VARIAN MEDICAL SYSTEMS CLINICAL PERSPECTIVES ACUROS XB Acuros XB advancd dos calculation for th Eclips tratmnt planning systm Grgory A. Failla 1, Todd Waring 1, Yvs Archambault 2, Stphn Thompson 2 1 Transpir Inc., Gig Harbor, Washington 2 Varian Mdical Systms, Palo Alto, California
TABLE OF CONTENTS Introduction 1 Background 2 Acuros XB in Eclips Sourc Modl 3 Acuros XB in Eclips Patint Transport and Dos Calculation 4 Comparison with Mont Carlo 7 Dos-to-watr and dos-to-mdium Elctron cutoff nrgy Implmntation diffrncs Acuros XB Calculation Options 9 Acuros XB Validation Exampls 10 Acuros XB Calculation Tims 16 Conclusion 18 Rfrncs 18 Appndix 19 Acuros XB solution mthods patint transport Matrial spcification Th LBTE Stp 1: Transport of sourc modl flunc into th patint Stp 2: Transport of scattrd photon flunc in th patint Stp 3: Transport of scattrd lctron flunc in th patint Discrtization mthods Stp 4: Dos calculation
Introduction Th Acuros XB advancd dos calculation (Acuros XB) algorithm was dvlopd to addrss two stratgic nds of xtrnal photon bam tratmnt planning: accuracy and spd. In xtrnal photon bam radiothrapy, htrognitis introducd by matrials such as lung, bon, air, and non-biologic implants may significantly affct patint dos filds, spcially in th prsnc of small or irrgular filds. Acuros XB uss a sophisticatd tchniqu to solv th Linar Boltzmann transport quation (LBTE) and dirctly accounts for th ffcts of ths htrognitis in patint dos calculations. Acuros XB provids comparabl accuracy to Mont Carlo mthods in tratmnt planning for th full rang of X-ray bams producd by clinical linar acclrators, 4 MV 25 MV with xcptional calculation spd and without statistical nois. Additionally, Acuros XB calculations ar minimally snsitiv to th numbr of filds in a plan such that calculation of th dos in a RapidArc radiothrapy tchnology plan is almost as fast as for a singl fild. Th ffct of which is that whil singl fild dos calculations ar somwhat slowr than with Eclips Analytical Anisotropic Algorithm (AAA), Acuros XB is significantly fastr for RapidArc. Acuros XB is fully intgratd into th Eclips distributd calculation framwork (DCF) as a nw dos calculation algorithm and uss th multipl-sourc modl originally drivd for AAA. Thrfor, th usr will also apprciat that AAA bam data can b importd in th Acuros XB bam modl and only rquirs rconfiguration bfor it is rady to b usd for dos calculations. 1 V A R I A N M E D I C A L S Y S T E M S
Background Th Boltzmann transport quation (BTE) is th govrning quation which dscribs th macroscopic bhavior of radiation particls (nutrons, photons, lctrons, tc.) as thy travl through and intract with mattr. Th LBTE is th linarizd form of th BTE, which assums that radiation particls only intract with th mattr thy ar passing through, and not with ach othr, and is valid for conditions without xtrnal magntic filds. For a givn volumtric domain of mattr, subjct to a radiation sourc, undr th abov conditions th solution to th LBTE would giv an xact dscription of th dos within th domain. Howvr, sinc closd form solutions (analytic solutions) to th LBTE can only b obtaind for a fw simplifid problms, th LBTE must b solvd in an opn form, or non-analytic, mannr. Thr ar two gnral approachs to obtaining opn form solutions to th LBTE. Th first approach is th widly known Mont Carlo mthod. Mont Carlo mthods do not xplicitly solv th LBTE; thy indirctly obtain th solution to this quation. Th scond approach is to xplicitly solv th LBTE using numrical mthods [rf. 1]. Mthods usd to xplicitly solv th LBTE quation, such as thos of Acuros XB, ar rlativly nw to th mdical physics community. Both Mont Carlo and xplicit LBTE solution mthods such as Acuros XB ar convrgnt. That is, with sufficint rfinmnt both approachs will convrg on th sam solution of th LBTE. Th achivabl accuracy of both approachs is quivalnt and is limitd only by uncrtaintis in th particl intraction data and uncrtaintis in th problm bing analyzd. In practic, nithr Mont Carlo nor xplicit LBTE solution mthods ar xact, and both mthods produc rrors. In Mont Carlo, rrors ar random and rsult from simulating a finit numbr of particls and following ach particl as it intracts with a mdium. Whn Mont Carlo mthods mploy tchniqus to acclrat solution tims or rduc nois (dnoising), systmatic rrors may b introducd. In th xplicit LBTE solution mthods, rrors ar primarily systmatic and rsult from discrtization of th variabls in spac, angl, and nrgy. Largr stps in th discrtization procss rsult in a fastr solution, but lss accuracy. In both mthods, a trad-off xists btwn spd and accuracy. Diffrncs btwn th two mthods may also rsult from th tratmnt of chargd particl Coulomb intractions. Modl- or corrction-basd algorithms such as pncil bam or collapsd con convolution ar only convrgnt undr th xact conditions in which thir dos krnls ar gnratd. Th imptus bhind th dvlopmnt of xplicit LBTE solution mthods was to provid a rapid altrnativ to Mont Carlo simulations, which ar known to b tim intnsiv. A scond bnfit of LBTE is th absnc of statistical nois. Many of th mthods containd within Acuros XB wr originally dvlopd in a prototyp solvr calld Attila, which was co-authord by th foundrs of Transpir, Inc. whil at Los Alamos National Laboratory [rf. 2, 3]. Th dvlopmnt of th Acuros XB xtrnal photon bam prototyp was fundd in part through an SBIR Phas II Grant from th National Cancr Institut. 2 V A R I A N M E D I C A L S Y S T E M S
Acuros XB in Eclips Sourc Modl Acuros XB in Eclips lvrags th xisting AAA machin sourc modl. This modl consists of four componnts: Primary sourc usr-dfind circular or lliptical sourc locatd at th targt plan which modls th brmsstrahlung photons cratd in th targt that do not intract in th tratmnt had. Extra focal sourc Gaussian plan sourc locatd at th bottom of th flattning filtr, which modls th photons that rsult from intractions in th acclrator had outsid th targt (primary in th flattning filtr, primary collimators, and scondary jaws). Elctron contamination rprsnts th dos dpositd in th build-up rgion not accountd for by th primary and xtra-focal sourc componnts. Photons scattrd from wdg rprsnts th scattr from hard wdgs, whr prsnt. Implmntd with a dual Gaussian modl, whr th width of th Gaussian krnl incrass with distanc from th wdg. A dtaild dscription rgarding ths sourcs can b found in th papr on AAA photon dos calculation by Sivinn t al. [rf. 8]. 3 V A R I A N M E D I C A L S Y S T E M S
Acuros XB in Eclips Patint Transport and Dos Calculation (Th Acuros XB solution mthods ar brifly dscribd hr, with a dtaild ovrviw providd in th Appndix.) Th Acuros XB patint transport consists of four discrt stps, which ar prformd in th following ordr: 1. Transport of sourc modl flunc into th patint 2. Calculation of scattrd photon flunc in th patint 3. Calculation of scattrd lctron flunc in th patint 4. Dos calculation Stps 1 through 3 ar prformd to calculat th lctron flunc in vry voxl of th patint. Onc th nrgy dpndnt lctron flunc is solvd, th dsird dos quantity (dos-to-mdium or dos-to-watr) is computd in Stp 4. Stp 1 is th only stp rpatd for ach bam, and Stps 2 through 4 ar prformd onc, rgardlss of th numbr of bams. In th cas of RapidArc, ach bam will hav a larg numbr of orintations, and Stp 1 is rpatd at ach orintation and Stps 2 through 4 ar prformd just onc. In Stp 1, th machin sourcs ar modld as xtrnal sourcs and ray tracing is prformd to calculat th uncollidd photon and lctron flunc distributions in th patint. In Stps 2 and 3, Acuros XB discrtizs in spac, angl, and nrgy, and itrativly solvs th LBTE. In Stp 4, th dos in any voxl of th problm is obtaind through applying an nrgy dpndnt flunc-to-dos rspons function to th local nrgy dpndnt lctron flunc in that voxl. Acuros XB supports two dos rporting options: dos-to-watr (D W ) and dos-to-mdium (D M ). Whn D M is calculatd, th nrgy dpndnt rspons function is basd on th matrial proprtis of that voxl. Whn D W is calculatd, th nrgy dpndnt flunc-to-dos rspons function is basd on watr. Thrfor, to calculat dos, Acuros XB must hav a matrial map of th imagd patint. Unlik convolution/suprposition algorithms, whr htrognitis ar gnrally handld as dnsity basd corrctions applid to dos krnls calculatd in watr, Acuros XB xplicitly modls th physical intraction of radiation with mattr. To do this accuratly, Acuros XB rquirs th chmical composition of ach matrial in which particls ar transportd through, not only th dnsity. To nabl this, Eclips provids Acuros XB with a mass dnsity and matrial typ in ach voxl of th imag grid. Th Acuros XB matrial library includs fiv biologic matrials (lung, adipos tissu, muscl, cartilag, and bon) and 16 nonbiologic matrials, with a maximum supportd dnsity of 8.0 g/cc (stl). 4 V A R I A N M E D I C A L S Y S T E M S
In Figur 1, an illustration of th diffrncs btwn D W and D M ar prsntd for a 5 x 5 cm 2 6 MV fild on a watr-bon-lung slab phantom. Also shown ar rsults for scald watr dnsity, in which th ntir phantom was assignd watr matrial, with varying dnsity according to th rgion. As shown, D W and D M ar idntical in watr voxls upstram of th bon, and ar narly idntical in th lung downstram of th bon. This is xpctd sinc in both cass, th lctron transport fild is idntical, and only th lctron nrgy dposition intraction is diffrnt. Howvr, for scald watr dnsity thr ar significant diffrncs in th build-up rgion bfor th bon, in th bon, and in th lung downstram of th bon. Ths diffrncs highlight th significanc of using th actual matrial composition as opposd to scaling th dnsity of watr matrial. Figur 1. Acuros XB dpth dos comparison btwn diffrnt dos rporting mods for a 5 x 5 cm 2 6 MV fild on a watr-bon-lung slab phantom. For scald watr dnsity, th ntir phantom consistd of watr matrial, but with th dnsity scald in ach rgion (1.85 g/cc in bon rgion, and 0.26 g/cc in lung). 5 V A R I A N M E D I C A L S Y S T E M S
In Figur 2, diffrncs btwn D W and D M ar prsntd for a 5 x 5 cm 2 18 MV fild for th biologic matrials in Acuros XB. Figur 2. Acuros XB dpth dos curvs for dos-to-watr (D W ) and dos-to-mdium (D M ) for a 5x5 cm 2 18 MV bam on a slab phantom containing: watr (1.0 g/cc), cartilag (1.1 g/cc), bon (1.85 g/cc), lung (0.26 g/cc), adipos tissu (0.92 g/cc), and muscl (1.05 g/cc) 6 V A R I A N M E D I C A L S Y S T E M S
Comparison with Mont Carlo Sinc Mont Carlo mthods ar wll known in th radiothrapy community, a usful way to undrstand th mthods in Acuros XB is to highlight whr and why diffrncs btwn Acuros XB and Mont Carlo can occur, which ar discussd blow. Dos-to-watr and dos-to-mdium Both Acuros XB and Mont Carlo mthods calculat D M basd on nrgy dposition, and as shown in th includd figurs, produc vry similar rsults. Howvr, whn calculating D W in non-watr matrials, Acuros XB and Mont Carlo mthods mploy diffrnt approachs. Acuros XB calculats th nrgy dpndnt lctron flunc using th matrial compositions of th patint, rgardlss of whthr D W or D M is slctd. Whn D W is slctd, in non-watr matrials this is analogous to calculating th dos rcivd by a volum of watr which is small nough to not significantly prturb th nrgy dpndnt lctron flunc. Du to th vry short rang of low nrgy lctrons, this volum may b much smallr than ithr th dos grid voxl siz or dtctors usd to xprimntally masur D W. This ffct is most significant for bon and non-biologic, high dnsity matrials such as aluminum, titanium, and stl. In such cass, whn comparing Acuros XB to xprimntal masurmnts of D W, it is rcommndd to xplicitly modl a small watr volum rprsnting th dtctor in Acuros XB. Mont Carlo mthods will gnrally calculat D M, and mploy stopping powr ratios to convrt D M to D W [rf. 5]. To illustrat th xpctd diffrncs btwn th approachs of Acuros XB and Mont Carlo in calculating D W, Figur 3 shows a comparison btwn nrgy dposition ratios (watr/mdium) [rf. 4] and collisional stopping powr ratios (watr/mdium) [rf. 6] in diffrnt biologic matrials as a function of lctron nrgy. Th nrgy dposition ratios (Figur 3 - lft) show th ratio of D W /D M which would b calculatd by Acuros XB, and th collisional stopping powr ratios (Figur 3 - right) show th ratio of D W /D M which would b calculatd by Mont Carlo mthods. Figur 3. (lft) Enrgy dposition ratios (watr/mdium) and (right) collisional stopping powr ratios (watr/mdium), as a function of lctron nrgy (MV). Th ratio btwn dos-to-watr and dos-to-mdium in Acuros XB is rflctd in th nrgy dposition ratios; and for Mont Carlo in th collisional stopping powr ratios. 7 V A R I A N M E D I C A L S Y S T E M S
Although Acuros XB and Mont Carlo us diffrnt mthods, calculating D W in a non-watr mdium is a thortical quantity, and thrfor nithr approach is corrct or incorrct. Elctron cutoff nrgy Acuros XB mploys an lctron cutoff of 500 kv (kintic nrgy, not including lctron rst mass nrgy). Elctrons passing blow this nrgy ar assumd to dump all of thir nrgy in th voxl in which thy ar locatd. Whn comparing Acuros XB and Mont Carlo in voxls containing vry low dnsity lung or air, th choic of lctron cutoff nrgy may rsult in diffrncs btwn th two solvrs. Howvr, such diffrncs will gnrally b isolatd to th low dnsity voxls, and will not significantly influnc th dos in adjacnt tissu. Implmntation diffrncs As discussd arlir, Acuros XB and Mont Carlo mthods ar uniqu in radiothrapy in that both mthods xplicitly solv for th lctron flunc, without th us of pr-calculatd dos krnls. As mntiond arlir, nithr mthod is xact and in practic diffrncs will occur. A simpl way to undrstand th diffrnt approachs btwn Acuros XB and Mont Carlo is as follows: Analog Mont Carlo mthods simulat a finit numbr of particls, and stochastic rrors rsult from a finit numbr of particls bing trackd. Acuros XB simulats an infinit numbr of particls, and systmatic rrors ar introducd by discrtization in spac, angl, and nrgy. In Acuros XB, th discrtization sttings ar spcifid intrnally to provid an optimal balanc of spd and accuracy for patint tratmnt planning conditions. This is analogous to a Mont Carlo cod which intrnally sts a limit on th statistical uncrtainty. 8 V A R I A N M E D I C A L S Y S T E M S
Acuros XB Calculation Options Th Acuros XB implmntation in Eclips is vry similar to that of AAA. A fw ky points rlatd to th Acuros XB implmntation ar summarizd blow, and svral diffrncs with AAA ar highlightd. Calculation grid voxl siz: Th Acuros XB calculation grid voxl siz can rang from 1 to 3 mm. AAA currntly supports a voxl siz rang btwn 1 and 5 mm. Dos rporting mod: In Acuros XB, D M or D W can b slctd as dos rporting options. This concpt dos not xist in AAA. Plan dos calculation: This is a uniqu option for Acuros XB. In Acuros XB, th calculation tim has a vry wak dpndnc on th numbr of filds, sinc th majority of th calculation tim is spnt calculating th scattrd photon and lctron flunc, which is prformd onc for all filds in th plan. Whn a sparat Acuros XB calculation is prformd for ach fild, th scattr calculation phas has to run for vry fild, which significantly incrass th calculation tim. Sinc fild wights cannot b ditd whn plan dos calculation is slctd, this option is wll suitd for rapidly calculating intnsity-modulatd radiation thrapy (IMRT) and RapidArc plans. Howvr, in 3D conformal planning whr fild wights may b individually changd during optimization, plan dos calculation would gnrally b turnd off. Matrial spcification: Matrial dtrmination is don in two ways for Acuros XB. Th dfault mthod usd to dtrmin th matrial composition of a givn voxl in a 3D imag is basd on th HU valu. Th HU valu in th voxl is convrtd to mass dnsity using th CT calibration curv. This curv can b configurd by th usrs for thir spcific CT scannr. Onc mass dnsity is known in a voxl, th matrial is dtrmind basd on a hard codd look up tabl stord in th Varian systm databas. This automatic convrsion is usd for all voxls with mass dnsity blow 3.0g/cc. Any voxl with dnsity highr than 3.0g/cc rquirs usr assignmnt. Furthrmor, th automatic matrial assignmnt only assigns biological matrials to voxls. Basd on thir mass dnsity, voxls will b assignd lung, adipos tissu, muscl, cartilag, or bon. Evn vry low dnsity rgions ar automatically assignd a matrial, ithr lung or air. Usrs hav th option to manually ovrrid th automatic matrial assignmnt. Configuration: Sinc Acuros XB uss th sam sourc modl as AAA, no additional bam data is ndd and th AAA configurd data can b importd into th Acuros XB modl dirctly. AAA bam data importd in Acuros XB will nd to b rconfigurd and all configuration stps will nd to b run again to optimiz th sourc modl for Acuros XB. Th prconfigurd bam data availabl for AAA is also availabl for Acuros XB. For vry DCF vrsion, prconfigurd bam data for AAA and Acuros XB is availabl. 9 V A R I A N M E D I C A L S Y S T E M S
Acuros XB Validation Exampls A brif sampling of Acuros XB validation cass with htrognitis ar providd blow. Not that in ordr to fully validat Acuros XB against Mont Carlo N-Particl Xtndd (MCNPX), th MCNPX computations wr run with a vry larg numbr of particls to crat rsults that wr vry smooth and without statistical uncrtaintis that may hav influncd th validations of Acuros XB. In practic typical Mont Carlo rsults ar much lss smooth and statistical uncrtaintis ar clarly visibl. Additional validation rsults can b found in th litratur [rf. 7]. Th Acuros XB matrial library includs 13 non-biologic matrials. Figur 4 and Figur 5 compar D M rsults from Acuros XB and MCNPX on a slab phantom containing 12 of th 13 non-biologic matrials for 6 MV and 20 MV. Figur 4. Dpth dos comparison (dos-to-mdium) btwn Acuros XB and MCNPX for a 6X 10 x 10 cm 2 fild on a multi-matrial slab phantom. Slab matrials ar as follows: (1) Polystyrn 1.05 g/cc, (2) Epoxy 1.04 g/cc, (3) Aluminum 2.7 g/cc, (4) PMMA 1.19 g/cc, (5) Titanium alloy 4.42 g/cc, (6) Radl 1.30 g/cc, (7) Wood 0.70 g/cc, (8) PEEK 1.31 g/cc, (9) PVC 1.38 g/cc, (10) Actal 1.42 g/cc, (11) PVDF 1.77 g/cc, (12) PTFE 2.20 g/cc. 1 0 V A R I A N M E D I C A L S Y S T E M S
Figur 5. Dpth dos comparison (dos-to-mdium) btwn Acuros XB and MCNPX for a 20X 10x10 cm 2 fild on a multi-matrial slab phantom. Slab matrials th sam as in Figur 4. Th highst dnsity matrial supportd in Acuros XB is stainlss stl, with a maximum dnsity of 8.0 g/cc. Figurs 6 through 8 prsnt an xtrm cas whr a 2 x 2 x 2 cm 3 stl implant (8.0 g/cc) is placd in a watr phantom insid an 18 MV 10 x 10 cm 2 fild. As shown, both cods ar in clos agrmnt, vn in th high gradint lctron disquilibrium rgions surrounding th implant. Figur 6. Phantom containing a 2 x 2 x 2 cm 3 8.0 g/cc stl implant stl implant usd in Figurs 7 and 8. Acuros XB dos contours shown (dos-to-mdium) for an18 MV 10 x 10 cm 2 fild. 1 1 V A R I A N M E D I C A L S Y S T E M S
Figur 7. Dpth dos comparison btwn Acuros XB and MCNPX for an 18 MV 10 x 10 cm 2 fild impinging on th stl insrt phantom shown in Figur 1. Dos-to-mdium shown in both cods with dos normalizd to 100% at dpth of 4.875 cm. Figur 8. Latral dpth dos comparison (dpth of 4.875 cm) btwn Acuros XB and MCNPX for an 18 MV 10 x 10 cm 2 fild impinging on th stl insrt phantom shown in Figur 1. Dos-to-mdium shown in both cods with dos filds normalizd to 100% at cntrlin dpth of 4.875 cm. 1 2 V A R I A N M E D I C A L S Y S T E M S
Figurs 9 through 11 prsnt comparisons btwn Acuros XB and Mont Carlo for a half cork (0.19 g/cc) phantom for 5 x 5 cm 2 filds and 6 MV and 15 MV bam nrgis. Figur 9. Phantom containing a half cork slab (.193 g/cc) usd in Figurs 10 through 13. Acuros XB dos contours shown (dos-to-mdium) for a 6 MV 5x 5 cm 2 fild. Figur 10. Dpth dos comparison btwn Acuros XB and MCNPX for a 5 x 5 cm 2 15 MV fild on th half cork slab phantom shown in Figur 9. Dpth dos lin locatd 1.125 cm OAX on th cork sid. Dos-to-mdium shown in both cods with dos normalizd to 100% at dpth of 4 cm. 1 3 V A R I A N M E D I C A L S Y S T E M S
Figur 11. Latral dos comparison btwn Acuros XB and MCNPX for cas shown in Figur 10, at dpths of 4.625, 17.875, and 21.125 cm. Figur 12 prsnts comparisons btwn Acuros XB and Mont Carlo for 2 x 2 cm 2 6 MV filds on a watr phantom containing a 2 x 2 x 10 cm 3 block of air, which simulats an sophagus. Figur 12. Dpth dos comparison btwn Acuros XB and MCNPX for a 2 x 2 cm 2 6 MV fild on a phantom containing a 2 x 2 x 10 cm 3 air block rprsnting an sophagus. Elctron nrgy cutoff for both Mont Carlo and Acuros XB is 500 kv. 1 4 V A R I A N M E D I C A L S Y S T E M S
Figur 13 shows th rsult of an Acuros XB RapidArc dos comparison with th Radiological Physics Cntr (RPC) had and nck phantom. TLD masurmnts ar within 2% of calculatd dos for th 3 mm x 3 mm calculation grid siz. Figur 13. Acuros XB RapidArc plan for Radiological Physics Cntr (RPC) Had and Nck Phantom (figur courtsy of Firas Mourtada, Ph.D., UT MD Andrson Cancr Cntr). Acuros XB, Masurd dos (cgy) Htrognity on ; TLD position Dos to mdium Tratmnt 1 Tratmnt 2 Tratmnt 3 Avrag Calculatd % Error dos (cgy) TLD_54_I 621.4 621.9 621.6 621.6 593.3 4.56% TLD_54_S 591.2 603.5 608 600.9 591 1.65% TLD_66_Iant 745.3 742.4 735.5 741.1 734.6 0.87% TLD_66_Ipost 735.3 744 751.4 743.6 739.9 0.49% TLD_66_Sant 723.9 736.9 736.5 732.4 726.2 0.85% TLD_66_Spost 728 733.7 738.4 733.4 726.4 0.95% TLD_CORD_I 355.3 360.9 362.1 359.4 349.7 2.71% TLD_CORD_S 357.3 357.2 357.6 357.4 346.4 3.07% Avragd prcntag rror (%) 1.89% Tabl 1. Masurd (TLD) vs calculatd doss for RapidArc plan shown in Figur 13 abov. Masurmnt is within 2% of calculation avragd ovr all TLD locations (courtsy of Firas Mourtada, Ph.D., UT MD Andrson Cancr Cntr). 1 5 V A R I A N M E D I C A L S Y S T E M S
Acuros XB Calculation Tims Calculations of a singl or fw filds ar longr with Acuros XB than AAA. For a 10 x 10 cm 2 6 MV fild on a 30 x 30 x 30 cm 3 watr phantom, Acuros XB will calculat th dos on a 2.5 mm voxl grid in about 85 sconds (Dll T5500 with dual quadcor Xon 2.27 GHz procssors and 24 GB DDR3 RAM). AAA will rquir approximatly 10 sconds for a similar cas. For a 5 x 5 cm 2 fild on th sam phantom, Acuros XB will rquir about 40 sconds. Largr filds and highr nrgis tak longr to calculat, as do phantoms containing larg amounts of bon. Most of th Acuros XB calculation tim is in solving for th scattrd photon and lctron fluncis, which ar prformd only onc for all bams in th plan. As a rsult, Acuros XB calculation tims scal vry wakly with th numbr of filds. Howvr, AAA calculation tims scal linarly with th numbr of filds. As a rsult, th rlativ calculation spd of Acuros XB incrass with incrasing numbrs of filds in th plan. For cass with largr numbrs of filds, i.., RapidArc, Acuros XB xploits spatial adaption to spd-up calculations in low dos, low gradint rgions. Acuros XB bcoms significantly fastr than AAA for cass with a larg numbr of filds, i.., RapidArc. As an xampl, th calculation tims for svral RapidArc cass ar providd in Tabl 2 blow, with scrnshots of th Acuros XB dos calculation for th lung and had/nck cas shown in Figurs 14 and 15. Cas Acuros XB AAA Improvmnt ovr AAA Lung (57 control points, Fig. 14) 1 min 26 s 3 min 35 s 2.5x Lung (114 control points, Fig. 14) 2 min 7 s 6 min 15 s 3.0x Had & Nck (89 control points, Fig. 15) 2 min 43 s 8 min 23 s 3.1x Had & Nck (178 control points, Fig. 15) 4 min 13 s 16 min 12 s 3.8x Prostat (89 control points, not shown) 2 min 7 s 5 min 46 s 2.7x Prostat (178 control points, not shown) 3 min 8 s 11 min 21 s 3.6x Tabl 2. Acuros XB and AAA calculation tims shown for rprsntativ RapidArc cass. All tims shown on a Dll T5500 (Hypr-thrading off for Acuros XB) with 2.5 mm voxl grids. Calculation tims includ both sourc modl and patint transport componnts. 1 6 V A R I A N M E D I C A L S Y S T E M S
Figur 14. Acuros XB dos fild (dos-to-mdium) for a 6 MV RapidArc lung cas. Total dos calculation tim, including sourc modl and patint transport, on a 2.5 mm voxl grid: 86 sconds (4 dgr sparation - 57 control points). Figur 15. Acuros XB dos fild (dos-to-mdium) from a 6 MV RapidArc had and nck cas. Total dos calculation tim, including sourc modl and patint transport, on a 2.5 mm voxl grid: 163 sconds (4 dgr sparation - 89 control points). It should b notd that Acuros XB is considrably fastr on th Varian Eclips Dll T5500 machins than on th Dll T5400 machins, vn with similar clock spds. Th limitd mmory bandwidth of th DDR2 mmory on T5400 machins prvnts Acuros XB from ffctivly scaling on all availabl cors (AAA dos not xhibit this bhavior). Howvr, this bottlnck is rmovd with th DDR3 mmory on T5500 machins, allowing Acuros XB to scal much mor fficintly on availabl cors. This rsults in almost a factor of two spd-up on th T5500 (compard with th T5400) for som cass. Additionally, it is rcommndd to run Acuros XB with hyprthrading turnd off, as hyprthrading may also dgrad prformanc. 1 7 V A R I A N M E D I C A L S Y S T E M S
Conclusion Th Acuros XB advancd dos calculation algorithm was dvlopd and implmntd in Eclips to addrss th accuracy and spd rquirmnt for modrn tchniqus in radiation thrapy including IMRT and RapidArc. Acuros XB provids comparabl accuracy in tratmnt planning conditions to bnchmarkd Mont Carlo mthods for th full rang of X-ray bams producd by clinical linar acclrators, 4 MV 25 MV. Validation has bn prformd to assur dos calculation accuracy in typical and challnging phantom and patint gomtris with xcllnt rsults. Rfrncs 1. Lwis EE, Millr WF, Computational mthods of nutron transport, Wily, Nw York, 1984. 2. Waring TA, McGh JM, Morl JE, Pautz SD, Discontinuous Finit Elmnt Sn Mthods on Thr-Dimnsional Unstructurd Grids, Nucl. Sci. Engr., Volum 138, Numbr 2, July 2001. 3. Waring TA, Morl JE, McGh JM, Coupld Elctron-Photon Transport Mthods on 3-D Unstructurd Grids, Trans Am. Nucl. Soc., Washington D.C., Vol 83, 2000. 4. Lornc L, Morl J, and Valdz G, "Physics Guid to CEPXS: A Multigroup Coupld Elctron-Photon Cross Sction Gnrating Cod," SAND89-1685, Sandia National Laboratory, 1989. 5. Sibrs JV, Kall PJ, Nahum AE, and Mohan R, Convrting absorbd dos to mdium to absorbd dos to watr for Mont Carlo basd photon bam dos calculations, Phys. Md. Biol. 45 (2000) 983-995. 6. http://www.nist.gov/physlab/data/star/indx.cfm 7. Vassiliv ON, Waring TA, McGh J, Failla G, Validation of a nw gridbasd Boltzmann quation solvr for dos calculation in radiothrapy with photon bams, Phys. Md. Biol. 55(3) 2010. http://iopscinc.iop.org/0031-9155/55/3/002/ 8. Sivinn J, Ulmr W, Kaissl W. AAA photon dos calculation modl in Eclips. Palo Alto (CA): Varian Mdical Systms; 2005. [RAD 7170B] 1 8 V A R I A N M E D I C A L S Y S T E M S
Appndix Acuros XB solution mthods patint transport Th Acuros XB patint transport consists of four discrt stps, which ar prformd in th following ordr: 1. Transport of sourc modl flunc into th patint. 2. Calculation of scattrd photon flunc in th patint. 3. Calculation of scattrd lctron flunc in th patint. 4. Dos calculation Stps 1 through 3 ar prformd to calculat th lctron flunc in vry voxl of th patint. Onc th nrgy dpndnt lctron flunc is solvd for, th dsird dos quantity (dos-to-mdium or dos-to-watr) is computd in Stp 4. Stp 1 is th only stp rpatd for ach fild orintation, and Stps 2 through 4 ar prformd onc, rgardlss of th numbr of orintations. Matrial spcification Prior to initiating Stp 1, Acuros XB must hav a matrial map of th imagd patint. Unlik AAA, whr htrognitis ar gnrally handld as dnsity-basd corrctions applid to dos krnls calculatd in watr, Acuros XB xplicitly modls th physical intraction of radiation with mattr. To do this accuratly, Acuros XB rquirs th chmical composition of ach matrial in which particls ar transportd through, not only th dnsity. To nabl this, Eclips provids Acuros XB with a mass dnsity and matrial typ in ach voxl of th imag grid. Th Acuros XB matrial library includs fiv biologic matrials (lung, adipos tissu, muscl, cartilag, and bon) and 16 non-biologic matrials, with a maximum supportd dnsity of 8.0 g/cc (stl). Th fundamntal matrial data usd by Acuros XB ar known as macroscopic atomic cross sctions. A macroscopic cross sction is th probability that a particular raction will occur pr unit path lngth of particl travl, so it has units of cm -1. Th cross sctions also dscrib th angular and nrgy bhavior probabilitis associatd with any givn intraction. Macroscopic cross sctions ar composd from two valus: th microscopic cross sction for a givn raction (gnrally givn in barns/atom = 10-24 cm 2 /atom and symbolizd by ) and th mass dnsity of th matrial (, givn in g/cm 3 ). Th xprssion for th macroscopic cross sction, whr N ρ σ = a ~ σ M σ ~ σ, is: ρ M = N a = Mass of th atom in atomic mass units (AMU) Avogadro s numbr 1 9 V A R I A N M E D I C A L S Y S T E M S
Acuros XB uss coupld photon-lctron cross sctions producd by CEPXS [rf. 4]. For photon intractions, CEPXS includs Compton scattr (also known as incohrnt scattr), th photo-lctric ffct, and pair production. CEPXS dos not account for Rayligh scattr (also known as cohrnt scattr), th ffct of which is insignificant for dos distributions at nrgis typical in photon bam radiothrapis. Th LBTE In Stps 1 through 3, Acuros XB solvs th tim-indpndnt thr-dimnsional systm of coupld Boltzmann transport quations (LBTE) shown blow (for brvity th dpndnt variabls hav bn supprssd in th quations): Eq. 1 ˆ Ψ Ω γ + σ γ γ γγ t Ψ = q + q γ, Eq. 2 Ψ + σ Ψ t E γ ( S Ψ ) = q + q + q, R whr y γ Ψ = Angular photon flunc (or flunc if not tim intgratd), Ψ E, ), as a function of position, r = ( x, y, z), nrgy, E, and dirction, Ω ˆ = ( μ, η, ζ ) Ψ = Angular lctron flunc, Ψ E, ) yy = Photon-to-photon scattring sourc, q yy q E, ), which is th photon sourc rsulting from photon intractions = Elctron-to-lctron scattring sourc, q q E, ), which is th lctron sourc rsulting from lctron intractions y = Photon-to-lctron scattring sourc, q y q E, ), which is th lctron sourc rsulting from photon intractions q y = Extranous photon sourc, q y ( E, ), for point sourc p, at position r p This sourc rprsnts all photons coming from th machin sourc modl. q = Extranous lctron sourc, q ( E, ), for point sourc p, at position r p This sourc rprsnts all lctrons coming from th machin sourc modl. y y σ t = Macroscopic photon total cross sction, σ t E), units of cm -1 σ t = Macroscopic lctron total cross sction, σ t E), units of cm -1 σ t = Macroscopic total cross sction, σ t E), units of cm -1 = Rstrictd collisional plus radiativ stopping powr, S R ( S r, E R ) 2 0 V A R I A N M E D I C A L S Y S T E M S
Th first trm on th lft hand sid of Equations 1 and 2 is th straming oprator. Th scond trm on th lft hand sid of Equations 1 and 2 is th collision or rmoval oprator. Equation 2 is th Boltzmann Fokkr-Planck transport quation, which is solvd for th lctron transport. In Equation 2, th third trm on th lft rprsnts th continuous slowing down (CSD) oprator, which accounts for Coulomb soft lctron collisions. Th right hand sid of Equations 1 and 2 includ th scattring, production, and th xtrnal sourc trms from th AAA γ sourc modul ( q and q ). Th scattring and production sourcs ar dfind by: Eq. 3 q γγ E, ) = 0 deʹ 4π dωʹσ γγ s Eʹ E, Ωʹ ˆ ) Ψ γ Eʹ, Ωʹ ˆ ) Eq. 4 q Eq. 5 q γ E, ) = E, ) = 0 0 deʹ deʹ 4π 4π γ dωʹσ Eʹ E, Ωʹ ˆ ) Ψ dωʹσ s s Eʹ E, Ωʹ ˆ ) Ψ Eʹ, Ωʹ ˆ ), whr γγ σ s = Macroscopic photon-to-photon diffrntial scattring cross sction σ γ s = Macroscopic photon-to-lctron diffrntial production cross sction σ s = Macroscopic lctron-to-lctron diffrntial scattring cross sction Th basic assumptions usd in Equations 1 and 2 ar brifly summarizd as follows: Both chargd pair production scondary particls ar assumd to b lctrons instad of on lctron and on positron. Also, th partial coupling tchniqu is assumd, whrby photons can produc lctrons, but lctrons do not produc photons. Rgarding th lattr, th nrgy from Brmsstrahlung photons is assumd to b ngligibl and is discardd. Ths assumptions hav only a minor ffct on th nrgy dposition fild, and ar similar to thos mployd in clinical Mont Carlo cods. A primary assumption of Equation 2 is that th Fokkr-Planck oprator (of which th CSD oprator is th first ordr trm), is usd for Coulomb, or soft, intractions that rsult in small nrgy losss. Catastrophic intractions that rsult in larg nrgy losss ar rprsntd with th standard Boltzmann scattring. This can b sn as th dtrministic quivalnt to lctron condnsd history modls in Mont Carlo. γ Eʹ, Ωʹ ˆ ), 2 1 V A R I A N M E D I C A L S Y S T E M S
To rprsnt th anisotropic bhavior of th diffrntial scattring and production sourcs, in a mathmatically practical mannr, th macroscopic diffrntial scattring cross sctions ar xpandd into Lgndr polynomials, P, whr = Ω ˆ l ( μ0 ) μ 0 Ωʹ. This xpansion allows th diffrntial scattring or production cross sction(s) to b xprssd as: Eq. 6 σ γγ / γ / s Eʹ E, Ωʹ ˆ ) = Σ l = 0 2l + 1 γγ / γ σ 4π s,l / Eʹ E) Pl ( μ0), Similarly, th angular flunc apparing in th scattring sourc is xpandd into sphrical harmonics momnts: Eq. 7 l ΣΣ Ψ Eʹ, Ωʹ ˆ ) = φ (, ʹ) ( ˆ l, m r E Yl, m Ωʹ), l= 0 m= l whr Y ˆ l, m ( Ω) = Sphrical harmonic functions l,m = Angular indics φl, m( r, Eʹ) = Sphrical harmonics momnts of th angular flunc, calculatd as: σ s 4π dω' Y * l, m ( Ωʹ ˆ ) Ψ( r, ',E), whr * dnots th complx conjugat = Macroscopic lctron-to-lctron diffrntial scattring cross sction Equations 6 and 7 ar xact. Additionally, for purly isotropic scattring, l = 0 is also xact. Howvr, Acuros XB sts a limit on th scattring ordr, l 7, and hnc th numbr of sphrical harmonic momnts kpt in th scattring/production sourc. Using th Lgndr addition thorm, th scattring and production sourcs bcom: Eq. 8 q γγ / γ / E, ) = 7 l ΣΣ l= 0 m= l 0 de' σ γγ / γ / s, l E' E) φ l, m E ') Y l, m ( ). 2 2 V A R I A N M E D I C A L S Y S T E M S
Stp 1: Transport of sourc modl flunc into th patint y Th xtrnal photon and lctron sourcs, q and q, ar modld as anisotropic point sourcs in Acuros XB. At ach static bam phas spac (i.. control point), a sparat point sourc xists for ach of th AAA sourcs. For th primary sourc, y th anisotropy of q is dscribd through a 2D flunc grid, in which both th particl flunc and nrgy spctra ar spatially variabl. For th xtra-focal and y wdg scattr sourcs, th anisotropy of q is dscribd through a 3D flunc grid, and th nrgy spctra is spatially constant. For th lctron contamination sourc, th anisotropy of q is dscribd through a 3D flunc grid, and th nrgy spctra is spatially constant. All point sourcs ar locatd at th targt for th rspctiv control point. ˆ r For a photon point sourc, q γ ( E, Ω) locatd at position, p, Equation 1 bcoms: Eq. 9 ˆ γ γ γ γγ γ Ω Ψ + σ Ψ = + (E, ) δ t q q ( r rp ), whr δ = Dirac-dlta function Th principl of linar suprposition may b usd to dfin th photon angular flunc as th summation of uncollidd and collidd flunc componnts, Eq. 10 Ψ γ Ψ + γ γ unc Ψ coll, whr γ Ψ unc = Uncollidd, or unscattrd, photon angular flunc. Rfrs to photons which hav not yt intractd with th patint/phantom. γ Ψ coll = Collidd, or scattrd, photon angular flunc. Rfrs to photons which wr producd or scattrd by a photon intraction in th patint/ phantom. Substituting Equation 10 into Equation 9, lads to th following quation for th uncollidd photon flunc: Eq. 11 ˆ γ γ γ γ Ω Ψ + σ Ψ = q (E, unc t unc ) δ ( r r p ), 2 3 V A R I A N M E D I C A L S Y S T E M S
γ A proprty of Equation 11 is that Ψ unc can b solvd for analytically. Doing so provids th following xprssion for th uncollidd photon angular flunc from a point sourc: Eq. 12 Ψ γ unc E, ) = δ γ ˆ τ ( r, rp ) (, ) ( ˆ ˆ q E Ω Ω Ω r, r ) 2 p 4π r r p, whr ˆΩ r, r p τ (, rp ) r r r p r r =, whr r p p and r ar th sourc and dstination points of th ray trac, rspctivly. = Th optical distanc (masurd in man-fr-paths) btwn r and r p. Equation 12 is solvd for ach primary, xtra focal, and wdg sourc in th calculation, to comput Ψ unc throughout th patint. Th lctron contaminant γ sourc is modld in a similar mannr, but with th inclusion of th CSD oprator to account for chargd particl intractions. Stp 2: Transport of scattrd photon flunc in th patint γγ Onc Equation 12 is solvd, q unc is calculatd according to Equation 8, and is γ considrd a fixd sourc in Equation 13, which is solvd to comput Ψ coll throughout th patint: Eq. 13 ˆ γ Ψ Ω coll + σ γ γ t Ψcoll γγ γγ = q coll + qunc, whr yy q unc yy q coll = First scattrd photon sourc. Rfrs to photons which ar cratd or scattrd from th first photon intraction insid th patint/phantom. = Scondary scattrd photon sourc. Rfrs to photons which ar cratd or scattrd from scondary photon intractions insid th patint/phantom. 2 4 V A R I A N M E D I C A L S Y S T E M S
Stp 3: Transport of scattrd lctron flunc in th patint y Onc Equation 13 is solvd, q coll is calculatd according to Equation 8, and is considrd a fixd sourc in Equation 14. Similarly, from th solution to Equation y 12, q unc is calculatd according to Equation 8, and is also considrd a fixd sourc in Equation 14. Equation 14 is solvd to comput Ψ throughout th patint: Eq. 14 Ψ + σ Ψ t S E R Ψ = q + q γ coll + q γ unc + q, whr y q unc y q coll = First scattrd lctron sourc. Rfrs to lctrons which ar cratd or scattrd from th first photon intraction insid th patint/phantom. = Scondary scattrd lctrons sourc. Rfrs to lctrons which ar cratd or scattrd from scondary photon intractions insid th patint/phantom. Discrtization mthods Acuros XB discrtizs in spac, angl, and nrgy to itrativly solv Equations 12 through 14, th mthods of which ar discussd blow. Spatial discrtization Th computational grid in Acuros XB consists of spatially variabl Cartsian lmnts, whr th local lmnt siz is adaptd to achiv a highr spatial rsolution insid th bam filds, with rducd rsolution in lowr dos, lowr gradint rgions outsid th bam pnumbra. Commonly rfrrd to as adaptiv msh rfinmnt (AMR), th msh is limitd to rfinmnt in factors of 2 (from on lvl to th nxt) in any dirction, allowing for localizd rfinmnt to rsolv aras of sharp gradints. Spatial discrtization is prformd through using a linar discontinuous Galrkin finit-lmnt mthod [rf. 1], providing a linar solution variation throughout ach lmnt, with discontinuitis prmittd across lmnt facs. Th first scattrd photon and first producd lctron sourcs, obtaind from solving Equation 12, ar also rprsntd as linar varying functions in ach lmnt, sinc ths sourcs ar usd for th linar discontinuous discrtization of Equations 13 and 14. To accuratly intgrat ths first scattrd sourcs, th analytic solution is computd at a dnsity insid th primary bam and pnumbras of at last 8 ray tracs pr output grid voxl. 2 5 V A R I A N M E D I C A L S Y S T E M S
Enrgy discrtization Enrgy discrtization is prformd through th standard multigroup mthod [rf. 1], which is usd in both th nrgy dpndnc of Equations 12 and 13 and th Boltzmann scattring in Equation 14. In nrgy, th nrgy drivativ of th continuous slowing down (CSD) oprator in Equation 14 is discrtizd using th linar discontinuous finit-lmnt mthod [rf. 3]. Th Acuros XB cross sction library includs 25 photon nrgy groups and 49 lctron nrgy groups, although not all groups ar usd for nrgis lowr than 20 MV. Angular discrtization For th spatial transport of th scattrd particl fild, th discrt ordinats mthod is usd to discrtiz in angl [rf. 1]. Th discrt ordinats mthod consists of rquiring Equations 13 and 14 to hold for a fixd numbr of dirctions, n. Ths discrt dirctions ar chosn from an angular quadratur st that also srvs to comput th angular intgrals in Equation 5 for th gnration of th scattring sourc. Squar-Tchbyshv lgndr quadratur sts ar usd and th quadratur ordr rangs from N=4 (32 discrt angls) to N=16 (512 discrt angls). Th angular quadratur ordr varis both by particl typ and nrgy. Highr nrgy particls hav longr man fr paths, or rangs for lctrons, and thus for ach particl typ, th angular quadratur ordr is incrasd with th particl nrgy. Spatial transport cutoff Acuros XB mploys a spatial cutoff for photon nrgis blow 1 kv and lctron nrgis blow 500 kv. Whn a particl passs blow th cutoff nrgy, any subsqunt intractions ar assumd to happn locally in that voxl. Additional rrors may also b prsnt from th intrnally st convrgnc tolrancs in Acuros XB. Ths tolrancs control how tightly th innr itrations in Acuros XB ar convrgd in nrgy group. Ths rrors will gnrally b on th ordr of 0.1% of th local dos in any voxl. 2 6 V A R I A N M E D I C A L S Y S T E M S
Stp 4: Dos calculation Onc Acuros XB solvs for th lctron angular flunc for all nrgy groups, th dos in any output grid voxl, i, of th problm is obtaind through th following: Eq. 15 D whr σ ED ρ i ˆ σ ED E) = de dω Ψ ρ( r ) 0 4π = Macroscopic lctron nrgy dposition cross sctions in units of MV/cm = Matrial dnsity in g/cm 3 E, ) Acuros XB supports two dos rporting options: dos-to-watr (D W ) and dosto-mdium (D M ). Whn D M is calculatd, σ ED and ρ ar basd on th matrial proprtis of output grid voxl, i. Whn D W is calculatd, σ ED and ρ ar basd on watr. Sinc Equation 15 is calculatd as an intrnal post procssing opration aftr th nrgy dpndnt lctron flunc is solvd, both D M and D W can b thortically obtaind from a singl transport calculation., 2 7 V A R I A N M E D I C A L S Y S T E M S
RAD 10156 2 9 V A R I A N M E D I C A L S Y S T E M S 10/10 (350)