accuracy of the phase space evolution dose calculation model

7/28/2019 Accuracy of the Phase Space Evolution Dose Calculation Model

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Phys. Med. Biol. 45 (2000) 29312945. Printed in the UK PII: S0031-9155(00)13708-X

Accuracy of the phase space evolution dose calculation model

for clinical 25 MeV electron beams

Erik W Korevaar, Abdelhafid Akhiat, Ben J M Heijmen andHenk Huizenga Daniel den Hoed Cancer Center, University Hospital Rotterdam, PO Box 5201,3008 AE Rotterdam, The Netherlands Joint Center for Radiation Oncology ArnhemNijmegen, University Medical Center Nijmegen,PO Box 9101, 6500 HB Nijmegen, The Netherlands

E-mail: [email protected]

Received 4 May 2000

Abstract. The phase space evolution (PSE) model is a dose calculation model for electronbeams in radiation oncology developed with the aim of a higher accuracy than the commonlyused pencil beam (PB) models and with shorter calculation times than needed for Monte Carlo(MC) calculations. In this paper the accuracy of the PSE model has been investigated for 25 MeVelectron beams of a MM50 racetrack microtron (Scanditronix Medical AB,Sweden) andcomparedwith theresults of a PBmodel. Measurementshavebeen performedfor testslike non-standard SSD,irregularly shaped fields, oblique incidence and in phantoms with heterogeneities of air, bone andlung. MC calculations have been performed as well, to reveal possible errors in the measurementsand/or possible inaccuracies in the interaction data used for the bone and lung substitute materials.Results show a good agreement between PSE calculated dose distributions and measurements. Forall points the differencesin absolute dosewere generally well within 3% and 3 mm. However,the PSE model was found to be less accurate in large regions of low-density material and errorsof up to 6% were found for the lung phantom. Results of the PB model show larger deviations,

with differences of up to 6% and 6 mm and of up to 10% for the lung phantom; at shortenedSSDs the dose was overestimated by up to 6%. The agreement between MC calculations andmeasurement was good. For the bone and the lung phantom maximum deviations of 4% and 3%were found, caused by uncertainties about the actual interaction data. In conclusion, using thephase space evolution model, absolute 3D dose distributions of 25 MeV electron beams can becalculated with sufficient accuracy in most cases. The accuracy is significantly better than for apencil beam model. In regions of lung tissue, a Monte Carlo model yields more accurate resultsthan the current implementation of the PSE model.

1. Introduction

At present the pencil beam (PB) method is the most widespread method for electron beam dose

calculations in radiotherapy (Hogstrom et al 1981, Brahme et al 1981). An alternative is theMonte Carlo (MC) method (Nelson et al 1985, Ma et al 1999), which has the advantage of ahigher accuracy but the disadvantageof relatively long calculation times. In thepast alternativeelectron beam dose calculation models have been developed in an attempt to compute the dosefaster than Monte Carlo yet retain an accuracy that is sufficient for application in radiationoncology. Examples of such modelsbased on theMonte Carlo methodare macro Monte Carlo,super Monte Carlo andvoxelMonte Carlo (Neuenschwanderand Born 1992, Neuenschwanderet al 1995, Keall and Hoban 1996, Kawrakow et al 1996). The subject of the present study is

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2932 E W Korevaar et al

the phase space evolution (PSE) model, also developed with the aims of increased calculationspeed and accuracy but being a numerical solution to the transport equation in phase space(Huizenga and Storchi 1989, Morawska-Kaczynska and Huizenga 1992, Janssen et al 1994,1997a, Korevaar et al 1996). The version of the phase space evolution model studied here

is a research version that was coupled to the CADPLAN treatment planning system (Varian-Dosetek version 6.0.4). Two main modifications were made in comparison to the modeldescribed by Janssen et al (1997a), i.e. the use of a diverging calculation grid and a new modelto derive the initial phase space from a few measured depth dose distributions in water (Janssenet al 1997b).

The aim of the present paper is to investigate the accuracy of the PSE model for clinical,high-energy electron beams and to compare the PSE results with the results of a commonlyused implementation of the pencil beam method. In this study it was investigated whether andto what extent, the accuracy of electron beam dose calculations can be improved by using thePSE model instead of a PB model. The pencil beam model studied here was the pencil beamalgorithm implemented in CADPLAN (version 3.1.2). This is a generalized Gaussian pencilbeam model (Hyodynmaa 1991); however, in the 2250 MeV energy range a single Gaussian

pencil beam model is used. Recent publications have already shown that the accuracy of thepencil beam model in CADPLAN is sufficient in many cases, with the exception of complexgeometries with heterogeneities (Samuelsson et al 1998, Ding et al 1999).

The accuracies of the PSE model and the PB model have been determined by comparisonsof calculated and measured results in water phantoms and in phantoms with heterogeneities.The test cases were selected to best reflect situations frequently encountered in clinicaltreatment planning and are similar to tests described by other authors (Shiu et al 1992,Muller-Runkel and Cho 1997, Blomquist et al 1996). For some geometries, Monte Carlosimulations have been done with the EGS4 code to verify the measured results. Furthermore,thecomparisonofcalculatedresultsfromthePSEandPBmodelswithMCcalculationsallowedanalysis of the whole 3D dose distribution, whereas the comparison with measured results waslimited to a few lines and/or planes for practical reasons.

2. Methods and materials

2.1. Measurements

Measurements have been done for 25 MeV, multileaf collimator shaped electron beams of anMM50 racetrack microtron (Karlsson et al 1992). The purpose of the measurements was tocollect input data for the configuration of the dose calculation models and to obtain benchmarkdata to assess the accuracy of the models. The measurements performed to collect data forconfiguration of the dose calculation models are specified in table 1.

To test the dose calculation models for the 25 MeV electron beam, measurements wereperformed in both water and heterogeneous phantoms (table 2). The standard sourcesurfacedistance (SSD) adopted was 900 mm. Non-standard SSD measurements were performed in

water as were the irregular field shape (figure 1(a)) and the oblique incidence (figure 1(b)). Fortests in heterogeneous phantoms three polystyrene based phantoms were used: one containingan air cavity, one containing a cavity with a bone substitute and one with a lung substitute(figures 1(c), 1(d) and 1(e)). To determine the accuracy of the dose calculation models forpolystyrene measurements were performed in a homogeneous polystyrene phantom. Thesemeasurements utilized a cylindrical ionization chamber inserted into a milled space within aslab (figure 1(c)). Depth dose curves were measured by variation of the depth of this slab.Ionization readings were converted to dose in polystyrene by multiplication with the air to


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Table 1. Measured input data to configure the 25 MeV electron beam for the PSE model and thePB model. (For the PSE model the depth dose curve of the blocked field was not used. For the PBmodel only the depth dose curves at 1000 mm SSD (sourcesurface distance) for the largest fieldsize and the blocked field were used.)

Name SSD (mm) Field sizes (mm2) Measurement

Depth dose curve 900, 310 400, 150 150, Diode in water tank1000 100 100, 50 50

Depth dose curve 1000 Completely blocked field Diode in water tanksvir 8001300 310 400 Ionization chamber in water tankx 8001500 100 100 Film in air

Scanditronix linear diode array in an RFA 300 scanning system. Distance from the virtual point source to the plane through the isocentre (ICRU 1984). svir has been determined bythe inverse square law method for the largest field size. NACP type ionization chamber. Root of the angular variance at the level of the multileaf collimator, derived from penumbra widths measured in airat various SSDs (van Battum and Huizenga 1999).

Table 2. Summary of tests. Field sizes are defined at the level of the isocentre. Measurement

positions have been specified in figure 1.

SSD (mm) Field size (mm2) Phantom Detector type

Non-standard SSD 850, 950 100 100 Water DiodeL-shaped field 900 40/80 100/150 Water DiodeOblique incidence (30) 1000 150 150 Water IC, filmPolystyrene phantom 900 100 100 PS ICPhantom with air cavity 900 140 140 PS/air IC, filmPhantom with bone 900 140 140 PS/bone IC, filmLung phantom 900 100 100 PS/cork/PS IC, film

IC = ionization chamber, type PTW Semiflex Tube, 0.125 cm3, Waterproof; T31002. PS = white polystyrene, density 1.05 g cm3. Bone = hard bone-equivalent material, density 1.92 g cm3 (SB5, ICRU 1989). Cork density = 0.23 g cm3.

polystyrene stopping power ratio at the corresponding depth (AAPM 1983). The stoppingpower ratios applied to measurements behind the air or bone heterogeneity were determinedby calculation of the equivalent thickness of the heterogeneity, i.e. the actual thickness of theheterogeneity times the density of air or bone. No measurements were done inside the bonematerial. The effective point of measurement was taken as half the chamber radius above thecentre of the chamber (ICRU 1984).

Film (Kodak X-omat V) was used to measure dose distributions in planes perpendicularto the beam axis in the phantom with the air cavity and the phantom with bone. The filmswere scanned using a Wellhofer densitometer (WP102) and optical densities were convertedto dose by application of density to dose curves (van Battum and Huizenga 1990). The dose

profiles measured with film have been normalized according to the depth dose curve measuredwith the ionization chamber.

Film was also used to determinea depth dose curve in thephantom with lung substitute, i.e.cork. Film was preferred over ionizationchamber measurements, since in low-density materialthe chamber wall (density1 g c m3) introduces fluence perturbations. It has been shown thatfilm with an orientation perpendicular to the beam axis can be used as a reliable dosimeter incork (El-Khatib et al 1992). After application of the optical density to dose conversion, doseto the photographic emulsion was known and the dose to the phantom material was determined


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Figure 1. Measurement geometries. (a) Beams-eye-view of the L-shaped field formed by theleaves of the multileaf collimator. Depth dose curves (at positions indicated by points) and doseprofiles (dotted lines) have been measured with a diode in water in a plane through the isocentre (1)and ina plane50 mmoff-axis(2). (b) Foroblique incidencethe beam axis andfield edges have beenindicated (dashed lines). Diode measurements have been done in water along the beam axis andalong lines at depths of 30 mm, 75 mm and 100 mm from the water surface (dotted lines). (c) Side

view of the phantom with T-shaped heterogeneity consisting of three slabs of polystyrene (30 mm),two slabs of polystyrene with a heterogeneity (20 mm) and 11 slabs of polystyrene (110 mm). Thedashed lines indicate the beam axis and the field edges. Ionization chamber measurements havebeen done along the beam axis (open circles) and dose profiles have been measured at depths of60 mm, 80 mm and 100 mm, using film (thick lines). (d) Beams-eye-view of the phantom withT-shaped heterogeneity with the field edges and x, y axes indicated by dashed lines. (e) Side viewof the lung phantom consisting of three slabs of polystyrene (30 mm), 10 slabs of cork (155 mm)and five slabs of polystyrene (50 mm). The beam axis and field edges have been indicated (dashedlines), as well as the positions of the films (thick lines).

by multiplication of the photographic emulsion to medium stopping power ratio. In this case,the stopping power ratios were calculated as the ratios of collisional stopping powers for themean energy at depth, assuming a linear decrease of energy with depth. This method was

described by Harder and is expected to be accurate to within 23% (Klevenhagen 1993).Since reliable stopping power ratios for cork were not available, this material was consideredas water equivalent with a density of 0.23 g cm3. For the calculation with the PSE and MCmodels the same assumption about cork was made to describe energy loss and scattering ofparticles in this medium (i.e. the interaction data for water with a modified density were used).Monte Carlo calculations have been done to estimate the magnitude of the error that is madeby using this assumption. The differences between depth dose curves calculated for a phantomwith water and for a phantom with cellulose (C6O5H5), both of density 0.23 g cm3, were


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within 2% (cellulose is the main component of cork). For the PB calculations the defaultmethod of heterogeneity correction was applied (section 2.3).

All measured dose values have been normalized to the same reference dose, i.e. thedose maximum in water (approximately 30 mm depth) of the 25 MeV electron beam with a

100 100 mm2 field size at an SSD of 1000 mm. The machine is calibrated to give 1 cGy permonitor unit (MU) as reference dose.

2.2. Phase space evolution model

The PSE model requires as input an initial phase space, i.e. directions and energies of electronsand photons in a plane in front of the surface of the phantom or patient. Initial phase spacescorresponding to the 25 MeV electron beams of the MM50 were determined using the beammodel of Janssen et al (1997b). This beam model consists of a main beam componentdescribing the direct electrons and photons and a secondary beam component describingelectrons and photons that have interacted with the collimating system (i.e. blocks or multileafcollimator for the MM50). The two additional beam components that normally model the

lower collimator scatter and transmission have not been used, since these components aresuperfluous due to the design of the treatment head of the MM50. (This accelerator utilizesa scanning beam with thin scattering foils and a double focused multileaf collimator.) Theenergy spectra of the beam components were derived by fitting a sum of PSE calculateddepth dose curves of monoenergetic beams to measured depth dose curves in square fields.The positional and directional distributions of the beam components were derived frommeasurements of the virtual source position and the angular variance (see table 1). Forthe configuration, depth dose curves measured at two SSDs were used. This method wasdeveloped for conventional treatment machines to model the variation in dose contributionfrom collimator scattered electrons with source to surface distance. For the MM50 the use oftwo SSDs in the configuration process is less important.

For the water and lung phantom synthetic CT images were created with aid of theCADPLAN treatment planning system, and for the air and bone phantom CT information wasacquired by using a CT scanner (Picker PQ5000V). Field sizes, beam angles and SSDs wereset in the CADPLAN planning system. The CT data files and the planning files with beampositions were used as input for the PSE calculations. The CT information was transferred tothe PSE coordinate system (with the z-coordinate along the beam axis) and down-sampled toa coarser grid of volume elements (voxels). Based on the average Hounsfield numbers in thevoxels, the materials in the voxels for the phantoms considered were defined to be either air,water, polystyrene, bone substitute or lung substitute material. A combination of two materialscould be chosen for voxels intersecting an interface. A correction to the default conversion ofHounsfield number to density was necessary for polystyrene, since the default formula resultedin the density being underestimated by about 5%. For the lung substitute, interaction data forwater were used with an adjusted density. The default resolution of the dose and calculationgrid was 5 5 5 mm3. For situations where it was expected that a higher resolution could

especially improve the accuracy, a resolution of 2.5 2.5 5 mm3 (x, y, z) was used as well.The resolution was increased in the directions perpendicular to the beam axis, because in thosedirections the steepest dose gradients occur for the cases considered.

2.3. Pencil beam model

The input for the pencil beam model in CADPLAN consisted of depth dose curves ofa large field and a completely blocked field, the virtual source position and the angular


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variance (table 1). Additionally, the mean electron energy at the surface has been specified andwas derived from the R50 of a measured depth dose curve (ICRU 1984). The depth dose curveof thefield blocked by themultileafcollimator wasused to determinethephoton contaminationin the electron beam, since the photon contribution is handled separately in the dose calculation

algorithm.The CADPLAN system was used for the specification of phantoms and the selection of

beam parameters, as described in section 2.2. The present implementation of the pencil beammodel in CADPLAN contains radiationinteraction data for fivematerials, i.e. air, lung, adiposetissue, muscle and bone. Heterogeneity corrections were based upon the interaction data forthese materials. The resolution of the calculation grid was 5 5 5 mm3.

2.4. Monte Carlo simulations

In the Monte Carlo simulations with EGS4 the positional and directional distributions ofelectrons in the incident beams were the same as in the initial phase spaces used in the

PSE calculations. An energy distribution was determined that resulted in the best agreementbetween themeasureddose distribution for thereferencefield andMC calculations. Theenergydistributions determined for the MC and PSE calculations were in close agreement but notidentical, due to the method of tuning the energy distribution until a good agreement betweencalculated and measured depth dose curves was found. Due to small differences between MCand PSE calculated depth dose curves for the same beam, slightly different energy spectraare needed for the MC and PSE calculations to obtain a good agreement between calculatedand measured depth dose curves. The fraction of dose deposited in the phantom by incidentphotons in the electron beam considered is small (


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Figure 2. Central axis depth dose curves in water for the reference beam; the 25 MeV electronbeam with 100 100 mm2 field size and an SSD of 1000 mm. Dose values have been normalizedto the central axis maximum dose.

3. Results

3.1. Configuration

After the configuration process was finished, the dose distribution in water of the referencefield, i.e. the 25 MeV electron beam with a 100100 mm2 field size and an SSD of 1000 mm,wascalculated using the PSE model, the PB model andthe Monte Carlo model. The agreementbetween the calculated depth dose curves and the measurement is well within 2% and 2 mm(figure 2). In the first 3 cm of the depth dose curve, the PB model underestimates the dose byabout 1.5% and for the remaining depth range the results of the three models are similar.

For off-axis positions the agreement between measurements, PSE calculations and MC

calculations are comparable with the results found for the central axis. The PSE calculatedpenumbrasaresomewhatbroader than themeasured ones, resultingin differences up to 2% and2 mm. The accuracy of the PSE calculated dose distribution in the penumbra region dependson the resolution of the dose grid. A higher resolution of 2.52.55 mm3 resulted in a betteragreement between calculation and measurement in the penumbra region. The PB calculatedpenumbras at depth are also too broad, resulting in maximum differences of 3% and 4 mm.

3.2. Influence of the source to surface distance

Measurements of depth dose curves at various SSDs show that the variation of central axismaximum dose with SSDfortheracetrackmicrotronis well describedby an inverse square law.ThePSE calculated dose distributionsexhibit thesameinverse squarelawandfor SSDs between850 mm and 1000 mm the central axis maximum dose agrees within 1% with measurements.

The PB model showed an overestimation of the dose at shortened SSDs. At an SSD of 850 mm,the maximum deviation found between PB calculation and measurement is 6% of the referencedose (see figure 3).

Figure 3 shows that the R50 range calculated with the PSE model is 2.5 mm less than themeasurement. A deviation that is 1 mm larger than that found for an SSD of 1000 mm. Thislarger deviation may be partly due to both measurement error and set-up variation. (Depthdose curves measured for the reference field on various days show variations in R50 of about1 mm. It is uncertain whether this variation is due to measurement accuracy limitations, or


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Figure 3. Central axis depth dose curves in water for the 25 MeV electron beam with a field sizeof 85 85 mm2 at an SSD of 850 mm. Dose values have been normalized to the central axismaximum dose of the reference beam.

Figure 4. Depth dose curves (a) and dose profiles (b) in the plane 50 mm off-axis in the y direction(plane (2) in figure 1(a)), for the L-shaped field of 25 MeV. The dose profiles are at depths of30 mm, 75 mm and 100 mm. The dashed line in (b) indicates the x position of the depth dosecurves shown in (a). Dose values have been normalized to the central axis maximum dose of thereference beam.

daily variation in accelerator behaviour, or a combination of both.) The depth dose curve for

an SSD of 850 mm calculated with the MC model is closer to the PSE calculated curve thanto the measured curve.

3.3. L-shaped field

For the L-shaped field, measured depth dose curves and dose profiles have been comparedwith calculated results in two planes, as indicated in figure 1( a). The agreement between themeasured depth dose curves and PSE calculations is within 1% and 1 mm, except for the tail


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Figure 5. Depth dose curves along the beam axis (a) and dose profiles in the central axis plane atdepths of 30 mm, 75 mm and 100 mm (b), for 30 oblique incidence. The field size and SSD ofthe 25 MeV electron beam were 150 150 mm2 and 1000 mm, respectively. Dose values havebeen normalized to the central axis maximum dose of the reference beam.

of the depth dose curve, where the difference is about 2% (see figure 4( a)). The resolution ofthe PSE calculation grid was 2.52.55 mm3. For the standard resolution of 555 mm3

the range is underestimated, resulting in a maximum deviation in the R50 of 3 mm. This isdue to the fact that the line along which the depth dose curve is considered is close to the fieldedge, so the dose distribution is influenced by the shape of the penumbra and calculations forthe standard resolution show a small overestimation of the penumbra width. The depth dosecurves calculated with the pencil beam model show an overestimation of the dose in the fall-offregion, resulting in a maximum difference between measurement and calculation of 6% in theregion around R90.

The measured dose profile in the off-axis plane at 30 mm depth shows an obliquenessthat was not found in the calculations, resulting in a difference between measurement and PSEcalculations of 3% (see figure 4(b)). This might be due to an accelerator beam asymmetry that

is neglected in the initial phase space used in the PSE calculation. The dose profiles calculatedwith the PB model show differences that are equal to or smaller than the maximum differenceof 6% found in the depth dose curves.

3.4. Oblique incidence

As shown in figure 5(a), the calculated PSE range is too small compared with the measurementand a maximum difference of about 3 mm is found in the region around R80. The results of thepencil beam model agree with the measurements within 2 mm in the dose fall-off region, butthe dose maximum is underestimated by 3% and in the build-up region a maximum differenceof 6% was found. Figure 5(b) shows that the obliqueness of the dose profiles is well predictedby the PSE model and the PB model. The differences in absolute dose between calculated andmeasured dose profiles is a result of the described differences in depth dose curves.

3.5. Homogeneous polystyrene phantom and phantom with air cavity

For the homogeneous polystyrene phantom, the measured central axis depth dose curve andresults of the PSE model, PB model and MC model all agree within 2 mm and 2%. Thisindicates that for the measurement the correct stopping power ratios have been applied andthat in the calculations the correct interaction data have been used. The material data set ofthe pencil beam model in CADPLAN does not contain data for polystyrene, but apparently


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Figure 6. Central axis depth dose curves (a), dose profiles at 60 mm depth (b) and dose profilesat 80 mm and 100 mm depth (c), for the polystyrene phantom with an air cavity. The field size ofthe 25 MeV electron beam was 126 126 mm2 at an SSD of 900 mm. Dose values have beennormalized to the central axis maximum dose in water of the reference beam, except for the pencilbeam results, for which a renormalization was necessary (see section 3.5).

the use of data for muscle and/or adipose tissue does not result in large errors for calculationof doses in polystyrene.

Figure 6(a) shows the depth dose curves for the polystyrene phantom with an air cavity.The effect of the air cavity on the depth dose curve is a reduction of the dose beyond the cavity,followed by a second build up. Another effect is an increase in the electron range of about20 mm. These effects are well predicted by the three dose calculation models. The agreement

between measurements and results of the PSE model and the MC model is within 2% and2 mm. However, the PB results were about 4% too high at all points along the central axis.This result can be explained in part by the PB models overestimation of dose at an SSD of900 mm. To separate out the contribution of the SSD to the difference, the dose distributionscalculated with the PB model for the phantoms with air, bone and lung were renormalized to96%. This is similar to considering electron beams at an SSD of 900 mm to be separate beamsin the PB model, which require a configuration independent of beams at an SSD of 1000 mm.After renormalization, the PB calculated depth dose curve for the phantom with an air cavityagrees within 2% and 2 mm with the measurement (figure 6( a)).

For off-axis positions the variation in dose due to the air cavity is well described by thePSE model, as shown by comparison with the measurement and the Monte Carlo calculations(figures 6(b) and 6(c)). The dose profiles calculated with the PB model demonstrate the wellknown fact that PB models underestimate the effect of air cavities at depth. By using the

PB model the dose lateral to the air cavity is overestimated by about 8%, and without therenormalization this would be an error of 12%.

3.6. Phantom with bone heterogeneity

The effect of the bone heterogeneity on the depth dose curve is a reduced electron range andthe results of the MC model show a reduction of dose in bone, followed by a second build up inbone (figure 7(a)). The two effects are also present in the PSE results, and the differences with


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Figure 7. Central axis depth dose curves (a), dose profiles at 60 mm depth (b) and dose profiles at80 mm and 100 mm depth (c), for the polystyrene phantom with a bone heterogeneity. The fieldsize of the 25 MeV electron beam was 126 126 mm2 at an SSD of 900 mm. Dose values havebeen normalized to the central axis maximum dose in water of the reference beam, except for thePB results, for which a renormalization was necessary (see section 3.5).

the measured depth dose curve and the MC calculation are within 2% and 2 mm. The depthdose curve calculated with the PB model shows that with this model the build up in bone isneglected, which leads to an underestimation of the dose in and beyond the bone heterogeneityof about 6%. Furthermore the results of the PB model show an underestimation of dose ofabout 8% that could not be explained from the PB theory, about 6 mm superficially from the

front side of the bone heterogeneity.For off-axis positions, maximum differences between dose values calculated with the PSEmodel and measurements were found of about 4% and 4 mm (figure 7(b)). A comparison ofthe results of the MC model and the measurements shows differences of the same order, andfor most positions the PSE calculations are in better agreement with the MC calculations thanwith the measurement. This might indicate that deviations between the PSE model and themeasurement are partly due to errors in the input data used in the calculations. The magnitudeof the lateral dose variations due to the bone heterogeneity is not underestimated when the PBmodel is used, but the steepness of the dose gradients is too small. Differences between PBcalculations and the measurement for off-axis positions have a maximum of 6% and 4 mm.

3.7. Lung phantom

As shown in figure 8, the low-density region has a large influence on the shape of the depthdose curves. In the low-density material the high-dose plateau is absent, instead the dosedecreases continuously with depth. The electron range is doubled compared with the depthdose curves in water (figure 2). Both effects are present in the measurement and the results ofthe dose calculation models. However, differences between measured and calculated resultswere larger than found in previous tests. The differences between measurement and MCcalculations are within 3%. Considering the uncertainties in interaction data and stoppingpower ratios, as discussed in section 2.1, these differences are acceptable. Compared with


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Figure 8. Central axis depth dose curves (a) and dose profiles at 45 mm, 90 mm and 155 mm depth(b) for thepolystyrene/lung phantom. Thefield size of the25 MeVelectronbeam was9090mm2

at an SSD of 900 mm. Dose values have been normalized to the central axis maximum dose in

water of the reference beam, except for the PB results, for which a renormalization was necessary(see section 3.5).

both the measurement and the MC calculation, the PSE calculated depth dose curve showsdose values that are too low in the distal part of the lung region. The maximum differenceis about 6% of the reference dose. The dose profiles show that the penumbra broadening inlung is well predicted when the PSE model is used. The PB model shows an overestimationof the dose in lung that increases with depth. Compared with both the measurement and theMC calculation, the difference has a maximum of about 9%. The dose profiles show that atdepth the PB calculated penumbras are sharper than the measured ones.

3.8. Calculation times

For the PSE model, the time to calculate a 3D dose distribution for the reference field with aresolution of the dose grid of 555 mm3 was about 30 min on an HP workstation (Hewlett-Packard 9000/B1000 with 1 Gb of internal memory). For the PB model in CADPLAN, thetime needed to calculate the complete 3D dose distribution, i.e. 30 slices with a slice distanceof 5 mm, was about 15 min. So with the PB model the calculation time was only a factor of 2shorter but the advantage of the PB model is the possibility of calculating the dose distributionin a single slice in a relatively short time of 30 s. The calculation times for the MC modeldepend on the magnitude of statistical noise that is allowed. Maximum errors were chosen tobe within 1% of the reference dose and the calculation time for the reference field was 12.5 hon the workstation mentioned above. Compared with the MC model, the calculation time forthe PSE model was about a factor of 25 shorter but for a still acceptable maximum error of 2%

in the MC calculations, the advantage in calculation times drops to about a factor of 6.

4. Discussion and conclusions

The majority of results show a good to excellent agreement between PSE calculations andmeasurements. A simplifiedsummary of theresults given above is given in table 3. Differencesin absolute dose larger than 3% of the reference dose (the central axis maximum dose of thereference field), or a shift in dose larger than 3 mm, were only found for limited volumes in the


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Table 3. Maximum differences found between measurements and calculations, in per cent of thereference dose (if the difference occurred in a region with a low dose gradient, see section 2.5) andin mm (if the difference occurred in a region with a high dose gradient). Values are for the centralaxis and maximum differences for off-axis positions are given in brackets if they exceed the centralaxis values.

PSE PB

Non-standard SSD 2 mm 6%L-shaped field 2% (3%) 6%Oblique incidence 3 mm 6%Air cavity 2 mm 2 mm (8%)Bone 1% (4 mm) 8%Lung phantom 6% 9%

phantom with the bone and in the phantom with the lung density material. For the phantomwith bone the agreement between the PSE results and the Monte Carlo calculations was betterthan the agreement with the measurement, which indicates the presence of measurement errors

and/or inaccuracies in the input data used for the PSE and MC calculations. The interactiondata used for the bone substitute material could be inaccurate, since it was found that thedifferences were smaller for the phantom with an air cavity where the interaction data for airare well known. The test in the lung phantom has shown that the PSE calculations are lessaccurate for large regions of low-density material.

When the PSE results are compared with the PB model, it is clear that the PSE model yieldsmore accurate results for the cases considered. However, for some of the results inaccuraciesmay be caused by the way the PB method is implemented, rather than due to the PB methoditself. For instance, in the test of oblique incidence and for the phantom with bone, the largestdifferences between calculation and measurement were found in the build up region, a regionwhere a poor accuracy of the PB method is not expected on theoretical grounds. Furthermoreit should be noted that the PB model tested here was a single Gaussian PB model and ageneralized Gaussian PB model might have given better results. Although tests of the lattermodel show results comparable with what has been presented here (Samuelsson et al 1998,Ding et al 1999).

In lung material, PSE calculated results are less accurate than MC calculations. Since theunderlying physics in the two models is largely the same, the difference in accuracy is probablydue to artefacts introduced by the discretization of phase space (space, energy and direction) inthe PSE model. For instance, for an accurate description of energy loss and directional spreadof electrons due to interactions in a voxel, the widths of energy and directional bins have tobe adopted to the number of interactions in a voxel. In the current version of the PSE model afixed discretization is used throughout the whole volume. An improvement to the PSE modelcould be a discretization adapted to local interaction properties, which could further increaseagreement between PSE and MC results.

These investigations have shown that it is difficult to measure dose distributions reliably

within12%,dueto both themeasurement techniqueand theday-to-dayvariation inacceleratorbehaviour. Since measurements were used for both configuration of the dose calculationmodels and for the determination of the accuracy, these errors should be taken into account inthe judgement of the accuracies of the models. Furthermore, the importance of inaccuraciesin calculated dose distributions depend not only on the magnitudes of the errors but also onthe magnitudes of the volumes in which the errors occur. To consider the effects of under andoverdosage, errors in the dose calculation should be judged against a typical steepness of thedose-effect relation of a 13% increase in tumour control probability (TCP) per 1% increase


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in equivalent uniform dose (EUD) in the tumor (gren Cronqvist 1995, Niemierko 1997).So, errors are clinically more relevant if they extend over larger parts of the target volume.Especially errors in the maximum dose of the central axis (the dose specification point) areimportant, since they relate linearly to the EUD.

At present the time for calculation of a 3D dose distribution using the PSE model is toolarge for interactive treatmentplanning. It was found that thePB model canbe used to calculatea relative dose distribution in a single slice with a reasonable accuracy and with a calculationtime that is acceptable for interactive treatment planning. The accuracy of the PSE model is(just) within clinically acceptable limits, except for large regions with low-density material,as in lung. In that case Monte Carlo calculations are more accurate, assuming that a modelis available to describe the incident electron beam. Furthermore, the present study shows thatif an adequate description of the initial phase space of the clinical electron beam is available,Monte Carlo calculations provide a reliable reference for the determination of the accuracy ofdose calculation models.

Acknowledgments

The authors wish to thank Jack Janssen for discussions and help with the phase space evolutionmodel, Leo van Battum for contributing his experience in film measurements, Erik Loeff andBenGobel for their help with measurements and Evert Woudstra and Joep Stroom for valuablecomments. This work was supported by a grant from the Dutch Cancer Society (NKB 96-1230).

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