APPLICATION OF GENERALIZED PERTURBATION THEORY TO SENSITIVITY ANALYSIS IN BORON NEUTRON CAPTURE THERAPY

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1 211 Internatonal Nuclear Atlantc Conference - INAC 211 elo Horzonte,MG, razl, October 24-28, 211 AOCIAÇÃO RAILEIRA DE ENERGIA NUCLEAR - AEN IN: APPLICAION OF GENERALIZED PERURAION HEORY O ENIIVIY ANALYI IN ORON NEURON CAPURE HERAPY Vanessa. Garca 1, Fernando C. lva 2, Ademr X. lva 3, Gustavo. Alvarez 4 1 Programa de Pós-Graduação em Modelagem Computaconal em Cênca e ecnologa- MCC (EEIMVR/UFF-RJ Av. dos rabalhadores, 42, Vla anta Cecíla Volta Redonda, RJ vanessa.sgarca@gmal.com 2 Programa de Engenhara Nuclear (COPPE /UFRJ-RJ Av. Horáco de Macedo, 23, loco G, ala Ro de Janero, RJ fernando@con.ufrj.br 3 Programa de Engenhara Nuclear (COPPE /UFRJ-RJ Av. Horáco de Macedo, 23, loco G, ala Ro de Janero, RJ ademr@con.ufrj.br 4 Departamento de Cêncas Exatas (EEIMVR/ UFF-RJ Av. dos rabalhadores, 42, Vla anta Cecíla Volta Redonda, RJ bentez.gustavo@gmal.com ARAC oron neutron capture therapy NC s a bnary cancer treatment used n bran tumors. he tumor s loaded wth a boron compound and subsequently rradated by thermal neutrons. he therapy s based on the 1 (n, α 7 L nuclear reacton, whch emts two types of hgh-energy partcles, α partcle and the 7 L nucle. he total knetc energy released n ths nuclear reacton, when deposted n the tumor regon, destroys the cancer cells. nce the success of the NC s lnked to the dfferent selectvty between the tumor and healthy tssue, t s necessary to carry out a senstvty analyss to determnate the boron concentraton. Computatonal smulatons are very mportant n ths context because they help n the treatment plannng by calculatng the lowest effectve absorbed dose rate to reduce the damage to healthy tssue. he objectve of ths paper s to present a determnstc method based on generalzed perturbaton theory (GP to perform senstvty analyss wth respect to the 1 concentraton and to estmate the absorbed dose rate by patents undergong ths therapy. he advantage of the method s a sgnfcant reducton n computatonal tme requred to perform these calculatons. o smulate the neutron flux n all bran regons, the method reles on a two-dmensonal neutron transport equaton whose spatal, angular and energy varables are dscretzed by the damond dfference method, the dscrete ordnate method and multgroup formulaton, respectvely. he results obtaned through GP are consstent wth those obtaned usng other methods, demonstratng the effcacy of the proposed method.

2 1. INRODUCION oron neutron capture therapy (NC s a bnary cancer therapy that s used especally for treatment of bran tumors. he tumor tssue s loaded wth a 1 -enrched compound, PA-f (boronophenylalanne-fructose, and subsequently rradated by neutrons wth epthermal energes between 1 ev and 1 kev [1], whch penetrate the tssue and are thermalzed (.253 ev, producng lttle effect on healthy tssue [15]. he therapy s based on the 1 (n, α 7 L nuclear reacton, whch emts two types of partcles wth hgh energy, alpha partcles and the 7 L nucle. he total knetc energy (2.79 MeV released n ths nuclear reacton, when deposted n the tumor regon, destroys the cancer cells. o plan NC treatment, n addton to consderng the proper dose of the 1 (n, α 7 L nuclear reacton, t s necessary determnate the dose deposted by neutrons scattered n the healthy tssue surroundng the tumor. For ths reason, t s mportant to have dfferent selectvty between the tumor and healthy tssue. In ths context, senstvty analyss consderng dfferent boron concentraton values s essental to reduce the damage to healthy tssue [7]. he objectve of senstvty analyss s to evaluate the effect of varyng the parameters of a model or phenomenon on the fnal result of the smulaton. In NC, the computatonal smulaton s very mportant; because t helps plannng the treatment by analyzng the absorbed dose rate and ts possble effects on healthy tssues, so as not to expose patents to unnecessary doses. he objectve of ths paper s to present a determnstc method based on generalzed perturbaton theory (GP to perform senstvty analyss of the 1 concentraton. hs senstvty analyss s mportant to determne of the absorbed dose rate of patents undergong NC. For ths purpose, the method reles on the neutron transport equaton to smulate the neutron flux n all bran regons. he spatal, angular and energy varables are dscretzed by the fnte dfference method (damond dfference-dd, the dscrete ordnate method and multgroup formulaton, respectvely. wo-dmensonal Cartesan geometry s appled. he generalzed perturbaton theory method s advantages, compared to methods tradtonally used, s that the former calculates the optmzed 1 concentraton and consequently the absorbed dose rate wth a sgnfcant reducton n computatonal tme. he results obtaned wth GP are consstent wth those obtaned usng the tradtonal methods [13], demonstratng the effcacy of the proposed method. 2.1 Generalzed Perturbaton heory (GP 2. MEHODOLOGY he generalzed perturbaton theory (GP s a heurstc mathematcal method used for senstvty analyss of some physcal phenomena. he ntal applcatons of the theory were concentrated n the reactor physcs feld. he GP formally uses the concept of an mportance functon and of conservaton of partcles, n relaton to lnear or lnearzed felds. he mportance functon concept n GP [3,4] corresponds to the contrbuton of a gven partcle, nserted at a gven tme t and a gven pont r of the phase space, to the response functon. he mportance functon equaton s obtaned drectly, whereas that the contrbuton response of a partcle, ntroduced nto a system of the phase space at a gven ntal tme t, s conserved INAC 211, elo Horzonte, MG, razl.

3 untl a fnal tme t F (mportance conservaton prncple. In ths paper, we consder the statonary problem based on [3], so that the response functon, whch represents the neutron absorpton rate by the 1, s defned for two-dmensonal geometry as: = (x, ϕ(x, de dωˆ da (1 AR4π where A R s a determned regon of the bran, ϕ ( x, s the neutron angular flux at the pont (x,y n A R, whch s the soluton of the lnear transport equaton, defned as, Lϕ (r, E, Ωˆ = (r, E, ˆ ext Ω (2 where (r, E, ˆ ext Ω s the external source and L the lnear neutron transport operator, such that, L = Ωˆ ( (r, E( (r, E' E, Ωˆ ' Ωˆ ( de' dωˆ ' (3 t 4π he parameters t (r, E and (r, E' E, ˆ ' ˆ Ω Ω are the total macroscopc cross secton and the dfferental scatterng macroscopc cross secton of neutrons at pont (x,y wth energy E and travelng n drecton Ωˆ that wll be scattered n an energy nterval de ' about E ' at a sold angle dω ' about ˆΩ ' [11]. he source term (x,, n the regon A R, s represented by: a (x, y, E para (x, y A R (x, = (4 para (x, y AR he value a (x, y, E s the absorpton macroscopc cross secton of 1 at a pont (x,y of A R, such that, (x, y, E N σ (x, y, E. (5 a he values of N and σ a (x, y, E are, respectvely, the atomc number densty and the absorpton mcroscopc cross secton of 1 at pont (x,y of A R. If a neutron wth energy E, travelng n drecton Ωˆ, at tme t, s ntroduced nto a system n poston r, an ncrease n angular neutron flux results n an ncrease δ of the absorpton rate consdered. hs varaton can be produced drectly for ths neutron or, n multplcatve systems, by ts descendents (mportance conservaton prncple. herefore, the ncrease δ s defned as the neutron mportance. hs mportance can be denoted by ψ (r, E, Ωˆ, whch s a functon that depends on the space, angle and energy, and s a soluton of the equaton: a L ˆ ψ (r, E, Ω = (r, E, Ω ˆ, (6 INAC 211, elo Horzonte, MG, razl.

4 where L s the adjont neutron transport operator, defned by: L = Ωˆ ( (r, E( (r, E E', Ωˆ Ωˆ ' ( de' dωˆ '. (7 t 4π y consderng the lnear system and the mportance concept t s possble to obtan the relatonshp known as the source recprocty relatonshp [4]: V 4π (r, E, Ωˆ ϕ(r, E, Ωˆ de dωˆ dv = = V 4π ψ (r, E, Ωˆ (r, E, Ωˆ de dωˆ dv, (8 where (r, E, Ω ˆ = (r, E, Ωˆ (r, E, Ωˆ, (9 ext Wth (r, E, Ωˆ = 4π (r, E' E, Ωˆ ' Ωˆ ϕ(r, E, Ωˆ de' dωˆ '. (1 he man nterest when performng senstvty analyss of physcal phenomenon s to evaluate the varaton δ of due to perturbaton δ p of the parameter set p ( = 1,2,...,I that are relevant to the phenomenon. he varaton δ can be defned as: t F δ = { δp δpδp j δpδp jδpk K,} (11 p 2! p p 3! p p p t,j j,j,k j k he frst-order senstvty coeffcent consdered n ths paper can be defned by: t p F β dt. (12 p t Each reacton of the neutrons and boron deposts energy due to the α partcle and 7 L atoms. hus, the senstvty analyss of the neutron absorpton rate by 1 serves to support treatment plannng for patents undergong NC, because t provdes an estmate of the number of 1 (n,α 7 L reactons per unt of tme. hs nformaton s mportant to calculate the absorbed dose rate due to the neutrons, whch accordng to [12] s calculated by the equaton: D(x, y = c φ (x, y σ NA(x, y E, (13 g a,g where D(x,y s the absorbed dose rate at pont (x,y n the medum n Gy/h, φ (x, g y s the thermal scalar neutron flux at pont (x,y n n/cm 2.s, NA(x,y s the number of the specfed INAC 211, elo Horzonte, MG, razl.

5 nucle/gr at pont (x,y, σa, g s the absorpton mcroscopc cross secton for thermal neutrons n cm 2, E s the energy released n the reacton n MeV and g s the energy group, whch s consdered to be thermal. he constant c = 5.76 x 1-7 s the converson coeffcent, convertng MeV/gr nto Gy/h. 2.2 enstvty Analyss n NC usng GP he senstvty analyss concernng the neutron absorpton rate by 1 n the bran (healthy and tumor tssue s performed usng the generalzed perturbaton theory (GP, consderng the boron concentraton as the prncpal parameter [5]. he equaton that represents the varaton of, equaton (1, accordng to the boron concentraton, s calculated as follows: N = (x, ϕ(x, (x, N A 4 R π y, E, ˆ ϕ(x, Ω N y, E, Ωˆ dω ˆ de da (14 From equaton (2, the followng equaton s defned, hen, ϕ(x, L L N N ϕ(x, =. (15 L ϕ(x, L = ϕ(x, (x, N N (16 wth σ (E' E ϕ(x,y,e', Ωˆ 'dωˆ 'de' 4π (x,y,e, Ωˆ = σt (E ϕ(x, para(x,y A para(x,y A R R (17 ased on the source recprocty relatonshp of GP, ˆ ˆ ϕ(x, y, E, Ω (x, y, E, Ω dωˆ deda = N AR 4π AR 4π ψ (x, (x,y,e, Ωˆ dωˆ deda (18 where ψ (x, y, E., Ωˆ s the mportance functon defned n equaton (6. he frst-order senstvty coeffcent β, based on equaton (12, s calculated as follows: INAC 211, elo Horzonte, MG, razl.

6 β so that { (E (x, y, E, ˆ (x, y, E, ˆ (x, y, E, ˆ } d ˆ σ a ϕ Ω ψ Ω Ω Ω de da N A 4 π, (19 δn δ = β. (2 N nce the GP method uses the mportance functon concept and conservaton prncples and also the source recprocty relatonshp, to perform the senstvty analyss, equatons (2 and (6, whch defne the values ˆ ϕ ( x, y, E, Ω and ψ (x, y, E., Ωˆ, are solved only once. hese values are then used to determne the values of and β, consderng a reference boron concentraton value. For any other boron concentraton value, δ s always calculated by equaton (2, whch s very smple. Calculaton of δ by the drect method uses the followng equaton: δ = F, (21 where s the neutron absorpton rate for an ntal boron concentraton (reference value and F s the neutron absorpton rate for another boron concentraton value consdered. hus, to perform senstvty analyss by the drect method t s necessary solve equaton (2 every tme the boron concentraton value s changed, makng the computatonal smulaton very tme consumng. he use of GP to perform senstvty analyss n NC s very advantageous, because t reduces the computatonal tme to calculate the response varatons of nterest when the parameters are altered. 3. REUL In ths paper, the senstvty analyss calculatons are based on generalzed perturbaton theory and consder the boron concentraton as the parameter of nterest. he geometrc shape of the spatal dstrbuton of thermal neutron rradaton of bran tumors allows up to 6 cm wthout surgcal procedures [6,8]. For ths reason, here we smulate a case wth a tumor of dmenson (6x6 cm 2, consderng the bran as beng a square wth dmenson (18x18 cm 2, energes (1 ev to 1 kev; mesh (4x4. he damond dfference method s used to solve the equaton (2 whch consders the 4 dscrete ordnate formulaton, P 3 method (ansotropc scatterng and vacuum boundary condtons. For numercal soluton of the fxed source problem s used the source teraton method [11], wth 1-4 as convergence crteron. he boron concentraton reference values are 1 ppm n the healthy tssue and 3 ppm n the tumor. o perform the senstvty analyss, we consdered 4 ppm, 1 ppm and 125 ppm n the tumor. Fgure 1 shows the geometrcal confguraton. esdes the tumor (R5 regon, regons R2, R4, R6 and R8 are also consdered, whch are healthy tssue adjacent to the tumor. INAC 211, elo Horzonte, MG, razl.

7 y O U R C E R3 R6 R9 1cm O F N E U R O N R2 R1 umor R4 R8 R7 6cm 2cm 2cm 6cm 1cm x Fgure 1. Confguraton of bran consderng a tumor measurng (6x6 cm 2 he computatonal code used to solve the neutron transport equaton by the damond dfference (DD numercal method [11] was developed by [1]. he nuclear data of the elements of human tssue and tumor were obtaned from the ENDF/-VI.8 database [14,2]. he atomc denstes (1g/cm3 of these elements are shown n able 1 and able 2 [12]. able 1. Atomc densty of the human tssue elements Elements W% Atomc Densty x 1 24 H O C N able 2. Atomc densty of the boron 1 Concentraton Atomc Densty x ppm 3ppm 4ppm 1ppm 125ppm 5.57E E E E E-6 he neutron absorpton rate for the reference values (1 ppm-3 ppm, tssue and tumor were calculated and the results of ths smulaton are presented n able 3. INAC 211, elo Horzonte, MG, razl.

8 able 3. he and D values for the R 2, R 4, R 6, R 8 regons and tumor Regons Neutron Absorpton Rate Absorbed Dose Rate D (Gy/h umor 1.93E E-13 R E E-11 R E-4 2.9E-13 R E E-14 R E E-15 he results obtaned show that the value of the absorbed dose rate s hgher n the R2 regon (whch represents healthy tssue, 1 ppm 1 than n the tumor regon. he consequence of ths fact s more neutrons absorbed n healthy tssue, whch results n severe damage and destructon of healthy cells. enstvty analyss s performed to ncrease the selectvty (tumor/tssue. In the frst calculaton, we determned the senstvty coeffcent β, whch was After that, we calculated the varatons δ consderng the followng boron concentraton ncreases n the tumor regon: 3ppm-4ppm, 3ppm-1ppm and 3ppm-125ppm, as shown n able 4. able 4. Result of senstvty analyss n tumor regon obtaned by GP Varaton of 3ppm to: Varaton of δ Neutron Absorpton Rate 4 ppm 1.E-3 2.9E-3 1 ppm 4.5E-3 6.4E ppm 6.1E-3 8.E-3 he results presented n able 4 show that to acheve hgher tumor selectvty, the 1 concentraton must be greater than 125ppm n ths regon, because to any value less than ths the neutrons absorpton rate wll be hgher n healthy tssue. However, the selectvty wth dfference 1:12.5 (1ppm n tssue and 125ppm n tumor s not yet acheved wth the boron compounds used. However, some studes n humans, as shown by [9], use a combnaton of dfferent compound, such as borocaptate sodum (H and PA-fructose, to ncrease the 1 concentraton n the tumor. hs would be a vable soluton n cases lke the one presented. Another mportant consderaton s related to the ncdent neutron energy. In the smulatons performed by [13], for neutrons orgnatng from a source wth energes between (1eV and 3keV, the absorbed dose rate n the tumor regon, wth 4 cm n dameter (sphercal geometry, was hgh only n the frst two centmeters of the tumor. o verfy the behavor observed by [13], we ran a smulaton consderng the frst two centmeters of the tumor, the 1 concentratons n the tssue (1ppm and tumor (3ppm and energes between 1eV and 1keV. able 5. values for dfferent tumor dmensons umor Dmenson Neutron Absorpton Rate 2cm x 6cm 1.25E-2 6cm x 6cm 1.93E-3 INAC 211, elo Horzonte, MG, razl.

9 Accordng to the result, shows n able 5, smlar behavor occurs. herefore, n ths case, the energy range between 1 ev and 1 kev s not suffcent for successful therapy. As most neutrons are absorbed n the frst two centmeters of the tumor some cancer cells cannot be destroyed. he greater the dstance between the neutron source and tumor, the greater must be the ncdent neutron energy so that the neutron absorpton by 1 s suffcent to destroy all the cancer cells. In ther smulatons [13], concluded that wth energes between 1 kev and 1 kev, the boron dose s more evenly dstrbuted n the tumor, whch provdes greater selectvty between the tumor and surroundng tssue. mulatng ths energy range n the example proposed n ths paper, we obtaned the results presented n able 6. able 6. Neutron absorpton rate n the tumor and R 2 regons for 1 kev to 1 kev Regons Neutron Absorpton Rate Absorbed Dose Rate (Gy/h umor 1.89E E-9 R E E-11 Agan the behavor s the same for the neutron absorpton rate as n the smulatons of [13], whch show that the generalzed perturbaton theory method s effectve to perform senstvty analyss n NC. 4. CONCLUION nce the radaton feld n NC conssts separately of the dose of multple components (dfferent physcal propertes and bologcal effects, the value of the dose due to the radaton component vares not only wth the boron concentraton n the tumor regon, but also wth the selectvty between tumor and tssue, the energy spectrum of ncdent neutrons and the tumor locaton n of the regon of nterest. For ths reason, the senstvty analyss of these parameters s mportant to plan proper treatment. he results of the studes performed n the adjacent regon to the tumor are mportant to mnmze the effects on healthy tssue because they allow estmatng parameter values relevant to therapy to ncrease the dfferental of dose deposton between tumor and tssue. hese results also show the compatblty of the results obtaned by the method proposed n ths work wth other prevous studes employng other methods. ACKNOWLEDGMEN he authors wsh to thank the razlan research fundng agences CAPE, FAPERJ and CNPq for ther support to ths work. REFERENCE 1. AREU, M. M. P., olução Numérca do Problema de ransporte de Nêutrons Independente do empo na Formulação de Ordenadas Dscretas, Programa de Engenhara Nuclear, COPPE/UFRJ, Ro de Janero, RJ, razl ( FERREIRA, C. R.; ANO, A; "Análse de Crtcaldade Utlzando-se os stemas NJOY, AMPX-II e KENO-IV", Anas do VII ENFIR, Encontro Naconal INAC 211, elo Horzonte, MG, razl.

10 de Físca de Reatores e ermohdráulca, Vol. 1, p , Recfe-PE, razl, ( GANDINI, A., J. Nucl. Energy, 21, p. 755, ( GANDINI, A., Generalzed Perturbaton heory (GP Methods. A Heurstc Approach. Advanced n Nuclear cence, 19: 25-38, J. Lewns and M. ecker Eds. Plenum Press ( GARCIA, V.., Estudos de ensbldade em NC Usando a eora de Perturbação Generalzada, Programa de Engenhara Nuclear, COPPE/UFRJ, Ro de Janero, razl, ( HARLING, O.K., ROER, K.A., MOULING, D.J., ROGU, R.D., Head phantoms for neutron capture herapy. Med Phys, 22: ( INERNAIONAL AONIC ENERGY AGENCY (IAEA, Current tatus of Neutron Capture herapy, In: IAEA ecdoc 1223, IAEA, Wen ( KAZ, D., reatment plannng for boron neutron capture therapy at the New England Medcal Center. (hess MI, Massachusetts ( KAWAAA,., MIYAAKE,., KAJIMOO, he early successful treatment of globlastoma patents wth modfed boron neutron capture therapy, Journal of Neuro- Oncology 65: ( LEE, C.L., ZHOU, X.-L., HARMON, J.F., et al., hermal Neutron Flux Mappng n a Head Phantom, Nuclear Instruments and Methods n Physcs Research A 422: 16 11, ( LEWI, E. E., MILLER JR., W. F., Computatonal Methods of Neutron ransport, Amercan Nuclear ocety, Inc, La Grange Park, Illnos, UA ( MARAHI, M. K., Analyss of Absorbed Dose Dstrbuton n Head Phantom n oron Neutron Capture herapy, Nuclear Instruments and Methods n Physcs Research, A 44: , ( NIEVAAR, V.A., pectral alorng for oron Neutron Capture herapy, Ph.D dssertaton, Delft Unversty of echnology, Amsterdam, Netherlands ( NUCLEAR ENERGY AGENCY (NEA, Avalable at: retreve n ( VAN RIJ, C. M., WILHEN, A. G., VAN LOREN, A. C., AUERWEIN, W. A. G., oron Neutron Capture herapy for Globlastoma Multforme. Pharm World c, 27: (25. INAC 211, elo Horzonte, MG, razl.