of the source and the encapsulation thickness, respectively. r, r and r are the distances traveled by photons within the source core, filter material and the water medium, respectively, and r 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 60Co source is positioned at the geometric center of the inactive metallic 60Co material

r = r + r +r      …..(18)

Motivated by our analytic model described for a bare cylindrical source, we consider that the total activity is concentrated at the center of the 60Co material instead of being distributed throughout its volume. The influence of the distribution of activity in the source on the dose-rate is taken into account separately by the line-source-based geometry function G(r,θ). According to our simplified model, the dose-rate Ḋ(r, θ) per S at a point (r, θ) can be written as follows:

where rs, rf and rw (= r - rs – rf) 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 rw. When θ = θ0, rs = t1, rf = t2. For the new design of the 60Co source, t1 = 0.025 mm, t2= 0.015 cm and t1+ t2 = 0.04 cm. Therefore, rw = r and f(rw) = f(r) for values of r larger than 2.5 mm. For θ = θ0, equation 19 can be written as:

Equation 20 is same as equation 15 when we set θ = θ0 in equation 15. For the calculation of the transverse axis dose-rate distribution, equation 20 is good enough. When we set r = r0 = 1 cm in equation 20,

Equation 21 is a general expression for the dose-rate constant of a monoenergetic photon source of active length L.

Dose-rate calculation for BEBIG source by use of the analytic tool

We have adapted the analytical tool described above for calculating dose-rate distributions in water around the new BEBIG 60Co 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. Dose-rate 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 1-cm 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 dose-rate values for the new BEBIG 60Co 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.[38] For example, for the regions close to the source axis, the analytically calculated dose-rates are higher by up to 10% when compared with the published Monte Carlo-based values.[38] 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 μs = 0.47/cm, μf = 0.43/cm and μw = 0.063/cm, all obtained at 60Co 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. TG-43[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. Self-absorption by the source core (μs) and attenuation by the filtration material (μf) 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 integral-based 137Cs dosimetry and derived best-fit parameters for μs and μf. Similarly, Pérez Calatayud et al.[16] derived best-fit parameters for μs and μf in their quantization method-based dosimetry study on CDC-type miniaturized 137Cs sources[15] and the best-fit parameter for μ for 192Ir wires.

Guided by the above-mentioned published work, we treated the parameters μs and μf as free-fit parameters. Instead of using the actual values of μs (= 0.47/cm) and μf (= 0.43/cm) at 60Co energies,[13] we used the fitted values μs = 0.25/cm and μf = 0.25/cm for the new BEBIG 60Co HDR source [Table 1]. For the above-described analytic method, computer software has been developed in C++ computer-programming language. The software generates a complete dosimetry dataset around the source. The data include TG-43 parameters and a 2D look-up table. For the calculation of g(r) and F(r,θ) we used G(r,θ). This is consistent with the updated TG-43 formalism.[6]

Table 1

Actual and fitted values of linear attenuation coefficients for source and filtration materials μs and μf, respectively, for the new BEBIG 60Co source

Results and Discussion

Table 2 compares the values of the dose-rate constants Λ of BEBIG sources to published values.[238] In the present study, this is 1.088 cGy/h/U (for both new and old designs). The GEANT4-based published values of Λ are 1.084 cGy/h/U and 1.087 cGy/h/U for the old and new source designs, respectively.[23] When r0 = 1 cm, equation 9 represents the dose-rate constant for a point source, Λp. The calculated value of Λp for a 60Co 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 EDKnrc-based 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 60Co sources

Table 3 compares the values of g(r) for BEBIG 60C sources calculated in the present study and the EGSnrc-based published work.[8] At r = 2.5 mm, the EGSnrc-based 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%. [Tables 5 and 6] present the 2-D dose-rate distribution in water (in cGy/h/U) for the new source model with 5-mm and 1-mm-length stainless-steel cable, respectively. For the 1-mm cable length, the analytical data agree with the data published by Granero et al.[3] within 1%, and for the 5-mm cable length, the agreement is within 3%. For regions where charged-particle equilibrium exists, a comparison of these data with the corresponding EGSnrc-based 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 Carlo-based published data of radial dose function, gL (r) L, of the new BEBIG 60Co HDR source

Table 4

Anisotropy function F(r,θ) values for the new (model Co0.A86) BEBIG 60Co HDR source.

Table 5

Dose rate distributions per unit air-kerma strength (cGy/h/U) around the new (model Co0.A86) BEBIG 60Co HDR source.

Table 6

Dose rate distributions per unit air-kerma strength (cGy/h/U) around the new (model Co0.A86) BEBIG 60Co HDR source.

Conclusions

We have proposed a point-source-based simple analytic method for calculating the dose-rate distribution in water in units of cGy/h/U for a BEBIG 60Co HDR source. Using this method, we calculated TG-43 parameters such as the dose-rate constant, radial dose function and anisotropy function. We also calculated a 2-D dose-rate table in Cartesian format. The proposed analytic method needed best-fit 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 treatment-planning dose calculations. For regions where electronic equilibrium exists, a comparison between the analytically calculated and published Monte Carlo-based data shows good agreement (for most calculation points, agreement was within 0.5%, and the maximum deviation was about 3%). The dose-rate data calculated with this method could be used for verifying the results of RTPS or directly as input data for radiotherapy treatment-planning dose calculations.