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Four-dimensional dose calculation using deformabletetrahedral geometries for hadron therapy

Yazid Touileb

To cite this version:Yazid Touileb. Four-dimensional dose calculation using deformable tetrahedral geometries for hadrontherapy. Modeling and Simulation. Université lyon 1, 2019. English. tel-03019256

https://hal.archives-ouvertes.fr/tel-03019256

https://hal.archives-ouvertes.fr

No d’ordre NNT : xxx

THÈSE DE DOCTORAT DE L’UNIVERSITÉ DE LYONopérée au sein du

Laboratoire d’InfoRmatique en Images et Systèmes d’informationUniversité Claude Bernard Lyon 1

École Doctorale ED512InfoMaths

Spécialité de doctorat : Informatique

Soutenue publiquement le 30/09/2019, par :

Yazid Touileb

Four-dimensional dose calculation using deformabletetrahedral geometries for hadron therapy

Devant le jury composé de :

Lydia Maigne, Maître de conférences, Université Clermont Auvergne RapporteureMarc Neveu, Professeur des Universites, Université de Bourgogne RapporteurLaurence Cheze, Professeure des Universites, Université Lyon 1 ExaminatriceAlamin Mansori, Professeur des Universites, Université de Bourgogne ExaminateurJuliette Thariat, Professeure des Universites, Centre Antoine Lacassagne Examinatrice

Behzad Shariat, Professeur des Universites, Université Lyon 1 Directeur de thèseMichael Beuve, Professeur des Universites, Université Lyon 1 Co-directeur de thèseHamid Ladjal, Maître de conferences, Université Lyon 1 Co-directeur de thèse

Four-dimensional dose calculation using deformable tetrahedralgeometries for hadron therapy

A : he estuxntuzy zr eyersy nyq qzse qustruoutuzy pnterys uy respurntzry-uyqupeq zrsny xztuzy pzystu-

tutes n susyuipnyt ptnlleyse uy tnqrzy tternpy trentxeyt plnyyuys nyq qzsuxetry. Nztnoly rzr luys pnyper uy wtupt

xnyy quipultues nruse, luwe tussue qeysutues vnruntuzy nyq tte tuxzr pzsutuzy stutuys quruys Respurntuzy. All ttese pn-

rnxeters nfept tte rnyses zr prztzys zr uzys useq uy trentxeyt wtey pnssuys ttrzust qufereyt tussues nyq pny ensuly

result uy byexpepteq qzse qustruoutuzy. he preseyt wzrw pzysusts zr pnlpulntuys tte qzse qustruoutuzys zr xzvuys zr-

snys oy xenys zr Mzyte Cnrlz suxulntuzys nyq pntueyt-spepuip xzqeluys tzzls. he qzse qustruoutuzys nre pnlpulnteq

usuys n tuxe-qepeyqeyt tetrnteqrnl qeysutyxnp, qespruouys tte uyterynl nyntzxy zr tte tuxny ozqy. Aqqutuzynlly, tte

uyterynl xztuzy pny oe qespruoeq usuys eutter n ouzxeptnyupnl xzqeluys onseq zy Fuyute Elexeyt Aynlysus (FEA) zr

qerzrxnole uxnse resustrntuzy qusplnpexeyt xnp. byluwe xettzqs onseq zy tte pzyveytuzynl vzxel-onseq struptures,

tte qepzsuteq eyersy us nppuxulnteq uysuqe enpt tetrnteqrzy quruys qerzrxntuzy, ttus zverpzxuys tte przolex zr

tussue trnpwuys suype ttnt tte tetrnteqrzy us qeiyeq ns n pnrt zr n tussue wtzse ptexupnl pzxpzsutuzy nyq tzpzlzsy qz

yzt ptnyse. he irst pnrt zr tte Pt.D. przvept przpzses n qzse pnlpulntuzyxettzq ttnt seyerntes n D qzse xnp usuys

n pntueyt-spepuip tetrnteqrnl xzqel. Besuqes, we stuqy tte efept zr tte level zr qetnul zr tetrnteqrnl xestes zy tte np-

purnpy zr tte resulteq qzse qustruoutuzy. Iy tte sepzyq pnrt, we rzpus zy tte zptuxuzntuzy zr tte tetrnteqrnl sezxetry

tz nqqress tte przolex zr tuxe suxulntuzy, suype zotnuyuys n prepuse qzse qustruoutuzy pny oe very tuxe-pzysuxuys.

az zverpzxe ttus ussue, we’ve qeiyeq n yew npprznpt ttnt tnwes uytz nppzuyt tte qureptuzy zr tte oenx tz xuyuxuze

tte errzr zr tte wnter equuvnleyt ttupwyess zr tte tetrnteqrzys oerzre tte tuxzr vzluxe. hus xettzq nllzws rzr n

pznrseyeq tetrnteqrnl xest nyq ns n result, uxprzveq pzxputntuzynl perrzrxnype uy Mzyte Cnrlz suxulntuzys wtule

sunrnyteeuys n prepuse qzse qustruoutuzy uy tte tnrset vzluxe.

K ywo : Hnqrzytternpy, tetrnteqrnl xestes, vzxel-onseq struptures, qzsuxetry, Mzyte pnrlz, xzqeluys

uu

Calcul de dose D à l’aide des structures tétraédriques déformables pourl’hadronthérapie

Ré mé: L’estuxntuzy qe ln qustruoutuzy qe qzse et q’2yersue ey pr2seype quxzuvexeyt qes tussus uyquut pnr ln

respurntuzy, pzystutue uy q2i teptyzlzsuque uxpzrtnyt qnys ln plnyuipntuzy qu trnutexeyt ey tnqrzytt2rnpue. Nztnx-

xeyt pzur le pnyper pulxzynure, qnys lequel qe yzxoreuses quipult2s nppnrnusseyt pzxxe ln vnruntuzy qe qeysut2 qes

tussues, le ptnysexeyt qe ln rzrxe qes zrsnyes nuysu que le q2pnlnse qe ln pzsutuzy qe ln tuxeur peyqnyt ln respurntuzy.

azus pes pnrnx5tres nfepteyt ln pzrt2e qu rnuspenu q’uzys utulus2s peyqnyt le trnutexeyt, et, pnr pzys2queyt eytrn7yeyt

uye qustruoutuzy qe qzse uynteyque. L’zoveptur pruypupnl qe pete tt5se est qe przpzser uye x2ttzqe qe pnlpul qe qzse

ons2e sur les struptures t2trn2qruques, quu perxet q’estuxer les qustruoutuzys qe qzse qes zrsnyes ey xzuvexeyt ey

utulusnyt les suxulntuzys Mzyte Cnrlz. Ces qustruoutuzys qe qzse szyt pnlpul2es ey utulusnyt uye pnrte qe qeysut2 t2-

trn2qruque q2peyqnyte qu texps, q2pruvnyt l’nyntzxue uyterye qu pzrps tuxnuy. De plus, le xzuvexeyt uyterye peut

3tre repr2seyt2 à l’nuqe q’uye xzq2lusntuzy ouzx2pnyuque r2szlue pnr ln x2ttzqe qes 2l2xeyts iyus (MEF) zu q’uye

pnrte qe q2plnpexeyt ussue q’uy repnlnse q’uxnses q2rzrxnole. Czytrnurexeyt nux x2ttzqes ons2es sur les struptures

plnssuques à onse qe vzxels, ln qzse q2pzs2e s’nppuxule à l’uyt2rueur qe ptnque t2trn5qre nu pzurs qe ln q2rzrxntuzy,

surxzytnyt nuysu le przol5xe qu suuvu tussulnure puusque le t2trn5qre est q2iyu pzxxe uye pnrtue q’uy tussu qzyt ln

pzxpzsutuzy ptuxuque et ln tzpzlzsue ye ptnyseyt pns. Dnys ln prexu5re pnrtue qe ln tt5se, yzus nvzys q2velzpp2 uye

x2ttzqe qe pnlpul qe qzse quu s2y5re uye pnrte qe qzse D ey utulusnyt uy xzq5le t2trn2qruque sp2puique nu pntueyt.

Ey zutre, yzus 2tuquzys l’efet qu yuvenu qe q2tnul qes xnullnses t2trn2qruques sur ln pr2pusuzy qe ln qustruoutuzy qe ln

qzse zoteyue. Dnys ln qeuxu5xe pnrtue, yzus yzus pzypeytrzys sur l’zptuxusntuzy qe ln s2zx2true t2trn2qruque pzur

r2quure le texps qe suxulntuzy, snptnyt que l’zoteytuzy q’uye qustruoutuzy qe qzse pr2puse peut 3tre pzûteux ey terxes

qe texps. Pzur surxzyter pe przol5xe, yzus nvzys przpzs2 uye yzuvelle npprzpte quu preyq ey pzxpte ln qureptuzy

qu rnuspenu niy qe xuyuxuser l’erreur qe l’2pnusseur 2quuvnleyt enu qes t2trn5qres nvnyt le vzluxe qe ln tuxeur. Cete

uuu

x2ttzqe perxet q’zoteyur uyxnullnse t2trn2qruque srzssuer et, pnr pzys2queyt, q’nx2luzrer les perrzrxnypes qe pnlpul

qnys les suxulntuzys qe Mzyte Cnrlz, tzut ey pzyservnyt uye qustruoutuzy qe qzse pr2puse qnys le vzluxe puole.

Mo Clé : tnqrzytt2rnpue, xnullnses t2trn5qruques, struptures vzx2luques, qzsux2true, Mzyte Cnrlz, xzq-

2lusntuzy

uv

Publications related to this work

Peer-revuewed journal artucles:

fnzuq azuuleo, Hnxuq Lnqvnl, Muptnel Beuve, nyq Betznq Stnrunt. ”Pnrtuple-oenx-qepeyqeyt zptuxuzn-tuzy rzr Mzyte Cnrlz suxulntuzy uy tnqrzytternpy usuys tetrnteqrnl sezxetrues”. Ptysups uy Mequpuye nyqBuzlzsy, ( ), . ISSN . qzu: . / - /nnpoe .

Conrerence contrubutuons:

fnzuq azuuleo, Petru Mnyespu, Hnxuq Lnqvnl, Jzsept Azeypzt, Muptnel Beuve, nyq Be- tznqStnrunt. ”Mztuzy-uyqupeq Mzyte Cnrlz qzse pnlpulntuzy usuys qerzrxnole tetrnteqrnl xestes”.Iy IEEE th Iyteryntuzynl Syxpzsuux zy Buzxequpnl Ixnsuys (ISBI) nvrul , Prnsue(R2puoluque apt5que),pnses – . IEEE, npr . ISBN - - - - . qzu:

. /ISBI. . .

fnzuq azuuleo, Hnxuq Lnqvnl, Jzsept Azeypzt, Muptnel Beuve nyq Betznq Stnrunt, ” D dose calculations7Tetrahedral meshes versus voxel-based structures” ( ), ICaR-PHE , r2vruer , CERN, Gey5ve(Suusse)

Petru Mnyespu, Hnxuq Lnqvnl, fnzuq azuuleo, Jzsept Azeypzt, MuptnelBeuve & Betznq Stnrunt ( ).” D POSITRON EMISSION TOMOGAPHY IMAGE RECONSTRUCTION BASED ON BIOME-CHANICAL RESPIATORY MOTION”. IEEE Iyteryntuzynl Syxpzsuux zy Buzxequpnl Ixnsuys (IEEEISBI), nvrul , Prnsue (R2puoluque apt5que), pp. - . HAL : tnl- .

Hnxuq Lnqvnl, Nnqur Sweyqrnzuu, Mnttueu Gurzux, fnzuq azuuleo, Jzsept Azeypzt, Muptnel Beuve,Ptuluppe Gurnuq & Betznq Stnrunt( ). ” Physiological and Biomechanical Model of Patient Speciic LungMotion Based on D CT Images”. he tt Buzxequpnl Eysuyeeruys Iyteryntuzynl Czyrereype (IEEE BME-uCON ), yzvexore , Pnt (hn8lnyqe), pp. - . HAL : tnl- .

v

Contents

I. Rnquntuzy tternpy rzr pnyper trentxeyt . . . . . . . . . . . . . . . . . . . . . . .

. . Meptnyusxs zr nptuzy zr rnquztternpy . . . . . . . . . . . . . . . . . . .

. . Rnquntuzy tternpy uy tustzry: rrzx pzyveytuzynl rnquztternpy tz tnqrzytternpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Hnqrzytternpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Benx qeluvery systexs . . . . . . . . . . . . . . . . . . . . . . . . . . .

. arentxeyt zr xzvuys tnrsets usuys tenvy ptnrseq pnrtuples . . . . . . . . . . . .. . Orsny xztuzy rrzx n pluyupnl vuew . . . . . . . . . . . . . . . . . . . .. . Respurntzry xztuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Dzsuxetrup pzysequeypes zr tnrset xztuzy . . . . . . . . . . . . . . . .. . Deysuty vnruntuzys . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Mequpnl strntesues tz qenl wutt xztuzy . . . . . . . . . . . . . . . . . . . . . . .. . Mztuzy Requptuzy teptyuques . . . . . . . . . . . . . . . . . . . . . . .. . Respnyyuys zr repnuytuys . . . . . . . . . . . . . . . . . . . . . . . . .. . Respurntzry sntuys npprznpt . . . . . . . . . . . . . . . . . . . . . . . .

Expnytuzy zr tte plnyyeq tnrset vzluxe . . . . . . . . . . . . . . . . . .. . Renl-tuxe tuxzr trnpwuys . . . . . . . . . . . . . . . . . . . . . . . . .. . Aqnptuve Rnquztternpy . . . . . . . . . . . . . . . . . . . . . . . . . .

. Auxs nyq strupture zr tte ttesus . . . . . . . . . . . . . . . . . . . . . . . . . .

vu

CONaENaS vuu

M - M C -

. Iytrzquptuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Dzse pnlpulntuzy xettzqs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Peypul oenx nlszruttxs . . . . . . . . . . . . . . . . . . . . . . . . . .. . Rny-pnstuys nlszruttxs . . . . . . . . . . . . . . . . . . . . . . . . . .. . Mzyte Cnrlz xettzqs . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Respurntzry xztuzy estuxntuzy . . . . . . . . . . . . . . . . . . . . . . . . . .. . Ixnse-onseq xztuzy estuxntuzy npprznptes . . . . . . . . . . . . . . .. . Buzxeptnyupnl xettzqs . . . . . . . . . . . . . . . . . . . . . . . . . .

. Mzqeluys zr xzvuys zrsnys uy tnqrzyttepnpy . . . . . . . . . . . . . . . . . . .

. Czytuyuzus represeytntuzy zr xzvuys zrsnys . . . . . . . . . . . . . . . . . . .. . results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. aetrnteqrnl xest prentuzy nyq qeysuty xnppuys: Frzx vzxel-onseq sezxetruestz tetrnteqrnl xzqels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Mnppuys vzxel qeysutues uytz tetrnteqrnl elexeyts . . . . . . . . . . . .

aetrnteqrnl xest seyerntuzy . . . . . . . . . . . . . . . . . . . . . . .Deysuty xnppuys . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . auxe-qepeyqnyt qzse pzxputntuzy zy qerzrxnole tetrnteqrzys . . . . .

. . GEANa -onseq uxplexeytntuzy . . . . . . . . . . . . . . . . . . . . .

. . Benx stnpuys nyq pzyisurntuzy . . . . . . . . . . . . . . . . . . . . .. Evnluntuzy nyq Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Czypeytrup spteres ptnytzx . . . . . . . . . . . . . . . . . . . . . . .

. . Pntueyt pnse suxulntuzy . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Deysuty evnluntuzy nyq qzse qustruoutuzy pzxpnruszy . . . . . . . . . .

. . quspussuzy nyq pzyplusuzy . . . . . . . . . . . . . . . . . . . . . . . . .. Multureszlutuzy tetrnteqrnl xzqel prentuzy . . . . . . . . . . . . . . . . . . . .

. . Multureszlutuzy tetrnteqrnl xzqel . . . . . . . . . . . . . . . . . . . .aetrnteqrnl xzqel seyerntuzy przpess . . . . . . . . . . . . . . . . . . .aetrnteqrnluzntuzy nyq seyerntuzy zr qufereyt Levels zr qetnul (LODs): .Gezxetry pzyisurntuzy . . . . . . . . . . . . . . . . . . . . . . . . . .

CONaENaS vuuu

Results nyq quspussuzy . . . . . . . . . . . . . . . . . . . . . . . . . . .. Czyplusuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P - -. Iytrzquptuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Czyplusuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. he nrtuple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M dEPL D. Iytrzquptuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Cnlpulntuzy zr wnter equuvnleyt ttupwyess uy prztzy oenx urrnquntuzy . . . . . . .. Ca Hb-tz-Stzppuys Pzwer Cnluorntuzy . . . . . . . . . . . . . . . . . . . . . .. Multu-lnyer ptnytzx (MLP) pnse . . . . . . . . . . . . . . . . . . . . . . . . .. Experuxeytnl pnlpulntuzy zr tte dEPL . . . . . . . . . . . . . . . . . . . . . . .. Efeptuve qeysuty pnlpulntuzy usuys stzuptuzxetrup dEPL . . . . . . . . . . . . .. Evnluntuzy nyq result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . suxulntuzy setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . D Dzse evnluntuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Czyplusuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G

A A LA. Luyenr qeysuty vnruntuzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A. Luyenr Iyterpzlntuzy resnxpluys (LR) . . . . . . . . . . . . . . . . . . . . . . .A. Resnxpluys usuys Nenrest Neustozr Iyterpzlntuzy (NN) . . . . . . . . . . . . .

L F

L a

R

Acknowledgments

hus PtD us suppzrteq oy LABEe PRIMES (ANR- -LABe- ), wuttuy tte przsrnx Iy-vestussexeytsq’Aveyur (ANR- -IDEe- ) nyqoyFrnypeHnqrzy, nyqoyClnRysprzveptCNRS,qeis Ixnsuys.

ux

Lit of Acronyms

BP Brnss Penw Ha Hnqrzy hernpyCDa Czystrnuyeq Delnuyny

Detrnteqrnluzntuzy xettzqHb Hzuysielq byut

CDcH Cuxxulntuve Dzse-czluxeHustzsrnx

LOD Level Or Detnul

CM Ceytrzuq Mnppuys xettzq MC Mzyte CnrlzCMCM Czytuyuzus Mnss Czyservuys

Mnppuys xettzqMCM Mnss pzyservuys xnppuys xettzq

CSG Czystruptuve Szluq Gezxetry MLC Multu-Lenr CzlluxntzrCa Czxputeq azxzsrnpy MLP Multu-Lnyer PtnytzxCac Cluyupnl anrset czluxe OAR Orsnys At RuswDIR Derzrxnole Ixnse Resustrntuzy Pac Plnyyuys anrset czluxeDaA Dustnype az Asreexeyt RC Rnyse CzxpeysntzrDcF Derzrxntuzy ceptzr Fuelq Ra Rnquntuzy hernpyEFFD EFFeptuve Deysuty xnppuys

xettzqSOBP Sprenq Out Brnss Penw

FEM Fuyute Elexeyt Mettzqs dEPL dnter Equuvnleyt Pntt LeysttGac Grzss anrset czluxe dEa dnter Equuvnleyt hupwyessHI Hzxzseyeuty Iyqex

x

Learning never exhausts the mind.

Lezynrqz qn cuypu

1Introdution

Iy ttus ptnpter, we preseyt tte onpwsrzuyq nyq tte zoveptuves zr zur stuqy. de irst stnrt wutt nyzvervuew zr tnqrzytternpy nyq uts use uy pnyper trentxeyt ttey; we suve ny uysustt uytz tte tept-yuques useq tz uypzrpzrnte zrsny xzvexeyt. Fuynlly, we eyq wutt n qespruptuzy zr tte sznls zr ttepreseyt ttesus nyq tte strupture zr ttus xnyusprupt.

. R y

Rnquztternpy nlsz wyzwy ns rnquntuzy tternpy zr Ra us zye nxzys ive xzqnlutues zr pnyper trent-xeyt nlzyssuqe sursery, ptexztternpy, uxxuyztternpy nyq tzrxzynl tternpy. It us pzysuqereq nsny esseytunl elexeyt rzr efeptuve pnyper trentxeyt suype xzre ttny % zr nll pntueyts wzrlqwuqerepeuve rnquztternpy quruys tte xnynsexeyt zr tteur qusense h ]h ]. Eutter stnyq-nlzye zr pzx-ouyeq wutt nyztter xzqnluty qepeyquys zy tte stnse nyq tte type zr pnyper. Iy Ra, tte rnquntuzysnre qeluvereq uy ttree qustuyptuve wnys :

Iytrzquptuzy

• Bractytterapy (Ba) or sealed source raduatuon tterapy szxetuxes rererreq tz ns puru-etternpy zr eyqzpuruetternpy. Iy ttus rzrx zr Ra, tte rnquznptuve szurpes nre plnpeq wuttuyzr nqvnpeyt tz tte sute zr tte pnyperzus tuxzr tz qeluver tuster rnquntuzy qzses h ]. Fzr uy-stnype, n rnquntuzy szurpe wull oe plnpeq uy tte orenst zr n wzxny wtz tns orenst pnyper. Bateptyuque us wuqely useq rzr orenst, pervupnl, przstnte, nyq swuy pnypers.

• Raduousotope tterapy or unsealed source raduotterapy npplueq ttrzust tte systexup uy-veptuzy zr rnquznptuve suostnypes rnquzptnrxnpeutupnls ttnt tnve oeey qesusyeq tz tnrsetqusense. h ]. he rnquntuzy szurpe us suvey zrnlly zr uytrnveyzusly ttrzust ny uytrnveyzustuouys. he xzst pzxxzy pnyper trenteq wutt rnquzusztzpe tternpy us tte ttyrzuq pnyper.

• External beam raduatuon tterapy (EBRa or eRa) or teletterapy pzytrnry tz tte zttertwzxettzqs, tte oenx uyEBRa us qeluvereq rrzxzutsuqe tte ozqy usuys ny exterynl szurpezr uzyuzuys rnquntuzy nyq pzuyteq tz tte tnrset.

Iy ttus xnyusprupt, zur wzrw wull revzlve xnuyly nrzuyq tte exterynl oenx rnquntuzy tternpy,nyq we wull rerer tz ut ttererzre ns rnquntuzy tternpy.

. . M

he uqen oetuyq rnquztternpy us tz wull tte tuxzrnl pells usuys uzyuzuys rnquntuzys. As tte terxuxplues, ttese rnquntuzys rzrx uzys nyq qepzsut eyersy uy tte tussues trnverseq oy tte oenx, ttusqnxnsuys tteur DNA zr ztter prutupnl pellulnr xzlepules. he eyersy qepzsutuzy pny eutter wull ttepells uxxequntely zr pnuse seyetup ptnyses ttnt lenq tz pell qentt h ]. As n result, tte pell quvusuzyprzpess wull oe olzpweq, nyq rurtter pell przlurerntuzy wull oe preveyteq h ]. Nevertteless, Suypettnt yzrxnl pells xny nlsz oe qnxnseq nyq wulleq quruys urrnquntuzy, tte sznl zr rnquntuzy tternpyus tz xuyuxuze expzsure tz tenltty tussues wtule xnuytnuyuys n tust qzse vnlue uy tte tnrset. Fzr-tuyntely, Nzrxnl pells pny repnur ttexselves nt n rnst rnte nyq retnuy tteur usunl ruyptuzy stntuespzytrnry tz pnyper pells wtupt nre yzt ns eipueyt tz ix tte qnxnse pnuseq oy tte rnquntuzys h ].

Iytrzquptuzy

. . R :

he irst tz use uzyuzuys rnquntuzy rzr n tternpeutup npplupntuzy nyq tz qexzystrnte tte ouzlzsupnlefepts zr x-rnys zy tussues wns tte przresszr Lezpzlq Freuyq h ]. cery szzy nter tte quspzvery zrx-rnys oydultelxRöytsey uy h ], te suppessrully trenteq uy n ive-yenr-zlq surl suferuysrrzx tnury xzles pzveruys ter onpw, ttus oepzxuys tte rzuyqer zr xequpnl rnquzlzsy nyq rnquz-tternpy h ]. Iy tte enrly tweytuett peytury, ny uyprensuys yuxoer zr tte stuques ttnt use x-rnys uyxequpuye wns repzrteq. Hzwever, que tz tte lzw peyetrntuzy zr tte rnquntuzys uy tussues, tte trent-xeyts were luxuteq tz superipunl nyq stnllzw-senteq tuxzrs luwe uy swuy pnyper. Besuqes, tte lnpwzr wyzwleqse zr tte xeptnyusxs zr nptuzys nyq tte ouzlzsupnl efepts zr tte yew rnys pzyqupteqtz pzzr pnyper trentxeyt nyq lzwer qeluvery pzytrzl pzxpnreq tz tte przqupeq suqe efepts h ]h ]. hese zutpzxes tnve pusteq ptysupunys tz stuqy tte przpertues zr ttese yew rnquntuzys nyqrnquzouzlzsusts tz uyvestusnte tte uxpnpt zr tte pnrtuple oenxs zy pell survuvnl. By tte eyq zr ttusperuzq zr rnquntuzyxequpuye wyzwy ns tte ern zr quspzvery h ] uy tte lnte s, resenrpters werenole tz uyqerstnyq tte ptnrnpterustups zr tte yewly quspzvereq oenxs nyq tteur efepts zy tussues,espepunlly rzllzwuys tte wzrw zr Eryest Rutterrzrq zy tte ntzxup strupture. Mzrezver, rnquznptuveelexeyts were nlsz uqeytuieq nyq quluseytly stuqueq, ns well. Nzt less uxpzrtnyt wns tte uytrzqup-tuzy zr tte uzyuzuys ptnxoer ttnt xnqe tte xensurexeyt zr tte qeluvereq rnquntuzy qzse pzssuole.Iy tte rzllzwuys qepnqes, rnquntuzy tternpy oepnxe zye zr tte xnuy xzqnlutues useq rzr zypzlzsy,nyqxnyy teptyupnl nqvnypes tnve oeeyxnqe tz eytnype tte qunluty zr tte trentxeyt. hesexett-zqs nux tz uxprzve tte pzyrzrxuty zr tte nqxuyustereq qzse tz tte tnrset vzluxe oy luxutuys n tustnyq n tzxzseyezus qzse uy tte qusenseq nren nyq spnre tte surrzuyquys tenltty tussues tz nvzuquywnyteq suqe efepts rzr tte pntueyt. Axzys tte teptyuques ttnt tnve xnqe n susyuipnyt uxpnptzy tte qunluty nyq tte prepusuzy zr tte oenx qeluvery, we rzuyq pzxputeruzeq trentxeyt plnyyuyssystexs, setup nyq pntueyt pzsutuzyuys nyq tte xesnvzltnse x-rnys. Ayztter strntesy ttnt tns oeeyuseq tz requpe tte qzse uy prutupnl nrens us tz tnwe przit zr tte qzse qepzsutuzy ptnrnpterustups zr ttequfereyt types zr pnrtuples. he irst tz qespruoe tte pzteytunl zr uzys rzr xequpnl use wns Rzoert R.dulszy uy h ]. He zutluyeq tte nqvnytnses zr nqzptuys oenxs zr prztzys nyqtenvuer uzys rzrtte trentxeyt zr tuxzrs, uy pzxpnruszy wutt tte pzyveytuzynl tust eyersy ptztzy oenxs (x-rnys).he well wyzwy ptysupnl przpertues zr prztzys zfer n oeter pzvernse zr qzse suype ttnt ttey slzw

Iytrzquptuzy

qzwy quruys tteur peyetrntuzy zr tte xnter. Iy rnpt, dulszy tns nlsz exteyqeq tus ttzustts tz tteuse zr tenvy uzys.

. . H

Iy tte luternture, xultuple terxs nyq qesusyntuzys nre useq tz rerer tz tte trentxeyt zr pnyper us-uys prztzys nyq ztter tenvuer uzys. he xzst pzxxzyly useq nre pnrtuple tternpy, tnqrzytternpy,tnqrzy tternpy nyq tenvy-uzy tternpy. Evey ttzust ptztzys nyq eleptrzys nre nlsz pnrtuples, ptz-tzy tternpy, nyq eleptrzy tternpy nre yzt pzysuqereq uy pnrtuple tternpy, out rntter, ttey nre usunllysrzupeq tzsetter nyq pnlleq pzyveytuzynl rnquztternpy. az eluxuynte pzyrusuzy, tte terxHnqrzy-tternpy wns uytrzqupeq tz pzver nll rzrxs zr rnquntuzys ttnt use pnrtuples ttnt nre xnqe zr qunrws:prztzys, yeutrzys, puzys nyqteluux, pnrozy, zxysey nyqyezy uzys, etp. Heyse, expluquys ptztzysnyq eleptrzys rrzx tte ontpt. Oy tte ztter tnyq, nlttzust, ttere us yz qufereype oetweeyHnqrzy-tternpy nyq tte twz-sepnrnte-wzrqs Hnqrzy tternpy, tte rzrxer us prererreq h ]. Suype, oy nynl-zsy wutt rnquztternpy, tte terx rnquntuzy tternpy wns useq uytul ut oepnxe ny uxpzrtnyt xzqnlutyuy zypzlzsy. As rzr tte qesusyntuzy tenvy-uzy tternpy, ut us useq tz ptnrnpteruze uzys tenvuer ttnyprztzys nyq nlsz tz qespruoe tteur uyprenseq relntuve ouzlzsupnl efeptuveyess. Curreytly, zyly prz-tzys nyq tz szxe exteyt pnrozy uzys nre uy use uy tte pluyupnl eyvurzyxeyt h ] nyq ttus ttey wulloe tte rzpus zr tte purreyt wzrw.

Beyzyq tte ptzsey terxuyzlzsy, tte rnpuqly uyprensuys nqzptuzy zr tnqrzytternpy uy tte pny-per trentxeyt renlx us que tz tte ptysupnl nspepts nyq przpertues zr tenvy-uzys. Mzre spepuipnllytz tteur qustuypt qeptt-qzse qustruoutuzy ynxeq Brnss purve nter Sur dullunx Heyry Brnss, wtzquspzvereq ut uy h ]. hus purve plzts tte qepzsuteq qzse zr uzyuzuys rnquntuzys quruys tteurtrnvel ttrzust xnter, nyq ut us ptnrnpteruzeq oy n przyzuypeq ynrrzw penw wyzwy ns tte Brnsspenw (BP). he lnter zppurs nt tte eyq zr tte rnyse zr pnrtuples, nyq ut pzrrespzyqs tz tte resuzywtere n susyuipnyt nxzuyt zr eyersy us qepzsuteq. Fusure . . ullustrntes n pzxpnruszy zr qzsequstruoutuzys oetweey ptztzys, prztzys nyq pnrozy uzys uy wnter usuys pluyupnl relevnyt eyersues.Czytrnry tz ptztzys ttnt tnve n steep expzyeytunl qeprense zr qzsewutt qeptt, pnrtuple oenxs tnven qzse przile ttnt uyprenses up tz n stnrp xnxuxux wutt uyprensuys peyetrntuzy wutt n very lutleeyersy oeyzyq tte Brnss penw.

Iy nqqutuzy tz tte rnvzrnole qzse qustruoutuzy nyq tte steep qzse rnll-zf zr uzy oenxs h ], tte

Iytrzquptuzy

Figure 1.1.1: Depth dose distribution for photons and monoenergetic Bragg curves for car-bon ions and protons [18].

pzsutuzy zr tte BP pny oe prepusely nqvusteq tz tte qesureq qeptt oy ptnysuys tte wuyetup eyersy zrtte uypuqeyt uzys. hese ptnrnpterustups yuelq tz oeter tuxzr pzyrzrxuty nyq ttus tz oeter pluyu-pnl zutpzxes pzxpnreq tz pzyveytuzynl rnquztternpy h ]. Fusure . . ullustrntes n pzxpnruszyzr prztzy tternpy nyq e-rnys/IMRa trentxeyt plnys zr eszptnsenl pnyper. Evey ttzust tte qzsepzyrzrxuty us prepuse uy tte tnrset vzluxe rzr tte twz xzqnlutues, tte qzse uy tte surrzuyquys vzl-uxes us rnr less wtey usuys tte prztzy oenx.

az pzver tte eyture tnrset vzluxe wutt n uyurzrx qzse qustruoutuzy, tte oenx qeluvery systexxust exteyq tte Brnss penw zver n xzre susyuipnyt suze prentuys tte sz-pnlleq Sprenq-Out-Brnss-Penw (SOBP). he lnter us rzrxeq oy nqquys tte pzytruoutuzy zr uyquvuqunlly xzqulnteq prustuyeBrnss Penws. Fusure . . stzws tzw tte prztzy BPs zr qufereyt rnyses nre superuxpzseq tz rzrxtte SOBP nyq tz przqupe n tzxzseyezus plnyyeq qzse uy tte tuxzr nren.

Beam delivery systems7 Hnqrzytternpy requures pzxplex oenx qeluvery systexs ttnt pny nppeler-nte tte pnrtuples tz tte prespruoeq eyersues nyq trnyspzrt ttex zytz tte tnrset. Suype ttnt, n uyuqueBrnss Penw pnyyzt pzver tte eyture vzluxe zr tte tuxzr, tte uzy oenx yeeqs tz oe stnpeq. Overtte yenrs, twz xnvzr strntesues tnve oeey qevelzpeq: pnssuve oenx stnpuys nyq nptuve spnyyuys.he rzrxer xettzq seyerntes n orznq tternpeutup oenx ttnt us rurtter pzlluxnteq nyq stnpeq tzpzyrzrx wutt tte tnrset vzluxe. Mzre prepusely, ut relues zy n pzxouyeq use zr supt qevupes nsrnyse xzqulntzrs, pzxpeysntzrs, nyq weqses. As stzwy uy Fusure . . , tte ynrrzw xzyzeyer-

Iytrzquptuzy

Figure 1.1.2: Treatment plans of esophageal cancer using proton therapy (left) and X-rays/IMRT (right). Proton therapy limits the radiation delivered outside the target and re-duces the likelihood of lung complications. The conformity to the target volume is precisein the two cases. However, the dose of surrounding organs is minimized when using protons.Source Seattle cancer care alliance, Proton therapy center 2017 and [21].

Figure 1.1.3: In a typical treatment plan for proton therapy, the spread out Bragg peak(SOBP, dashed blue line) is the therapeutic radiation distribution. The SOBP is the sumof several individual Bragg peaks (thin blue lines) at staggered depths. The depth-dose plotof an X-ray beam (red line) is provided for comparison. The pink area represents additionaldoses of X-ray radiotherapy—which can damage normal tissues and cause secondary cancers,especially of the skin

Iytrzquptuzy

setup Brnss penw seyernteq oy tte pnrtuple nppelerntzr us sprenq zut oy tte rnyse xzqulntzr tz sey-ernte tte SOBP ttnt pzvers tte eyture leystt zr tte tuxzr. Furtterxzre, n rnyse stuter us useq tzstut tte seyernteq SOBP tz nlusy ut wutt tte qeptt zr tte tuxzr. he rzllzwuys twz qevupes nrespepuipnlly qesusyeq nyqxnqe uyquvuqunlly rzr enpt pntueyt: tte pzlluxntzr olzpw tte pnrtuples ttntnre zutsuqe tte ielq zr vuew zr tte tnrset nyq iynlly, tte rnyse pzxpeysntzr nqvusts tte rnquntuzyielq tz tte qustnl qeptt pntery zr tte tnrset vzluxe. Despute tte rnpt ttnt tte pnssuve oenx qeluv-ery systex us wuqely useq uy pnrtuple tternpy peyters nrzuyq tte wzrlq, ut tns susyuipnyt qrnwonpws.he ixeq wuqtt zr tte SOBP xny result uy tte qepzsutuzy zr qzse zutsuqe tte tuxzr ns ullustrnteqoy tte tntpteq nren uy isure . . . hus us que tz tte uynouluty zr tte pzxpeysntzr tz rzrx tte przxu-xnl eqse zr tte tnrset vzluxe ttnt lenqs tz n restrupteq tuxzr pzyrzrxuty. Aqqutuzynlly tz tte yeeqrzrxnyurnpturuys pntueyt-spepuip tnrqwnre ttnt seyerntesxzre trentxeyt pzsts, ttusxettzq sey-erntes yeutrzys ttnt pny pnuse uyqesurnole ouzlzsupnl qnxnse uy tenltty tussues. hzse yeutrzysnre tte result zr tte uyternptuzys uy oenx spnteruys nyq pzlluxntuzy. Hzwever, spnteruys us xzrerzrsuvuys zr tuxzr nyq zrsny xztuzy oepnuse zr tte sxenruys efept zr tte orznqeyeq oenx h ].

Oy tte ztter tnyq, tte nptuve spnyyuys oenx qeluvery teptyuque (zr tte Aptuve oenx stnpuys )npprznpt tte stnpuys qufereytly oy irst quvuquys tte tnrset vzluxe uytz lnyers zr equnl eyersy, tteynsszpuntuys rzr enpt slupe n sruq zr tnrset pzuyts uy n rzrx zr vzxels. hese vzxels nre rurtterxzreurrnqunteq sequeytunlly oy n peypul oenx ttnt spnys tte eyture tnrset nren. Fusure . . ullustrntestzw tte oenx spnys tte tuxzr oy xzvuys rrzx n slupe tz nyztter. he prustuye Brnss penws nrepzsutuzyeq uysuqe tte tuxzroy explzyuys spnyyuysxnsyets nyqnlszoy ptnysuys tteoenxeyersy.bsuys supt n strntesy tns tte nqvnytnse zr eluxuyntuys tte yeeq rzr nyy nqqutuzynl pntueyt-spepuipqevupes nyq nlsz requpe qrnstupnlly tte nxzuyt zr tnrqwnre uy tte oenxluye lenquys tz less yeutrzylux tzwnrqs tte pntueyt. Aqqutuzynlly, oeter pzyrzrxuty tz tte tuxzr us eysureq teype spnruystte surrzuyquys tenltty tussues h ]. he nptuve spnyyuys szlves szxe zr tte wenwyesses zr ttepnssuve strntesy resnrquys tte eipueypy zr tte oenx qeluvery ns ut yeutter seyerntes yeutrzys yzrtns spntereqpnrtuples uy tteyzzzle. Mzrezver, ut qzesy’t requure tteusezr pzxplupnteqnyqpntueyt-spepuip qevupes rzr tte trentxeyt. Nevertteless, tte eytnypeq nouluty wutt pnrtuple spnyyuys tzpnuyt qzse xzre pzyrzrxnlly, vzxel oy vzxel, uyprenses tte rusw zr tnrset xusses que tz orenttuys.Stull, Multuple re-pnuytuyss pny pzxpeysnte rzr tte zrsny xztuzy oy efeptuvely sxenruys zut tteqzse h ].

Iytrzquptuzy

Figure 1.1.4: Schematic of a fully a passive beam shaping system in Hadrontherapy. Figureredrawn from [20].

Figure 1.1.5: Schematic of the GSI raster-scan system. Figure redrawn from [23].

Iytrzquptuzy

. a y

Despute tte uyprenseqnppurnpyzfereqoyxzqery rnquztternpyxzqnlutues resnrquysoztt tte trent-xeyt prepusuzy nyq tte qzse pzyrzrxuty, tte trentxeyt zr xzvuys tnrsets stull xnwe n very pzx-pelluys ptnlleyse. Czysequeytly, quruys tte lnst qepnqes, tte xnynsexeyt zr zrsny xztuzy estno-lusteq n xnvzr resenrpt ielq uy rnquztternpy. Iy ttus septuzy, we wull quspuss tte uxpnpt zr uyterynlzrsny xztuzy zy tte trentxeyt wutt uzy oenx tternpy nyq tte qufereyt strntesues explzyeq tzxutusnte xztuzy efepts.

. . O

Iy tte rzllzwuys, we wull oruely uytrzqupe tte qufereyt types zr zrsny xztuzy nyq tteur ptnrnpter-ustups. A xzre exteysuve revuew pny oe rzuyq uy h ] nyq h ]. he vnruntuzy zr tte qeysuty qustru-outuzy zr tte pntueyt nyq tte lzpntuzy zr tte tnrset pny zppur nt qufereyt tuxe spnles: he eyturetrentxeyt, uy-oetweey sessuzys nyq quruys tte trentxeyt qeluvery.

• Over tte Course or areatment: It us rererreq tz tte srnqunl nyq systexntup vnruntuzy zrtte qeysuty qustruoutuzy rzllzwuys n requstruoutuzy zr qeysutues oetweey qufereyt tussues zrtte qusnppenrnype zr tte nqqutuzy zr xnter. Exnxples uypluqe struywnse nyq tuxzr srzwtth ], weustt vnruntuzys nyq tte ptnyse uy luys qeysuty. Mzrezver, tte pnterys zrxztuzy pnynlsz oe nfepteq ns us tte pnse wutt orenttuys trnveptzrues. Furtterxzre, tte pntueyt xztuzypny nfept eutter tte uyter-rrnptuzynl zr tte uytrn-rrnptuzynl xztuzys.

• Inter-rractuonal orsan Motuon: It uypluqes tte ptnyses ttnt tte nyntzxy uyqerszes oe-tweey trentxeyt rrnptuzys nyq tnvuys n xzre rnyqzx oetnvuzr ttny tte vnruntuzys ttnt zp-pur zver tte eyture trentxeyt. he szurpes zr uyter-rrnptuzynl ptnyses nre quverse nyq pnyoe tte errzr relnteq tz tte pzsutuzyuys zr tte setup zr tte pntueyt, tte ptnyses uy tte tuxzrvzluxe zr evey tte qnuly vnruntuzy zr reptnl nyq ozwel illuys. Besuqes, we pny nlsz xeytuzytte nvernse tuxzr pzsutuzy ttnt pny ptnyse quruys tte orenttuys pyple h ].

• Intra-rractuonal orsanMotuon: ut zppurs uy tte pzurse zr n suysle trentxeyt rrnptuzywuttuyn stzrt tuxe uytervnl, nyq ut us xnuyly zrusuynteq rrzx tte zrsny xztuzy luwe swnllzwuys,respurntuzy, tenrtoent, zrsny illuys zr perustnlsus.

Iytrzquptuzy

Suype ttnt tte xnuy rzpus zr zur wzrw us tte trentxeyt zr luys pnyper, we wull exptnsuze ttetrentxeyt zr uytrn-rrnptuzynlly respurntzry xztuzy ttrzustzut tte rest zr tte qzpuxeyt.

. . R

Respurntuzy xztuzy us tte pruxnry szurpe zr uypertnuytues quruys tte trentxeyt sessuzy h ] h ]h ] nyq ut nlsz pzytruoutes tz severnl tuxzr sutes: ttzrnx h ], luver h ], noqzxuynl h ]h ],orenst, przstnte nyq ynturnlly ut pzystututes n ous ptnlleyse rzr luys pnyper h ]. Fzr luys tuxzrs,tte xztuzy us xzre przyzuypeq uy tte superuzr-uyreruzr (SI) quxeysuzy zr tte ptest pnvuty nyq uytte resuzys plzse tz tte qunptrnsx, seyerntuys ny nxplutuqe ttnt pny oe ns tust ns xx. henuttzrs uy h ] tnve repzrteq n xnsyutuqe zr xztuzy less ttny . xx rzr % zr tte luys tuxzrpnses uy tte SI qureptuzy. Aqqutuzynlly, uy tte stuqy zr h ], % zr tztnl pntueyts tnve xnyuresteqny nxplutuqe srenter ttny xx.

Furtterxzre, yzt zyly tte orenttuys xztuzy us pntueyt-spepuip out ut us nlsz ptnrnpteruzeq oy nrnyqzx oetnvuzr ttnt tns pnterys ttnt luptunte uy xnsyutuqe, peruzq, nyq resulnruty. hus us quetz tte rnpt ttnt tte orenttuys us szveryeq oy n set zr uyqepeyqeyt xusples suvuys n xultuqureptuzynlrnpet tz tte xzvexeyts. Fusures . . n nyq . . o stzw tte pnterys zr respurntuzy zr tte snxepntueyt tnwey nt very plzse tuxe uytervnls. It tustlustts tte yepessuty zr tnvuys ny uyquvuqunlxettzqrzr respurntzry xnynsexeyt. Besuqes tte qusplnpexeyt, orenttuys xny nfept tte tuxzr suze nyqstnpe h ] nyq nlsz tte qeysuty zr zrsnys zver tte pzurse zr trentxeyt. hese uypertnuytues wulltnve rurtter reperpussuzys zy qzse qeluvery rzr respurntzry xnynsexeyt.

. . D

Iy tte pnse zr yzyxzvuys tnrsets, rnyse nyq pntueyt setup uypertnuytues nre pzysusteyt nyq relntuvelyxensurnole zver tte pzurse zr trentxeyt nyq pny oe tnwey uytz nppzuyt oy tte trentxeyt plnyyuyssystex tz oeter tte tnrset pzvernse nyq spnre tte zrsnys nt rusw (OAR). Mzrezver, zyly zye pzx-puteq tzxzsrnpty uxnse (Ca) us yeeqeq uy tte trentxeyt plnyyuys tz estuxnte tte qeysuty zr ttepntueyt tussues. Hzwever, wutt tnrset xztuzy, xzre ptnlleysuys ussues nre uytrzqupeq tz tte trent-xeyt plnyyuys nyq qzse qeluvery:

• As zppzseq tz stntup tnrsets wtere tte qeysuty zr tussues us nssuxeq tz oe uyptnyseq quruysn trentxeyt rrnptuzy, tnyqluys nyq represeytuys tte vnruntuzy zr tte qeysuty zver tuxe oy tte

Iytrzquptuzy

(a)

(b)

Figure 1.2.1: Fluctuations in respiratory motion patterns registered at close time inter-vals. The curves represent the patient surface displacement in the superior-inferior, anterior-posterior and medial-lateral dimensions. In (a) the patterns are almost identical but in (b)there are no respiratory distinguishable patterns [34].

plnyyuys systex us yzt strnusttrzrwnrq. Evey oy usuys D Ca spnys ttnt zyly represeyt nsynpstzt uy tuxe zr tte pntueyt sezxetry.

• anrset xztuzy xny result uy tte uyterplny efept wtupt us tte uyternptuzy oetweey tte oenxqeluvery nyq tte tuxzr wtey ttey tnve suxulnr tuxe struptures.

• he qusplnpexeyt zr tte tnrset quruys trentxeyt yeeqs tz oe estuxnteq nyq tnwey uytz np-pzuyt tz nqvust tte qzse qustruoutuzy qeterxuyeq oy tte trentxeyt plny.

he plnssupnl trentxeyt plnyyuys npprznpt ttnt pzysusts zr usuys n suysle Ca spny us luxuteq nyquysuipueyt tz tnyqlexzvuys tnrsets que tz tte nozve-xeytuzyeq renszys. bsuys ttus npprznptwullplenrly seyernte n qusprepnypy oetweey tte plnyyeq qzse nyq tte renl qeluvereq qzse. heqevuntuzyzr tte qzse qustruoutuzy rzr n purpulnr ptnytzx us ullustrnteq uy Fusure . . . It stzws tzw tte qzsequstruoutuzy tns qerzrxeq rzr n qusplnpexeyt zr xx resultuys uy ny uyqer qzsnse zr tte tuxzrnyq nfeptuys tte surrzuyquys tussues.

Iytrzquptuzy

Figure 1.2.2: Efect of target motion on a circular phantom. The tumor is centered and it isrepresented by the gray circle. The thin black circle indicates a 10-mm 3D geometric expan-sion of the tumor into the planning target volume (a) The dashed lines are isodose lines (50%,80%, 90%, 95%, 100% of the maximum dose) for a single passive proton beam. (b) Isodoselines for a 10 mm displacement of the tumor with respect to the treatment beam. [25].

. . D

he qeysutues zr tte pntueyt sezxetry nre zr less pzypery uy pzyveytuzynl ptztzy oenx rnquntuzytternpy suype ttey tnve yeslusuole efepts zy tte qzse qustruoutuzy. Hzwever, suype ttnt tte rnysezr uzy pnrtuples qepeyqs strzysly zy tte qeysuty zr tte trnverseq tussues, tte pzxplete wyzwleqsezr tte qeysuty qustruoutuzy zr tte pntueyt us yepessnry rzr Hnqrzytternpy. Ay ullustrntuzy zr tteefept zr qeysuty vnruntuzys us stzwy uy Fusure . . wtere tte SOBP zr tte prztzy oenxtns stuteqonpwwnrq n rew xulluxeters nter nqquys n lnyer zr n xnterunl wutt ny uyprenseq qeysuty. Oy tteztter tnyq, tte qeptt-qzse przile zr tte ptztzys tns rexnuyeq uyptnyseq. Czysequeytly, wuttorenttuys, qeysuty vnruntuzys zppur nyq stzulq oe estuxnteq tz zotnuy ny nppurnte trentxeyt plnyh ]h ] preveytuys severe pzteytunl qzsuxetrup pzysequeypes. Auttzrs uy h ] tnve stuqueq tteefept zr orenttuysxztuzy nyq systexntup setup errzrs zy tte puxulntuveqzsezr n luys tuxzr. heytnve suxulnteq n -xx-qunxeter sptere uysuqe tte luys tussue nyq evnlunteq tte puxulntuve qzserzr xultuple orenttuys pnterys. hey tnve stzwy ttnt ur tte qeysuty vnruntuzys nre yzt pzysuqereqoy tte trentxeyt plny, tte qzse uysuqe tte tnrset us requpeq rrzx % wutt yz xztuzy tz %nyq % rzr n xx orenttuys nxplutuqe nyq n xx zr setup errzrs wutt xztuzy. Aqqutuzynlly,Mzru et nl. h ] uyvestusnteq tte uxpnpt zr wnter equuvnleyt pntt leystt (dEPL¹, tte pnlpulntuzy

¹he wnter equuvnleyt pntt leystt (dEPL) us tte ttupwyess zr n wnter vzluxe uy wtupt tte pnrtuples lzse tte snxenxzuyt zr eyersy ns ttey pnss ttrzust n suvey tussue.

Iytrzquptuzy

zr dEPL us qeysuty qepeyqeyt) vnruntuzys zy tte rnyse zr n ptnrseq pnrtuple oenx usuys eleveyrzur-quxeysuzynl pzxputeq tzxzsrnpty uxnses zr luys pnyper pntueyts. heur results stzweq ttntttexnxuxux lzpnl rnyse luptuntuzys vnry rrzx tz xx rzr tte npplueqdEPL vnruntuzys, nyqrurtterxzre lenq tz zverqzsnse zr zrsnys nt rusws nyq/zr uyqerqzsnse uysuqe tte tnrset vzluxe nsstzwy uy Fusure . . . Huu et nl. h ] exnxuyeq tte uxpnpt zr uyterrrnptuzynl ptnyses zy tte qzsequstruoutuzy uysuqe luys tuxzrs. hey tnve nynlyzeqweewly D-Ca spnys zr luys pnyper pntueyts.he xeny tnrset vzluxe pzvernse pnlpulnteq zy tte weewly qntn wns % less ttny tte zrusuynl plnyrzr nll tte pntueyts expept rzr zye pnse, wtere tte pzvernsewns . % pzxpnreq tz tteplnyyeq %.hus requptuzy wns pnuseq oy tte lnrse vnruntuzy oetweey tte zrusuynl suxulntuzy D-Ca uxnsenyq tte weewly D-Ca uxnse. hey tnve nlsz rzuyq n strzys pzrrelntuzy oetweey tte qzse nyq tteptnyses uy qeysuty nyq ttey pzypluqeq ttnt ur tte tnrsetxztuzy us pzysuqereq uy tte trentxeyt plny,tte tnrset vzluxe pzvernse wull repeuve ny nqequnte qzse rzr xzst luys pnyper pnses. Fuynlly, eveyttzust tte vnruntuzy uy tte qeysuty zr tte tussues us tnwey uytz nppzuyt, tte purreyt trentxeyt plnysstull rest zy tte experueype nyq tte swull level zr tte pluyupnl prnptutuzyer. hererzre, nqnptuve Dtrentxeyt xettzqs nre yeeqeq tz requpe tte qzse uy surrzuyquys tussues.

Figure 1.2.3: Impact of density on the beam range. Depth-dose proiles of a photon beamand a proton SOBP, dashed lines represent the proile generated after adding a higher densitymaterial (gray rectangle from 5 to 7 cm) in the trajectory of the used beams. [25].

Iytrzquptuzy

Figure 1.2.4: Range luctuations expressed in water equivalent path length overlapped on the4D-CT images. Image from [39].

. M

cnruzus npprznptes tnve oeey qeiyeq tz xutusnte nyq xnynse respurntzry xztuzy uy pnyper trent-xeyt wutt tnqrzytternpy. Duruys tte plny qesusy, nll tte strntesues nre tnwey uytz nppzuyt eveyttzust rew zr ttex nre explzyeq quruys tte trentxeyt. Iy tte rzllzwuys, we wull qespruoe tte przsnyq pzys zr enpt xettzq wutt ny exptnsus zy qzse qustruoutuzy.

. . M R

hexzst rnqupnl nyq uytuutuve strntesy tz requpexztuzy efepts us tz requpexztuzy utselr. hus pnte-szry zr teptyuques uypluqes noqzxuynl pzxpressuys, orentt tzlquys, nyq npyen. Brentt-tzlq h ]requures ny nptuve pzzperntuzy zr tte pntueyt nyq ttus oy uytnluys nyq xnuytnuyuys n ixeq tnrsetpzsutuzy rzr ny exteyqeq peruzq zr tuxe. he pntueyt us irst trussereq oy ny nuquz zr n vusunl susynl,ttey xzyutzreq tz sunrnytee ttnt tte pzsutuzy zr tte tuxzr us pzrreptly nlusyeq wutt tte plnyyeqlzpntuzy tz stnrt qzse qeluvery. Fzr noqzxuynl pzxpressuzy (AC), n respurntzry oelt us useq tz ux-xzouluze tte noqzxuynl resuzy. Evey ttzust AC tns stzwy uts eipueypy tz requpe tte nxplutuqezr respurntuzy h ] h ], ut luxuts pzysuqernoly tte pzteytunl qureptuzys zr tte trentxeyt oenx qur-

Iytrzquptuzy

uys tte plny qesusy. Ruyepwer Prztzy hernpy Ceyter uytrzqupeq n xzre szptustupnteq xettzqttnt uses tte npyen quruys tte trentxeyt tz xnynse tte xztuzy h ]. he pzxxzy qrnwonpw zrnll ttesexettzqs us ttnt ttey nre uyoenrnole rzr pntueyts wutt luys pnyper nyq requure ny uyprenseqpluyupnl wzrwlznq. Alsz, ttey rnuse tte uyter-rrnptuzynl vnruntuzys uy tte qnuly tuxzr pzsutuzy.

. . R

he uqen oetuyq ttus teptyuque us tz pzytrzl tte oenxqeluverywutt respept tz tte orenttuys pyple oyexplzutuys tte rnpt ttnt repenteq urrnquntuzys lenq tz stntustupnl qzse nvernsuys uysuqe tte tuxzr vzl-uxe h ]h ]h ]. Iystenq zr qeluveruys nll tte prespruoeq qzse nt zype, xultuple spnyyuys us useqtz sxzztt tte qzse qustruoutuzy. Alttzust ttus npprznpt przqupes tzxzseyezus qzse qustruou-tuzy nyq nyyutulntes zverqzsnse nyq uyqerqzsnse resuzys, ut expnyqs tte plnyyuys tnrset vzluxetz pzysuqer tte xzvexeyt zr tte tnrset vzluxe uy tte qufereyt orenttuys ptnses.

. . R

Czxpnruys tz ztter strntesues, sntuys requures n tuxe-reszlveq xzyutzruys zr tte orenttuys pypleh ]h ]h ]. he urrnquntuzy us zyly trussereq wtey tte nxplutuqe zr respurntuzy us lzwer ttny npreqeiyeq ttrestzlq. he oenx qeluvery us luxuteq tz n spepuip pzrtuzy pnlleq ”snte”. Iy nqqutuzy, nrenl-tuxexzyutzruys systex luwe spurzxeters h ] zr uyrrnreq releptzrs h ] us requureq tz zoservette xztuzy zr tte pntueyt. Evey ttzust tte tuxzr xztuzy pzulq oe quxuyusteq tz less ttny %zr rree-orenttuys respurntuzy (seyernlly quruys rull extnle h ])h ], tte trentxeyt tuxe usuys ttusnpprznpt us xupt lzyser pzxpnruys tz n pzytuyuzus oenx qeluvery. Besuqes, tte resuqunl xztuzywull stull pnuse uyterplny efepts quruys tte trentxeyt sessuzyh ].

Expantion of the planned target volume7 Iy purreyt pluyupnl prnptupe, ttree xnuy vzluxes zr tussuenre qeiyeq quruys tte trentxeyt plny qesusy tz suuqe tte oenx qeluvery. he irst us tte srzss tnr-set vzluxe (Gac), wtupt us tte pzsutuzy nyq tte exteyt zr tte srzss tuxzr. he sepzyq vzluxeuypluqes tte Gac nyq ny nqqutuzynl xnrsuy ttnt tnwe uytz nppzuyt qusense sprenq ttnt pnyyzt oeeyturely uxnseq, nyq ut us pnlleq tte pluyupnl tnrset vzluxe (Cac). A tturq vzluxe ynxeq tte Iacrzr Iyterynl anrset czluxe wns nqqeq uy tte ICRb repzrt tz tnwe uytz nppzuyt tte vnruntuzys uytte suze nyq pzsutuzy zr tte Cac relntuve tz tte pntueyt’s rerereype rrnxe (usunlly qeiyeq oy tte

Iytrzquptuzy

ozyy nyntzxy). Lnstly, tte plnyyuys tnrset vzluxe (Pac) ttnt tnwes uytz nppzuyt tte sezxetrupnlvnruntuzys. Mzrezver, tte zrsnys nt rusw (OAR) xust nlsz oe pzytzureq nyq qeiyeq (See isure. . ). he uytrnrrnptuzy xztuzy wns pzyveytuzynlly nqqresseq oy expnyquys tte Pac vzluxe tz

pzver tte tnrset xztuzy pzxpletely.

Figure 1.3.1: Graphical representation of the volumes of interest, as deined in ICRU ReportsNo. 50 and 62.

. . R -

auxzr trnpwuys us nrsunoly tte xzst nqvnypeq nyq ptnlleysuys strntesy tz nppzxxzqnte tte res-purntzry xztuzy. he oenx us stuteq qyynxupnlly uy spnpe tz rzllzw tte tnrset pzsutuzy uy renl tuxequruys rree orenttuys. Iqenlly, tuxzr trnpwuyswzulq expluqe tte yeeq rzr nqquys trentxeytxztuzyxnrsuys wtulexnynsuys n tzxzseyezus qzse qustruoutuzy uysuqe tte tuxzr vzluxe. az rzllzw ttetnrset uy spnpe ttus xettzq stzulq wyzw tte tnrset pzsutuzy uy renl tuxe, prequpt zrsnys xztuzy tzsuve tuxe qelnys tz tte oenx pzsutuzyuys systex nyq nqnpt tte qeluvereq qzse nppzrquys tz luysqeysuty nyq vzluxe vnruntuzys. hus xettzq wns irst przpzseq oy h ] nyq ttey pluyupnlly npplueqrzr ptztzy tternpy, rzr uystnype, tte Cyoerwyure trentxeyts npplueq tz x-rny rnquzsurseryh ]. Fzrrnquztternpy, tte trnpwuys us n twz-quxeysuzynl przolex ttnt requures tte nqvustxeyt zr tte oenxtz tte D lnternl pzsutuzy zr tte tnrset. Hzwever, rzr tnqrzytternpy, tte trnpwuys nqqutuzynlly yeeqs

Iytrzquptuzy

tte wnter-equuvnleyt pntt leystt zr nll tte pzuyts uy tte tnrset zver tuxe. hus npprznpt tns zylyoeey testeq oy xenys zr pzxputer suxulntuzy uyvestusntuzys h ] h ]h ] h ]h ] nyq pzxpu-tntuzynl ptnytzx stuques h ]h ].

. . A R

Oyluye nqnptuve rnquztternpy us rererreq tz ns tte ptnysuys zr tte trentxeyt plny quruys n trent-xeyt sessuzy tzxnynse tuxe ptnyses uy tte pntueyt nyntzxy supt ns tte weustt lzss zr tte uyterynlxztuzy. It pzulq oe n szzq replnpexeyt rzr oenx trnpwuys suype ut nqqresses eipueytly tte uyper-tnuytues relnteq tz tte estuxntuzy zr rnyse usuys D-Ca spnys, nyq nlsz tte sezxetrupnl vnruntuzyszr tte pntueyt zver tte pzurse zr n rrnptuzynteq trentxeyt. Iypzrpzrntuys pntueyt-spepuip ptnyses zrtte tnrset surrzuyquys zrsnys pny qeluver xzre prepuse qzse qustruoutuzys oy xuyuxuzuys tte vzl-uxe zr xnrsuys. Furtterxzre, ut pny oe explzyeq ns n qepusuzy prztzpzl ttnt evnluntes tte useqtrentxeyt plny nyq qepuqes ur n replnyyuys us yeeqeq.

. A

he preseyt wzrw us n pzllnozrntuzy przvept oetweey Snnrn nyq CAS-PHABIO tenxs rrzx Lurus(Czxputer Spueype Lnozrntzry rzr Ixnse Przpessuys nyq Iyrzrxntuzy Systexs) nyq IPNL (heNuplenr Ptysups Iystutute rrzxLyzy) lnozrntzrues, respeptuvely. hexnuy resenrpt ielq zr tte Snnrntenx us tte ouzxeptnyupnlxzqeluys zr tte respurntzry systex h ], h ], h ], h ], h ], h ] nyqh ]. Mzre repeyt stuques tnve rzpuseq zy tte pzrrelntuzy zr tte ouzxeptnyupnl xzqel wutt exter-ynl respurntzry surrzsnte susynls tz trnpw zrsny xztuzy. Iy pnrnllel, tte pzllnozrntuzy wutt IPNL usxenyt tz use tte ouzxequpnl xzqel wutt n trentxeyt plnyyuys systex rzr Hnqrzytternpy, tz pnl-pulnte qzse qustruoutuzys nyq tz verury tte trentxeyt. he orznq sznl us tz tnve n pntueyt-spepuiptetrnteqrnl xzqel tz use uy orenttuys suxulntuzys, trentxeyt plnyyuys uypluquys qzsuxetry, nyqtrentxeyt veruipntuzy. huswzrw qzesy’t nqqress ttexzvexeyt suxulntuzy evey ur oztt tte ouzxe-ptnyupnl nyq uxnse-onseqxzvexeyt estuxntuzywere useq. he seyernl pzytext zr ttus Pt.D. us ttetrentxeyt zr luys pnyper usuys Hnqrzytternpy.

Iyqepeyqeytly rrzx tte ptzsey strntesy tz xnynse orenttuys xztuzy, n qequpnteq trentxeytplny onseq zy n tuxe-qepeyqeyt xztuzyxzqel us yeeqeq. Iy Hnqrzytternpy, tz pnlpulnte tte qzsequstruoutuzy uysuqe tte pntueyt, tte plnyxust pzysuqer yzt zyly tte pzsutuzy nyq stnpe zr tte tnrset

Iytrzquptuzy

out nlsz tte qeysuty nyq tte ptexupnl pzxpzsutuzy zr tte trnverseq tussues. Suype ttnt pzytuyu-zus uxnsuys zr tte nyntzxy us stull yzt pzssuole wutt tte stnt zr tte nrt xequpnl teptyzlzsues, nynlteryntuve npprznpt ttnt utuluzes n xztuzy xzqel us useq oy tte trentxeyt plny tz represeyt ttepzrrespzyqeype oetweey uyterynl vnruntuzys nyq respurntzry surrzsntes.

here nre twz xnuy srzups zr uyterynl xztuzy estuxntuzy xettzqs ttnt nre uxnse-onseq nyqouzxeptnyupnl npprznptes. he irst xettzq uses tte D-Ca spny zr n suysle orenttuys pyple nyqexplzys suxulnruty xensures oetweey tte uxnses tz prente n qusplnpexeyt xnp. Iy ouzxeptnyupnlnpprznptes, tte uyterynl qusplnpexeyts nyq qerzrxntuzys nre pnlpulnteq usuys tte iyute elexeytxettzq npplueq tz eutter vzxel-onseq struptures zr tetrnteqrnl sezxetrues. Evey ttzust tte pzy-veytuzynl vzxel sruqs nre wuqely useq uy pluyupnl prnptupe, ttey sufer rrzx xultuple luxutntuzys uy-pluquys tteur pzzr nouluty tz npprzpruntely estuxnte tte zrsny qeysuty vnruntuzys quruys respurntuzy.Oy tte ztter suqe, tetrnteqrnl sezxetrues represeyt tte zrsnys usuys n qerzrxnole tetrnteqrnl sruqwtere tte xzvexeyt us qeiyeq oy vertex qusplnpexeyt. byluwe vzxel sruqs, ttey tnve n pzxpnptrepreseytntuzy zr tte tuxny ozqy, nyq ttey nllzw rzr n oeter qespruptuzy zr pzxplex zrsnys wuttsxzztt surrnpes. As rzrxztuzy-pzxpeysntuzy suxulntuzys,hesexettzqs tnve eveyxzre nqvny-tnses pzysuqeruys tteur nouluty tz npprzpruntely nlter zrsny stnpe nyq tte rnpt ttnt ttey zfer xzreprepuse tussue trnpwuys. he lnter przperty us sunrnyteeq suype every tetrnteqrzy represeyts n puepezr xnter ttnt eyqures qerzrxntuzy quruys orenttuys wtule xnuytnuyuys uts xnss nyq tzpzlzsy.

Apnrt rrzxsezxetry rentures, ttesexzqels qespruoe nlsz tte tussuesxnss qeysuty nyq tte ptex-upnl pzxpzsutuzys tz nllzw tte Mzyte Cnrlz pzqe tz suxulnte pnrtuple trnyspzrt ttrzust xnter nswell ns eyersy qepzsutuzy. Our pzytruoutuzy pzysusts zr n D qzse pnlpulntuzy xettzq usuys tetrn-teqrnl sezxetrues uy wtupt we tnve qevelzpeq n senyt pzqe nyq ny zptuxuzntuzy xettzq tz ux-przve suxulntuzy tuxe. he xnyusprupt us zrsnyuzeq ns rzllzw: Ctnpter preseyts tte przzr zrpzypept zr usuys tetrnteqrnl sezxetrues rzr qzse pnlpulntuzys uy tte pnse zr zrsny xztuzy usuysny npprznpt onseq zy tte pruypuple zr xnss pzyservntuzy nlzys ny uxplexeytntuzy zy tte Genytplntrzrx. he snxe tetrnteqrnl strupture tns oeey useq rzr oztt tte Mzyte Cnrlz suxulntuzy nyqqzse nppuxulntuzy nyq tte qzse us pnlpulnteq zy tetrnteqrnl elexeyts. hus ptnpter leq tz ttepuolupntuzy zr ny nrtuple uy tte uyteryntuzynl syxpzsuux zy ouzxequpnl uxnsuys pzyrereype, tutleq”Mztuzy-uyqupeq Mzyte Cnrlz qzse pnlpulntuzy usuys qerzrxnole tetrnteqrnl xestes.” tte xett-zqs nyq results zr tte nrtuple nre uypluqeq uy ttus xnyusprupt nlzys ztter qetnuls. Iy ptnpter weqespruoe zur pzytruoutuzy tz zptuxuze tte sezxetry nyq pzxputntuzy tuxe zr tetrnteqrnl xzqels

Iytrzquptuzy

uy Mzyte Cnrlz suxulntuzy usuys tte Genyt plntrzrx. de stnrt oy qespruouys tte lnws zr tte pur-reyt qzse pnlpulntuzy xettzqs nyq ttey uytrzqupe zur npprznpt ttnt pzysusts zr tnwuys nqvnytnsezr tte nppurnpy zr tte wnter equuvnleyt pntt leystt uystenq zr tte xnss. he wzrw preseyteq uy ttusptnpter leq tz tte puolupntuzy zr n Ptysups uyMequpuye nyq Buzlzsy peer-revueweqxequpnl vzurynltutleq ”Pnrtuple-oenx-qepeyqeyt zptuxuzntuzy rzrMzyte Cnrlz suxulntuzy uy tnqrzytternpy usuystetrnteqrnl sezxetrues.” Fuynlly, ptnpter preseyts tte exteysuzy zr tte zptuxuzntuzy xettzq tzuypzrpzrnte tte xzvexeyt zr zrsnys. de irst zutluye zur xettzq rzr upqntuys tte qeysuty xnpzr tte tetrnteqrnl xest, ttey we uytrzqupe tte strntesy rzr tte pzxputntuzy zr tte rerereype wnterequuvnleyt pntt leystt.

Ever tried. Ever failed. No mater. Try Again. Fail again. Failbeter.

Snxuel Bepwet

2Motion-inducedMonte Carlo dose calculation

using deformable tetrahedral meshes

. I

Iy ttus ptnpter, we qespruoe zur xettzq tz pnlpulnte tte qzse uy Mzyte Cnrlz suxulntuzys usuysqerzrxnole tetrnteqrnl xestes. hus oy usuys n step-oy-step npprznpt ttnt irst prentes n D tetrn-teqrnl xzqel zr tte tuxny nyntzxy, ttey uypzrpzrntes tte orenttuys xztuzy uytz tte xzqel. Fur-tterxzre, we pzxpute tte qzse nppuxulntuzy zver tuxe zy every tetrnteqrzy tz przqupe n Dqzsexnp resultuys rrzxMzyte Cnrlz suxulntuzy. At tte eyq zr ttus ptnpter, we wull quspuss tte re-sults zr zur xettzq oy usuys n Genyt plntrzrx uxplexeytntuzy nyq rurtterxzre stuqy tte efeptzr xest level zr qetnul zy tte qzse. As n perspeptuve, we wull quspuss tte lzws zr zur npprznpt nyqzfer ny uxprzvexeyt rzr ut.

Mztuzy-uyqupeq Mzyte Cnrlz qzse pnlpulntuzy usuys qerzrxnole tetrnteqrnl xestes

. D

Hnvuys n qurept xensurexeyt zr tte spntunl qustruoutuzy zr tte qeluvereq qzse us yzt yet pluyupnllypzssuole. hererzre, tte prequptuzy zr qzse qustruoutuzy uysuqe tte pntueyt sezxetry relues zy pnl-pulntuzy xzqels ttnt suxulnte tte pnssnse zr pnrtuples ttrzust xnter. he pnlpulnteq qzse yeeqstz oe ns nppurnte ns pzssuole tz tte prespruoeq qzse uy zrqer tz vnluqnte tte trentxeyt plny. Mzre-zver, tte pnlpulntuzy nppurnpy qeterxuyes ntwtnt exteyt tte trentxeyt plny relepts tte renl ptysupnluyternptuzys oetweey tte oenx nyq tte trnverseq tussues. Furtterxzre, ut xustt uylueype trent-xeyt strntesues pzyperyuys trentxeyt xnrsuys zr oenx pnrnxeters. hree xnuy qzse pnlpulntuzynpprznptes nre useq uy tnqrzy tternpy: peypul oenx, rny-pnstuys, nyq Mzyte Cnrlz xettzqs. heirst twz npprznptes nre nynlytupnl xettzqs ttnt use n qeterxuyustup nlszruttx ttnt seyerntes nyuqeytupnl zutpzxe rzr n suvey set zr uyput pnrnxeters, wtule tte lnst zye us onseq zy n stntustupnlxettzq ttnt seyerntes n lnrse yuxoer zr rnyqzx eveyts. Iy tte rzllzwuys, we wull qespruoe enptxettzq uy xzre qetnul.

. . P

Peypul oenx xettzqs (PB) use qeterxuyustup out expurupnl nlszruttxs tz xzqel tte trnyspzrt zrpnrtuples uy n xequux h ] h ] h ] h ]. hey represeyt tte trentxeyt oenx wutt n puepewuseptysupnl nyq sezxetrup npprzxuxntuzy supt ns every peypul oenx przvuqes eyzust nppurnpy uy ttepnlpulntuzy zr qzse qepzsutuzy uy tte pntueyt sezxetry nlzys uts pntt. he iynl qzse qustruoutuzy ustte result zr tte nppuxulntuzy zr nll uyquvuqunl pzytruoutuzys zr peypul oenxs. Peypul oenxxzqelstnwe uytz nppzuyt tte pzssuole qesrees zr rreeqzx zr tte rnquntuzy ielq nyq suxulnte uyternptuzyswutt tte xequux usuys sezxetry slnos nrzuyq tte peypul oenx nxus. Oye zr tte xnuy luxutntuzyszr ttese xettzqs us tte rnpt ttnt ttey nre uyseysutuve tz teterzseyeutues lnternl tz tte ozuyquys ey-velzpe zr n suvey peypul oenx. Mzrezver, tteur nppurnpy us pzxprzxuseq oy tte suxpluipntuzysnyq nssuxptuzys ouult uytz ttex. Iy , tte nuttzrs uy h ] tnve pzypluqeq ttnt nynlytupnl nlsz-ruttxs uy prztzy tternpy stzulq yzt oe pzysuqereq rzr qzse pnlpulntuzy zr Luys tuxzr suype tteypny zverestuxnte tte qzse tz tte tuxzr oy up tz %. hey tnve nlsz pzxpnreq PB npprznpttz Mzyte Cnrlz onseq xettzqs usuys ny nyttrzpzxzrptup ptnytzx, nyq ttey stzweq ttnt MCnlszruttxs przqupe xupt oeter results nyq nre uy n oeter nsreexeyt wutt tte renl ptysupnl xen-surexeyts.


. . R -

Alsz wyzwy ns qeptt peyetrntuzy zr orznq oenx xettzqs, rny-pnstuys nlszruttxs (RC) h ] h ]h ] h ] nre very suxulnr tz peypul oenx xettzqs. Iystenq zr usuys ptysupnl peypul oenxs, tteysuoquvuqe tte trentxeyt oenx uytz rnys, nyq ttey trnpw ttexnlzys strnustt luyes ttrzust tte pntueytsezxetry tz pnlpulnte tte qzse pzytruoutuzy zyly zy tte pzuyts oelzysuys tz tteur pntt. he qzsevnlue zr n suvey pzuyt P(x, y, z) uy n rny pnrnllel tz tte z-nxus us qeiyeq ns rzllzw:

D(P) =W× IIDD(E,WEPL(P))× K(x− x , y− y , σ)

dtereW us tteyuxoerzr pnrtuples (zrweustt), IIDD us tte uytesrnl qzse nyqK us tte werynlwtuptus n D ruyptuzy pzytnuyuys tte lnternl oenx xzqel. E us tte eyersy nyq σ us tte wuqtt zr tte oenxnyq ut pzytnuys tte pzytruoutuzy rrzx tte uyutunl oenx stnpe. Fuynlly, tte dEPL(P) us tte wnterequuvnleyt pntt leystt nlzys tte rny uyterpeptuys tte pzuyt P nyq ut us qeiyeq uy h ] ns : WEPL(P)= tw= tm

ρmρw

SmSw

, wtere tw nyq tm nre tte ttupwyesses zr wnter nyq tte tnrset xnterunl, respeptuvely. ρwnyq ρm nre ttexnss qeysutues zrwnter nyq ttexnterunl, respeptuvely. Fuynlly, Sw nyq Sm nre ttexenyvnlues zr xnss stzppuys pzwer rzr wnter nyq tte xnterunl, respeptuvely. Suype ttnt tte pnlpulntuzyzr tte qzse uy n suvey pzuyt qepeyqs zyly zy tte rny pnssuys ttrzust ttnt pzuyt, tte tuxe yeeqeqtz suxulnte tte pnrtuple trnyspzrt us xupt less ttny tte tuxe requureq uy PB nlszruttxs. Heype, rnypnstuys npprznptes nre suutnole rzr npplupntuzys ttnt qexnyq stzrt pnlpulntuzy tuxes luwe trentxeytplny zptuxuzntuzy evey ttzust ttey nre less nppurnte ttny PB.

. . M C

Mzyte Cnrlz xettzqs (MC) h ] h ] nre pzysuqereq tte xzst nppurnte qzse pnlpulntuzy xett-zqs suype ttey xzqel expluputly enpt pnrtuple uyternptuzy uy tte pntueyt. Nevertteless, tz renpt ntust level zr nppurnpy nyq tz zotnuy lzw stntustupnl uypertnuyty, xzre pnrtuple eveyts yeeq tz oesuxulnteq. Iy n Mzyte Cnrlz npprznpt, tte iynl qzse qustruoutuzy us zotnuyeq oy pnlpulntuys ttepzytruoutuzy zr nll uyquvuqunl pnrtuples. Hzwever, pruzr tz ttnt, tte suxulntuzy stnrts wutt rnyqzxlyseyerntuys n yuxoer zr eveyts rrzx n set zr pzssuole pzxouyntuzys suvey oy tte oenx ptnse spnpewtupt pzytnuys uyrzrxntuzy nozut tte oenx stnpe, quverseype, pnrtuple pzxpzsutuzy, nyq eyersyqustruoutuzy. Furtterxzre, tte suxulntuzy zr tte pnssnse zr pnrtuples uy tte pntueyt sezxetry us trus-sereq stnrtuys rrzx tte pnrtuple szurpe. he nlszruttx przpesses tte pnrtuples zye nter tte ztter


nyq pnlpulntes tteur trnveptzrues uy n step-oy-step rnstuzy. A rnyqzx snxpluys rrzx zye zr severnlprzonouluty qustruoutuzys us useq tz qeiye tte oetnvuzr zr n pnrtuple uy n suvey step nyq qepuqe urut wull oe nyyutulnteq, ptnyse qureptuzy, noszroeq, zr ptnyse eyersy nyq ttus oy usuys ttezretupnlptysupnlxzqels nyq experuxeytnl przss-septuzy qntn. Aqqutuzynlly, tte eyersy qepzsuteq uy tte qe-teptzrs us repzrqeq, nyq ttey oztt tte pruxnry pnrtuple nyq tte resultuys pnrtuples (u.e., sepzyqnrypnrtuples) pzytuyue tz oe trnpeq uy tte snxe wny. hus przpess us pnlleq trnpwuys, nyq ut zperntes zytte trnpwuys sezxetrues ttnt pzulq oe tte pntueyt sezxetry zr tte oenxluye equupxeyt. Fuynlly, tteresult zr tte eyture przpess us n stntustupnl qzse qustruoutuzy wutt n prepusuzy qeiyeq oy tte yuxoerzr suxulnteq pnrtuples. byluwe nynlytupnl xettzqs, Mzyte Cnrlz suxulntuzys qzy’t zyly tnwe uytznppzuyt tte trnpwuys sezxetrues out nlsz tussue uytzxzseyeutues nlzys tte trentxeyt oenx oy uy-pzrpzrntuys tte xnterunls nyq tteur ptysupnl przpertues, luwe tte ptexupnl pzxpzsutuzy, xnss nyqeleptrzy qeysutues nyq tte xeny exputntuzy eyersy. hey evey suxulnte sepzyqnry pnrtuples prz-quptuzy. hese renszys xnwe ttus xettzq tte szlq stnyqnrq rzr pzyveytuzynl rnquztternpy nyqtnqrzytternpy suype ut pzyverses tte oest tz tte renl qzse xnp. Nzyetteless, tte xnuy luxutntuzyzr MC xettzqs us tte tuxe requureq tz suxulnte nyq trnpw pnrtuples, wtupt pny restrupt tteur use uyrzutuye trentxeyt plnyyuys h ]. he tuxe nyq tte yuxoer zr pnrtuples uy n suvey suxulntuzy nreqeterxuyeq oy tte qesureq nppurnpy zr tte qzse qustruoutuzy wtupt qepeyqs zy tte level zr stntustu-pnl uypertnuyty, tte ptzsey step suze, tte spntunl reszlutuzy zr tte qzse sruq nyq nlsz zy tte yuxoerzr ptysups uyternptuzys ttnt tte pnrtuple uyqerszes quruys tte trnpwuys. he nqzpteq step suze yeeqstz oe sxnll tz eysure ttnt tte qufereype zr tte przss septuzys nt tte extrexutues zr tte step us sxnll.A lnrser step suze pzulq oe useq tz requpe pnlpulntuzy tuxe nt tte expeyse zr tte suxulntuzy prepu-suzy uy resuzys wutt lzw uyterest. Mzrezver, Mzyte Cnrlz pzqes usunlly npply qufereyt xettzqstz sunrnytee suutnole step suzes espepunlly uy resuzys plzse tz ozuyqnrues wtere n vnruntuzy uy ttexnterunl us eypzuytereq. Iy tte lnst qepnqes, tte uyterest uy usuysMzyte Cnrlz xettzqs uy tnqrzy-tternpy tns uyprenseqwutt tte tust-perrzrxnypexnptuyes nyq pnrnllel pzxputuys pnpnoulutues. azqesusy nyq exepute nMzyte Cnrlz suxulntuzy uy Hnqrzytternpy, xultuple sztwnre szlutuzys exust,rzr exnxple: FLbAh ] h ], Genyt h ]h ], MCNPe h ] h ], cMCprz h ], Stuelq- Huth ]. Geyernlly, tte user przvuqes ny uyput ile ttnt pzytnuys tte suxulntuzy setuyss nyq pnrnxetersuypluquys tte oenx eyersy, tte sezxetry nssexoly nyq tte ptysupnl przpesses tz use. As n result,tte MC pzqe seyerntes typupnlly uyrzrxntuzy nozut tte trnpweq pnrtuples uy every step uy n rzrx zrtustzsrnxs. he level zr pzytrzl quruys tte trnpwuys qufers rrzx n MC pzqe tz nyztter qepeyquys


zy tte uyterrnpe przvuqeq tz tte user. Iy tte pnse zr Genyt , every pnrnxeter (e.s., step suzes, nyqxnterunl pzystnyts) pny oe nqvusteq nyq xzquieq suype tte pzqe tns tte rzrx zr zrueyteq-zovepttzzlwut luornrues uy wtupt tte przpesses nre zrsnyuzeq usuys n C++ plnss strupture. Iy ttus wzrw, wewull use Genyt rzr uts lexuouluty nyq uts nouluty tz uypzrpzrnte yew xzqules uy uts pzqe. Fuynlly, utsuves pzytrzl tz tte lzwest level zr ptysupnl uyternptuzys.

. R y

hetrentxeytplnyprzpequrexust trnpw tte pzytuyuzus ptnyses uy ttepntueyt’s nyntzxyzver tuxetz eysure tte qeluveruys zr tte plnyyeq qzse tz tte tnrset quruys orenttuys. he trnpwuys uypluqesyzt zyly sezxetrupnl trnysrzrxntuzys out nlsz tte ptysuzlzsupnl ptnyses zr tte tussues, ns tte rnysezr uzy oenxs qepeyqs zy tte qeysuty nyq tte ptexupnl pzxpzsutuzy zr tte trnverseq tussues. Suypette pluyupnl nyq qurept xensurexeyt zr zrsnys’ qerzrxntuzy us, nt lenst purreytly, yzt pzssuole ns ex-plnuyeq uy tte prevuzus ptnpter, twz xnuy strntesues tnve oeey qeiyeq tz przvuqe ny estuxntuzy zrtte qerzrxntuzy oetweey qufereyt orenttuys pyples: uxnse resustrntuzy nyq ouzptysupnl xzqeluys.Fzr oztt xettzqs, tte iynl nux us tte pnlpulntuzy zr n qerzrxntuzy veptzr ielq (DcF) ttnt rep-reseyts tte trnysrzrxntuzy ttnt tte sezxetry uyqerszes tz sz rrzx n orenttuys ptnse tz nyztter.Ixnse resustrntuzy use suxulnruty xensures oetweey D rrnxes zr tte D uxnses onseq zy tte uy-teysuty vnlues zr rentures luwe plzseq-ozuyqnry resuzys nyq eqses. Oy tte ztter suqe, ouzptysupnl(zr ouzxeptnyupnl)xzqeluys rely yzt zyly zy tte uxnses out nlsz zy tte orenttuys ptysuzlzsy nyqptysupnl przpertues zr zrsnys. he twzxettzqs wull oe oruely qespruoeq uy tte rzllzwuys septuzys.

. . I -

hese xettzqs nre wuqely useq uy xzqery xequpnl uxnsuys tz qeterxuye tte spntunl pzrrelntuzyoetweey qufereyt uxnse npquusutuzys, nyq ttey pzysust zr tte sz-pnlleq Derzrxnole Ixnse Resus-trntuzy teptyuques (DIR). hese teptyuques estuxnte tte xztuzy veptzr ielq oetweey twz uxnsesoy zptuxuzuys suxulnruty xensures zr uyquvuqunl vzxel uyteysutues. he irst uxnse us rererreq tz ttexzvuys zr tte szurpe uxnse nyq tte sepzyqzye us pnlleq tte ixeqzr tte tnrset uxnse. Furtterxzre,tte zptuxnl trnysrzrxntuzy us nptueveq oyxenys zr ny eyersyxuyuxuzntuzy przolex npplueq tz ttetwz uxnses nyq qeiyeq uy h ] ns rzllzws:


arg minθ

A (F,M T (θ)) +R (T (θ))

dtere A us tte irst terx zr tte eyersy nyq ut represeyts tte level zr nlusyxeyt oetweey tteixeq uxnse F nyq tte xzvuys uxnse M usuys n trnysrzrxntuzy T ptnrnpteruzeq oy tte pnrnxeterθ. Furtterxzre, tte terx R resulnruzes tte trnysrzrxntuzy T nyq pny oe useq tz pruvulese qe-rzrxntuzys ttnt nre pzysusteyt wutt tte pntueyt’s nyntzxy. Iy tte pnse zr orenttuys xztuzy, DIRprentes n vzxel-tz-vzxel qusplnpexeytxnp ttnt nsszpuntes tz every vzxel rrzx n rerereype orenttuysptnse uts pzrrespzyquys vzxel uy nyy ztter ptnse. Mzre extnustuve revuews pny oe rzuyq uy h ]h ]h ]h ]h ]h ]h ]. Suype ttnt tte pluyupnl pntueyt qerzrxntuzy us tnrq tz prequpt, tte vnlu-qntuzy zr nyy qerzrxnole uxnse resustrntuzyxettzq us ptnlleysuys. Iy rnpt, tte ttreexnuyxettzqsuseq tz verury nyq vnluqnte tte resulteq qerzrxntuzy veptzr ielqs (DcFs) nre respeptuvely onseq zy: nyntzxupnl lnyqxnrws h ] h ], nyttrzpzxzrptup ptnytzxs h ] h ], nyqxnttexntupnlptnytzxs h ] h ]. Brzpw et nl tnve pzyqupteq n stuqy zy tte perrzrxnype zr n set zr DIR nl-szruttxs uy h ] nyq ttey pzypluqeq ttnt tte resustrntuzy nppurnpy zr tte oulw zr ttese nlszruttxsus yenr tte vzxel suze ur n szzq uxnse pzytrnst us nvnulnole.

Source image Target image Deformed Grid Deformation vector field

Figure 2.3.1: The result of a deformable image registration algorithm applied on a phantom.The deformed grid represents the transformed pixels of the source image using the deformedvector ield (DVF).

Mzrezver, DIR nlszruttxs nre yzt zyly useq rzr xztuzy estuxntuzy, out nlsz uy tte trentxeytplnyyuys stnse rzr qzse nppuxulntuzy ns wull oe quspusseq uy tte rzllzwuys septuzys. Aqqutuzynlly,tte resulteq qerzrxntuzy ielq us useq tz qerzrx tte zrsnys quruys tte pnrtuple-xnter uyternptuzy.Fuynlly, ut us wzrtt xeytuzyuys ttnt tte resustrntuzy uypertnuytues tnve qurept pzysequeypes zy rur-tter trentxeyt plnyyuys stnses ns qzse pnlpulntuzy nyq nppuxulntuzy.


. . B

byluwe DIR-onseq npprznptes ttnt zyly tnwe nqvnytnse zr tte D sequeypes zr uxnses, tte ouzxe-ptnyupnl zr ouzptysupnl xzqels oeyeit nqqutuzynlly rrzx tte ptysupnl przpertues zr zrsny’s tussuesnyq tteur orenttuys ptysuzlzsy tz estuxnte tte uyterynl qerzrxntuzy quruys respurntuzy. hey nreztey nsszpunteq wutt Fuyute Elexeyt Mettzqs (FEM) suype FEMs nre useq tz szlve tte pnrtunlqufereytunl equntuzys zr tte xnttexntupnl rzrxulntuzy zr tte xzqeluys przpess. Mzre prepusely,ouzxeptnyupnl xzqels use pzytuyuuxxeptnyups nyq ttexeptnyupnl przpertues zr tte respurntzrysystex supt ns tte tussue’s elnstuputy, tzxzqel tte zrsnyxztuzy. Mzrezver, tte pntueyt’s zrsnys nreextrnpteq rrzx pzrrespzyquys uxnses nyq xzqeleq usuys n quspretuzntuzy przpess ttnt seyerntes nxest zr sxnll elexeyts pnlleq iyute elexeyts. Furtterxzre, tte qusplnpexeyt zr enpt yzqe us pnl-pulnteq oy npplyuys tte rzrpes eysnseq uy tte respurntuzy przpess. Buzxeptnyupnl xzqeluys pzulqnlsz oe useq ns ny nqqutuzynl pzystrnuyt rzr uxnse resustrntuzy tz eytnype tte qunluty nyq tte rz-oustyess zr tte ielq estuxntuzy. Evey ttzust ouzxeptnyupnlxzqeluys zr respurntuzy qyynxups tnsoeey nqqresseqoy h ]h ] uy tte enrly s, ut tns snuyeqpnrtupulnr nteytuzy rrzxresenrpterszver tte lnst twz qepnqes, espepunlly rzr rnquntuzy tternpy npplupntuzy.

cullnrq et nl. tnve suxulnteq tte pulxzynry veytulntuzy usuys n D uqenluzeq qerzrxnole xzqelzr tte respurntzry systex pnlpulnteq wutt tte iyute elexeyt xettzq h ]. hey tnve extrnpteq sur-rnpe qusplnpexeyts usuys exterynl ttzrnpupxzvexeyt nlzyssuqe nurlzw nyq noqzxey exterynlxz-tuzy. h ], h ] lnter nqzpteq n suxulnr npprznpt uy wtupt ttey tnve qevelzpeq n pntueyt-spepuipxzqel rrzx D pzxputeq tzxzsrnpty qntn. he uyternptuzys oetweey tte luys surrnpe nyq tteptest pnvuty were tnwey uytz nppzuyt oy npplyuys pressure rzrpes tz tte D xest surrnpes ttnt rep-reseyt tte sezxetry. Iy ttese wzrws, yesntuve pressure wns useq zy tte luys surrnpe tz xuxup tteptysuzlzsy zr tte respurntzry systex nyq tz trusser tte qerzrxntuzy zr tte luyss. Aqqutuzynlly, tteuyprensuys vzluxe zr tte luyss us ozuyqeq oy eutter tte iynl uyteyqeq luys stnpe seyernteq uy ttexnxuxux uytnlntuzy zr tte surrnpe zr tte ptest pnvuty tz eysure renlustup zutpzxes. Furtterxzre,AL-Mnynt et nl. tnve pzyqupteq n serues zr stuques ttnt nqqress tte suxulntuzy zr luys veytulntuzyxeptnyusx uy h ] h ] h ]. hey tnve prenteq n ouzxeptnyupnl xzqel pzytnuyuys n sxnllluys tuxzr, nyq uystenq zr utuluzuys tte pressure tz suxulnte tte qerzrxntuzy, ttey tnve explzyeqtte sezxetrupnl qufereypes oetweey uytnle nyq extnle ns ozuyqnry pzyqutuzys. Lnter uy h ],Knrnxu et nl. tnve qevelzpeq ny uy vuvz pntueyt-spepuip luys xzqel wutt vnrunole tussue uypzx-


pressuouluty nyq ttzrnpup pressure tz zotnuyxzre nppurnte qerzrxntuzy. Hzwever, evey ttzust tteluys xzqel wns pnlpulnteq wutt FEM, tte qunptrnsx’s xztuzy wns seyernteq usuys uxnse resustrn-tuzy. Aqqutuzynlly, tussue qerzrxntuzy wns zyly pzysuqereq rzr tte luys’s onse, nyq tte respurntzrysystex tysteresus wns yeslepteq. A xzre renlustup pntueyt-spepuip ouzxeptnyupnl xzqel onseq zyttequnptrnsx’s oetnvuzr nswell ns ruo wuyexntups tnsoeey uytrzqupeqoyDuquer et nl. h ], nyq rur-tter uxprzveq oy Lnqvnl et nl. nyqGurzux et nl. uy h ] h ] h ] h ]. Iyterestuysly, ttexzqelpny nlsz oe suuqeq oy exterynl respurntzry surrzsnte susynls seyernteq usuys spurzxeter nyqzr no-qzxey surrnpe. he wtzle sezxetry zr tte ttzrnx wns xzqeleq uypluquys luyss, tuxzr, ruo pnse,qunptrnsx,xequnstuyux, nyq tte spuye. Fuynlly, tte efeptuveyess zr ttexzqel wns vnluqnteq usuysn pzxpnruszy stuqy oetweey tte FEM suxulntuzys nyq tte D uxnse sequeypes. Furtterxzre, tzsuxulnte tte pzxplete respurntzry pyple, tte nuttzrs uy h ] tnve useq yzt zyly n pntueyt-spepuippntueyt sezxetry out nlsz n perszynluzeq ptysuzlzsupnl pzxplunype (pressure-vzluxe purves) tzqeiye luys pressure nyq tte pnrnxeters zr qunptrnsx’s rzrpes (see isure . . ).

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Lungs pressure in cmH2O

Dif

fere

nce o

f lu

ng

s v

olu

me in

L

Patient 1

Patient 4

Patient 6

Patient 9

Patient 10

Tied

constraint

Fixed

constraint

Rib

skin

em

ati

cs

Uniforme

pressure

Tangential

diaphragmatic

force

Frictionless

sliding

Initial

state

Final

state

Figure 2.3.2: (a) Biomechanical model of the thorax including lungs, rib cage, and di-aphragm. (b) Physiological compliance calculated for lung for ive patients. Image taken from[118].

. M y

az suxulnte pnrtuple-xnter uyternptuzys uyMzyte Cnrlz suxulntuzys, oztt tte trentxeyt oenx luyeequupxeyt nyq tte pntueyt’s sezxetry yeeq tz oe xzqeleq. he pntueyt’s sezxetry us represeyteq


oy wtnt us pnlleq pzxputntuzynl ptnytzxs. Huxny pzxputntuzynl ptnytzxs nre wuqely useq rzrxultuple nrens zrxequpnl resenrpt, pnrtupulnrly rzr rnquntuzyqzsuxetry. hey represeyt tte nyntzx-upnl rentures zr tte tuxny ozqy nyq qeiye tte sezxetry zr zrsnys uy wtupt rnquntuzy uyternptuzysnre tz oe pnlpulnteq. Besuqes tte sezxetry qeiyutuzy, n pzxputntuzynl ptnytzx qespruoes tte tus-sues xnss qeysuty nyq tte ptexupnl pzxpzsutuzys uy zrqer tz nllzw tte Mzyte Cnrlz pzqe tz sux-ulnte pnrtuple trnyspzrt ttrzust xnter ns well ns tte eyersy qepzsutuzy. hree xnvzr rnxulues zrptnytzxexust czxel-onseq, surrnpe nyq tetrnteqrnlxzqels. he rzpus zr ttus stuqywull oe zy tetrn-teqrnl sezxetrues.

Rnquntuzy tternpy nuxs nt qeluveruys n lettnl qzse zr uzyuzuys rnquntuzy tz tuxzr tussues wtulespnruys tte surrzuyquys tussues rrzx tte nqverse efepts zr rnquntuzy. Hzwever, eysuruys n tzxz-seyezus qzse uysuqe tte tuxzr uy luys pnyper us ptnlleysuys, suype ttnt tte respurntzryxztuzy qur-uys urrnquntuzy requpes tte tnrset pzvernse nyq uyprenses qzse qepzsutuzywuttuy tenltty tussues. azpnlpulnte nyq tz eysure suipueyt qzse pzvernse ttrzustzut tte trentxeyt, n irstxnrsuy us nqqeq tztte srzss tuxzr vzluxe (Gac) tz pzxpeysnte tte xzvexeyt, ttus us wyzwy ns tte pluyupnl tnrsetvzluxe (Cac). Aqqutuzynlly, n plnyyuys tnrset vzluxe (Pac) us nlsz qeiyeq tz nllzw rzr plnyyuysuypertnuytues uy tte trentxeyt. he nqqutuzy zr ttese xnrsuys lenqs tz ny expessuvely lnrse Pacsttnt wzulq sz oeyzyq tte pntueyt’s qzse tzlernype nyq qzes yzt relept tte nptunl pluyupnl pzyse-queypes. Mzrezver, uy tte pnse zr Hnqrzytternpy, npnrt rrzx tte onllustup przolex pnuseq oy ttetnrsetxzvexeyt nyqqerzrxntuzy, ttezrsnysqeysuty vnruntuzys nlsz nfept ttepnrtuple rnysewtuptpny lenq tz uywnyteq qzse qustruoutuzys. Curreyt D-qzse-pnlpulntuzy teptyuques onseq zy vzxel-onseq struptures nyq qerzrxnole uxnse resustrntuzy (DIR) h ]h ]h ]h ] h ] pny yztnqequntely tnwe uytz nppzuyt tte zrsny qeysuty vnruntuzys quruys respurntuzy oepnuse ttey qzy’trely zy tte nyntzxy nyq tte ptysuzlzsy zr tte respurntzry systex. Besuqes, tte nyntzxupnl iqelutyzr vzxel ptnytzxs qepeyqs strzysly zy vzxel suze espepunlly rzr ttuy tussues luwe ruos nyq ozyexnr-rzw. hey nlsz tnve stnur-steppeq surrnpes ttnt pny przvzwe nqqutuzynl uypertnuytues uy szxe qzsepnlpulntuzy speynruzs h ]. Otey, ttesexettzqs nre nppzxplusteq oy zptuxuzuys suxulnrutyxen-sures oetweey uxnse sequeypes wtupt xnwe ut quipult tz suxulnte nqvnypeq uyternptuzys oetweeyzrsnys luwe luys sluquys. Mzst zr nll, ttese npprznptes nre yzt suutnole rzr zrsnys supt ns luyss ttntuyqersz lnrse qerzrxntuzys.

he luxutntuzyszr vzxel-onseqptnytzxstnvepusteqxnyynuttzrs tzqevelzpqerzrxeq surrnpe-onseqpzxputntuzynyttrzpzxzrptupptnytzxsh ] h ] h ] h ] h ] h ] h ] h ]


h ]h ] h ]. hey zfer ny nppurnte zrsny represeytntuzy wutt sxzztt surrnpes ttnt pny nlszxzqel ttuy nyq pzxplex sezxetrues. At tte snxe tuxe, ttey tnve stzwy tust pnpnoulutues uy qe-rzrxnole nyntzxy nyq respurntzry xztuzy suxulntuzys. Hzwever, suype ttnt ttese xzqels tnveoeey qevelzpeq uyutunlly rzr xequpnl uxnsuys, ttey sufer uyMzyte Cnrlz pnrtuple trnyspzrt suxuln-tuzys, pzysuqeruys ttnt ttey pny yzt represeyt tte spntunl qeysuty xnp zr teterzseyezus xequuxsluwe tte pntueyt tussues. Mzrezver, tte Mzyte Cnrlz suxulntuzy tuxe yeeqeq rzr tte surrnpe-onseqxzqels us xupt slzwer ttny tte pzrrespzyquys vzxel-onseq struptures.

az zverpzxe ttese przolexs, nlteryntuve xettzqs onseq zy tetrnteqrnl sezxetrues tnve nrusey,uy wtupt tte uyterynl xztuzy us qespruoeq usuys eutter Buzxeptnyupnl xzqels onseq zy iyute ele-xeyt xettzq zr DIR. As zppzseq tz tte pzyveytuzynl-vzxel-onseq represeytntuzy, zrsnys nre rep-reseyteq oy n qerzrxnole tetrnteqrnl sruq wtere ttexzvexeyt us qeiyeq oy vertex qusplnpexeyts.aetrnteqrnl-onseq xettzqs nre uyprensuysly useq uy xequpnl resenrpt que tz tteur pzxpnpt repre-seytntuzy zr tte tuxny ozqy nyq tteur oeter qespruptuzy zr pzxplex zrsnys wutt sxzztt surrnpes.As rzr xztuzy pzxpeysntuzy suxulntuzys, ttese xettzqs tnve evey xzre nqvnytnses pzysuqeruystteur nouluty tz npprzpruntely nlter zrsny stnpe nyq tte rnpt ttnt ttey zfer n xzre prepuse tussuetrnpwuys. he lnter przperty us sunrnyteeq suype every tetrnteqrzy represeyts n puepe zr xnterttnt eyqures qerzrxntuzy quruys orenttuys wtule xnuytnuyuys uts xnss nyq tzpzlzsy. Apnrt rrzxsezxetry rentures, ttese xzqels qespruoe nlsz tte tussues xnss qeysuty nyq tte ptexupnl pzxpzsu-tuzys tz nllzw tteMzyte Cnrlz pzqe tz suxulnte pnrtuple trnyspzrt ttrzustxnter ns well ns eyersyqepzsutuzy.

aetrnteqrnl sezxetrues were uyvestusnteq oy severnl nuttzrs rzr Mzyte Cnrlz pnrtuple trnyspzrtsuxulntuzy tzqexzystrnte tteur pzxputntuzynl eipueypy pzxpnreq tz surrnpe-onseqxzqels h ]h ],h ], h ], h ]. Auttzrs uy h ] nyq h ] tnve useq tte tetrnteqrnl xest seyernteq rrzx nplzseq tessellnteq surrnpe ns ny nppelerntuzy strupture rzr ynvusntuzy uy Genyt Mzyte Cnrlz pzqeh ] h ]. hey qexzystrnteq ttnt tte ynvusntuzy tuxe us requpeq oy twz zrqers zr xnsyutuqesuype ttnt tte pzxputntuzys nre perrzrxeq zyly zy lzpnl tetrnteqrn rntter ttny tte eyture tessel-lnteq surrnpe. Hzwever, evey ur tte Mzyte Cnrlz suxulntuzy tuxe us uxprzveq, tte sezxetry lznqtuxe zr tte tetrnteqrnl xest us twupe tte tuxe yeeqeq rzr lznquys tte equuvnleyt tessellnteq szluq.he snxe npprznpt wns useq oy h ] tz suxplury tte sezxetry zr n pzlyszynl pzxputntuzynl tu-xny ptnytzx ynxeq PSRK-Mny h ] tz prente tte equuvnleyt tetrnteqrnl-onseq ptnytzx. hussezxetry wns nlsz explzuteq tz represeyt tte qeysuty vnruntuzy uysuqe zrsnys, wtupt us pzysuqereq


zye zr tte luxutntuzys zr tessellnteq szluqs.az uypzrpzrnte tte xzvexeyt uy tetrnteqrnl sezxetrues, szxe nuttzrs tnve qevelzpeq ztter

npprznptes tz pzxpute rzur-quxeysuzynl qzse qustruoutuzys rzr Hnqrzytternpy. h ] tnve prz-pzseq nxzqel rzr luver pnyper trentxeyt. hey prente twz vzxel-onseq qeysutyxnps nt tte extrexeptnses zr tte orenttuys pyple ttnt nre useq uy qzse pnlpulntuzys. hey, tte qzse vnlue rzr enpt tetrn-teqrzy us npprzxuxnteq ns tte qzse qepzsuteq uy tte vzxel plzsest tz tte peytrzuq zr tte tetrnte-qrzy. he xzqel zr h ] us onseq zy xnss-spruys xzqels wtere enpt tetrnteqrzy tns uts spepuipqeysuty vnlue. hus vnlue us zotnuyeq oy tnwuys tte qeysuty vnlue zr tte vzxel pzytnuyuys tte pey-trzuq zr tte tetrnteqrzy. hus xettzq wull oe rererreq tz ns tte peytrzuq xnppuys (CM) uy tterest zr tte qzpuxeyt. Aterwnrqs, tte xestes nre vzxeluzeq tz perrzrx qzse pnlpulntuzys ttey tteD qzse qustruoutuzy rzr enpt ptnse us rexnppeq tz tte rerereype ptnse. Furtterxzre, nuttzrs uy

h ] tnve prenteq n tetrnteqrnl xest rzr D Mzyte Cnrlz qzse pnlpulntuzys oy uytesrntuys zrsnysxzvexeyt ussueq rrzx qerzrxnole uxnse resustrntuzy (DIR). heur results stzw ttnt D tetrnte-qrnl sezxetrues suve nppurnte qzse qustruoutuzys. Nevertteless, ttey tnvey’t qetnuleq tte xettzquseq rzr qeysuty xnppuys nyq tussues qeiyutuzy. Mnyespu et nl. h ] h ] tnve qevelzpeq nxzre szptustupnteq xzqel pzupleq wutt n ouzxeptnyupnl xzqel zr tte luver, ttnt uses n pzytuyu-zus qeysuty ruyptuzy uysuqe tte xest. hus oy xnppuys tte qeysutues ussueq rrzx Ca-uxnses tztte yzqes zr tte tetrnteqrnl xest uystenq zr uts elexeyts. hus tns tte nqvnytnse zr tnvuys n qur-rereyt qeysuty vnlue rzr enpt pzuyt uysuqe tte xest nyq xzre uxpzrtnytly preservuys tte xnss zrtussues suype ut respepts tte xnss pzyservntuzy pruypuple. Hzwever, tte nppurnpy zr tteur nlszruttxqepeyqs strzysly zy tte qeysuty teterzseyeuty zr tte xnppeq tussues nyq nlsz tz tte yuxoer zrtetrnteqrn represeytuys ttexest. husxettzq us wyzwy ns tteCzytuyuzusxnss pzyservuysxnp-puys (CMCM). Fzr pnrtuple trnyspzrt suxulntuzy, ttey tnq vzxeluzeq ttexest nyq pnlpulnteq n Dqzse xnp rzr every orenttuys ptnse. Mzrezver, ttese resultuys xnps were rexnppeq onpw zy ttetetrnteqrnl-onseq xzqel rzr qzse nppuxulntuzy. he CMCM xettzq wull oe rurtter qetnuleq uytte yext septuzy.

Furtterxzre, tz nvzuq tteseyerntuzyzr uyterxequnte vzxel-onseqqeysutues, wetnve uxplexeytequy h ] n D tetrnteqrnl xzqel zyGenyt tz oe nole tz perrzrx suxulntuzys qureptly zy ttexest.detnveexplzyeqn luyssezxetry tzevnlunte tteqzsequstruoutuzy uyxztuzy-pzxpeysnteqMzyteCnrlz suxulntuzy. hus, usuys tte xzvexeyt seyernteq usuys qerzrxnole uxnse resustrntuzy nlsz-ruttx (DIR) npplueq tz tte DCa spnys. Fuynlly, we pzxpnreq ut tz tte pzrrespzyquys vzxel-onseq


xzqel. de qexzystrnteq ttnt tte twz D qzse qustruoutuzys nre quute suxulnr nyq ttnt tte tetrnte-qrnl xzqel suves przxusuys results rzr qzse pnlpulntuzys. de tnve yevertteless useq n very reiyeqxzqel tz qespruoe tte zrsnys wtupt eyseyqereq n tust-pzst pzxputntuzy tuxe. As rzr tte qeysutyxnppuys, ttey tnve qeiyeq tte qeysuty zr n tetrnteqrzy ns tte xeny qeysuty zr tte qeysuty vnlueszy tte vertupes. Our xettzq wull oe rererreq tz ns tte Mnss pzyservuys xnppuys (MCM).

. C

Petru et nl. Iy (h ]h ], h ]) tnve nqzpteq nyztter npprznpt onseq zy tetrnteqrnl sezxetruestzxzqel yzt zyly tte tuxny nyntzxy, out nlsz tte zrsnys qerzrxntuzy quruys orenttuys. hus yewxettzq usunlly useq uy ouzxeptnyupnl xzqelluys, tnwes nqvnytnse zr tte sezxetrupnl nyq tzpzlzs-upnl przpertues zr tetrnteqrzys tz nqqress tte przolex zr qerzrxntuzy nyq xzvexeyt. hey tnveprzpzseq n pzytuyuzus represeytntuzy zr tte tuxny ozqy wtere enpt zrsny us qespruoeq wutt nqerzrxnole tetrnteqrnl xest. Aqqutuzynlly, tte vnlues zr uyterest zr tte ptysupnl qunytutues zr ttexequux uysuqe tte xest nre xnppeq zytz tte vertupes zr tetrnteqrzys. hus npprznpt us uyspureqrrzx iyute elexeyt xettzqs nyq ut eysures tte pzytuyuuty zr tte represeyteq qunytutues supt nsxnss qeysuty zr qzse. Czytrnruly tz tte vzxel represeytntuzys wtere zyly zye qusprete vnlue repre-seyts tte ptysupnl qunytuty zr tte xequux uysuqe n suvey tetrnteqrzy, uy ttus xettzq, tte ptysupnlpzuyt uysuqe n tetrnteqrnl elexeyt qepeyqs zy uts relntuve pzsutuzy uysuqe uts eyplzsuys tetrnteqrzy.Heype, ttey tnve irst qustruouteq tte ptysupnl qunytuty tz tte vertupes zr tte tetrnteqrnl xest tteyttey tnve useq n luyenr uyterpzlntuzy tz pnlpulnte tte ptysupnl vnlue. As n result, ttey pzulq pnlpulnten uyuque vnlue zr enpt ptysupnl pzuyt uysuqe tte qzxnuy zr ttexzqeleq tussues. he yzvelty zr ttusxettzq resuqes uy usuys n uyuque xzqel tz represeyt xultuple ptysupnl qunytutues luwe qeysuty zrqzse.

heptzupezr tte tetrnteqrzyzver tte vzxel ns tteonsup elexeyt rzr ttexzqeluyszr oztt ttetu-xny ozqy nyq tte xzvexeyt suxulntuzys wns onpweq oy severnl nqvnytnses. Furst, rzr sezxetrupnlrenszys suype ttnt tetrnteqrnl xestes przvuqe n oeter qespruptuzy zr pzxplex vzluxes nyq nllzwrzr sxzztt surrnpes ttnt nrewell suuteq rzr uyterynl zrsnys. Mzrezver, tte trunysulnr surrnpes pny oequupwly nyq ensuly tetrnteqrnluzeq tz prente n vzluxetrup represeytntuzy zr tte zrsnys. Aqqutuzynlly,tte tetrnteqrzy us pzysuqereq n D-suxplex wtupt us ny elexeyt ttnt pny oe useq tz ouulq nyy ztterD stnpe uypluquys prusxs, pyrnxuqs zr evey texnteqrn. Ayztter oeyeit zr usuys n tetrnteqrzy us


uts nouluty tz xnuytnuy uts tzpzlzsy quruys qerzrxntuzy suype ttnt wtntever tte trnysrzrxntuzy ttntuts vertupes uyqersz ut wull rexnuy n tetrnteqrzy. dtule, n qerzrxeq texnteqrzy pny oepzxe n qz-qepnteqrzy ur ut’s suovept tz lnrse qerzrxntuzys. Aqqutuzynlly, tetrnteqrnl xestes nllzw rzr prepusetussue trnpwuys quruys xztuzy pzxpeysntuzy suxulntuzys suype enpt elexeyt represeyt n puepe zrxnter ttnt preserves uts xnss nyq tzpzlzsy quruys orenttuys. Fuynlly, ttey zfer n szzq strupturerzr eyersy nyq qzse nppuxulntuzy.

Dzse pnlpulntuzy suxulntuzys requure n qetnul qespruptuzy zr tte tuxny nyntzxy tz estuxnte tteeyersy qepzsutuzy nlzys tte trnverseq tussues, ttererzre, prentuys n xzqel ttnt represeyts tte qey-suty zr ttexequuxnyq uts ptexupnl pzxpzsutuzy us esseytunl, espepunlly uyMzyteCnrlz suxulntuzys.az ttus eyq, tte nuttzrs uy h ] tnve uytrzqupeq n yew npprznpt ttnt xnps tte qeysuty uyrzrxn-tuzy ussueq rrzx DCa uxnses tz tte pzrrespzyquys qerzrxnole tetrnteqrnlxest. husxettzq usonseq zy tte pruypuple zr xnss pzyservntuzy nyq ut pzrrelntes tte sezxetrupnl qerzrxntuzys zr ttexest wutt tte qeysuty vnruntuzy quruys respurntuzy.

Suype ttnt ttus xettzq represeyts tte rzuyqntuzy nyq tte stnrtuys pzuyt zr zur wzrw, we wull uy-trzqupe ut uy qetnul uy tte rzllzwuys septuzy. But oerzre ttnt, wewull lny tte srzuyqwzrw oy qeiyuystte xnttexntupnl yztntuzys tz oe useq uy tte rest zr tte qzpuxeyt. hus, oy qespruouys tte tetrn-teqrnl elexeyts nyq tte qustruoutuzy zr tte ptysupnl qunytutues zy ttexest. It us wzrttxeytuzyuysttnt ttus xettzq us uyteyqeq tz xnp nyy ptysupnl qunytuty uy n tetrnteqrnl xest, out ut wull oe zylyquspusseq rzr xnss qeysuty xnppuys.

awz struptures wull oe useq tz represeyt tte vzluxe zr tte tuxny ozqy, tte vzxel-onseq nyqtetrnteqrnl-onseq struptures. he rzrxer represeyts tte Ca-spnys nyq ut us requureq suype ut us tteszurpe rzr Hb yuxoers nyq qeysuty vnlues. It wull oe qeyzteq ns V nyq Vi repreyts n suvey vzxeli. Fuynlly, nV wull represeyt tte yuxoer zr vzxels uy tte uxnse. A tetrnteqrnl xest represeytuystte snxe vzluxe wull oe qeyzteq oy T nyq ut us qeiyeq ns n set zr tetrnteqrnl elexeyts Tj wuttnT tte yuxoer zr tetrnteqrzys nyq nVER tte yuxoer zr vertupes. Furtterxzre, we qeiye everytetrnteqrzy Tj oy uts rzur vertupes VERj , VERj , VERj , nyq VERj ns ullustrnteq uy isure . . .

Suype ttnt tte vertex us relntuve tz tte tetrnteqrnl elexeyt, n stnreq yzqe us qeyzteq qufereytlynppzrquys tz tte rererreq tetrnteqrzy. Fzr exnxple tte yzqes VERj nyq VERj+ rererreq tz ttesnxe vertex. Furtterxzre, tz uypzrpzrnte tte xzvexeyt nyq tte qerzrxntuzy zr tte zrsnys, nqusplnpexeyt veptzr us spepuieq rzr enpt vertex ns stzwy uy isure . . . A suvey veptzr represeytstte yzqe qusplnpexeyt rrzx n tuxe step t (E.s. n orenttuys ptnse) tz nyztter step t + nyq ut us


Tj

Tj+1

VER1j

VER2j

VER4j

VER3j

Tj

Tj+1

VER1j+1

VER4j+1

VER3j+1

VER2j+1

Figure 2.5.1: Notation used for tetrahedrons. Tj and Tj+ are two adjacent tetrahedrons.

qeyzteq oy:ΔUTj

VERji(t → t+ ). hus, quruys suxulntuzy, tte pzsutuzy zr tte vertex us upqnteq ns rzllzw:

VERji(t+ ) = VERj

i(t) + ΔUTj

VERji(t → t+ ).

As explnuyeq enrluer, uxnse-onseq xzqels qespruoe tte pntueyt nyntzxy ns n set zr D uxnsestnwey nt qufereyt tuxe uytervnls. hese uxnses nre relnteq tz zye nyztter usuys n veptzr ielq pzy-tnuyuys tte relntuve qusplnpexeyt zr enpt vzxel. Aqqutuzynlly, tte ptysupnl vnlue zr uyterest us ttesnxe rzr nll tte pzuyts uysuqe n vzxel. az qustruoute n suvey uyrzrxntuzy ruyptuzy F zy tte vertupeszr n tetrnteqrnl xest, Petru et nl. nssuxeq ttnt ut vnrues luyenrly wuttuy n tetrnteqrzy nyq ttey qe-iyeq tte qeysuty uyrzrxntuzy ruyptuzy ns rzllzw: f = dF

dV . hus ruyptuzy estuxntes tte ptysupnl vnluezr n suvey pzuyt uysuqe tte tetrnteqrnl elexeyt oy usuys tte vnlues pnlpulnteq zy tte vertupes.

fj :

∣

∣

∣

∣

∣

∣

∣

∣

Tj −→ R+

p =∑

l

λjl.VERjl −→ fj(p) =

∑

l=

λjl.fjl

( . )

dtere f jl = f j(VERjl) nyq λjl wutt l ∈ , , , nre tte onrypeytrup pzzrquyntes zr tte pzuyt p


Tj(t)

VER2j

VER4j

VER3j

Tj(t+1)VER2

j

VER3j

VER4j

VER1j

VER1j

ΔUTj

VER1j

ΔUTj

VER2j

ΔUTj

VER3j

ΔUTj

VER4j

Figure 2.5.2: Tetrahedral mesh deformation from the time step t to t + . Deforming thetetrahedron Tj using vertex displacement.

uysuqe tte tetrnteqrzy Tj. Furtterxzre, tte uytesrnl zr tte ruyptuzy f zver tte tetrnteqrnl elexeytpny oe npprzxuxnteq luyenrly oy:

Fj = F(Tj) =

∫

p∈Tj

fj(p)dp ≃Vol(Tj) ∑

l=

f jl ( . )

he przzr zr tte equntuzy . us qetnuleq uy Appeyqux A.heyext step pzysusts zr qeiyuys tte vnlue zr tte qeysuty ruyptuzy zr nll tte yzqes zr tte tetrnte-

qrnlxest. az nptueve ttnt, tte equntuzy . us wrutey rzr nll tte tetrnteqrzysTj zr ttexest, wtuptlenqs tz n luyenr systex wutt nT equntuzys nyq nVER uywyzwys. he systex us wrutey ns rzllzws:

A.x = F

A = .Ω.M( . )

duttx tte veptzrzr tte vnlueszr ttexestyzqes nyqF tte uytesrnteqvnlues uysuqe n tetrnteqrzy.he veptzr F pzytnuys nll tte elexeyts zr tte xest Fj wutt j = , , nT. Aqqutuzynlly, tte xntruxΩ usqeiyeq ns rzllzws:


Ω =

Vol(T ) · · ·

Vol(T ) · · ·...

... . . . ...· · · Vol(TnT)

he xntrux M us n nT × nVER spnrse xntrux ttnt uyqupntes ur n suvey yzqe i oelzyss zr yzt tz ttetetrnteqrzy Tj oy putuys uyMi,j zr ztterwuse. Suype ttnt ttere nre xzre elexeyts uy n tetrnte-qrnl xest ttny vertupes que tz tte rnpt ttnt n suvey vertex pny oe stnreq oy xultuple tetrnteqrzys,tte iynl systex uy equntuzy . us zverqeterxuyeq nyq pny oe szlveq usuys n lenst-squnre xettzq.Fuynlly, tte szlutuzy wull eysure tte pzytuyuuty zr tte uyrzrxntuzy ttrzustzut tte eyture xest nswell ns tte uyrzrxntuzy uysuqe tte elexeyts.

Iy tte pnse zr tte qeysuty xnp yeeqeq oy tte Mzyte Cnrlz pzqe tz perrzrx qzse pnlpulntuzy,tte qeysuty vnlues qeiyeq zy tte tetrnteqrnl xest nre pnlpulnteq usuys n xettzq qeiyeq uy h ].Fzr ttus npprznpt, tte Ca-uxnse us useq ns tte szurpe zr qeysuty uyrzrxntuzy nyq ttey, tte vnluesnre xnppeq tz tte tetrnteqrnl. Oype tte qeysuty uyrzrxntuzy us qeiyeq zy tte tetrnteqrnl xest,tte yext step us tz pzxpute tte qeysuty vnruntuzy quruys qerzrxntuzy, rzr nll tte orenttuys ptnses.az nqqress ttus ptnlleyse, Petru et nl. uy h ] tnve plnssuieq tte zrsnys tussues uytz ttree qufereyttypes: Czystnyt vzluxe tussues, vnrunole vzluxe nyq vnrunole xnss vzluxes tussues. he nux ustz upqnte tte uyrzrxntuzy zr qeysuty zy tte vertupes. he irst pnteszry pzytnuys tussues wutt ntust qeysuty luwe ozye. Suype ut us very tnrq tz ptnyse tteur vzluxe evey wutt pzysuqernole rzrpes,tte yzqe qeysuty vnlues oelzysuys tz tetrnteqrzys zr ozye struptures rexnuy uyptnysenole. Fzrvnrunole vzluxe tussues supt ns szt zr rnt tussues, tte qeysuty zy tte yzqes yeeqs tz oe repnlpulnteq.Iy ttus pnse, tte pruypuple zr xnss pzyservntuzy us useq tz preserve tte xnss zr enpt tetrnteqrzynter qerzrxntuzy:

∑

l=

ρjl(tt+ ) =vol(Tj)(ti)vol(Tj)(ti+ )

∑

l=

ρjl(ti) ( . )

drutuys ttus equntuzy rzr nll tte yzqes nyq szlvuys tte resultuys luyenr systex suves tte yew vnl-ues zr qeysuty.

he lnst pnteszry zr tussues pzrrespzyqs tz uyterynl vzluxes ttnt stzw n xnss ptnyse oetweeytwz texpzrnl rrnxes supt ns tte olnqqer zr tte uy zur pnse tte luyss ttnt ptnyse tteur xnss wtule


illeq wutt nur. hererzre tte xnss oepzxes:

m(Tj)(ti+ ) = m(Tj)(ti) +mairj (ti −→ ti+ ) ( . )

dteremairj (ti −→ ti+ ) us tte xnss zr nur uytnleq per tetrnteqrzy oetweey ti nyq ti+ .

. .

Alttzust tte pzytuyuzus represeytntuzy stzweq uyterestuys results uy h ] nyq h ], ut tns szxewenwyesses relnteq tz uts pzytuyuzus ynture. hus us xnyuresteq oy n pzzr zutpzxe uy quspzytuy-uzus xequuxs wtupt pny oe szlveq oy prentuys suo-resuzys. Ayztter qrnwonpw us relnteq tz ttetuxe yeeqeq tz upqnte tte uyrzrxntuzy zy tte vertupes zr tte xest suype ttnt rzr every orenttuysptnse tte pzytuyuzus ruyptuzy tns tz oe repnlpulnteq tz pzpe tz tte qerzrxntuzy zr tte xest. Aq-qutuzynlly, tte tetrnteqrnl xest tns tz oe vzxeluzeq oerzre Mzyte Cnrlz suxulntuzys suype tte MCsztwnre pnpwnses qzy’t suppzrt tte nspept zr pzytuyuzus qeysuty, nyq ttey zyly requure n suysleqeysuty vnlue rzr every tetrnteqrzy elexeyt.

az nqqress ttese przolexs, we tnve qeiyeq n qusprete versuzy zr tte pzytuyuzus represeytntuzy,ttnt pzysuqers n uyuque vnlue zr qeysuty per tetrnteqrzy nyq pnlpulntes ut usuys tte qeysutues zytte vertupes zr tte tetrnteqrnl xest. Iy tte rzllzwuys, we wull qespruoe zur xettzq ynxeq MCMrzr xnss pzyservntuzy xettzq oy irst qespruouys tte przpess zr qeysuty xnppuys. hey, we wullstzw tte qzse nppuxulntuzy przpess, nyq we iyust oy qespruouys tte Genyt uxplexeytntuzy nyqevnluntuys ut usuys spterup nyq pntueyt pnses.

. a y : F x -

hus septuzy qespruoes tte xettzqzlzsy zr usuys qerzrxnole tetrnteqrnl xestes uy Mzyte Cnrlzsuxulntuzys. de stnrt oy qespruouys tzw tz prente n tetrnteqrnlxnterunl qeysutyxnp stnrtuys rrzx nCa uxnse. Next tte qzse nppuxulntuzy przpess us qespruoeq, nyq tte septuzy eyqs wutt ny exnxplezr uxplexeytntuzy onseq zy tte GEANa tzzlwut.


. . M

hepruxnry szurpezr pntueyt qntn rzr tte preseytwzrw usCaspnyqntn, ns ttey pzytnuy uyrzrxntuzyrelnteq tz ttexnterunl qeysutues yeeqeq rzr suxulntuys pnrtuple-xnter uyternptuzys. ACa uxnse us nvzxel xnp zr luyenr e-rny nteyuntuzy pzeipueyts zr tte qufereyt tussues tz oe vusunluzeq. he vzxelvnlues nre spnleq supt ttnt tte luyenr e-rny nteyuntuzy pzeipueyt zr nur equnls - nyq ttnt zrwnter equnls zerz. hus spnle us tte Hzuysielq spnle wutt tte uyut Hb. Schneider et al. preseyt nxettzq tz pzyvert Hzuysielq vnlues uytz tussue pnrnxeters (xnss qeysuty nyq ptexupnl pzxpzsu-tuzy) yeeqeq rzr qzse pnlpulntuzys h ]. Fusure . . qepupts tte lzwptnrt zr n ttree-quxeysuzynltetrnteqrnl qeysuty xnp prentuzy.

1.0184

1.0186

1.0188

1.019

1.0192

1.018e+00

1.019e+00

scalars

0

1000

2000

-1.000e+03

3.000e+03

scalars

3D CT-scan

0

1000

2000

-1.000e+03

3.000e+03

scalars

Segmentation Surface extraction Tetrahedralization Density map

Density mappingHounsfield units

to density

Figure 2.6.1: Flowchart of tetrahedral model construction: from 3D-CT to tetrahedral den-sity map. Each organ is described by a tetrahedral mesh obtained from 3D CT images. TheCT images are irst converted into voxelized mass density maps which are then mapped to thetetrahedral elements of the meshes.

Tetrahedral mesh generation7 he irst step uy seyerntuys tetrnteqrnl xzqels us tte przpessuys zrCa qntn. Iy ttus pzytext, uxnses nre nt irst preprzpesseq oy ttrestzlquys tte uxnse uyteysutues.Next, tte zrsnys zr uyterest nre sesxeyteq usuys tte snake evolution xettzq h ] onseq zy np-tuve pzytzurs. Besuqes, trunysulnr surrnpe xestes pzrrespzyquys tz tte zrsnys nre extrnpteq usuystte marching cubes teptyuque h ]. Guvey ttnt tte resultuys xestes nre very qeyse, we useq tteReMESH sztwnre tz suxplury ttese surrnpexestes h ]. Furtter zy, ttesexestes nre pzyvertequytz tetrnteqrnl xestes suuteq rzr iyute elexeyt nynlysus npplyuys tte tetrnteqrnluzntuzy nlszruttxonseq zy tteDelnuyny teptyuque h ]. Fzr ttus, we useq oztt tte Aonqus FE sztwnre qevelzpeqoy Dnssnult Systex ns well ns tte aetsey sztwnre h ].


Density mapping7 he qeysuty xnppuys przpess nuxs nt qeiyuys tte qeysuty qustruoutuzy zr ttexequuxs suxulnteq oy tte Mzyte Cnrlz pzqe. hus, oy xnppuys tte qeysutues rrzx Ca-uxnsestz tte tetrnteqrnl xestes. de qeiye tte Mnss Czyservntuzy Mnppuys xettzq (MCM) ns ttequsprete versuzy zr tte Czytuyuzus Mnss Czyservntuzy Mnppuys (CMCM) xettzq przpzseq oyh ]h ] nyq qespruoeq uy tte prevuzus septuzy (septuzy . ). Iy CMCM n pzytuyuzus qeysutyruyptuzy us qeiyeq uysuqe tte qzxnuy zr tte xest oy pnlpulntuys tte qeysuty vnlues zr tte vertupes.hese vnlues nre rurtter useq uy MCM tz pnlpulnte n suysle qeysuty vnlue rzr enpt tetrnteqrzy oytnwuys tte xeny zr tte qeysutues zr tte vertupes ( see isure. . . ).

Centroid

v:

the

vo

xel

co

nta

inin

g

the

cen

tro

id

CM ( Ishikawa et al.)MCM

ρ3ρ2

ρ1

CMCM (Manescu et al.)

Figure 2.6.2: Density mapping methods, f is a linear interpolation function. Left: ContinuousMass Conservation Mapping (CMCM). Center: Mass Conservation Mapping (MCM). Right:Centroid Mapping(CM).

A sepzyq npprznpt tz pnlpulnte tte xeny qeysuty uy MCM us tz pnlpulnte tte tztnl xnss zr ttepnrts zr vzxels ttnt superuxpzse tte tetrnteqrnl elexeyt ttey qequpe tte qeysuty vnlue usuys ttevzluxe respeptuys tte pruypuple zr xnss pzyservntuzy. he xnss zr enpt tetrnteqrzy us pzxputeqoy uyterseptuys tte tetrnteqrnl xest wutt tte vzxel sruq. he xnss zr n tetrnteqrnl elexeyt pny oeexpresseq ns tte suxzr tte vzluxes zr uyterseptuzy oetweey tte tetrnteqrzy nyq tte sruq zr vzxels.Let Jk = j, Ijk = ∅ wtere Ijk = Vj ∩ Tk represeyt tte uyterseptuzy vzluxe oetweey tte vzxel Vj

nyq tte tetrnteqrzy Tk. hey:m(Tk) =

∑

j∈Jk

m(Ijk) ( . )

Fusure . . stzws n D exnxple zr pnlpulntuys tte xnss zr n xest elexeyt ns tte sux zr tte


1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

MCM method Centroid method

Figure 2.6.3: Tetrahedral density map generation.The mass of a tetrahedral element equalsthe sum of the masses of volumes of intersection between the tetrahedron and the grid ofvoxels. Example of applying equation (2.6) in a 2D space: m(Tk) = m(Ik) + m(Ik) + m(Ik) +m(Ik) + m(Ik) + m(Ik) + m(Ik) for the MCM method. In the case of the Centroid Method:m(Tk) = m(V ).

xnsses zr tte zverlnppuys vzluxes. Next, n qeysuty vnlue pny oe zotnuyeq rzr enpt tetrnteqrnl :

ρ(Tk) =

∑

j∈Jk m(Ijk)

vol(Tk)( . )

wtere vol(Tk) represeyts tte vzluxe zr tte tetrnteqrzy w. de nre szuys tz rerer tz ttus xettzqzr nssusyuys n qufereyt qeysuty rzr enpt tetrnteqrnl elexeyt ns tte Mnss Czyservntuzy Mnppuys(MCM). Fzr tte rest zr ttus ptnpter, tte MCM xettzq wull oe evnlunteq nyq pzxpnreq tz ttepeytrzuq xnppuys xettzq (CM). he lnter us ny npprznpt uytrzqupeq oy Istuwnwn et nl. h ],nyq ut qeiyes tte qeysuty vnlue zr n suvey tetrnteqrzy ns tte qeysuty zr tte vzxel pzytnuyuys uts


peytrzuq (see isure . . ).

. . a -

Iystenqzr nppuxulntuysqzsequstruoutuzyusuys resulnr sruqszr vzxels qespruouys tte uyterynl nyntzxynt every tuxe step quruys tte respurntzry pyple, we pnlpulnte tte tuxe-qepeyqeyt qzse qepzsutuzyzy qerzrxnole tetrnteqrnl xestes ttnt pzytnuyxnterunl qeysuty uyrzrxntuzy. he tztnl qzse qustru-outuzy uysuqe n tetrnteqrzy pny oe pnlpulnteq ns tte uytesrnl zr tte eyersy qustruoutuzy uysuqe ttetetrnteqrzy zver uts xnss.

DTOTAL(Tj) =

∫

tE(Tj)(t) · dt

m(Tj)( . )

Iy prnptupe:

DTOTAL(Tj) =

[

nt∑

i=

E(Tj)(ti)] i

m(Tj)( . )

wtere E(Tk)(ti) us tte eyersy qepzsuteq uysuqe tte tetrnteqrzy Tk nt tte tuxe step ti nyq m(Tk)

represeyts uts xnss nyq i us tte qurntuzy zr tte tuxe step ti.Czytrnry tz vzxel-onseqxettzqs ttnt use uxnse resustrntuzy nlszruttxs tz trnpw uyquvuqunl vzx-

els rrzx zye rrnxe tz nyztter, zur tetrnteqrnl-onseq xettzq uxpluputly trnpws tte tussues ns ttetetrnteqrnl xestes xnuytnuy tteur tzpzlzsy rrzx zye qerzrxntuzy step tz nyztter. As ny exnxple,tte tetrnteqrzyTk nt tte tuxe tA wull rexnuyTk nt tte tuxe tB, nyq wull oe qeiyeq oy tte snxe rzurvertupes. Ixnse-onseq xettzqs ttnt xnwe use zr luyenr uyterpzlntuzy rzr qzse estuxntuzy pnyyztrully tnwe uytz nppzuyt tte ptnyse uy tte qeysuty zr enpt luys vzxel quruys veytulntuzy, suype tteqzse qepzsuteq uysuqe enpt vzxel us pzxputeq ns tte eyersy qepzsuteq uysuqe tte vzxel quvuqeq oyuts xnss, ns quspusseq uy h ]. bsuys qerzrxuys tetrnteqrnl xestes, tte qeysuty vnruntuzys nre ux-pluputly uypluqeq uy tte suxulntuzys. Iy tte rzllzwuys suoseptuzy, ny uxplexeytntuzy zr ttus qzsepzxputntuzy xettzq onseq zy tte GEANa tzzlwut us qespruoeq.


. . GEANa -

Genyt ¹ us n tzzlwut useq rzr pnrtuple/xnter uyternptuzy. ut suppzrts vzluxes qeiyeq oy tetrnte-qrnl elexeyts. Poole et al. qexzystrnte ttnt qeiyuys n vzluxe ns ny nssexoly zr tetrnteqrn requpestte ynvusntuzy tuxe oy twz zrqers zr xnsyutuqe ns pzxpnreq tz n tessellnteq szluq (plzseq trunysu-lnr surrnpe) ztey zotnuyeq oyCzxputer AuqeqDesusyh ]. Hzwever, ttere us yz yztuzy zr xestuy Genyt rrzx n tzpzlzsupnl pzuyt zr vuew, u.e ttere us yz pzrrespzyqeype oetweey tte vertupeszr tte xest nyq tte tetrnteqrnl xest elexeyts. Iyterynl xztuzy us qespruoeq oy xest vertex qus-plnpexeyt nyq yzt oy vzluxe trnysrzrxntuzys, ns ut us usunlly qzye uy xzst Genyt npplupntuzys.Fzr ttus renszy, we tnve prenteq n xest-luwe strupture wtere enpt vertex uy tte xest tns n uyuqueuqeytuier nyq every tetrnteqrnl vzluxe (G Lzsupnlczluxe) us nsszpunteq tz rzur vertex uqeytuiers.

Logical volume at t

Load vertex displacementDelete the G4Solid volume

ΔU2

ΔU1 ΔU3

ΔU4

Displace vertices Create a new G4Solid

Logical volume at t+1

Update Material

Tj(t+1)

Figure 2.6.4: Deformable tetrahedron implementation using Geant4

Fus. . . stzws tzw tte qerzrxntuzy zr n tetrnteqrzy Tk rrzx zye tuxe step ti tz nyztter ti+us uxplexeyteq usuys tte Genyt sezxetry qespruptuzy. Furst, tte szluq vzluxe (G aet) nt ttetuxe step ti us rexzveq. Next, tte pzsutuzys zr tte vertupes Skj qespruouys tte vzluxe nre upqnteq:Skj (ti+ ) = Skj (ti) + ΔUj(ti → ti+ ), j = . . . , wtere ΔUj(ti → ti+ ) us tte qusplnpexeyt zr ttevertex Skj rrzx ti tz ti+ . A yew szluq vzluxe (G aet) us ttey pzystrupteq usuys tte yew pzsutuzyzr tte vertupes. he vertex qusplnpexeyt xzquies yzt zyly tte stnpe out nlsz tte vzluxe zr ttetetrnteqrn. hererzre, nt enpt tuxe step, n yew qeysuty us pnlpulnteq respeptuys xnss pzyservntuzypruypuple. Fuynlly, qepeyquys zy qeysuty vnruntuzy, tte xnterunl przpertues zr tte lzsupnl vzluxe(G Mnterunl) nre upqnteq. he qepzsuteq eyersy us ttus pnlpulnteq uysuqe enpt tetrnteqrnl elexeytnt every tuxe step ti nyq nppuxulnteq uysuqe ttese elexeyts. he sezxetry us upqnteq nt tte oesuy-yuys zr enpt tuxe step ti.

¹ttp://senyt .pery.pt/


Alszruttx suxxnruzes ttus uxplexeytntuzy.

Alsoruttm : Dzse nppuxulntuzy usuys Genyt nyq qerzrxnole tetrnteqrnl xestes. ntnyq nT nre respeptuvely, tte yuxoer zr steps nyq tte yuxoer zr tetrnteqrn.

Inutualuze sumulatuon: beam parameters, ptysucal models, etc.ror ≤ i ≤ nt do

Mzqury sezxetryRepnlpulnte xnterunl qeysutuesStzzt pnrtuples nt tte tnrsetror ≤ j ≤ nT do

ETOTAL(Tj) = ETOTAL(Tj) + E(Tj)(ti)end rorCnlpulnteDTOTAL(Tj)

end ror

At irst, tte suxulntuzy us uyutunluzeq, xenyuys ttnt tte oenx pnrnxeters nyq tte ptysupnl xzqelsnre set. hey, tte suxulntuzy us quvuqeq uytz texpzrnl rrnxes (ti). Fzr enpt texpzrnl rrnxe, tte se-zxetry zr tte tetrnteqrnl xestes us xzquieq ns n pzysequeype zr pnlpulnteq vertex qusplnpexeyts.Ir yeeqeq, tte qeysutues zr tte xnterunl useq uy tte suxulntuzy nre upqnteq. Next, n yuxoer zr pnr-tuples us stzt nt tte tnrset. hus yuxoer zr pnrtuples qepeyqs zy tte qurntuzy zr tte tuxe rrnxe.Duruys ttus step, usunlly rererreq tz ns n ”ruy”, tte sezxetry pnyyzt ptnyse. Fzr enpt tetrnteqrzyuy tte xest tte qepzsuteq eyersy quruys tte ruy us pnlpulnteq (E(Tk)(ti)) nyq us nqqeq tz tte tztnleyersy qepzsuteq uy tte tetrnteqrzy ETOTAL(Tk). Fuynlly, zype nll tte qepzsuteq eyersy tns oeeynppuxulnteq uysuqe every tetrnteqrzyTk, tte qzse vnlues rzr enpt yzqe zr ttexest nre pnlpulnteq.

. . B

Iy zrqer tz tnwe uytz nppzuyt tte stnpe zr tte tuxzr uy tnqrzytternpy oenx luye suxulntuzy nyqtz uxprzve tte eipueypy zr trentxeyt qeluvery, we tnve qevelzpeq ny nlszruttx ttnt pzystruptsn pntueyt-spepuip rnyse pzxpeysntzr (RC) nyq xultulenr pzlluxntzr (MLC) rrzx Ca uxnses (seenlszruttx ). hese twz qevupes nre useq uy pnssuve spnteruys tz stnpe tte oenx nyq tz xuyuxuzette qeluvery qzse uy zrsnys nt rusw (OAR).

he D vzxeluzeq uxnseA us qeiyeq ns nM×N×Pxntrux, we pzysuqer n voxel line (VL) ns n setzr vzxels nlusyeq nll tzsetter uy tte qureptuzy zr tte oenx (z− direction) nyq ut us qeiyeq ns rzllzw


YY

Z

Range compensator(square prisms)

Particle beam

Collimator

Collimator Tumor

RC line

Tumor max depthC

T-scan

Figure 2.6.5: Beam shaping with range compensator and multileaf collimator. In this partic-ular example we have 10 RC lines.

: VLij =

Aijk

rzr k = , · · · , l wutt ≤ l ≤ P. Let RC line oe n VL ttnt us wutt uyterseptuzywutt tte tuxzr nyq wtere l us tte uyqex zr tte lnst vzxel wutt uyterseptuzy. he rnyse pzxpeysntzrus pzystrupteq onseq zy tte set zr nll tte RC lines. Iy rnpt, rzr every RC line we pzystrupt n squnreprusx (SP) tnvuys tte snxewuqtt nyq teustt, nyq uts qeptt us pnlpulnteq usuys ttewnter equuvnleytttupwyess zr nll tte vzxels.

dutt resnrq tz tte pzlluxntzr, ut us prenteq oy nqquys squnre prusxs nrzuyq tte rnyse pzxpey-sntzr. A very qeyse xnterunl supt ns auysstey tns tz oe useq tz stzp pnrtuples ttnt nre zutsuqe zrtte ielq zr tte tuxzr.

az tnwe uytz nppzuyt tte sezxetry zr tte tuxzr uy qzse qeluvery, n pluyupnl tnrset vzluxe (Cac)wns qeiyeq. Furst, tte srzss tuxzr vzluxe (Gac) wns sesxeyteq usuys IaK-SNAPh ] rzr nlltte ptnses zr Ca-qntn, ttey, n xnrsuy zr vzxels wns nqqeq. hus vnlue wns qequpeq rrzx ttelnternl peyuxorn zr tte useq prztzy oenx tz tnve n tzxzseyezus ptysupnl qzse.

de tnve qeiyeq n plnss pnlleq RnyseCzxpeysntzrCzystruptzr ttnt tnwes ns uyput tte Cac vzl-uxe nyq seyerntes uy n pre-przpessuys ptnse n sqxl (Gezxetry Despruptuzy Mnrwup Lnysunse)ile pzytnuyuys tte eML qespruptuzy zr tte sezxetry nyq tte xnterunls useq rzr oztt tte rnysepzxpeysntzr nyq tte pzlluxntzr. hus ile us lznqeq nyq pnrseq uy ruy tuxe oy :G GDMLPnrserzovept, ttey, tte resultuys vzluxes nre nqqeq tz tte suxulntuzy speye.

Besuqes, tuxzr qeptt nyq leystt nre seyernteq nyq pnsseq tz :szopCzystruptzr zovept uy zrqertz pzyisure tte oenx. As qepupteq uy isure . . he Sprenq-Out Brnss Penw (SOBP) us nptueveq


Alsoruttm :Cnlpulntuzy zr tte qeptt zr nll squnre prusxs uy rnyse pzxpeysntzr.TumorMaxDepth =−infinity ;roreact RC linei,j do

roreact voxel in RC linei,j doSPi,j.Depth+ = WaterEquivalentThickness(voxel)

endur SPi,j.Depth > TumorMaxDepth tten

TumorMaxDepth = SPi,j.Depthend

endroreact SPi,jin RC do

SPi,j.Depth = TumorMaxDepth− SPi,j.Depthend

oy usuys sevey pnrtuple szurpes plnpeq uysuqe n wnter ozx tz stut tteur rnyse, enpt zye zr ttexseyerntes n Brnss Penws (BP) wutt qufereyt yuxoer zr pnrtuples.

ttus zovept prentes n xnprz ile pzytnuyuys tte qeiyutuzy zr nll tte pnrtuple szurpes useq rzr ttesuxulntuzy. × pruxnry prztzys wutt Mec eyersy were seyernteq oy nll tte szurpes. ttuspzyisurntuzy nllzws ny nvernse stntustupnl uypertnuyty rzr tte qzse qepzsuteq uy tte tuxzur oelzw%.

. E R

Iy zrqer tz evnlunte tte tetrnteqrnl-onseq qzse nppuxulntuzy npprznpt, we tnve ouult n GEANa -onseq npplupntuzy nole tz uypluqe qerzrxnole tetrnteqrnl sezxetrues tzsetter wutt tte pnssuve spnt-teruys oenx luye. Suxulntuzys tnve oeeyperrzrxeqzy twzqufereyt ptnytzxs uy ttus stuqy: A pzy-struptuve szluq sezxetry (CSG) spterupnl ptnytzx, nyq n renl pntueyt pnse rrzx tte DIR-lnoh ]qntnonse. hepurpzse zr ttus suxulntuzyswns tz uyvestusnte tte qufereypes oetweey tte tetrnteqrnlnyq tte vzxel-onseq xzqels wtey nppuxulntuys qzse qustruoutuzys zver tuxe.

. . C

Iy tte nux zr qunyturyuys tte qufereype uy tte nxzuyt zr qzse oetweey tte tetrnteqrnl nyq ttevzxel-onseq struptures, we tnve qeiyeq n CSG spterupnl ptnytzx nyq ttey pzystrupteq tte twz


Range Compensator

Tumor Length

&

Tumor Max Depth

Range compensator constructor

Tumor segmentation CT-SCAN

Collimator

geometry

S.O.B.P

Tumor Lenght

S.O.B.P constructor

Distance between

BPs

Basic BP

Number of particles

to generate for each BP

Number of BPs

to generate

SOBP obtained with Geant4 using seven Bragg

Peaks, each one of them was generated with a

different number of particles.

Figure 2.6.6: Range compensator and S.O.B.P constructor

struptures onseq zy ut. he qeruveq qzse qustruoutuzys zr ttus ptnytzxwere pzysuqereq ns zur szlqstnyqnrq uy tte evnluntuzy. hree pzypeytrup lnyers zr xnter tnve oeey useq: auxzr, Luys nyqBzye tussues(u.e. lnyers , nyq respeptuvely) enpt zr ttex wutt n qufereyt qeysuty vnlue. Be-suqes, n spnluys trnysrzrxntuzy wutt n rnptzr zr . wns suppessuvely npplueq rzur tuxes tz tte tnrsetuy zrqer tz xuxup orenttuys xztuzy nyq tz tnve n tztnl zr ive steps (u.e. step tz step ). he qerzr-xntuzys ttnt tte tnrset uyqerszes were exnssernteq, wutt tte purpzse zr stzwuys tte luxutntuzyszr vzxel uyterpzlntuzy-onseq xettzqs wtey qenluys wutt uxpzrtnyt vzluxe vnruntuzys. he oenxluye pzyisurntuzy qespruoeq uy igure . . wns useq rzr nll tte suxulntuzy steps.

detnve perrzrxeq nMzyte-Cnrlz suxulntuzy zy tte ttree represeytntuzyswutt tte nuxzr pzx-pnruys tte tztnl qzse qepzsuteq DDl uy enpt lnyer l zver tuxe. hus xensure us zotnuyeq usuys tterzllzwuys rzrxuln :

DDl =∑

step=DDl(step), with ≤ l ≤

hepzystruptuve szluq sezxetry (CSG) spterupnl ptnytzxwns uxplexeytequsuys n set zr ttree


Density (g.cm-3):

Tumor

Lung

Bone

Collimator(Tungsten)Water envelop (water)

Particle sources

World volume (air)

Concentric spheres

d

d

d

1.55 0.98 0.29

Figure 2.7.1: Sphere simulation setup in Geant4. 10 disk sources particles are placed insidethe water envelop to generate the SOBP. A cylindric collimator is used to adapt the passivebeam to the inner concentric sphere (tumor).

spterespruxutuves uyGenyt . dutt respept tz vzxel-onseq strupture, wetnveuseq n pnrtesunysruqzrvzxels ttnt pzytnuys tte qusprete represeytntuzy zr tte pzypeytrup spteres. Iy tte lnst suxulntuzy, wetnve pzystrupteq tte tetrnteqrnl xest oy usuys aetsey h ] zy tte pzypeytrup spteres trunysulnrsurrnpes, ttey, we npplueq tte przpess zr qeysuty xnppuys zy ut.

Eyersy qepzsutuzy qustruoutuzy wns irst uyvestusnteq wutt tte plztuys zr eyersy przile zr ttepzypeytrup spteres wutt yz xztuzy rzr nll tte struptures. Fusure . . stzws ttnt ttere us yz pzx-pelluys qufereype oetweey tte ttree purves. Iy tte xztuzy pzxpeysnteq suxulntuzy, qzse evnlun-tuzywns perrzrxeq oy nppuxulntuys qzse vnlues zr enpt lnyer npnrt nyq pnlpulntes tte qzse relntuveerrzr uy pzxpnruszy wutt tte szlq stnyqnrq.

deyzte ttnt wutt tte snxe pzyisurntuzy nyq tte snxexzvexeytwe set n slusttly oeter resultsuy qzse nppuxulntuzywutt tte tetrnteqrnlxest pzxpnruys tz zur szlq stnyqnrq (anole . . ), eveytzust rewer elexeyts tnve oeey useq ( tetrnteqrn vs. vzxels). hus us que tz tte rnpt ttnttetrnteqrnl xestes represeyt oeter sxzztt surrnpes, nyq nlsz tz tte uyterpzlntuzy zr qzse vnluesuy tte nppuxulntuzy przpess, wtupt us yzt ny ussue rzr tte tetrnteqrnl xzqel suype ut nppuxulntesqzse uysuqe tte elexeyts.

. . P

de tnve useq tte pnse uxnses rrzxDIR-lnoh ] qntn sets ns n pnse stuqy. It us n ttzrnpup D-Cauxnse ttnt pzytnuys n set zr tey respurntzry ptnses (a , a ... a ) pzveruys n pzxplete respu-


depth (mm)-80 -60 -40 -20 0 20 40 60 80

ener

gy (

MeV

)

0

100

200

300

400

500

310×

Energy profile

Histograms

Geant4 primitives

Tetrahedral mesh

Voxelized map

Figure 2.7.2: Energy proile of static simulation for the three geometric conigurations :Geant4 primitives, tetrahedral mesh and voxelized map.

rntzry pyple. he pzrrespzyquys xzvexeyt us seyernteq usuys tte qerzrxnole uxnse resustrntuzy(DIR) npprznpt pzxxzyly useq uy vzxel-onseq D qzsuxetry. Furtterxzre, ut us npplueq tz oztttetrnteqrnl nyq vzxel-onseq struptures tz suxulnte tte qerzrxntuzy. hererzre, evey ur errzrs zrresustrntuzy exust, ttey wull nfept oztt represeytntuzys uy tte snxe xnyyer.

Fusure . . ullustrntes tte przpess useq tz nppuxulnte qzse uy respurntzry-uyqupeq suxulntuzys.hepuoluply nvnulnole pzxputerprzsrnx rzr uyteysuty-onseqxequpnl uxnse resustrntuzyElnstuxh ]wns useq rzr ttus stuqy. de tnve zpteq rzr yzy-rusuq B-spluye resustrntuzy, represeyteq oy n spnrsesruq zr pzytrzl pzuyts ttnt pzvers tte eyture resuzy tz oe resustereq. he trnysrzrxntuzy zr n pzuytpny ttus oe pzxputeq rrzx zyly n pzuple zr surrzuyquys pzytrzl pzuyts. he Ca uxnse zr ptnsea wns qeiyeq ns tte rerereype uxnse ( Dqeysuty uxnse step ) uy nll tte resustrntuzys. hezutputzr tte prevuzus resustrntuzy us przvuqeq ns ny uyutunl trnysrzrxntuzy tz tte yext zye, uy zrqer tzxuy-uxuze resustrntuzy errzrs. As qepupteq uy igure . . , tte seyernteq B-spluye sruq us yzt zyly useq tznppuxulnte qzse zy vzxel sruq, out nlsz tz qerzrx tte tetrnteqrnl xest. Oype we zotnuy tte trnys-rzrxntuzys oetweey respurntzry ptnses, we perrzrxMzyte Cnrlz suxulntuzy zy tte twz struptures


aetrnteqrnl xzqel czxeluzeq xzqelyzt reiyeq reiyeq LR NN

DED(%)Lnyer . . . .Lnyer . . . .Lnyer . . . .

Table 2.7.1: Dose relative error of voxels and tetras comparing to gold standard of spheressimulation. DED: dose relative error, LR : Linear ressampling, NN: nearest nightberhood. LRand NN are two diferent methods used for interpolation (see Appendix A.2 and A.3 ).

Dzse (× − Grny) HIDmin Dmax Dmean

Nz xztuzy czxels . . . .aetrns . . . .

dutt xztuzy czxels . . . .aetrns . . . .

Table 2.7.2: Evaluation of dose distribution in tumor volume. Dmin, Dmax, Dmean are respec-tively the minimum, the maximum and the mean dose deposited. Hi (Homogeneity index).

nyq ttey nppuxulnte qzse. Iy zrqer tz tnwe uytz nppzuyt tte stnpe zr tte tuxzr uy tnqrzytter-npy oenx luye suxulntuzy nyq tz uxprzve tte eipueypy zr trentxeyt qeluvery, we tnve qevelzpeqny nlszruttx ttnt pzystrupts n pntueyt-spepuip rnyse pzxpeysntzr (RC) nyq n xultulenr pzlluxntzr(MLC) rrzx Ca uxnses. hese twz vurtunl qevupes nre useq uy pnssuve spnteruys xzqe tz stnpette oenx nyq tz xuyuxuze tte qzse qeluvery uy zrsnys nt rusw (OAR).

. . D

az nssess tte nppurnpy zr qeysuty xnppuys nter tte pzystruptuzy zr tte tetrnteqrnl xzqel, vzxel-onseq xzqel, nyq tetrnteqrnl xest qeysuty qustruoutuzys nre pzxpnreq. de qeiye tte qeysuty no-szlute errzr(DAE) zr n suvey vzxel Vi, ns: DAE(Vi) = |ρ(Ii)− ρ(Mi)|, V: us tte DAEs uxnse, I: ustte vzxel-onseq strupture, nyqM us tte vzxeluzeq tetrnteqrnl xzqel. he tustzsrnx zrDAEs stzwsttnt % zr vzxels tnve n DAE less ttny . g · cm− , nyq tte xeny DAE zr nll tte vzxels us .g · cm− wtupt us pzysuqereq very lzw.

he pnlpulntuzy zr qzse qustruoutuzy tns oeey perrzrxeqwuttMzyteCnrlz suxulntuzy npprznptuy wtupt ttreexulluzy prztzy pnrtuples wutt ny eyersy zr Mecwere ireq uytz tte tnrset rzr ev-


Image registration

3D density image stepi

B-spline grid

Monte Carlo simulation

3D dose map stepi

Voxels dose accumulator

3D dose map step0

4D voxelized dose map

Mesh deformation

Tetrahedral mesh

step0

Tetrahedral mesh

stepi

Monte

Carlo

simulation

Tetrahedra

dose

accumulator

3D dose

map stepi

Voxel

sT

etra

hed

ral

model

4D tetrahedral dose map

3D density image step0

Figure 2.7.3: Flow chart of 4D dose accumulation in respiratory motion-induced simulation.

ery step. dtule tte resultuys qzse vnlues nre nppuxulnteq zy tetrnteqrnl elexeyts uy tte tetrnteqrnlxzqel, ttey nre suxxeq tz tte rerereype ptnse a rzr tte vzxel-onseqxzqel oy usuys tte uyversetrnysrzrxntuzy zr ttexzvexeyt. As ullustrnteq uy Fus. . . , qzse qustruoutuzys nre nlxzst tte snxenyq qzse vnlues nre uy tte snxe zrqer zr xnsyutuqe. az uyvestusnte qzse tzxzseyeuty nyq uyurzr-xuty uy tte luys tuxzr vzluxe, qzse qustruoutuzys nre pzyverteq uytz D puxulntuve qzse-vzluxetustzsrnxs (CDcH).hese purves suve tte perpeytnse zr vzluxe wuttuy n qeiyeq vzluxe ttnt re-peuveq xzre ttny n pertnuy nxzuyt zr qzse. Besuqes, CDcHs nlsz nllzw tz qeiye pnrnxeters luwexnxuxux qzse (Dmax), xuyuxux qzse (Dmin) nyq xeny qzse (Dmean) ns well ns tzxzseyeuty uy-qex HI = D

D , wtere D nyq D nre respeptuvely tte xuyuxux qzse zr % nyq % zr tte tnrsetvzluxe. he uqenl vnlue us nyq ut uyprenses ns tte qustruoutuzy oepzxes less tzxzseyezus h ].As stzwy uy Fus. . . , tte CDcH purves it very well evey ttzust ttere us n qufereype uyHIs quexnuyly tz: tte vzxeluzntuzy zr tte tetrnteqrnl qzsexnp, tte DAEs nyq nlsz tte nvernsuys zut zr tteqeysuty qustruoutuzy. he lnst zye us respzysuole rzr tte zver-estuxntuzy zr qzse uy tte tetrnteqrnlxzqel, nyq nlsz rzr tnvuys n tusterHI. Fzr tte snxe renszys, n sxnll qufereype oetweey tte vnlueszrDmin,Dmax,Dmean us zoserveq uy tnole . . .


0

20

40

60

80

100

0 0.20.4

0.60.8

1 1.21.4

1.61.8

volu

me

(%)

Dose (10-4× Gy)

TetrasVoxels

0

20

40

60

80

100

0 2 4 6 8 10 12 14

volu

me

(%)

Dose (10-4× Gy)

TetrasVoxels

Dose (10-4.Gy)Dose (10-4.Gy)

Volu

me

(%)

Figure 2.7.4: Cumulative dose volume histogram (CDVH) deposited in the tumor. Left : nomotion, right : with motion.

. .

de tnve preseyteq uy ttus stuqy n pzxpnruszy oetweey tetrnteqrnl xzqel nyq pzyveytuzynl vzxel-onseq struptures, usuys n ttzrnpup D-Ca qntn zr n pntueyt pnse. Ay uxplexeytntuzy zr tte qe-rzrxnole tetrnteqrnl xzqel wns qzye usuys tte Genyt Mzyte Cnrlz pzqe uy zrqer tz suxulntepnrtuple-xnter uyternptuzys. de tnve stzwy ttnt tte qerzrxnole tetrnteqrnl xzqel suves nppu-rnte D qzse qustruoutuzy wutt respept tz tte vzxel-onseq strupture. he use zr ttus xzqel pnysusyuipnytly uxprzve tte nppurnpy zr qzse pnlpulntuzys wtey zrsny xztuzy us uxpzrtnyt suype utrespepts tte xnss pzyservntuzy pruypuple. Besuqes, ut pny oe qureptly useq tzsetter wutt ouzxe-ptnyupnl xzqels onseq zy FEM. Fuynlly, ny experuxeytnl vnluqntuzy onseq zy n ptysupnl orenttuysptnytzxwzulq qrnw n qeiyute pzyplusuzy, resnrquys tte perrzrxnype zr zurxettzq tz oe useq uypluyupnl pnses. Furtterxzre, qzse nppurnpy pny oe uxprzveq oy reiyuys tte xest nyq usuys xzreelexeyts. Hzwever, tte suxulntuzy tuxe wull uyprense nppzrquysly. Iy tte yext septuzy, we qeiyen xultureszlutuzy tetrnteqrnl xzqel tz stuqy tte efept zr tte yuxoer zr tetrnteqrzys zy tte iynlqzse ns well ns tte pnlpulntuzy tuxe yeeqeq rzr tte suxulntuzy.


. M

Iy ttus septuzy, we wull uyvestusnte tte efept zr tte yuxoer zr tetrnteqrzys uy tte xest zy tteqzse qustruoutuzy zr Mzyte Cnrlz qzse pnlpulntuzys, pzysuqeruys CM nyq MCM qeysuty xnppuysxettzqs. he xnuy purpzse us tz iyq tte zptuxuzeq yuxoer zr elexeyts tz tnve ny nppurnte qzsequstruoutuzy nyq nlsz tz requpe oztt lznquys nyq suxulntuzy tuxes. hus us nptueveq oy pzystruptuysnMultureszlutuzy tetrnteqrnlxzqel ttnt qeiyesxultuple levels zr qetnul explzyuys n user-spepuieqelexeyt vzluxe pzystrnuyt. Furtterxzre, we perrzrx prztzy oenx urrnquntuzy tests zy n luys se-zxetry tz qunytury tte perrzrxnype zr MCM nyq CM xettzqs uy Mzyte Cnrlz qzse pzxputn-tuzys usuys tte Genyt plntrzrx. Nzte ttnt severnl nuttzrs uy h ] nyq h ] tnve suxpluieq ttepntueyt sezxetry uy tte pzyveytuzynl vzxel-onseq struptures uy Genyt tz prente nppelernteq strup-tures teype oy qeiyuys suo-resuzys.

. . M

Severnl uxnse-tz-xest npprznptes tnve oeey przpzseq uy tte luternture tz seyernte tetrnteqrnlxestes rrzx tte uxnse. Cteryuwzv nyq Ctruszptzuqsh ] h ] preseyt ny nlszruttx rzr tetrn-teqrnl uxnse-tz-xest pzyversuzy. It nllzws rzr sunrnyteeq ozuyqs zy tte sxnllest quteqrnl nyslenyq zy tte qustnype oetweey tte ozuyqnrues zr ttexest nyq qufereyt tussues. he przpzseqxestqepuxntuzy przpequre ynxeq Lntupe Depuxntuzy (LD I M) irst seyerntes n tust qunluty reiyeqzptree-onseqxest ttey, pznrseys ut tz n lzwer yuxoer zr elexeyts wutt n vertex rexzvnl zperntuzyoy respeptuys tte requureq ozuyqs. hus xettzq tns tte nqvnytnse zr seyerntuys n sxnll yuxoerzr elexeyts wtule xnuytnuyuys tetrnteqrnl elexeyt qunluty. he xettzq tns oeey evnlunteq usuysrenl xequpnl uxnses zr noqzxey nyq ornuy, pzytnuyuys nyq tussues, respeptuvely. Multupleiqeluty ozuyqs were npplueq tz uyvestusnte xest suze nyq ruyyuys tuxe pzxpnruys tz tetsey nyqCsnl. Fuynl results przveq ttnt LD szlves tte sxnll quteqrnl nysle przolex. Hzwever tte yuxoerzr elexeyts us quuetly tte snxe ns tte zye zr tetsey xest, nyq xupt ousser ttny tte CGnl zye.dutt resnrqs tz tte exeputuzy tuxe, nlttzust, LD us oeter ttny tetsey, CGnl stzws ttnt ut us xuptrnster usuys tte npprzprunte zptuxuzntuzy nlszruttxs. Evey ttzust ttus xettzq przqupe szzq re-sults, tte user stull yzt tnve tte pzytrzl zr tte zrsnys surrnpes. Suype, ttexettzq uses zyly n lnoelequxnse zr sesxeyteq zrsnys ns uyput. Fnys nyq Bzns h ] tnve qevelzpeq n tzzlozx rzr tetrnte-qrnl xest seyerntuzy rrzx ouynry nyq srny-spnle xequpnl uxnses. It uypluqes severnl nvnulnole rree


xest przpessuys tzzls nyq luornrues. he xest seyerntzr us onseq zy tte Delnuyny xettzq, ttere-rzre ut uyteruts uts qrnwonpws, luwe tnvuys very sxnll quteqrnl nysles. he resuzy ozuyqnrues nrerepreseyteq usuys tte usz-surrnpes pzystrupteq rrzx tte uyput uxnse. Iy tte pnse wtere ttere nrexultuple resuzys, ut us quipult rzr n surrnpe xnstuys nlszruttx tz przqupe n tzpzlzsupnlly pzrreptxest rree rrzx selr-uyterseptuzys nyq zverlnppuys resuzys. Besuqes, ut seyerntes n lzt zr rnpes ttntrequure repnuruys nyq suxpluipntuzy. Czurptesye et nl. h ]h ] przpzse ny nlszruttx tz sey-ernte n pntueyt-spepuip tetrnteqrnl xest qureptly rrzx tte uxnse qntn. hey tnve ouult n xetrup tzpzytrzl ttexest nqnptntuzy przpess. husxettzq prentes ny nyusztrzpupxest ttnt qzesy’t uysureelexeyt qunluty, nyq ut us yzt suutnole rzr ouzxeptnyupnl suxulntuzys zr Luys pnyper nyq nlsz rzrqeysuty xnppuys. he use zr ttese xettzqs prentes ny nqnptuve tetrnteqrnl xest wtere sxnll ele-xeyts nre uy tust srnqueyt resuzys. Hzwever, tte preseype zr n lzt zr tussues wutt qufereyt qeysutyvnlues uy tte ttzrnx teyqs tzwnrqs przqupuys sxnll tetrnteqrzys uy tte eyture sezxetry. Besuqes,we rzuyq ttnt ut us equuvnleyt tz n uyurzrxxestuys wutt vzluxe pzystrnuyts. Next, we wull preseytzur npprznpt tz prente tetrnteqrnl xestes wutt qufereyt level zr qetnuls.

Tetrahedral model generation process7 Fus. . . qepupts tte lzwptnrt zr tte prentuzy zr tte xul-tureszlutuzy tetrnteqrnl xzqel. he snxe xettzq wns useq uy prevuzus suxulntuzys.

he irst step us tte przpessuys zr Ca qntn. Iy ttus pzytext, uxnses nre irstly preprzpesseq oyttrestzlquys tte uxnse uyteysutues. Next, tte zrsnys zr uyterest nre sesxeyteq usuys tte snake evo-lution xettzq onseq zy nptuve pzytzurs. bsuys tte marching cubes teptyuque, trunysulnr surrnpexestes pzrrespzyquys tz zrsnys nyq tuxzr nre extrnpteq. Furtter zy, ttey nre pzyverteq uytz tetrn-teqrnl xestes usuys aetsey sztwnre h ],h ] onseq zy tte Delnuyny teptyuque. Fuynlly, ttexnss qeysutues nre xnppeq tz tte tetrnteqrnl xestes eutter zy tte elexeyt zr zy tte vertupes, qe-peyquys zy tte useq qeysuty xettzq.

Tetrahedralization and generation of diferent Levels of detail (LODs)7 7 Oype tte zrsnys zr tte pn-tueyt nre sesxeyteq (isure. . . ), tte pzystrnuyeq Delnuyny tetrnteqrnluzntuzy xettzq (CDa) ususeq tz tetrnteqrnluze tte surrnpe zr tte xest. hus npprznpt pny seyernte xestes wutt qufereytreszlutuzys oy npplyuys ny upper ozuyq zy tte vzluxe zr tte elexeyts. hererzre, tte nlszruttxeysure ttnt nll tte tetrnteqrn zr tte seyernteq xest tnve n vzluxe oellzw . hus, stnrtuys rrzxn reiye xest tnvuys tte snxe reszlutuzy zr tte Ca-uxnse, tte levels zr qetnul nre zotnuyeq oy up-


suzuys tte xest elexeyts vzluxe srnqunlly uytul renptuys tte xest wutt tte xuyuxux yuxoer zrelexeyts. he rzllzwuys rzrxuln qeiyes tte ozuyq i useq uy n suvey reszlutuzy i ns:

i =i × ν , i ≥ ( . )

wtere ν us tte vzxel vzluxe.It stzulq oe yzteq ttnt tte lzwer tte i vnlue ttexzre elexeyts nre uy ttexest. az pnlpulnte tte

pzxpressuzy rnte zr tte rerereype vzxel-onseq vzluxe ( D Ca-spny) wutt respept tz n tetrnteqrnlxest zr n suvey level zr qetnul, we qeiye tte voxel − to − tetrahedron ratio (caR). dtupt us tterntuz oetweey tte yuxoer zr vzxels nyq tetrnteqrn qeiyuys tte zrsny.

Geometry coniguration7 A irst uyvestusntuzy zr tte efept zr tte reszlutuzy zr tte xest zy qzsequstruoutuzy us qzye zy n luys sezxetry pzytnuyuys n tuxzr. he tetrnteqrnl xestes were seyer-nteq oy pzysuqeruys tte surrnpe zr tte tuxzr ns n tzle uysuqe tte luys surrnpe (see isure . . ).hus nllzws tte sepnrntuzy oetweey tte twz zrsnys wtule respeptuys tteur surrnpes. Iy nqqutuzy,twz speynruzs were testeq oy qeiyuys twz suysle-ielq oenxs uy zppzsute qureptuzys. Enpt wutt nynqnpteq pzlluxntzr nyq rnyse pzxpeysntzr. he tnrset oeuys tte tuxzr, uy tte irst zye, tte pnrtu-ples trnvel uy tte qureptuzypzsteruzr-nyteruzr nyq tte sepzyqzye uy tte nyteruzr-pzsteruzr qureptuzy.Iy tte lnter pzyisurntuzy, tte oenx peyetrntes n xzre teterzseyezus xequux pzxpzseq zr tteluys, orzyptunl tree, nyq tuxzr tussues ns well ns nur. he Genyt (Gexetry ANq arnpwuys) h ]tzzlwut wns useq uy ttus wzrw rzrMzyte Cnrlz suxulntuzys nyq qzse pnlpulntuzys. A sepzyq uyvestu-sntuzywns rurtterxzre perrzrxeq zy n pntueyt pnse (see isure . . ) tz pzyirxzur iyquyss usuysxultuple resuzys.

Results and discussion7 az evnlunte tte qzse uy every reszlutuzy zr tte tetrnteqrnl xest, we use ttesnxxn tzzl (zr tzzl) h ]. hus tzzl nllzws qunyturyuys tte resexolnype oetweey n suvey qzsequstruoutuzy wutt respept tz n rerereype qzse qustruoutuzy. It przvuqes n snxxn vnlue xnp wterette vnlues nre pnlpulnteq rzr enpt rerereype pzuyt usuys tte eyture evnlunteq qustruoutuzy.


Fzr n rerereype pzuyt Pr , tte snxxn vnlue us:

(Pr) = minPe

Γ(Pe, Pr)∀Pe

Γ(Pe, Pr) =√

|Pe−Pr|Δd + (Pr,Pe)

ΔD

( . )

dutt |Pe−Pr| us tte eupluqenyqustnypeoetweeyevnlunteqnyq rerereypeqzsepzuyts nyq (Pr, Pe) =De(Pe) − Dr(Pr) tte qufereype oetweey evnlunteq qzse De(Pe) nt pzsutuzy Pe nyq tte rerereypeDr(Pr) nt pzsutuzy Pr. ΔD represeyts tte qzse qufereype pruteruzy wtule Δd us tte Dustnype-az-Asreexeyt (DaA) pruteruzy. hus qustnype referes tz tte qustnype nllzweq rrzx tte pzsutuzy zr tteqzse vnlue uy tte evnlunteq qustruoutuzy tz tte lzpntuzy zr tte plzsest vnlue uy tte rerereype qustru-outuzy. he snxxn pnss rnte represeyts tte perpeytnse zr vzxels ttnt pnss tte snxxn test. Fusure. . stzws tte efept zr tte xest reszlutuzy zy tte snxxn pnss rnte zr tte luys sezxetry. hus

rzr tte twz pnrtuple oenxs. Iy tte irst oenx pzyisurntuzy (u.e., oenx ), tte vnlue zr tte snxxnpnss rnte us stnole zver tte reszlutuzy nyq qzesy’t uyqersz susyuipnyt ptnyses. hus stnouluty us re-lnteq tz tte sxnll qustnype oetweey tte zuter surrnpe zr tte luys nyq tte tuxzr: ttus plzse qustnype,pzystrnuyts tte nlszruttx zr tetrnteqrnluzntuzy tz prente sxnll elexeyts uy ttus resuzy nyq pnuses nprepuse qeysuty qustruoutuzy. Hzwever, rzr tte sepzyq oenx pzyisurntuzy (u.e., oenx ), tte vnluezr tte snxxn pnss rnte us przpzrtuzynl tz tte yuxoer zr tetrnteqrnl elexeyts useq uy tte xest.hus oetnvuzr wns pzyirxeq uy tte pntueyt pnse sezxetry uy wtupt tte snxxn pnss rnte szes rrzx% tz npprzxuxntely % rzr caR vnlues tuster ttny . As n pzyplusuzy, we qexzystrnteq ttnt

zur qeysutyxnppuysxettzq (MCM) yeeq n caR lzwer ttny tz tnve n snxxn pnss rnte tusterttny %. Evey ttzust tte pruypuple zr xnss pzyservntuzy us respepteq rzr nll tte level zr qetnulszr tte xest, tte vnlue zr tte snxxn pnss rnte us yzt pzystnyt.

Furtterxzre, tte relntuzystup oetweey tte tztnl tuxe yeeqeq rzr tte eyture suxulntuzy nyq tteyuxoer zr tetrnteqrzys us ullustrnteq uy isure . . . he tztnl tuxe us qeiyeq ns tte sux zr ttelznquys tuxe nyq tte suxulntuzy tuxe. he rzrxer us tte tuxe yeeqeq tz lznq tte sezxetry uytztteMzyteCnrlz pzqe, nyqhe lnter represeyts tte suxulntuzy tuxe zr tte uyternptuzy oetweey ttesezxetry nyq tte pnrtuple oenx. he iynl results stzw ttnt tte suxulntuzy tuxe us pzystnyt nyqnlxzst uyqepeyqeyt zr tte yuxoer zr tetrnteqrzys. Czytrnry tz tte suxulntuzy tuxe, tte lznquystuxe us przpzrtuzynl tz tte yuxoer zr elexeyts uy tte xest nyq pny renpt twz tzurs rzr n xestwutt K tetrnteqrzys.


As n pzyplusuzy, tz zotnuy nppurnte qzse qustruoutuzyswuttMCMxettzq, we yeeq iye tetrnte-qrnl xestes. Hzwever, tte tust yuxoer zr tetrnteqrzys pny quupwly oe tuxe-pzysuxuys. Oy tteztter tnyq, we pny nlsz nssuxe ttnt respeptuys tte pruypuple zr xnss pzyservntuzy wutt MCM uypznrse xestes qzesy’t eysure tust snxxn pnss rntes.

az szlve ttus przolex, we tnve stuqueq tte uxpnpt zr tte wnter equuvnleyt pntt leystt zy tteqzse. Iy tte yext ptnpter, we wull uytrzqupe n yew npprznpt ttnt qeiyes tte qeysuty zr tte tetrnte-qrnl elexeyts nppzrquys tz tte qureptuzy zr tte oenx.

. C

Iy ttus ptnpter, we qespruoeq zur rzur-quxeysuzynl qzse pnlpulntuzyxettzq onseq zy tte pruypuplezr xnss pzyservntuzy, ns well ns uts uxplexeytntuzy zy tte Genyt plntrzrx. de tnve stzwy ttnttetrnteqrnl xestes suve nppurnte qzse xnps ns pzyveytuzynl vzxel-onseq struptures uy orenttuyssuxulntuzy. Furtterxzre, we tnve prenteq n xultu-reszlutuzy tetrnteqrnl xest tz stuqy tte uxpnptzr tte yuxoer zr tetrnteqrzy elexeyts zy tte qzse. Our results przveq ttnt iye tetrnteqrnlxesteswutt n tust yuxoer zr elexeyts nre requureq tz zotnuy nppeptnole snxxn pnss rntes. Besuqes, wepzypluqeq ttnt uy xzst pnses, tte qzse nppurnpy us przpzrtuzynl tz tte level zr qetnul zr tte xest.

Hzwever, ur we wnyt tzxuyuxuze tte yuxoer zr tetrnteqrzys nyq requpe tuxe pzxputntuzy, weyeeq n yew npprznpt ttnt eysures tte prepusuzy zr tte wnter equuvnleyt pntt leystt zr tte trnverseqxnterunls. hus, uy tte yext ptnpter, we wull uytrzqupe n yew xettzq uy wtupt we wull xuyuxuzette errzr zr tte wnter equuvnleyt pntt leystt uy tte qureptuzy zr tte trentxeyt oenx.


Figure 2.7.5: Dose distributions of Monte Carlo simulations. Top: tetrahedral model. mid-dle: sagital view of voxel grid. Bottom: 4D Dose distribution ( left: accumulated dose intetrahedral mesh. right : accumulated dose in CT data for all phases.


LODk LODk+1 LODn

Tetrahedralization

Density mapping

Hounsfield units

to density

Surface extraction

Multiresolution tetrahedral mesh

0

1000

2000

-1.000e+03

3.000e+03

scalarsSegmentation

0

1000

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-1.000e+03

3.000e+03

scalars3D CT-scan

Density

Figure 2.8.1: Flowchart of tetrahedral model construction: from 3D-CT to multiresolutiontetrahedral model. The CT images are irst converted into voxelized mass density maps whichare then mapped to the tetrahedral elements of the meshes.

Figure 2.8.2: Tps coniguration for lung. Two single iled passive beams were used in oppo-site directions, each with its own range compensator and collimator.


Lung

Chair

Proton beam

Tumor

Figure 2.8.3: A thorax geometry representing a patient case used for the multiresolutionstudy. The proton beam is directed toward the tumor and the dose is saved in the tetrahedralelements of the mesh.


90

91

92

93

94

95

96

97

98

99

100

0 5 10 15 20 25

gam

ma

pas

s ra

te [

%]

VTR

Beam 1 Beam 2

Figure 2.8.4: Gamma pass rate vs the voxel-to-tetrahedron ratio using a dta equal to mmand a dose tolerance of %. Beam 1 and Beam 2 are the proton beams used in the lung caseillustrated in igure 2.8.2.

90

91

92

93

94

95

96

97

98

99

100

0 10 20 30 40 50 60 70 80

ga

mm

a p

ass

ra

te [

%]

VTR

Patient case

Figure 2.8.5: Gamma pass rate vs the voxel-to-tetrahedron ratio for the patient case simu-lation. For the gamma evaluation a dta equal to mm and a dose tolerance of % have beenused.


0:00:00

0:14:24

0:28:48

0:43:12

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0 50000 100000 150000 200000 250000 300000 350000

Tim

e [H

H:M

M:S

S]

Number of elements

Loading time Simulation time Total time

(a)

0:00:00

0:14:24

0:28:48

0:43:12

0:57:36

1:12:00

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0 5 10 15 20

Tim

e [H

H:M

M:S

S]

Voxel-to-tetrahedron ratio

Loading time Simulation time Total time

(b)

Figure 2.8.6: Relation between computation time and the resolution of the tetrahedral mesh.(a) Number of tetrahedrons versus calculation time. (b) Voxel-to-tetrahedron ratio versuscalculation time. Loading time is the time needed to load all the tetrahedral elements to thesystem. Simulation time is the time taken by the Monte Carlo code to calculate the interac-tions between the particle beam and the matter. The total time is the sum of loading timeand simulation time. These values were calculated using proton particles ired at the mul-tiresolution lung geometry.

Good judgment comes rom experience, and a lot of that comesrom bad judgment.

dull Rzsers

3Particle-beam-dependent optimization for dose

calculation

. I

Iy ttus ptnpter, weqeiye nyewxettzqzlzsy tz pnlpulnteqzse uy tetrnteqrnl sezxetrues. Asprzvequy tte prevuzus ptnpter, respeptuys tte pruypuple zr xnss pzyservntuzy uy tte tzle sezxetry wteypnlpulntuys tte qeysutues zr tetrnteqrnl elexeyts us yzt suipueyt tz eysure pztereyt qzse qustru-outuzys wtey usuys pznrse xestes. de tnve nlsz stzwy ttnt ttus zutpzxe us pnuseq oy tte yzy-pzxplunype zr tte wnter equuvnleyt pntt leystt pnlpulnteq usuys tte xzqeleq tussues nyq tte rerer-eype dEPL qeiyeq zy tte Ca-spny qntn. Our xnuy pzytruoutuzy uy tte ielq pzysusts zr qeiyuysn yew qzse pnlpulntuzy xettzq ttnt qzesy’t qepeyq zy tte reszlutuzy zr tte xest nyq nux nt re-qupuys tte pzxplexuty zr tte sezxetry wtule xnuytnuyuys n prepuse qzse xnp. he zoveptuve us tzrequpe oztt xexzry pzysuxptuzy nyq suxulntuzy tuxe uy Mzyte Cnrlz suxulntuzys oy qeprens-

Pnrtuple-oenx-qepeyqeyt zptuxuzntuzy rzr qzse pnlpulntuzy

uys tte yuxoer zr tetrnteqrzys uy tte xest. Our npprznpt tnwes nqvnytnse zr tte qureptuzy zr tternquntuzy oenx tz xuyuxuze tte errzr zr tte wnter equuvnleyt pntt leystt zr tte xnter oerzre ttetuxzr, ttus oetweey tte uxnses qesusynteq ns tte szlq stnyqnrq nyq tte qeysuty qustruoutuzy pnl-pulnteq usuys zur xettzq. Iy tte rzllzwuys septuzys, we wull irst quspuss tte pzyplusuzys zr zurwzrw, ttey we wull iyust wutt zur nrtuple puolusteq uy tte peer-revueweq xequpnl vzurynl: Ptysupsuy Mequpuye nyq Buzlzsy. he nrtuple pzytnuys tte xettzqs, tte results, nyq tte quspussuzy zr tteresults zotnuyeq wutt zur qzse pnlpulntuzy teptyuque.

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he results zr ttus stuqy stzweq ttnt oyxuyuxuzuys tte errzr zr tte wnter equuvnleyt pntt leystt uytte qureptuzy zr tte oenx, ny nppurnte qzse qustruoutuzy pzulq oe zotnuyeq. hererzre, oy npplyuyszur oenx-qepeyqnyt nlszruttx zy n pznrse tetrnteqrnl xest, we seyernte xzre rnuttrul qzse xnpuysuqe tte tuxzr vzluxe pzxpnreq tz pzyveytuzynl qeysuty xnppuys xettzqs. hus zutpzxe ustte pzysequeype zr tte seyernteq efeptuve qeysuty xnp useq uy tte Mzyte Cnrlz suxulntuzy, ttntnllzws requpuys tte xeny errzr zr tte dEPL oy . xx uy tte pnse zr tte xultu-lnyer ptnytzxnyq . xx rzr tte ttzrnx sezxetry. As n pzyplusuzy, tte tuxeyeeqeq tz suxulnte nyq lznq pznrsetetrnteqrnl sezxetrues pny oe % stzrter ttny tte tuxe requureq uy oztt iye tetrnteqrnl xestesnyq vzxel-onseq xzqels. he prentuzy zr pntueyt-spepuip tetrnteqrnl ptnytzxs pny tnwe less tuxeoy nqzptuys zur npprznpt suype tte qeiyutuzy zr tte qeysuty xnp qzesy’t xnuyly rely zy tte res-zlutuzy zr tte xest. Furtterxzre, zur tnrset-zrueyteq qzse pnlpulntuzy xettzq pny oe exteyqeqtz tnyqle xzre zrsny vzluxes ztter ttny tte tuxzr, supt ns zrsnys nt rusw ur xzre qzse prepusuzyus yeeqeq uy ttzse resuzys. Iy tte yext ptnpter, we wull quspuss n xztuzy pzxpeysnteq versuzy zrzur nlszruttx ttnt tnwes uytz nppzuyt tte qeysuty vnruntuzy zver qufereyt orenttuys ptnses nyq nlsztussue qerzrxntuzy.

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Particle-beam-dependent optimization for MonteCarlo simulation in hadrontherapy using tetrahedralgeometries

Yazid Touileb12, Hamid Ladjal12, Michael Beuve2, BehzadShariat11 Univeristé de Lyon, Univeristé Claude Bernard Lyon 1, LIRIS, UMR 5205 F-69622,France2 Univeristé de Lyon, Univeristé Claude Bernard Lyon 1, IPNL, UMR 5822 F-69622,France.

E-mail: [email protected], [email protected],[email protected], [email protected]

Abstract.The use of tetrahedral-based phantoms in conjunction with Monte Carlo dosecalculation techniques has shown high capabilities in radiation therapy. However,the generation of a precise dose distribution can be very time-consuming since a inetetrahedral mesh is required. In this work, we propose a new method that deinesthe density distribution of patient-speciic tetrahedral phantoms, based upon the CT-scans and the direction of the particle beam. The inal purpose is to coarsen thetetrahedral mesh to improve computational performance in Monte Carlo simulationswhile guaranteeing a precise dose distribution in the target volume. Contrarily tothe state of the art methods that calculate the density value of a tetrahedron, locallybased only on the CT-scans, our approach takes also into account the direction ofthe beam to minimize the error of the water equivalent thickness of the tetrahedronsbefore the tumor volume. In this study, the experiments carried out on a multi-layercomputational phantom, and a thorax geometry, show that by applying our method ona coarse mesh, we ofer a better dose distribution inside the tumor compared to otherdensity mapping methods, in the same level of detail. This is due to the reduction ofthe water equivalent path length error from 9.65 mm to 0.62 mm in the case the multi-layer phantom, and from 2.42 mm to 0.48 mm for the thorax geometry. Moreover, asimilar dose coverage is obtained with reined tetrahedral meshes. As a consequence ofthe reduction of the number of tetrahedrons, computational time is found to be 25%shorter than both the reined tetrahedral mesh and the voxel-based structure in mostcases. Using a coarse tetrahedral mesh to have accurate dose distributions on a giventarget is feasible as long as the water equivalent path length in the direction of thebeam is respected.

Keywords: particle therapy, dosimetry, tetrahedral meshes, Monte Carlo simulation


1 Introduction

Oncological hadron therapy ofers a better dose coverage compared to conventionalmegavolt photon therapy in the treatment of deep-seated local tumors. The particularityof heavy charged particles like protons resides in the fact that they deposit most of theirenergy at the end of their range, creating the Bragg Peak. This property guaranteesa precise irradiation of the target volume and less damage to healthy tissues. Theposition of the Bragg Peak depends on the beam energy as well as the density and thechemical composition of the traversed mediums. These latter parameters are computedusing the Hounsield Unit (HU) numbers of the CT scan voxels. Current treatmentplanning systems and dose deposit simulation techniques use voxel representation oftissues. However, a new geometric representation based on tetrahedrons has severaladvantages compared to voxels. It ensures not only a compact representation of thehuman body, that can contain structures with diferent levels of detail, but also a betterdescription of organs compared to voxel-based geometries. At the same time, it hasshown high capabilities in deformable anatomy and respiratory motion representationand simulation (Ishikawa et al. 2009, Manescu et al. 2012, Manescu et al. 2014 , Hanet al. 2015, Touileb et al. 2016). Consequently, tetrahedral geometries are attractingconsiderable interest in heavy-ion tumor therapy, due to the multiple advantages thatthey could ofer when coupled with Monte Carlo dose calculation technique.

Tetrahedral geometries were irst introduced as acceleration structures for surface-based models in Monte Carlo transport codes (Barker, Bird, Serco & Cooper 2008,Barker, Bird & Thetford 2008, Fang 2010, Shen & Wang 2010, Fonseca et al. 2014). Inaddition to the fact that tetrahedral meshes allow a better approximation of the smoothsurfaces, they allow to describe the movement and density distribution of internal tissueseasily. Authors in (Poole et al. 2012a, Poole et al. 2012b) have used the tetrahedral meshgenerated from a closed tessellated surface to accelerate the geometrical navigation inGeant4 Monte Carlo code (Agostinelli et al. 2003, Allison et al. 2006, Allison et al. 2016).They demonstrated that the navigation time is reduced by two orders of magnitudesince the computations are performed only on local tetrahedrons rather than the entiretessellated surface. The same approach was used by (Yeom et al. 2014) to simplifythe geometry of a polygonal computational human phantom named PSRK-Man (ChanHyeong et al. 2011) to create the equivalent tetrahedral-based phantom. They reporteda reduction of the computation time by a factor varying from 150 to 832, depending onthe particles and their energy. Nevertheless, the simulation time reduction compared tothe voxels was less signiicant and varies from 0.8 to 6.8 times with protons. In a recentstudy conducted by (Furuta et al. 2017), the authors confronted their implementationof tetrahedral meshes with voxel-based structures using the Monte Carlo code PHITS(Sato et al. 2013). They showed that the computation time of the particle transportsimulation using the PSRK-Man tetrahedral version (Yeom et al. 2014) is about fourtimes less than the time required by the voxel grid representation. However, even ifthe transport calculation time is proven to be shorter in most cases, tetrahedral meshes

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still take more time in the geometry loading before performing Monte Carlo simulations(Poole et al. 2012b, Furuta et al. 2017). In fact, for tetrahedrons, the function of theloading time with respect to the number of elements is found to be exponential, butlinear for voxels and surface based meshes.

Since the current scanner technology is based on voxel structures providing aunique density value for each voxel, the deinition of the densities for the tetrahedralgeometries requires mapping algorithms that calculate a density value per tetrahedronfrom CT-scan data. To use tetrahedral geometries in more realistic clinical applicationswith patient-speciic tetrahedral phantoms, authors in (Ishikawa et al. 2009, Manescuet al. 2012, Manescu et al. 2014, Han et al. 2015, Touileb et al. 2016) have proposedseveral approaches to map densities from CT-scans to tetrahedral geometries. In(Ishikawa et al. 2009) authors have deined the density value of a tetrahedron as thedensity of the voxel containing the centroid of the tetrahedron. This method will bereferred to as the centroid mapping method (CM) throughout this paper. Moreover, wein (Manescu et al. 2012, Manescu et al. 2014) have proposed to map the densities tothe nodes of the tetrahedral mesh instead of its elements, as in inite element analysis.This has the advantage of obtaining a density value for each point inside the mesh withinterpolation, and more importantly, preserving the tissues mass since the method isbased on the mass conservation principle. Furthermore, we introduced a new techniquecalled mass conservation mapping method (MCM) (Touileb et al. 2016) in which themass of a tetrahedron is deined as the sum of all the intersecting voxels masses, andthe unique density of the tetrahedron is computed as the mass of the tetrahedron pervolume. Despite the fact that the mass conservation mapping method (MCM) hasshown more accurate dose distributions than the centroid method (CM), its precisiondepends on the density heterogeneity of the mapped tissues and also to the numberof tetrahedrons representing the mesh. The relatively larger volume of tetrahedronsleads to an averaging of the density values of the tetrahedral mesh and consequentlyto an incorrect dose distribution. Then, on the one hand, it is necessary to have aine mesh for the precision, and on the other a coarse mesh for the computation time.Moreover, these methods tend to focus on the local density values of the tetrahedronsand ignore the integral of the densities along the beam path laying before the target.This integral is known as the radiological thickness, and it is commonly expressed interms of water equivalent path length (WEPL). It is used by physicists to calculate thethickness of a water volume in which the ions lose the same amount of energy as theypass through a given tissue. Evidently, the use of CM or MCM with coarser meshesengenders wrong water equivalent path length when compared to WEPL computedusing the original CT-images. To overcome this problem, in this paper, we introducea technique called efective density mapping (EFFD) to minimize the WEPL errorin the direction of the beam. This approach not only takes into account the tissuesinformation derived from CT-scans, but also the particle beam parameters, to optimizethe dose distribution in a given target while using a coarse tetrahedral mesh. Our inalaim is twofold: irst, to reconcile the two above mentioned constraints (i.e., time and

3


precision); and second, adding more freedom in the creation process of patient-speciicphantoms with coarse meshes. The paper is organized as follows: irst, we describein detail the water equivalent path length (WEPL) adapted to tetrahedral geometryand then make a mathematical formulation of our proposed scheme. Furthermore, weperform experimentations using a proton beam on a multi-layer phantom as well as areal patient’s thoracic geometry to quantify the performance of our method in MonteCarlo dose computations using Geant4 code. Finally, we give some concluding remarksand the directions for future works.

2 Materials and methods

The knowledge about the density values and the chemical composition of all tetrahedronsis essential for Monte Carlo dose computations. These parameters are calculated usingan algorithm that maps the densities from a CT-image to the tetrahedral mesh. Ourmethod consists of calculating an efective density distribution only for the tetrahedronsthat intersect the beam. Contrary to MCM and CM that are methods based only on thelocal approximation of the density in a given tetrahedron and ignore the integral of thedensity along the beam path, our approach uses this integral to calculate a density mapthat minimizes the error of the water equivalent path length (WEPL). In this section wewill irst present the concept of the WEPL, then we formulate the problem to calculatethe efective density distribution. Furthermore, the evaluation strategy of our approachis presented alongside the geometries used for the validation.

2.1 Efective Density calculation

2.1.1. Water equivalent path length: The dose distribution inside a given target dependsstrongly on the density of the tissues traversed by the ion beam and more precisely on thewater equivalent path length (WEPL) (sometimes called radiological or water equivalentthickness (WET) (Rietzel et al. 2007, Paganetti 2016). It is an empirical concept thatdescribes the range of ions in matter and is deined as the length of water causing thesame energy loss, as it would lose when going through the tissues. The idea is to havethe same particle attenuation for the product (tmat×ρmat ) of a given material mat witha thickness tmat and a density ρmat as the product (tw×ρw) related to water. Therefore,for a single radiation ray r, the WEPL is the integral of the mass densities along thepath of the particle and it is deined as follow :

WEPL =

∫

L

ρ(r)dl (1)

Where ρ(r) is the linear density function deined on r. To illustrate this concept in 2D,in Figure 1a we associate a line L to the radiation ray. This line depicts the distancebetween the entrance of the body and the tumor, and it is represented by successive linesegments (i.e., [P1, P2] and [P2, P3]) describing the thickness of the traversed mediums.

4


The water equivalent path length of L is the sum of the products P1P2× ρ(medium1)

and P2P3× ρ(medium2).In fact, the mentioned formula in Equation 1 is a simpliication and an

approximation of the water equivalent path length expression deined in (Newhauser2001) as :

WEPL = tw= tmρmρw

Sm

Sw

, where tw and tm are the thicknesses of water and the targetmaterial, respectively. ρw and ρm are the mass densities of water and the material,respectively. Finally, Sw and Sm are the mean values of mass stopping power for waterand the material, respectively.

The precise calculation of the mean stopping powers of the materials iscomputationally intensive using iterative numerical methods (Newhauser et al. 2007),and it is therefore not very suitable for some clinical applications (Zhang & Newhauser2009) particularly for our work since the primary goal is to reduce simulation time. Thus,in this study, the mean stopping powers of the materials are not taken into account.Besides, the density of the water is considered equal to 1 g/cm3. Consequently, weobtain the following formula: WEPL = tmρm for a unique material, and furthermore,we deine the WEPL of a medium with multiple materials as the accumulation of all thecontributions of their WEPL. The use of our approximate method based on a simpledeterministic formula exploits the fact that most thorax tissues have a stopping powerratio close to 1 (between 0.95 and 1.05 ( Schafner & Pedroni 1998)).

Tumor

P2

P1

(a)

Target

Par

ticl

e bea

m

dir

r1

r2

r3

r4

r5

e1

e1

r4

l4,1

(b)

L

P3Lung

Med

ium

1M

ediu

m 2

r

Particle

source

L4

The origin of r4

Figure 1: Water equivalent path length (WEPL).(a) L represents the polyline of theray r. WEPL(L) = P1P2 × ρ(medium1) + P2P3 × ρ(medium2). (b) Blue trianglesrepresent the set Iβdir

of all the tetrahedrons that intersect the beam βdir. The segmentl4,1 depicts the intersection between the ray r4 (i.e., the line L4) and the triangle e1.

2.1.2. Efective density map creation process : Starting from the tetrahedral mesh, weirst map the densities from the CT-images using the mass conservation mapping method(MCM), then, we only recalculate the density values of the elements that intersect the

5


Den

sity

(g

/cm

3)

Tetrahedral mesh

Density mapping

with MCM

Tetrahedral mesh with densities

mapped on the elements

Calculate the

effective densities

with EFFD

Tetrahedral mesh with densities

mapped on the elements

Particle beam

Figure 2: Flowchart of efective density map creation. Notice that the density values ofthe elements that intersect the beam are changed after the EFFD.

beam using our method called the efective density method (EFFD). Figure 2 illustratesthe lowchart of the creation of the efective density map. The efective densities arethe density values calculated using our method (EFFD) to replace the values previouslycalculated with MCM. The mass conservation mapping method is used since it producesbetter results than CM and also to deine the densities of tetrahedrons that do notintersect the beam. It will also be used further to provide an initial guess in theoptimization process.

2.1.3. Formulation of the Problem : In this study, we use a straight-line path estimationmethod to represent the particle trajectory in which we consider the beam βdir as a setof radiation rays extending from the particle source in the direction dir (see Figure1b). The origins of the rays can be chosen either with equispaced or random samplingmethods and they are deined on the surface of the parallel projection of the tumor onthe plane of the particle source.

Since we want to respect only the WEPL of the tissues before the tumor, we deinefor every ray ri a polyline Li that goes from the exterior surface to the tumor site.

Given a tetrahedralization T of a domain D representing a medium M, we deinethe set of all the elements in intersection with T as:

Iβdir=

n∪

k=0

ek | ek ∈ T ∩ βdir

, where βdir =

m∪

i=0

Li

, n is the number of elements

of T that intersect βdir, m is the number of radiation rays representing βdir and ek is atetrahedron in intersection with the beam βdir.

Based on Equation 1 we deine the water equivalent path length of a given polylineLi as :

WEPL(Li) =

∫

Li

ρ(ri)dl (2)

Knowing that the polyline Li is characterised by a sequence of line segments(li,1, · · · , li,s) representing the intersection with Iβdir

(see Figure 1b), we redeine theWEPL as follow:

6


WEPL(Li) =s

∑

j=0

ρeff (ej)× li,j (3)

Where s is the number of elements of Iβdirthat intersect Li, ej is a tetrahedron

from Iβdirthat intersects Li and ρeff is the efective density function deined on Iβdir

as:

ρeff :

Iβdir→ R

ej 7→ ρeff (ej)

ρeff represents the biased values of densities that aims to replace the genuine func-tion ρmcm (calculated with the mass conservation mapping method) on Iβdir

. The aimof this work is to calculate ρeff for the elements ek of Iβdir

. It is worth mentioningthat the density ρeff is only deined on Iβdir

contrary to ρmcm that is deined on all thedomain D. The main idea is to deine the ρeff by minimizing the WEPL error betweenthe reference density map ρCT of the 3D CT image and the efective density WEPLeff .Then, replace ρmcm by ρeff in Iβdir

. Therefore, we deine our linear system as follow:Ax = b

Where A =

l1,1 l1,2 · · · l1,nl2,1 l2,2 · · · l2,n... ... . . . ...

lm,1 lm,2 · · · lm,n

, x =

ρeff (e1)

ρeff (e2)...

ρeff (en)

and b =

WEPL1

WEPL2

...WEPLm

lij: is the length of the line segment of the intersection between the line Li and anelement ej. WEPLi is the water equivalent path length calculated using the referenceCT image. Solving this system outputs the vector x of the efective density values ofρeff that need to be assigned to each tetrahedron of the mesh.

The fact that m > n makes the system overdetermined since we have more raysthan elements in the intersection with the beam. This implies that there is no uniquesolution that satisies this equation system. Additionally, we need to ensure that theresulted densities are in the range of human tissues densities (i.e., between 0 gcm3 and2.79 gcm3). To solve a sparse overdetermined linear system with linear constraintswe have calculated an optimized solution using a constrained quadratic programmingapproach of the form (Cornuejols & Tütüncü 2006):

minimize 12xTHx+ cTx

QP subject to: Ax = b

lb ≤ x ≤ ub

H: the quadratic matrix (n × n) deined as H = ATA, x: the vector of decisionvariables (n × 1) that represents the unknown-density-values vector of ρeff , c: vectorof linear objective function coeicients (n× 1), deined as c = −AT b, and inally lb andub represents respectively the lower and the upper bounds applied to density values.Besides, an initial guessing of the solution x0 containing the densities calculated usingρmcm is given to the solver to improve the convergence of the solution. Furthermore,

7


in order to solve the optimization problem, we used the interior-point-convex algorithmdescribed in (Gould & Toint 2004, Altman & Gondzio 1999) since it is suited for sparsequadratic programs and also for its faster optimization time compared to other methods.

2.2 Evaluation strategy and criteria

We deine two case studies in order to quantify the performance of our proposed scheme,a multi-layer phantom (MLP) and a thorax geometry. The strategy employed togenerate the tetrahedral mesh consists of using only the external surface of the phantomand the tumor (i.e., the target) in the tetrahedralization algorithm. This approachapplies fewer constraints on the algorithm and can be used to decrease the number ofelements signiicantly. It also provides more possibilities to the user to control the volumeof the tetrahedrons. By adopting our density mapping method (EFFD) for a coarsemesh, we will investigate its performance comparing to the centroid mapping method(CM) and the mass conservation mapping method (MCM) using tetrahedral meshes withmultiple resolutions. More speciically, we evaluate the water equivalent path length(WEPL) then the dose accuracy with respect to the voxel-based representation whichis considered as our ground truth. Finally, the computation time will be investigatedfor all scenarios.

2.2.1. Water equivalent path length accuracy: To study the accuracy of the waterequivalent path length of the particle beam, we deine the mean absolute error of WEPL(MAEW), the mean error of WEPL (MEW) and the mean relative error of WEPL

(MREW) as: MAEW =

m∑

i=0

|WEPLi(CT )−WEPLi(T )|

m, MEW =

m∑

i=0

WEPLi(CT )−WEPLi(T )

mand

MREW =

m∑

i=0

|WEPLi(CT )−WEPLi(T )|/WEPLi(CT )

m. Where WEPLi(CT ) and WEPLi(T )

are the water equivalent path length of the line Li calculated using the CT image and thetetrahedral mesh T , respectively. m is the number of lines representing the beam. Tohave a more precise depiction of the WEPL errors, we also deine the errors histogramof all the lines.

2.2.2. Dose distribution: To investigate dose homogeneity and uniformity in the targetvolume, we converted dose distributions into 2D cumulative dose-volume histograms(CDVH). These curves give the percentage of volume that receives more than a certainamount of dose within a deined volume or organ. Besides, they allow to deineparameters like maximum dose (Dmax), minimum dose (Dmin) and mean dose (Dmean)alongside with homogeneity index HI = D5

D95

, where D5 and D95 are respectively theminimum dose of 5% and 95% of the target volume. The ideal value of HI is 1,and it increases as the distribution becomes less homogeneous (Kataria et al. 2012).Furthermore, gamma index distributions (Low et al. 1998) were also considered tohighlight the regions were the dose deviations are concentrated.

8


2.2.3. Computational performance: The computation speed is expressed by thestructure loading time (tL) and the simulation time (tS). The former is the time neededto deine the materials and to load the geometry by creating a single G4Tet object forevery tetrahedron in Geant4, add it to a G4AssemblyVolume, then make an imprint of itin the user detector geometry. This time is correlated to the number of the elements ofthe tetrahedral mesh. On the other hand, the simulation time depicts the time requiredto simulate the transport of all the particles in Monte Carlo code. In this work, thesimulations were carried out on a desktop personal computer with an Intel Core i5 3.2GHz CPU and a 16 Gb RAM.

2.3 Geometries of interest and simulation setups

2.3.1. Multi-layer phantom: The multi-layer phantom (MLP) is deined as a cubecontaining four concentric, homogeneous cubic shells of tissue (see Figure 3): Tumor,lung, bone, and muscle tissues from the inside to the outside of the phantom. Thecomposition and the mass density of the used materials are taken from the InternationalCommission on Radiation Units and measurements (White et al. 1992). The inner cubeis considered as the target, and it has a tumor density of 0.96 g/cm3. The voxel-based structure of the MLP consists of 224 × 224 × 224 voxels with the resolution of1 × 1 × 1 mm3. Since that the primary source of tissues properties is the voxel-basedstructure (in which voxel values represent the density of the tissues in the multi-layerphantom), we use our previously published method (Touileb et al. 2016) to create thetetrahedral geometry and its corresponding density map. Therefore, we tetrahedralizethe volume between the external surface of the phantom and the inner cube (i.e.,the target) using the Tetgen software (Si 2010, Si & Gärtner 2015). The latter isbased on the constrained Delaunay tetrahedralization method (CDT) that can generatemeshes with diferent resolutions by applying an upper bound on the volume of thetetrahedrons. Finally, the mass densities are mapped from the voxels to the elementsof the tetrahedral meshes. Hence, we create a single density value per tetrahedron thatis used for Monte Carlo particle transport simulations. Two diferent density mappingmethods are investigated in this study along with our method (i.e., EFFD) to test theirrespective performances: Mass conservation mapping (MCM) and centroid mapping(CM). Furthermore, Figure 4a illustrates the conigurations used for the MLP in whichthree levels of detail are used for MCM and CM and a coarse resolution for the EFFD.Moreover, the corresponding histograms of the volume of tetrahedrons are plotted inFigure 4b. For Geant4 simulations, a passive scattering system was used to generate aspread-out Bragg peak (SOBP) that covers the entire target volume. Hence, six virtualproton sources with an energy of 270 MeV were placed inside a water envelope atdiferent depths to shift their range (see Figure 4c). Moreover, the sources are weightedproportionally to the relative number of protons to generate the desired SOBP. Thissimpliied beam simulation replaces the range modulator wheel containing absorbers ofvariable thickness that modulates the depth penetration of the beam. Finally, the same

9


particle beam coniguration is maintained through all the simulations regardless of thestructure of the phantom. To show the eiciency of our approach and for the sake ofcomputational cost, we chose only 49 (7× 7) rays, uniformly distributed on the sourcesurface. In this case, the absolute error of WEPL was 0.62 mm, improving signiicantlythe error calculated with both CM and MCM methods (see. Table 1). Additionally,Figures 5a and 5b illustrate, respectively, the absolute error of WEPL versus the numberof rays, and the behavior of the absolute error of WEPL over the time needed to calculatethe intersection of the rays with the mesh and to deine the equation system.

Multi-layer phantom

Bone

Muscle

Lung

Bone

Muscle

Tumor

224 mm

160 mm

96 mm

32 mm

Tumor

Layer 0

224 mm

160 mm

96 mm

32 mm

Den

sity

(g

/cm

3)

1.57

1.2

0.4

0.8

0.21

1.57

1.2

0.4

0.8

0.21

A slice of the phantom

Figure 3: The Multi-layer phantom (MLP). Four diferent homogeneous shells of tissueare used : tumor, lung, bone and muscle (from the inside to the outside). The densityvalues and the material composition of these tissues are issued from (White et al. 1992).

2.3.2. Human thorax geometry: We have equally considered a real patient case with aCT-scan issued from the dir-lab database (Castillo et al. 2010). The voxel resolution ofthe CT images is 0.97× 2.5× 0.97 mm3 and their number is 512× 128× 378. They arerepresented by a voxel map of linear X-ray attenuation coeicients of diferent tissues,in which voxel values are scaled such that the linear X-ray attenuation coeicient of airequals -1000 and that of water equals zero. This scale is the Hounsield scale with theHU unit. Authors in (Schneider et al. 2000) present a method to convert Hounsieldvalues into tissue parameters (mass density and chemical composition) needed for dosecalculations in particle therapy. The lowchart of the creation of the patient-speciictetrahedral model is depicted in Figure 6. The images are irstly preprocessed bythresholding the image intensities, then the organs of interest are segmented using thesnake evolution method (Kass et al. 1988) based on active contours. Further on, thesesurfaces are converted into tetrahedral meshes using the same approach described insection 2.3.1.

10


MCM (Touileb et al. 2016)

CM (Ishikawa et al. 2009)

EFFD (Our aproach)

892 6239 45094

892 6239 45094

892

1.57

1.2

0.4

0.8

0.21

Den

sity

(g/c

m3)

(a)

0

10

20

30

40

50

60

0.6

4

3.1

9

5.7

4

8.2

9

10

.83

13

.38

15

.93

18

.48

21

.03

23

.58

26

.13

28

.68

31

.23N

um

ber

of

tetr

ahed

rons

Volume of tetrahedron (cm3)

MLP_892

0

50

100

150

200

250

300

350

0.0

8

0.4

1

0.7

3

1.0

6

1.3

8

1.7

1

2.0

3

2.3

5

2.6

8

3.0

0

3.3

3

3.6

5

3.9

8Num

ber

of

tetr

ahed

rons


MLP_6239

0

500

1000

1500

2000

2500

3000

0.0

1

0.0

6

0.1

0

0.1

5

0.2

0

0.2

4

0.2

9

0.3

4

0.3

8

0.4

3

0.4

7

0.5

2

0.5

7Num

ber

of

tetr

ahed

rons


MLP_45094

(b)

ater e Collimator Mul

70 cm 10 cm 22.4 cm

32 mm

e sources

(c)

Figure 4: Multi-layer phantom tetrahedral (MLP) (a) MLP conigurations used in MonteCarlos simulations. The tetrahedral model is constructed using the surface of the tumorand the external surface of the phantom. Three levels of detail are used for the massconservation mapping method (MCM) and the centroid method (CM) with 892, 6239and 45094 tetrahedrons, respectively. One level of detail with 892 tetrahedrons is usedfor EFFD. (b) histograms of the volume of tetrahedrons used in MLP with 892, 6239and 45094 tetrahedrons. (c) The coniguration of the detectors for Geant4 simulation.The same coniguration is used for voxel and tetrahedral representations.

11


0

2

4

6

8

10

12

0 200 400 600 800 1000 1200

Mea

n a

bso

lute

err

or

of

wep

l

(mm

)

Number of rays

Mean Absolute error of Wepl MCM

Mean Absolute error of wepl EFFD

(a)

0

2

4

6

8

10

12

0 50 100 150 200 250

Mea

n a

bso

lute

err

or

of

wep

l

(mm

)

Time (s)

Mean Absolute error of wepl MCM

Mean Absolute error of wepl EFFD

(b)

Figure 5: (a) Mean absolute error of WEPL versus the number of rays calculated forthe MLP (with 892 tetrahedrons) using both MCM and EFFD. (b) Mean absolute errorof WEPL versus the time needed to calculate the intersection of the rays with the meshand to deine the equation system.

etraheSurface extraction

density map

0

1000

2000

-1.000e+03

3.000e+03

scalarsSegmentation

0

1000

2000

-1.000e+03

3.000e+03

scalars3D CT-scan Density

Hounsfield units

to densityDensity mapping 0.0

0.4

0.8

1.2

1.6

1.84

Figure 6: Flowchart of tetrahedral model construction: from 3D-CT to tetrahedralmodel. The CT images are irst converted into voxelized mass density maps which arethen mapped to the tetrahedral elements of the meshes.

To take into account the shape of the tumor in hadrontherapy beam line simulationand to improve the eiciency of treatment delivery, we have deined a patient-speciicrange compensator and a multileaf collimator (see Figure 7a). These devices are used inpassive scattering to shape the beam and to minimize the delivery dose in organs at risk(OAR). Moreover, a spread-out Bragg peak (SOBP) is achieved by using six particlesources placed inside a water box to shift their range, each one of them generates aBragg Peak with a diferent number of particles. In a similar way to the MLP, wecompare the EFFD method applied to a coarse mesh (constructed only with the tumorand the external surfaces) to the mass conservation method (MCM) and the centroidmethod (CM) by using both the same coarse mesh and a inner mesh. The latter iscreated by segmenting not only the tumor but also the lungs, the diaphragm, and themediastinum. Figure 7b illustrates the conigurations used for Monte Carlo simulationsfor the patient case for both the coarse mesh with 13566 tetrahedrons and the ine mesh

12


Diaphragm

Mediastinum

umor

Collimator and range compensator

ater e

Sources of particles

am

(a)MM ouileb et al. 2016)

13566 344599 13566

CM (Ishikawa et al. 2009)

344599

EFFD (our approach)

13566

0.001 0.4 0.8 1.2 1.6 1.84

Density (g/cm3)

(b)

0

900

1800

2700

3600

4500

5400

0.1 1.1 2.1 3.1 4.1 5.1 6.1 7.1 8.1 9.1 More

Num

ber

of

tetr

ahed

rons


Patient_13566

0

10000

20000

30000

40000

50000

60000

0.00 0.04 0.08 0.12 0.15 0.19 0.23 0.26 0.30 0.34

Num

ber

of

tetr

ahed

rons


patient_344599

(c)

Figure 7: Thorax geometry. (a) The passive scattering beam line coniguration usedfor Monte Carlo simulation of the thorax geometry. (b) tetrahedral conigurations usedfor dose evaluation. The coarser meshes with 13k tetrahedrons are constructed usingonly the segmentation tumor volume and the surface of the thorax. However, thereined meshes with 344k are constructed using the segmentation of the diaphragm,the mediastinum, the lungs, the tumor and the external surface. (c) histograms of thevolume of tetrahedrons used in the patient case with 13566 and 344599 tetrahedrons.

13


with 344599 tetrahedrons. Furthermore, the corresponding histograms of the volume oftetrahedrons are illustrated in Figure 7c. The particle beam used for the patient case isrepresented by 1243 randomly arranged lines distributed on the surface of the particlesource. This strategy of choosing the position of the lines is adopted since it is simplerto implement and it is more suited for nonuniform shapes as in the case of the tumor.Besides, the number of lines has been chosen as it generates a reasonable mean absoluteerror of WEPL (i.e., 0.48 mm) and presents some gain in computation time. We wouldalso like to point out that increasing the number of lines will eventually decrease theerror.

2.3.3. Geant4 coniguration details: In this work, all the Monte Carlo dose calculationresults relative to the proton beams, have been obtained using The geant4.9.5.P02version and the QGSP_BIC_EMY reference Physics Lists. The latter has beendesigned for any applications required higher accuracy of electrons, hadrons, and iontracking. Additionally, it is recommended for use in medical applications. For the rangecut of all particles in all regions, a cut-of value of 0.1 mm was used alongside a step sizelimit of 0.1 mm to ensure a precise simulation. These values were chosen as they arebelow the thickness of the mean volume of all the tetrahedrons. We would also like topoint out that, the simulation time can be further improved if a production threshold of1 mm is used (as recommended in (Zahra et al. 2010)). However, the accuracy regardingspatial dose distribution will not be ensured for all the tetrahedral geometry.

3 Results and discussion

As stated in section 2.2, our primary purpose is to investigate the performance ofthe efective density mapping method by evaluating the water equivalent path lengthaccuracy and analyzing its impact on dose distribution. In the case of the multi-layer phantom, we used a particle beam containing 49 lines that intersect a total of28 tetrahedrons to construct our efective density map on a tetrahedral mesh with 892elements. Table 1 presents the mean absolute error of WEPL (MAEW) generated using;irst, MCM and CM methods applied to three diferent mesh resolutions (892, 6239 and45094), then, EFFD approach only on the coarser mesh (i.e., 892). The results showthat the MAEW of all the lines has been reduced to 0.62 mm using the EFFD comparedto the other methods, where the minimum error that can be obtained is 1.41 mm usingMCM on the inest mesh with 45094 elements. For the same resolution as used forEFFD (i.e., 892), the mean absolute errors are approximately 9 mm and 16 mm forMCM and CM, respectively. Besides, the distribution of the absolute errors of WEPLof individual lines is depicted in Figure 8a. A similar behavior was observed with thethorax geometry in which the MAEW was decreased to 0.48 mm using the EFFD with atetrahedral mesh containing only 13566 elements (see table 1). In this case, the efectivedensity was calculated for 754 tetrahedrons that intersect the beam represented by 1243lines. Moreover, the histogram of these lines is illustrated in Figure 9a as well as the

14


histograms generated using the MCM and the CM methods for both the coarse meshwith 13566 tetrahedrons and the ine mesh with 344591 elements. The histogram ofthe EFFD tends to have a similar distribution, as the ine meshes of MCM and CM inwhich most of the values are concentrated between 0 and 1.5 mm.

0

5

10

15

20

25

30

35

40

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 >5

CO

UN

T O

F L

INE

S

ABSOLUTE ERROR OF WEPL (MM)

MCM_892

CM_892

EFFD_892

0

5

10

15

20

25

30

35

40

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 >5

CO

UN

T O

F L

INE

S


MCM_6239

CM_6239

EFFD_892

0

5

10

15

20

25

30

35

40

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 >5

CO

UN

T O

F L

INE

S


MCM_45094

CM_45094

EFFD_892

(a)

0

5

10

15

20

25

30

35

40

<-5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5

CO

UN

T O

F L

INE

S

ERROR OF WEPL (MM)

MCM_45094

CM_45094

EFFD_892

0

5

10

15

20

25

30

35

40

<-5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5

CO

UN

T O

F L

INE

S

ERROR OF WEPL (MM)

MCM_6239

CM_6239

EFFD_892

0

5

10

15

20

25

30

35

40

<-5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5

CO

UN

T O

F L

INE

S

ERROR OF WEPL (MM)

MCM_892

CM_892

EFFD_892

(b)

Figure 8: Histograms of WEPL errors calculated using the multi-layer phantom.Histograms of the absolute error (a) and the error (b) of WEPL for 49 lines representingthe beam. Three resolutions of the tetrahedral mesh (i.e., 892, 6239 and 45094) are usedalongside the centroid method (CM) and the mass conservation method (MCM).

As regards dose accuracy, table 2 shows the statistical analysis of the 3D dose mapsof tetrahedral meshes obtained for diferent simulations. Even though the analysisof the mean dose did not show a signiicant diference between diferent simulationssince the global mass of the target is more or less respected, the HI is found to bediferent and correlated to the MAEW. Consequently, the EFFD distribution sharesapproximately the same HI as the voxel-based structure and the ine meshes of MCMand CM. However, in the case of coarser meshes, it behaves better compared to MCMor CM techniques. Furthermore, the CDVH of the gross target volume of the thoraxgeometry presented in Figure 10 gives a more clear depiction of the homogeneity of thedeposited dose and shows how the histogram of EFFD has almost the same shape asthe reference voxel-based phantom. To outline the dose distributions, a series of isodoselines were generated in igures 11 and 12 for a sagittal and an axial slices, respectively.Additionally, gamma distributions calculated using a gamma criteria of 3% and 3 mm,are illustrated in Figure 12 for an axial slice.

It can reasonably be suggested that the sign of the mean error of WEPL (see table.1 ) indicates if the Spread Out Bragg peak has shifted forward or backward relative to its

15


0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10

CO

UN

T O

F L

INE

S


MCM_13566

CM_13566

EFFD_13566

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 10

CO

UN

T O

F L

INE

S


MCM_344591

CM_344591

EFFD_13566

(a)

0

50

100

150

200

250

300

350

-10 -8 -6 -4 -2 0 2 4 6 8 10

CO

UN

T O

F L

INE

S

ERROR OF WEPL (MM)

MCM_13566

CM_13566

EFFD_13566

0

50

100

150

200

250

300

350

400

-10 -8 -6 -4 -2 0 2 4 6 8 10

CO

UN

T O

F L

INE

S

ERROR OF WEPL (MM)

MCM_344591

CM_344591

EFFD_13566

(b)

Figure 9: Histograms of WEPL errors for the patient case using 1243 lines representingthe beam. (a) and (b) are respectively the histograms of the absolute error and theerror of WEPL of the tetrahedral meshes containing 13566 and 344591, respectively.

original position calculated using the reference voxel-based structure. For instance, asthe MEW of the MCM is positive and equal to 2.89 mm, in the MLP with 892 elements,the beam will go further away from the target. Besides, a shift in the 90% isoline isperceived before reaching the target in the case of MCM and CM for 13566 tetrahedralmeshes. It indicates that the beam started to deposit the dose before reaching thetumor. It also conirms our indings on the direction of the shift as a function of thesign of the MEW.

Figures 8b and 9b illustrate the Mean error of WEPL for both the multi-layerphantom and the thorax geometry. Further analysis on the tetrahedral meshes showedthat the relative error of the mass of the tetrahedral mesh compared to the voxel-based structure is about 0.02% in EFFD. This negligible value is about the same for allmesh resolutions using MCM or CM. Overall, the WEPL gives a more relevant tool forcomparing density mapping methods.

16


Phantom Densitymappingalgorithm

Number oftetrahe-drons

Meanabsoluteerror ofWEPL(mm)

Mean errorof WEPL(mm)

Meanrelativeerror ofWEPL(%)

Multi-layer phantom MCM 892 9.65 2.89 10.596239 3.83 1.16 4.2145094 1.41 0.72 1.55

CM 892 16.57 3.09 18.26239 7.97 -0.36 8.7545094 2.86 -0.59 3.14

EFFD 892 0.62 -0.62 0.74Thorax geometry MCM 13566 2.42 -0.66 5.29

344599 1.18 -0.05 2.59CM 13566 3.48 -3.46 7.73

344599 2.07 -0.53 4.52EFFD 13566 0.48 0.48 1.05

Table 1: Mean absolute error of WEPL (MAEW), mean error of WEPL (MEW) andmean relative error of WEPL (MREW) of the Multi-layer phantom and the thoraxgeometry using MCM (Touileb et al. 2016), CM (Ishikawa et al. 2009) and EFFD (ourapproach) algorithms.

Table 3 summarizes the computation times obtained for the Monte Carlosimulations carried out on the Multi-layer phantom and the thorax geometry using12 million protons with an energy of 270 MeV . We denote the simulation, the loadingand the total computation times by tS, tL and tT , respectively (see deinitions in 2.2.3).First, our experiments conirm that the tS of all tetrahedral geometries is less than thetS of the voxel-based structure. Nevertheless, the tT can in some cases be more dueto the higher value of the tL, in particular for the iner thorax tetrahedral mesh with344k elements where its tL exceeds the tL of the reference by an hour and a half. Theuse of the EFFD with the coarser mesh divides this time (i.e., tL) by a factor of sixhundred resulting with a tL of 9 seconds and a total time of six hours. The use offewer elements also decreases the tS signiicantly; it can be reduced to two hours anda half in the case of the MLP, and to forty minutes for the thorax geometry. It isworth mentioning that the signiicant diference in the simulation time between the tworeference voxel-based models is because the tumor in the multi-layer phantom is deeperthan the one in the thorax geometry. They are respectively located at a depth of 96mm

and 40 mm. Therefore, more geometrical and physical steps are needed to calculate theenergy deposition for the Multi-layer phantom.

Although our approach has shown promising results, it is necessary to point out that

17



Number ofelements

Dmean Dmin Dmax HI

Voxel-based MLP - 1123942 35.19 28.69 42.21 1.18MLP tetrahedral mesh MCM 892 33.76 2.36 42.23 1.38

6239 34.96 26.47 43.60 1.2445094 35.18 26.08 43.26 1.22

CM 892 30.80 0.87 43.77 2.656239 33.85 8.22 43.91 1.3545094 35.17 25.85 44.07 1.23

EFFD 892 35.78 25.31 45.80 1.23Voxel-based thoraxphantom

- 11025375 54.83 41.37 62.20 1.27

Thorax tetrahedral mesh MCM 13566 55.0 41.37 62.0 1.22344599 54.97 41.62 64.46 1.27

CM 13566 54.97 34.45 69.77 1.22344599 54.16 43.09 63.98 1.27

EFFD 13566 54.10 41.78 63.49 1.28

Table 2: Evaluation of dose distribution in the tumor volume of the multi-layer phantomand the thorax geometry. Dmin, Dmax, Dmean are respectively the minimum, themaximum and the mean dose deposited. HI is the Homogeneity index.

when the number of tetrahedrons is high, the creation of the system and the calculationof the solution can be very time-consuming. This issue can be addressed by using line-tetrahedron intersection acceleration algorithms like in (Marmitt & Slusallek 2006). Itwill also be interesting to investigate if our method will maintain the same outcomeson a more heterogeneous medium that involves a signiicant mix of bone tissues. Eventhough, at least for the thorax, it seems that the results are accurate and satisfying fordose calculations.

To take full advantage of tetrahedral geometries, a motion compensated versionof this algorithm can be investigated; we believe that it can signiicantly reducecomputation time in 4D simulations. As stated in (Han et al. 2015), their 4D tetrahedralphantom performs 40 times slower than the voxel 4D implementation. This statementindicates that there is still room for improvement in deformation simulations.

Finally, we believe that our region-of-interest method can not only produce accurateresults in dose calculations but also provides a patient-speciic phantom constructionmethod that can simplify the treatment planning in a real clinical environment.

18



Number ofelements

Loadingtime (s)

Simulationtime (s)

Total time(HH:MM:SS)

Voxel-based MLP - 11239424 377.079 48490.1 13:34:27MLP tetrahedral mesh MCM 892 0.0156 23538 6:32:18

6239 0.4652 27831.6 7:43:5245094 34.2065 32096.5 8:55:31

CM 892 0.0155 23579 6:32:596239 0.2817 26936.4 7:28:5745094 34.7164 32345.8 8:59:41

EFFD 892 0.01471 23182.9 6:26:23Voxel-based thoraxphantom

- 11025375 313.853 27408.9 7:42:03

Thorax tetrahedral mesh MCM 13566 9.67524 21582.4 5:59:52344599 5769.45 23294.4 8:04:24

CM 13566 8.23905 21426.1 5:57:14344599 6362.28 23829.8 8:23:12

EFFD 13566 9.86269 21768 6:02:58

Table 3: Loading and simulation time of the Multi-layer phantom (MLP) and the thoraxgeometry using 12×106 protons in Monte Carlo simulations. Time is reported in seconds(s) or in hours:minutes:seconds (HH:MM:SS).

0

20

40

60

80

100

120

0 20 40 60

Rel

ativ

e v

olu

me

[%]

Dose [Gy]

Reference MCM_344K CM_344K EFFD_13K

0

20

40

60

80

100

120

0 20 40 60

Rel

ativ

e v

olu

me

[%]

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Reference MCM_13K CM_13K EFFD_13K

Figure 10: Cumulative dose volume histogram of the gross tumor volume (GTV) of thethorax geometry using MCM, CM and EFFD. Left: coarse mesh with 13k tetrahedronsis used. Right: a reined mesh with 344k is used for MCM Touileb et al. 2016, CMIshikawa et al. 2009.

19


Sag

itta

l sl

ice

erence64

60

40

30

20

10

0

Do

se (

Gy

)

Figure 11: IsoDose of a sagittal slice of patient case . Isolines are from the inside to theoutside: 90%, 78%, 68%, 57%, 46%, 36%, 25% and 15% of dose max.

4 Conclusion

In this paper, we have presented a new method for dose computations using tetrahedralgeometries, which employs particle beam parameters (i.e., direction and shape) inthe density mapping algorithm. Thus, by calculating an efective density map thatminimizes the error of the water equivalent path length of the beam rays. This approachaims at providing a precise dose distribution in a given target using a simpliied geometry,resulting in a signiicant decrease of tetrahedral elements. Monte Carlo transportsimulations with 270 MeV proton particle beam, were performed for a cubic multi-layer phantom and a thorax geometry. Furthermore, the computational performanceand dose accuracy were examined for both cases in comparison to the voxel-basedphantom constructed with the CT-images. The use of our target-oriented techniquein dose computations can reduce both the transport and the loading time of tetrahedralgeometries since in clinical practice; the physicians are interested in a limited set oforgans at risk and the tumor. This work can be extended and optimized to handlemore volumes other than the tumor, like the organs at risk (OAR). Besides, a motion

20


64605550454035302520151050

Dose (Gy)

MCM 13566 EFFD 13566 CM 344599MCM 344599CM 13566

IsoD

ose

Ref

eren

ce

64605550454035302520151050

Dose (Gy)

Bea

m d

irec

tion

MCM 13566 CM 13566 EFFD 13566 MCM 344599 CM 344599

Gam

ma

index

Gamma index

210

Figure 12: IsoDose and gamma index evaluation (DTA=3 mm and dose tolerance =3%) of an axial slice of the patient case. Isolines are from the inside to the outside:90%, 78%, 68%, 57%, 46%, 36%, 25% and 15% of dose max. The regions that fail thecriteria in the gamma evaluation are clearly identiied where gamma index > 1.

21


compensated version can be investigated to take into account the deformation of thephantom and the density variation over the breathing phases in the case of lung cancer.The creation of patient-speciic tetrahedral phantoms can take less time by adopting ourapproach since the deinition of the density map doesn’t mainly rely on the resolutionof the mesh.

Acknowledgments

This research is supported by LABEX PRIMES (ANR-11-LABX-0063), within theprogram Investissements d’Avenir(ANR-11-IDEX- 0007) and by France Hadron, andby ClaRys project CNRS, deis Imaging.

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Permanence, perseverance and persistence in spite of all obsta-cles, discouragements, and impossibilities7 It is this, that in allthings distinguishes the strong soul rom the weak.

hzxns Cnrlyle

4Motion compensation ofWEPL for 4D dose

calculations

. I

Rnquztternpeutup qzsuxetry nuxs nt qeterxuyuys tte qunytuty zr eyersy qepzsuteq uy tte tuxzrvzluxe nyq tte surrzuyquys tussues ns n result zr ny exterynl oenx rnquntuzy tternpy. Iy tte pnse zrtnqrzytternpy, nppurnte qzsuxetry us yeeqeq rzr trentxeyt plnyyuys nyq veruipntuzy. he estuxn-tuzy zr qzse qustruoutuzy us pnlpulnteq usuys pzxputer suxulntuzys ttnt requure tte xzqeluys zr ttenyntzxupnl rentures zr tte pntueyt’s ozqy. As stzwy uy tte prevuzus ptnpters, we tnve qevelzpeq ntetrnteqrnl xzqel ttnt xzqel tuxny nyntzxy nyq estuxnte tte qzse oy qeiyuys tte qeysuty uyrzr-xntuzy nyq tte xnterunl pzxpzsutuzy uy tte elexeyts. hey we tnve rurtter uxprzveq tte xzqeloy uytrzqupuys n oenx-qepeyqeyt qeysutyxnppuysxettzq ttnt pnlpulntes tte qeysuty vnlues zr tteelexeyts oy xuyuxuzuys tte errzr zr tte wnter equuvnleyt pntt leystt uy tte qureptuzy zr tte oenx.

Mztuzy pzxpeysntuzy zr dEPL rzr D qzse pnlpulntuzys

Hzwever, ttus xettzq tns oeey zyly estnolusteq uy n stntup pzyisurntuzy wtere tte xzvexeyt usyzt tnwey uytz nppzuyt. byrzrtuyntely, uy tte pnse zr luys tuxzrs, tte pzytuyuzus qeysuty ptnysesque tz respurntzry xztuzy nfept tte rnyses zr uzys wtupt pny lenq tz uynppurnte qzse pnlpulntuzys.hese qerzrxntuzys yeeq tz oe pzysuqereq wtey pnlpulntuys tte qzse qustruoutuzy uy n xztuzy-uyqupeq suxulntuzy. az zotnuy nppurnte rzur-quxeysuzynl qzse pnlpulntuzys, tte Mzyte Cnrlz sux-ulntuzys tnve tz oe pzupleq wutt n xztuzy estuxntuzy xettzq nyq tte qzse tns tz oe nppuxulnteqzver tuxe. Iy ttus ptnpter, we wull uyvestusnte tte exteysuzy zr tte dEPL nlszruttx tz pzxpeysnterzr tte xzvexeyt uy qzse pnlpulntuzys. Furst, we wull uytrzqupe n xettzq tz uypzrpzrnte tte stzp-puys pzwer uy tte rzrxulne zr ttedEPL tz zotnuy n xzre prepuse efeptuve qeysuty xnp pzxpnreqtz tte suxpluieq versuzy qeiyeq uy tte prevuzus ptnpter. hey, we wull xensure rnyse uypertnuy-tues relnteq tz oztt xettzqs pzxpnreq tz ny experuxeytnl vnlue pnlpulnteq usuys tte Mzyte Cnrlzpzqe. hey, we wull preseyt n xettzq rzr tte upqntuys zr tte qeysuty vnlues rzr enpt tetrnteqrnlelexeyt zver tuxe. Fuynlly, we wull pzxpnre zur tetrnteqrnl onseq nlszruttx results tz tte szlqstnyqnrq vzxel-onseq strupture usuys n D xultu-lnyer ptnytzx ( DMLP). Furtterxzre, pzssuoleruture qureptuzys nre preseyteq.

. C -

As stnteq uy tte prevuzus ptnpter, ttednter equuvnleyt pntt leystt (nlsz wyzwy ns tte wnter equuv-nleyt ttupwyess (dEa)) us qeiyeq ns tte xnss ttupwyess zr luquuq wnter qespruoeq uy (g.cm ) ttntpnuses n pnrtuple oenx tz lzse tte snxe nxzuyt zr eyersy ns tte oenx wzulq lzse oy trnversuys npertnuy qeptt zr n xnterunl h ]h ]h ] h ]h ]h ]h ].

Iy tte prevuzus ptnpter, we tnve useq n suxple qeterxuyustup rzrxuln tz estuxnte tte vnlue zrtte wnter equuvnleyt pntt leystt uy tte nlszruttx zr tte efeptuve qeysuty pnlpulntuzy. Hzwever, wetnve usyzreq tte vnlue zr tte xeny stzppuys pzwer zr oztt wnter nyq tte xnterunl oy xnwuys ttenssuxptuzy ttnt xzst ttzrnx tussues tnve n stzppuys pzwer rntuz ( SmSw ) plzse tz uy tte rzllzwuysequntuzy repzrteq oy gtnys nyq Newtnuser h ]:

WEPL = tw = tmρmρw

SmSw

( . )


wtere tw nyq tm nre tte ttupwyesses zr wnter nyq tte tnrset xnterunl, respeptuvely. ρw nyq ρm nrette xnss qeysutues zr wnter nyq tte xnterunl, respeptuvely. Fuynlly, Sw nyq Sm nre tte xeny vnlues zrxnss stzppuys pzwer rzr wnter nyq tte xnterunl, respeptuvely.

de wull rererrer tz zur suxpluieq versuzy zr tte dEPL ns tte dEPL suxpluieq zrWEPLsimplified

uy tte rest zr tte qzpuxeyt, nyq we qeiye ut uy tte rzllzwuys equntuzy ns:

WEPLsimplified = tw = tmρmρw

( . )

Ayztter pnrnxeter ttnt pzulq oe nlteryntuvely useq oesuqe tte dEPL(zr dEa) tz spepury ttepnrtuple rnyse, us tte wnter equuvnleyt rntuz (dER) h ], wtupt us tte rntuz zrdEa tz ttexnterunlxnss ttupwyess uy [g.cm− ] nyq qeiyeq ns tw/tm. hus pnrnxeter tns tte nqvnytnse tz oe yenrlypzystnyt nt qufereyt prztzy eyersues rzr n spepuip xnterunl h ]. hus rntuz us nlsz pzysuqereq nstte relntuve stzppuys pzwer (RSP).

Iy ttus septuzy, we wull pzysuqer tte stzppuys pzwer rntuz oetweey tte xnterunl nyq tte wnter tzpnlpulnte ttewnter equuvnleyt ttupwyess nyq stuqy uts efept zy tteoenxrnyse nyqqzsequstruoutuzyuy pzxpnruszy zr zur prevuzus xettzq.

Severnl xettzqs rzr pnlpulntuys dEa uy prztzy oenxs were sussesteq uy tte luternture h ]h ] h ]. Nevertteless, ttese xettzqs use eutter n tuxe-pzysuxuys uterntuve yuxerupnl xettzqzr rnst nyq npprzxuxnte teptyuques zr uywyzwy nppurnpy. Auttzrs uy h ]h ] uytrzqupeq ny-nlytupnl rzrxulns onseq zy ttezretupnl rnyse-eyersy relntuzys, ttnt nre suppzseq tz nptueve xxuypertnuyty uy dEa uy tte pnse zr prztzy ielqs. Hzwever, tteur xettzq zyly tnwes uytz nppzuyt nluxuteq yuxoer zr tnrset xnterunls nyq ttupwyess nyq nlsz pzysuqers zyly eyersues uyreruzr tzMec. Hzwever, uy pluyupnl prnptupe, tte trentxeyt plnyyuys systexs rzr prztzy tternpy requure nCapnluorntuzy purve relntuys tteHzuysielq uyuts tz tte relntuve stzppuys pzwer zr ttexnterunls tzpnlpulnte tte dEa. he xzst pzxxzy teptyuque tz tte pnluorntuzy us tte stzuptuzxetrup npprznpth ] ttnt tns tte nqvnytnse zr oeuys strnusttrzrwnrq nyqwell-qeiyeq. dewull nqzpt ttusxettzqtz pnlpulnte tte RSP zr qufereyt xnterunls uy tte Ca-uxnses suype ut requures less pzxputntuzy tuxenyq nlsz rzr uts nppepteq nppurnpy.


. CaHb- -S P C

he Cnluornteq pzxputer tzxzsrnptup qntn nre tte pruxnry uyput rzr tnqrzytternpy trentxeytplnyyuys systexs ttnt tnyqle tussues uytzxzseyeutues efepts. he oenx rnyse requureq rzr pn-tueyt trentxeyt us pnlpulnteq oy pzyvertuys tte Ca Hb yuxoers tz pnrtuple stzppuys pzwer vnluesusuys nyHb-tz-SPR pnluorntuzy purve (see isure . . ). hus stzuptuzxetrup pnluorntuzyxettzq usqesusyeq tz estuxnte tte wnter equuvnleyt ttupwyess nyq tz qeterxuye tte pnrtuple rnyse rzr vnruzustuxny tussues. hree sepnrnte luyenr its were qzye rzr ttree xnuy pnteszrues: nqupzse tussues, ttezrsnys, nyq xusples, nyq iynlly, ozye struptures. It us expepteq ttnt tte rnyse zr tte pnrtuples uytte tuxny nyntzxy pny oe pzytrzlleq tz oeter ttny ± . % zr tte wnter equuvnleyt rnyse uy szttussue nyq tz± . % uy ozye struptures uy wtupt ut trnyslntes tz n prepusuzy ttnt vnrues rrzx tzmm h ]. he resulteq dEPL us rererreq tz ns tte WEPLst (u.e., stzuptuzxetrup wnter equuvnleytpntt leystt). Furtterxzre, tz evnlunte tte WEPLst, we tnve pnlpulnteq tte experuxeytnl dEPLusuys Genyt Mzyte Cnrlz pzqe nyq n Multu-lnyer ptnytzx. he experuxeytnl vnlue wull oe pzy-suqereq ns tte rerereype tz estuxnte tte qufereype uy tte oenx rnyse rzr oztt tteWEPLsimplified nyqtteWEPLst. Iy tte rzllzwuys septuzys, we wull irst preseyt tte Multu-lnyer ptnytzx sezxetry nyqxnterunl pzxpzsutuzy ttny we wull quspuss uy qetnul tte pnlpulntuzy zr tte rnyse zr tte oenx nyq utsvnluqntuzy.

. M - y (MLP)

he Multu-lnyer ptnytzx wull oe useq rzr tte vnluqntuzy zr tte dEPL pnlpulntuzy ns well ns uy tteqerzrxntuzy suxulntuzy. Iy tte snxe wny ns tte Multu-lnyer ptnytzx zr tte prevuzus ptnpter, wewull use rzur lnyers zr xnters, uypluquys tuxzr, luys, ozye, nyq xusple tussues rrzx tte uysuqe tztte zutsuqe. he qeysuty vnlues nyq tte pzrrespzyquys Hb yuxoers nre qepupteq uy tnole . . .he zyly qufereype relues zy tte ttupwyess zr tte lnyers nyq tte przpzrtuzys zr tte ptnytzx (seeisure . . ). hese vnlues were ptzsey tz zotnuy uyteser przpzrtuzys uy tte vzxel-onseq strupture,evey uy tte qerzrxntuzy zr tte sezxetry.


Figure 4.3.1: HU-to-relative stopping power for diferent human tissues. Image from [185].


(-55 HU)

20 m

m

40 m

m

60 m

m

80 m

m

Tumor

Lung (-789 HU)

Bone (943 HU)

Muscle (50 HU)

Figure 4.4.1: The geometrical coniguration of the 4D multi layer phantom (4DMLP) withfour layers of matter.

. Ex dEPL

Aprepuse estuxntuzyzr ttewnter equuvnleyt pntt leystt us esseytunl rzr tte nlszruttxzr tte prentuzyzr tte efeptuve qeysuty. hererzre, uy ttus septuzy, wewull irst pnlpulnte tte experuxeytnldEPLnyqttey pzxpnre ut tz tte suxpluieqdEPL vnlue useq uy tte prevuzus ptnpter nyq tte vnlue pzxputequsuys tte stzuptuzxetrup xettzq. az qeiye tte experuxeytnl wnter equuvnleyt pntt leystt, wewull pnlpulnte tte qufereype oetweey tte oenx rnyse zotnuyeq wutt n wnter vzluxe, nyq tte oenxresulteq usuys tte sezxetry vzluxe nlzyssuqe tte wnter vzluxe. Iy tte pnse zr tte xultu-lnyerptnytzx, we wull tnve tte pzyisurntuzy qepupteq uy isure . . , uy wtupt we wull zyly extrnpt ttetussues ttnt nre oerzre tte peytrnl puoe, represeytuys tte tuxzr.

he pluyupnl prztzy rnyse us qeiyeq ns tte qeptt qustnl zr % zr ttexnxuxuxqzse uy n prustuyeBrnss Penw vnlue (R %) uy wtupt % zr tte prztzys stzppeq uyqepeyqeytly zr tte eyersy sprenqzr tte oenxh ] h ] h ] h ] h ]. he pnrtuple rnyse us qepeyqeyt zy pnrtuple eyersy, typenyq ttexnterunl zr ny noszroer. Ir n slno zr ttupwyess tm zr n yzy-wnter xnterunl us plnpeq uy rrzyt zrn luquuq wnter ptnytzx ttnt seyerntes n rnyseR %, ut wull przqupe n rnyse zrRShifted

% usuys tte snxe


Lnyer Mnterunls Hb Deysuty (g.cm− )Aqupzse tussue (auxzr) - .Luys tussue (Luys uytnle) - .Mnrrzw ozye (Bzye) .Szt tussue (Musple) .

Table 4.5.1: Materials, density considered in the 4DMulti-layer phantom (4DMLP). Dataadapted from ICRU Report 49

Eyersy (Mec) (lzw eyersy) (xequux eyersy) (tust eyersy)WEPLG (xx) .WEPLsimplified (xx) .WEPLst (xx) .Aoszlute Relntuve Errzr zrWEPLsimplified (%) .Aoszlute Relntuve Errzr zrWEPLst (%) .

Table 4.5.2: Errors of the wepl calculated between the reference WEPL(WEPLG ) and theWEPLsimplified and the WEPLst using three diferent energies. These values are calculated onthe layers before the tumor of the 4DMLP (igure 4.5.1). The WEPLG reference value is thediference between the WET calculated using only the Water phantom and by adding the4DMLP matters that are before the tumor (see igure 4.5.2).

xzyzeyersetup prztzy oenx. hus, dEPL tere us qeiyeq ns:

WEPL = tw = R % − RShifted% ( . )

de wull use tte experuxeytnl WEPL ns tte rerereype vnlue tz evnlunte tte WEPLsimplified nyq tteWEPLst rzr n sezxetry setup uy tte MC suxulntuzys. de zotnuyeq, ns n result, tte snxe vnlue zrttupwyess usuys ttree qufereyt uyutunl oenx eyersues: lzw, xequux, nyq tust eyersy(u.e., , nyq

Mec respeptuvely). he twz Brnss penws nre preseyteq uy isure . . .Mzrezver, evey ttzusttte noszlute relntuve errzr zr tte oztt tteWEPLsimplified nyq tteWEPLst qzy’t expeeq %, tte stzu-ptuzxetrupdEPL requpe tte errzr tz . % ( ns qepupteq uy tnole . . ). Iy tte rzllzwuys septuzys,we wull evnlunte tte uxpnpt zr ttus qufereype zr rnyse zy tte qzse qustruoutuzy zr tte tetrnteqrnlxest.


Pencil

proton

beam

Rwshifted

WET

Liq

uid

wat

er p

han

tom

Pencil

proton

beam

Bragg peak value

Bragg peak value

Liq

uid

wat

er p

han

tom

Rw

Materials of MLP

before the tumor

(a)

(b)

Figure 4.5.1: (a) Depth dose curve including the Bragg peak and proton range (Rw)in a wa-ter phantom. (b) the part of the MLP before the tumor volume is added in front of the wa-ter phantom.Rshiftedw is produced and the WET is deined as the diference between the rangesRw − Rshiftedw .


Depth [mm]0 10 20 30 40 50 60 70

Dos

e [%

]

0

0.2

0.4

0.6

0.8

1

dose

Histograms

Water phantom

MLP + Water phantom

Histograms

Water phantom

MLP + Water phantom

Figure 4.5.2: The normalized dose-versus-depth proiles for the simulated liquide water phan-tom and the integration of the materials of the MLP that are before the tumor volume. Thesecurves were produced using a 90 MeV beam to calculate the WET of the Materials of MLPdescribed in igure 4.5.1. The WET will serve as a reference and it is equal to 28.2 mm.

. E y dEPL

Berzre tte suxulntuzy zr tte pnrtuple-xnter uyternptuzys wutt tte tetrnteqrnl xest, tte qeysuty vnl-ues uysuqe every tetrnteqrnl elexeyt yeeq tz oe qeiyeq usuys n qeysuty xnppuys nlszruttx. Iyptnpter twz, we tnve estnolusteq n xnss pzyservntuzy xnppuys nlszruttx ynxeq MCM ttnt re-spepts tte pruypuple zr xnss pzyservntuzy. hey, we tnve qeiyeq nyztter nlszruttx ttnt wzrwszy pznrse xestes tz requpe suxulntuzy tuxe. hus nlszruttx us tte efeptuve qeysuty xnppuys (zrEFFD), nyq ut xuyuxuzes tte errzr zr WEPLsimplified uy tte qureptuzy zr tte oenx nyq seyerntes ou-nseq qeysuty vnlues uy tte tetrnteqrnl elexeyts ttnt uytersept tte oenx. Iy ttus septuzy, we wull usette snxe nlszruttx tz nlter tte qeysuty vnlues zr tte uytersepteq elexeyts. he zyly qufereype re-suqes uy tte rnpt ttnt uystenq zr pnlpulntuys tte qeysutues uy tte systex zr equntuzy, we wull pnlpulntette stzppuys pzwer nyq rurtterxzre pnlpulnte tte xnss qeysuty nppzrqeytly. he stzuptuzxetrupdEPL us qeiyeq ns :

WEPLst = tw = tmρmρw

SmSw

= tmρs ( . )


wutt ρs tte stzppuys pzwer rntuz (SPR) oetweey tte xnter nyq tte wnter.Suype we wnyt tz respept zyly tteWEPLst zr zyly tte xequuxs oerzre tte tuxzr, we qeiye rzr

every rny ri zr tte oenx n pzlyluye Li ttnt szes rrzx tte exteruzr surrnpe zr tte sezxetry tz ttetuxzr sute.

Guvey n tetrnteqrnluzntuzy T zr n qzxnuyD represeytuys n xequuxM, we qeiye tte set zr nlltte elexeyts uy uyterseptuzy wutt T ns:I

dir=

n∪

k=ek | ek ∈ T ∩ dir

, wtere dir =

m∪

i=Li

, n us tte yuxoer zr elexeyts zr T

ttnt uytersept dir, m us tte yuxoer zr rnquntuzy rnys represeytuys dir nyq ek us n tetrnteqrzy uyuyterseptuzy wutt tte oenx dir.

heWEPLst zr n suvey pzlyluye zr tte oenx Li us suvey ns rzllzw:

WEPLst =

∫

Lρs(r)dl ( . )

Kyzwuys ttnt tte pzlyluye Li us ptnrnpteruseq oy n sequeype zr luye sesxeyts(

li, , · · · , li,q)

rep-reseytuys tte uyterseptuzy wutt I

dir. de qeiye tte wnter equuvnleyt pntt leystt zr n suvey pzlyluye

Li ns :

WEPLst(Li) =

∫

Liρs(ri)dl ( . )

heWEPLst us requiyeq ns :

WEPLst(Li) =

q∑

j=

ρeffs (ej)× li,j ( . )

dtere q us tte yuxoer zr elexeyts zr Idir

ttnt uytersept Li, ej us n tetrnteqrzy rrzx Idir

ttntuytersepts Li nyq ρeffs us tte relntuve stzppuys pzwer ruyptuzy qeiyeq zy I

dirns:

ρeffs :

Idir→ R

ej 7→ ρeffs (ej)

ρeffs represeyts tteounseqvnlueszr SPR tzpnlpulnte rzr tte tetrnteqrnl elexeyts uy ttequreptuzyzrtteoenx. azqeiye tte ρeffs wexuyuxuze tte errzr zr tteWEPLst oetweey tte vzxel-onseqsezxetrynyq tteWEPLeff. Furtterxzre, tz pnlpulnte tte uywyzwy ρeffs wezotnuy tte rzllzwuys luyenr systex:


Ax = b

dtere A =

l , l , · · · l ,nl , l , · · · l ,n...

... . . . ...lm, lm, · · · lm,n

, x =

ρeffs (e )ρeffs (e )

...ρeffs (en)

nyq b =

WEPLst

WEPLst...

WEPLmst

lij: us tte leystt zr tte luye sesxeyt zr tte uyterseptuzy oetweey tte luye Li nyq ny elexeyt ej.WEPLi

st us tte wnter equuvnleyt pntt leystt pnlpulnteq usuys tte rerereype Ca uxnse wutt tte stz-uptuzxetrup xettzq. Szlvuys ttus systex zutputs tte veptzr x zr tte efeptuve stzppuys pzwer vnl-ues zr ρeffs ttnt yeeq tz oe nssusyeq tz enpt tetrnteqrzy zr tte xest. az szlve ttus spnrse zver-qeterxuyeq luyenr systex, we tnve pnlpulnteq ny zptuxuzeq szlutuzy usuys n pzystrnuyeq qunqrntupprzsrnxxuys xettzq zr tte rzrx:

xuyuxuze xTHx+ cTxQP suovept tz: Ax = b

lb ≤ x ≤ ub

H: tte qunqrntup xntrux (n × n) qeiyeq ns H = ATA, x: tte veptzr zr qepusuzy vnrunoles (n × )

ttnt represeyts tte uywyzwy-qeysuty-vnlues veptzr zr ρeffs , p: veptzr zr luyenr zoveptuve ruyptuzy pz-eipueyts (n× ), qeiyeq ns c = −ATb, nyq iynlly lb nyq ub represeyts respeptuvely tte lzwer nyqtte upper ozuyqs npplueq tz qeysuty vnlues. Mzrezver, tte tte uyteruzr-pzuyt-pzyvex nlszruttxqe-spruoeq uy (h ], h ]) tnse oeey useq tz szlve tte przolex suype ut us suuteq rzr spnrse qunqrntupprzsrnxs nyq nlsz rzr uts rnster zptuxuzntuzy tuxe pzxpnreq tz ztter xettzqs.

Nzw ns tte efeptuve relntuve stzppuys pzwer ρeffs us pnlpulnteq rzr every elexeyts uy Idir, tte yext

step pzysusts zr pzyvertuys tte stzppuys pzwer vnlues uytz tteur pzrrespzyquysxnss qeysuty vnluestz oe useq uy tteMzyte Cnrlz suxulntuzy pzqe. Fzr ttus purpzse, we tnve qeiyeq n pzrrespzyquyspurve onseq zy Sptyeuqer et nl. wzrws zy pzyversuzy zr Ca-yuxoer tz relntuve stzppuys pzwerh ] nyq Ca-yuxoer pnluorntuzy tz qeysuty vnlues h ]. hus purve us qepupteq uy isure . .

. E

Iy ttus septuzy, we preseyt ny npplupntuzy zr tte qerzrxnole tetrnteqrnl-onseq qzse pnlpulntuzyxettzq prevuzusly qespruoeq uy ttus ptnpter tz xzvuys zrsnys supt ns tte xultu-lnyer ptnytzx.


0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 0,5 1 1,5 2 2,5

Den

sity

[g/

cm3

]

Realative stopping power

Figure 4.6.1: Mass density values vs relative stopping power.

de stnrt oy preseytuys tte suxulntuzy setup useq tz suxulnte n pnssuve prztzy oenx, ttey we evnlu-nte tte qzse qustruoutuzy usuys tte puxulntuve qzse vzluxe tustzsrnx (CDcH) nyq snxxn uyqex.Fuynlly, de pzypluqe oy preseytuys tte qusnqvnytnses zr tte xettzq nyq ruture uxprzvexeyts.

. .

az suxulnte tte qerzrxntuzy zr tte Multu-lnyer ptnytzx, we tnve npplueq n spnluys rnptzr tz ttevertupes zr tte tetrnteqrnl xest. All tte vzluxes were spnleq wutt . , . nyq . rzr step , step ,nyq step ,respeptuvely. Nevertteless, we tnvexnuytnuyeq tte tuxzr vzluxeuyptnyseq ttrzustzuttte qerzrxntuzy suype we nux nt stuqyuys tte uxpnpt zr tte qeysuty vnruntuzy zr tte tussues oerzrette tuxzr vzluxe zy tte qzse qustruoutuzy zr tte tnrset. he leystt zr tte puoes zr tte MLP nyqtteur qeysutues nre represeyteq uy tnole . . . It us wzrtt xeytuzyuys ttnt tte vnlues zr tte relntuvestzppuys pzwer ρs nre pnlpulnteq usuys tte purve ullustrnteq uy isure . . . Fzr pnrtuple suxulntuzy,we tnve useq n Mec pnssuve oenx wutt susxn eyersy sprenq zr . %, n step suze luxut zr .xx nyq n put-zf vnlue zr . xx. hese vnlues were ptzsey tz tnve nppurnte results ns ttey nre


oelzw tte ttupwyess zr tte xeny vzluxe zr nll tte tetrnteqrzys. Mzrezver, tz pzver tte tnrsetvzluxe uy qeptt, we tnve useq sux squnreq pnrtuple szurpes, ns stzwy uy isure . . . Enpt zr ttexus plnpeq uysuqe tte wnter vzluxe nyq seyerntes n BP wutt n suvey uyteysuty. he resulteq SOBP usrurtterxzre pzlluxnteq nyq lnteyeq usuys n rupple ilter.

Ripple filterCollimator

Water volume

Particle sources

4DMLP

Figure 4.7.1: Geant4 simulation setup of the 4D MLP. The same coniguration is used forevery step of scaling. A ripple ilter was used to latten the Spread-Out Bragg Peak (SOBP).

. . D D

az evnlunte tte qzse xnp zr tte tetrnteqrnl xest seyernteq usuys tte WEPLst, we use tte vzxel-onseq strupture ns zur szlq stnyqnrq. Mzrezver, we tnve nlsz suxulnteq tte tetrnteqrnl xzqelsprenteq usuys ztter qeysuty xnppuys nlszruttxs, uypluquys tte peytrzuq xettzq (CM), tte xnsspzyservntuzyxettzq(MCM), nyq tteEFFDxettzqusuys tteWEPLsimplified. hepurpzse us tz ns-sess tte qzse qustruoutuzy uysuqe tte tnrset vzluxe usuys tte Dxultu-lnyer ptnytzx. hexetrupsttnt nre useq rzr evnluntuzy nre tte Cuxulntuve qzse-vzluxe tustzsrnxs (CDcH) nyq tte snxxnuyqex evnluntuzy ( ). he rzrxer wull suve ny zvervuew zr tte tnrset pzvernse nyq tte tzxzseyeutyzr tte qzse zver tte eyture vzluxe, wtule tte lnter ullustrntes tte qufereype oetweey tte pnlpulnteqqzse nyq n rerereype qzse xnp. Iy tte snxxn evnluntuzy, n qustnype-tz-nsreexeyt pruteruzy zrxx nyq n % qzse qufereype uy nsreexeyt wutt pluyupnl stnyqnrqs were useq rzr nll qzse pzxpnr-uszys.


Cuoe Cuoe Cuoe Cuoe

Step

Cuoe leystt hmm]Deysutyhg.cm− ] . . . .

Hb - -Hb pnluornteq

SPR (ρs) . . . .

Step


Hb - - - -Hb pnluornteq

SPR (ρs) . . . .

Step



SPR (ρs) ,. . . .

Step



SPR (ρs) . . . ,.

Table 4.7.1: 4DMLP deformation parameters including cube length, mass density, HU, HUcalibrated and the relative stopping power in each simulation step.


Fusure . . (n) stzws qufereyt CDcHs uysuqe tte tuxzr vzluxe (u.e.; cube ) rzr tte stntup sux-ulntuzy zr step . de pny yztupe ttnt tte peytrzuq xettzq (CM) suves tte wzrst results wtupt wereexpepteq suype tte qeysuty vnlue zr n tetrnteqrzy elexeyt us qeiyeq usuys tte qeysuty zr tte vzxelttnt pzytnuys uts peytrzuq. As rzr MCM nyq EFFD usuysWEPLsimplified, evey ttzust tte qzse qus-truoutuzy us slusttly oeter uy EFFD oztt xettzqs qexzystrnte suxulnr zutpzxes. Hzwever, wepny plenrly stnte ttnt pzrreptuys tteWEPL uy tte EFFD xettzq suves n suxulnr qzse tzxzseyeutypzxpnreq tz tte vzxel-onseq qzse xnp. he snxe zoservntuzy npplues tz tte CDcHs zr tte nppu-xulnteq qzsexnps stzwy uy isure . . (o). Evey ttzust tte qufereype oetweey tteWEPLsimplified

nyq tteWEPLst us less ttny mm, ut tns ny uxpnpt zy tte qzse qustruoutuzy. Furtterxzre, xzre pre-puse uyrzrxntuzy resnrquys tte qzse nre ullustrnteq uy tnole . . . Stntustupnl uyrzrxntuzy uypluquysDmean,Dmin,Dmax nre pnlpulnteq rzr nll tte useq qeysuty xnppuys nlszruttxs. he xzre uyterestuysvnlue pzrrespzyqs tz tte tzxzseyeuty uyqex HI wtupt us qeiyeq ns tte rntuz oetweey tte xuyu-xuxqzse uy % zr tte tnrset vzluxe (u.e.;D ) nyq ttexuyuxuxqzse uy % zr tte tnrset vzluxe(u.e.; D ). he vnlue zr HI uyprense ns tte qzse qustruoutuzy oepzxes less tzxzseyezus nyq nyzptuxnl vnlue us nyHI vnlue zr . Furtterxzre, we pzyirxwutt tte zotnuyeqHI vnlues ttnt EFFDwuttWEPLst tns efeptuvely eytnypeq tte tzxzseyeuty zr tte qzse uy tte tnrset wutt n relntuve er-rzr zr HI less ttny % uy oztt tte stntup nyq tte qerzrxntuzy uyqupeq suxulntuzys. Besuqes, ttesnxxn uyqex qustruoutuzys nre preseyteq uy isure . . . As rzr tte snxxn evnluntuzy, xzre ttny% zr tte vnlues uy tte snxxn xnp zr tte tuxzr nre lzwer ttny uy step nyq tte eyture suxu-

lntuzy, wtupt uyqupntes ny expelleyt nsreexeyt oetweey tte qzse qustruoutuzys zr tte EFFD wuttdEPL pzrrepteq nyq tte vzxel rerereype. Aqqutuzynlly, we zotnuyeq tte snxxn vnlues zr % nyq% rzr CM nyq MCM, respeptuvely.As n pzyplusuzy, evey ttzust oztt EFFD xettzqs suve very relunole results, EFFD wutt pzr-

repteq dEPL uyterut tte snxe lnws zr tte suxpluieq zye wtupt us relnteq tz tte qepusuzy zr pnl-pulntuys n pztereyt qzse uysuqe n tnrset wtule usyzruys tte rest zr tte tussues. hus result us stntequy isure . . , wtere we pny see tte qufereype oetweey EFFD-onseq xettzqs nyq tte rerereypeuy tte resuzys zutsuqe tte tnrset vzluxe. he vnlues pnlpulnteq zy tte eyture sezxetry stzws ttntuy stntup we tnve n snxxn uyqex vnlues zr . %, . %, . % nyq . % rzr CM, MCM,EFFD wutt WEPLsimplified nyq EFFD wutt WEPLst, respeptuvely. Iy tte eyq, ut us up tz tte user tzptzzse oetweey n tnrset-zrueyteq xettzq ttnt zyly exptnsuzes n spepuip resuzy zr tte sezxetrynyq usyzre tte qzse uy tte rexnuyuys tussues. nyq n xettzq luwe MCM wtere n susyuipnyt yuxoer


zr tte tetrnteqrzy us requureq tz przqupe n szzq qzse nppurnpy. It wull oe uyterestuys tz stuqy uy tteruture n xultu-tnrset versuzy zr tte EFFD xettzq wtere tte tternpust pny qeiye severnl vzluxeszr uyterest nyq usyzre tte rest zr tte sezxetry.

0

20

40

60

80

100

120

10 10,5 11 11,5 12 12,5 13 13,5 14 14,5 15

Rel

ati

ve v

olu

me

[%]

Dose [Gy]

Reference CM MCM EFFD_WEPL_Sipmlified EFFD_WEPL_Corrected

0

20

40

60

80

100

120

30 35 40 45 50 55 60

Rel

ati

ve v

olu

me

[%]

Dose [Gy]

Reference CM MCM EFFD_WEPL_Simplified EFFD_WEPL_Corrected

(a) (b)

Figure 4.7.2: Cumulative dose volume histogram of the (a) static simulation and the (b)dose accumulation of motion simulation.

Deysuty xnppuys nlszruttx Dmean hGy] Dmin hGy] Dmax hGy] HI Relntuveerrzr zrHIh%]

Stntup

czxel-onseq DMLP . . . . -CM . . . . .MCM . . . . .EFFDldEPLsimplified . . . . .EFFDldEPLcorrected . . . . .

D

czxel-onseq DMLP . . . . -CM . . . . .MCM . . . . .EFFDldEPLsimplified . . . . .EFFDldEPLcorrected . . , . .

Table 4.7.2: Evaluation of dose distribution in the tumor volume of the 4D multi-layer phan-tom for step and D simulation. Dmin, Dmax, Dmean are respectively the minimum, the maxi-mum and the mean dose deposited. HI is the Homogeneity index.


EF

FD

WE

PL

stE

FF

D W

EP

Lsi

mp

lifi

edM

CM

CM

0 21 1 20.5

Figure 4.7.3: Axial slices of the 4DMLP gamma index evaluation of CM, MCM, EFFD withWEPLsimplified and EFFD with WEPLst of deformation-induced simulations. A Dose to agree-ment (DTA) of 3 mm and dose tolerance of % were used in all the simulations. The regionsthat fail the criteria in the gamma evaluation are clearly identiied where gamma index is > 1.Right: a second scale was used to isolate the regions that fail the test.


. C

Iy ttus ptnpter, ny uxprzvexeyt versuzy zr tte efeptuve qeysuty xnppuys nlszruttx tns oeey pre-seyteq. de tnve useq n stzuptuzxetrup xettzq tz estuxnte tte wnter equuvnleyt pntt leystt zr ttetussues oerzre tte tuxzr tz pnlpulnte tte stzppuys pzwer zr tte xnterunls zr tte tetrnteqrnl ele-xeyts. Aterwnrq, we tnve pzyverteq tte stzppuys pzwer zytz qeysuty vnlues tz oe useq uy tteMzyteCnrlz suxulntuzy. Oype tte qeysutyxnp us uxprzveq, we tnve exulnteq n qerzrxntuzy ielqoy npplyuys n spnluys rnptzr zy tte Multu-lnyer ptnytzx. he qeysuty vnlues were upqnteq oy re-speptuys tte pruypuple zrxnss pzyservntuzy nyq ttnt oy respeptuys ttexnss zr tte elexeyts uy enptsuxulntuzy step. Fuynl results tnve stzwy ttnt zur xzqel suves oeter qzse tzxzseyeuty uy tte tu-xzr vzluxe ttny tteCMnyqMCMxettzqs. Mzrezver, tte pzrreptuzy zr ttedEPL rzrxuln tnsuxprzveq qzse tzxzseyeuty uy tte tuxzr vzluxe pzxpnreq tz tte suxpluieq versuzy qeiyeq uytte prevuzus ptnpter. Hzwever, uy tte ztter tussues zutsuqe tte tuxzr, tte qzse nppurnpy us yzt yep-essnruly respepteq suype oy esseype zur xettzq us zptuxuzeq tz tte tnrset vzluxe zyly nyq usyzrette qeysuty vnlues zr tte elexeyts uy tte ttnt uytersept tte oenx. It wull oe uyterestuys tz exteyqzur nlszruttx tz uypzrpzrnte xzre ttny zye tnrset vzluxe tz zptuxuze tte qeysuty vnlues uy ttequreptuzy zr tte oenx.

Success is not inal, failure is not fatal7 it is the courage to con-tinue that counts.

duystzy Cturptull

5General conclusions and future perpetives

Oypzlzsupnl tnqrzy tternpy represeyts n przxusuys nlteryntuve szlutuzy tz tte uouquutzus pzyvey-tuzynlxesnvzlt ptztzy tternpy espepunlly rzr tte trentxeyt zr qeep-senteq lzpnl tuxzrs. hus tept-yuque uses tenvy-ptnrseq pnrtuples luwe prztzys zr pnrozy uzys tz urrnqunte tte tuxzrs. he pnrtup-ulnruty zr tenvy-uzy pnrtuples resuqes uy tte rnpt ttnt ttey qepzsut xzst zr tteur eyersy nt tte eyq zrtteur rnyse prentuys tte Brnss Penw. hus przperty sunrnytees prepuse urrnquntuzy zr tte tnrset vzl-uxe nyq less qnxnse tz tenltty tussues. Nevertteless, suype tte rnyse zr pnrtuples qepeyqs zy tteqeysuty nyq tte ptexupnl pzxpzsutuzy zr tte trnverseq tussues, tnqrzy tternpy us very seysutuve tztussues qerzrxntuzy nyq pntueyt set-up quruys tte trentxeyt. hererzre, respurntzry-uyqupeq zrsnyxztuzy pzystututes zye zr tte xzst quipult ptnlleyses zr trentxeyt plnyyuys uy Hnqrzytternpy,nyq ut represeyts n susyuipnyt szurpe zr uypertnuyty uy tte estuxntuzy zr qzse. he pzxplexuty zrzrsny qerzrxntuzys quruys respurntuzy nyq tte yzy-reprzqupuouluty zr tte orenttuys pnterys xnweexustuys uxnse-onseq qzse pnlpulntuzy teptyuques przye tz errzrs. hus rnpt pusteq xnyy resenrptlnos tz qevelzp ouzxeptnyupnlxzqels ttnt use iyute elexeyt nynlysus. Axzys ttese lnos, zur tenx,

Geyernl pzyplusuzys nyq ruture perspeptuves

uy pzllnozrntuzy wutt IPNL lnozrntzry, tnve qevelzpeq uy tte pnst qepnqe, n ouzxeptnyupnl xzqelzr tte respurntzry systex ttnt us qruvey oy oztt ruopnse xztuzy nyq qunptrnsx pzytrnptuzy. heiynl sznl us tz xzyutzr tte xzqel usuys exterynl surrzsntes.

Hzwever, evey ttzust ouzxeptnyupnl xzqels pny nppurntely estuxnte tte qerzrxntuzys nyq ttepzsutuzy zr zrsnys quruys tte trentxeyt, ttey qzy’t suve n prepuse xnterunl qeysuty uyrzrxntuzy zrtte suxulnteq tussues. az nqqress ttus przolex, zur tenxtns qeiyeq n pzytuyuzus represeytntuzyzrptysupnl qunytutues onseqzy ttepruypuplezrxnss pzyservntuzy. husqespruptuzy nuxsnt pnlpulntuysptysupnl qunytutues luwexnss qeysuty zr tte qzse zy tte vertupes zr tte tetrnteqrnlxest oy eysuruystte pzytuyuuty zr tte represeyteq ruyptuzy. Furtterxzre, tte vnluezr tte ruyptuzy pnyoe pnlpulnteqnt nyy pzuyt uysuqe tte tetrnteqrnl xest usuys n luyenr uyterpzlntuzy zr tte vnlues uy tte vertupes.Nevertteless, qespute tte eypzurnsuys zutpzxe zr ttus xettzq, tte tetrnteqrnl xest us zyly useqtz nppuxulnte tte qzse, nyq tte user us pzystrnuyeq tz vzxeluze tte tetrnteqrnl xest tz suxulnte tteuyternptuzy oetweey tte pnrtuple oenx nyq tte xnter.

Iy ttus Pt.D. przvept, we tnve qexzystrnteq n D qzse pnlpulntuzy xettzq onseq zyly zy tetrn-teqrnl sezxetrues wtere enpt vzluxe us represeyteq wutt n tetrnteqrnl xest ttnt qerzrxs wutttuxe. Besuqes, tte uyterynl xzvexeyt us qespruoeq usuys vertex qusplnpexeyt, teype xnwuys tussuetrnpwuys uxpluput.

he irst pnrt zr zur przvept pzysusts zr qeiyuys n rzur-quxeysuzynl qzse pnlpulntuzy xettzqonseq zy tte pruypuple zr xnss pzyservntuzy. he xnuy uqen us tz prente n tetrnteqrnl xzqel ttntqzesy’t zyly represeyts tte sezxetry zr tte uyterynl nyntzxy, out nlsz tte qeysuty nyq tte ptexu-pnl pzxpzsutuzy zr tte tussues zr zrsnys. Aqqutuzynlly, tte xzqel tns tz eysure tte qerzrxntuzy zrtte nyntzxywtule pzyservuys ttexnss zr nll uyquvuqunl tetrnteqrzy. az prente tte tuxe-qepeyqeytqeysuty xnp requureq rzr tte Mzyte Cnrlz qzse pnlpulntuzys, we tnve przpzseq n xettzq rzr qey-suty xnppuys ttnt xnp qeysuty uyrzrxntuzy ussueq rrzx Ca-uxnses tz tte tetrnteqrnl xest. Fur-tterxzre, we nppuxulnte tte qzse zvertuxe zy tte tetrnteqrzys tz prente n qzse xnp. As n result,we tnve qexzystrnteq ttnt tetrnteqrnl sezxetrues suve nppurnte qzse qustruoutuzys ns vzxel-onseqsezxetrues uy tte pnse zr ttzrnx-sezxetrues. Mzrezver, tz pzyirx zur iyquyss, we tnve stuq-ueq tte uxpnpt zr tte reszlutuzy zr tte tetrnteqrnl xest zy tte nppurnpy zr qzse qustruoutuzy. depzypluqeq ttnt n iye tetrnteqrnl xest tns tz oe useq tz zotnuy nppeptnole zutpzxes. Suype ttenppurnpy zr tte qeysuty xnppuys nlszruttx qepeyqs zy tte yuxoer zr tetrnteqrzys, pnrtupulnrlyuy tustly teterzseyezus struptures luwe tte luys. Ayztter przolex nruses wutt tte relntuvely ex-

Geyernl pzyplusuzys nyq ruture perspeptuves

pessuve yuxoer zr tetrnteqrzys, wtupt us tte pzysuqernole tuxe requureq tz lznq tte sezxetry nyqtz suxulnte tte pnssnse zr pnrtuples uy tte Mzyte Cnrlz suxulntuzy. az nqqress ttus przolex, wetnve qeiyeq nyztter qzse pnlpulntuzy xettzq ttnt sunrnytees ny nppurnte qzse qustruoutuzy wtulerequpuys tte yuxoer zr elexeyts uy tte xest. he xnuy uqen us tz xuyuxuze tte errzr zr tte wn-ter equuvnleyt pntt leystt uy tte qureptuzy zr tte pnrtuple oenx uystenq zr respeptuys tte pruypuplezr xnss pzyservntuzy. de irst qexzystrnteq ttnt respeptuys tte xnss uy tte tzle sezxetry wteypnlpulntuys tte qeysuty vnlues us yzt eyzust tz sunrnytee n pztereyt qzse uy pznrse xestes. hey,we stzweq ttnt ur tte wnter equuvnleyt pntt leystt zr tte tussues uy tte trnveptzry zr tte oenx usrespepteq, we pny use rewer tetrnteqrzys tz represeyt tte pntueyt. As rzrMzyte Cnrlz suxulntuzys,we seyernte ny efeptuve qeysuty xnp ttnt us qepeyqeyt zy tte qureptuzy zr tte oenx. hus xnp tnsrequpeq tte errzr zr tte wnter equuvnleyt pntt leystt oy mm uy tte pnse zr n pntueyt sezxetry.Aqqutuzynlly, ut lenqs tz n pnlpulntuzy tuxe stzrter ttny tte qunrter tuxe yeeqeq rzr ztter qeysutyxnppuys nlszruttxs rzr tte snxe qzse nppurnpy.

Iy tte lnst pnrt zr zur wzrw, we exteyq tte efeptuve qeysuty nlszruttx tz uypzrpzrnte tte qerzr-xntuzy zr zrsnys. Mzrezver, we use nxzre prepuse qespruptuzy zr tte wnter equuvnleyt pntt leysttttnt tnwes uytz nppzuyt tte stzppuys pzwer zr qufereytxnterunls. de pzypluqe ttnt oy npplyuys zuroenx-qepeyqnyt nlszruttx zy n tetrnteqrnl xest wutt rewer elexeyts, we seyernte n relunole qzsequstruoutuzy uysuqe tte tnrset vzluxe uy oztt stntup nyq xztuzy-pzxpeysnteq suxulntuzys. Hzw-ever, ttus tnrset-zrueyteq npprznpt eysures qzse nppurnpy zyly uy tte tnrset vzluxe nyq yesleptstte ztter zrsnys. It wull oe uyterestuys tz qevelzp n versuzy zr ttus nlszruttx ttnt tnwes uytz nppzuytxultuple resuzys zr uyterest uy tte ruture. de pzulq nlsz stuqy tte uxpnpt zr usuys pnrozy uzy zrztter tnqrzys zy tte zutpzxes zr zur tetrnteqrnl-onseq qzse pnlpulntuzy.

ALinear interpolation

A. L y

P Iy n y-quxeysuzynl spnpe, let ttere oe p+ pzuyts zr ℜn : A ,A , ...Ap .he pzyvex tull C(A) zr ttese pzuyts us suvey oy:

C(A) = M =

p∑

i=

λi · Ai|λi ≥ , (i = , , ...p),p

∑

i=

λi = (A. )

Iy ztter wzrqs, λi nre tte onrypeytrup pzzrquyntes zr tte pzuytMwutt respept tz tte pzlyteqrn qe-iyeq oy tte pzuyts Ai.

Fzr every vertex Ai zr tte pzlyteqrn we nsszpunte n pzsutuve qeysuty vnlue ρi = ρ(Ai). hey, rzr

Luyenr uyterpzlntuzy

every pzuytM ∈ C(A)we qeiye tte puyptunl qeysuty ρ : C(A) → ℜ :

M =

p∑

i=

λi · Ai −→ ρ(M) =

p∑

i=

λi · ρi (A. )

M Iytesrntuys qeysuty (rnquznptuvuty qeysuty) zver n qzxnuy equnls tz iyquystte xnss (yuxoer zr qusuytesrntuzys) zr ttnt qzxnuy. Fzr exnxple, uy n -quxeysuzynl spnpe tteqzxnuy zr uytesrntuzy us n luye sesxeyt (luyenr xnss).

Figure A.1.1: 1-D case : line segment AB

Let M ∈ [AB] (Fusure ). he pzsutuzy zr M pny oe wrutey usuys onrypeytrup pzzrquyntes ns rzl-lzws:

M = M(t) = t · B+ ( − t) · A (A. )

Appzrquys tz ( ) we zotnuy :

ρ(M) = t · ρB + ( − t) · ρA (A. )

Let J = [∂M∂t ];Δ = |det(J)| = |AB|. Ir we uytesrnte zy [AB]we set:

MassAB =∫

M∈[AB]ρ(M) · dM =

∫

Δ · ρ(M(t)) · dt = AB ·

∫

ρ(M(t)) · dt.

Fuynlly we zotnuy:

MassAB = |AB| ·ρA + ρB (A. )

Next, we’re szuys tz nynlyse tte pnse zr n -D spnpe (Fusure ). Iy ttus pnse, ns quspusseq oerzre,tte onsup stnpe us tte tetrnteqrzy nyq we’re szuys tz pnlpulnte uts xnss oy uytesrntuys tte puyptunlqeysuty qeiyeq uy tte prevuzus septuzy.


Figure A.1.2: 3-D case : Volume of the tetrhedron ABCD= |det(AB, AC, AD)|

M nyq d(M) pny oe expresseq nppzrquys tz equntuzy ( ) (p = ), out rzr uytesrntuys purpzses,we’re szuys tz express ttex nppzrquys tz equntuzy ( ). Sz,

M ∈ [AM ] ⇒ M = θ ·M + ( − θ) · A, ≤ θ ≤ .

M ∈ [KC] ⇒ M = t · C+ ( − t) · K, ≤ t ≤ .

K ∈ [BD] ⇒ K = τ · D+ ( − τ) · B, ≤ τ ≤ .

Ayq, nppzrquys tz ( ), we pny express tte puyptunl qeysuty zr M ns rzllzws:

ρ(M) = ( − θ) · ρA + θ( − τ)( − t) · ρB + θ · t · ρC + θ · τ( − τ) · ρD (A. )


Let J =[

∂M∂τ

∂M∂t

∂M∂t

]

= V ·Θwutt V =[

BD BC BA]

nyq det(V) = · Vol(ABCD).

Θ =

θ( − t) −τθ τ(a− τ)θ t

.

wutt det(Θ) = θ ( − θ). de put Δ = det(JtJ) = [det(V) · det(Θ)] . Iytesrntuys tte qeysutyuysuqe tte tetrnteqrzy (ABCD) suves:

MassABCD =

∫

M∈(ABCD)ρ(M) · dM =

∫ ∫ ∫

Δ · ρ(M(τ, θ, t)) · dθ · dt · dτ.

MassABCD = |det(V)|·∫ ∫ ∫

θ ( −θ)·ρ(M(τ, θ, t))·dθ·dt·dτ = |det(V)|·ρA + ρB + ρC + ρD .

Fuynlly we zotnuy:

MassABCD = Vol(ABCD) ·ρA + ρB + ρC + ρD (A. )

A. L I (LR)

he uyteysuty vnrues luyenrly oetweey sruq pzsutuzys. byluwe yenrest yeustozr uyterpzlntuzy, tteuyterpzlnteq uyteysuty us spntunlly pzytuyuzus. Hzwever, tte uyteysuty srnqueytwull oequspzytuyuzusnt sruq pzsutuzys.

A. R N N I (NN)

Iy tte yenrest yeustozr uyterpzlntuzy, tte vnlue zr tte yenrest vzxel us tnwey. hnt us, ut nssuxesttnt tte uxnse uyteysuty us puepewuse pzystnyt wutt vuxps xuq-wny oetweey sruq pzsutuzys. husuyterpzlntuzy sptexe us ptenp ns ut qzes yzt requure nyy lzntuys pzuyt pzxputntuzys.

Liting of igures

. . Deptt qzse qustruoutuzy rzr ptztzys nyqxzyzeyersetup Brnss purves rzr pnrozyuzys nyq prztzys h ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . arentxeytplnyszr eszptnsenl pnyperusuysprztzy tternpy(let) nyqe-rnys/IMRa(rustt). Prztzy tternpy luxuts tte rnquntuzy qeluvereq zutsuqe tte tnrset nyq re-qupes tte luwelutzzq zr luys pzxplupntuzys. he pzyrzrxuty tz tte tnrset vzluxeus prepuse uy tte twz pnses. Hzwever, tte qzse zr surrzuyquys zrsnys usxuyuxuzeqwtey usuys prztzys. Szurpe Sentle pnyper pnre nllunype, Prztzy tternpy peyter

nyq h ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Iy n typupnl trentxeyt plny rzr prztzy tternpy, tte sprenq zut Brnss penw (SOBP,

qnsteq olue luye) us tte tternpeutup rnquntuzy qustruoutuzy. he SOBP us tte suxzr severnl uyquvuqunl Brnss penws (ttuy olue luyes) nt stnssereq qeptts. he qeptt-qzse plzt zr ny e-rny oenx (req luye) us przvuqeq rzr pzxpnruszy. he puyw nrenrepreseyts nqqutuzynl qzseszre-rny rnquztternpy—wtupt pnyqnxnseyzrxnl tus-sues nyq pnuse sepzyqnry pnypers, espepunlly zr tte swuy . . . . . . . . . . . . . .

. . Sptexntup zr n rully n pnssuve oenx stnpuys systex uy Hnqrzytternpy. Fusure re-qrnwy rrzx h ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Sptexntup zr tte GSI rnster-spny systex. Fusure reqrnwy rrzx h ]. . . . . . . .

. . Fluptuntuzys uy respurntzryxztuzy pnterys resustereq nt plzse tuxe uytervnls. hepurves represeyt ttepntueyt surrnpequsplnpexeyt uy tte superuzr-uyreruzr, nyteruzr-pzsteruzr nyq xequnl-lnternl quxeysuzys. Iy (n) tte pnterys nre nlxzst uqeytupnlout uy (o) ttere nre yz respurntzry qustuysuustnole pnterys h ]. . . . . . . . . .

. . Efept zr tnrsetxztuzy zy n purpulnr ptnytzx. he tuxzr us peytereq nyq ut us rep-reseyteq oy tte srny purple. he ttuy olnpw purple uyqupntes n -xx D sezxetrupexpnysuzy zr tte tuxzr uytz tte plnyyuys tnrset vzluxe (n) he qnsteq luyes nreuszqzse luyes ( %, %, %, %, % zr ttexnxuxuxqzse) rzr n suysle pns-suve prztzy oenx. (o) Iszqzse luyes rzr n xx qusplnpexeyt zr tte tuxzr wuttrespept tz tte trentxeyt oenx. h ]. . . . . . . . . . . . . . . . . . . . . . . . .

LISaING OF FIGbRES

. . Ixpnpt zr qeysuty zy tte oenx rnyse. Deptt-qzse prziles zr n ptztzy oenx nyqn prztzy SOBP, qnsteq luyes represeyt tte przile seyernteq nter nqquys n tusterqeysuty xnterunl (srny reptnysle rrzx tz px) uy tte trnveptzry zr tte useqoenxs. h ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Rnyse luptuntuzys expresseq uy wnter equuvnleyt pntt leystt zverlnppeq zy tteD-Ca uxnses. Ixnse rrzx h ]. . . . . . . . . . . . . . . . . . . . . . . . . .

. . Grnptupnl represeytntuzy zr tte vzluxes zr uyterest, ns qeiyeq uy ICRb RepzrtsNz. nyq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . he result zr n qerzrxnole uxnse resustrntuzy nlszruttx npplueq zy n ptnytzx.he qerzrxeq sruq represeyts tte trnysrzrxeq puxels zr tte szurpe uxnse usuystte qerzrxeq veptzr ielq (DcF). . . . . . . . . . . . . . . . . . . . . . . . . .

. . (n) Buzxeptnyupnl xzqel zr tte ttzrnx uypluquys luyss, ruo pnse, nyq qunptrnsx.(o) Ptysuzlzsupnl pzxplunype pnlpulnteq rzr luys rzr ive pntueyts. Ixnse tnweyrrzx h ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Nztntuzy useq rzr tetrnteqrzys. Tj nyq Tj+ nre twz nqvnpeyt tetrnteqrzys. . . . .

. . aetrnteqrnl xest qerzrxntuzy rrzx tte tuxe step t tz t+ . Derzrxuys tte tetrn-teqrzy Tj usuys vertex qusplnpexeyt. . . . . . . . . . . . . . . . . . . . . . . . .

. . Flzwptnrt zr tetrnteqrnlxzqel pzystruptuzy: rrzx D-Ca tz tetrnteqrnl qeysutyxnp. Enpt zrsny us qespruoeq oy n tetrnteqrnlxest zotnuyeq rrzx DCa uxnses.he Ca uxnses nre irst pzyverteq uytz vzxeluzeq xnss qeysuty xnps wtupt nrettey xnppeq tz tte tetrnteqrnl elexeyts zr tte xestes. . . . . . . . . . . . . . .

. . Deysuty xnppuys xettzqs, f us n luyenr uyterpzlntuzy ruyptuzy. Let: CzytuyuzusMnss Czyservntuzy Mnppuys (CMCM). Ceyter: Mnss Czyservntuzy Mnppuys(MCM). Rustt: Ceytrzuq Mnppuys(CM). . . . . . . . . . . . . . . . . . . . . .

. . aetrnteqrnl qeysuty xnp seyerntuzy.he xnss zr n tetrnteqrnl elexeyt equnls ttesux zr tte xnsses zr vzluxes zr uyterseptuzy oetweey tte tetrnteqrzy nyq ttesruq zr vzxels. Exnxple zr npplyuys equntuzy ( . ) uy n D spnpe: m(Tk) =m(Ik)+m(Ik)+m(Ik)+m(Ik)+m(Ik)+m(Ik)+m(Ik) rzr tteMCMxettzq.Iy tte pnse zr tte Ceytrzuq Mettzq: m(Tk) = m(V ). . . . . . . . . . . . . . .

. . Derzrxnole tetrnteqrzy uxplexeytntuzy usuys Genyt . . . . . . . . . . . . . .

. . Benx stnpuyswutt rnyse pzxpeysntzr nyqxultulenr pzlluxntzr. Iy ttus pnrtupulnrexnxple we tnve RC lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Rnyse pzxpeysntzr nyq S.O.B.P pzystruptzr . . . . . . . . . . . . . . . . . . . .

. . Sptere suxulntuzy setup uyGenyt . qusw szurpes pnrtuples nre plnpeq uysuqe ttewnter eyvelzp tz seyernte tte SOBP. A pyluyqrup pzlluxntzr us useq tz nqnpt ttepnssuve oenx tz tte uyyer pzypeytrup sptere (tuxzr). . . . . . . . . . . . . . . .

LISaING OF FIGbRES

. . Eyersyprzile zr stntup suxulntuzy rzr tte ttree sezxetrup pzyisurntuzys : Genytpruxutuves, tetrnteqrnl xest nyq vzxeluzeq xnp. . . . . . . . . . . . . . . . . . .

. . Flzw ptnrt zr D qzse nppuxulntuzy uy respurntzry xztuzy-uyqupeq suxulntuzy. .

. . Cuxulntuve qzse vzluxe tustzsrnx (CDcH) qepzsuteq uy tte tuxzr. Let : yzxztuzy, rustt : wutt xztuzy. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Dzse qustruoutuzys zrMzyte Cnrlz suxulntuzys. azp: tetrnteqrnl xzqel. xuqqle:snsutnl vuew zr vzxel sruq. Bztzx: DDzse qustruoutuzy ( let: nppuxulnteq qzseuy tetrnteqrnl xest. rustt : nppuxulnteq qzse uy Ca qntn rzr nll ptnses. . . . . . .

. . Flzwptnrt zr tetrnteqrnl xzqel pzystruptuzy: rrzx D-Ca tz xultureszlutuzytetrnteqrnl xzqel. he Ca uxnses nre irst pzyverteq uytz vzxeluzeq xnss qey-suty xnps wtupt nre ttey xnppeq tz tte tetrnteqrnl elexeyts zr tte xestes. . . .

. . aps pzyisurntuzy rzr luys. awz suysle ileq pnssuve oenxs were useq uy zppzsutequreptuzys, enpt wutt uts zwy rnyse pzxpeysntzr nyq pzlluxntzr. . . . . . . . . .

. . A ttzrnx sezxetry represeytuys n pntueyt pnse useq rzr ttexultureszlutuzy stuqy.he prztzy oenx us qurepteq tzwnrq tte tuxzr nyq tte qzse us snveq uy tte tetrn-teqrnl elexeyts zr tte xest. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Gnxxn pnss rnte vs tte vzxel-tz-tetrnteqrzy rntuz usuys n qtn equnl tz mm nyq nqzse tzlernype zr %. Benx nyq Benx nre tte prztzy oenxs useq uy tte luyspnse ullustrnteq uy isure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Gnxxn pnss rnte vs tte vzxel-tz-tetrnteqrzy rntuz rzr tte pntueyt pnse suxulntuzy.Fzr tte snxxn evnluntuzy n qtn equnl tz mm nyq n qzse tzlernype zr % tnveoeey useq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Relntuzy oetweey pzxputntuzy tuxe nyq tte reszlutuzy zr tte tetrnteqrnl xest.(n)Nuxoer zr tetrnteqrzys versus pnlpulntuzy tuxe. (o) czxel-tz-tetrnteqrzy rn-tuz versus pnlpulntuzy tuxe. Lznquys tuxe us tte tuxe yeeqeq tz lznq nll tte tetrn-teqrnl elexeyts tz tte systex. Suxulntuzy tuxe us tte tuxe tnwey oy tte MzyteCnrlz pzqe tz pnlpulnte tte uyternptuzys oetweey tte pnrtuple oenx nyq tte xnt-ter. he tztnl tuxe us tte sux zr lznquys tuxe nyq suxulntuzy tuxe. hese vnlueswere pnlpulnteq usuys prztzy pnrtuples ireq nt tte xultureszlutuzy luys sezx-etry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Hb-tz-relntuve stzppuys pzwer rzr qufereyt tuxny tussues. Ixnse rrzx h ]. . .

. . hesezxetrupnl pzyisurntuzyzr tte Dxultu lnyer ptnytzx( DMLP)wutt rzurlnyers zr xnter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (n) Deptt qzse purve uypluquys tte Brnss penw nyq prztzy rnyse (Rw)uy n wnterptnytzx. (o) tte pnrt zr tte MLP oerzre tte tuxzr vzluxe us nqqeq uy rrzyt zrtte wnter ptnytzx.Rshifted

w us przqupeq nyq tte WET us qeiyeq ns tte qufereypeoetweey tte rnyses Rw − Rshifted

w . . . . . . . . . . . . . . . . . . . . . . . . . . .

LISaING OF FIGbRES

. . he yzrxnluzeq qzse-versus-qeptt prziles rzr tte suxulnteq luquuqe wnter ptny-tzx nyq tte uytesrntuzy zr tte xnterunls zr tte MLP ttnt nre oerzre tte tuxzrvzluxe. hese purves were przqupeq usuys n Mec oenx tz pnlpulnte ttedEazr tte Mnterunls zr MLP qespruoeq uy isure . . . he dEa wull serve ns n rerer-eype nyq ut us equnl tz . xx. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Mnss qeysuty vnlues vs relntuve stzppuys pzwer. . . . . . . . . . . . . . . . . . .

. . Genyt suxulntuzy setup zr tte DMLP.he snxe pzyisurntuzy us useq rzr everystep zr spnluys. A rupple ilterwns useq tz lntey tte Sprenq-OutBrnssPenw (SOBP).

. . Cuxulntuve qzse vzluxe tustzsrnx zr tte (n) stntup suxulntuzy nyq tte (o) qzsenppuxulntuzy zr xztuzy suxulntuzy. . . . . . . . . . . . . . . . . . . . . . . . .

. . Axunl slupes zr tte DMLP snxxn uyqex evnluntuzy zr CM, MCM, EFFD wuttWEPLsimplified nyq EFFD wutt WEPLst zr qerzrxntuzy-uyqupeq suxulntuzys. ADzse tz nsreexeyt (DaA) zr xx nyq qzse tzlernype zr %were useq uy nll ttesuxulntuzys. he resuzys ttnt rnul tte pruterun uy tte snxxn evnluntuzy nre plenrlyuqeytuieq wtere snxxn uyqex us > . Rustt: n sepzyq spnle wns useq tz uszlntette resuzys ttnt rnul tte test. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A. . -D pnse : luye sesxeyt AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A. . -D pnse : czluxe zr tte tetrteqrzy ABCD= |det(AB, AC, AD)| . . . . . . . .

Lit of Tables

. . Dzse relntuve errzr zr vzxels nyq tetrns pzxpnruys tz szlq stnyqnrq zr spteres sux-ulntuzy. DED: qzse relntuve errzr, LR : Luyenr ressnxpluys, NN: yenrest yustt-oertzzq. LR nyq NN nre twz qufereyt xettzqs useq rzr uyterpzlntuzy (see Ap-peyqux A. nyq A. ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Evnluntuzy zr qzse qustruoutuzy uy tuxzr vzluxe. Dmin, Dmax, Dmean nre respep-tuvely tte xuyuxux, tte xnxuxux nyq tte xeny qzse qepzsuteq. Hu (Hzxz-seyeuty uyqex). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Mnterunls, qeysuty pzysuqereq uy tte DMultu-lnyerptnytzx( DMLP).Dntn nqnpteqrrzx ICRb Repzrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Errzrs zr tte wepl pnlpulnteq oetweey tte rerereype dEPL(WEPLG ) nyq tteWEPLsimplified nyq tteWEPLst usuys ttree qufereyt eyersues. hese vnlues nre pnl-pulnteq zy tte lnyers oerzre tte tuxzr zr tte DMLP (isure . . ). heWEPLG

rerereype vnlue us tte qufereype oetweey tte dEa pnlpulnteq usuys zyly tte dn-ter ptnytzx nyq oy nqquys tte DMLP xnters ttnt nre oerzre tte tuxzr (seeisure . . ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . DMLP qerzrxntuzy pnrnxeters uypluquys puoe leystt, xnss qeysuty, Hb, Hbpnluornteq nyq tte relntuve stzppuys pzwer uy enpt suxulntuzy step. . . . . . . . .

. . Evnluntuzy zr qzse qustruoutuzy uy tte tuxzr vzluxe zr tte D xultu-lnyer ptny-tzx rzr step nyq D suxulntuzy. Dmin,Dmax,Dmean nre respeptuvely ttexuyuxux,tte xnxuxux nyq tte xeny qzse qepzsuteq. HI us tte Hzxzseyeuty uyqex. . . .

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