Impact of the differential fluence distribution of brachytherapy sources on the spectroscopic dose-rate constant Martha J. Malin,a) Laura J. Bartol,a) and Larry A. DeWerda) Department of Medical Physics, University of Wisconsin - Madison, Madison, Wisconsin 53705

(Received 15 December 2014; revised 6 March 2015; accepted for publication 4 April 2015; published 16 April 2015) Purpose: To investigate why dose-rate constants for 125I and 103Pd seeds computed using the spectroscopic technique, Λspec, differ from those computed with standard Monte Carlo (MC) techniques. A potential cause of these discrepancies is the spectroscopic technique’s use of approximations of the true fluence distribution leaving the source, ϕfull. In particular, the fluence distribution used in the spectroscopic technique, ϕspec, approximates the spatial, angular, and energy distributions of ϕfull. This work quantified the extent to which each of these approximations affects the accuracy of Λspec. Additionally, this study investigated how the simplified water-only model used in the spectroscopic technique impacts the accuracy of Λspec. Methods: Dose-rate constants as described in the AAPM TG-43U1 report, Λfull, were computed with MC simulations using the full source geometry for each of 14 different 125I and 6 different 103Pd source models. In addition, the spectrum emitted along the perpendicular bisector of each source was simulated in vacuum using the full source model and used to compute Λspec. Λspec was compared to Λfull to verify the discrepancy reported by Rodriguez and Rogers. Using MC simulations, a phase space of the fluence leaving the encapsulation of each full source model was created. The spatial and angular distributions of ϕfull were extracted from the phase spaces and were qualitatively compared to those used by ϕspec. Additionally, each phase space was modified to reflect one of the approximated distributions (spatial, angular, or energy) used by ϕspec. The dose-rate constant resulting from using approximated distribution i, Λapprox,i , was computed using the modified phase space and compared to Λfull. For each source, this process was repeated for each approximation in order to determine which approximations used in the spectroscopic technique affect the accuracy of Λspec. Results: For all sources studied, the angular and spatial distributions of ϕfull were more complex than the distributions used in ϕspec. Differences between Λspec and Λfull ranged from −0.6% to +6.4%, confirming the discrepancies found by Rodriguez and Rogers. The largest contribution to the discrepancy was the assumption of isotropic emission in ϕspec, which caused differences in Λ of up to +5.3% relative to Λfull. Use of the approximated spatial and energy distributions caused smaller average discrepancies in Λ of −0.4% and +0.1%, respectively. The water-only model introduced an average discrepancy in Λ of −0.4%. Conclusions: The approximations used in ϕspec caused discrepancies between Λapprox,i and Λfull of up to 7.8%. With the exception of the energy distribution, the approximations used in ϕspec contributed to this discrepancy for all source models studied. To improve the accuracy of Λspec, the spatial and angular distributions of ϕfull could be measured, with the measurements replacing the approximated distributions. The methodology used in this work could be used to determine the resolution that such measurements would require by computing the dose-rate constants from phase spaces modified to reflect ϕfull binned at different spatial and angular resolutions. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4918325] Key words: brachytherapy, dose-rate constant, spectroscopic dose-rate constant, differential fluence distribution, Monte Carlo

1. INTRODUCTION Clinical dosimetry of low-energy, photon-emitting brachytherapy sources currently uses the methodology of the American Association of Physicists in Medicine Task Group 43 report (TG-43) and its update.1,2 TG-43 utilizes clinically measured air-kerma strengths, SK , and source-modeldependent dose-rate constants, Λ, to convert tabulated relative dose distributions to dose rates. Λ is defined as the ratio 2379

Med. Phys. 42 (5), May 2015

of the dose rate in water at one centimeter from the source along the perpendicular bisector of the source to SK .2 While measurements with thermoluminescent dosimeters (TLDs) and simulations using Monte Carlo (MC) are the predominant techniques used to determine Λ, a technique, devised by Chen and Nath, that combines spectral measurements with analytical calculations has also been used.3–6 The spectroscopic dose-rate constant, Λspec, is computed using the following equation:

0094-2405/2015/42(5)/2379/10/$30.00

© 2015 Am. Assoc. Phys. Med.

2379

2380

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

 Λspec =

µ en i Ei · n(i) · ( ρ )air,i · Λmono(Ei ) ,  µ en i Ei · n(i) · ( ρ )air,i

(1)

where the sum is taken over all the peaks of the energy spectrum measured along the perpendicular bisector of the source, Ei is the energy of peak i, (µen/ρ)air,i is the mass energy-absorption coefficient for air at the energy of peak i, and n(i) is the fluence of peak i. Λmono(Ei ) is the doserate constant computed for a monoenergetic photon source of energy Ei using an isotropically emitting line or dual point source geometry. Λmono(Ei ) is called the monoenergetic doserate constant.3 The spectroscopic technique approximates the differential photon fluence leaving the encapsulation of the source. The approximated differential photon fluence used to compute ⃗ Λspec is denoted by ϕspec(⃗r ,Ω,E), where ⃗r is the spatial coordinate of the line or dual point source at which the photon ⃗ is the direction at which the photon is initially is generated, Ω going, and E is the energy of the photon. The variation of ϕspec with respect to position is given by ϕspec,r (⃗r ),   ⃗ ϕspec,r (⃗r ) = ϕspec(⃗r ,Ω,E) dΩdE. (2) E Ω

When Λspec is computed using a line source, ϕspec,r (⃗r ) will be constant across the length of the line source and zero outside of the length. When Λspec is computed using a dual point source, ϕspec,r (⃗r ) will have a nonzero value at the locations of the two point sources and will be zero elsewhere. The angular distribution at ⃗r = r is given by  ⃗ = ϕspec(⃗r = r,Ω,E) ⃗ ϕspec,Ω(⃗r = r,Ω) dE. (3) E

⃗ will have an isotropic distribution at all ⃗r . ϕspec,Ω(⃗r = r,Ω) ⃗ = Ω, the energy spectrum is given by Finally, at ⃗r = r and Ω ⃗ = Ω,E) = ϕspec(⃗r = r,Ω ⃗ = Ω,E). ϕspec, E (⃗r = r,Ω

(4)

⃗ = Ω,E) will be the same for all ⃗r and Ω, ⃗ namely, ϕspec, E (⃗r = r,Ω which was measured along the perpendicular bisector of the source. ϕspec is only an approximation of the fluence leaving the encapsulation of the actual source. The differential photon fluence leaving the encapsulation of the source is denoted by ⃗ ϕfull(⃗r ,Ω,E), where, here, ⃗r is the position on the encapsulation ⃗ is the angle with respect from which the photon is emitted, Ω to the encapsulation that the photon is emitted at, and E is the energy of the particle when it leaves the encapsulation. As will be shown in this paper, the spatial variation in ϕfull, given by   ⃗ ϕfull,r (⃗r ) = ϕfull(⃗r ,Ω,E) dΩdE, (5) E Ω

is not uniform with position on the source, as approximated by ϕspec,r . The angular distribution of the fluence at ⃗r = r is given by  ⃗ ⃗ ϕfull,Ω(⃗r = r,Ω) = ϕfull(⃗r = r,Ω,E) dE. (6) E

⃗ as will be shown, does not have an isotropic ϕfull,Ω(⃗r = r,Ω), distribution as approximated by ϕspec,Ω. Additionally, the Medical Physics, Vol. 42, No. 5, May 2015

2380

shape of ϕfull,Ω depends on the value of r. Finally, the energy spectrum emitted from the source is given by ⃗ = Ω,E) = ϕfull(⃗r = r,Ω ⃗ = Ω,E). ϕfull, E (⃗r = r,Ω

(7)

⃗ = Ω,E) can vary with the value r The shape of ϕfull, E (⃗r = r,Ω and Ω, unlike the approximation used in ϕspec, E . This work studies if approximating ϕfull by ϕspec degrades the accuracy of Λspec. Specifically, the approximations used in ϕspec,r , ϕspec,Ω, and ϕspec, E were studied. Even though ϕfull is only approximated by ϕspec, Λspec values agree with those determined using TLDs or MC to within the uncertainty of each technique (the uncertainties of those determinations were approximately 4% for the spectroscopic method, 8% for TLD measurements, and 3% for MC simulations at k = 1).2,5 Rodriguez and Rogers were the first to directly study the impact of approximating ϕfull on the accuracy of Λspec.7 They reduced the relative uncertainty between MC-computed and spectroscopically determined dose-rate constants for 20 LDR models by simulating, rather than measuring, the spectrum used to determine Λspec from a MC model of each source. The dose-rate constant computed using the full source model, Λfull, was also determined using MC. As the same cross sections were used for Λspec and Λfull, the relative uncertainty between the two determinations of Λ was reduced to the statistical component of the uncertainties of the simulations (0.2% at k = 1). Using this approach, they found that Λspec differed from Λfull by up to 4.6% with an average offset of approximately 3%. This discrepancy is within the uncertainty of prior spectroscopic determinations but likely stems from a systematic source of error not reported in the stated uncertainty budget. Rodriguez and Rogers hypothesized that the discrepancy between Λspec and Λfull was caused by approximating ϕfull,Ω with ϕspec,Ω. Though not discussed by Rodriguez and Rogers, the approximations used for ϕfull,r and ϕfull, E may also contribute to this discrepancy. When computing Λmono, the line or dual point source is placed directly in water (for the dose rate in water calculation) or vacuum (for the air-kerma strength calculation) with none of the components of the real source included in the geometry. As the spectrum used to weight Λmono is measured after leaving the encapsulation of the source, interactions between decay photons and the components of the source are still accounted for in the spectroscopic technique. Additionally, as a full water medium is used for the dose rate calculation, scatter within the water medium is largely accounted for. One scatter condition that is not accounted for in the spectroscopic technique is any photons that scatter in the water and are directed back into the source. When computing Λmono, these photons will continue to interact with a water medium, as the source is not present. In the full MC simulation methodology, these photons will interact with the source. This difference in medium has the potential to alter the dose rate in water and, therefore, may be a contributing factor to the difference between Λfull and Λspec. As air-kerma strength is defined in vacuum, this photon trajectory will only occur when computing the dose in water. This work investigates if the approximations used in the spectroscopic technique contribute to the discrepancies

2381

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

between Λspec and Λfull that were uncovered by the work of Rodriguez and Rogers. The effect on the accuracy of Λspec from using the approximated ϕfull,r , ϕfull,Ω, and ϕfull, E distributions was determined, with the effect of each approximation isolated. The impact on the accuracy of Λspec from not including the components of the source in the dose calculation was also determined. Identification of which of these effects impact the accuracy of Λspec can improve brachytherapy dosimetry. By either using more accurate approximations of the fluence leaving the source or by measuring the required fluence distributions, a modified spectroscopic technique may compute dose-rate constants that converge to TLD-measured and MC-computed values.

2. METHODS 2.A. Monte Carlo code and seed geometry modeling

The Monte Carlo N-Particle Code 5 (5) version 1.60 was used to characterize the differential fluence distribution leaving the encapsulation of the source and to compute the dose in water per starting particle (D w /sp) and the air-kerma per starting particle (Kair/sp) for each seed model studied.8 Photon-only transport was used. Coherent and incoherent scattering, photoelectric capture, and fluorescence derived from photoelectric capture were modeled with cross section data from the MCPLIB84 photon cross section library.9,10 A 5 keV energy cutoff was used for the air-kerma simulations in accordance with the TG-43 definition of air-kerma strength. A 1 keV energy cutoff was used for all other simulations. Simulations were run until the relative errors reported in the 5 output files were less than 0.2%. The same 20 sources that Rodriguez and Rogers studied (6 103Pd and 14 125I) were modeled in this work.7 Source model geometries were based on the descriptions provided by Taylor and Rogers and the works cited therein.11 Renderings of several of the source models used in this study are shown in Fig. 1. The models can be roughly broken into three categories: those with radioactive material on a single, cylindrical internal

component [Figs. 1(a), 1(c), and 1(d)]; those with radioactive material on multiple internal components near a central radioopaque marker [Figs. 1(b) and 1(f)]; and those with the radioactive material on multiple internal components but without a central radio-opaque marker [Fig. 1(e)]. Initial photon spectra for both 103Pd and 125I sources were obtained from the AAPM TG-43U1 report.2 The accuracy of the source models was verified by computing the TG-43U1 Λ using Monte Carlo simulations that utilized the full source geometry (rather than the spectroscopic technique) for each source and comparing to the literature values.2 The air-kerma strength computed for Λ was tallied using a geometry that was equivalent to the 7.6◦ half-angle of the aperture used by the National Institute of Standards and Technology (NIST) wide-angle free-air chamber (WAFAC) which has an 8 cm in diameter defining aperture located 30 cm from the source.12 The simulation methodology described by Kennedy et al. was used.13 For each source model, Λ computed for this work was compared to Λ reported in the 2014 paper of Rodriguez and Rogers.14 A direct comparison to Taylor and Rogers was not used as the 2014 paper of Rodriguez and Rogers indicated several discrepancies in the original work of Taylor and Rogers which were amended in the 2014 paper of Rodriguez and Rogers.14 Agreement between the dose-rate constants determined in this work and those determined by Rodriguez and Rogers was within 1.2% for all sources with the exception of the STM, Implant Model STM1251, which was within 1.6% 2.B. Determining the differential fluence distribution

⃗ and energy (E) The underlying spatial (⃗r ), angular (Ω), ⃗ distributions of ϕfull(⃗r ,Ω,E) were studied. As the sources modeled had cylindrical symmetry, ⃗r was uniquely defined by the distance from the center of the source along the source’s ⃗ was given by the spherical polar and longitudinal axis (z). Ω azimuthal angles (α, β). Along the cylindrical portion of the encapsulation (shown as dotted lines in Fig. 2), emission parallel to the positive z-axis of the source was defined as having a polar angle of emission of zero radians. Along

F. 1. Rendering of six of the source models used in this work. Medical Physics, Vol. 42, No. 5, May 2015

2381

2382

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

F. 2. The coordinate system used in this work. The dotted lines illustrate the cylindrical portion of the encapsulation. The differential fluence, ϕ, is a function of position along the encapsulation, z, and angle of emission from the encapsulation. Angle of emission is denoted by the spherical coordinates (α, β).

the cylindrical portion of the encapsulation, an azimuthal angle of zero was defined such that the direction given by (α, β) = (π/2,0) was perpendicular to the surface of the encapsulation at the point of interest. Figure 2 shows a diagram of the coordinate system used to describe the positional and angular variations of the fluence in this work. The shapes of the underlying spatial and angular distributions of ϕfull and ϕspec were qualitatively compared. A phase space of photons exiting the cylindrical portion of the encapsulation of each source was generated. The source model was placed in vacuo for the simulation used to determine ϕfull. Using the data in the phase space, ϕfull,r and ϕfull,Ω were binned, plotted, and compared to the corresponding distribution from ϕspec. ϕfull,r was binned at a resolution of 0.05 mm from z = 0 mm to the end of the cylindrical portion of the encapsulation of the source. The polar angular fluence distribution, ϕfull,α , which was found to vary as a function of z, was binned separately for each z-bin used to determine ϕfull,r . The polar angle bins were 0.01 radians wide and were normalized by the solid angle of each bin. Polar bins ranged from 0◦ to 180◦. The azimuthal distribution, ϕfull, β , which was found to vary as a function of position and polar angle of emission, was binned at several z coordinates (0.1 mm-wide bins) and several polar angles (0.02 radian-wide bins) with 0.02 radian-wide azimuthal bins. The increased width of the bins in the z direction was needed to reduce the noise in the binned azimuthal distributions. Azimuthal bins ranged from −90◦ to 90◦. 2.C. Dose-rate constant simulations

Λfull was determined by simulating both D w /sp and Kair/sp using a full model of each source. For the simulation, the source was placed in the center of a spherical water phantom (mass density of 0.998 g/cm3) with a radius of 30 cm. The tally Medical Physics, Vol. 42, No. 5, May 2015

2382

cell consisted of a subsection of a spherical shell concentric with the center of the source. The spherical shell had inner and outer radii of 0.995 and 1.005 cm, respectively. Dose was tallied using the part of the shell with z between −0.018 and +0.018 cm. D w /sp was approximated as collision kerma per starting particle and tallied with a F6:p energy deposition tally. Kair/sp was simulated with the source centered in a 110 cm radius vacuum sphere. Kair/sp was tallied at 100 cm from the center of the source in a ring-shaped tally cell with a thickness of 0.05 cm that subtended a 15.2◦ region centered on the transverse axis of the source. The 15.2◦ angle was used so that the simulation geometry used the same halfangle as the NIST WAFAC. A modified energy fluence tally (∗F4:p) was used to determine Kair/sp. The energy fluence tally was modified by multiplying the tally contribution by the appropriate mass energy-absorption coefficient for air. The mass energy-absorption coefficients were determined using the procedure described by Kennedy et al.13 The tally output was multiplied by the squared radial distance from the center of the source to the tally cell. This distance-corrected air kerma was divided into the dose per starting particle to determine the dose-rate constant. 2.D. Spectroscopic dose-rate constant

Similar to the work of Rodriguez and Rogers, the spectroscopic dose-rate constant was determined from a simulated, rather than measured, spectrum for each of the 20 source models studied. The spectrum was tallied in vacuo in a ring-shaped cell (0.05 cm-thick and 7.6◦ half-angle) with a front face 10 cm away from the source along the source’s perpendicular bisector. The 10 cm distance to the surface of the tally cell was used to replicate the work of Rodriguez and Rogers. The energy spectrum was tallied with a F4:p fluence tally and was binned into 0.2 keV bins. Using the simulated spectrum, Λspec was determined for each of the 20 sources studied and compared to the MCcomputed dose-rate constant using the full source geometry, Λfull. The full simulated energy spectrum instead of the primary spectrum alone was used in this work as Rodriguez and Rogers showed that the inclusion of the scattered photon spectrum does not alter Λspec to within 0.5%.7 Instead of summing MC-generated Λmono values to determine Λspec, the dose in water and air kerma were simulated for either a line or dual point source isotropically emitting the tabulated spectrum using the methods described in Sec. 2.C. This technique for computing Λspec is equivalent to that used by Chen and Nath with the weighting and summing of the Λmono values completed internally by the MC code. The same source models and cross sections were used to simulate the spectrum and compute both Λspec and Λfull. Thus, the relative uncertainty between the spectroscopic and full dose-rate constants was reduced to the statistics of the simulations. 2.E. Impact of approximations on Λspec

The extent to which approximating ϕfull by ϕspec,r , ϕspec,Ω, and ϕspec, E affects the accuracy of Λspec was studied. The

2383

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

impact of using ϕspec,Ω was studied for both the polar distribution, ϕspec,α , and azimuthal distribution, ϕspec, β . ϕfull was tallied for each source model by generating a phase space of the fluence leaving the encapsulation of each full source model. To isolate the effect of each approximation, i, ϕfull,i in the phase space was resampled using the cumulative density function of ϕspec,i , while the other distributions from ϕfull were left unaltered. The dose-rate constant when using approximation i, Λapprox,i , was computed from the modified phase space using the tally methods described in Sec. 2.C and was compared to Λfull. When computing Λapprox,i , the components of the source were fully modeled in the simulation geometry to provide the appropriate scatter conditions. This process was repeated for each of the approximated distributions. The four ϕspec,i distributions used when rewriting the phase spaces were defined to closely match those used in the spectroscopic technique but were modified where needed to account for the three-dimensional nature of the phase space files. The discrepancy introduced to Λspec by using ϕspec, E was determined by resampling ϕfull, E in the phase space using the energy distribution tallied along the perpendicular bisector of the source as described in Sec. 2.D. To rewrite the phase space to reflect ϕspec,α or ϕspec, β , the appropriate angle was sampled isotropically. Photons with resampled directions pointing into the source were rejected and resampled. To rewrite the phase space to reflect ϕspec,r , the z-coordinate was resampled from either a uniform distribution with the same extent as the effective length of the source (for the line source models) or equally from one of the two values (for the dual point sources). The other spatial coordinates were not altered. When resampling using ϕspec,r , ϕspec,α , or ϕspec, β , the unaltered phase space only included photons leaving the cylindrical portion of the encapsulation. This was done to simplify the coordinate system used for the work as the ends of the sources studied have a wide variety of shapes, each of which would have required a different coordinate system to enable the alterations to the phase spaces described below. For these cases, Λapprox,i was compared to the dose-rate constant computed using photons leaving only the cylindrical portion of the encapsulation, Λcyl. The unaltered phase space used when resampling using ϕspec, E contained photons leaving both the cylindrical and ends of the encapsulation. Λapprox, E was compared to Λfull.

2383

There is a nontrivial dependence of ϕfull on z, E, α, and β (for example, the spatial and angular dependence of the energy spectrum). Due to these complex relationships, resampling a specific ϕspec,i may lead to unintentional resampling of the remaining distributions. These unintentional resamplings could potentially alter Λapprox,i , therein masking the effect of the ϕspec,i under investigation. To prevent this, distributions that would be unintentionally altered were resampled using the distributions discussed below and summarized in Table I. ⃗ As discussed in Sec. 1, ϕfull, E varies with both ⃗r and Ω. The spatial and angular dependencies of ϕfull, E mean that when ϕspec,r , ϕspec,α , or ϕspec, β are used to resample ϕfull, ϕfull, E will also be unintentionally changed. This was prevented by also resampling ϕfull, E with ϕspec, E when altering the phase spaces using either ϕspec,r , ϕspec,α , or ϕspec, β . As will be shown in Sec. 3.C, resampling with ϕspec, E introduced minimal changes to Λ. As the error from using ϕspec, E was known, the error from using ϕspec,r , ϕspec,α , or ϕspec, β could be determined. Unintentional resampling will also alter ϕfull, β , which is a function of α and ⃗r (see Sec. 3.A), when resampling with ϕspec,α or ϕspec,r . This was prevented by also resampling ϕfull, β by a distribution called ϕsingle, β when resampling with ϕspec,α or ϕspec,r . ϕsingle, β is the azimuthal distribution emitted within 0.01 radians of α = π/2 and within 1 mm of the z-coordinate centered on one of the radioactive elements. Initial simulations showed that Λ computed from a phase space modified to have this distribution agreed with Λfull to within the statistics of the simulation (0.2% at k = 1). Finally, the ⃗r dependence of ϕfull,α will cause that distribution to be unintentionally altered when resampling with ϕspec,r . For sources modeled with a uniform z distribution, this was prevented by also resampling ϕfull,α with a distribution denoted by ϕbinned,α . ϕbinned,α is a collection of polar distributions (bin width of 0.01 radians) binned both as a function of angle and as a function of z (bin width of 0.05 mm). Dose-rate constants calculated using a phase space resampled with ϕbinned,α agreed with Λfull to within the statistics of the simulation (0.2% at k = 1) for 17 of the 20 sources studied. Three source models had larger discrepancies: the Theragenics Model 200 (0.4%), the IBt 1032P (0.8%), and the IsoAid IAPd-103A (0.5%). As the error from using ϕbinned,α was known, the additional error from using ϕspec,r could be determined. For sources modeled with

T I. Distributions used to resample the phase space (columns two through seven) when determining the discrepancy to Λ introduced by each of the four approximations studied (column one). ϕ spec, E is the use of the approximate energy distribution. ϕ single, β is the use of the single azimuthal distribution. ϕ spec, β or ϕ spec, α is isotropic emission in the azimuthal or polar directions, respectively. ϕ binned, α is the use of the binned polar distribution. ϕ spec, z is either uniform emission along the effective length of the source or from the two z-coordinates of the dual point source, depending on the source being modeled. Distributions used to modify phase space Approximate distribution Energy Angular: azimuthal Angular: polar Spatial

Medical Physics, Vol. 42, No. 5, May 2015

ϕ spec, E

ϕ single, β

ϕ spec, β

ϕ spec, α

ϕ binned, α

ϕ spec, z

   

— —  

—  — —

— —  —

— — — 

— — — 

2384

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

emission from two rings, the polar distribution was sampled from a single distribution binned at the same z-coordinate as the center of one of the internal radioactive elements. The effect of not modeling the source components when determining Λspec was also studied by computing Λ using the phase space of the fluence leaving the entire encapsulation of the source. The phase space was left unmodified, but the source components were replaced with either water (for the dose in water simulation) or vacuum (for the airkerma simulation). The dose-rate constant computed with this geometry, Λapprox,scatter, was compared to Λfull. As the phase spaces used in this work recorded the fluence leaving the encapsulation of the sources, they had threedimensional extents; however, the spectroscopic technique utilized a two-dimensional source model (i.e., either a line source or two point sources). Because of this, Λapprox,i cannot be directly compared to Λspec. Instead, the dose-rate constant was computed from a phase space with the same fluence distribution as ϕspec, except the fluence was emitted from a cylinder or two rings rather than a line or two points. The dose-rate constant computed from this altered phase space, ΛspecCyl, was compared to Λfull to determine the magnitude of the discrepancy expected when all the approximations studied in this work were applied to the three-dimensional phase spaces.

2384

F. 4. ϕ full, α of the Model 6711 at five z locations on the encapsulation. The solid angle was normalized to the bin with the highest intensity. The statistical component of the uncertainty ranges from 0.5% in regions with a high intensity to approximately 3% in regions with lower intensities.

The spatial and angular distributions of ϕfull were more complex than those of ϕspec. ϕfull,r for a subset of the source models studied is shown in Fig. 3. For all the models studied, the intensity of ϕfull,r varied with z. The shape of ϕfull,r was strongly dependent on the internal geometry of the source. The line source approximation used by ϕspec,r most closely approximated ϕfull,r when the radioactive material of the source was coated on a single, cylindrical internal component (e.g., Model 6711 in Fig. 3). ϕfull,r for sources with a central radio-opaque marker had two peaks as approximated by the

dual point source model of Λspec, but the spatial extent of each peak was larger than the point sources used in ϕspec (e.g., Model 200 in Fig. 3). ϕfull,Ω was not isotropic as assumed by ϕspec,Ω. ϕfull,α is shown in Figs. 4 and 5 for a subset of the seeds studied. ϕfull,α was found to be anisotropic for all sources studied with model-dependent distributions (Fig. 5) that varied with z (Fig. 4). ϕfull, β for several source models is shown in Fig. 6. The shape of ϕfull, β was model dependent. As seen in Fig. 7, the shape varied slightly with z and α for all source models studied. While both the polar and azimuthal distributions were model dependent, models with similar internal structures had similar angular distributions. The nonuniform and nonisotropic nature of ϕfull is largely due to the nonuniform distribution of radioactive material in the source and the varying amounts of material that different photon trajectories encounter while leaving the source. ϕfull,r is most intense at z values of the internal, radioactive component(s). Photons are emitted from z values beyond the extent of the internal, radioactive component(s) because photons are

F. 3. ϕ full, r as a function of position (z) along the encapsulation for three source models (normalized to the bin with the highest intensity). The statistical component of the uncertainty is less than 0.1%.

F. 5. ϕ full, α between z = [0.0, 0.05] mm for four source models representing two different internal geometries. The intensity per unit solid angle was normalized to the polar bin with the highest intensity. The statistical component of the uncertainty ranges from 0.5% in regions with a high intensity to approximately 3% in regions with lower intensities.

3. RESULTS AND DISCUSSION 3.A. Differential fluence distribution

Medical Physics, Vol. 42, No. 5, May 2015

2385

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

F. 6. ϕ full, β for several source models with different internal geometries. The intensity per unit solid angle was normalized to the azimuthal bin with the highest intensity. The distributions were tallied with a 0.2 mm-wide bin centered on one of the radioactive components within the model and with a polar bin width of 1◦ centered at 90◦. The statistical component of the uncertainty ranges from 3% in regions with a high intensity to approximately 5% in regions with lower intensities.

emitted at oblique angles from these components and from the ends of the radioactive component(s) [see Fig. 8(a)]. The nonisotropic nature of ϕfull,α is likely caused by the increased amount of titanium encapsulation photons emitted at oblique angles must transverse compared to those that are emitted perpendicularly to the encapsulation. Additionally, the two smaller peaks on the polar distribution of the Model 6702 seen in Fig. 5 correspond to the angles with a direct line of sight, at the z-value plotted, to the two off-center radioactive spheres (see Fig. 1). The slowly changing shape of the polar distribution of the 6711 when z is less than the effective length (Fig. 4) is caused by the changing angles of emission that correspond to a direct line of sight to the internal radioactive component as illustrated in Fig. 8(b). All arrows shown in

F. 7. ϕ full, β leaving the encapsulation of the Model 6711 binned at several different polar angles and positions along the encapsulation (normalized to the azimuthal bin with the highest intensity). The distributions were tallied with a 0.2 mm-wide bin centered at the z-coordinate listed with polar bin widths of 1◦ centered at the polar angle listed. The statistical component of the uncertainty ranges from 3% in regions with a high intensity to approximately 5% in regions with lower intensities. Medical Physics, Vol. 42, No. 5, May 2015

2385

F. 8. Diagrams illustrating the cause for the spatial and angular fluence distributions. (a) Photons leaving the internal component at an oblique angle [arrow (ii)] and photons generated at the end of the internal component [arrow (i)] will exit the encapsulation with z-values greater than the active length of the source. (b) For a given polar angle, z-coordinates with no direct line of sight to the internal component [arrow (i)] will have a reduced intensity. (c) The shape of the polar distribution changes at the polar angle where photons from the end of the internal component reach the encapsulation [arrow (ii)]. (d) A cross sectional view of the source with arrows denoting the line of sight for two azimuthal bins.

Fig. 8(b) have the same value of α, but arrow (i) is not in the line of sight of the radioactive component. Fewer photons will leave with the trajectory given by arrow (i) than with the trajectories given by the other arrows because of this. The z coordinate at which the angle loses its direct line of sight will depend on the angle but will occur at a smaller z value for more oblique angles. The abrupt peaks in the shape of the polar distribution for the 6711 at z = 1.225 mm and z = 1.625 mm (Fig. 4) occur at the angle where there is a direct line of sight to the end of the cylindrical internal component of the source [see arrow (ii) for Fig. 8(c)]. At angles smaller than that labeled (ii) in Fig. 8(c), the majority of the fluence will come from the side of the internal component [arrow (iii) in Fig. 8(c)] and will have a similar intensity to that emitted with the same angle at other, smaller, z values. Nonisotropic emission in the azimuthal direction (Fig. 6) is caused by the distribution of the radioactivity on the internal component. The radioactive material for the Models 6711 and 200 is coated on the surface of the internal component. At certain azimuthal angles, more of the radioactive material has a direct line of sight to the azimuthal bin of interest, and the intensity of that bin is greater than the other bins. This condition is illustrated in Fig. 8(d). In the figure, angle (i) has a direct line of sight to more radioactive material than angle (ii) and so will have a higher intensity. The azimuthal distribution of the Model IAPd-103A is representative of the azimuthal distribution when the radioactive material is uniformly distributed throughout the internal component. For these models, angle (ii) in Fig. 8(d) has a direct line of sight to more radioactive material than angle (i) and so has the higher intensity. 3.B. Spectroscopic dose-rate constant

Table II shows Λfull, Λspec, and ΛspecCyl for the 20 source models studied by Chen and Nath.5 The separation distances between the dual point sources were taken from Rodriguez and Rogers.7 The discrepancies between Λspec and Λfull identified in this work were similar to those observed by Rodriguez

2386

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

2386

T II. Comparison of the full source model dose-rate constant (Λfull), the dual point/line source dose-rate constant (Λspec), the spectroscopic dose-rate constant computed from photons leaving a cylinder or two rings (ΛspecCyl), and the dose-rate constant of the full source model only computed with photons leaving the cylindrical portion of the encapsulation (Λcyl). The ratio of Λspec to Λfull found by Rodriguez and Rogers is shown in parentheses for comparison. Source models marked with an asterisk are still being manufactured as of January 2015. Model

Λfull

Λcyl

Λspec

ΛspecCyl

Λspec/Λfull (Rodriguez and Rogers)

ΛspecCyl/Λfull

I-125 GE 6711* Imagyn IS-12051 MBI SL-125 6733 IsoAid IAI-125A* Nucletron 130.002* Draximage LS-1 Implant Sciences 3500 Bebig/Thera I25.S06* OncoSeed 6702 NASI MED3631 Best 2301* STM 1251* IBt 125 1L

0.924 0.915 0.930 0.919 0.915 0.917 0.925 0.993 0.999 1.006 0.993 0.998 0.976 0.991

0.922 0.912 0.928 0.915 0.909 0.910 1.215 0.989 0.994 1.002 1.030 0.988 0.971 0.985

0.955 0.946 0.953 0.944 0.951 0.949 0.960 1.016 1.017 1.020 1.013 1.016 1.017 1.010

0.981 0.971 0.980 0.966 0.974 0.925 0.984 1.043 1.045 1.048 1.033 1.040 1.044 1.036

1.033 (1.033) 1.033 (1.035) 1.024 (1.024) 1.027 (1.021) 1.040 (1.034) 1.035 (1.040) 1.038 (1.043) 1.023 (1.012) 1.018 (1.008) 1.014 (1.017) 1.020 (1.021) 1.018 (1.026) 1.042 (1.028) 1.019 (1.026)

1.061 1.061 1.053 1.051 1.064 1.065 1.064 1.050 1.046 1.042 1.040 1.042 1.069 1.045

Pd-103 Theragenics 200* NASI MED3633 Best 2335* IBt 1032P Draximage PD-1 IsoAid IAPd-103A*

0.683 0.662 0.654 0.667 0.621 0.659

0.681 0.694 0.696 0.663 0.911 0.656

0.679 0.674 0.668 0.664 0.661 0.676

0.705 0.699 0.693 0.688 0.684 0.701

0.994 (0.985) 1.018 (1.008) 1.022 (1.014) 0.996 (0.990) 1.064 (1.046) 1.027 (1.015)

1.033 1.056 1.059 1.032 1.102 1.064

and Rogers. In this work, the discrepancy averaged +2.7% for 125I sources (range of +1.4% to +4.2%) and +2.0% for 103 Pd sources (range of −0.6% to +6.4%). Positive percentages indicate that Λspec was larger than Λfull. These discrepancies were similar to those found by Rodriguez and Rogers which averaged +2.6% for 125I source and +1.0% for 103Pd sources. The differences in the discrepancies are likely caused by differences in the MC models of the sources. The discrepancy between Λfull and ΛspecCyl was larger than that seen with Λspec. The average discrepancy was +5.4% for 125 I sources (range of +1.0% to +6.9%) and +5.7% for 103Pd sources (range of +1.03% to +10.2%). The difference between ΛspecCyl and Λspec is caused by the nonzero radius of the cylinder from which the particles were emitted when determining ΛspecCyl. The majority of the difference was due to differing dose tallies, which were larger when the cylinder phase space was used. Both the cylinder and the line sources produced air-kerma tallies that agreed to within 0.4%. The difference in the dose was traced to an increased fluence through the tally cell when the cylinder is used and is likely caused by the differences in inverse squared divergence between the line and the cylinder. The value of ΛspecCyl converged to that of Λspec as the radius of the cylinder used in ΛspecCyl was decreased. Also shown in Table II are the values of Λcyl, the doserate constant computed from the full source model using only those photons emitted from the cylindrical portion of the encapsulation. Λcyl and Λfull were closest in value when most of the photons were emitted from the cylindrical portion of the source. For 15 of the 20 source models, Λcyl and Λfull were within 1% of each other, suggesting that the value of Medical Physics, Vol. 42, No. 5, May 2015

Λfull was dominated by the fluence leaving the cylindrical portion of these sources. For three of the other sources, the differences were within 5%. Two outliers, the Draximage LS1 and Draximage PD-1, had significantly different values of Λcyl and Λfull. The spatial and angular distribution studies in Sec. 3.C used only the fluence leaving the cylindrical portion of the source. When Λcyl and Λfull were close in value, the spatial and angular distribution studies were reflective of the impact the approximation had on the entire source. For the two Draximage models where this was not the case, the results from the spatial and angular distributions were not necessarily reflective of the results from the full source model. 3.C. Impact of approximations on Λspec

Table III shows the impact of each approximation used in ϕspec on the calculated dose-rate constant, Λapprox,i . The assumption of isotropic emission, particularly in the polar direction, Λapprox,α , had the largest impact on the calculated dose-rate constant. Excluding the Draximage LS-1 and Draximage PD-1, both outliers which are discussed below, Λapprox,α was systematically high with an average discrepancy of +5.3% and a range of +1.2% to +7.8%. The other approximate distributions caused smaller discrepancies in Λapprox,i . In decreasing order of impact, the approximate distributions and the average discrepancy introduced by their use are isotropic azimuthal distribution (Λapprox, β ) at −1.1%, approximate spatial distribution (Λapprox,z) at −0.4%, lack of source components (Λapprox,scatter) at −0.4%, and approximate energy distribution (Λapprox, E ) at +0.1%.

2387

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

2387

T III. Impact of the approximations used in dual point/line source model on the dose-rate constant (Λ). Λfull refers to the dose-rate constant computed for the full source model. Λcyl refers to the dose-rate constant computed using only the photons emitted from the cylindrical portion of the source. The approximation used in each simulation is listed in the table header. Source models marked with an asterisk are still being manufactured as of January 2015. Λapprox, i /Λfull

Λapprox, i /Λcyl

Energy

Scatter

z

Azimuthal

Polar

I-125 GE 6711* Imagyn IS-12051 MBI SL-125 6733 IsoAid IAI-125A* Nucletron 130.002* Draximage LS-1 Implant Sciences 3500 Bebig/Thera I25.S06* OncoSeed 6702 NASI MED3631 Best 2301* STM 1251* IBt 125 1L

1.000 0.999 1.000 1.000 0.998 0.998 1.001 1.002 1.001 1.001 1.000 1.001 0.998 1.005

1.007 1.005 1.004 1.005 1.006 0.998 1.004 1.007 1.004 1.003 1.003 1.006 1.003 1.005

1.004 0.981 0.968 1.000 1.030 1.028 0.714 0.993 0.984 1.029 0.908 0.977 1.041 1.107

0.989 0.990 0.988 0.986 0.987 0.989 0.982 0.994 0.991 0.992 0.990 0.988 0.993 0.992

1.064 1.068 1.060 1.067 1.078 1.075 0.821 1.055 1.055 1.050 1.012 1.063 1.075 1.050

Pd-103 Theragenics 200* NASI MED3633 Best 2335* IBt 1032P Draximage PD-1 IsoAid IAPd-103A*

1.003 1.005 1.005 1.002 1.003 1.006

1.004 1.005 1.006 0.998 1.003 1.005

1.010 0.929 0.926 1.004 0.801 1.013

0.990 0.989 0.987 0.988 0.978 0.988

1.039 1.011 1.009 1.052 0.762 1.076

Model

The differing directions of the disagreement between Λfull and Λapprox,i partially counteract each other when all are applied in ΛspecCyl (Table II). Of the six source models where Λapprox, E was greater than 0.2%, Λapprox, z , Λapprox,α , and Λapprox, β disagreed by at least twice the percentage of the discrepancy introduced by Λapprox, E indicating that using the approximate energy distributions in these simulations was not the sole cause of the disagreement when the spatial or angular approximation was studied. Similarly when using the binned polar distribution for the Theragenics Model 200 and the IsoAid IAPd-103A, the discrepancy introduced by the approximated spatial distribution was more than twice that introduced by the binned polar distribution, indicating that the approximate spatial distribution introduced an additional error into the dose-rate constant. For the IBt 1032P, the discrepancy when using the binned polar distribution and when testing the approximate spatial distribution agreed to within the statistics of the simulation. Thus, for this source model, it was not possible to tell if the spatial approximation introduced an additional discrepancy into the dose-rate constant. The Draximage LS-1 and Draximage PD-1 were outliers in this analysis. The discrepancies introduced by ϕspec,α and ϕspec,r were large for both models: −17.9% and −28.6%, respectively, for the LS-1 and −23.8% and −19.9%, respectively, for the PD-1. The radioactive material for both of these sources is positioned on two spheres. Unlike the other sources used in this work that were modeled as dual-points Medical Physics, Vol. 42, No. 5, May 2015

in the spectroscopic technique, the spheres of the LS-1 and the PD-1 were positioned such that they extended beyond the cylindrical portion of the encapsulation into the hemispherical ends of the source as seen in Fig. 1(f). This geometry caused approximately 50% of the fluence to leave from the end welds of the source rather than the cylindrical portion of the encapsulation. As this work only considered the fluence leaving the cylindrical portion of the encapsulation when studying the impact of assuming ϕspec,r and ϕspec,α , the majority of the fluence leaving these sources was not considered in the analysis. The fraction that was considered represented the fluence leaving the encapsulation at a small subset of the actual range of polar angles with which photons were emitted from the source. Specifically, those angles directed toward the cylindrical portion of the encapsulation were sampled, while those directed to the ends of the source were not. Thus, Λcyl was computed from a very different fluence distribution than that of ϕfull, ϕspec,r , and ϕspec,α and differed greatly from them, causing the large discrepancies for these sources.

4. CONCLUSIONS The differential fluence distribution leaving the encapsulation of low-energy, photon-emitting brachytherapy sources is more complex than ϕspec. Similar to Rodriguez and Rogers, this work found discrepancies between Λfull and Λspec that

2388

Malin, Bartol, and DeWerd: Impact of fluence distribution on spectroscopic dose-rate constant

averaged +2.7% and +2.0% for 125I and 103Pd sources, respectively. The impact the approximate spatial, angular, and energy distributions of ϕspec have on the accuracy of Λspec was determined. Assuming isotropic polar emission introduced the largest error into Λapprox,i with an average discrepancy of +5.3%, though all the approximations studied altered Λapprox,i for at least a subset of the sources. The results of this work show that ϕfull,r , ϕfull,Ω, and, in some cases, ϕfull, E would need to be more accurately modeled by ϕspec to ensure that Λspec agreed with Λfull as measurement uncertainties are reduced. One of the benefits of using the spectroscopic technique to compute Λ is that it does not require knowledge of the geometry of the source. At the low energies of these sources, small inaccuracies in the modeled source geometry can have a large impact on MC-computed dosimetric quantities. To preserve this feature of the spectroscopic technique, ϕfull would need to be measured rather than simulated. The spatial and angular resolutions necessary for measurements of ϕfull will determine if this approach is feasible. For example, if accurate Λspec determinations required measurements of ϕfull,r with a spatial resolution of micrometers, the measurements would likely not be feasible with current spectroscopic measurement techniques. The methodology of this work could be used to determine this resolution. The phase spaces used in this work could be resampled from binned distributions of ϕfull,r , ϕfull,Ω, and ϕfull, E . The bin width would represent the resolution of the measurement for each distribution. Dose-rate constants computed from these altered phase spaces could be compared to Λfull. This process could be repeated with different bin widths until a bin width was found which produced agreement between Λspec and Λfull. ACKNOWLEDGMENTS The authors wish to thank Wesley Culberson, Joshua Reed, Samantha Simiele, and Michael Lawless for their helpful suggestions and conversations regarding this paper. The authors also thank the customers of the University of Wisconsin Radiation Calibration Laboratory and University of Wisconsin Accredited Dosimetry Calibration Laboratory, whose calibrations help support ongoing research at the

Medical Physics, Vol. 42, No. 5, May 2015

2388

University of Wisconsin Medical Radiation Research Center. Finally, the authors wish to thank the reviewers of this paper for their helpful suggestions. a)Electronic

addresses: [email protected]; [email protected]; and [email protected]. 1R. Nath, L. L. Anderson, G. Luxton, K. A. Weaver, J. F. Williamson, and A. S. Meigooni, “Dosimetry of interstitial brachytherapy sources: Recommendations of the AAPM Radiation Therapy Committee Task Group No. 43,” Med. Phys. 22, 209–234 (1995). 2M. J. Rivard, B. M. Coursey, L. A. DeWerd, W. F. Hanson, M. S. Huq, G. S. Ibbott, M. G. Mitch, R. Nath, and J. F. Williamson, “Update of AAPM Task Group No. 43 Report: A revised AAPM protocol for brachytherapy dose calculations,” Med. Phys. 31, 633–674 (2004). 3Z. J. Chen and R. Nath, “Dose rate constant and energy spectrum of interstitial brachytherapy sources,” Med. Phys. 28, 86–96 (2001). 4Z. J. Chen and R. Nath, “Photon spectrometry for the determination of the dose-rate constant of low-energy photon-emitting brachytherapy sources,” Med. Phys. 34, 1412–1430 (2007). 5Z. J. Chen and R. Nath, “A systematic evaluation of the dose-rate constant determined by photon spectrometry for 21 different models of low-energy photon-emitting brachytherapy sources,” Phys. Med. Biol. 55, 6089–6104 (2010). 6J. Usher-Moga, S. M. Beach, and L. A. DeWerd, “Spectroscopic output of 125I and 103Pd low dose rate brachytherapy sources,” Med. Phys. 36, 270–278 (2009). 7M. Rodriguez and D. W. O. Rogers, “On determining dose rate constants spectroscopically,” Med. Phys. 40, 011713 (10pp.) (2013). 8X-5 Monte Carlo Team, —A general Monte Carlo N-particle transport code, version 5, Technical Report No. LA-UR-03-1987 (Los Alamos National Laboratory, Los Alamos, 2003). 9M. C. White, Further notes on MCPLIB03/04 and new MCPLIB63/84 Compton broadening data for all versions of 5, Technical Report No. LA-UR-12-00018 (Los Alamos National Laboratory, Los Alamos, 2012). 10D. Cullen, J. Hubbell, and L. Kissel, EPDL97: The evaluated photon data library, Technical Report No. UCRL-50400 (Lawrence Livermore National Laboratory, Livermore, Rev. 5, 1997), Vol. 6. 11R. E. P. Taylor and D. W. O. Rogers, “An EGSnrc Monte Carlo-calculated database of TG-43 parameters,” Med. Phys. 35, 4228–4241 (2008). 12S. M. Seltzer, P. J. Lamperti, R. Loevinger, M. G. Mitch, J. T. Weaver, and B. M. Coursey, “New national air-kerma-strength standards for 125I and 103Pd brachytherapy seeds,” J. Res. Natl. Inst. Stand. Technol. 108, 337–358 (2003). 13R. M. Kennedy, S. D. Davis, J. A. Micka, and L. A. DeWerd, “Experimental and Monte Carlo determination of the TG-43 dosimetric parameters for the model 9011 THINSeed™ brachytherapy source,” Med. Phys. 37, 1681–1688 (2010). 14M. Rodriguez and D. W. O. Rogers, “Effect of improved TLD dosimetry on the determination of dose rate constants for 125I and 103Pd brachytherapy seeds,” Med. Phys. 41, 114301 (15pp.) (2014).

Impact of the differential fluence distribution of brachytherapy sources on the spectroscopic dose-rate constant.

To investigate why dose-rate constants for (125)I and (103)Pd seeds computed using the spectroscopic technique, Λ spec, differ from those computed wit...
1MB Sizes 0 Downloads 6 Views