Fitted dosimetric parameters of high dose-rate 192 Ir sources according to the AAPMTG43 formalismF. Lliso, J. Pérez-Calatayud, V. Carmona, F. Ballester, J. L. Lluch, M. A. Serrano, Y. Limami, and E. Casal Citation: Medical Physics 28, 654 (2001); doi: 10.1118/1.1359438 View online: http://dx.doi.org/10.1118/1.1359438 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/28/4?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Technical note: Fitted dosimetric parameters of high dose-rate 192 Ir sources according to the AAPM TG43formalism Med. Phys. 30, 651 (2003); 10.1118/1.1561621 Calibration of new high dose rate 192 Ir sources Med. Phys. 29, 1483 (2002); 10.1118/1.1487860 Erratum: “Fitted dosimetric parameters of high dose-rate 192 Ir sources according to the AAPM TG43 formalism”[Med. Phys. 28(4), 654–660 (2001)] Med. Phys. 28, 1964 (2001); 10.1118/1.1398562 Monte Carlo dosimetry of a new 192 Ir high dose rate brachytherapy source Med. Phys. 27, 2521 (2000); 10.1118/1.1315316 Monte Carlo and TLD dosimetry of an 192 Ir high dose-rate brachytherapy source Med. Phys. 25, 1975 (1998); 10.1118/1.598371

Fitted dosimetric parameters of high dose-rate 192Ir sources accordingto the AAPM TG43 formalism

F. Lliso, J. Perez-Calatayud, and V. CarmonaPhysics Section, Radiation Oncology Department. ‘‘La Fe’’ University Hospital, Avda. Campanar 21,E-46009 Valencia, Spain

F. Ballester,a) J. L. Lluch, M. A. Serrano, and Y. LimamiDepartment of Atomic, Molecular and Nuclear Physics (University of Valencia) and Instituto de Fı´sicaCorpuscular (IFIC), C/ Dr. Moliner 50, E-46100 Burjassot, Spain

E. CasalDepartment of Atomic, Molecular and Nuclear Physics (University of Valencia), C/ Dr. Moliner 50,E-46100 Burjassot and Centro Nacional de Dosimetrı´a, Avda. Campanar 21, E-46009 Valencia, Spain

~Received 30 June 2000; accepted for publication 5 February 2001!

The purpose of this study is to find fitted functional forms to the anisotropy function,F(r ,u), andthe radial dose function,g(r ), in order to characterize dose-rate distributions around all the high-intensity 192Ir sources currently in use. Dosimetry data are at present available as tables for: themicroSelectron HDR~‘‘classic’’ and ‘‘new’’ design models!, the PDR source, and the VariSourceHDR source, expressed in terms of the AAPM Task Group No. 43 recommendations. There is onlyone paper out which introduces a functional form to fit the anisotropy function, but only forsymmetric sources with respect to the transverse axis. However, dosimetric data of the HDR andPDR sources mentioned above cannot be reproduced with these functional forms. In our studyF(r ,u) and g(r ) published data are fitted with functional forms in such a way that appropriatelimits are reached for both functions and the maximum fit error approaches the data uncertainty.The average fit error is less than 1% in all cases. These functional forms make handling data easierwithin the treatment planning system, avoiding the use of tabulated data. ©2001 AmericanAssociation of Physicists in Medicine.@DOI: 10.1118/1.1359438#

Key words: dosimetry, brachytherapy, treatment planning systems, HDR sources

I. INTRODUCTION

As indicated in the AAPM Radiation Therapy CommitteeTask Group No. 56~TG56!,1 accurate dose-rate distributiontables of each source type used in clinical practice based onrealistic geometrical and mechanical characteristics areneeded to verify and provide input data for treatment plan-ning systems.

There are a number of single-stepping source pulsed~PDR! and high dose-rate~HDR! remote afterloader devicescurrently in use, and publications about dosimetric data ofthe concerned sources are available. Williamsonet al.2 usesa Monte Carlo code to calculate dose-rate distributionsaround the Nucletron HDR192Ir source~‘‘classic model’’!and the PDR192Ir source. The results are presented as 2-DCartesian lookup tables and in the formalism recommendedby the TG43.3 Wanget al.4 calculated basic dosimetry datafor a VariSource HDR192Ir source and Daskalovet al.5 car-ried out a similar study for the ‘‘new design’’ microSelectronHDR 192Ir source.

The difficulties in the implementation of the TG43 recom-mendations clinically arise from the fact thatg(r ) andF(r ,u) are provided in the form of tables, and their usepresents, in some cases, interpolation and extrapolation prob-lems. Furhang and Anderson6 have discussed the advantagesof using functional fits for the anisotropy function and theradial dose function. They useg(r ), F(r ,u) andfan(r ) data

of the sources addressed by the TG433 (192Ir, 125I and103Pd).The functional forms that they introduce for the anisotropyfunction cannot be used for the sources addressed in thisstudy, first because they only apply to sources which exhibitsymmetry with respect to the transverse axis, and secondthey cannot reproduce the appropriate limits for the wholerange of the data. Moss7 also introduces an analytical fit tothe radial dose function for the same sources.

In this paper we present closed forms forF(r ,u) andg(r )functions obtained by fitting the existing HDR192Ir sourcesdosimetry data mentioned above. The fit parameters of thefunctional forms have been chosen to satisfy the boundaryconditions at the limits; hence, they avoid extrapolationproblems. The proposed functional forms can reproduceF(r ,u) and functions for nonsymmetrical sources giving ac-curate values of these functions in a simple form suitable foruse in treatment planning systems. Using the TG43 formal-ism with these functional forms, the published dose-rate datareported as 2-D Cartesian lookup tables for these sources canbe reproduced very satisfactorily.

II. MATERIAL AND METHODS

All the HDR 192Ir sources in use in clinical practice forwhich there are published dosimetric data available havebeen studied in this paper, and a summary of their character-istics is presented in Table I.

654 654Med. Phys. 28 „4…, April 2001 0094-2405 Õ2001Õ28„4…Õ654Õ7Õ$18.00 © 2001 Am. Assoc. Phys. Med.

A. TG43 dose calculation formalism

The TG43 formalism establishes that the absorbed doserate in a medium at a distancer from the source center and atan angleu relative to the longitudinal axis should be ex-pressed as

D~r ,u!5SkLG~r ,u!

G~r 0 ,u0!g~r !F~r ,u!, ~1!

whereSk is the source air-kerma strength,L is the dose-rateconstant,G(r ,u) is the geometry factor that accounts for thedistribution of the radioactive material,F(r ,u) is the anisot-ropy function that accounts for the angular dependence ofphoton absorption and scattering andg(r ) is the radial dosefunction that accounts for radial dependence of photon ab-sorption and scattering in the medium along the transverseaxis (u5p/2). The reference point (r 0 ,u0) is r 051 cm andu05p/2. After Sk and D(r ,u) are calculated,L, g(r ), andF(r ,u) can be formulated as follows:

L5D~r 0 ,u0!

Sk,

g~r !5D~r ,u0!

D~r 0 ,u0!

G~r 0 ,u0!

G~r ,u0!, ~2!

F~r ,u!5D~r ,u!

D~r ,u0!

G~r ,u0!

G~r ,u!.

B. Functional forms for F„r ,u… and g „r …

For the anisotropy function the general functional formused in this study is

F~r ,u!5k~r !1a~r !~u/p!e(r )

11b~r !~u/p!e(r )

1a8~r !~12u/p!e8(r )

11b8~r !~12u/p!e8(r ), ~3!

which is able to reproduce the data fromu50 to u5p. Theradial dependence of the parameters is

k~r !5k1r k21k3r 1k4

a~r !5a1r a21a3r 1a4 a8~r !5a18ra281a38r 1a48

b~r !5b1r b21b3r 1b4 b8~r !5b18rb281b38r 1b48 ~4!

e~r !5e1r e21e3r 1e4 e8~r !5e18re281e38r 1e48.

For the radial dose function the general functional form is

g~r !5hri

11 j r k. ~5!

C. Fitting procedure

For a fixed radius,r, the shape of the anisotropy functioncurve ~from u50 to u5p/2) is very similar to the magne-tization curve in ferromagnetic materials.8 This curve is re-produced with the last term in Eq.~3!. In order to reproducethe anisotropy function fromu50 to u5p, another equiva-lent term was introduced here@second term in Eq.~3!# thattakes into account that these sources are not symmetricalwith respect to the transverse source axis. Moreover, the con-stant termk(r ) in Eq. ~3! shifts the functional to the adequatevalues. A set of parameters was found for eachr-value andtheir radial dependence was analyzed@Eq. ~4!#. Finally, theparameters for the general formF(r ,u) were fitted. Theshape ofg(r ) can be considered as the mirror of the magne-tization curve, so the same functional forms are valid.

For the fitting process of the tabulated data an objectivefunction has been defined as follows:

X25(i 51

N

Wi@ f fit~a0 ,a1 , . . . . . .ap ,xi !2 f data~xi !#2, ~6!

Wi being a weighting factor,f fit((a0 ,a1 , . . . . . .ap ,xi) theselected functional form for F(r ,u) and g(r ),(a0 ,a1 , . . . . . .ap) the parameters,xi the variablesr andu,and f data(xi) the dosimetric data to which the functional formmust be fitted.

By changing the weighting factor, using an iterative pro-cess, a minimum set of parameters, which minimize the ob-jective function Eq.~6!, was chosen. These parameters pro-vide absolute values of relative deviations between the dataand the predicted value that are closest to the data uncertain-ties and also result in appropriate limit values. The param-eters were chosen in such a way that the boundary conditions

TABLE I. Characteristics of the HDR192Ir sources analyzed in this study. All the physical dimensions are inmillimeters.

MicroSelectron‘‘Classic’’ design

MicroSelectron‘‘New’’ design

MicroSelectronPDR VariSource

Active length 3.5 3.6 0.6 10Distal source face to tipdistance

0.35 0.2 0.55 1

Active diameter 0.6 0.65 0.6 0.35Total diameter 1.1 0.9 1.1 0.61Encapsulation stainless steel stainless steel stainless steel nickel/titaniumManufacturer Nucletron Nucletron Nucletron Varian Oncology

Systems

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are satisfied:~i! F(r ,p/2)51 andg(1)51, ~ii ! the anisot-ropy function approaches 1 at larger, and ~iii ! g(r ) givesreasonably small values for larger. Initially the weightingfactor of each data was fixed as the inverse of its relativeerror. To satisfy the boundary conditions the data corre-sponding toF(r ,p/2) and g(1) receive a heavier penalty~larger weighting factors!.

III. RESULTS AND DISCUSSION

A. Anisotropy function F„r ,u…

Fit parameters of the functional form are presented inTable II. By weighting appropriately the data as explained

above, the parameters have been chosen in order to obtainabsolute values of relative deviations between the anisotropyfunction Monte Carlo data,Fdata, and the predicted value,Ffit , that are as close as possible to the reported data uncer-tainties for each source. The Nucletron sourcesF(r ,u) ex-hibit uncertainties ranging from 1% to 3.5%2,5 ~near to farfrom the sources! and for the Varisource4 the uncertaintiesare within 1% for most values and between 1% and 5% forvalues close to the long axis. Fit functional forms for eachsource are plotted in Figs. 1–4 along with Monte Carlo data.It can be seen that the results are within the desired uncer-tainties except for some isolated points. The absolute values

TABLE II. Fitted parameters of the anisotropy function for the sources under study. Zero values are representedby dashes.

Nucletron HDR ‘‘classic’’ designi ki ai bi ei

1 2.613331022 – 5.8123 1.0882 21.44644 – 21.0123 22.14631022

3 3.91831022 21.960 4.6031021 –4 1.291731021 35.863 64.82 7.44631021

ai8 bi8 ei8

1 – 1.33431025 21.62931024

2 – 23.572 23.8273 26.516931021 3.447431021 –4 25.220 65.552 1.9574

Nucletron HDR ‘‘new’’ designi ki ai bi ei

1 1.811131022 – 3.072 2.184631021

2 21.48814 – 21.5753 25.56531021

3 3.535231022 21.2851 21.207631021 –4 1.631731021 29.523 69.696 1.74

ai8 bi8 ei8

1 – 7.281431022 27.27731024

2 – 23.137 22.5463 21.227 21.214931021 –4 29.941 65.54 1.8305

Nucletron PDRi ki ai bi ei

1 21.11182 – 20.054 2.77762 23.561331022 – 1.22231021 28.3331023

3 1.35631022 25.64531021 25.7731021 –4 1.55792 16.958 11.714 21.0391

ai8 bi8 ei8

1 – 21.7092 1.45002 – 3.36331023 1.00193 23.90731023 24.991431022 21.39044 5.83831022 222.3700 5.65031021

Varian HDR Varisourcei ki ai bi ei

1 9.712531021 17.542 106.858 22.1631021

2 6.11331022 21.07331021 1.367 3731021 22.0853 3.08631022 4.9608 211.16535 24.83631022

4 21.254083 50.245 23.406 1.6151ai8 bi8 ei8

1 26.0619 28.561 28.70931022

2 1.105 337 25.317231021 23.6463 239.7765 26.7652 25.85431022

4 73.554 77.124 1.5911

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FIG. 1. Comparison of the anisotropy function data~Table IV in Ref. 2! for the Nucletron classic designHDR 192Ir source with the fit anisotropy function pro-posed in this article at different radial distancesr.

FIG. 2. Comparison of the anisotropyfunction data~Table II in Ref. 5! forthe Nucletron new design HDR192Irsource with the fit anisotropy functionproposed in this article at different ra-dial distancesr.

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FIG. 3. Comparison of the anisotropyfunction data~Table III in Ref. 2! forthe Nucletron PDR192Ir source withthe fit anisotropy function proposed inthis article at different radial distancesr.

FIG. 4. Comparison of the anisotropyfunction data~Table IV in Ref. 4! forthe VariSource HDR192Ir source withthe fit anisotropy function proposed inthis article at different radial distancesr.

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of relative deviations,u12 Ffit /Fdatau, are 0.6% for theNucletron ‘‘classic’’ design and the ‘‘new’’ design sources,0.4% for the PDR Nucletron sources and 0.7% for the Vari-Source.

B. Radial dose function g „r …

Fit parameters for the radial dose functional form are pre-sented in Table III and the fit functional form for each sourcealong with Monte Carlo data are presented in Fig. 5. Theaverage absolute values of relative deviations,u12 gfit /gdatau, are 0.4% for the Nucletron~‘‘classic’’ designand the ‘‘new’’ design! sources and for the PDR Nucletronsources, and 0.5% for the VariSource. For large values ofrthe limit of the function was set to 0. Theg(r ) data for thePDR source increase when close to zero, in contrast to theother sources for which the data decrease in the same range.The functional form Eq. 5 cannot reproduce this behavior ofthe PDR source data.

C. Test of the fit functional forms

In order to verify the accuracy of our fit functional forms,we reproduced published data of dose-rate distributions ofthe sources under study in the form of away-along tables.

This was done using the TG43 formalism with our fit func-tional forms for F(r ,u) and g(r ), the publishedL valuesand the geometrical factor of the TG43 for linear sources. Inthe following, y-values are taken away from the transversesource axis andz-values along the longitudinal source axis.

1. Classic nucletron

For this source the differences between Monte Carlo doserate data~Table V in Ref. 2! and our calculations are within1% for uzu<5 cm andy.0.1 cm @the range where MonteCarlo data are available forF(r ,u) is 0.25 cm<r<5 cm#.Very close to the source (uzu<0.25 cm andy<0.1 cm! thedifferences are'10%. Far away from the source (uzu.5cm! the differences are within 2% showing that extrapolatedF(r ,u) values of our fit function give reasonable values atlarge r.

2. New nucletron

For this source the differences between Monte Carlodose-rate data~Table III in Ref. 5! and our calculations arewithin 1% for uzu<4 cm andy.0.25 cm@the range whereMonte Carlo data are available forF(r ,u) is 0.25 cm<r<5 cm#. Very close to the source (uzu<0.25 cm andy<0.25 cm! the differences are'5%. Far away from thesource (uzu>5 cm! the differences are within 3% showingthat extrapolatedF(r ,u) values of our fit function give rea-sonable values at larger

3. PDR nucletron

For this source the differences between Monte Carlodose-rate data~Table VI in Ref. 2! and our calculations arewithin 1% for uzu<5 cm andy.0.1 cm @the range whereMonte Carlo data are available forF(r ,u) is 0.25 cm<r<5 cm#. Very close to the source (uzu<0.1 cm andy<0.1cm! the differences are'5%. Far away from the source(uzu>6 cm! the differences are within 3% showing that ex-trapolatedF(r ,u) values of our fit function give reasonablevalues at larger

4. VariSource

For this source the differences between Monte Carlodose-rate data~Table I in Ref. 4! and our calculations arewithin 1% for the whole range except for point near thelongitudinal source axis where the differences are'5%. Forthis source the range where Monte Carlo data are availablefor F(r ,u) is 1 cm<r<10 cm.

IV. CONCLUSION

In this paper we present closed functional forms forF(r ,u) andg(r ) functions obtained by fitting the publishedHDR 192Ir sources dosimetry data. The fitted parametershave been chosen to satisfy the boundary conditions at thelimits hence they avoid extrapolation problems. The pro-posed functional forms can reproduceF(r ,u) andg(r ) func-tions for nonsymmetrical sources giving accurate values ofthese functions in a simple form suitable for use in treatment

FIG. 5. Radial dose function for the HDR192Ir sources studied in this paper.Points represent Monte Carlo results~Classic Nucletron: Table II in Ref. 2;New Nucletron: Table I in Ref. 5; PDR: Table II in Ref. 2; VariSource:Table III in Ref. 4! and the curve is the result of the fit as explained in thetext.

TABLE III. Parameters for the fit of the radial dose functiong(r ).

Sources ‘‘Classic design’’ ‘‘New design’’ PDR Varisource

h 1.000 21 1.0011 1.0001 1.0001i 5.94231023 2.35431022 7.3031023 9.4831023

j 2.130731024 1.07431023 8.9431025 1.36131024

k 2.9038 2.3650 3.235 3.0681

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planning systems. These functional forms make easier datahandling within the treatment planning system, making itpossible to avoid the use of tabulated data. Using the TG43formalism with these functions, the published dose-rate datareported as 2-D Cartesian lookup tables for these sources canbe reproduced satisfactorily.

ACKNOWLEDGMENTS

This work has been partially supported by ‘‘GeneralitatValenciana’’ under Project No. GV98-12-122 and by Minis-terio de Educacio´n y Cultura ~Spain! under Project No.FEDER 1FD97-0609.

a!Electronic mail: Facundo.Ballester@uv.es1R. Nath, L. L. Anderson, K. J. A. Meli, A. J. Olch, J. A. Stitt, and J. F.Williamson, ‘‘Code of practice for brachytherapy physics: Report of theAAPM Radiation Therapy Committee Task Group No. 56,’’ Med. Phys.24, 1557–1598~1997!.

2J. F. Williamson and Z. Li, ‘‘Monte Carlo aided dosimetry of the mi-croselectron pulsed and high dose-rate192Ir sources,’’ Med. Phys.22,809–819~1995!.

3R. Nath, L. L. Anderson, G. Luxton, K. A. Weaver, J. F. Williamson, andA. S. Meigooni, ‘‘Dosimetry of interstitial brachytherapy sources: Rec-ommendations of the AAPM Radiation Therapy Committee Task GroupNo. 43,’’ Med. Phys.22, 209–234~1995!.

4R. Wang and R. Svoboda, ‘‘Monte Carlo dosimetry of the Varisourcehigh dose-rate192Ir source,’’ Med. Phys.25, 415–423~1998!.

5G. M. Daskalov, E. Lo¨ffler, and J. Williamson, ‘‘Monte Carlo aideddosimetry of a new high dose-rate brachytherapy source,’’ Med. Phys.25, 2200–2208~1998!.

6E. Furhang and L. Anderson, ‘‘Functional fitting of interstitial brachy-therapy dosimetry data recommended by the AAPM Radiation TherapyCommittee Task Group 43,’’ Med. Phys.26, 153–160~1999!.

7D. C. Moss, ‘‘Technical note: Improved analytical fit to the TG-43 radialdose function, g~r!,’’ Med. Phys.27, 659–661~2000!.

8G. F. T. Widger, ‘‘Representation of magnetization curves over extensiverange by rational-fraction approximations,’’ Proc. IEE116, 156–160~1969!.

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