Dosimetry of 192 Ir wires for LDR interstitial brachytherapy following the AAPM TG-43dosimetric formalismP. Karaiskos, P. Papagiannis, A. Angelopoulos, L. Sakelliou, D. Baltas, P. Sandilos, and L. Vlachos Citation: Medical Physics 28, 156 (2001); doi: 10.1118/1.1339885 View online: http://dx.doi.org/10.1118/1.1339885 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/28/2?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Dosimetric characteristics of a linear array of γ or β-emitting seeds in intravascular irradiation: Monte Carlostudies for the AAPM TG-43/60 formalism Med. Phys. 30, 403 (2003); 10.1118/1.1538229 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 Beta versus gamma dosimetry close to Ir-192 brachytherapy sources Med. Phys. 28, 1875 (2001); 10.1118/1.1395038 Fitted dosimetric parameters of high dose-rate 192 Ir sources according to the AAPM TG43 formalism Med. Phys. 28, 654 (2001); 10.1118/1.1359438 Functional fitting of interstitial brachytherapy dosimetry data recommended by the AAPM Radiation TherapyCommittee Task Group 43 Med. Phys. 26, 153 (1999); 10.1118/1.598497

Dosimetry of 192Ir wires for LDR interstitial brachytherapyfollowing the AAPM TG-43 dosimetric formalism

P. Karaiskos, P. Papagiannis, A. Angelopoulos, and L. Sakellioua)

Nuclear and Particle Physics Section, Physics Department, University of Athens, Panepistimioupolis, Ilisia,157 71, Athens, Greece

D. BaltasDepartment of Medical Physics and Engineering, Strahlenklinik, Kliniken, Offenbach, 63069 Offenbach,Germany and Institute of Communication and Computer Systems, National Technical University ofAthens, Zografou, 157 73 Athens, Greece

P. Sandilos and L. VlachosDepartment of Radiology, Medical School, University of Athens, Areteion Hospital, 76 Vas. Sofias Avenue,115 28, Athens, Greece and Medical Physics Department, Hygeia Hospital, Kiffisias Avenue, ErythroyStavrou, Marousi, 151 23 Athens, Greece

~Received 4 May 2000; accepted for publication 3 November 2000!

Implementation of the AAPM Task Group 43 dosimetric formalism for192Ir wires used as inter-stitial sources in low dose-rate~LDR! brachytherapy applications is investigated. Geometry factors,dose-rate constant values, radial dose functions, and anisotropy functions, utilized in this formalism,were calculated for various lengths of all commercially available wire source designs by means ofa well-established Monte Carlo simulation code and an improved modification of the Sievert inte-gral method. Results are presented in the form of look up tables that allow interpolation fordose-rate calculations around all practically used wire lengths, with accuracy acceptable for clinicalapplications. ©2001 American Association of Physicists in Medicine.@DOI: 10.1118/1.1339885#

Key words: brachytherapy, dosimetry, Monte Carlo, Sievert,192Ir, wires

I. INTRODUCTION

192Ir for interstitial low dose-rate~LDR! brachytherapy pur-poses is most commonly used in the form of Pt encapsulatedwires of 0.3, 0.5, and 0.6 mm outer diameters. These wiresare usually delivered in standard length of 14 cm with airkerma strengths of about1 140 mGy m2 h21 and cut down tosmaller lengths, mostly in the order2 of 4–6 cm, to facilitatea specific application.

Despite their wide use, dosimetry of wire sources is con-sidered beyond the scope of the AAPM Task Group 43 in-vestigation of dosimetry in interstitial brachytherapyapplications3 and therefore relative values of the quantitiesrequired for the implementation of the proposed dosimetricformalism are not supplied.

Furthermore, there are only a small number of papers pre-senting relative dosimetric data1,4–6 with the Task Group 43dose calculation formalism applied only in the works of Bal-lester et al.1 and Perez-Calatayudet al.6 for selected wirelengths of 1 and 5 cm.

The aim of this work is to investigate the dependence ofthe dosimetric quantities proposed by the AAPM TaskGroup 43 on wire length and diameter and provide look uptables that allow interpolation for dose rate calculations withaccuracy acceptable for clinical applications.

Monte Carlo photon transport calculations are accepted asthe most efficient method of detailed 3D dosimetry in the

field of brachytherapy. The only disadvantage is that suchcalculations are rather time consuming, especially for thenumerous wire dimensions investigated, and the fine calcu-lation grid needed in this study in order to provide resultsthat support accurate dose-rate calculations for points aroundall practically used wire lengths.

An attractive alternative is the Sievert integral methodcombining simplicity with reasonable computational times,provided that it produces results with acceptable accuracy.7,8

We recently reported on an improved modification ofWilliamson’s7 isotropic scattering model that proved to offeraccuracy comparable to the more rigorous Monte Carlo cal-culations for192Ir high dose-rate~HDR! sources.9

In the present work, both methods were utilized for thecalculation of absolute dose rates around various lengths ofall commercially available192Ir wires used as interstitialsources in LDR brachytherapy. For Monte Carlo calculationsour own, well-established and experimentally verified MCsimulation code10–14 was used.

II. MATERIALS AND METHODS

A. Investigated sources

Calculations were performed around numerous lengths,L,of wire sources with 0.1-, 0.3-, and 0.4-mm active core di-ameters composed of 25% Ir and 75% Pt and encased in a

156 156Med. Phys. 28 „2…, February 2001 0094-2405 Õ2001Õ28„2…Õ156Õ11Õ$18.00 © 2001 Am. Assoc. Phys. Med.

0.1-mm Pt sheath. Thus these wires presented an outer diam-eter,d, of 0.3, 0.5, and 0.6 mm to simulate all commerciallyavailable192Ir source wire designs.6

B. Monte Carlo simulation code

A well-established and experimentally verified MC simu-lation code10–14 was used to calculate the dose-rate distribu-tion in water around the192Ir wires. The code incorporatesthe construction details and dimensions of the investigatedsource designs, including their encapsulation. A detailedtracking is performed for every primary photon initiated in arandom position and emitted in a random direction within theactive core. Primary and secondary produced photons aresampled individually in direct analogy to the main processes,namely photoabsorption, coherent, and incoherent scattering.The tracking and interactions of photons are based on a set ofself-consistent total, partial and differential crosssections15–18 taking into account binding corrections for theincoherent scattering process. The192Ir photon spectrum wastaken from Glasgow and Dillman.19 A photon history is ter-minated, with all the residual energy deposited on spot, if itsenergy falls below a selected transport cutoff threshold of 10keV or if its spatial coordinates lie outside the boundaries ofthe phantom used.

The 192Ir wire sources were positioned at the center of aliquid–water spherical phantom of 30 cm in diameter~usu-ally used in similar studies12,13,20,22! to ensure photon back-scatter conditions. The phantom sphere was divided into dis-crete concentric spherical shells of 0.1 cm thickness, eachsplit into angular intervals of 1° both with respect to polarangleu ~0, 180°! and azimuthal anglew ~0, 360°!. However,as all dosimetric quantities involved are isotropic with re-spect to the azimuthal anglew for the sources investigatedhere, this three-dimensional segmentation can be simplifiedto a two-dimensional one, only with respect to coordinatesrandu. Furthermore, all dosimetric quantities are equal for the~0, 90°! and~90°, 180°! polar angle regions due to symmetryexistent for the investigated source designs, and results willbe presented for the~0, 90°! polar angle region. For theenergies considered in this work electronic equilibrium existsfor all scoring voxels at radial distancesr .1 mm from thesource and therefore dose and water kerma for these voxelsare equal. As a result of the above, and in order to speed upcalculations, dose was calculated from water kerma, byweighting the photon energy fluence, at a specificr, u sur-face, with the corresponding mass-energy absorption coeffi-cient, since results in this study refer to radial distances ofclinical interestr .1 mm. Thus calculated dosimetric datarefer to exact radial distances and are not the result of aver-aging within a scoring volume where steep dose gradientsmay occur. Results in this study refer to 108 primary photonsyielding statistical errors of less than 1% along the transversebisector of the sources. Further detail and analysis on thecode and its implementation for192Ir sources can be foundelsewhere.10,14

C. Sievert integration model

Our Sievert integration model9 follows the primary andscattered radiation separation technique of Williamson’s7

isotropic scatter model to calculate full 3D dose-rate distri-butions around brachytherapy sources with increased accu-racy and reasonable computational times. However, for thedetermination of the primary dose component narrow beamattenuation coefficients, directly calculated by the initiallyemitted192Ir spectrum, are used instead of Monte Carlo de-rived thickness-dependent filtration coefficients. As far as thescatter radiation dose component is concerned, the assump-tion of isotropy is not justified for elongated brachytherapysource designs. As active source core lengthL increases, thescatter distribution presents increasing anisotropy and there-fore we proposed modeling of its contribution to dose ratealong the source’s transverse axis@scatter to primary dose-rate ratio at radial distancer, SPR(r )# through use of asimple equation and application of an empirical correction@C(r ,u)# for deviation of scatter from isotropy.9 This Sievertmodel proved to offer accuracy comparable~maximum de-viations less than 2% for 10°,u,170°) to the more rigor-ous, but time-consuming, Monte Carlo computations for thecommonly used VariSource and microSelectron192Ir highdose-rate sources~active core lengthsL of 1 and 0.35 cmrespectively!. However, in this study comparison with MonteCarlo results revealed that for source lengthsL.1 cm ourSievert model presents deviations up to 5% due to its inabil-ity to handle scatter radiation with increased accuracy.Therefore, based on our Monte Carlo calculations, we re-evaluated the scatter corrections, SPR(r ) and C(r ,u) forsources of lengthL.1 cm, according to:

SPR~r !5~0.02910.0024L10.000 65L2!10.126r

10.0034r 2, ~1!

C~r ,u!5 H12~7.6E25!~u290°! for r ,L12~3.0E29!~u290°! for r>LJ . ~2!

The above corrections ensure dosimetric calculations withaccuracy acceptable for clinical applications as overall rootmean square~rms! errors of less than 3% are met forL.1 cm and less than 5% for usual HDR sources ofL,1 cm @for the definition of rms errors see Eq.~10! in Sec.III #.

Sievert calculations were performed around192Ir wires ofvarious lengths of all three diameters centered in a liquidwater spherical phantom of 30 cm in diameter. The grid ofcalculation points has its origin on the sources geometricalcenter and spreads within polar anglesu of 0.5° to 179.5°and radiir of 0 cm to 10 cm, in 1° and 1-mm step intervalscorrespondingly. The 90° angle refers to the sources trans-verse axis. Due to symmetry existent for the investigatedwire sources, all dosimetric quantities for the 0°–90° and the90°–180° regions are equal~note that this symmetry is notvalid for HDR sources mentioned earlier!. Therefore, in thisstudy, results will be presented only for the~0, 90°! polarangle region.

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D. Dose calculation formalism

The dose calculation formalism proposed by the AAPMRadiation Therapy Committee Task Group 43~TG-43! wasfollowed.3 According to this formalism, dose in a medium atpoint (r ,u) should be expressed as:

D~r ,u!5SKLG~r ,u!

G~1 cm,90°!g~r !F~r ,u!, ~3!

where r is the radial distance with respect to the source’scenter,u is the polar angle relative to the source’s longitudi-nal axis, andSK is the source’s air kerma strength in units ofU~1 U51 mGy m2 h2151 cGy cm2 h21!.

L is the source dose-rate constant, defined as:

L5D~1 cm,90°!/SK . ~4!

G(r ,u) is the geometry factor accounting for the source’sradioactive material distribution, defined as:

G~r !5EV

r~r 8!dV8

ur2r 8u2 Y EVr~r 8!dV8, ~5!

with r(r 8) representing the density of radioactivity at pointr 8 within the source,V denoting integration over the activesource volume withdV8 referring to the volume elementlocated atr 8.

In practice, the point or line source approximation is uti-lized instead of Eq.~5!. However, we reported recently22 thatuse of point source approximation to calculate the geometryfactors of an elongated brachytherapy source design is validonly for radial distancesr greater than about twice the lengthL of the active core of the source. For smaller radial dis-tances it is unacceptable since significant errors~.50%! areintroduced.

On the contrary, use of the line source approximation forthe determination of geometry factors of elongated brachy-therapy sources was found to introduce errors,1%, for ra-dial distancesr .L/2. At shorter radial distances errors de-pend on the ratio of the sources active core diameter tolength, d/L, and increase as this ratio increases due to thefact that line source approximation assumes the source as 1Dline segment and ignores its radial dimensiond.

In the present study, since clinically relevant wire lengthsare in the order of 4 to 6 cm theird/L ratio is small and theline source approximation is employed following the equa-tion:

G~r ,u!5b

Lr sinu, ~6!

where b is the angle subtended by the active source withrespect to calculation point (r ,u), as presented in Fig. 1.

Based on simple algebra and the similarity of the twotriangles formed in Fig. 1, a scaling may be produced be-tween the geometry factors of different source lengths. It canbe seen that:

GL~r ,u!

GL8~r 8,u!5

b/Lr sinu

b/L8r 8 sinu5

L8r 8

Lr5S L8

L D 2

,

wherer 8

r5

L8

L. ~7!

g(r ) is the radial dose function accounting for radial de-pendence of photon absorption and scatter in the mediumand is given by the equation:

g~r !5D~r ,90°!/G~r ,90°!

D~1 cm,90°!/G~1 cm,90°!. ~8!

F(r ,u) is the source’s anisotropy function accounting forthe angular dependence of photon absorption and scatter andis calculated by the equation:

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

D~r ,90°!/G~r ,90°!. ~9!

Given the strong dependence of the above TG-43 dosim-etric quantities on the encapsulated source geometry and thewide length range of the investigated wire sources, care hasbeen taken in providing sufficient data to ensure reliable in-terpolations. In any case, however, dose-rate calculations bydirect application of the proposed Sievert integration methodprovides clinically acceptable accuracy in reasonable compu-tational times.

III. RESULTS AND DISCUSSION

As stated in the Introduction, clinically relevant wirelengths are most usually between 4 and 6 cm. However, cal-culations by means of our Sievert integration model wereperformed around numerous lengths, varying from 0.5 to 12cm, of all three outer wire diameters to investigate the de-pendence of the dosimetric quantities proposed by theAAPM TG-43 on wire lengthL and diameterd providingguidance for interpolations.

Monte Carlo calculations were performed for selectedwire lengths of 1, 3, 5, 7, and 8 cm of the smaller outerdiameterd50.3 mm, in order to evaluate the accuracy of ourSievert integration model.

Sievert results accuracy relative to Monte Carlo, for anycalculated quantityX, can be quantified by the root meansquare error given by equation:

RMS5A(i 51

N

~~XSievert/XMonteCarlo!21!2/N3100%,

~10!

FIG. 1. Schematic diagram for the calculation of the geometry factor,G(r ,u).

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with N denoting the total number ofX values available.The 5-cm-long source will be used as the basis for the

presentation of our results, since it mediates the practicallyused wire length range.

A. Geometry factor

Geometry factorG(r ,u) is the simplest dosimetric quan-tity to calculate. However, it is of paramount importance fordosimetric accuracy since it presents the strongest depen-dence on source lengthL, and its values vary significantlywith calculation point position (r ,u) with respect to thesource.

The line source approximated geometry factors of the5-cm-long192Ir wire are presented in Table I in a grid appro-priate to support interpolations. The aforementioned depen-dence ofG(r ,u) on calculation point (r ,u) is evident as,generally,G(r ,u) decreases as one moves away from thesource~r increases! or toward the source’s transverse bisec-tor ~u tends to 90°! and large variations are observed atpoints close to the source.

The accuracy of tabulated values was checked by com-parison with the exact geometry factors around the 5-cm-long 192Ir wire as calculated by Eq.~5!. Errors were found tobe less than 2% even at points very close to the source.Shaded cells refer to points that are either within the sourcestructure or at distances smaller than 1 mm from the sourcewhere dosimetric accuracy cannot be claimed due to elec-tronic disequilibrium and dose contribution by electronsemitted by the source. These values, deprived of physicalmeaning, should be ignored for the source length ofL55 cm but are presented to facilitate scaling to any othersource length~see below!. The radial distance scale rises upto 10 cm since forr .2L the point source approximationmay be safely applied as stated before.

The scaling introduced in Sec. II D may be applied to

produce the corresponding table of geometry factors for anyother wire length. If, for example, the geometry factor of anL853.5-cm-long wire source is needed then, according toEq. ~7!, (L8/L)50.7, (L8/L)250.49 and one has to:~a! di-vide all tabulated geometry factor values for theL55-cm-long wire by (L8/L)2 and~b! replace the radii scaleof r 50.1,..,10 cm byr 85(L8/L) r 5(0.7) r 50.07,..,7 cm.For points at radial distancesr .2L point source approxima-tion may be applied.

In Fig. 2, line source approximated geometry factorsG(1 cm,90°) at a distance 1 cm along the source’s transversebisector, where the dose-rate constant is defined, are plottedagainst source length. By definition,G ~1 cm, 90°!, also pre-sented in Table II for wire source lengthsL of 0.5–12 cm,

FIG. 2. Line source approximated geometry factorsG(1 cm,90°) at radialdistancer 51 cm along the sources transverse bisector (u590°), as a func-tion of source lengthL.

TABLE I. Geometry factorsG(r ,u) of the L55-cm wire. Shaded values refer to points within the source structure or at distances less than 0.1 cm from thesource.

u ~°!

Geometry factorG(r ,u)

Radial distancer ~cm!0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2 2.5 3 4 5 6 7 10

1 205.4 179.9 119.8 89.8 71.8 59.8 51.3 44.8 39.8 35.811 23.751 17.557 7.200 0.363 0.103 0.053 0.034 0.023 0.0112 102.5 89.9 59.9 44.8 35.8 29.8 25.6 22.3 19.8 17.813 11.753 8.561 3.601 0.359 0.102 0.053 0.034 0.023 0.0113 68.3 59.9 39.9 29.9 23.8 19.8 17.0 14.8 13.2 11.815 7.754 5.566 2.401 0.354 0.102 0.053 0.034 0.023 0.0114 51.1 44.9 29.9 22.4 17.9 14.8 12.7 11.1 9.83 8.817 5.756 4.073 1.801 0.347 0.102 0.053 0.034 0.023 0.0115 40.9 35.9 23.9 17.9 14.3 11.6 10.1 8.833 7.83 7.019 4.558 3.181 1.442 0.339 0.101 0.053 0.034 0.023 0.0117 29.1 25.6 17.0 12.7 10.2 8.42 7.19 6.267 5.53 4.966 3.191 2.171 1.031 0.321 0.101 0.053 0.033 0.023 0.011

10 20.3 17.9 11.9 8.88 7.07 5.86 5.00 4.346 3.84 3.430 2.171 1.431 0.724 0.291 0.099 0.052 0.033 0.023 0.01115 13.5 12.0 7.93 5.91 4.69 3.88 3.30 2.858 2.52 2.241 1.386 0.883 0.486 0.246 0.094 0.051 0.033 0.023 0.01120 10.2 9.03 5.96 4.43 3.51 2.89 2.45 2.122 1.86 1.653 1.002 0.629 0.367 0.210 0.089 0.050 0.032 0.023 0.01125 8.16 7.27 4.79 3.55 2.81 2.31 1.95 1.685 1.48 1.306 0.780 0.487 0.297 0.183 0.085 0.049 0.032 0.023 0.01130 6.85 6.12 4.03 2.98 2.35 1.93 1.63 1.400 1.22 1.079 0.637 0.399 0.251 0.163 0.080 0.047 0.031 0.023 0.01035 5.93 5.32 3.49 2.58 2.03 1.66 1.40 1.200 1.05 0.921 0.539 0.339 0.219 0.147 0.076 0.046 0.030 0.022 0.01040 5.26 4.73 3.10 2.28 1.79 1.47 1.23 1.055 0.917 0.807 0.469 0.297 0.195 0.134 0.072 0.044 0.030 0.022 0.01045 4.75 4.28 2.80 2.06 1.62 1.32 1.11 0.946 0.821 0.721 0.418 0.266 0.178 0.124 0.068 0.043 0.029 0.022 0.01050 4.36 3.94 2.57 1.89 1.48 1.20 1.01 0.862 0.747 0.656 0.379 0.242 0.164 0.116 0.066 0.042 0.029 0.021 0.01060 3.82 3.47 2.26 1.65 1.29 1.05 0.88 0.747 0.647 0.566 0.327 0.211 0.145 0.105 0.061 0.040 0.028 0.021 0.01070 3.50 3.18 2.07 1.51 1.18 0.96 0.80 0.679 0.587 0.513 0.296 0.192 0.134 0.098 0.058 0.038 0.027 0.020 0.010

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approaches unity for point sources and tends to zero whensource length tends to infinity.

B. Dose-rate constant

Accurate determination of the dose-rate constant value ofa source involved in a specific application is crucial since thedose-rate constant constitutes the only absolute quantity ofthe AAPM TG-43 dosimetric formalism.

The Sievert and Monte Carlo calculations of the dose-rateconstants per unit source air kerma strength,L, of all inves-tigated source lengthsL are presented in Table II in units ofcGy h21 U21. In the same table, corresponding geometry fac-torsG(1 cm,90°) at distance 1 cm along each source’s trans-verse bisector, whereL is defined, and the dose-rate constantper unit source air kerma strength values reported by Ball-esteret al.1 are also presented.

Tabulated results reveal that Sievert calculated dose-rateconstant values agree within 1.5% with Monte Carlo resultsfor all investigated wire lengths while a very good agreement~1%! with Monte Carlo results of Ballesteret al.1 is alsoobserved constituting our modified Sievert model’s adequacyfor dosimetry of192Ir wires.

It is also evident that the dose-rate constant does not de-pend heavily on the source diameter as differences for thesame wire lengths are within 3%. This behavior indicatesthat dose-rate constantL is not affected by scattering andabsorption within source and encapsulation. This is due tothe fact that the TG-43 dosimetric formalism demands thatthe encapsulated sources be calibrated in terms of referenceair kerma rateK. Thus for calculation of the dose-rate con-stant, the effect of the encapsulated source geometry on thescattering and absorption of photons is cancelled out. Thisbehavior has also been observed in previous studies9,14 and isexpected to apply for radial dose functiong(r ) as well sinceit is defined along the transverse axis of the source.

In Fig. 3, Sievert calculations of dose-rate constant valuesare plotted against source length for thed50.3-mm outerdiameter wire and a strong dependence on source length canbe observed. The dose-rate constant decreases with increas-ing source length with a trend similar to that observed in Fig.2. The implied, almost linear, relation between dose-rateconstantL and geometry factorG(1 cm,90°) is presented inFig. 4. The percentage differences in the dose rate constantwere found to follow the percentage differences between thegeometry functions of different source lengths.

Based upon the above findings, the dose-rate constant ofany 192Ir source of known active lengthL can be calculatedwith respect to that of a reference source length~in this casethe 5-cm-long wire source is used! following the equation:

FIG. 3. Sievert calculated dose rate constant valuesL of outer diameterd50.3-mm wires, as a function of source lengthL.

TABLE II. Calculated dose-rate constant valuesL for all investigated wire source designs. Monte Carlo calculations reported by Ballesteret al. ~Ref. 1! arepresented for comparison.

Source lengthL ~cm!

Dose rate constantL ~cGy h21 U21!

Monte Carlo~Source diameterd50.03 cm)

Sievert

G ~1,90°! This work Ballesteret al. ~Ref. 1!Source diameter

d50.03 cmSource diameter

d50.05 cmSource diameter

D50.06 cm

0.5 0.980 - - 1.109 1.109 1.1101 0.927 1.040 1.047 1.044 1.043 1.0431.5 0.858 - - 0.979 0.976 0.9752 0.785 - - 0.890 0.886 0.8843 0.655 0.724 - 0.733 0.727 0.7254 0.554 - - 0.614 0.607 0.6045 0.476 0.521 0.521 0.525 0.518 0.5256 0.416 - - 0.458 0.451 0.4487 0.369 0.403 - 0.407 0.396 0.3978 0.331 0.361 - 0.366 0.359 0.3569 0.300 - - 0.334 0.327 0.324

10 0.275 - - 0.307 0.301 0.29911 0.253 - - 0.285 0.280 0.27712 0.234 - - 0.267 0.262 0.259

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LL5GL~1 cm,90°!

GL55cm~1 cm,90°!LL55cm, ~11!

where:LL55 cm50.521 cGy h21 U21 andGL55cm~1 cm,90°!50.476.

Use of the above equation has proved to provide dose-rateconstant values that agree with Monte Carlo calculationswithin ;2% for all investigated wire lengths. Thus Eq.~11!can be safely used as an alternative to dose-rate constantcalculations by interpolation through Table II.

Furthermore, the above equation yields dose-rate con-stantsL with errors less than 3% even for the microSelectronHDR source ~active core length 0.35 cm, L51.116 cGy h21 U21)12 and both old and new VariSourceHDR sources~active core length 1 cm,L51.043 cGy h21

U21 and active core length 0.5 cm,L51.101 cGy h21 U21,respectively!.13,14

C. Radial dose function

The Monte Carlo calculations of radial dose function val-ues g(r ) for selected lengthsL51, 5, and 10 cm ofd50.3-mm outer wire diameter are presented in Fig. 5 alongwith Monte Carlo calculations reported by Ballesteret al.1

and corresponding Sievert calculations for the reference wirelength of 5 cm.

It can be seen that our Monte Carlo and Sievert results arein excellent agreement~62%!. Compared to Monte Carlocalculations reported by Ballesteret al.1 differences greaterthan 3% are observed for radial distancesr greater than 5 cm.These differences are due to different backscatter conditionsarising from the different phantom dimensions used~in thisstudy a spherical phantom of 30 cm in diameter, commonlyused in similar Monte Carlo studies,12–14,20,21was utilized asopposed to the cylindrical phantom of 40 cm in height anddiameter used in the study of Ballesteret al.1!.

Plotted results also reveal that the radial dose functionincreases with increasing length for radial distancesr>1 cm. However, overall variation for wire lengths between0.5 and 12 cm is less than 2%.

Radial dose function valuesg(r ) of all investigatedsource lengthsL were also calculated for all three outersource diametersd by means of the proposed Sievert integra-tion model. Our results confirmed12–14 that source dimen-sions do not significantly affect radial dose function valuesasg(r ) increases slightly with increasing outer wire diameterat large radial distances but values for all three outer wirediameters were found to agree within 3%, at expected ac-cording to the analysis in Sec. III A.

The above findings can be observed in Table III whereMonte Carlo and Sievert calculations of radial dose functionvalues for the typicalL55-cm wire length, as selected radialdistancesr are presented.

Since the radial dose function is only a relative dosimetricquantity that does not vary significantly with source dimen-sions, straightforward use of the Monte Carlo calculated val-ues for the 5-cm-long wire, included in Table III, would beadequate for any practically used wire lengthL.

D. Anisotropy function

The anisotropy function is a measure of the dose angularvariation due to oblique filtration within the source structurewith values depending on source geometry and dimensionsand varying significantly with calculation point (r ,u).

Monte Carlo calculations of anisotropy function valuesfor the typical wire length ofL55 cm andd50.3 mm outerdiameter are presented in Table IV. In Fig. 6, these data areplotted against polar angleu at selected radial distancesr,along with corresponding Sievert calculated values andMonte Carlo values reported by Ballesteret al.1 In the samefigure, Sievert calculated data ford50.5 mm and d

FIG. 4. Dose-rate constantL, as a function of geometry factorG(1 cm,90°)at radial distancer 51 cm along the source’s transverse bisector (u590°).

FIG. 5. Monte Carlo calculated radial dose function valuesg(r ) of wirelengthsL51 andL55 cm and outer diameterd50.3 mm, as a function ofradial distancer. Monte Carlo calculations reported by Ballesteret al. ~Ref.1! and Sievert calculations for wire lengthL55 cm are presented for com-parison.

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50.6 mm are also presented to investigate the dependence ofanisotropy function values on source diameter.

It can be seen that Sievert data are generally in goodagreement with our Monte Carlo calculations presentingRMS errors of less than 3% thus proving capable of predict-ing anisotropy at all points around investigated wire sourcedesigns. Anisotropy values reported by Ballesteret al.1 agreewithin 2% for polar anglesu.20°. For polar anglesu,20° large differences are observed which increase signifi-

cantly as one moves closer to the long axis of the source.Especially at polar angles;1° differences are greater than100% with data of Ballesteret al.1 being out of the presentedscale. The Sievert calculated anisotropy function values pre-sented generally increase with increasing outer wire diam-eter, as one intuitively expected, but this decrease is small~,3%! at all radial distances.

Overall, based on Fig. 6, we can conclude that anisotropyis of importance only for points at radial distancesr .L/2

TABLE III. Monte Carlo and Sievert calculated radial dose function values for wire lengthL55 cm, at selectedradial distancesr.

Radialdistancer ~cm!

Radial dose Functiong(r )

Monte Carlo Sievert

d50.03 cm d50.03 cm d50.05 cm d50.06 cmL55 cm L55 cm L55 cm L55 cm

0.2 0.949 0.944 0.923 0.9150.3 0.964 0.959 0.942 0.9350.5 0.980 0.976 0.966 0.9620.7 0.989 0.988 0.982 0.9801 1.000 1.000 1.000 1.0001.5 1.011 1.015 1.020 1.0222 1.017 1.024 1.032 1.0363 1.020 1.030 1.042 1.0474 1.017 1.025 1.038 1.0445 1.010 1.013 1.026 1.0326 0.997 0.994 1.008 1.0147 0.981 0.971 0.984 0.9908 0.956 0.944 0.958 0.9649 0.930 0.915 0.928 0.934

10 0.897 0.884 0.896 0.902

TABLE IV. Monte Carlo calculated anisotropy function,F(r ,u), for wire lengthL55 cm and wire diameterd50.3 cm. Shaded values refer to points whereanisotropy function values present large deviations~.10%! from unity.

Polar angleu ~degrees!

Anisotropy functionF(r ,u)

Radial distancer ~cm!0.5 1 2 3 4 5 6 7 8 10

0.5 0.526 0.355 0.378 0.418 0.462 0.488 0.5472.5 0.829 0.577 0.547 0.561 0.577 0.595 0.6435.5 0.941 0.901 0.752 0.710 0.709 0.717 0.731 0.7567.5 0.944 0.945 0.918 0.813 0.778 0.773 0.783 0.789 0.808

10.5 0.953 0.956 0.937 0.867 0.840 0.838 0.835 0.836 0.84915.5 0.963 0.964 0.965 0.956 0.917 0.899 0.895 0.893 0.893 0.89920.5 0.976 0.968 0.972 0.966 0.942 0.931 0.927 0.924 0.925 0.93025.5 0.985 0.976 0.976 0.973 0.958 0.951 0.950 0.950 0.950 0.95230.5 0.990 0.980 0.983 0.977 0.968 0.963 0.963 0.961 0.963 0.96135.5 0.991 0.985 0.985 0.982 0.978 0.972 0.972 0.973 0.973 0.97340.5 0.992 0.986 0.987 0.986 0.983 0.981 0.978 0.977 0.980 0.97845.5 0.991 0.989 0.991 0.990 0.985 0.984 0.984 0.984 0.983 0.98450.5 0.992 0.991 0.992 0.992 0.990 0.990 0.990 0.987 0.989 0.98955.5 0.993 0.992 0.996 0.995 0.994 0.992 0.993 0.992 0.990 0.98960.5 0.994 0.993 0.996 0.997 0.997 0.994 0.994 0.993 0.995 0.99665.5 0.994 0.996 0.998 0.997 0.999 0.997 0.998 0.997 0.998 0.99870.5 0.995 0.992 0.998 0.999 0.997 0.998 0.999 0.997 0.998 0.99875.5 0.998 0.996 0.999 1.001 0.999 0.998 0.999 0.997 0.998 0.99580.5 1 0.998 0.999 1 1.001 0.998 0.999 1 0.998 0.99985.5 1 1 0.998 1 1 0.999 0.999 0.998 0.999 0.99890 1 1 1 1 1 1 1 1 1 1

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and polar anglesu,30°, namely only for points away fromboth the geometrical center of the source and the source’stransverse bisector. This can also be observed in Table IVwhere shaded entries refer to points with anisotropy functionvalues presenting deviations from unity greater than 10%.

In Fig. 7 we present Monte Carlo calculated anisotropyfunctions for various wire source lengthsL of d50.3-mmouter wire diameter, at selected radial distancesr to furtherinvestigate the dependence of anisotropy function on sourcelength.

It can be observed that for polar angles 30°,u,90° an-isotropy function values are almost unity for any radial dis-tance r and/or source lengthL. However, a strong depen-dence on both source length and radial distance is observedfor polar anglesu,30°. In this small polar angle region,anisotropy function values are almost unity forr ,L/2. Atradial distancer;L/2 a discontinuity is observed as anisot-ropy function values set off decreasing significantly forr.L/2. This is, partly, due to the fact that for a point close to

the source the main dose contributor is the source segmentclosest to that point while for points away from the source,the whole source contributes and anisotropy function de-creases for polar angles close to the long axis of the source(u,30°) due to oblique filtration within the source structure.This behavior has also been observed in previous studies forthe VariSource HDR192Ir brachytherapy source~1-cm activesource length!.12,20

The above findings may prove useful for brachytherapyapplications since the anisotropy function was found to be ofdosimetrical importance only at points of radial distancesr.L/2 and polar anglesu,30°. For all other points (r ,u)deviations of anisotropy function values from unity aresmall. We estimated that these deviations are less than 3%for the most practically used wire length range ofL54 – 6 cm.

Detailed anisotropy function values for any wire length~actually, full 3D dose distributions! can be calculated withreasonable computational times~;10 min on a PIII-500

FIG. 6. Monte Carlo and Sievert calculated anisotropy functionsF(r ,u) for wire lengthL55 cm of outer diameterd50.3 mm, as a function of polar angleu, relative to the long axis of the source, at selected radial distancesr 50.5, 1, 3, 5, 7, and 10 cm. Corresponding Monte Carlo values reported by Ballesteret al. ~Ref. 1! as well as Sievert calculated values for wire diametersd50.5 mm andd50.6 mm are presented for comparison.

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equivalent PC! by means of our Sievert model providingaccuracy acceptable for clinical applications~RMS errors;3%!. Moreover, Monte Carlo calculated data for wirelengths of 3 and 7 cm are presented in Tables V and VI anddata for the commonly used wire lengths ofL53 – 7 cm canbe produced by interpolation through Tables IV–VI based onanisotropy function behavior presented here.

IV. CONCLUSIONS

The implementation of the AAPM TG-43 dosimetric for-malism for 192Ir, Pt encapsulated wires used as interstitialsources in brachytherapy applications was investigated.

Dosimetric quantities utilized in the AAPM TG-43 dosi-metric formalism were calculated for lengthsL50.5– 12 cm of all three outer wire diameters and their depen-dence on source dimensions and calculation point was exam-ined using a well-established Monte Carlo simulation code

and our improved modification of the Sievert integral model.This Sievert integration model proved adequate for detaileddosimetry at all points around investigated wire sources pro-viding acceptable accuracy~RMS errors;3%! and reason-able computational times~;10 min on a PIII-500 equivalentPC!.

All dosimetric quantities agreed within 3% for all threecommercially available outer wire diameters of 0.3, 0.5, and0.6 mm.

The line source approximation proved applicable forsource lengths considered in this work. The geometry factorsof theL55-cm wire were provided and a scaling was intro-duced to provide the corresponding geometry factors of anyother source length.

The dose-rate constantL was found to decrease with in-creasing wire length with percentage differences followingthe percentage differences of the geometry factorG ~1 cm,

FIG. 7. Monte Carlo calculated anisotropy functionsF(r ,u) for wires of lengthL51, 3, 5, 7, and 10 cm and outer diameterd50.3 mm, as a function of polarangleu relative to the long axis of the source, at selected radial distancesr 50.5, 1, 3, 5, 7, and 10 cm.

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90°! at distancer 51 cm and angleu590°. Results are pre-sented in a grid appropriate to support accurate calculation ofL, for any wire lengthL and outer diameterd, through inter-polation. Alternatively, an estimation of the dose-rate con-stant of investigated wires, with errors less than 2%, may beprovided by the formula:

LL5GL~1 cm,90°!

GL55cm~1 cm,90°!LL55cm.

Radial dose function valuesg(r ) were found to increasewith increasing length for radial distancesr>1 cm. How-

TABLE V. Monte Carlo calculated anisotropy functions,F(r ,u), for wire lengthL53 cm and wire diameterd50.3 cm. Shaded values refer to points whereanisotropy function values present large deviations~.10%! from unity.

Polar angleu ~degrees!

Anisotropy functionF(r ,u)

Radial distancer ~cm!0.5 1 2 3 4 5 6 7 8 10

0.5 0.365 0.371 0.467 0.505 0.522 0.545 0.5592.5 0.921 0.946 0.560 0.493 0.505 0.525 0.538 0.556 0.581 0.6145.5 0.946 0.950 0.731 0.683 0.684 0.686 0.688 0.701 0.721 0.7457.5 0.952 0.953 0.813 0.748 0.737 0.753 0.779 0.782 0.783 0.801

10.5 0.958 0.959 0.869 0.820 0.810 0.812 0.826 0.825 0.829 0.84815.5 0.964 0.972 0.921 0.883 0.880 0.885 0.890 0.891 0.893 0.89720.5 0.968 0.978 0.944 0.922 0.920 0.914 0.925 0.925 0.927 0.93525.5 0.972 0.982 0.963 0.943 0.944 0.945 0.950 0.952 0.953 0.96230.5 0.976 0.984 0.966 0.955 0.957 0.959 0.962 0.960 0.963 0.97035.5 0.979 0.985 0.973 0.968 0.970 0.963 0.969 0.971 0.968 0.97240.5 0.982 0.983 0.981 0.975 0.972 0.973 0.973 0.974 0.975 0.97945.5 0.985 0.987 0.987 0.981 0.981 0.982 0.984 0.981 0.980 0.98050.5 0.987 0.991 0.991 0.983 0.981 0.985 0.988 0.983 0.986 0.99155.5 0.988 0.993 0.995 0.987 0.993 0.991 0.992 0.993 0.987 0.99260.5 0.990 0.995 0.997 0.988 0.989 1.001 1.000 0.995 0.995 0.99665.5 0.992 0.995 0.998 0.990 0.995 0.998 0.998 1.000 0.995 0.99770.5 0.994 0.998 0.997 0.994 0.996 1.000 0.999 1.001 1.000 1.00075.5 0.995 1.000 1.001 0.996 0.997 0.998 0.999 1.001 0.999 1.00080.5 0.997 1.000 1.000 0.999 1.001 1.001 1.000 0.999 1.000 0.99985.5 0.998 0.998 0.999 1.001 1.001 1.000 1.000 1.000 0.998 1.00090 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

TABLE VI. Monte Carlo calculated anisotropy functions,F(r ,u), for wire lengthL57 cm and wire diameterd50.3 cm. Shaded values refer to points whereanisotropy function values present large deviations~.10%! from unity.

Polar angleu ~degrees!

Anisotropy functionF(r ,u)

Radial distancer ~cm!0.5 1 2 3 4 5 6 7 8 10

0.5 0.480 0.520 0.417 0.460 0.489 0.5032.5 0.928 0.927 0.915 0.928 0.703 0.596 0.600 0.593 0.600 0.6245.5 0.951 0.946 0.952 0.953 0.840 0.757 0.747 0.748 0.749 0.7677.5 0.957 0.951 0.964 0.959 0.881 0.828 0.808 0.811 0.812 0.824

10.5 0.962 0.956 0.973 0.964 0.924 0.872 0.863 0.856 0.862 0.87215.5 0.968 0.961 0.982 0.970 0.952 0.922 0.922 0.920 0.914 0.91520.5 0.972 0.966 0.986 0.974 0.960 0.956 0.952 0.949 0.951 0.95125.5 0.975 0.969 0.990 0.977 0.968 0.971 0.968 0.970 0.963 0.96530.5 0.978 0.972 0.992 0.979 0.974 0.975 0.978 0.979 0.971 0.97235.5 0.980 0.976 0.994 0.982 0.978 0.977 0.980 0.981 0.983 0.98540.5 0.983 0.979 0.996 0.985 0.982 0.981 0.983 0.986 0.988 0.99045.5 0.985 0.982 0.998 0.987 0.984 0.987 0.990 0.992 0.991 0.99350.5 0.988 0.985 1.000 0.989 0.987 0.991 0.992 0.993 0.993 0.99555.5 0.990 0.988 0.997 0.991 0.993 0.993 0.995 0.995 0.996 0.99760.5 0.993 0.991 0.998 0.994 0.998 0.998 0.997 0.996 0.998 0.99965.5 0.993 0.994 1.000 0.996 0.995 0.997 1.001 0.996 0.999 0.99970.5 0.995 0.997 0.999 0.998 0.997 0.999 1.000 1.001 1.001 1.00175.5 0.997 0.998 1.000 1.000 1.000 0.998 1.000 0.999 1.000 1.00080.5 0.998 1.000 1.000 1.000 0.998 0.999 0.999 1.000 0.999 0.99885.5 1.000 1.000 1.000 0.998 0.999 0.998 1.000 1.001 1.000 1.00090 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

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ever, overall variation was less than 2% and straightforwarduse of the Monte Carlo calculated values for the 5-cm-longwire, presented in this work, is suggested for any practicallyused wire lengthL.

Anisotropy function of all investigated sources was foundto deviate significantly~.10%! from unity only for points atradial distancesr .L/2 and polar anglesu,30°. Anisotropyfunction values for the commonly used wire length range ofL53 – 7 cm can be calculated by interpolation throughMonte Carlo calculated data for wire lengths of 3, 5, and 7cm presented in this work.

ACKNOWLEDGMENTS

The authors wish to thank H. Papanikolaou for her valu-able technical assistance. This work was supported in part bythe Special Research Account of the University of Athens.

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