Is the ‘fictitious’ virtual source a redundant concept for scanned therapeutic electron beams

7/28/2019 Is the fictitious virtual source a redundant concept for scanned therapeutic electron beams

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Phys. M ed. Biol., 1989, Vol. 34, No 3, 369-378. Printed n the U K

Scient@ note

Is the fictitious virtual source a redundant concept for scannedtherapeutic electron beams?G A Sandisonit and W Hudatt Department of M edical Physics, M anitoba Cancer T reatment and Research Foundation,Winnipeg, M anitoba, R3E OV9, Canada$ Department of Physics, University of M anitoba, W innipeg, M anitoba, CanadaReceived 19 J anuary 1988, in final form 1 November 1988

1. IntroductionThe fictitious virtual source is an empirical aid to treatment planning but does nothave any firm definition. Most commonly it is defined such that, over a limited range,the inverse square law may be applied to predict beamutput with actual source-surfacedistance (Briot and Dutreix1976, Cecatti et a1 1983). Alternatively, but less frequently,the fictitious virtual source ocation may be determined from the full width at halfmaximum ( F W H M ) of a beam profile (Schroder-Babo 1983). The location of a fictitiousvirtual source is problematical since it is not fixed but depends on many parameters(Sweeney et a1 1981), most notable of which are field size and electron beam energy.The empirical method normally used to locate the fictitious virtual source position isa linear extrapolation of data to the source-chamber distance (SCD) axis in a plot ofthe square root of the inverse of an ionisation reading on the central axis of the beamagainst SCD. Normally the linear extrapolation is performed over a range of distancelarge compared with the range of SCD for which the data are collected. The fictitiousvirtual source position determined by this method moves toward the reference measure-ment point (typically 100 cmSCD) as the electron beam energy is decreased for a fixedfield size and similar behaviour is exhibited for a fixed beam energy as the field sizeis decreased.Fermi-Eyges transport theory (Rossi and Griesen 1941, Eyges 1948) has previouslybeenappliedtotherapeutic electronbeams opredictbeam output with changingpatient reatmentgeometry Mills et a1 1982, Meyer et a1 1984, Mills et a1 1985).However, hese approachesdonotseparatethe influence of backscatter rom hecollimation system intothemonitorchamber(PattersonandShragge1981),whichcannotbepredicted by Fermi-Eyges heory, rom he nfluence of lateralscatterdisequilibrium in the electron field due to thecollimation system, which can. Magneti-cally scannedacceleratorelectronbeams have recentlybeenmodelled in airusingFermi-Eyges heory by a collimated sotropicsource Huizenga and Storchi 1987,Sandison and Huda 1988). The purpose of the present paper is to use the collimatedisotropic source model to evaluate the predictive power of Fermi-Eyges theory forboth beam output and beam profile FWHM with changing beam energy and SSD. Thisevaluation uses rectangular fields defined by the photon collimation system alone.Additional collimation is usually ntroduced for patient treatment and therefore theprocedure outlined in this paper is not appropriate for direct application to treatmentplanning.0031-9155/89/030369+ lOS02.50 0 1989 IOPublishing Ltd 369


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370 G A Sandison and WHuda2. TheoryThe spatial distribution of electron fluence F(z, x, y ) for a rectangular field producedby magnetic scanning may be modelledy a collimated isotropic source ands computedon the basis of Fermi-Eyges theory (Sandison and Huda 1988) as

(1)where z is the distance in air from the isotropic point source located at the acceleratorexit window, a ( z ) s the half-field width at distance zdefined by the collimation systemin the x direction, b(z) is the half-field width at distance z defined by the collimationsystem in the y direction, x and y are lateral coordinates, a, is the root mean squarespatial spread of a pencil beam at distance z when the pencil beam is defined at thelevel of the x coll imation, and a>,s the root mean square spatial spread of a pencilbeam at distance z when the pencil beam is defined at the level of the y collimation.The appropriate value of c, (and similarily ay)may be computed from the followingrelation (Huizenga and Storchi 1987, Sandison and Huda 1988)

k(z- 2,)'6

where f is an empirical fitted constant, k is the linear angular scattering power of airat the beam energy used and z, is the distance from the source to the level of the xcollimation.The effect of additional scattering by the monitor chamber at typical treatmentdistances is negligible. This can be understood by imagining the monitor chamber tobe represented as a thin metallicoil surrounding the isotropic source. Theoil-coveredsource still emits electrons sotropically and at large distances downstream appearsidentical in effect to an uncovered isotropic source.

2.1. Variation of output with distance in air from the isotropic point sourceFermi-Eyges theory alone cannot account for the variation of beam output with fieldsize since it does not consider the component of output variation due to backscatter( BS) from the top of the collimation system to the monitor chamber (Patterson andShragge 1981), or the small component of output variation due to forward scatter fromthe collimation system (L ax and Brahme 1980). However, this isotropic source modelemploying Fermi-Eyges scattering theory is able to account forateral scatter disequili-brium (L SD) effects at therapeutic beam energies. I f the collimator scatter is assumedto be a small effect compared with the backscatter component then the output factor(OF) in air a distance z from the source, on the central axis of a rectangular field, isgiven by

where the subscriptr represents the parameters defined n equation (1) forhe referencefield and


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Scanned therapeuticlectron beams 371I n addition, the reference field size may be chosen to be a very large square field(36x36 cm) forcing the error function values to unity and the reference source-chamber distance may be set equal to 100cm. Equation (3) then becomes

Equation (3) indicates that for a fixed field size setting ( ax b) cm2, the outputfactor OF varies with distance in air z and lateral scatter disequilibrium LSD, ince theratio BS (a,b)/ss(a,, b,) is constant. Any geometric dependence of OF due to forwardscattering off the photon collimation system has been ignored.Lateral scatter disequilibrium for a rectangularield occurs when the ield penumbrawidth is greater than or approximately equal toalf the corresponding field dimension(e.g. a6 U, and/or b6 U , , ) . This results in the LSD term n equation (5) decreasingbelow unity and reduces the luenceon the centralaxis of the beam below that expectedon the basis of the simple inverse square law.2.2. Variation of F W H M with distance in air from the isotropic point sourceA further consequence of lateral scatter disequilibrium is the increase in full width athalf maximum (FWHM) of the x axis profile because of the decreasing central axisfluence value F(z ,0,O) for the off-axis ratio F(z ,x,0)/F(z,0,O)

The value of the FWHM is given by 2x, when x is the solution to equation (6) forthe ratio F(z, x, O)/F(z,0,O) equal to 1/2. This value of x is obtained iteratively fromthe equation

3. M aterials andmethodThe radiation output factor is defined here as the ratio of the ionisation in air on thecentral axis of a field-of-interest at 100 cm source-chamber distance (SCD) to that ofa reference field size. The reference field size adopted in this work was a 36x36 cm2square field. For fixed field size settings, the variation of ionisation readings on thecentral axis of the beam with SCD are normalised to the 100 cmSCD reading. All in-airionisation measurements were performed using a Capintec Farmer typeR-06C ionisa-tion chamber of 0.65 cm3 sensitive volume, inner axial length.2 cm and inner diameterof 0.7 cm. Selected beam energies of 7, 10, 16, 22 and 32 MeV were investigated forsquare field sizes in the range 3X 3 cm to 36X 36 cm2 andSCD in the range 80-120 cmfor fields formed by the photon collimation system (figure 1). I n the smallest fieldsinvestigated there is a rapid variation in intensity of the beam across the active volumeof the chamber. This leads to an underestimation of the ionisation ratio in the worstcase of 7.5% which occurs in a 5X 5 cm2 field at 80cmSCD for a 7 MeV beam (figure3(a)). The stem effect measured for the chamber was less than 0.4%. These types oferrors are not accounted for in the present data.


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372 G A Sandison and W HudaQuadrupolemagnets

[XI=B m wlndow

l IIPrlrnary j

cofllrnator'

Photoncolflmators

105 cml 1I

IsocentreFigure 1. Schematic of the Sagittaire Therac0 linear accelerator showing the magnetic scanning quadrupoleand photon collimation system.

In-air beam profile measurements were obtained perpendicularly to the central axisof the therapy electron beam using K odak Industrex M film pinned to a wooden boxframe.heowestoseatevailableorhecceleratorasmployed(-50 cGy min") to ensure a uniform fluence distribution on the film for the scannedbeam. For convenience, the film was retained within the manufacturer's opaque papercassettewhenused. An opticaldensity-dosecalibrationcurvewasobtainedat hedepth of dose maximum in a tissue equivalent phantom composed of Temex (Staceyet af 1961), for each electron beam energy. T he film was found to be li near within2%for doses in the range5-30 cGy. To ensure greatest sensitivity in this linear region forpenumbra measurements, the maximumof each profile was made to correspond to anopticaldensity ntherange 1.5-2.0. Films romthesamebatchwereused neachspecified experiment and developed under identical conditions using hand processing.An unexposed film was included wi th each batch to assess the background fog level.The films were henreadmanuallybyaSargent-Welchdensitometer Densichronmodel PPD) using a 1mm diameter l ight aperture.4. Results4. . F ixed jield size settingThe effects of lateral scatter disequilibrium (LSD) on beam output in air were investi-gated by fixing the field size setting ( ax b) cm2. Equation (3) shows that measurementof the variation in ionisation along the central axis of a small square field with SCD,normalised to the value obtained at100cm SCD, may be used to investigate deviationsfrom the inverse square law due toSD. Figure 2 shows the results of such measurementsalong the central axis of a 6X 6cm2 field (defined at 100cm SCD) with SCD for beam


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Scanned therapeutic electron beams 373

c 1 . 6ac-

1 . 2

0. 8

0 . 480 85 90 95 100 10510 115 120

S o u r c e - chamber dis tance ( cm1Figure 2. Variation of ionisation readings with SC D on the central axisf a 6 cmx 6 cm field at beam energiesof 7 ( 0 ) , 1 6 ( 0 ) , 22(A) MeV. Brokencurve ndicates he nversesquare awvariation. Full curvesaretheoretical predictions ncluding ateral scatter disequili brium.energies of 7, 16 and 22 MeV. The predicted inverse square law varation is shown asa broken curve and the full curves indicate the predicted variation using equation (3)with lateral scatter disequilibriumeffects included. The experimental data for a2 MeVbeam is not shown in figure 2 but, as is expected, follows the inverse square law overthe SCD range of measurements. The 22 MeV beam is predicted to depart from aninverse square law variation only at extended SCD (>100cm) due the value of LSDfalling below unity. The 16 MeV and 7 MeV beam predicted variations deviate fromthe inverse square law overthewhole SCD range of measurements.There is goodagreement between the experimental and predicted variation especially at extendedSCD. The measured variation in ionisation ratio along the central axis of a 7 MeVbeam with SCD for square field size settings of 5x 5 cm2, l ox 10cm2 and 15x 15cm2are compared in figure 3(a) to the theoretical predictions of equation ( 1 ) which areshown as full curves. The predicted nverse square law is shown as a broken curve.Similar measurements were repeated for a 10MeV beam and the results are shown infigure 3(b) . Notice that all predicted curves in figures 3(a, b ) deviate from the inversesquare law except for the 10 MeV beam with 15 X 15cm2 field size at values of SCDbelow about 110cm.4.2. Variable jield size settingThe beam output factor OF was measured in air at 100cm SCD for square field sizesin the range 4x4 cm2 to 36 x 36 cm2 at electron beam energies of 7, 10, 22 and 32 MeV.These results are shown in figure 4 and demonstrate that the reduction in output withdecreasing square field size is strongly dependent on beam energy. Thecollimatedisotropic source model employing Fermi-Eyges scattering theory can only predict theinverse square law and lateral scatter disequilibrium effects. The full curves in figure5 show the predictions of Fermi-Eyges theory in air at 100cm SCD for the variationin the output factor OF with square field size for the beam energies of 7 and 32 MeV.Differences between the measured and predicted data of figure 5 are due to electrons


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3742.8 1

G A Sandison and WHudal a I

80 85 90500 105 11015 120Source-chamber dlstance (cm1

2.4

2.0

1 . 60c.-e5 1.2-c0C.-I 0. 8

0. 4 I , I I l80 85 90 95 10005 110 115 120Source-chamber distance ( cm

Figure 3. Variation of ionisation readings with SCD for ( a )7 MeV beam ( b ) 10MeV beam on the centralaxis of square fields of 5 X S cm2 (O ) , o x 10cm2 ( 0)nd 15x 15cm2 ( A ) . Broken curve ndicates theinverse square aw variation. Full curves are the theoretical predictions ncluding ateral scatter disequil i-brium.backscattering from the photon collimators and entering the monitor chamber plusany differences in scatter off the photon collimators reaching the ionisation chamber.4.3. Beam projile F W H MA further effect of ateral scatter disequilibrium is to increase the full width at halfmaximum FW HM ) of anormalisedbeam profi le above hatpredicted on purelygeometrical grounds. Table 1 presents measurements of beam profile FWHM at 100cmSSD for square fields formed by the photon collimators. The field size is set accordingto the light field measurements and is varied over the range 3x3cm to 20x20 cm2for beam energies of 7, 16, 22 and 32 MeV. Notice that at the smallest energy andfield size the measured FWHM of the normalised beam profile is over twice as large aswould be expected basedn the geometrical set-up. Tablealso compares the measureddata with the FWHM of the beams predicted using equation (7).


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Scanned therapeutic electron beams 375

loo

0 1 I0 4 8 126 20 24 28 3 2 3 6Square f i e l d l eng th lcml

Figure4. V ariation in output at100cm source-chamber distance with field size for beam energies of 7 MeV(0),0 MeV (A) , 22 MeV ( 0 ) nd 32 MeV ( W ) . Field is defined by the photon collimators. Broken curvesact only as a guide to the eye.

loo

/ ,

I Oo ! I 1 l 10 4 8 12 16 2 0 24 2 82 36

Square ie ld l eng th Icm)Figure 5. Variation in output at 100cm SCD with field size for beam energies of 7 MeV (0) nd 32 MeV(H) . heoretically predicted curves shown as full curves. Broken curves act only as a guide to the eye.5. DiscussionThe prediction of beam profile shapeis an important component f treatment planning.Knowledge of the change n beam outputwith treatment geometry s equally important.Both are related to ateral scatter disequilibrium. Figure 2 shows that for a ixed squarefield size of 6x6 cm2 and beamenergies of 16 and 22 MeV the change in beam outputwith SSD is predicted within 270, y the inverse square law and lateral scatterdisequili-brium effects predicted by equation (3) at values of SCD between 100 cm and 120 cm.The correspondence between the measured and predicted data is equallygood orother field sizes greater than 6 X 6 cm2, as demonstrated n figure 3(a) for a 7 MeVbeam and figure 3(b) for a 10MeV beam. The agreement between the measured and


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376 G A Sandisonnd WHudaTable 1. Comparisonofmeasured valuesof F W H M (cm)with predicted valuesfor square field beam profiles.Field size(cm?) at 7 Electron energy (M eV)16 22 32100cmSSD M P M P M P M P20x 20 20.1 20.1 20.0 20.0 20.0 20.0 20.0 20.015 x 15 15.3 15.3 15.0 15.0 15.0 15.0 15.0 15.0lox 10 10.9 11.1 10.0 10.0 10.0 10.0 10.0 10.06 x 6 8.4 8.5 6.2 6.1 6.0 6.0 6.0 6.05 x 5 7.7 7.6 5.3 5.3 5.0 5.0 5.0 5.04 x 4 7.1 7.0 4.5 4.5 4.1 4.1 4.0 4.03 x 3 6.6 6.2 3.8 3.8 3.3 3.3 3.1 3.0M , measured (error k0.2 cm); P, predicted.

predicted data although good is not perfect, especially for the 7 and 10 MeV beamsat field sizes less than or equal to 6 X 6 cm2. These discrepancies between predictionand measurement can in part be explained by the variation n ntensity of the beamacross the chamber in small fields but may also possibly derive from assumptions inthe accelerator model used becoming less valid at small values of SCD, beam energyand field size. Namely that the effect of the monitor chamber is negligible, or theimplicitassumption nequation (3) of ignoring hesmallvariation with z in thecontribution of forward scatter from the collimation system to the ionisation chamber.Figure 4 shows that the reduction of beam output with a reduction in field size isstrongly dependenton beam energy. Figure 5 compares the measured output variationfor the7 MeV and 32 MeV beams with that predicted by equation (3). The discrepancybetween the measured and predicted data in figure 5 indicates there is another effect,in addition o lateral scatter disequilibrium, which reduces beamutput with a reductionin field size. This is the effect of radiation backscattering from the topof the collimatorsystem into the monitor chamber and any collimator scattereffects into the ionisationchamber. The gun current and hence the electron output is controlled by a feedbackmechanism based on the signal from the monitor chamber. As the field size is reducedthere is an increase in collimator surface available for the electrons to backscatter intothe monitor chamber. This causes an increase in the signal and therefore a correspond-ing decrease in the actual electron output (Patterson and Shragge 1981).Often noted clinically is the increase in FWHM of normalised electron beam profilesabove that normally expected on geometrical grounds. This increase is most noticeablefor low beam energies and small field sizes. Table 1 demonstrates that i f the valuesof a,(z) are predicted theoretically or measured, then equation (7) may be used toaccurately predict the FWHM of rectangular beam profiles for any energy and field size.A comparison of the measured and predicted results clearly shows that the ncreasein FWHM of beam profiles above hatgeometricallyexpectedcanbesatisfactorilyexplained by lateral scatter disequilibrium effects.Agreement between the predicted and measured results for the beam output at afixed field size and the FWHM data is achieved simply by correctly parametrising thebeam in air onthe basis of Fermi-Eyges theory (Huizenga and Storchi 1987, Sandisonand Huda 1988) and then accounting for ateral scatter disequilibrium effects. Theseresults obviate the concept of a fictitious virtual source for magnetically scanned broadelectron beams since both variations in beam output due to lateral electron disequili-


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Scannedherapeutic electron beams 377brium and beam profile shape (FWHM) can be predicted on the basis of Fermi-Eygestheory.

6. ConclusionThe application of Fermi-Eyges theory to electron beams n air is capable of predictingthe changes n beam output for a fixed field size with SCD or beam energy due tolateral scatter disequil ibrium effects. However, overall changes n beam output withfield size arenotaccuratelypredicted dueto Fermi-Eyges heorybeingunable toaccount for the backscattering of electrons from the top of the photon collimators intothemonitorchamberor orward scattering rom the collimators o he onisationchamber.The ncrease n FWHM of beam profi les above he FWHM expected ongeometrical grounds at mall field sizes and low beam energies s satisfactorily explainedby lateral electron disequilibrium effects and is adequately predicted using the colli-mated sotropicsourcemodel of scanningaccelerators.Theaccuracy of thebeamoutput and FWHM predictions demonstrates that the use of a fictitious virtual sourceconcept for magnetically scanned therapeutic electron beams is unnecessary.

AcknowledgmentsWe would like to take this opportunity to express our thanks to the referees for theirvery helpful comments on this manuscript. This work was supported by a grant fromtheM anitobaHealth ResearchCouncil and in part by agrant rom heNationalCancer Instistute of Canada.

ReferencesBrahme A 1983 Geometric parameters of clinical electron beams Acra Radial. Suppl. 364 11-9Briot E and Dutreix A 1976 Dosimetre des Faisceaux delectrons de haute energie dun accelerateur l ineaireCecatti E, Goncalves J , Cecatti S and de Penha Silva M 1983 Effect of the accelerator design on the positionEyges L 1948 M utliple scattering with energy loss Phys. Ret.. 74 1534-5Huizenga H and Storchi P 1987The in-air scattering of cl inical electron beams as produced by acceleratorsICRU 1984 I nternational Commission on Radiation Units and Measurements, Radiation Dosimetry, E lectronLax I and Brahme A 1980 Collimation of high energy electron beams Acta Radial. Oncol. 19 199-207M eyer J , Palta J and Hogstrom K 1984 Demonstration of relatively new electron dosimetry measurementMills M , Hogstrom K and A lmond P 1982 Prediction of electron beam output factors M ed. Phys. 9 60-8Mills M , Hogstrom K and Fields R 1985 Determination of electron beam output factors for a 20-M eV linearPatterson M and Shragge P 1981 Characteristics of an 18 MV photon beam from a Therac 20 medical linear

Polman H and van der L inden P 1987a Determination of three parameters describing the uncoll imated- 987b Description of the uncollimated electron beam n air by means of a directional pencil beamRossi B and Griesen K 1941 Cosmic ray theory Ret.. M ed. Phys. 134250-70

J . Radial. E lectrol. 57 447-54of the effective electron source M ed. Phys. 10 683-6

with scanning beams and diaphragm col limators Phys. M ed. Biol. 32 1011-29beams with Energies between 1 and 50M eV IC RU Report 35 (Bethesda, M D: ICR U)

techniques on the mevatron 80 M ed. Phys. 11 670-7

accelerator 1985 M ed. Phys. 12473-6accelerator M ed. Phys. 8 312-8electron beam in air Phys. M ed. Biol. 32 345-53model Phys. M ed. Biol. 32 355-63


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378 G A Sandison and W HudaSandison G and Huda W 1988 A pplication of Fermi scattering theory to a magnetically scanned electronSchroder-Babo P 1983 Determination of the virtual electron sourceof a betatron Acta Radio/.Suppl.3647-10Stacey A , Bevan A and Dickens C 1961 A new phantom material employing depolymerised natural rubberSweeney L, G ur D and Bukovitz A 1981 Scatter component and its effect on virtual source and electron

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Er. J . Radio/.34 510-5beam quality I nr . J . Radial. Onco/. Bioi. Phys. 7 967-71

Is the ‘fictitious’ virtual source a redundant concept for scanned therapeutic electron beams

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