A CT-based analytical dose calculation method for HDR 192Ir brachytherapyEmily PoonMedical Physics Unit, McGill University, 1650 Cedar Avenue, Montreal, Quebec H3G 1A4, Canada

Frank Verhaegena�

Medical Physics Unit, McGill University, 1650 Cedar Avenue, Montreal, Quebec H3G 1A4, Canadaand Department of Radiation Oncology (MAASTRO), GROW, University Hospital Maastricht, Maastricht6229ET, The Netherlands

�Received 15 February 2009; revised 14 May 2009; accepted for publication 1 July 2009;published 12 August 2009�

Purpose: This article presents an analytical dose calculation method for high-dose-rate 192Irbrachytherapy, taking into account the effects of inhomogeneities and reduced photon backscatternear the skin. The adequacy of the Task Group 43 �TG-43� two-dimensional formalism for treat-ment planning is also assessed.Methods: The proposed method uses material composition and density data derived from computedtomography images. The primary and scatter dose distributions for each dwell position are calcu-lated first as if the patient is an infinite water phantom. This is done using either TG-43 or adatabase of Monte Carlo �MC� dose distributions. The latter can be used to account for the effectsof shielding in water. Subsequently, corrections for photon attenuation, scatter, and spectral varia-tions along medium- or low-Z inhomogeneities are made according to the radiological paths deter-mined by ray tracing. The scatter dose is then scaled by a correction factor that depends on thedistances between the point of interest, the body contour, and the source position. Dose calculationsare done for phantoms with tissue and lead inserts, as well as patient plans for head-and-neck,esophagus, and MammoSite balloon breast brachytherapy treatments. Gamma indices are evaluatedusing a dose-difference criterion of 3% and a distance-to-agreement criterion of 2 mm. PTRAN_CT

MC calculations are used as the reference dose distributions.Results: For the phantom with tissue and lead inserts, the percentages of the voxels of interestpassing the gamma criteria �P��1� are 100% for the analytical calculation and 91% for TG-43. Forthe breast patient plan, TG-43 overestimates the target volume receiving the prescribed dose by 4%and the dose to the hottest 0.1 cm3 of the skin by 9%, whereas the analytical and MC results agreewithin 0.4%. P��1 are 100% and 48% for the analytical and TG-43 calculations, respectively. Forthe head-and-neck and esophagus patient plans, P��1 are �99% for both calculation methods.Conclusions: A correction-based dose calculation method has been validated for HDR 192Ir brachy-therapy. Its high calculation efficiency makes it feasible for use in treatment planning. Becausetissue inhomogeneity effects are small and primary dose predominates in the near-source region,TG-43 is adequate for target dose estimation provided shielding and contrast solution are notused. © 2009 American Association of Physicists in Medicine.�DOI: 10.1118/1.3184695�

Key words: 192Ir, brachytherapy, scatter, shielding, inhomogeneity corrections

I. INTRODUCTION192Ir is the most commonly used radionuclide in high-dose-rate �HDR� brachytherapy. At 192Ir photon energies, Comp-ton scattering is the predominant interaction in tissue, andthe attenuation of primary photons is nearly offset by thebuildup of scattered radiation within the first 4–5 cm fromthe source.1 The dose is thus largely characterized by aninverse square falloff with distance,2 while the effects of tis-sue composition variations play a secondary role in 192Ir do-simetry.

All modern HDR brachytherapy planning systems complywith the water-based Task Group 43 �TG-43� dose calcula-tion formalism.3 Although computed tomography �CT� is in-creasingly being used for three-dimensional �3D� image-

based treatment planning, the tissue composition and densityinformation derived from CT images is not yet used clini-cally for brachytherapy dose calculations.4

The objectives of this paper are twofold. Firstly, we willintroduce an efficient CT-based analytical dose calculationmethod for HDR 192Ir brachytherapy. Secondly, we will as-sess the adequacy of TG-43 for treatment planning. MonteCarlo �MC� dose calculations are used as a benchmark.Given that tissue inhomogeneity effects of 192Ir are smallcompared to those of lower energy sources,5,6 we will focuson cancer sites near the lungs, air cavities, bones, and radio-graphic contrast solution. In light of some simplificationsmade in this method, we will point out its limitations andintended clinical applications. Advantages over alternativedose calculation methods will be discussed.

3982 3982Med. Phys. 36 „9…, September 2009 0094-2405/2009/36„9…/3982/13/$25.00 © 2009 Am. Assoc. Phys. Med.

II. MATERIALS AND METHODS

II.A. Algorithm overview

This correction-based dose calculation method uses a pri-mary and scatter dose separation approach proposed byWilliamson7,8 and Russell et al.9,10 The basic idea is to firstcalculate the primary and scatter doses for each dwell posi-tion as if the source is in an infinite water phantom. Then, theradiological path between a given point of interest �POI� andthe source, determined by ray tracing, is used to correct forphoton attenuation and scatter along tissue inhomogeneities.Lastly, the scatter dose is scaled by a scatter correction �SC�factor to account for the lack of a full scatter environmentnear the skin. This factor depends on the distances betweenthe source, the POI, and the body contour, as was demon-strated in our previous study.11 The SC method is useful forclinical cases where the dwell positions are off center and thenearby skin boundary has an irregularly smooth curvature.To report the absorbed dose to medium Dm, the dose is mul-tiplied by an effective medium-to-water mass energy absorp-tion coefficient ratio.

The photon energies of 192Ir correspond to secondaryelectron ranges that are short enough for the absorbed dose tobe equal to collision kerma.12 In this paper, primary photonsrefer to photons created inside the encapsulated source andhave not undergone interactions outside the source. Primarydose represents the collision kerma resulting from the inter-actions of primary photons, and scatter dose results from allother photon interactions.

II.B. Algorithm implementation and preprocessing

The analytical algorithm is incorporated intoBrachyGUI,13 an in-house brachytherapy planning systemdeveloped in MATLAB �version 7.7, MathWorks, Natick,MA�. The microSelectron v2 HDR 192Ir source model�Nucletron, Veenendaal, The Netherlands�14 and the NuDat2.0 192Ir photon spectrum15 are used.

We import the CT images of the patient into BrachyGUIto create 3D material and density matrices for tissue inhomo-geneity corrections. The elemental tissue compositions aretaken from the International Commission on Radiation Unitsand Measurements �ICRU� Reports 44 and 46.16,17 The massdensities are derived from the CT Hounsfield units. We cre-ate a 3D distance map for looking up the closest distancebetween a given voxel and the skin, which will be needed forthe SC method. The distance map is created by a preprocess-ing routine,11 which delineates the body contour and per-forms a 3D Euclidean distance transform of the contour. �Y.Mishchenko, 2007, “3D Euclidean distance transform forvariable data aspect ratio,” MATLAB Central File Exchange,The MathWorks, Natick, MA, http://www.mathworks.com/matlabcentral/fileexchange�

II.C. Dose calculations in an infinite water phantom

In the first step of the algorithm, the primary and scatterdoses in an infinite water phantom �Dprim,wat and Dscat,wat� arecalculated. This can be done using the TG-43 two-

dimensional �2D� formalism3 with the radial dose functionand anisotropy function broken down into primary and scat-ter components.11 When high-atomic-number �high-Z� mate-rials such as shielding are present, we use a database of 3Dprimary and scatter dose data18,19 instead to account for theirdosimetric effects in water. To generate this database, allpossible arrangements of single dwell positions with respectto each shielding type are to be simulated by the MCmethod. Coordinate transformations are applied to the appro-priate 3D dose data in accordance with the source orientationand the patient coordinate system to calculate Dprim,wat andDscat,wat.

II.D. Correcting for tissue inhomogeneities and finitepatient dimensions

II.D.1. Primary dose calculation

The primary dose Dprim,med at a POI in the medium med iscalculated as follows:

Dprim,med = Dprim,wat

�exp��i�− � �̄

�

prim,medi

�i + �̄prim,watri��� �̄en

�prim

wat

med

. �1�

The exponential term will be referred to as the attenuationcorrection factor. It corrects for the difference in photon at-tenuation caused by inhomogeneities along the radiologicalpath d between the source and the POI. We ray trace throughthe patient’s density matrix to determine d, which comprisesmultiple segments i of density �i and length ri,

d = �i

�iri. �2�

The ray tracing procedure will be described in Sec. II D 4.The effective mass attenuation coefficient of the medium

��̄ /��med, the effective linear attenuation coefficient for wa-ter �̄wat, and the effective medium-to-water mass energy ab-sorption coefficient ratio ��̄en /��wat

med are given the subscriptprim. Each quantity is averaged over the primary 192Ir pho-ton energy fluence spectrum. We use ��̄en /��wat

med to convertfrom absorbed dose to water Dw to Dm. For nontissue media�e.g. air, radiographic contrast solution, and shielding�, weset ��̄en /��wat

med to unity because they are not of clinical inter-est.

II.D.2. Scatter dose calculation

The scatter dose Dscat,med is calculated as follows:

Dscat,med = Dscat,wat

�exp��i�− � �̄

��d4,d3�

scat,medi

�i

+ ��̄�d4,d3��scat,wat�ri

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� � �̄en

�scat�d4,d3�

wat

med

fscat�d1,d2,d3�

�SPRwat�d4,d�

SPRwat�d4,d3�wat�. �3�

Five distance variables are used here: d represents theradiological path between the source and the POI; d1 refersto the distance between the source and the patient’s skin; d2

is the distance between the POI and the skin; d3 is the dis-tance between the POI and the source; and d4=d2+d3, whichis the distance from the source to the skin through the POI.Both d1 and d2 are determined from a 3D distance map.

The subscript scat in ��̄ /��scat,med, �̄scat,wat, and��̄en /�scat�wat

med denotes their association with the scatteredphotons. They are functions of the distances d4 and d3,whose effects on the spectral energy variations in tissueshave been quantified systematically using the approach de-scribed in Sec. II D 3.

The SC factor fscat at a given point is a ratio of the scatterdose in bounded water to the scatter dose in unboundedwater.11 Its value is retrieved from a SC table derived by MCcalculations. The indices for table lookup come from thedistances d1, d2, and d3. In general, fscat decreases with de-creasing d1 and d2, and increasing d3. The minimum value offscat for the 192Ir source is 0.177.

According to Anagnostopoulos et al.,1 the scatter to pri-mary dose ratio �SPR� at a distance d3 and radiological dis-tance d from an 192Ir point source in tissue materials underfull scatter conditions can be approximated by SPRwat�d�,which is the SPR at radiological distance d in an infinitewater phantom of density �wat=1 g /cm3. We apply theirfinding to correct for the scatter dose altered by inhomoge-neities. Given that SPRwat is different in a finite volume, weinclude the extra variable d4 in Eq. �3� to indicate that SPRwat

is calculated in a water sphere of radius d4. The dose correc-tion thus entails taking the SPR ratio, which is SPRwat�d4 ,d�divided by SPRwat�d4 ,d3�wat�.

We calculated SPRwat at various distances from an HDR192Ir source in spherical water phantoms with radii of 5, 7.5,10, 12.5, 15, and 50 cm using the GEANT4 MC code �version9.1�.20 The low-energy electromagnetic physics package,20

the EPDL97 photon cross section library,21 and the mass en-ergy absorption coefficients of Hubbell and Seltzer22 wereused. We scored the primary and scatter dose using a lineartrack length estimator.23 The SPRwat corresponding to eachphantom radius was fitted as a quadratic function of radio-logical distance. To determine the SPR ratio for a POI, welook up the SPRwat function according to the associated d4

value by a nearest neighbor search.

II.D.3. Derivations of �̄ /� and �̄en/� ratios

GEANT4 was used to simulate the 192Ir source in the centerof spherical water phantoms with radii of 5, 7.5, 10, 15, and50 cm. The primary and scatter energy fluences were talliedat 5 mm radial distance intervals. The primary componentsof �̄ /� and ��̄en /��wat

med are derived from the primary energyfluence of the largest phantom. Since the values vary slowly

with distance in tissue, they are set as constants for dosecalculations. The scatter components were fitted using qua-dratic functions for the five phantoms of different radii. Thefunctions related to the phantom radius closest to d4 are usedto find ��̄ /��scat,med, �̄scat,wat, and ��̄en /�scat�wat

med at a POI.

II.D.4. Ray tracing in 3D patient body

We ray trace through the patient body in the sphericalcoordinate system. For each dwell position, 271 paths at 2°separation are traced along the azimuth and zenith directionsin 2 mm steps, i.e., ��=��=2°, for 0° �358°, 0° �180°, and �r=2 mm. Let the term inside the exponentialfunction of Eq. �1� or Eq. �3� be called t. We calculate eachattenuation correction factor for a POI in a Cartesian dosegrid by taking the exponential of the mean of the eight near-est t values determined by ray tracing. There may be fewerthan eight t values if the POI is near the grid boundary. TheSPR ratio and fscat are calculated similarly. We take the meanrather than using an eight-point interpolation method24 toreduce the CPU time. To evaluate the robustness and effi-ciency gain of this method, a comparison was made with acalculation that ray traced in the 3D Cartesian grid using animproved version25 of Siddon’s algorithm.26

On the other hand, Dprim,wat and Dscat,wat include alreadythe effects of shielding, if they are present. A separate raytracing through shielding is then not needed, and the voxelsoccupied by shielding are replaced by water with a density of1 g /cm3.

II.E. MC vs analytical and TG-43 calculations

The analytical and TG-43 calculations in phantom andCT-based geometries were compared to PTRAN_CT

27 calcula-tions. PTRAN_CT is an extended version of the PTRAN MCphoton transport code.23,28 We used a phase space file togenerate the primary photons13,19 and an exponential tracklength estimator to score the dose.23 The DLC-146 photoncross section library29 and the mass energy absorption coef-ficients of Hubbell and Seltzer22 were used. The dose perparticle history was converted to dose per unit air kermastrength.19 Above 10% of the prescribed dose Dref, the maxi-mum 1– statistical uncertainties are 0.6% for the phantomcalculations and 1.5% for the patient calculations.

Dose differences were quantified by the 3D gamma evalu-ation method of Wendling et al.,30 using PTRAN_CT calcula-tions as the reference dose distributions. We set the dose-difference criterion to 3% of Dref, the distance-to-agreementcriterion to 2 mm, the sample step size to 0.5 mm, and themaximum search distance to 6.67 mm. The maximum inten-sity projections of the gamma indices ��MIP� along the axialplane were generated for the patient plans. We reported themean gamma ��mean�, the 99th percentile ��1%�, and the per-centage of points with gamma indices below unity �P�1�.These three quantities were evaluated for regions receiving�20% of Dref as calculated by the MC method, excludingvoxels assigned as air, contrast solution, or lead.

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II.E.1. Phantom calculations

Figures 1�a� and 1�b� show the two phantom calculationsetups. In both cases, the 192Ir source was centered in a phan-tom consisting of 2�2�2 mm3 voxels. We set the airkerma strength to 40 000 U �1 U=1 �Gy m2 h−1�, thedwell time to 100 s, and Dref to 3 Gy.

The first setup was used to study tissue inhomogeneityeffects in an environment lacking full photon backscatter.The soft tissue phantom contained lung, adipose, spongiosa,and cortical bone inserts. Cortical bone represents one of thedensest and least water-equivalent tissue media. Spongiosa iscomposed of cortical bone, red marrow, and yellow marrowin equal proportion by mass. We also did a calculation usingthe SC method alone.11

The second setup was designed to study tissue inhomoge-neity effects near shielding. The water phantom containedeight tissue inserts and one lead insert. The MC dose in thisphantom with only a lead insert was also calculated. More-over, we examined if ray tracing along the lead insert couldbe done to account for its effects on photon attenuation andscatter.

II.E.2. Patient calculations

Three clinical treatment plans were calculated. The firstone was for a head-and-neck cancer patient undergoing na-sopharyngeal brachytherapy. Two catheters and 52 dwell po-sitions were used to deliver 6 Gy/fraction to the target. Thevoxel dimensions were 1.05�1.05�3 mm3.

The second plan was for an esophageal cancer patient. Afilm-based treatment plan was created to deliver 5 Gy/fraction at 1 cm from the catheter. Seventeen dwell positionsat 5 mm step intervals were used. The same plan was thendone retrospectively using CT images. The voxels were 1.8�1.8�5 mm3.

The last plan was for a breast cancer patient treated with aMammoSite balloon applicator �Cytyc Corporation, Marlbor-ough, MA�.31 The Dref was 3.4 Gy/fraction prescribed at 1cm from the balloon’s surface. After the injection of dilutediodine contrast solution �Omnipaque, GE Healthcare, UK�,the balloon diameter expanded to around 43 mm. One dwellposition was used. The voxels were 2�2�2 mm3. The con-trast solution was assigned, according to the proportion ofwater mixed with the contrast solution, a concentration of

TABLE I. 3D gamma statistics for the analytical and TG-43 calculations in phantom and patient geometries. Thereference dose distributions are calculated using PTRAN_CT. Voxels assigned as air or contrast solution areexcluded in the statistics.

Calculation

�mean �1%

P�1

�%�

Analytical TG-43 Analytical TG-43 Analytical TG-43

Phantom with tissue inserts 0.08 0.42 0.35 0.85 99.9 99.6Phantom with tissue and lead inserts 0.05 0.51 0.50 6.25 100.0 90.7Head-and-neck patient 0.17 0.30 0.70 1.03 99.7 98.9Esophagus patient 0.13 0.24 0.52 0.51 99.6 99.7Breast patient 0.18 1.06 0.63 2.11 100.0 48.3

FIG. 1. �a� A 15�15�10 cm3 soft tissue phantom without shielding. Four 3�1.5�1.5 cm3 tissue inserts are placed inside. �b� A 30�30�25 cm3 waterphantom with lead shielding. It contains eight 2�2�2 cm3 tissue inserts. The center of mass of a 2�0.2�2 cm3 lead insert is 0.8 cm away from thephantom’s center. The numbers correspond to the listed materials and densities. The HDR 192Ir source is placed in the center �marked by a dot�.

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FIG. 2. Primary and scatter components of the effective mass attenuation coefficients of water and six different tissue materials versus distance from the HDR192Ir source. The values are calculated according to the photon spectra at various radial distances in spherical water phantoms of radii r.

FIG. 3. Primary and scatter components of the effective medium-to-water mass energy absorption coefficient ratios versus distance for six different tissuemedia. The values are calculated according to the photon spectra at various radial distances in spherical water phantoms of radii r.

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50 mg I /ml and a density of 1.056 g /cm3. Two additionalcalculations, regarding the patient body to be composed ofwater, were done using the SC method and the MC methodtaking only density variations into account. The results forthe latter will be denoted as MCden.

The following were calculated for the breast patient plan:The target volumes receiving at least 90%, 100%, 150%, and200% of Dref �V90, V100, V150, and V200�; the minimum dosesto 90% and 100% of the target �D90 and D100�; and the mini-mum doses to the hottest 0.1, 1, and 10 cm3 of the skin,lung, and chest wall �D0.1 cc, D1 cc, and D10 cc�.

III. RESULTS AND DISCUSSION

III.A. �̄ /�, „�̄en/�…watmed, and SPR

Figures 2�a�–2�d� show �̄ /� versus distance from the 192Irsource for water and six tissue materials derived from theprimary and scattered photon spectra in water phantoms ofdifferent sizes. In Figs. 3�a�–3�d�, ��̄en /��wat

med for six tissuematerials are shown. The primary components of both quan-tities exhibit a slow decrease with increasing distance be-

cause of beam hardening. The scatter components are af-fected more by spectral changes, especially for bones in thelarger phantoms.

Figure 4 shows the SPRwat for the 192Ir source calculatedin water spheres of six different radii. For the phantoms ofradii 7.5 cm, SPRwat is always below unity and the pri-mary dose at any POI is always higher than the scatter dose.In the larger phantoms, the primary and scatter dose contri-butions become nearly the same at 7 cm.

III.B. 3D gamma evaluation

The statistics of the gamma indices for the phantom andpatient calculations are summarized in Table I. The values of�mean for the analytical calculations are consistently lowercompared to TG-43, indicating a better general agreementwith MC calculations.

III.C. Phantom studies

III.C.1. Phantom with tissue material inserts

Figures 5�a�–5�c� show the first phantom calculation re-sults. The TG-43 dose is higher and the overestimate be-comes increasingly obvious near the phantom surface. A bet-ter agreement is seen after applying the SC method despitethe higher density �1.06 g /cm3� of this soft tissue phantom.The dose is nearly unperturbed by the adipose and spongiosainserts. Below the 30% isodose level, the SC dose is lowerby up to 15% behind the lung insert and higher by up to 17%behind the cortical bone. The analytical method underesti-mates the dose behind the lung insert by up to 7%, but isotherwise in agreement with the MC calculation within 3%.

The primary and scatter dose distributions calculated bythe analytical method are compared to the corresponding MCdistributions in Figs. 6�a� and 6�b�. The primary dose distri-butions agree better than 2%. However, the analyticalmethod underestimates the scatter dose behind the lung in-sert by up to 20% and slightly overestimates the dose aroundit. The reverse happens on a smaller scale around the corticalbone, and the maximum scatter dose error is 7%. The differ-ences demonstrate that the analytical method cannot account

FIG. 4. Scatter to primary dose ratio as a function of radiological distancefrom the HDR 192Ir source calculated in water spheres of different radii �50,15, 12.5, 10, 7.5, and 5 cm�.

FIG. 5. Isodose distributions around an HDR 192Ir source in a soft tissue phantom with four tissue inserts. The MC isodose is shown in all panels in solid lines.The dashed isodose distributions are calculated using �a� TG-43, �b� scatter correction method, and �c� analytical method.

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for the lateral scatter near regions of markedly lower orhigher densities. Nonetheless, the total dose error is smallerand P�1=99.9% because the primary dose predominates.

III.C.2. Phantom with tissue and shieldinginserts

Figure 7�a� illustrates tissue inhomogeneity effects nearlead shielding. The analytical method, which uses precalcu-lated 3D dose data to account for shielding perturbations, isable to partially correct for such effects �see Fig. 7�b��. As itneglects the lateral scatter altered by the tissue inserts, thereare small local differences of up to 5% and 10% behind thelung and cortical bone regions on the shielded side, respec-tively. The differences occur in the low-dose region andP�1=100.0%.

The total, primary, and scatter dose components of theMC calculations are compared to the analytical results inFigs. 8�a�–8�c�. In this particular analytical calculation, wecorrected for both shielding and tissue inhomogeneity effectsvia ray tracing. Since lead absorbs more of the lower energyphotons and yet a constant value is used for �̄ /�prim, theprimary dose is underestimated by up to 18%. The methodalso fails to predict the increase in the SPR behind the lead,

because the increased importance of photoelectric effect in-validates the determination of the SPR ratio based on SPRwat

and density scaling.1 The values of �mean, �1%, and P�1 are0.36, 3.70, and 90.9%, respectively. The improved Siddonalgorithm was used here because the phantom contains a leadinsert, and our ray tracing method will cause discretizationartifacts �see Fig. 8�d��. Nonetheless, the isodose lines forboth ray tracing methods are indistinguishable on the un-shielded side because the 192Ir spectrum, fscat, and SPR ratiosvary slowly over the sampling interval for lower-Z media.The efficiency gain over the improved Siddon method is sev-enfold when calculating the dose in 153�153�128 voxels.

III.D. Patient studies

III.D.1. Head-and-neck cancer patient

The isodose, axial �MIP image, and dose-volume histo-gram �DVH� comparisons for the head-and-neck patient cal-culations are shown in Figs. 9�a�–9�f�. The isodose linesabove 50% of Dref for all calculations are practically thesame because the inverse square law has a greater dosimetricimpact than tissue inhomogeneity effects in the near-sourceregion. The gamma indices for voxels of �2 mm from thedwell positions apply to a steep dose-gradient region and

FIG. 6. �a� Primary and �b� scattercomponents of the isodose distribu-tions around an HDR 192Ir source in asoft tissue phantom with four tissueinserts. In both panels, the solid iso-dose lines are MC calculations, andthe dashed lines are analytical calcu-lations.

FIG. 7. Isodose distributions around anHDR 192Ir source in a water phantomwith lead and tissue inserts. The MCisodose is in solid lines in both panels.The dashed lines in �a� represent theMC isodose which only includesshielding effects. The dashed lines in�b� represent the isodose calculated bythe analytical method.

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their values are above unity, but such differences may not beclinically important. As the primary dose predominates, thetarget coverage is immune to the influence of scatter despiteits proximity to the skin, nasal cavities, and bones. TheTG-43 dose is higher near the skin, while the analytical andMC results are in better agreement. TG-43 also slightly over-estimates the dose to the brain stem, which is lightlyscreened by bones. The DVHs show that the three methodsare comparable in predicting the dose to organs at risk�OARs�.

III.D.2. Esophagus cancer patient

Figures 10�a�–10�f� show the calculation results for theesophageal patient plan. Changes in the scatter conditionscaused by the lung and spinal cord are somewhat localized.Even though the slight discrepancy around the lung cannotbe resolved by analytical means, there is close agreement inthe OAR DVHs. Our results are similar to those of Anag-nostopoulos et al.32 who developed their own analyticalalgorithm.1,33 They used MC and analytical methods to cal-culate an esophagus treatment plan with a phantom thatmimics the upper thorax.

III.D.3. Breast cancer patientThe MammoSite patient calculations in Figs. 11�a�–11�f�

show a larger discrepancy between the TG-43 and MC cal-culations. Since the target is �2 cm away from the 192Irsource, the scatter dose contribution is more important com-pared to the other two patient cases. Also, the balloon appli-cator pushes the target closer to the skin where the photonbackscatter is reduced. Furthermore, the contrast solutioncauses more photon attenuation.

Table II shows the dose-volume indices for the target,skin, ipsilateral lung, and chest wall. TG-43 overestimatesthe dose to all structures by �5%. The errors are reduced bythe SC method. The analytical method further improves theaccuracy, although small differences are seen around thelung. For the MCden calculation, the scatter dose is overesti-mated while the primary beam hardening has minimal ef-fects. The high-dose volumes �target V150 and V200� arelarger, and the dose to OAR is higher by 2%. Previous MCstudies34–36 corroborate our finding that making adjustmentsfor density variations alone is not enough to correct for theattenuation effects of contrast solution. A phantom study byYe et al.34 shows that contrast solution may reduce the doseby 1.0%–4.8%, depending on its concentration. The totaldose error ranges from 4% to 10% when the smaller back-scatter near the skin is also neglected.34

FIG. 8. Isodose distributions around anHDR 192Ir source in a water phantomwith lead and tissue inserts. The �a�total, �b� primary, and �c� scatter com-ponents of the dose calculated by theMC method �solid lines� are comparedto the analytical results. In this analyti-cal calculation, the improved Siddonalgorithm is used to correct for bothshielding and tissue inhomogeneity ef-fects. In �d�, the analytical calculationusing our proposed ray tracing method�solid lines� is compared to the calcu-lation using the improved Siddonmethod �dashed lines�.

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III.E. Calculation efficiency

The codes for the TG-43 and analytical algorithms werecompiled into MATLAB callable C executables using the LCC-

WIN32 compiler. We did all calculations on a 2.0 GHz pro-cessor running a 32-bit Windows operating system. On aver-age, the analytical method took 3.6 times longer to run thanTG-43. Its efficiency depends on the dose grid and is in-versely proportional to the number of dwell positions. Typi-cal CPU times range from 1 s for a MammoSite patient planto a few minutes for a multicatheter breast patient plan with�200 dwell positions. This is different than the MC method,whose efficiency depends on the scattering volume and isless influenced by the number of dwell positions. Also, theMC CPU time is related to the statistical uncertainties. Itcould be one to three orders of magnitude slower than theanalytical method.

III.F. Algorithm assumptions and limitations

The analytical algorithm is partly based on the work ofAnagnostopoulos et al.1 They found that by using �̄ /� and��̄en /��wat

med weighted over the 192Ir spectrum, one can collec-tively account for the contributions from every emission lineof the polyenergetic source �cf. Ref. 37� with better than 2%

accuracy.1,33 In tissue media under full scatter conditions, theapproach of Anagnostopoulos et al. to find the SPR by usingSPRwat and density scaling gives a maximum error of 2% inthe total dose.1

As the makeup of a human body is complex, approxima-tions are used to make dose calculations efficient and appli-cable for various treatment sites. We make two suppositionsin determining �̄ /�scat and ��̄en /�scat�wat

med: �1� The scatteredphoton spectrum at the POI has not been greatly altered byinhomogeneities and �2� the scatter conditions are nearlyequivalent to having the POI positioned in a water sphere ofradius d4. The latter also applies to the determination of theSPR ratio. Provided shielding is not present, the errors aris-ing from the breakdown of these conditions are small be-cause changes in the 192Ir spectrum do not strongly influencethe dose in tissue materials.6 Furthermore, the attenuationcorrection factor depends on the difference between �̄scat,med

and �̄scat,wat, which is less error prone than their respectivevalues. Similarly, an error in estimating the scattering me-dium does not influence the SPR ratio as much as the indi-vidual SPR values.

The inability to account for lateral and backscatter radia-tion near inhomogeneities is a limitation of this analyticalmethod. It manifests itself mainly in scatter dose errors

FIG. 9. Isodose distributions, axial �MIP, and OAR DVHs for a head-and-neck patient plan. The isodose distributions are normalized to the Dref of 6Gy/fraction. In the upper panels, MC �solid lines� and TG-43 �dashed lines� calculations are compared. The lower panels compare the MC �solid lines� andanalytical �dashed lines� calculations.

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FIG. 10. Isodose distributions, axial �MIP, and OAR DVHs for an esophageal patient plan. The isodose distributions are normalized to the Dref of 5 Gy/fraction.In the upper panels, MC �solid lines� and TG-43 �dashed lines� calculations are compared. The lower panels compare MC �solid lines� and analytical �dashedlines� calculations.

FIG. 11. Isodose distributions, axial �MIP, and DVHs for a MammoSite breast patient plan. The isodose distributions are normalized to the Dref of 3.4Gy/fraction. In the upper panels, MC �solid lines� and TG-43 �dashed lines� calculations are compared. The lower panels compare MC �solid lines� andanalytical �dashed lines� calculations.

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around low density regions such as large air cavities and thelungs. The errors around bones in realistic patient geometriesare small. As shown in the patient calculations, this limita-tion is unlikely to affect the target coverage because the near-source region is dominated by primary dose.

The phantom calculations show that, in principle, usingprecalculated MC dose data improves the dosimetric accu-racy near high-Z materials. We have not demonstrated thetechnique for patient dose calculations with shielding. Propermaterial and density segmentations may require correctionsof CT metal streaking artifacts38 as well as dual energy CTimaging.39

III.G. Other dose calculation methods

The MC method in radiation transport is a well-established approach for accurate dose calculations. SeveralMC codes27,40,41 allow for efficient simulations of complexbrachytherapy seeds together with the patient body derivedfrom CT data. The target dose in low-energy seed implantsapplications can be calculated in less than 1 min.41 A longerCPU time is needed for 192Ir brachytherapy owing to a moregradual energy loss primarily by Compton scattering com-pared to sources in the low-energy range where the photonmean free paths are shorter.4 In fact, the CPU time to calcu-late the dose to the OARs by the MC method could be con-siderable when low statistical uncertainties are desired. Thisis particularly so when the organs are a few centimeters awayfrom the treatment site where the scatter dose becomes im-portant. On the contrary, the efficiency of the analyticalmethod is independent of the scattering volume. The CPUtime could be further reduced by only calculating the dose toselected organs. The analytical method is hence more suit-able for anatomy-based inverse treatment planning.

The collapsed cone superposition technique is a promis-ing dose calculation method, but the long CPU time for 192Irbrachytherapy is still an unresolved issue.42–44 Techniques

have also been proposed for tissue inhomogeneity correc-tions in phantoms of cylindrical symmetry5 and for scatterdose calculations around inhomogeneities of knowndimensions,45–47 but they are not adaptable for CT-based ge-ometries. For brachytherapy of superficial skin lesions thatextend over a large region, the SC method alone is faster andcan adequately correct for the reduced photon backscatternear the skin.11

The one-dimensional path length correction method8,48

was designed for 137Cs brachytherapy with a shielded appli-cator. Since 137Cs photons are higher in energy, the dose iscontributed mostly by primary photon interactions and is un-affected by spectral energy variations. This method is lessaccurate for 192Ir applications and therefore we did not useray tracing to account for shielding effects. As shown in thiswork and elsewhere,49,50 dose errors are mainly on theshielded side.

Deterministic approaches of solving the 3D Boltzmanntransport equation such as the discrete ordinates method havebeen investigated for brachytherapy applications.51–53 Sinceall geometries are modeled as composites of meshes, rayeffects originating from brachytherapy sources and applica-tors will be noticeable unless small meshes are used. Giffordet al.53 showed that deterministic transport parameters can beoptimized for a faster calculation speed, but the method hasnot been applied for CT-based dose calculations yet.

IV. CONCLUSIONS

A correction-based analytical dose calculation algorithmhas been developed for HDR 192Ir brachytherapy to accountfor the effects of tissue inhomogeneities, shielding, and thereduced photon backscatter near the skin. Although themethod neglects the lateral and backscatter around inhomo-geneities, it is a major improvement over the TG-43 formal-

TABLE II. Dose-volume indices for MammoSite breast patient calculations.

Structure Dose-volume index MC

Ratio

TG-43 / MC SC / MC Analytic / MC MCden / MC

Target D90 �%� 97.9 1.06 1.03 1.00 1.02D100 �%� 59.8 1.05 1.02 1.01 1.01V90 �%� 94.3 1.02 1.01 1.00 1.01V100 �%� 88.6 1.04 1.02 1.00 1.02

V150 �cm3� 31.5 1.20 1.11 1.01 1.08V200 �cm3� 7.0 1.40 1.25 1.02 1.18

Skin D0.1 cc �%� 81.1 1.09 1.03 1.00 1.02D1 cc �%� 73.3 1.10 1.02 1.00 1.02D10 cc �%� 52.0 1.12 1.02 1.00 1.02

Lung D0.1 cc �%� 68.0 1.09 1.06 1.03 1.02D1 cc �%� 59.9 1.09 1.07 1.04 1.02D10 cc �%� 44.5 1.09 1.06 1.04 1.02

Chest wall D0.1 cc �%� 94.3 1.06 1.04 1.02 1.01D1 cc �%� 82.7 1.07 1.04 1.02 1.02D10 cc �%� 53.9 1.08 1.05 1.03 1.02

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ism. It is of interest especially for brachytherapy with ashielded applicator, or breast brachytherapy with a balloonapplicator injected with contrast solution.

This work also shows that TG-43 is adequate for 192Irtreatment planning when shielding and contrast solution arenot used. It gives the correct target dose even for treatmentsites near the lungs, air cavities, and bones. This is becauseof the small tissue inhomogeneity effects and the predomi-nance of primary dose in the near-source region.

ACKNOWLEDGMENTS

The authors would like to thank Professor J. F. William-son and Dr. Yi Le for providing the PTRAN_CT code, andDr. Brigitte Reniers for creating the esophageal treatmentplan. E.P. is a CIHR Strategic Training Fellow in the Excel-lence in Radiation Research for the 21st Century Program.Financial support from Nucletron is gratefully acknowl-edged. F.V. was supported by the Fonds de la recherche ensanté du Québec.

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