Converting absorbed dose to medium to absorbed dose to water for Monte Carlo based photon ...

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Converting absorbed dose to medium to absorbed dose to water for Monte Carlo based
photon beam dose calculations
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2000 Phys. Med. Biol. 45 983
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Phys.Med.Biol.45(2000)983–995.PrintedintheUK PII:S0031-9155(00)09047-3
Converting absorbed dose to medium to absorbed dose to
water for Monte Carlo based photon beam dose calculations
JVSiebers†,PJKeall†,AENahum‡andRMohan†
†DepartmentofRadiationOncology,MedicalCollegeofVirginiaHospitals,
VirginiaCommonwealthUniversity,POBox980058,Richmond,VA,USA
‡JointDepartmentofPhysics,RoyalMarsdenHospitalandInstituteofCancerResearch,Sutton,
Surrey,UK
E-mail:[email protected]
Received27October1999,infinalform20January2000
Abstract. Currentclinicalexperienceinradiationtherapyisbasedupondosecomputationsthat
reporttheabsorbeddosetowater,eventhoughthepatientisnotmadeofwaterbutofmanydifferent
typesoftissue.WhileMonteCarlodosecalculationalgorithmshavethepotentialforhigherdose
accuracy,theyusuallytransportparticlesinandcomputetheabsorbeddosetothepatientmedia
suchassofttissue,lungorbone. Therefore,fordosecalculationalgorithmcomparisons,orto
reportdosetowaterortissuecontainedwithinabonematrixforexample,amethodtoconvert
dosetothemediumtodosetowaterisrequired. Thisconversionhasbeendevelopedhereby
applyingBragg–Graycavitytheory.Thedoseratiofor6and18MVphotonbeamswasdetermined
bycomputingtheaveragestoppingpowerratiofortheprimaryelectronspectruminthetransport
media.Forsofttissue,thedifferencebetweendosetomediumanddosetowaterisapproximately
1.0%,whileforcorticalbonethedosedifferenceexceeds10%.Thevariationinthedoseratioasa
functionofdepthandpositioninthefieldindicatesthatforphotonbeamsasinglecorrectionfactor
canbeusedforeachparticularmaterialthroughoutthefieldforagivenphotonbeamenergy.The
onlyexceptiontothiswouldbefortheclinicallynon-relevantdosetoair. Pre-computedenergy
spectrafor60Coto24MVareusedtocomputethedoseratiosforthesephotonbeamsandto
determineaneffectiveenergyforevaluationofthedoseratio.
1. Introduction
Conventionaldosecalculationsforphotonbeamradiationtherapytypicallyreporttheabsorbed
dosetowater(D ). ThereportingofD isduepartlytothehistoricaldevelopmentoftreatment
w w
planningalgorithmsaswellasthefactthattreatmentmachinesarecalibratedintermsofthe
absorbeddosetowater. Theinputdatausedfortreatmentplanningalgorithmsaregenerally
doseprofilesandoutputfactorsmeasuredinwaterphantoms. Thesedataaremanipulatedin
various ways by the algorithm used, but converting the in-phantom measured dose to water
to dose to medium is not considered. The assumption that the body is water is a good first
approximationaswatermakesupthebulkofthevolumeofcellsandbodyfluid(withafew
exceptionssuchasboneandtoothenamel)(TortoraandGrabowski1993).
MonteCarlo(MC)dosecalculationalgorithmsarebeingintroducedintoroutinetreatment
planning practice (Hartmann-Siantar et al 1994, 1995, Mohan 1997, DeMarco et al 1998,
Solbergetal1998,Wangetal1998,Fippel1999,Maetal1999). WhenusingMonteCarlo,
theabsorbeddoseiscomputedtothemediumcontainedinthedosevoxel(D ). Thesecan
med
bearbitrarymaterials,andgenerallyarerepresentativeofthosefoundinthebody. However,
0031-9155/00/040983+13$30.00 ©2000IOPPublishingLtd 983
984 JVSiebersetal
in order to compare MC algorithms with conventional D algorithms, the dose comparison
w
shouldbetothesamemedium. ThereareseveralotherreasonswhyD shouldbeconverted
med
toD ,whichwillbeexpandeduponinthediscussionsection.
w
Thepurposeofthispaperisto(a)determinethefactorsrequiredforconvertingabsorbed
dosetomediumtoabsorbeddosetowaterforphotontherapybeams,(b)determineifuniversal
conversionfactorscanbeappliedthroughoutthepatientvolume,(c)determineifconversion
factorsdependuponthematerialdensityand(d)provideamethodfordeterminingconversion
factorsforarbitrarypatientmaterialsandenergies.
2. Background
AbsorbeddoseD canbeconvertedtoD forphotonbeamirradiationusingBragg–Gray
med w
cavitytheory(Bragg1912,Gray1936,Attix1986,Brahmeetal 1988). Bragg–Graycavity
theoryanditsextensiontheSpencer–Attixcavitytheory(SpencerandAttix1955)havebeen
successfully applied to ionization chamber dosimetry (AAPM TG-21 1983, ICRU-35 1984,
Attix1986). Bragg–Graycavitytheoryisapplicablewhenthecavitymaterial(waterinthis
case)doesnotperturbthefluenceofchargedparticlesthatwouldhaveexistedifthecavitywere
composedofthesurroundingmaterial. Inthecaseofphotons,thisrequiresthattherangesof
thesecondaryelectronsaremuchgreaterthanthedimensionsofthecavityi.e.thatthecavity
doesnot‘perturb’thiselectronfluence,whichisthenentirelycharacteristicofthesurrounding
medium. Inpractice,forphotonsatmegavoltageenergies,theonlyrealdosimeterthatfulfills
this requirement is the gas-filled ionization chamber (Ma and Nahum 1991). The original
Bragg–Graytheoryrequiresthatdelta-ray(secondaryelectron)equilibriumexistsorthatthe
energytransferredtodeltaraysisdeposited‘onthespot’;itisevaluatedbyusingtheprimary
electron fluence and the unrestricted collision stopping powers (see below). Spencer–Attix
theory (Spencer and Attix 1955, ICRU-35 1984) removed this limitation in an approximate
fashionbyincludingdeltaraysinthefluenceofchargedparticlesenteringthecavity,andthen
usingthestoppingpowerrestrictedtolosseslessthan1, wherethiscutoffenergyisrelated
to the size of the cavity. For small cavities such as ion chambers 1 is generally set equal
to10keV(Nahum1978,ICRU-351984). Itcanbenotedthat,somewhatparadoxically,the
greatestdeviationbetweentheBragg–GrayandSpencer–Attixvaluesforthestopping-power
ratiooccurforthesmallestvaluesof1i.e.forthesmallestcavities.
To circumvent the unnecessary issues introduced by the selection of the Spencer–
Attixcutoffenergy, weconsiderahypotheticalBragg–Graywatercavityinwhichdelta-ray
equilibriumisestablished. Thustheconversioncanproceedasfollows.
Using Bragg–Gray cavity theory, the absorbed dose to water is related to the absorbed
dosetomediumby
D D s (1)
w med w,med
=
wheres istheunrestrictedwater-to-mediummasscollisionstoppingpowerratioaveraged
w,med
overtheenergyspectraofprimaryelectrons,(8 ) . Theso-calledprimaryelectronsdonot
E m
includeknock-onelectronsor�-rays,astheircontributionstoenergydepositionareaccounted
forintheunrestrictedstoppingpowers. Thestoppingpowerratioaveragedovertheprimary
electronspectrumiscalculatedusing
Emax Emax
s (8 ) (S/⇢) dE (8 ) (S/⇢) dE (2)
w,med E m w E m med
=
Z0 �Z0
where(S/⇢) and(S/⇢) aretheunrestrictedmasscollisionstoppingpowerforthewater
w med
andtransportmediumrespectively,andE isthemaximumenergyinthe(8 ) distribution
max E m
(NCRP-271961).
Convertingabsorbeddosetomediumtoabsorbeddosetowater 985
To evaluate the Bragg–Gray stopping power ratio for photon beams, knowledge of the
electronfluenceinthemediaisrequired. Presently,onlyMC-baseddosecalculationalgorithms
arecapableofdeterminingthisquantity.
3. Materialsandmethods
In order to determine s , the Monte Carlo code MCNP (Briesmeister 1997) was used
w,med
to transport photons and accompanying electrons through various homogeneous and simple
heterogeneousphantoms. ThemajorityofthecalculationsusedfullMonteCarlosimulations
ofthetreatmentheadtodescribetheoutputbeamsfromthelinearaccelerator. Detailsofthe
transportthroughthetreatmentheadaregivenelsewhere(Siebersetal1999),andarereviewed
herebriefly. Electronswithenergyequaltothenominalaccelerationpotentialofthetherapy
acceleratorwereincidentupontheacceleratorbremsstrahlungtarget. Photonsandelectrons
exiting the target were transported through the beam-line components (primary collimator,
flatteningfilterandionizationchamber)andhadtheirenergy,positionanddirectionstoredinto
aphase-spacedescriptionfile. Thesesavedparticleswerethensampledandtransportedinthis
studythroughthebeam-definingjawstoaphantomwhosesurfacewasattheisocentreplane
location. Particlesweretransportedthroughaphantomandtheprimaryelectronfluenceinthe
phantommedium((8 ) )wasscored(toscoreonlyprimaryfluence, theMCNPparameter
E m
RNOKwassetto0). Thelowenergyelectroncut-offenergywassetto7.5keVfortheMonte
Carlo simulation in this study. Below this energy, the electron fluence was estimated to be
equaltothefluenceat7.5keVandthemeanstoppingpowerfortheseelectronswasestimated
tobeequaltotheresidualenergydividedbytheCSDArange. Theuncertaintyintroducedby
thistechniqueisaddressedintheresultssection.
Monte Carlo simulations were performed for 6 MV and 18 MV photon beams from a
Varian2100Caccelerator. Thefieldsizewassetto10 10cm2at100cmSSDincidentupon
⇥
a 50 50 50 cm3 phantom. The electron fluence in each scoring region was determined
⇥ ⇥
using the track length estimate of the cell flux (which in steady state conditions equals the
cell fluence). In each region of the phantom, the electron fluence was used to compute the
fluence-averagedmediumandwaterstoppingpowervalues,andthestoppingpowerratio(using
equation(2)). Stoppingpowerratioswithrespecttowaterweredeterminedforair(ICRU-44
1988),lung(ICRU-441988),ICRUtissue(ICRU-331980),softbone(spongiosa)(ICRU-44
1988)andcorticalbone(ICRP-231975). Thedensitiesandcompositionsofthesematerials
arelistedintable1. Althoughairisnotapatientmaterial,itisincludedinthisworksoresults
canbecomparedwithAndreoandBrahme(1986)whoevaluateds usingSpencer–Attix
w,air
theoryforionizationchamberdosimetryasfunctionsofdepthandfieldsize.
The variation of the dose ratio as a function of off-axis position and depth was studied
by evaluating s at multiple depths in a pure water phantom both inside and outside
w,med
the 10 10 cm2 field. For the ‘inside the field’ evaluation, (8 ) was evaluated inside a
E m
⇥
4 4cm2areacentredaboutthecentralaxis,whilethe‘outsidethefield’evaluationconsisted
⇥
of evaluating (8 ) for electrons outside a 12 12 cm2 area centred on the central axis.
E m
⇥
1 108 phase-spaceparticlesweretransportedfromthephase-spacedefinitionplaneinthis
⇥
portionofthestudy.
The electron spectra produced in photon interactions will depend upon the material in
whichthetransporttakesplace. Toinvestigatetheinfluenceofthedifferingelectronenergy
spectra produced by various patient-like materials on s , the field-positional tests above
w,med
wererepeatedwiththewaterphantomreplacedwithacorticalboneandalung-tissuephantom.
Thestoppingpowerratiowasalsoevaluatedattheinterfacebetweenlungandboneslabs
toinvestigateiftheratiovariesinsituationsofelectronicdisequilibrium. Twophantomswere
986 JVSiebersetal
Table 1. The atomic percentage composition by relative mass of the materials used in these
calculationsaswellasthephysicaldensity,andthedensityrangeoverwhichthematerialisused
inourMonteCarlodosecalculations.
Density
[densityrange]
Material (gcm�3) H C N O Other
Water 1.0 11.2 88.8
Air 1.2⇥10�3[0–0.08] 0.0124 75.5 23.2 Ar1.28
(ICRU-441988)
Lung 0.26[0.08–0.5] 10.3 10.5 3.1 74.9 Na0.2,P0.2,S0.3,Cl0.3,
(ICRU-441988) K0.2
ICRUtissue 1.0[0.5–1.1] 10.1 11.1 2.6 76.2
(ICRU-331980)
Softbone 1.18[1.1–1.4] 8.5 40.4 2.8 36.7 Na0.1,Mg0.1,P3.4,S0.2,
(ICRU-441988) Cl0.2,K0.1,Ca7.4,Fe0.1
Corticalbone 1.85[1.4–2.5] 4.72 14.4 4.20 44.6 Mg0.22,P10.5,S0.315,
(ICRP-231975) Ca21.0,Zn0.01
configured. Oneconsistedofa5cmthicksemi-infinitelung(⇢ 0.25gcm 3)slabfollowed
�
=
by a 5 cm thick semi-infinite cortical bone (⇢ 1.85 g cm 3) slab; the other phantom
�
=
was similar, except the bone slab preceded the lung slab. In this case, the electron fluence
crossingthelung–boneinterfacewithin4cmofthebeamcentralaxiswasscoredandusedto
computethefluenceaveragedstoppingpowerratioswithrespecttowaterforeachmaterial.
Thesesimulationswereperformedfor6MVand18MVphotonsusing1 109 phase-space
⇥
particles.
Additional calculations were performed using pre-computed energy spectra incident on
thephantomtodetermines forphotonbeamscommonlyusedinradiotherapy. The60Co
w,air
beamenergyspectrumfromRogersetal(1988)wasused,whichincludestheeffectofphotons
scatteredinthesourceandbythebeamcollimators. Energyspectrafor4,6,10,15and24MV
acceleratorswereusedfromthecalculationsofMohanetal (1985). Inthesecases,a10cm
diameterinputbeamwasnormallyincidentuponthephantom. Thetrack-lengthaverageofthe
primaryelectronfluenceinthecentre8cmdiameterregionwasscoredasafunctionofdepth
inwater,lungandbonephantoms. Theseprimaryelectronfluenceswereusedtocomputethe
water-to-mediumstoppingpowerforeachmateriallistedintable1. Thegeometricdifferences
inthebeamorientationandthescoringgeometrybetweenthesecalculationsandthefullMonte
Carlosimulationsareexpectedtobenegligible.
InmostpatienttreatmentplanningMonteCarlodosecalculations,dataforagivenmaterial
(mass-stoppingpowersandcross-sections)arepre-calculatedatagivendensity,andthenused
overarangeofmaterialdensities. Inreality, thestoppingpowerofthematerialvarieswith
the material density due to the density effect (Swann 1938, ICRU-37 1984). To study the
effectofmaterialdensityonstoppingpowerratioevaluation,stoppingpowerratiovalueswith
respecttowaterwereevaluatedattheminimum,maximumandnominaldensityvaluesshown
intable1.
Inordertoextendtheconceptslistedinthispapertomaterialsandenergiesnotcoveredin
thispaper,aneffectiveenergy(E )forthestoppingpowerratiolook-upisdefinedastheenergy
eff
at which the mono-energetic stopping power equals that produced by (8 ) . The effective
E m
energyforeachpre-computedenergyspectrawasdeterminebyleastsquaresminimizationof
thedifferencebetweentheMonteCarlodeterminedaveragestoppingpowerratioandthatat
the sample effective energy for each material. A linear fit was then applied to the nominal
Convertingabsorbeddosetomediumtoabsorbeddosetowater 987
accelerationpotentialversuseffectiveenergyresultstoallowinterpolationtodifferentnominal
accelerationpotentials. AlthoughotherbeamqualityspecifierssuchasTPR20/10arebetter
indicatorsofthebeamquality,useofthenominalacceleratingpotentialisjustifiedheregiven
thatthedependenceofs onbeamqualityisweak. ItcanbenotedthatneitherE nor
w,med eff
TPR20/10allowsaccuratesimpleinterpolationofs .
w,air
Theeffectofapplyingthedose-to-mediumcorrectionfactorsonpatientdosecalculationsis
demonstratedonaheadandnecktreatmentplan. Here,aMonteCarlopatientdosecalculation
was performed using MCV, an in-house developed Monte Carlo dose calculation system
directlylinkedtotheADACPinnacle†treatmentplanningsystem. Isodoseanddose–volume
histogramsaregivenforbothdose-to-mediumanddose-to-waterpatientdosecalculations.
4. Results
Thedependenceofs onenergyformono-energeticelectronsfortypicalpatientmaterials
w,med
is shown in figure 1. (A similar plot evaluated at the mean energy of primary electrons for
electronbeams,butlikelyapplicabletoelectronenergyspectraindependentoftheirsource,can
befoundinBrahmeetal(1988).) Stoppingpowervalueswerecomputedusingthecomputer
codeESTAR(Berger1993)whichusestheICRU-37(1984)formalismanddata. From0.01to
20MeV,thestoppingpowerratioformono-energeticelectronschangessubstantially(>4%)
forair,corticalboneandlung,whileforICRUtissueandsoftbone,theratiovarieslittle.
Figure1.Water-to-mediummasscollisionstoppingpowerratiosasafunctionofelectronenergy.
Materialsindicatedareasspecifiedintable1.
Inordertodetermines foraphotonbeam,firsttheprimaryelectronfluence((8 ) )
w,med E m
over which the stopping-power ratio is calculated must be determined. Figure 2 displays
(8 ) computedatdepthsof1.5and45cminawaterphantomfor6MVand18MVphoton
E m
† ADACLaboratories,Milpitas,CA95035,USA.
988 JVSiebersetal
Figure2. Fluenceofprimaryelectronscomputedatdepthsof1.5cmand45cmfor6MVand
18MVphotons.
beams. Thedepthswerechosenfordisplayastheyshowthemaximumdeviationin(8 )
E m
for the depths computed. For each incident photon beam, the electron spectrum shifts to
higherenergiesasthedepthincreases. Fortheinstanceofthe6MVphotonbeam,themean
electronenergyat1.5cmdepthis1.00MeVwhileat45cmdepth,themeanelectronenergy
is1.16MeV.
The stopping power ratio computed using the 6 MV and 18 MV electron energy as a
function of depth in a water phantom is shown in figure 3. At both energies, the water-to-
mediumstoppingpowerratiosarenearlyconstantasafunctionofdepthinthephantomfor
the materials of interest. Values at 10 cm depth in the phantom are given in table 2. The
statisticaluncertaintiesinthes valuesarelessthan0.1%. Thesystematicuncertaintiesof
w,med
stoppingpowervaluesandstoppingpowerratiosarediscussedinICRU-37(1984)andAndreo
(1990). Theeffectofestimatingtheelectronfluencebelow7.5keVasbeingthesameasthat
at7.5keVhadlittleeffectonthecalculationasignoringtheelectronfluencebelow7.5keV
changeds byonly0.1%.
w,med
Evaluationofthestoppingpowerratioinacorticalbonephantomandlungphantom(using
theelectronenergyspectraproducedinthesematerials)revealedresultssimilartothewater
phantomresults. Valuesat10cmdepthinthesephantommaterialsaresummarizedintable2.
Fortypicalpatientmedia, s isinsensitivetovariationsintheelectronspectraproduced
w,med
byphotontransportinthatmaterial.
Otherlocationswhere(8 ) ands mightbeexpectedtovaryareatmaterialinterfaces
E m w,med
andoutsidethetreatmentfield. Table2alsosummarizesresultsfromtheevaluationofs
w,med
outside the treatment field area and at lung–bone and bone–lung interfaces. For the patient
materials(neglectingair),s isquiteconstantfortheconditionsstudied. Themaximum
w,med
deviationof1.1%seenforcorticalboneisduethefactthattheoutsidethefield,theelectron
Convertingabsorbeddosetomediumtoabsorbeddosetowater 989
(a)
(b)
Figure 3. Water-to-medium stopping power ratios as a function of depth in the phantom for
(a)6MVphotonsand(b)18MVphotons.Thephantommaterialforthiscaseiswater.
energyspectrumisshiftedtolowerenergiesduetoapredominanceofscatteredphotons. The
bone s changes more than the other patient materials due to its higher effective atomic
w,med
number. Theseresults,combinedwiththepreviousresults,showthatforpatientmedia,s
w,med
990 JVSiebersetal
Table2. Averagewater-to-mediumstoppingpowerratioscalculatedforthevarietyofconditions
andphantomsstudied. Theincidentbeamwas10 10cm2atthephantomsurface. Thein-field
resultsevaluatethedoseratiowithinthecentral4 ⇥4cm2areaofthisfieldwhiletheoutside-field
resultsevaluatetheratiooutsidea12 12cm2⇥area. Thein-fieldandoutside-fieldresultsare
⇥
evaluatedatadepthof10cmdepth,whiletheothersareatthedepthoftheinterface.
ICRU Soft Cortical
Location Air Lung tissue bone bone
6MV
Infield(water) 1.117 0.999 1.010 1.035 1.116
Infield(bone) 1.117 1.000 1.010 1.035 1.117
Lung–bone 1.120 1.000 1.010 1.035 1.117
Bone–lung 1.118 1.001 1.010 1.035 1.116
Outsidefield 1.132 1.006 1.010 1.035 1.127
18MV
Infield(water) 1.085 0.985 1.010 1.035 1.110
Infield(bone) 1.086 0.986 1.010 1.035 1.111
Lung–bone 1.092 0.989 1.010 1.035 1.111
Bone–lung 1.086 0.987 1.010 1.035 1.110
Outsidefield 1.117 0.999 1.010 1.035 1.112
Table3.Calculatedwater-to-mediumstoppingpowerratiosat10cmdepthforthebeamenergies
specified. AlsoincludedistheSpencer–Attix(sSA )stoppingpowerratiocomputedforeachof
w,air
thesespectrabyAndreoandBrahme(1986).
Air
(Andreo) ICRU Soft Cortical
Beamenergy (sSA ) Air Lung tissue bone bone
w,air
60Co 1.133 1.131 1.008 1.010 1.034 1.119
4MV 1.126 1.123 1.003 1.010 1.034 1.117
6MV 1.120 1.117 0.999 1.010 1.035 1.116
10MV 1.108 1.103 0.992 1.010 1.035 1.113
15MV 1.097 1.090 0.988 1.010 1.035 1.111
24MV 1.078 1.071 0.981 1.010 1.035 1.108
isaratherinsensitiveparameter. Thus,undertheassumptionthatBragg–Graycavitytheory
isvalid,forphotonbeamsasinglecorrectionfactorcanbeusedforeachmediumforagiven
photonenergywhenconvertingabsorbeddosetomediumtoabsorbeddosetowater. Inother
words,forphotonbeams,avoxel-by-voxelevaluationofthedosetomediumanddosetowater
isnotrequired.
Water-to-mediumstoppingpowerratiosevaluatedusingpre-computedpublishedenergy
spectraaresummarizedintable3. Theresultsat6MVagreewiththevaluesobtainedusingthe
fullMonteCarlosimulation(table2),andthe15and24MVresultsbracketthe18MVresults
fromthefullsimulation,confirmingthatthegeometricapproximationsmadeforthissimulation
are negligible. ICRU tissue and soft-bone s values do not depend upon the photon
w,med
energyfrom60Coto24MV,whilelungandcorticalbonevaluesdecreaseby2.7%and1.1%
respectively. Theairs change(from1.131to1.071)issignificant, andagreeswiththe
w,med
Spencer–AttixstoppingpowerratiosreportedbyAndreoandBrahme(1986)towithin0.7%.
Evaluations of s at the minimal, nominal and maximum material densities (from
w,med
table1)foragivenmaterialforan18MVbeamareshowninfigure4. Notethatthedifference
between lung and ICRU tissue stopping power ratios is almost entirely due to the material
density. Thestoppingpowerratiodeviatesfromthatevaluatedatthenominaldensitybyupto
Convertingabsorbeddosetomediumtoabsorbeddosetowater 991
Figure4. Thewater-to-mediumstoppingpowerratioscomputedusingstoppingpowersatthe
physicalmaterialdensity, andalsoatthelimitsofthedensityrangesshownintable1, forthe
18MVbeam.
Table4.Effectiveenergyforlookingupthestoppingpowerratioforeachmaterialat10cmdepth.
Effectiveenergyisdefinedastheenergyatwhichthemono-energeticstoppingpowerratioequals
thatobtainedusingtheenergyspectra. Foreachmaterial,thedeviationfromtheMCdetermined
stoppingpowerratioisalsogiven.
Nominal ICRU Soft Cortical
energy Eeff Air Lung tissue bone bone
60Co 0.098 0.2% 0.1 0 0.1 0
� �
4MV 0.731 0 0.1 0 0.3 0.1
� �
6MV 0.905 0 0 0 0.2 0
�
10MV 1.245 0.3% +0.3 0 0.3 0
� �
15MV 1.578 0.4% +0.6 0 0.3 0
� �
24MV 2.364 0.3% +0.9 0 0.2 0
� �
0.6%forICRUtissue,0.3%forlung,0.2%forsoftboneand0.5%forcorticalboneat6MV,
whileat18MVthedifferencesare1.3%forICRUtissue,1.0%forlung,0.4%forsoftbone
and 0.8% for cortical bone. Thus, for Monte Carlo codes that evaluate the stopping power
at the local density of each voxel, the use of a single s for all densities of the material
w,med
introduces a maximum error of 1.3%. For Monte Carlo codes that use stopping power data
fromanominaldensityatalldensitiesofagivenmedium,s isproperlyevaluatedatthe
w,med
nominaldensity.
Theeffectiveenergies(E )forlook-upofs foreachpre-computedenergyspectra
eff w,med
arelistedintable4. Usingthistable, s valuesformaterialsnotlistedinthispapercan
w,med
becomputedbylookingupthemono-energetics attheE forthenominalacceleration
w,med eff
potentialslisted. Themaximumerrorobservedusingthismethodtodetermines isless
w,med