INFORMS Journal on Computng Vol. 27, No. 4, Fall 215, pp. 788 83 ISSN 191-9856 (prnt) ó ISSN 1526-5528 (onlne) http://dx.do.org/1.1287/joc.215.659 215 INFORMS Optmzaton of Radaton Therapy Fractonaton Schedules n the Presence of Tumor Repopulaton Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. Thomas Bortfeld Department of Radaton Oncology, Massachusetts General Hosptal and Harvard Medcal School, Boston, Massachusetts 2114, tbortfeld@partners.org Jagdsh Ramakrshnan Wsconsn Insttute for Dscovery, Unversty of Wsconsn Madson, Madson, Wsconsn 53715, jramakrshn2@wsc.edu John N. Tstskls Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, Massachusetts 2139, jnt@mt.edu Jan Unkelbach Department of Radaton Oncology, Massachusetts General Hosptal and Harvard Medcal School, Boston, Massachusetts 2114, junkelbach@partners.org We analyze the effect of tumor repopulaton on optmal dose delvery n radaton therapy. We are prmarly motvated by accelerated tumor repopulaton toward the end of radaton treatment, whch s beleved to play a role n treatment falure for some tumor stes. A dynamc programmng framework s developed to determne an optmal fractonaton scheme based on a model of cell kll from radaton and tumor growth n between treatment days. We fnd that faster tumor growth suggests shorter overall treatment duraton. In addton, the presence of accelerated repopulaton suggests larger dose fractons later n the treatment to compensate for the ncreased tumor prolferaton. We prove that the optmal dose fractons are ncreasng over tme. Numercal smulatons ndcate a potental for mprovement n treatment effectveness. Keywords: dynamc programmng: applcatons; healthcare: treatment Hstory: Accepted by Allen Holder, Area Edtor for Applcatons n Bology, Medcne, & Health Care; receved June 214; revsed December 214, Aprl 215, May 215; accepted May 215. Publshed onlne December 9, 215. 1. Introducton Accordng to the Amercan Cancer Socety, at least 5% of cancer patents undergo radaton therapy over the course of ther treatment. Radaton therapy plays an mportant role n curng early stage cancer, preventng metastatc spread to other areas, and treatng symptoms of advanced cancer. For many patents, external beam radaton therapy s one of the best optons for cancer treatment. Current therapy procedures nvolve takng a pretreatment computed tomography (CT) scan of the patent, whch provdes a geometrcal model of the patent that s used to determne ncdent radaton beam drectons and ntenstes. In current clncal practce, most radaton treatments are fractonated;.e., the total radaton dose s splt nto approxmately 3 fractons that are delvered over a perod of sx weeks. Fractonaton allows normal tssue to repar sublethal radaton damage between fractons and thereby tolerate a much hgher total dose. Currently, the same dose s delvered n all fractons, and temporal dependences n tumor growth and radaton response are not taken nto account. Bologcally based treatment plannng, amng at optmal dose delvery over tme, has tremendous potental, as more s beng understood about tumor repopulaton and reoxygenaton, healthy tssue repar, and redstrbuton of cells. In ths paper, we study the effect of tumor repopulaton on optmal fractonaton schedules,.e., on the total number of treatment days and the dose delvered per day. We are prmarly motvated by accelerated tumor repopulaton toward the end of radaton treatment, whch s consdered to be an mportant cause of treatment falure, especally for head and neck tumors (Wthers et al. 1988, Wthers 1993). Our man concluson s that accelerated repopulaton suggests larger dose fractons later n the treatment to compensate for the ncreased tumor prolferaton. 1.1. Motvaton Radaton therapy treatments are typcally fractonated (.e., dstrbuted over a longer perod of tme) so that normal tssues have tme to recover. However, such tme between treatments allows cancer cells to 788
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 789 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. prolferate and can result n treatment falure (Km and Tannock 25). The problem of nterest then s the determnaton of an optmal fractonaton schedule to counter the effects of tumor repopulaton. Usng the bologcal effectve dose (BED) model, a recent paper (Mzuta et al. 212) mathematcally analyzed the fractonaton problem n the absence of repopulaton. For a fxed number of treatment days, the result states that the optmal fractonaton schedule s to delver ether a sngle dose or an equal dose on each treatment day. The former schedule of a sngle dose corresponds to a hypo-fractonated regmen, n whch treatments are deally delvered n as few days as possble. The latter schedule of equal dose per day corresponds to a hyperfractonaton regmen, n whch treatments are delvered n as many days as possble. The work n ths paper further develops the mathematcal framework n Mzuta et al. (212) and analyzes the effect of tumor repopulaton on optmal fractonaton schedules. We are nterested n optmzng nonunform (n tme) dose schedules, motvated prmarly by the phenomenon of accelerated repopulaton,.e., a faster repopulaton of survvng tumor cells toward the end of radaton treatment. 1.2. Related Work There has been pror work on the optmzaton of nonunform radaton therapy fractonaton schedules (Almqust and Banks 1976; Swan 1981, 1984; Marks and Dewhrst 1991; Yakovlev et al. 1994; Yang and Xng 25), some of whch also ncludes tumor repopulaton effects. However, these works have ether not used the BED model or have prmarly consdered other factors such as tumor reoxygenaton. It has been shown that effects such as reoxygenaton, redstrbuton, and sublethal damage repar can result n non-unform optmal fractonaton schemes (Yang and Xng 25, Bertuzz et al. 213). Prevous works have consdered the case of exponental tumor growth wth a constant rate of repopulaton (Wheldon et al. 1977, Jones et al. 1995, Armpla et al. 24). Other tumor growth models, e.g., Gompertzan and logstc, have also been consdered although mostly n the context of constant dose per day (Usher 198, McAneney and O Rourke 27). There s a sgnfcant amount of lterature, especally from the mathematcal bology communty, on the use of control theory and dynamc programmng (DP) for optmal cancer therapy. Several of these works (Zetz and Ncoln 27, Pedrera and Vla 1991, Ledzewcz and Schättler 24, Salar et al. 214) have looked nto optmzaton of chemotherapy. For radaton therapy fractonaton, some studes (Hethcote and Waltman 1973, Almqust and Banks 1976, Wen et al. 2) have used the DP approach based on determnstc bologcal models, as n ths paper. However, these works have not carred out a detaled mathematcal analyss of the mplcatons of optmal dose delvery n the presence of accelerated repopulaton. Usng magng nformaton obtaned between treatment days, dynamc optmzaton models have been developed to adaptvely compensate for past accumulated errors n dose to the tumor (Ferrs and Voelker 24, de la Zerda et al. 27, Deng and Ferrs 28, Sr et al. 212). There also has been work on onlne approaches that adapt the dose and treatment plan based on mages obtaned mmedately pror to treatment (Lu et al. 28; Chen et al. 28; Km et al. 29, 212; Km 21; Ghate 211; Ramakrshnan et al. 212). Perhaps the closest related work s Wen et al. (2), whch consders both faster tumor prolferaton and reoxygenaton durng the course of treatment. Although a dose ntensfcaton strategy s also suggested n Wen et al. (2), the prmary ratonale for ncreasng dose fractons s dfferent: t s concluded that because of the ncrease n tumor senstvty from reoxygenaton, larger fracton szes are more effectve at the end of treatment. Our work, however, suggests dose ntensfcaton (.e., larger doses over tme) as a drect consequence of a model of accelerated tumor repopulaton durng the course of treatment. 1.3. Overvew of Man Contrbutons The prmary contrbutons of ths paper are the development of a mathematcal framework and the analyss of optmal fractonaton schedules n the presence of accelerated repopulaton. We gve qualtatve and structural nsghts on the optmal fractonaton scheme, wth the hope that t can gude actual practce. Specfcally: 1. We formulate a problem that ncludes general tumor repopulaton characterstcs and develop a DP approach to solve t. We choose to model accelerated repopulaton mplctly by usng deceleratng tumor growth curves, where a larger number of tumor cells results n slower growth. Thus, faster growth s exhbted toward the end of radaton treatment, when there are fewer cells. 2. We prove that the optmal doses are nondecreasng over tme (Theorem 3), because of the deceleratng nature of tumor growth curves. Ths type of result remans vald even when we allow for weekend and holday breaks (Corollary 1). 3. We analyze the specal structure of the problem for the case of Gompertzan tumor growth and show that t s equvalent to maxmzng a dscounted verson of the BED n the tumor ( 2.3.2), whch results n a smplfed DP algorthm. 4. We show that when there s repopulaton, the optmal number of dose fractons s fnte (Theorem 4).
79 INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. 5. We fnd through numercal smulatons that the optmal fracton szes are approxmately proportonal to the nstantaneous prolferaton rate, suggestng larger dose fractons later n the treatment to compensate for the ncreased tumor prolferaton. 1.4. Organzaton In 2, we present the model, formulaton, and DP soluton approach. We also analyze the specal structure of the problem for the case of Gompertzan tumor growth. In 3, we dscuss both prevously known results and the man results of ths paper. Our prmary concluson s that the optmal dose fractons are nondecreasng over tme. In 4, we present and dscuss numercal results under exponental or Gompertzan growth models. In 5, we provde further remarks about the model under other assumptons and dscuss the results n relaton to pror work. Fnally, n 6, we summarze our man fndngs and the most mportant mplcatons. 2. Model, Formulaton, and Soluton Approach 2.1. Model of Radaton Cell Kll In ths secton, we descrbe the radaton cell kll model wthout any tumor growth dynamcs. We use the lnear-quadratc (LQ) model (Fowler 1989) to relate radaton dose and the fracton of survvng cells. Ths model s supported by observatons from rradatng cells n vtro. The LQ model relates the expected survval fracton S (n the absence of tumor growth) after a sngle delvered dose d, n terms of two tssue parameters Å and Ç, through the relaton S = exp4é4åd + Çd 2 55 Thus, the logarthm of the survval fracton conssts of a lnear component wth coeffcent Å and a quadratc component Ç (see the dotted curve n Fgure 1). Ths LQ model assumes two components of cell kllng by radaton: one proportonal to dose and one to the square of the dose. The respectve tssue-specfc proportonalty constants are gven by Å and Ç. It s possble to nterpret these two components of kllng as they relate to the probablty of exchange aberratons n chromosomes (see Hall and Gacca 26). We llustrate ths cell kll effect by plottng the logarthm of the survval fracton n Fgure 1. For N treatment days wth radaton doses d 1d 1 1 1d N É1, the resultng survval fractons from each ndvdual dose can be multpled, assumng ndependence between dose effects. The resultng relaton s NX É1 S = exp É 4Åd k + Çd 2 k 5 k= Survvng fracton S 1 1 1 1 2 1 3 Sngle dose Multple doses Lnear component 1 4 2 4 6 8 1 12 Dose d (Gy) Fgure 1 (Color onlne) Illustraton of the Fractonaton Effect Usng the LQ Model Notes. The dotted lne represents the effect of both the lnear and quadratc terms resultng from applyng a sngle total dose of radaton. The sold lne corresponds to the effect of the same total dose, f t s dvded nto multple ndvdual doses, whch results n a much hgher survval fracton when the quadratc Ç term s sgnfcant. Fnally, the dashed lne shows the total effect of the lnear term, whether doses are appled as sngle or multple ndvdual fractons. The effect of the quadratc factor Ç, n the above equaton, s that the survval fracton s larger when splttng the total dose nto ndvdual dose fractons (Fgure 1). Thus, there s an nherent trade-off between delverng large sngle doses to maxmze cell kll n the tumor and fractonatng doses to spare normal tssue. A common quantty that s used alternatvely to quantfy the effect of the radaton treatment s the BED (Barendsen 1982, Hall and Gacca 26, O Rourke et al. 29). It s defned by BED4d5 = 1 Å 4Åd + Çd2 5 = d 1 + d Å/Ç 1 (1) where Å/Ç s the rato of the respectve tssue parameters. Thus, the BED n the above defnton captures the effectve bologcal dose n the same unts as physcal dose. A small Å/Ç value means that the tssue s senstve to large doses; the BED n ths case grows rapdly wth ncreasng dose per fracton. Note that BED s related to the LQ model by settng BED = É ln4s5/å. In the BED model, only a sngle parameter, the Å/Ç value, needs to be estmated; e.g., n Mralbell et al. (212) the Å/Ç value s estmated for prostate cancer from radotherapy outcomes of thousands of patents. Whenever nonstandard fractonaton schemes are used n a clncal settng, the BED model s typcally used to quantfy fractonaton effects. In ths paper, we frequently swtch between the cell nterpretaton n the LQ model and the effectve dose nterpretaton n the BED model, as they
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 791 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. both provde alternatve and useful vews based on context. Based on the relaton gven n Equaton (1), we defne BED T 4d5 as the BED n a tumor when a dose d s delvered, where 6Å/Ç7 T s the Å/Ç value of the tumor. We also defne the total BED n the tumor from delverng doses d 1d 1 11d N É1 as NX É1 BED T = k= NX É1 BED T 4d k 5 = k= d k 1 + d k 6Å/Ç7 T In ths paper, we consder a sngle dose-lmtng rado-senstve organ-at-rsk (OAR); ths assumpton s approprate for some dsease stes (e.g., for prostate cancer, the rectum could be taken as the dose-lmtng organ). We assume that an OAR receves a fracton of the dose appled to the tumor. Thus, let a dose d be appled to the tumor result n a dose of Éd n the OAR, where É s the fractonal constant, also referred to as the normal tssue sparng factor, satsfyng <É<1. Implctly, ths assumes a spatally homogeneous dose n the tumor and the OAR as n Mzuta et al. (212). The generalzaton to a more realstc nhomogeneous OAR dose dstrbuton (Unkelbach et al. 213), whch leaves the man fndngs of ths paper unaffected, s detaled n 5.1. The value of É wll depend on the treatment modalty and the dsease ste. For treatment modaltes provdng very conformal dose around the tumor and dsease stes wth the OAR not closely abuttng the prmary tumor, the OAR wll receve less radaton and thus É would be a smaller. Usng Éd as the dose n the OAR and 6Å/Ç7 O as the OAR Å/Ç value, we can defne the assocated OAR BEDs, BED O 4d5, and BED O n the same way as was done for the tumor BED: NX É1 BED O = k= NX É1 BED O 4d k 5 = k= Éd k 1 + Éd k 6Å/Ç7 O 2.2. Tumor Growth Model In ths secton, we descrbe tumor growth models that wll be used later to formulate a fractonaton problem. We model the growth of the tumor through the ordnary dfferental equaton (Wheldon 1988): 1 dx4t5 = î4x4t551 (2) x4t5 dt wth ntal condton x45 = X, where x4t5 s the expected number of tumor cells at tme t. In the Equaton (2), î4x5 represents the nstantaneous tumor prolferaton rate. We assume that î4x5 s nonncreasng and s contnuous n x (for x>), whch mples that the soluton to the above dfferental equaton exsts and s unque for any X >. By choosng an approprate functonal form of î, we can descrbe a varety φ(x) tumor growth rate (days 1 ).5.4.3.2.1.2.4.6.8 1. x(t)/x normalzed number of cells Gompertz Exponental Fgure 2 (Color onlne) Tumor Growth Rate vs. Number of Tumor Cells Note. The Gompertz equaton models slower growth for larger number of cells whle the exponental model assumes a constant growth rate. of tumor repopulaton characterstcs relevant for radaton therapy: 1. We can model exponental tumor growth (Wheldon 1988, Yorke et al. 1993) wth a constant prolferaton rate ê by choosng î4x5 = ê. In ths case, the soluton x4t5 of the dfferental equaton wth ntal condton x45 = X s x4t5 = X exp4êt51 where X s the ntal number of cells and ê> s the prolferaton rate. 2. We represent accelerated repopulaton by choosng î4x5 to be a decreasng functon of x. In ths case, the nstantaneous tumor prolferaton rate ncreases when, toward the end of treatment, the number of remanng tumor cells decreases. The Gompertz model (Lard 1964, Norton et al. 1976, Norton 1988) s one such deceleratng tumor growth curve (Fgure 2). For Gompertzan growth, we would smply set Xà î4x5 = b ln 1 x where X s the ntal number of tumor cells, X à s the carryng capacty or the maxmum number of tumor cells, and b s a parameter that controls the rate of growth. The soluton to the dfferental Equaton (2), wth ntal condton x45 = X, s x4t5 = X exp4ébt5 X 1Éexp4Ébt5 à (3) Ths equaton models slower repopulaton for larger tumor szes and vce versa (see Fgure 2). 2.3. Formulaton In ths secton, we combne the LQ model from 2.1 and the tumor growth model from 2.2 and formulate
792 INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. a fractonaton problem. The am of radaton therapy s to maxmze the tumor control probablty (TCP) subject to an upper lmt on the normal tssue complcaton probablty (NTCP) n the OAR (O Rourke et al. 29). A sgnfcant amount of research has been conducted to determne approprate and better models of TCP (Brahme and Agren 1987, O Rourke et al. 29) and NTCP (Kutcher and Burman 1989, Lyman 1985). A common way to model NTCP s as a sgmodal functon of BED O (Kutcher et al. 1991). Snce a sgmodal functon s monotonc n ts argument, t then suffces n our model to mpose an upper lmt on BED O. Though some studes have rased concerns (Tucker et al. 199), TCP has been wdely modeled usng Posson statstcs (Munro and Glbert 1961; Porter 198a, b), under whch TCP = exp4éx + N É1 51 where X + N É1 s the expected number of tumor cells survvng after the last dose of radaton. In ths case, maxmzng the TCP s equvalent to mnmzng X + N É1. We now defne Y + N É1 = ln4x+ N É1 5/Å T 1 where Å T s a tumor tssue parameter assocated wth the lnear component of the LQ model. Note that the defnton of Y s analogous to the defnton of the BED. It has unts of radaton dose; thus dfferences n Y can be nterpreted as dfferences n effectve BED delvered to the tumor. We choose to prmarly work wth ths logarthmc verson because of ths nterpretaton. For the rest of the paper, we focus on the equvalent problem of mnmzng Y + N É1 subject to an upper lmt on BED O. The problem s stated mathematcally as mnmze Y + N É1 s.t. BED O apple c1 (4) 8d 9 where c s a prespecfed constant. There s no guarantee of the convexty of the objectve, and thus, ths problem s nonconvex. Note that the feasble regon s nonempty because a possble feasble schedule s a dose of zero for all treatment sessons. We clam that the objectve functon n (4) attans ts optmal value on the feasble regon. As a functon of the dose fractons d k, t can be seen that Y + N É1 s contnuous. Furthermore, the constrant on BED O ensures the feasble regon s compact. Thus, the extreme value theorem ensures that the objectve attans ts optmal value on the feasble regon. We now descrbe the dynamcs of the expected number of tumor cells durng the course of treatment (Fgure 3). We assume that a sequence of N doses d 1d 1 11d N É1 s delvered at nteger tmes;.e., tme y(t) = ln(x(t))/ T = ln(cell number)/ T Y Y 1 BED T (d ) + Y BED T (d 1 ) Y 1 + 1 2 Y + N 2 Y N 1 Y + N 1 N 2 N 1 BED T (d N 1 ) Tme (days) Fgure 3 Schematc Illustraton of the Expected Number of Tumor Cells Over the Course of Treatment Note. The effect of radaton dose d s a reducton, proportonal to BED T 4d5, n the log of the number of cells. s measured n days. The survval fracton of cells from delverng these radaton doses s descrbed by the LQ model n 2.1. If X É and X + are the numbers of tumor cells mmedately before and after delverng the dose d, we wll have X + = X É exp4é4å T d + Ç T d 255. For the logarthmc versons Y + nteger tmes and Y É Y + = Y É É BED T 4d 5, we have for For nonnteger tmes n 61N É 17, the tumor grows accordng to the dfferental Equaton (2) wth prolferaton rate î4x5, as descrbed n 2.2. For convenence, we denote by F4 5 the resultng functon that maps Y É to Y + when usng the growth dfferental Equaton (2). Thus, we have for = 1 111N É 2. Y É +1 = F4Y+ 5 2.3.1. Exponental Tumor Growth wth Constant Prolferaton Rate. For the exponental tumor growth model, where î4x5 = ê, the rate of repopulaton ê does not change wth tumor sze (see Fgure 2). Assumng ê represents a measure of growth per unt day (or fracton), the number of tumor cells s multpled by a factor of exp4ê5 after every fracton. Or, equvalently, a constant factor s added to the logarthmc cell number because of tumor growth n between treatment days. Snce there are N dose fractons and N É 1 days of repopulaton n between, t can be seen that the optmzaton problem (4) smplfes to mnmze 8d 9 Y + 1 Å T 4N É 15ê É BED T s.t. BED O apple c Here the effect of tumor repopulaton s captured n the term 4N É 15ê/Å T. (5)
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 793 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. 2.3.2. Gompertzan Tumor Growth. For the case of Gompertzan tumor growth, where î4x5=bln4x à /x5, we can also sgnfcantly smplfy the formulaton. As shown n 2.4, ths results n a smplfed DP approach to determne an optmal fractonaton schedule. We wll now derve an explct expresson for the expected number of tumor cells at the end of treatment. We clam that Y + = 1 Å T ln4x455 É X exp6éb4 É k57 BED T 4d k 51 (6) k= for = 1 11 1N É 1, where x4t5 represents the expected number of tumor cells n the absence of radaton treatment, and whch s gven by the expresson n Equaton (3). (In partcular, x4t5 should not be confused wth the number of tumor cells when the tumor s treated.) Equaton (6) holds for = because x45 = X and Y + = ln4x 5/Å T ÉBED T 4d 5. For the nductve step, suppose that Equaton (6) holds true. From the Gompertzan growth Equaton (3), and assumng that the tme nterval between fractons s one day (t = 1), we can wrte the functon F that maps Y + to Y+1 É as Y É +1 = F4Y+ 5 = exp4éb5y + + 41 É exp4éb55 ln4x à5 Å T Incorporatng the growth and the radaton dose from d +1, we fnd Y + +1 = F4Y+ 5 É BED T 4d +1 5 = 1 Å T ln4x45 exp4éb5 X 1Éexp4Éb5 à 5 X exp6éb4 + 1 É k57 BED T 4d k 5 +1 É k= = 1 X+1 ln4x4 + 155 É exp6éb4 + 1 É k57 BED Å T 4d k 51 T k= completng the nductve step. Ths results n the optmzaton problem 1 mnmze ln4x4n É 155 8d 9 Å T X exp6éb4n É 1 É k57 BED T 4d k 5 N É1 É k= s.t. BED O apple c The nterestng aspect of the above optmzaton problem s that t s essentally a maxmzaton of a dscounted sum of the terms BED T 4d k 5. Because the weghtng term gves larger weght to later fractons, we can conjecture that the optmal fractonaton scheme wll result n larger fracton szes toward the end of treatment. Ths s n contrast to the exponental tumor growth model for whch there s no accelerated repopulaton, and the BED T 4d k 5 terms are weghted equally. 2.4. Dynamc Programmng Approach To get from the ntal Y to Y + N É1, one recursvely alternates between applyng a dose d and the growth functon F. That s, Y + N É1 takes the form Y + N É1 = F4 F4F4Y É BED T 4d 55 É BED4d 1 55 5 É BED4d N É1 5 (7) Such a recursve formulaton lends tself naturally to a DP approach. We can solve the optmzaton problem by recursvely computng an optmal dose backward n tme. Note that although nonlnear programmng methods can also be used, there s no guarantee of the convexty of (7), and thus, such methods mght only provde a local optmum. A global optmum, however, s guaranteed f a DP approach s used. To determne the dose d k, we take nto account Y + ké1 and the cumulatve BED n the OAR from delverng the pror doses, whch we defne as ké1 X z k = BED O 4d 5 = Here, Y + ké1 and z k together represent the state of the system because they are the only relevant peces of nformaton needed to determne the dose d k. We do not nclude a cost per stage; nstead, we nclude Y + N É1 n a termnal condton. To ensure that the constrant on BED O s satsfed, we also assgn an nfnte penalty when the constrant s volated. The Bellman recurson to solve the problem s ( J N 4Y + N É1 1z Y + N É1 N 5= 1 f z N applec3 à1 otherwse1 J k 4Y + ké1 1z k5=mn Jk+1 4F 4Y + ké1 5ÉBED T 4d k 51 d k z k +BED O 4d k 55 1 for k = N É 11N É 2111. The ntal equaton for tme, gven next, s slghtly dfferent because there s no pror tumor growth (see Fgure 3): J 4Y É 1z 5 = mn d J1 4Y É É BED T 4d 51 z + BED O 4d 55 For the exponental and Gompertzan growth cases, the DP approach smplfes and only requres the sngle state z k. Next, we dscuss the approach for the Gompertzan case only; for the exponental case, as wll be dscussed n 3.1, an optmal fractonaton scheme can be characterzed n closed form. For smplcty, we use addtve costs per stage ths tme. The smplfed algorthm s ( 41/Å J N 4z N 5 = T 5 ln4x4n É 1551 f z N apple c3 à1 otherwse1 J k 4z k 5 = mn d k É exp6éb4n É 1 É k57 BEDT 4d k 5 + J k+1 4z k + BED O 4d k 55 1 (8) (9)
794 INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. for k = N É11NÉ211. For numercal mplementaton, the state varables need to be dscretzed and the tabulated values stored. For evaluatng the cost-to-go functon J k at any nondscretzed values, an nterpolaton of approprate dscretzed values can be used for ncreased accuracy. 3. Propertes of an Optmal Fractonaton Schedule In 3.1, we dscuss prevously known results. These prmarly concern the characterzaton of optmal fractonaton schedules n the absence of accelerated repopulaton. The purpose of ths subsecton s to provde the essental results that are scattered n the lterature n dfferent papers and formalze them n the framework of ths paper. In 3.2, we summarze our man results. The proofs of the results are provded n the onlne supplement (avalable as supplemental materal at http://dx.do.org/1.1287/joc.215.659). 3.1. Prevously Known Results The set of all optmal solutons to the fractonaton problem n the absence of repopulaton s characterzed n the followng theorem, publshed n Mzuta et al. (212). Theorem 1. Let N be fxed. In the absence of repopulaton (.e., î4x5 = ), an optmal fractonaton schedule can be characterzed n closed form. If 6Å/Ç7 O É6Å/Ç7 T, an optmal soluton s to delver a sngle dose equal to "s # d j = 6Å/Ç7 O 1 + 4c É 1 (1) 2É 6Å/Ç7 O at an arbtrary tme j and delver d = for all 6= j. Ths corresponds to a hypofractonaton regmen. If 6Å/Ç7 O < É6Å/Ç7 T, the unque optmal soluton conssts of unform doses gven by d j = 6Å/Ç7 O 2É "s 1 + # 4c É 1 N 6Å/Ç7 O 1 (11) for j = 1 111N É 1. Ths corresponds to a hyperfractonaton regmen,.e., a fractonaton schedule that uses as many treatment days as possble. Ths theorem states that f 6Å/Ç7 O É6Å/Ç7 T, a sngle radaton dose s optmal;.e., the optmal number of fractons N s 1. However, f 6Å/Ç7 O s small enough.e., the OAR s senstve to large doses per fracton, so that 6Å/Ç7 O < É6Å/Ç7 T t s optmal to delver the same dose durng the N days of treatment. Because takng larger N only results n extra degrees of freedom, N!à n ths case. However, ths s clearly not realstc and s an artfact of modelng assumptons. We wll show that ncludng tumor repopulaton results n a fnte N. In the followng remark, we comment on the result and the model assumptons. Remark 1. Our ultmate goal s to understand the effect of accelerated repopulaton over the duraton of treatment. Thus, we are prmarly nterested n the hyperfractonaton case, wth 6Å/Ç7 O < É6Å/Ç7 T, whch wll hold for most dsease stes. The case where 6Å/Ç7 O É6Å/Ç7 T needs careful consderaton snce the valdty of the model may be lmted f N s small and the dose per fracton s large. The condton 6Å/Ç7 O É6Å/Ç7 T would be satsfed for the case of an early respondng OAR tssue (see 5.4 for further detals); however, we have not ncluded repopulaton and repar effects nto the BED model, whch could play an mportant role for such tssue. Another aspect that requres consderaton s whether doses should be fractonated to permt tumor re-oxygenaton between treatments. Thus, wthout extensons to our current model, t may be better to set a mnmum number of fractons (e.g., fve) n the case where 6Å/Ç7 O É6Å/Ç7 T. In the next remark, we dscuss how Theorem 1 can be generalzed n the case of exponental tumor growth. Remark 2. For the problem ncludng exponental tumor growth wth î4x5 = ê, there s only an addtve term 4N É 15ê/Å T n objectve (5), whch s ndependent of the dose fractons. Thus, for a fxed N, the result from Theorem 1 stll holds for the exponental growth case. It turns out that one can also characterze the optmal number of fractons n closed form for the exponental tumor growth case. The result s consstent wth and smlar to the work n a few papers (Wheldon et al. 1977, Jones et al. 1995, Armpla et al. 24), though we nterpret t dfferently here. Our statement below s a slght generalzaton n that we do not assume unform dose per day a pror. Theorem 2. The optmal number of fractons N for exponental growth wth constant prolferaton rate ê (.e., î4x5 = ê) s obtaned by the followng procedure: 1. If 6Å/Ç7 O É6Å/Ç7 T, then N = 1. 2. If 6Å/Ç7 O < É6Å/Ç7 T, then (a) Compute N c = A4 p 4ê + B5 2 /4ê4ê + 2B55 É 15, where A = 2c/6Å/Ç7 O, B = 4Å T 6Å/Ç7 O /2É541 É 6Å/Ç7 O / É6Å/Ç7 T 5. (b) If N c < 1, then N = 1. Otherwse, evaluate the objectve at èn c ê and ën c í, where è êand ë íare the floor and celng operators, respectvely, and let the optmum N be the one that results n a better objectve value. Ths result also makes sense n the lmtng cases. When approachng the case of no repopulaton.e., ê! we see that N c!à, and the optmum N approaches nfnty. When ê!à, we see that N c!, meanng that the optmum N s a sngle dose. Recall that f 6Å/Ç7 O < É6Å/Ç7 T, N!à n the absence of
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 795 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. repopulaton. For the case of exponental repopulaton wth constant but postve rate, the above result shows that the optmum N s fnte. Indeed, even for general tumor growth characterstcs, we wll show nextthat as long as there s some repopulaton, the optmal number of fractons wll be fnte. 3.2. Man Results of Ths Paper We begn wth two lemmas. The frst one smply states that the constrant on BED O s bndng; ntutvely, ths s because of the assumpton that î4x5 > (mplyng the growth functon F4 5 s ncreasng) and the fact that the BED functon s monotone n dose. Lemma 1. Assume that î4x5 > for all x>. Then the constrant on BED O n (4) wll be satsfed wth equalty. Lemma 2. Assume that î4x5 > for all x> and that î4x5 s a nonncreasng functon of x. Suppose that <j and that we apply the same sequence of doses d +1 11d j, startng wth ether Y + or Ỹ +. If Y + < Ỹ +, then Y + j < Ỹ + j and Ỹ + j É Y + j apple Ỹ + É Y +. Assumng that the same sequence of doses are appled, Lemma 2 states a monotoncty and a contracton type property when mappng the expected number of cells from one pont n tme to another. Next, we state the man theorem. Theorem 3. Let us fx the number of treatment days N. Assume that there s always some amount of repopulaton.e., î4x5 > for all x> and that the nstantaneous tumor growth rate î4x5 s nonncreasng as a functon of the number of cells x. If 6Å/Ç7 O É6Å/Ç7 T, then t s optmal to delver a sngle dose equal to (1) on the last day of treatment. Ths corresponds to a hypofractonaton regmen. If 6Å/Ç7 O < É6Å/Ç7 T, then any optmal sequence of doses s nondecreasng over the course of treatment. That s, these doses wll satsfy d apple d 1 apple appled N É1. For the case where 6Å/Ç7 O É6Å/Ç7 T, we take note of Remark 1 agan. It s reasonable that an optmal soluton uses the most aggressve treatment of a sngle dose, as ths s the case even wthout repopulaton. The next remark gves an ntutve explanaton for why n the fxed N case t s optmal to delver only on the last treatment day and shows the optmal number of fractons N s 1 when 6Å/Ç7 O É6Å/Ç7 T. Remark 3. A sngle dose delvered on the last treatment day s optmal when 6Å/Ç7 O É6Å/Ç7 T because t s better to let the tumor grow slowly durng the course of treatment rather than to stmulate accelerated growth by treatng t earler. However, ths does not mean t s optmal to wat too long before treatng the patent. Startng wth a gven ntal number of tumor cells, t s clear that treatng a sngle dose wth a smaller N results n a better cost. Ths s because the tumor grows for a shorter tme. Thus, t follows that when 6Å/Ç7 O fractons s N = 1. É6Å/Ç7 T, the optmal number of When 6Å/Ç7 O < É6Å/Ç7 T, the doses must ncrease over tme. Intutvely, because of the decreasng property of î4x5 as a functon of x, the tumor grows faster when ts sze becomes smaller over the course of treatment; hgher doses are then requred to counter the ncreased prolferaton. An nterchange argument s used to prove the above theorem. The next theorem states that as long as the repopulaton rate cannot decrease to, the optmal number of fractons N s fnte. Theorem 4. Suppose that there exsts r> such that î4x5 > r for all x >. Then there exsts a fnte optmal number of fractons N. Typcally n a clncal settng, dose fractons are not delvered durng weekend and holday breaks. The followng remark explans how such breaks can be ncluded n our formulaton. Remark 4. We can adjust the fractonaton problem by settng N to be the total number of days, ncludng weekend and holday breaks. For days n whch a treatment s not admnstered, the dose fracton s set to. For all other days, the DP algorthm (8) s used as before to determne the optmal fractonaton schedule. Corollary 1 shows that the structure of an optmal soluton s stll smlar to that descrbed n Theorem 3 even when ncludng holdays n the formulaton and/or fxng some dose fractons. Corollary 1. Includng breaks and/or fxng the dose n some fractons does not change the structure of the optmzed dose fractons as descrbed n Theorem 3. Fx N. Assume that î4x5 > for all x> and that î4x5 s nonncreasng n x. If 6Å/Ç7 O É6Å/Ç7 T, then an optmal soluton s to delver a sngle dose on the last delverable day that s not a break and not a fxed dose fracton. If 6Å/Ç7 O < É6Å/Ç7 T, then an optmal sequence of doses, excludng breaks and fxed dose fractons, s nondecreasng over the course of treatment. Thus, accordng to ths corollary, f 6Å/Ç7 O < É6Å/Ç7 T and we ntroduce weekend breaks, the dose on Monday wll n general be larger than the dose on the Frday of the prevous week. To understand why ths s so, consder what would happen f the Frday dose were n fact larger. The tumor would then be growng at a faster rate over the weekend; then t would make better sense to delver a hgher Monday dose nstead. Ths result s dfferent from the numercal results n Wen et al. (2). Of course, the model was dfferent n that paper because the prmary motvaton was to determne the effect of the varyng
796 INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. tumor senstvty over the course of treatment, not to counter accelerated repopulaton. When we assume an exponental growth model, we can fully characterze the optmal soluton, lke n Remark 5. Remark 5. For the exponental growth model where î4x5 = ê, f 6Å/Ç7 O < É6Å/Ç7 T, we clam that the optmal soluton would be to delver unform doses at each treatment sesson,.e., on days that are not breaks and that do not have fxed dose fractons. Let us adjust the number of days N approprately and set the dose fractons to for the breaks. Then, for a fxed N, as seen from the objectve n (5), the only relevant term that determnes the dose fractons s ÉBED T. Thus, for a fxed N, the arguments from Theorem 1 hold and unform doses are optmal. Now we dscuss the effect of breaks on the optmal number of fractons. If the number of breaks s smply a fxed number, the optmal number of days N, ncludng holday breaks, would be stll gven by the expresson n Theorem 2. Ths s because the dervatve of the objectve gven n (5) as a functon of N would reman unchanged. If the number of breaks s nondecreasng n the number of treatment days (as would be the case wth weekend breaks), the expresson for N gven n Theorem 2 would not necessarly hold. Dervng a closed form formula for N for ths case appears to be tedous, f not mpossble. We suggest, therefore, exhaustvely evaluatng the objectve gven n (5) for reasonable values of N to obtan a very good, f not optmal, choce of the number of treatment days. For a gven break pattern, we could also optmze the startng day of treatment by approprately evaluatng the objectve n (5). 4. Numercal Experments In 4.1, we calculate the optmal treatment duraton whle assumng exponental tumor growth wth varyng rates of repopulaton. In 4.2 and 4.3, we model accelerated repopulaton usng Gompertzan growth and numercally calculate the resultng optmal fractonaton scheme. Fnally, n 4.4, we evaluate the effect of weekend breaks. For all of our numercal smulatons, we used MATLAB on a 64-bt Wndows machne wth 4GB RAM. 4.1. Faster Tumor Growth Suggests Shorter Overall Treatment Duraton We use realstc choces of radobologcal parameters to assess the effect of varous rates of tumor growth on the optmal number of treatment days. Here, we assume exponental growth wth a constant rate of repopulaton. We use 6Å/Ç7 T = 1 Gy, 6Å/Ç7 O = 3 Gy, and Å T = 3 Gy É1 for the tssue parameters; these are approprate standard values (Guerrero and L 23, Hall and Gacca 26). We consder a standard fractonated treatment as reference,.e., a dose of 6 Gy delvered to the tumor n 3 fractons of 2 Gy. For the above choce of Å/Ç values and É = 7, ths corresponds to an OAR BED of 61.6 Gy, whch we use as the normal tssue BED constrant c. To choose approprate values for the prolferaton rate ê, we relate t to the tumor doublng tme í d. Snce í d represents the tme t takes for the tumor to double n sze, we set exp4êt5 = 2 t/í d, resultng n the followng relaton: ê = ln425 í d For human tumors, the doublng tme can range from days to months (Wthers et al. 1988, Km and Tannock 25), dependng on the partcular dsease ste. We observe that for the parameters assumed above, the optmal number of treatment days s smaller for faster growng tumors (Fgure 4). The objectve value plotted n Fgure 4 s BED T É 1 Å T 4N É 15ê For the reference treatment (N = 3) and Å T = 3, the decrease n the tumor BED from the second term 4N É 15ê/Å T evaluates to about 13 Gy for a slowly prolferatng tumor wth doublng tme í d = 5. Ths s small compared to BED T = 72. For a fast prolferatng tumor wth doublng tme í d = 5, the correcton 4N É 15ê/Å T s about 134 Gy and becomes more mportant. Thus, smaller values of N are suggested for faster prolferatng tumors. 4.2. Accelerated Repopulaton Suggests Increasng Doses Toward the End of Treatment One way to model accelerated repopulaton s to use decreasng tumor growth curves (see Fgure 2). We model ths behavor usng the Gompertzan tumor growth model and solve the fractonaton problem by usng the smplfed DP Equaton (9). For numercal mplementaton of the DP algorthm, we dscretze the state nto 5 ponts for each tme perod. When evaluatng the cost-to-go functon for values n between dscretzaton ponts, we use lnear nterpolaton. We llustrate optmal fractonaton schemes for both slow and fast prolferatng tumors. For a slowly prolferatng tumor, we choose the parameters X = 4 1 6, X à = 5 1 12, and b = exp4é6925 by fxng X à (to be on the order of the value gven n Norton (1988) for breast cancer) and manually varyng X and b so that the doublng tme for the reference treatment starts at 5 days n the begnnng and decreases to 2 days at the end of treatment. For a fast prolferatng tumor, we adjust the parameters accordngly so that the doublng tme goes from 5 to 5 days: X = 6 1 11,
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 797 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. Objectve 8 7 6 5 4 3 2 1 18.7 19 34.9 35 6. 6 112.6 113 N N * d = 5 days d = 1 days d = 2 days d = 5 days Fgure 4 (Color onlne) Dependence of the Optmal Value of the Objectve Functon for Dfferent Choces of the Number of Fractons, Assumng Exponental Tumor Growth Notes. The optmal number of treatment days s smaller for faster growng tumors. The expresson n Theorem 2 was used to generate ths plot. The objectve was evaluated at the floor and celng of the contnuous optmum N to obtan the actual optmum N. X à = 5 1 12, and b = exp4é535. Such fast growth s not atypcal; clncal data on squamous cell carcnomas of the head and neck show that t s possble for the doublng tme to decrease from 6 days to four days, although there could be an ntal lag perod of constant repopulaton (Wthers et al. 1988). As before, we use the parameters 6Å/Ç7 T = 1 Gy, 6Å/Ç7 O = 3 Gy, Å T = 3 Gy É1, c = 616 Gy, and É = 7. As shown n Fgure 5, for a fast prolferatng tumor, the sequence of radaton doses ncreases from 1 Gy to 3 Gy, whch s a sgnfcant dfference from the standard treatment of 2 Gy per day for 3 days. For a slow prolferatng tumor, the doses closely resemble standard treatment and only ncrease slghtly over the course of treatment. Note that the optmal fractonaton scheme dstrbutes the doses so that they are approxmately proportonal to the nstantaneous prolferaton rate î4x5. The plotted î4x5 n Fgure 5 s the resultng nstantaneous prolferaton rate after the delvery of each dose fracton. For the reference treatment of 2 Gy per day wth N = 3, n the case of a fast prolferatng tumor, the objectve Y + N É1 s 263. The objectve Y + N É1 for the optmal fractonaton scheme n plot (b) of Fgure 5 s 2541. Ths s a change of about 24% n Y + N É1 and 17% change n X+ N É1 n comparson to the reference treatment. It s not straghtforward to make a meanngful statement about the mprovement n tumor control smply based on these values. However, we can say that even a small mprovement n tumor control for a specfc dsease ste can make a sgnfcant mpact because of the large number of patents treated wth radaton therapy every year. In prncple, runnng the DP algorthm for every possble value of N would gve us the optmal number of fractons. We choose to run the algorthm for N = 11 2111. For each run of the DP algorthm for a fxed N, t takes on average about 7 seconds usng MATLAB on a 64-bt Wndows machne wth 4GB RAM and an Intel 7 2.9 GHz chp. We fnd N = 79 results n the best cost for the slowly prolferatng tumor and N = 38 for the fast prolferatng tumor. For the slowly prolferatng tumor, a very small fracton of the tumor s cells remans, regardless of whether N = 79 or N = 3; for the fast prolferatng tumor, settng N = 3 nstead of N = 38 results only n a change of 7% n Y + N É1. Thus, for practcal purposes, t s reasonable to set N = 3 because more dose fractons could mean, among other factors, patent nconvenence and further cost. 4.3. Smaller Å/Ç Value of the Tumor Results n Larger Changes n the Fractonaton Schedule We use a smaller value for the Å/Ç value of the tumor and rerun the calculatons from the prevous subsecton. The parameters of the Gompertzan growth reman the same for the slow and fast prolferatng tumor. As before, we use the parameters 6Å/Ç7 O = 3 Gy, Å T = 3 Gy É1, c = 616 Gy, and É = 7, wth the only change beng that 6Å/Ç7 T = 57 Gy. Note that the condton 6Å/Ç7 O = 3 < É6Å/Ç7 T = 4 s satsfed, meanng that hypofractonaton s not optmal (see Theorem 3). We run the DP algorthm for N = 11 2111, and fnd N = 42 and N = 17 result n the best cost for the slowly and fast prolferatng tumors, respectvely. However, for practcal purposes, we agan set the maxmum number of fractons as 3 for the slowly prolferatng tumor because a very small fracton of the tumor s cells remans, regardless of whether N = 42 or N = 3, and there s a change of only 7% n Y + N É1. As seen n part (b) of Fgure 6, for the fast prolferatng tumor, the sequence of radaton doses ranges from approxmately 1 Gy to 55 Gy for the 17 days of treatment. Ths change n the fractonaton schedule results n the objectve Y + N É1 beng 1542, compared to 1778 for the reference treatment of 2 Gy per day for 3 days. Ths s a sgnfcant change of 133% n Y + N É1 and about 57% change n X+ N É1 n comparson to the reference treatment. Smlar to 4.2, the fracton doses are approxmately proportonal to the nstantaneous prolferaton rate. We can nfer that a smaller Å/Ç value of the tumor suggests usng larger changes n the fracton szes and shorter overall treatment duraton; ths results n larger gans n the objectve value and hence n overall tumor control. Low values of Å/Ç have been observed for dsease stes such as prostate cancer (Mralbell et al. 212). Numercal experments also ndcate a smlar effect when varyng the normal
798 INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. (a) Dose (Gy) (b) Dose (Gy) 3 2 1 1 3 2 1 1 Treatment day Treatment day 2 3 2 3 Fgure 5 (Color onlne) Optmal Fractonaton for Accelerated Repopulaton Notes. Plot (a) shows the optmal fractonaton schedule for a slowly prolferatng tumor and plot (b) for a fast one. The doublng tme for the reference treatment begns at í d = 5 days and decreases approxmately to (a) 2 and (b) 5 days, respectvely, at the end of treatment. The plotted î4x5 s the resultng nstantaneous prolferaton rate after the delvery of each dose fracton. tssue sparng factor É. That s, a smaller É results n larger changes n the fractonaton schedule. Intutvely, ths s because better sparng of normal tssue allows a more aggressve treatment wth hgher tumor control. Of course, f the Å/Ç value or sparng factor É s very small, hypofractonaton would be optmal (see Theorem 3). 4.4. Effect of Weekend Breaks We smulate the effect of weekend breaks usng 3 treatment sessons (startng Monday and endng Frday) wth approprately nserted weekend days, resultng n N = 4 days (6 weeks, 5 weekends). We set 6Å/Ç7 T = 1 and assume a fast prolferatng tumor, wth Gompertzan growth parameters gven n 4.2. Thus, the parameters are the same as those used n part (b) of Fgure 5, wth the only dfference beng the ncluson of weekend breaks. The optmal fractonaton scheme n ths case s shown n Fgure 7. Note that the dose fractons range from approxmately.9 Gy n the frst fracton to 3.5 Gy n the last fracton. The dose ncrease toward the end of treatment s larger than n Fgure 5 part (b), whch can probably be attrbuted to the lengthened overall treatment tme from weekend breaks. Note that î4x5 s slghtly decreasng durng the weekend nterval; ths.2.15.1.5.2.15.1.5 s because a growng tumor results n a smaller prolferaton rate. In ths numercal example, the decrease n î4x5 s small so that the dose fractons reman approxmately proportonal to î4x5. φ(x(t)) growth rate (days 1 ) φ(x(t)) growth rate (days 1 ) 5. Dscusson and Further Remarks 5.1. Nonunform Irradaton of the OAR Although we have assumed throughout the paper that a dose d results n a homogeneous dose Éd to the OAR, n realty the OAR receves nonunform rradaton. The tumor, however, s generally treated homogeneously. In Unkelbach et al. (213) and Keller et al. (213), the basc result stated n Theorem 1 s generalzed to arbtrary nhomogeneous doses n the OAR. The arguments n Unkelbach et al. (213) are also applcable to the case of repopulaton as consdered n ths paper. We can defne an effectve sparng factor É eff and an effectve upper lmt c eff on the BED O, and the results n ths paper wll stll hold. The results dffer for the case of parallel OAR and seral OAR. A parallel organ could reman functonal even wth damaged parts; a seral OAR, n contrast, remans functonal only when all of ts parts reman functonal. For the case of a parallel OAR (e.g., lung), assumng that É d represents the dose n the th voxel
INFORMS Journal on Computng 27(4), pp. 788 83, 215 INFORMS 799 Downloaded from nforms.org by [18.9.61.111] on 14 October 216, at 21:28. For personal use only, all rghts reserved. (a) Dose (Gy) (b) Dose (Gy) 6 5 4 3 2 1 1 6 5 4 3 2 1 1 Treatment day Treatment day 2 3 2 3 Fgure 6 (Color onlne) Optmal Fractonaton for Accelerated Repopulaton n the Case of 6Å/Ç7 T = 57 Gy Notes. Plot (a) shows the optmal fractonaton schedule for a slowly prolferatng tumor and plot (b) for a fast one. The doublng tme for the reference treatment begns at í d = 5 days and decreases approxmately to (a) 2 and (b) 5 days, respectvely. Dose (Gy) 4 3 2 1 1 2 Treatment day 3 4 Fgure 7 (Color onlne) Effect of Weekend Breaks on Optmal Fractonaton Note. We use 3 treatment sessons and 6Å/Ç7 T = 1 Gy. The doublng tme for the reference treatment begns at í d = 5 days and decreases approxmately to 5 days. (or spatal pont) n the OAR, the ntegral BED n the OAR s gven by BED O = NX É1 k= X É d k 1 + É d k 6Å/Ç7 O After some algebrac manpulatons, we obtan the same form for the normal tssue constrant as n ths paper: NX É1 k= É eff d k 1 + É effd k = c 6Å/Ç7 eff 1 O.2.15.1.5.2.15.1.5.2.15.1.5 where É eff = P É 2/ P É and c eff = cé eff / P É. For the seral case (e.g., spnal cord), only the maxmum dose to the OAR matters, resultng n É eff = max É and c eff = c. Addtonal detals can be found n Unkelbach et al. (213). When the OAR s nether completely parallel nor completely seral, a good approxmaton of the BED may, e.g., be a weghted combnaton of the BED for the parallel and seral cases. 5.2. Evoluton of Instantaneous Prolferaton Rate In ths work, we modeled accelerated repopulaton as Gompertzan growth. For the numercal examples φ(x(t)) growth rate (days 1 ) φ(x(t)) growth rate (days 1 ) φ(x(t)) growth rate (days 1 )