Monte Carlo study of fluence perturbation effects on cavity

Monte Carlo study of fluence perturbation effects on cavity

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Monte Carlo study of fluence perturbation effects on cavity dose response in clinical protonbeams

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Monte Carlo study of fluence perturbation effects on cavity

Phys.Med.Biol.43(1998)65–89.PrintedintheUKPII:S0031-9155(98)86156-3

MonteCarlostudyof uenceperturbationeffectsoncavitydoseresponseinclinicalprotonbeams

HugoPalmansandFrankVerhaegen

DepartmentofBiomedicalPhysics,UniversityofGent,Proeftuinstraat86,B-9000Gent,Belgium

Received21July1997,in nalform4September1997

Abstract.Currentprotocolsforclinicalprotonbeamdosimetryhavenotimplementedanychamber-dependentcorrectionfactorsforabsorbeddosedetermination.ThepresentworkinitiatesaMonteCarlostudyofthesefactorswithemphasisonproton uenceperturbationeffectsandpreliminarycalculationsofperturbationeffectsfromsecondaryelectrons.

TheprotonMonteCarlocodePTRANwasmodi edtoallowsimulationofprotontransportinnon-homogeneousgeometriesofbothunmodulatedandmodulatedbeams.Thedosetowaterderivedfromthedosecalculatedinanaircavityagreeswellwithresultsfromanalyticalcalculationsassumingadisplacementofthepointofmeasurement.Forunmodulatedbeamssmalldifferences,limitedto0.8%,couldbepartiallyattributedtoprotonmultiplescattering.Effectsofreplacingwateraroundthecavitywithwallmaterialareexplainedbytheintroductionofawater-equivalentwallthickness.Formodulatedbeamsnosigni cantperturbationeffectsarise.Secondaryelectronspectraarecalculatedanalytically.PreliminaryelectrontransportcalculationswithEGS4showthatwallperturbationsoftheorderof1%couldresult.

Perturbationeffectscausedbytheenergytransportofsecondaryparticlesfrominelasticnuclearinteractionshavenotbeenstudiedhere.Inclusionofinelasticnuclearenergytransfersinthecavitydose,assumingtotallocalabsorption,indicatethatseparatescalingofthiscontributionwiththeratiooftotalinelasticnuclearcrosssectionscouldbeimportant.

1.Introduction

CurrentdosimetryprotocolssuchastheAAPMTG16protocol(AAPM1986)andtheECHEDprotocol(Vynckieretal1991,1994)fordosetowaterdeterminationwithionizationchambersinclinicalprotonbeamswithanenergyrangeof50–250MeVhavenotincludedanychamber-dependentcorrectionfactors.

Medinetal(1995)proposedanexpressionfortheabsorbeddosetowaterDw,QatthereferencepointinaprotonbeamcorrespondingtotheNDformalismoftheIAEACodeofPracticeforphotonandelectronbeamdosimetry(IAEA1987)inwhichaperturbationcorrectionfactorpQisde ned:

Dw,Q=MQND,Q0

(Wair)Q

(sw,air)QpQ

(Wair)Q0

(1)

inwhichQandQ0denotetheprotonbeamqualityandthecalibrationbeamqualityrespectively,MQistheionizationchamberreading,correctedforatmosphericnon-standardconditions,forrecombinationandforpolarityeffects,ND,Q0istheabsorbeddosetoaircalibrationfactorfortheionizationchamber,(Wair)Q/(Wair)Q0istheratioofthemeanenergyrequiredtoproduceanionpairinthetwobeamqualitiesand(sw,air)Qisthewater

c1998IOPPublishingLtd0031-9155/98/010065+25$19.50

65

Monte Carlo study of fluence perturbation effects on cavity

66HPalmansandFVerhaegen

toairmassstoppingpowerratiointheprotonbeam.Inprinciple,restrictedstoppingpowersshouldbeusedtoaccountforthedistinctionbetweenenergylossestosecondaryelectronsthatareabletotransferenergyawayfromthegenerationpointandthosethatarenot,asforphotonbeams(SpencerandAttix1955).Thisraisesthequestionofthecut-offenergyandofthedealingwithtrack-endswhichwewillnotdiscusshereasitisoutofthescopeofthepresentwork.ThetotalperturbationcorrectionfactorpQconsistsoftheproductoffactorsforthenon-waterequivalenceofthewall(pwall),forthecentralelectrodeeffect(pcel gbl),fortheperturbationofelectron uenceduetoinsertionoftheaircavityinwater(pcav)andoptionally,ifthegeometricalcentreofthechamberisusedasreferencepoint,forthedisplacementeffect(pdispl).Wecanremarkthatthede nitionofpcavcouldbegeneralizedto uenceperturbationofallsecondarychargedparticles.

Regardingthe(Wair)pvalueforprotonsthereisstillarelativelylargeuncertaintyduetothedif cultyofdeterminingthisfactor.RecentcomparisonsofionizationchamberdosimetryandwatercalorimetrybySiebersetal(1995)andPalmansetal(1996)indicatethatthevalueadoptedbytheAAPMprotocol(AAPM1986)isclosertorealitythanthevaluerecommendedbytheECHEDprotocol(Vynckieretal1991,1994).However,onlytheproductof(Wair)pand(sw,air)pcanbedeterminedaccuratelybyuseofwatercalorimetry.

Regarding(sw,air)pforclinicalprotonbeams,importantworkhasbeendonebyMedinandAndreo(1992,1997a).Theyshowedthatthecontributionsfromsecondaryprotonstosw,airarelessthan0.1%,whereasthecontributionfromsecondaryelectronscouldchangethewatertoairstoppingpowerratiobyupto0.6%forprotonsintheclinicalenergyrange.

ItisgenerallyassumedthattheperturbationcorrectionfactorpQisveryclosetounityorthat,atleast,itsdeviationfromunityiswellwithinthetotaluncertaintyofthe naldose.Regardingprimaryproton uenceperturbations,thiscanbearguedtobeduetothelowscatteringcharacteristicsofprotons.Perturbationeffectscausedbyanincompletesecondaryelectronequilibriumarealsocommonlyneglected.Themaximumenergyofsecondaryelectronsisthreeordersofmagnitudesmallerthantheprotonenergysotherangeinwaterofmostsecondaryelectronsisverysmallanditisassumedthatonlysmalleffectsaretobeexpected.Furthermore,perturbationeffectscausedbysecondaryparticlesoriginatingfrominelasticnuclearinteractionsarealsoneglected.Forthemajorityofthechargedsecondaryparticlesthestoppingpowersaremuchhigherthanforprimaryprotonsresultinginveryshortranges.Onlyneutronsandsecondaryprotonshaverangeslargeenoughtotransportasigni cantpartoftheenergyawayfromthegenerationpoint.Thetransportofsecondaryprotonshasasigni cantin uenceonthedepthdosedistributionasshownbyMedinandAndreo(1997a).ThoughrecentexperimentalinvestigationsbyMedinetal(1995)andPalmansetal(1996)showthatthereiseveryreasontotakeintoaccountchamber-dependentperturbationcorrectionfactors,thepotentialperturbationeffectsshouldbesmallbecauseionizationchamberdosimetryinclinicalprotonbeamscanbeconsistentlyperformedwithin1%to2%usingdifferentchambertypeswithouttakingintoaccountperturbationcorrectionfactors.

Somecalculationshavealreadybeendoneonthestudyofchamber-dependentperturbationeffectsandtheimportanceofconsideringthein uenceofsecondaryelectronsonperturbationcorrectionfactorsandstoppingpowerratios.Bichsel(1995)studiedthein uenceofcavityshapeandcavitymaterialontheshapeoftherecordeddepthdosedistributions,especiallyaroundtheBraggpeak.Thein uenceofsecondaryelectronshasbeenstudiedbeforebyLaulainenandBichsel(1972).Theycalculatedthatfor50MeVprotonstraversinga1mgcm 2thickfoil,about10%oftheenergylostbytheprotonsescapesfromthefoilbysecondaryelectronsandthatforaplanedetectortheenergytransferredbysecondaryelectronsfromthefrontcavitywalltothecavitymediumandfrom

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Cavitydoseresponseinclinicalprotonbeams67

thecavitytothebackwallisnotinequilibrium.MedinandAndreo(1992)alsocommentedontheeffectofsecondaryelectrons,andremarkedthattheassumptionofsecondaryelectronenergydepositionatthesiteoftheirproductionmightcauseinaccuraciesintheabsorbeddosedeterminationsincetherangeofthemostenergeticsecondaryelectronsinairisofthesameorderofmagnitudeasthedimensionsofanionizationchamber.

Inthepresentworkwehavestudiedsomeaspectsoftheperturbationcorrectionfactorsdiscussedinthepreviousparagraphsbothinunmodulatedandmodulatedprotonbeams,fordifferentcavityshapesanddimensions,wallmaterials,wallthicknessandprotonenergies.Forthispurpose,theprotonMonteCarlotransportcodePTRAN(Berger1993a)isused.Inpreviousworkthetransportofprotonsinmaterialsotherthanwaterwasimplementedinthiscode(PalmansandVerhaegen1997).Thisallowssimulationofmodulatedprotonbeams.InthepresentworktheimplementationofcylindricalandsphericalgeometriesinPTRANisrealized.Theeffectofinsertinganaircavityinhomogeneouswaterontheprimaryproton uenceisinvestigatedandassociatedwithareplacementcorrectionfactorpdisploradisplacementofthepointofmeasurement.Asemi-analyticalmodelforthecalculationofthiseffectivemeasuringpointisproposedandcomparedwiththeMonteCarloresults.Theeffectofreplacingwateraroundtheaircavitywithwallmaterialwhichgivesrisetoacontributiontothewallcorrectionfactorpwallisalsocalculated.

Apreliminarystudyofthecontributionofsecondaryelectron uenceperturbationstopcavandpwallhasbeenstartedusingEGS4electrontransportcalculations.Thebalancebetweenenergydepositedinthewallwhichistransferredtothecavitybysecondaryelectronsandviceversaisevaluatedusingelectronspectrathatareanalyticallyderivedfromtheprotonspectra.

Fluenceperturbationeffectsbysecondaryheavychargedparticlesoriginatingfrominelasticnuclearinteractionsarenotevaluatedhere.Themainreasonisthatexceptwhenoxygenisthetargetparticle(Seltzer1993),thecrosssectionsofformationfortheseparticlesarenotknown.However,itisnotunlikelythattheseparticlescouldhaveimportanteffectsoncavitydoses,especiallyatthehigherenergies.MedinandAndreo(1997a)showedthatfora200MeVbeamanimportantfractionofthesechargedsecondariesistransportedoveraconsiderablerange,therebyin uencingthedepthdosedistributionsigni cantly.Furtherinvestigationofthiseffectiswarranted.

Withthisstudywedonotintendtocalculateperturbationcorrectionfactorsforrealionizationchambersbutweintendtoindicatehowwaterabsorbeddoseshouldbeevaluatedfromionizationchambermeasurementsinprotonbeamswithrespecttopotentiallyexistingperturbationeffects.2.Materialsandmethods2.1.TheprotonMonteCarlocode

ThetransportcodeusedinthisworkisbasedontheprotonMonteCarlocodePTRAN(Berger1993a).Thiscodesimulatesmonodirectionalmonoenergeticpencilprotonbeamsinhomogeneouswaterwithouttransportofsecondaryparticles(electronsandsecondariesfrominelasticnuclearinteractions).ItcalculatesprotontransportusingaclassI schemefollowingBerger’sclassi cationofMonteCarloparticletransportschemes(Berger1963).Theparticlestepsaredeterminedinaprecalculatedstep-sizegridwhichislogarithmicinthelowerenergyrangewherethestoppingpowerincreasesmorerapidlywithenergydegradationthaninthehigherenergyrange.Thecodecalculatesdepthdosedistribution,radialdistributionsandenergy uencespectrainslabsde nedinahomogeneouswater

Monte Carlo study of fluence perturbation effects on cavity

68HPalmansandFVerhaegen

geometryformonoenergeticpencilbeams(Berger1993a).Inpreviousworkwedescribedmodi cationsandextensionsoftheoriginalcodeinordertocalculatedepthdosepropertiesforlow-Zmaterialsotherthanwater(PalmansandVerhaegen1997).Thehomogeneousslabgeometryoftheoriginalcodewasmodi edtoallowdifferentslabmaterialstobeusedinonesimulation.

2.2.Implementationofmodulatorwheels

Forthepresentwork,additionalchangesaremadetoallowsimulationofmodulatedprotonbeams.Inclinicalpracticethemodulationofprotonbeamsisachievedusingamodulatorwheelwithavaryingthicknessthatisrotatedbetweentheincidentprotonbeamandthepatientorwaterphantomtoobtaina atdosepro leindepthbyspreadingouttheBraggpeak.ThemodulatorwheelisimplementedintheprotonMonteCarlotransportcodeasanadditionallayerofmodulatorwheelmaterialinfrontofthewaterphantom.Foreveryincidentprotonthethicknessofthislayerissampledfromthedistributionofthicknessesthatoccurinthewheel.2.3.Implementationofcavities

Thecalculationofdosedepositedincavitiesrequiredsigni cantchangestotheoriginalcodeanditsconcepts.Onehastodealwithseveraldif cultiestypicalofthiskindofproblemandcomparabletothoseencounteredinelectrontransportcalculations,suchasthecrossingofboundariesbetweendifferentmedia,thescoringofenergydepositedalongaparticletrack,calculationof uencespectraetc.Thefollowingparagraphsdealwiththesolutiontotheseproblems.Thesimulatedgeometriesarelimitedtoconcentricsphericalorcoaxialcylindricalcavities.Furthermoretheincidentbeamgeometryisextendedtobroadbeams(square,rectangularorcircularlateraldistribution)aswellasbeamswithacertainenergydistribution.

Thecrossingofboundariesbetweendifferentmediaisdealtwithasfollows.RoutinesareaddedtoPTRANthatcalculatetheintersectionofthestraightlineconnectingthestartandendpointofthesampledparticlestepandtheboundary.Themodularprogrammingoftheseroutinesallowsessentiallyanyboundaryshape.Thestraightlineiscalculatedusingitsparameterrepresentation.Theintersectionpointofthestraightlinewithaboundaryisdeterminedtocheckwhethertheparticletraversestheboundarywithinthesampledstep.Ifaboundarycrossingbetweendifferentmediaoccurs,thestepoftheprotoniscutattheboundary.Fortheenergyevaluationtheenergydecreasealongthestepisassumedtobelinear.Anewstepandscatteranglearethensampledforthenewmedium.Withthissimpli edapproachanumberoferrorsareintroduced.First,theactualcrossingpointisnotnecessarilyonthestraightline.Duetothesmalllateraldisplacementsthatprotonsundergo,wecanassumethatthiscausesnegligibleerrorsmainlybecauseofthesmallscatteringangles.Weveri edthattheaveragelateraldisplacementperstepintheMonteCarloprotontransportvariesfromabout0.1%ofthesteplengthat200MeVtoabout1%atenergiesbelow1MeV.Thesteplengthsat200MeVand1MeVareabout10 3gcm 2and10 6gcm 2respectively.Secondly,themultiplescatteringangleisnotcorrectedafterthecut-offofastep.IfwewouldreassesstheMoli`eredistributionfortheshorterstep,itcouldbeincorrectforveryshortstepsasshownbyAndreoetal(1993).Again,weassumethattheintroducederrorsarenegligibleforthesamereasonsasinthe rstpoint.Furthermore,theenergyisactuallynotdepositedequallyalongeachprotontracksegmentand,inconsequence,theassumptionoflinearenergydecreasealongthestepisonlyan

Monte Carlo study of fluence perturbation effects on cavity

Cavitydoseresponseinclinicalprotonbeams69

approximation.Astheenergylossespersteparesmall,bychoosingalogarithmicstepsizegridinPTRANforthelowerenergieswherethestoppingpowersshowthelargestvariationswithenergy,theintroducederrorsareminimized.

Speci callyforthecylindricalandsphericalcavitiesinthesimulationsofthisworkthefollowingmethodsareusedtosavecomputingtime.A‘geometryinterrogationreduction’methodisusedtoavoidunnecessarytesting.Thereforeaslabregionperpendiculartotheincidentbeamisde nedinwhichcavitiesarepresent.Whenaparticleentersthisregionatestisactivatedtoevaluateiftheparticlecrossesacavityborder.ThetransportinallotherslabregionsremainsessentiallythesameasintheoriginalPTRANcode.Forthesimulationofwallperturbationeffects,atwo-cavitygeometryissimulated.Theinnercavityissettoair,theregionbetweeninnerandoutercavityissettowallmaterialorwater.A‘correlated-sampling’technique(see,forexample,MaandNahum1994)wasusedtosimulatedifferentwallmaterials.Thismeansthatwhenaparticleenterstheoutercavity,theenergy,thespatialcoordinatesanddirectionalcoordinatesoftheprotonarestoredaswellastherandomnumberofthesimulationatthatpoint.Afterthetransportofthisprotonis nished,thetransportsimulationisrestartedwiththestoredconditionsbutwiththewallsettoanothermaterial.Thisapproachnotonlysavescomputingtimebutalsoimprovesthestatisticalcorrelationofthedosesinthecavitywithandwithoutwalls.

TheenergydepositedinthecavityisevaluatedasthesumoftheenergiesaprotonloseswhilepassingthroughthecavitybyCoulombinteractions(includingelasticnuclearscattering)andbytheenergytransfertosecondaryparticlesbyinelasticnuclearinteractions.Inanycasetheenergyaprotonlosesisassumedtobelocallyabsorbed.Whenappropriate,adiscussionisincludedwherethisassumptionmightin uencetheresults.Inaddition,apreliminarystudyofpossibleeffectsofsecondaryelectronswitharangelargeenoughtotransportenergyfromoneregiontoanotherisperformed.Theseeffectsareevaluatedseparatelyasexplainedinsection2.6.Anexplanationoftheobservedeffects,includingthepotentialrelationtoparticularperturbationcorrectionfactorsisdiscussedinsection3.Protonspectrainthedifferentcavitiesarecalculatedtoenablethesubsequentcalculationofsecondaryelectronspectra.Thisisdonebyscoringthetracklengththattheprotontraversesinthecavityandbydistributingthistracklengthovertheenergyandangularbinsofthespectrum.Uptonowonlycylinders(optionallywithtopandbottomplane)andspheresareimplemented.

2.4.Simulatedgeometryandcases

Forthecalculationofcavity-dependenteffectsondosedeterminationthegeometryof gure1isused.The(air- lled)cavityvolume,thewallthicknessandthewallmaterialarevariedforcylindricalandsphericalcavities.Theradiiofthecavitiesweretakenas0.25and0.50cmandtheheightofthecylindricalaircavitywastakenas1.0cm.Wallmaterialsusedarewater,graphite,PMMA(Lucite,Perspex),polystyreneandtissue-equivalentA150.Toevaluatetheeffectofprotonmultiplescatteringseparately,calculationsarealsodonewithscatteringturnedoff.Thecalculationsareperformedformonoenergeticprotonswithenergiesof70,100and200MeVfortheunmodulatedbeamsandwithenergiesof85,100and200MeVforthemodulatedbeams.Fortheunmodulatedbeamsthecavitycentreswereplacedatashallowdepth(1.0cmforthe70MeVand100MeVbeams,2.0cmforthe200MeVbeam)andatdepthscorrespondingtoabouthalfthecontinuous-slowing-downrangeRcsda(2.0cm,3.4cmand13.0cmrespectively).ForthemodulatedbeamspolystyrenemodulatorwheelswereusedresultinginaspreadoutBraggpeak(SOBP)fromabout0.3 Rcsdato1.0 Rcsda.Thecavitieswereplacedinthecentreofthe atdoseregion

Monte Carlo study of fluence perturbation effects on cavity

70HPalmansandFVerhaegen

(3.4cmfor85MeV,4.5cmfor100MeVand15.1cmfor200MeVbeam).Thethicknessofthecavitywallvariedfrom0.025cmto0.8cm.Forthecasewherethecavitywastooclosetotheentranceplaneofthephantomtoallowforthe0.8cmwallthickness,0.4cmwasusedasthemaximumvalue.Thesewallthicknessesaremuchlargerthanthoseoccurringinionizationchambersforclinicalprotondosimetry,buttheiruseallowsamoreaccurateinvestigationofperturbationeffectsandmakestheinterpolationtorealisticwallthicknesseseasier.Formosteffectsonlysomeoftheabovedescribedsituationsaresimulatedextensively(forexample,onlyforcylindricalcavities)assimilareffectsariseforother

situations.

Figure1.Geometrysimulatedtostudypotentialperturbationeffectsincavitydoseresponse.Thetwoconcentricalsphericalorco-axialcylindricalcavitiesaresurroundedbyhomogeneouswater.Theinnercavitycontainsair,theregionbetweenthetwocavitiesconsistsofwallmaterial.Twodottedlinesde nethewaterlayercontainingthecavityinwhichtheboundarycrossingtestisswitchedon.Forthesimulationofmodulatedbeamsanadditionallayerwithvariablethickness(sampledfromthewheelthicknessdistribution)thesimulatedwaterphantomisaddedupstream.Thebeamissuf cientlylaterallyextendedtoavoidparticlesfromthebeamedgereachingthecavities.

2.5.Primaryproton uenceperturbationwithasemi-analyticalmethod

Thesimulationsarecomparedwithasemi-analyticalmethod.Inthis,theeffectivewaterdepthofthecavityisapproximatedanalyticallyassumingthatnoscatteringoccurs(seeappendix).Foracylindricalcavityitiscalculatedas:

x=+R1

2(1 F)I8FR122

withI= (R2 x2)(R1 x2)dx(2)zeff=d

πR13πx= R1

forasphericalcavity:

r=+R1

3(1 F)I3FR122

(R2 r2)(R1 r2)rdr(3)withI=zeff=d 4R1r=0wheredisthedepthofthecentreofthecavity,R1istheradiusofthecavity,R2isthe

outerradiusofthewallandF=Rcsda,water/Rcsda,wallfortheincidentprotonenergy.Foratheoreticalaircavitywithoutwallsinwaterthepreviousexpressionsbecome

8R1

zeff=d (4)

3π

Monte Carlo study of fluence perturbation effects on cavity

Cavitydoseresponseinclinicalprotonbeams

forcylindricalcavitiesand

71

3R1

(5)

4

forsphericalcavitiesandzeffbecomesananalyticalestimateforthedepthoftheeffectivepointofmeasurement.

BothatthedepthofthecavitycentreandattheeffectivewaterdepthzeffthedoseisevaluatedfromthedepthdosedistributioncalculatedwithPTRANinhomogeneouswaterwhichexplainswhythemethodistermedsemi-analytical.Thedosethusobtainediscomparedwiththedosecalculatedintheaircavity,scaledwiththewatertoairstoppingpowerratioand,ifnecessary,forapartwiththeratiooftotalinelasticnuclearcrosssections(seesection3.2.1).

zeff=d

2.6.Effectofsecondaryelectronperturbation

2.6.1.Generationofsecondaryelectrons.ThecalculationofsecondaryelectronspectrausedinthisworkisdonewiththeclassicalRutherfordcrosssectionforcollisionbetweenanincidentprotonwithafreeelectron.AmoreaccurateapproachusingtheBahbacrosssectionresultsincrosssectionswhichareonlyslightlydifferentfromtheRutherfordcrosssectionsasindicatedbyMedinandAndreo(1997b).Weassumethatthemoresimpli edapproachinourworkdoesnotin uencetheresultsqualitatively.

TheclassicalRutherfordcrosssectiondifferentialinenergyisgivenby(see,forexample,Ruddetal1992):

24πr0Rdσ=(6)dWTW2

whereWisthesecondaryelectronenergy,σthecrosssectionperelectronandperincidentproton,r0theBohrradius,RtheRydbergenergyandTthekineticenergyofanelectronthatwouldhavethesamevelocityastheincidentproton(T=m/MT0,withmtheelectronmass,MtheprotonmassandT0theprimaryprotonenergy).IntheRutherfordapproximationacertainsecondary-particleenergycorrespondswithexactlyonescatterangleθ relativetotheincidentparticleaxis,whichcanbeevaluatedbyclassicalmechanics:

1MW

.(7)cosθ =

2mT0

Themaximalsecondaryelectronenergycorrespondstothesmallestanglesandisgivenby

m

Wmax=4T0.(8)

M

ThenumberofelectronsNthatiscreatedperunitofproton uence(cm 2)inavolumedVisthen

Z

(9)N=NAρσdV

A

whereNAisAvogadro’snumber,ρthemassdensityofthemediumandZ/Atheaverageratioofatomicnumberandatomicmassofthemedium.

Foreachbinoftheprotonspectrumthesecondaryelectronspectrumiscalculatedaccordingtoequation(6)usingenergybinsof1keVandalowercut-offenergyof10keV.Thecorrespondingelectronemissionanglesθ relativetotheprotondirection(thez -axisof gure2)arecalculatedusingequation(7).Thenumberofsecondaryelectronsineachbinisuniformlydistributedafterwardsovertheazimuthalangle between0and2πaround

Monte Carlo study of fluence perturbation effects on cavity

72HPalmansandF

Verhaegen

Figure2.Geometricalsituationofthegenerationofsecondaryelectrons,indicatingtheabsoluteandrelativeangulardirectionoftheincidentprotonandsecondary

electron.

Figure3.FiveregionsinwhichtheprotonspectraarecalculatedduringtheMonteCarlosimulation.IneachoftheseregionsthesecondaryelectronspectraarederivedanalyticallyandtransportedwithEGS4tocalculatetheenergytranfersbyelectronsfromoneregiontoanother.

theprotondirection.Theelectronemissionanglewithrespecttothecentralaxisoftheprimarybeam(thez-axisof gure2)iscalculatedusingequation(10):

cosθ=cosθ cosθ0 sinθ sinθ0cos

whereθ0representstheangleoftheproton’s ightdirectionrelativetothez-axis.2.6.2.Transportofsecondaryelectrons.Thetwo-dimensionalelectronspectracalculatedfromtheprimaryprotonspectrawereusedasinputforelectrontransportsimulationswiththeMonteCarlocodeEGS4(Nelsonetal1985).Separateinputspectrawereusedforelectronsthatwerecreatedbyprotonsindifferentregions.Fiveregionswerede ned:

(10)

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Cavitydoseresponseinclinicalprotonbeams73

theaircavity,thewallandthewatershellaroundthewallofwhichthewallandwatershellweresubdividedinafront(upstream)andback(downstream)sectorasshownin gure3.Electronsweretransporteddowntoanenergycut-offECUTof1keV.CrosssectionsforelectroninteractionswereaccordinglygeneratedintheEGSpreprocessorstage(AE=1keV).TheelectronstepsizealgorithmPRESTA(BielajewandRogers1987)wasinvokedtoimproveelectrontransportacrossmediumboundaries.TheEGS4algorithmfortransportoflow-energyelectrons,developedbyMaandNahum(1992)wasused.Rogers(1993)pointedoutthatPRESTAshouldbeusedincombinationwithanESTEPEvalueaslowas1%toachievereliableresultsfromMonteCarlosimulationsofionizationchamberresponse.InthisworksimulationsweredonewithESTEPEvaluesof1%and50%.InthelattercasePRESTAtakesfullcontrolovertheelectrontransport.Separaterunswereperformedforthedifferentwallmaterials(A150,graphite,PMMA,polystyrene,water)forsphericalcavities.Foreachrun,theenergydepositionbytheelectronswasscoredin15contributions:fortheelectronsgeneratedinthe veregionspreviouslyde ned,theenergydepositioninthethreeregionstakenbytheaircavity,wallandwatershellarescored.Toobtainasmallstatisticalvariance,inatypicalrun20to25timesmoreelectronsweregeneratedinthewallandwatershellthaninthecavity.Thisvariancewasestimatedbysplittingupacalculationinto10batches.TheobtainedelectronenergydepositionswerethenusedtocorrecttheenergydepositionofprotonscalculatedbyPTRANinthefollowingway:

2

(Ewall,i→cavNwall,i+Ewater,i→cavNwater,i)Ecorr=Eproton+

i=1

Ecav→wallNcav Ecav→waterNcav

(11)

whereEcorristhemeanenergydepositioninthecavitycorrectedforsecondaryelectron

uenceperturbation,Eprotonthemeanenergydepositionduetoprimaryprotonsperincidentprotonassumingnosecondaryelectronperturbation,Ea→bthemeanenergyperelectrongeneratedinregionathatisdepositedinregionb,Nathenumberofelectronsgeneratedinregionaperincidentprotonandianindexoverthenumberofsectors(frontandbackofbothwallandwatershell).Aperturbationcorrectionfactorduetoelectroneffectspeisthencalculatedaspe=Ecorr/Eproton.3.Resultsanddiscussion

3.1.Simulationofmodulatedprotonbeams

Figures4(a)and4(b)showcalculateddepthdosedistributionsformodulatedprotonbeamsofdifferentincidentenergiesE0withoutandwithtakingintoaccountinelasticnuclearinteractions.Forbothdatasetstheenergydistributionofprotonsafterpenetratingthemodulatorwasbasedonthedesignofpolystyrenemodulatingwheelswith80sectorsofvaryingthickness.Thethicknessdistributionwasdeterminedinsuchawaythattheresultingmodulatedprotonbeamproducesa atdepthdosedistributioninwateriftherecentstoppingpowerdataforwaterpublishedbytheICRU(1993)areusedandinelasticnuclearinteractionsareneglected(forananalyticalmethodtocalculatethethicknessdistributionofmodulatorwheels,seethepublicationofBortfeldandSchlegel(1996)).

Asisobviousfrom gure4(a),thedepthdosedistributionsforinitialprotonenergiesof50,85,100,150,200and250MeVcalculatedwithouttakingintoaccountinelasticnuclearinteractionsarereally atoveralargedepthregionofabout70%oftheprotoncontinuous-slowing-downrangeRcsda.Asshownby gure4(b),however,thedosecandeviate

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74HPalmansandF

Verhaegen

Figure4.Normalizeddepthdosedistributionsformodulatedprotonbeamswithenergiesrangingfrom50MeVto250MeV(a)withoutnon-elasticnucleardosecontributionsand(b)withnon-elasticnucleardosecontributions.Themodulatorwheeldesignwasbasedonthe(Coulomb)stoppingpowersonly.

considerablyfromaconstantvalueiftheenergylossofprotonscausedbyinelasticnuclearinteractionsandtransferredtosecondarychargedparticleswhichareassumedtobelocallyabsorbed(Berger1993b)aretakenintoaccountinwater.Fora250MeVmodulatedprotonbeamthedeviationfromthedesired atdosepro leismorethan15%.Becauseofthisfact,inelasticnuclearinteractionsmustbeincludedinthedesignofprotonbeammodulators.

Themodulatorwheelsusedinthefurthersimulationsofthisworkarethereforedesignedbyalsotakingintoaccountinelasticnuclearinteractionsexceptifotherwiseindicated.TherelativecontributionsofthewheelthicknessesarechangediterativelystartingfromthemostpenetratingBraggcurve.A atdosepro http://paredwiththewheeldesignusedtocalculatethedepthdosedistributionsof gure4,therelativecontributionshadtobechangedbymorethan2%toobtain atdosepro leswithinelasticnuclearinteractionsincluded.

Monte Carlo study of fluence perturbation effects on cavity

Cavitydoseresponseinclinicalprotonbeams

75

Figure5.Spectralproton uencedistributionsfroma100MeVmodulatedprotonbeamatdepthsinwaterof0.3 Rcsda,0.6 Rcsda,0.9 Rcsdaand1.0 Rcsdanormalizedperincidentproton.

Figure5showsprimaryprotonenergyspectracalculatedfora100MeVmodulatedprotonbeamhavinga attotaldosepro lefrom0.25Rcsdato0.95Rcsdaatfourdepths;oneatthebeginningofthe atdoseregion,oneinthemiddle,oneneartheendandonebehindthe atdoseregion.Inthis gurewecanseethatbesidestheapproximatelyGaussianenergyspectrumfromthemonoenergeticbeam,alongtailoflower-energycontributionsduetothesuperpositionofdifferentthicknessesofthewheeloccursinthisspectrum.

Wecanremarkthatbecausetherelativecontributionfromtheselowenergiesincreasesforlargerdepths,thelinearenergytransferandmicrodosimetriclinealenergywillincreaseforthesedepthsasshownbyVerhaegenandPalmans(1997).Thereforeitcanbeexpectedthatalthoughthedosepro leis at,thebiologicalconsequencesofthisdosewillvaryoverthe atdoseregion.Thisis,forexample,shownbyPaganettiandSchmitz(1996)whostudiedthein uenceofthebeammodulationtechniqueonthevariationofrelativebiologicaleffectivenessalongtheSOBP.Ifinthefuturesuf cientknowledgebecomesavailableaboutthemicrodosimetricandbiologicaleffectsofprotonradiationonhumantissueasafunctionofenergy,thewheeldesignshouldbeimprovedinordertoobtaina atbiologicalresponseoveralargedepth.

3.2.Effectsofprimaryproton uenceperturbation

3.2.1.Effectoftheaircavityinwater.Firsttheeffectofinsertinganaircavityisevaluated.Intheory,whentheBragg–Grayprincipleholdsfortheprimaryparticles(contrarytohigh-energyphotonswhereBragg–Grayisappliedtothesecondaryelectrons),theratioofthedoseintheaircavitytothedoseinwatershouldbeequaltothereciprocalofthewater-to-airstoppingpowerratioaveragedovertheprotonspectruminthecavity.

ThedosetowaterDwcalculatedinhomogeneouswaterbothatthecavitycentreandattheeffectivemeasuringpointobtainedwithequation(4)iscomparedwiththedosetowaterDw,acderivedfromtheenergydepositioninthecylindricalaircavityforthedifferentincidentprotonenergies,depthsandcavitysizes.TheratioDw/Dw,accouldberegardedasapdisplperturbationcorrectionfactorintheformalismreferredtointheintroduction.

DwatthedepthofthecavitycentreandattheeffectivemeasuringpointisdeterminedfromthedepthdosedistributionobtainedwithPTRANinhomogeneouswaterforapencilbeam.Thisdistributioniscalculated(Berger1993a)astheaverageenergylossperunit

Monte Carlo study of fluence perturbation effects on cavity

76HPalmansandFVerhaegen

depthinMeVg 1cm2(whichis,duetothereciprocitytheorem,alsotheaverageenergylossperunitdepthonthecentralaxisofanin nitelybroadbeam).Sothedoseatacertainpointonthecentralaxisforacertain eldsizeisobtainedbydivisionofaverageenergylossperunitdepth(expressedasmassperarea)withtheareaoftheincident eldprovidedthatthebeamissuf cientlyextendedlaterallytoavoidprotonsfrombeyondthelateraledgeofthebeambeingabletoreachtheaxisbyscattering.

ToobtainthedosetowaterDw,acdetectedbythecavity,thecavitydoseismultipliedwiththewatertoairstoppingpowerratiooftheprotonspectruminthecavity.Forunmodulatedmonoenergeticbeamsthestoppingpowerratioistakenattheeffectiveenergywhichisderivedfromtheresidualrange.ThelatterrangeiscalculatedasRcsdaoftheincidentprotonsminustheeffectivedepthofthecavityascalculatedbyequation(4).Thisapproachisjusti edbythesmallenergyspreadoftheprotonsinthecavityandbythesmalldependenceofthewatertoairstoppingpowerratioonenergy.ForthemodulatedbeamsthisapproachintroduceserrorsasshownbyMedinandAndreo(1992)becausetheprotonenergyspectraarebroadenedbyspreadingouttheBraggpeak.Table1showsthestoppingpowerratioscalculatedattheeffectiveenergynexttotheratioofstoppingpowersintegratedoverthecalculatedspectraatdifferentdepthsontheSOBP,asforexampleshownin gure5.WeseethatintheperipheralareaoftheSOBPdifferencesofupto0.4%occurandinitscentredifferencesofupto0.2–0.3%.AsshownbyMedinandAndreo(1997a),thein uenceonthestoppingpowerratioofneglectingsecondaryprotonsislessthan0.1%forunmodulatedmonoenergeticprotonbeamsinthestudiedenergyrange.Thisresultalsoholdsforamodulatedbeam,asitcanberegardedasasuperpositionofunmodulatedbeams.

Table1.Water-to-airstoppingpowerratiosatdifferentdepthsdinwaterformodulatedprotonbeamsofinitialenergyEandresidualrangeRresidualatdepthdcorrespondingtoaneffectiveenergyEeff:(S/ρ)wa(Eeff)iscalculateddirectlyfromthemassstoppingpowersforwaterand

wiscalculatedfromthe¯airatenergyEeffusingthetablespublishedbyICRU(1993);(S/ρ)a

protonspectrumatdepthdinwateralsousingtheICRUtables.E(MeV)858585100100100200200200

Depth(cm)1.753.455.202.324.656.987.7115.523.4

Rresidual(cm)4.032.330.585.393.070.7418.210.52.60

Eeff(MeV)69.551.323.781.859.827.2163.0119.054.4

w

Sa

(Eeff)

w

¯Sa

1.1321.1331.1361.1321.1331.1351.1301.1311.1331.1351.1361.1401.1341.1351.1391.1321.1331.136

Table2showsthecalculatedDw/Dw,acratiosforthedifferentincidentprotonenergiesE,depthsofthecavitycentredandcavitydiameters,besidestheeffectivemeasuringdepthzeffandtheeffectiveenergyEeff.Thesixthandseventhcolumnsoftable2showtheratiosDw(d)/Dw,acandDw(zeff)/Dw,acrespectivelyifDwiscalculatedatthedepthofthecavitycentreoratthedepthoftheeffectivemeasuringpoint,andonlyenergydepositedinCoulombinteractionsisincludedinDwandDw,ac.

TheeighthcolumnshowsDw(zeff)/Dw,acwhenenergydepositionsinCoulombinteractionsandinelasticnuclearinteractionsaretakenintoaccount.Thenextcolumnshowsthestatisticaluncertaintiesofthevaluesincolumn6,7and8with5×106histories

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Cavitydoseresponseinclinicalprotonbeams77

Table2.RatioDw/Dw,acofabsorbeddosetowaterDwinhomogeneouswaterandabsorbeddosetowaterDw,acforacylindricalaircavityasafunctionofincidentprotonenergyE,cavitydepthdandcavityradius.Incolumn6and7onlyenergydepositioninCoulombinteractionsisconsideredwithDwcalculatedatdepthdofthecavitycentreandattheanalyticallycalculatedzeffrespectively.Incolumn8energydepositionininelasticnuclearinteractionsisalsoincluded.Column9givesthestatisticaluncertaintyoftheabsorbeddoseratios.Column10showsDw(zeff)/Dw,acwhenmultiplescatteringisturnedoffinthecalculationofDw,ac.AlsoshownarethedepthzeffoftheeffectivemeasuringpointandtheeffectiveenergyEeff.

1

Energy,E

(MeV)70701001002002007070100100200200

2

Depth,d(cm)1.002.041.003.862.0013.01.002.041.003.862.0013.0

3

Cavityradius(cm)0.250.250.250.250.250.250.500.500.500.500.500.50

4zeff(cm)0.7881.8280.7883.6471.78812.770.5761.6160.5763.4351.57612.56

5Eeff(MeV)62.150.394.169.9192.0136.064.352.995.771.9193.0137.0

6

Dw(d)w,ac

7

Dw(zeff)w,ac

8

Dw(zeff)w,ac

910

Dw(zeff)w,ac

(C)(C)(C+N)0.9720.9760.9590.9700.9190.9420.9720.9750.9600.9680.9200.942

SD0.0010.0010.0010.0010.0010.0020.0010.0010.0010.0010.0010.002

(noscattering)0.9970.9970.9970.9960.9990.9990.9990.9971.0001.0001.0001.000

Unmodulatedbeams

1.0211.0331.0071.0160.9991.0001.0471.0721.0181.0351.0021.005

0.9940.9920.9950.9960.9970.9960.9950.9920.9970.9940.9980.996

Modulatedbeams8510020085100200

3.374.5015.13.374.5015.1

0.250.250.250.500.500.50

3.164.2914.92.954.0814.7

54.763.5122.057.165.7124.0

0.9981.0000.9980.9960.9971.002

1.0010.9990.9990.9980.9981.004

0.9840.9840.9630.9820.9800.967

0.0020.0020.0030.0010.0020.002

1.0011.0021.0000.9980.9991.003

fortheunmodulatedbeamsand107historiesforthemodulatedbeams.ThelastcolumnshowsagainDw/Dw,acwhenmultiplescatteringisturnedoffinthecavitydosecalculationswithastandarddeviationthatwaslessthan0.1%forallsimulations.

Startingthediscussionfortheunmodulatedbeams,itisclearfromcolumns6and7thatforthe70MeVand100MeVbeamsDw(d)issigni cantlyhigherthanDw,ac,whereasDw(zeff)agreesmorecloselywithDw,ac.Forthe200MeVbeam,differencesaresmaller,butgenerallyDw,acagreesbetterwithDw(zeff)thanwithDw(d).NeglectinginelasticnuclearscatteringandcomparingthevaluesofDw(zeff)/Dw,acforunmodulatedprotonbeamswithandwithouttakingintoaccountprotonmultiplescattering(columns7and10),weseethatfortheunscatteredbeamtheagreementofDwandDw,acisformostcasesbetterthanforthecompletesimulationwhereperturbationsadditionaltothedisplacementofthemeasuringpointrangingfrom0.2%to0.8%areobtained.Apartoftheseperturbationscanbeattributedtoscattering.Theobservationthatinallcasesthedoseinthecavityislargerthanthedoseinwaterattheeffectivepointofmeasurementcanbeexplainedbecauseparticlesthatundergoanimportantnumberofscatteringinteractionstravelalongerdistancebeforetheyenterthecavity,thushavingalowerenergyforwhichthestoppingpowerishigher.ThesmalldeviationsfromunityforDw(zeff)/Dw,acwith

Monte Carlo study of fluence perturbation effects on cavity

78HPalmansandFVerhaegen

scatteringturnedoffindicatethattheassumptionoflocallineardependenceofDwonzandtheneglectingofprotonenergydegradationovertheaircavityinthesemianalyticalmodel(seeappendix)isnotcompletelyjusti ed.Neglectingenergydegradationunderestimatesthedoseattheeffectivemeasuringpointwhichisconsistentwiththeresultsofcolumn10intable2.

Whencomparingcolumns7and8animportantdifferenceshowsupduetotheinclusionofenergydepositionfrominelasticnuclearinteractions.ThereasonisthattheratioofthestoppingpowersinwaterandairisnotequaltotheratiooftotalinelasticnuclearcrosssectionswhichweadoptedfromJanni(1982).Soitcannotbejusti edtoscalethecontributionofinelasticnuclearenergydepositionswiththestoppingpowerratio.Thiscontributioninaircanberescaledtowaterseparatelywiththeratiosoftotalinelasticnuclearcrosssectionsasfollows:

w σ wS

+Dair(N)(12)Dw,ac=Dair(C)

ρaAawithDair(C)thedosedepositedinCoulombinteractionsintheaircavity,(S/ρ)wathewater

toairmassstoppingpowerratio,Dair(N)thedosedepositedininelasticnuclearinteractionsintheaircavityand(σ/A)watheratiooftotalinelasticnuclearcrosssectionspernucleoninwaterandair.Applyingequation(12)theDw/Dw,acratioscalculatedbytakingintoaccountinelasticnuclearinteractionsbecomeexactlythesameasthoseofcolumn7.Thisindicatesthattheseparatescalingofthiscontribution,whichincreaseswithincreasingprotonenergies,doesnotin uencetheperturbationeffectsstudied.Nevertheless,itshouldbeconsideredforpurposesofaccurateabsoluteionizationchamberdosimetry.Furthermore,itshouldbeclearthattheeffectweshowforinelasticnuclearcontributionsisprobablyoverestimatedasonlyafractionofthisinelasticnuclearenergytransfergoestosecondarychargedparticlesandcontributestothelocallydepositeddose.Inwater,asshownbySeltzer(1993),animportantpartistransferredtolong-rangeneutronsthatdonotcontributetothelocaldose.Itisnotunlikelythatforair,consistingmainlyofnitrogen,thesituationissigni cantlydifferent.AnotherremarkisthatthetotalinelasticnuclearcrosssectionsfornitrogenareinterpolatedfromthedatagivenbyJanni(1982)assumingasmoothdependenceofthistotalcrosssectiononatomicnumber.Further,thechargedsecondaryparticlesoriginatingfrominelasticnuclearinteractions,especiallytheimportantfractionofsecondaryprotons,canalsobetransportedfromthecavitytothewallandviceversa,therebypossiblycompensatingtheincreasedenergytransferredtoaircomparedtowater.Theeffectsarereducedto50%to70%(equaltotheenergyfractiontransferredtochargedsecondaries(Seltzer1993))if(i)theinterpolationprocedureconcerningthetotalcrosssectionsiscorrect,(ii)theratioofenergytransferredtochargedandunchargedparticlesininelasticnuclearreactionsiscomparableforairandwaterand(iii)theassumptionisapplicablethatthetotalenergytransferredtochargedsecondariesduetoinelasticnuclearinteractionsisdepositedlocally.Recentlyanumberofcomparisonsofionizationchamberdosimetryandwatercalorimetryinbothmodulatedandunmodulatedclinicalprotonbeamsbetween85MeVand250MeVhavebeenperformed(Schulzetal1992,VatnitskyandSiebers1994,Vatnitskyetal1996,Palmansetal1996).Theseexperimentsshowthationizationchamberdosimetryisconsistentwithwatercalorimetrywithin2%overthecompleteclinicalenergyrangewhenapplyingthe(Wair)precommendedbytheAAPMprotocol(AAPM1986).Thisprovesthattheactualeffectsofinelasticnuclearenergydepositionandespeciallytheirdependenceonprotonenergymustbelessthanshownintable2.However,theycouldalsobeimportantinthediscussionofthe(Wair)pvaluewhenconclusionsregardingitwouldbedrawnfromcomparisonsbetweencalorimetrymeasurementsandionizationchambermeasurements.

Monte Carlo study of fluence perturbation effects on cavity

Cavitydoseresponseinclinicalprotonbeams79

Formodulatedbeams,inelasticnuclearinteractionswereneglectedandthedoseratioswerecalculatedwithmodulatorwheelsgivinga atdosepro leforonlyCoulombinteractions,i.e.thoseusedfor gure4.Dw(d)/Dw,ac,Dw(zeff)/Dw,acandDw(zeff)/Dw,acwithscatteringturnedoffareallclosetounityandthesmalldeviationsfromunitycouldbeattributedtolocal uctuations(‘ripples’)withamagnitudeofsometenthsofapercentonthedepthdosedistributionduetothesuperpositionofindividualBraggcurvesintheSOBPregion.ForDw(zeff)/Dw,acwithinelasticnuclearcontributionsincluded,othermodulatorwheelsgivinga atdosepro leforthetotalenergydepositionweresimulatedasindicatedinsection3.1.Regardingthefractionofenergydepositedininelasticnuclearinteractionssimilar,butsmaller,effectsasforunmodulatedbeamsareobserved.

Table3.Similartotable2butforsphericalcavitiesandonlyforunmodulatedmonoenergeticbeams.

1

Energy,E

(MeV)70701001002002007070100100200200

2

Depth,d(cm)1.002.041.003.862.0013.01.002.041.003.862.0013.0

3

Cavityradius(cm)0.250.250.250.250.250.250.500.500.500.500.500.50

4zeff(cm)0.8131.8530.8133.6721.81312.790.6251.6650.6253.4841.62512.61

5Eeff(MeV)61.950.093.969.719213563.852.395.471.5193136

6

D(d)w,ac

7

Dw(z)w,ac

8

Dw(z)w,ac

910

Dw(z)w,ac

(C)(C)(C+N)0.9740.9800.9610.9730.9210.9460.9730.9780.9590.9710.9200.947

SD0.0010.0010.0010.0010.0010.0010.0020.0010.0010.0010.0030.002

(noscattering)0.9980.9970.9990.9991.0011.0010.9980.9960.9990.9991.0011.001

Unmodulatedbeams

1.0201.0331.0081.0161.0001.0041.0421.0661.0161.0321.0011.009

0.9960.9970.9980.9980.9981.0000.9960.9950.9950.9970.9991.002

Table3showsthesamedataasintable2calculatedforsphericalcavitiesfortheunmodulatedbeams.DuetotheshapeofthecavitytheratiosDw/Dw,acareslightlydifferentfromthoseforthecylindricalcavities,butthetendenciesdescribedinthepreviousparagraphscanagainbeobserved.

3.2.2.Effectofreplacingwaterwithwall.Figure6showscalculatedratiosofthedoseinthecentralaircavitywithandwithoutwallforamonoenergeticunmodulated100MeVprotonbeamatadepthof3.859cm(=0.5 Rcsda)fordifferentwallmaterialsasafunctionofwallthicknessforacylindricalcavitywithinnerradiusof0.25cm.Wecanobservethatinthiscasearelativelylargedoseincreaseiscausedbyexchangingwaterwithwallmaterialaroundthecavity(about0.5%permmwallthicknessforgraphite).Thiseffectcanbeexplainedbythedecreaseofeffectivewaterdepthwithincreasingwallthicknessasisshownanalyticallyintheappendixanddiscussedinthefollowingparagraphs.

Table4givesanoverviewofourresultswithrespecttothepercentagecavitydosechangepermmwallthicknessforfourdifferentwallmaterialsasafunctionofprotonenergy,measuringdepthandcavitydiameter.Theywerederivedbylinear ttingofdatacorrespondingtothoseshownin gure6havingstatisticaluncertaintiessmallerthan0.2%foreachwallthickness.Thenumberofhistoriesrequiredvariedfrom7×105forthe