2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011

Transcription

1 011 Amercan Control Conference on O'Farrell Street, San Francsco, CA, USA June 9 - July 01, 011 Modelng the Effect of emozolomde reatment on Orthotopc Models of Gloma Francsco G. Vtal-Lopez, Costas D. Maranas, and Antonos Armaou, Member, IEEE Abstract We present a hybrd cellular-tumor level model of bran tumor progresson. he model descrbes tumor progresson as the collectve outcome of ndvdual tumor cells, the behavor of whch s governed by the nterplay of ntracellular sgnalng pathways (.e., MAPK pathway) and the spatal-temporal dstrbuton of key bochemcal cues (e.g., oxygen, growth factors). he model s deployed to smulate the effect of dfferent schedule-dose combnatons of a chemotherapeutc agent (.e., temozolomde) on tumor growth n murne orthotopc models of gloma. Smulaton results are n good quanttatve agreement wth expermental measurements. In addton, the model s used to predct the outcome of alternatve treatment strateges. Model smulatons can be helpful for desgnng more effcent treatment strateges. P I. INRODUCION rmary bran tumors are a vared group of ntracranal neoplasms orgnatng from dfferent tssues of the central nervous system wth dfferent degrees of malgnancy. he most common of these tumors are the glomas, whch account for nearly 50% of all cases [1]. A type of gloma, known as globlastoma multforme or just globlastoma, s the most frequent and lethal of the bran tumors [] and t s estmated that represents approxmately 80% of the malgnant bran tumors [3]. Globlastomas are characterzed by a hgh rate of uncontrolled prolferaton. In general they exhbt necrotc regons, marked angogeness, asymmetrcal nfltratng nvasveness and they are hghly refractory to rado/chemotherapy. Current globlastoma treatments nclude supportve care to allevate symptoms of the dsease (e.g., cerebral edema, sezures, cogntve dysfunctons, etc.) and local and/or systemc therapes to ablate the tumor. Ant-tumor therapes tradtonally nvolve surgcal resecton followed by radotherapy and chemotherapy. However, almost all globlastoma patents relapse after ntal therapy and the medan overall survval s about 15 months, only modestly mprovng over the last 5 years [4]. emozolomde (MZ) s a cytotoxc drug approved by the FDA for the treatment of anaplastc astrocytoma and globlastoma [5]. Clncal trals documented a statstcally Manuscrpt receved September 7, 010. hs work was supported n part by the Penn State Insttute for CyberScence Seed Fundng and the Natonal Scence Foundaton (NSF-CAREER award #CBE ). A. Armaou s wth he Pennsylvana State Unversty, Unversty Park, PA 1680 USA (correspondng author; phone: ; fax: ; e-mal: armaou@ engr.psu.edu). F. G. Vtal-Lopez s wth he Pennsylvana State Unversty, Unversty Park, PA 1680 USA (e-mal: fvtallopez@engr.psu.edu). C. D. Maranas s wth he Pennsylvana State Unversty, Unversty Park, PA 1680 USA (e-mal: costas@engr.psu.edu). sgnfcant survval beneft of the use of temozolomde n combnaton wth radotherapy for the treatment of newly dagnosed globlastomas [6]. However, the emergence of varous resstant mechansms have lmted the effcacy n the management of hgh-grade glomas. hs has motvated a number of studes to ncrease the effcacy of the treatment. hese studes reported that the effcacy of MZ s hghly schedule-dose dependent [7]. A major factor n treatment falure s the dffuse nfltraton of hghly nvasve tumor cells nto the surroundng tssue from the early stages of tumor development, generally resultng n recurrence just a few months after surgery [4, 8]. hs has strred consderable efforts to elucdate the underlyng mechansms of the pervascular mgraton of cancer cells at the molecular [9, 10], cellular [11, 1] and tumor [13] levels. hese studes have provded mportant nsghts about the tumor cell nvason process. However, the decrypton of tumor nvasveness s stll ongong and t requres that tumor cell mgraton be nvestgated n concert wth other bologcal processes such as cellular prolferaton, necross, host vessel co-opton and angogeness, and external factors such as treatments. Mathematcal modelng of tumor progresson has been an actve area over the last years. A large number of models have been publshed wth focus on vared aspects of the tumor progresson and wth dfferent levels of detal. Some models consder only tumor growth whle others ncorporate other processes such as angogeness or the effect of a therapeutc agent. From the mathematcal pont of vew, models can be classfed accordng to the modelng approach they deploy. wo major categores are contnuum based and dscrete based models whereas ther combnaton gves rse to an mportant thrd category of hybrd contnuum-dscrete models. A comprehensve revew of the abundant lterature n ths feld s out of the scope of ths work. he reader may refer to revews focused on modelng of tumor growth [14-16], contnuum based models [17], dscrete based models [18], tumor-nduced angogeness[19-], tumor therapy [3] and bran tumors [4-8] for more nformaton. In ths work, we develop a hybrd mult-scale agent based model to smulate the progresson of a bran tumor (.e., globlastoma). We descrbe tumor progresson as the outcome of the evoluton n space and tme of a collecton of tumor cells that dynamcally nteract wth ther envronment. he model ntegrates the dynamcs of key bologcal processes occurrng at the cellular and tumor levels. We deploy the model to nvestgate the effect of temozolomde /11/$ AACC 969

2 (a chemotherapeutc agent) dosng and schedulng n the progresson of the tumor. Frst we valdate the model aganst data from experments wth orthotopc models of gloma [7] and then we use the model to predct the outcome of alternatve therapeutc strateges. II. MODEL DESCRIPION he model conssts of two nterdependent components, whch descrbe processes at the cellular and tumor levels, shown pctorally n Fg. 1. At the cellular level, the state of ndvdual tumor cells s governed by a set of rules dependng on ther local envronment (.e., concentratons of oxygen, VEGF, and GFα) and ntracellular sgnalng pathways (.e., MAPK sgnalng pathway). he tumor level component determnes the spato-temporal dstrbuton of the key bochemcal cues. he two components are connected through the nterchange of nformaton requred to solve the whole model. Specfcally, the local concentraton of bochemcal cues for every tumor cell s obtaned from the soluton of the tumor level component whereas the producton and consumpton terms for the tumor level component are determned by cellular level component. A. umor level model component he tumor level model captures the spato-temporal dstrbuton of extracellular speces and tumor cells wthn the smulaton doman. he profles of the chemcal speces are descrbed by a set of PDEs. umor cells are treated as dscrete enttes (.e., agents). We follow a lattce-free approach to determne the locaton of the tumor cells. he smulaton doman (Ω) s a cubc regon of the whte matter of dmenson mm 3, whch s chosen large enough to mnmze the effect of the boundary condtons on chemcal speces concentratons. Durng the smulatons, we record the spato-temporal dstrbuton of oxygen, GFα, VEGF, and temozolomde as well as the state of every tumor cell. he state of each tumor cell s defned by ts phenotype, locaton, cellular mass and the actvaton level of ts MAPK pathway (.e., phosphorylaton level of ERK (ERK act )). he concentraton of extracellular speces s consdered to be contnuous felds descrbed by a set of PDEs: D C + K C Ω, ( z)( C, v wth boundary condtons: C ) + S ( z, C ntra ) k ( z) C = 0, n ( D C ) = 0, C Γ, () where for speces, C s ts extracellular concentraton, D s ts dffuson coeffcent, K s ts supply rate from the blood vessels, and k s ts consumpton rate constant (assumng frst order process for all the speces). he source term S ( ) refers to the producton of GFα and VEGF by tumor cells and depends on the actvaton level of the (1) MAPK pathway and the metabolc state of the tumor cells at the locaton z. Ω s defned as the computatonal doman of the PDEs and Γ s the boundary of Ω whle n s the normal vector to Γ. No-flux boundary condtons are assumed. he parameters of the tumor level model are collected from the open lterature when avalable or estmated to approxmate reported levels n the bran of the chemcal speces consdered. he locaton of tumor cells s determned by solvng the followng optmzaton problem: mn z = w1 d w ( rc + rv dk ) 3 k d k + w j> ( r d ) j u( r + r r ), c v c d j k u ( r d ) where d, d j, and d k are the dstance between tumor cell and ts target poston, tumor cell j and blood vessel k, respectvely. r c and r v are the nomnal radus of tumor cells and blood vessels, respectvely. u ( ) s the step functon and w' s are constant weghts. he target poston for quescent and prolferatng cells s ther current postons whereas the target poston for the mgratng cells s determned by ther chemotaxs response and s computed as: x = x 0 + a d, (4) umor level component Determne of extracellular speces dstrbutons Source VEGF Vasculature remodelng: Vessel occluson Angogeness Determne tumor cell locaton Cellular level component Mgratng drecton Determne cell phenotype Consumpton/producton rates of extracellular speces Fg. 1. Components of the model of bran tumor progresson. 0 where x s the current poston of mgratng tumor cell and d s the drecton (unt) vector and a mgratng dstance. he drecton vector s computed from a weghted sum of the oxygen and VEGF (as a surrogate of a chemorepellent) gradents. he mgratng dstance s derved from the ndvdual tumor cell velocty, whch s sampled from a normal dstrbuton n accordance wth expermental observatons [9]. B. Cellular level model component At the cellular level, tumor cell phenotype and mgratory c j + (3) 970

3 behavor s determned by ts local envronment. umor cells requre a mnmum level of nutrents to thrve whereas the transducton of sgnalng cues regulates ther phenotype (.e., mgratory or prolferatve). It has been observed that the growth factor-nduced phosphorylaton of a downstream component of the MAPK sgnalng pathway (.e., ERK) correlates wth the mgratory and prolferatve behavor of tumor cells [30]. he MAPK sgnalng pathway can be trggered by several dfferent growth factors, ncludng GFα. he cellular level model conssts of a set of rules governng the behavor of tumor cells (Fg. ). In bref, tumor cell phenotype depends on the actvaton level of the MAPK pathway and the avalablty of nutrents, whereas the mgraton drecton depends on the response of mgratng cells to chemcal gradents. Furthermore, we assume that the tumor cells do not sense the chemotactc gradents wth 100 % certanty. hus we determne the mgratng drecton randomly from the half-space defned by the exact mgratng drecton and the perpendcular plane passng through the cell poston as: d = d + p, (5) e e g, o b g, v d = +, (6) e where d and d are the exact and actual mgratng drectons, p s a random perturbaton, g, o and g, v are the gradents of oxygen and VEGF at the locaton of mgratng tumor cell and b s a constant. C. Effect of temozolomde on tumor cells We use the data from the n vtro experments of the effect of MZ concentraton on the prolferaton of C6/lacZ cells [7] to estmate the kllng rate as a functon of the MZ concentraton. o determne the relaton between MZ concentraton and the kllng rate, we assume that the n vtro growth of both cell lnes s descrbed by the followng system of ODEs [31]: dp = k1q k p β ( C dt dq = k p k1q, dt MZ ) p, where p and q are the number of prolferatng and quescent cells, respectvely. k 1, k and β ( C MZ ) are the transton rate from quescent to prolferatve, the mtoss rate and the kllng rate as a functon of the MZ concentraton, respectvely. We use ths two populaton model nstead of the total populaton model to take nto account that MZ only affects prolferatng cells. he parameters k1 and k where estmated [31] by fttng the model to expermental data [3] assumng a constant rato of the prolferatve to quescent cells [33]. β ( C MZ ) s assumed to be a sgmodal functon (Gompertz functon). he model (Eq. 7) was ftted to the data of the effect of MZ on tumor cell prolferaton to estmate the parameters of β C ). III. SIMULAION RESULS ( MZ Km et al., [7] evaluated the anttumor effect of dfferent schedule-dose combnatons of MZ through n vvo orthotopc rat models of gloma. C6/lacZ tumor cells, a tumor rat-derved cell lne that shows nfltratve growth n the bran (attc.org), were njected n to the whte matter of Sprague-Dawly rats. he mplanted tumors were treated wth dfferent schedule-dose combnatons and the volume of the tumors was reported. We use the model to smulate the effect of MZ on the progresson of orthotopc models of gloma. Subsequently, we deploy the model to predct the outcome of alternatve treatment strateges. A. MZ treatments of C6/lacZ tumors We smulate the effect of three dfferent schedule-dose combnatons on the growth of C6/lacZ tumors. he schedule-dose combnatons are gven n able I. Fg. 3 shows the expermental and predcted tumor volume at day 17 after mplantaton. As can be seen, the model predctons are n good quanttatve agreement wth the expermental data. Most mportantly, the model s able to capture the relatve effcacy of the dfferent treatment modaltes. Moreover, the smulatons provde nsghts regardng the progresson of the tumors. Fg. 4 shows the tme evoluton of the treated tumors. At the early stages, the three treatments show a sgnfcant (7) ABLE I SIMULAED MZ REAMENS reatment Schedule a MZ dose (mg/kg) Fg.. Phenotype transtons tumor cells. Dce ndcate stochastc processes. A0 Control A1 1:16 1 A 1:16 A3 7:11 7 a Intal and last day of consecutve treatment snce tumor mplantaton. 971

4 ABLE II SIMULAED MZ REAMENS reatment Schedule a MZ dose (mg/kg) C1 1:5 7 C 1:16 7 C3 3, 6, 9, 1, 15 7 C4 b 7:11 7 C5 c 7: a Intal and last day of consecutve treatment snce tumor mplantaton. b Smulaton wth no-angogeness. c wo doses daly. Fg. 3. Expermental and predcted volume of C6/lacZ tumors treated wth MZ. Expermental values represent the mean and the standard error [7]. A0 to A3 are the dfferent treatments. See able I for specfcs. retardaton of tumor growth. However, the rato of the volume of treated tumors to the untreated one eventually stablzes for treatments A1 and A. hs result has a noteworthy mplcaton f ths trend would contnuo for longer tme. A smplfed analyss, based on the observaton that the smulated tumors approxmately follow an exponental growth, a constant rato of the volume of the treated to the untreated tumor, mples the followng relatonshp between the growth rate constants: μ = μ + 1 MZ 0 ln a t, (8) where μ MZ and μ 0 are the growth rate constants of the treated and untreated tumors and a s the constant volume rato at later tmes. hs relaton (Eq. 8) suggests that as tme ncreases the growth rate of the treated tumors approaches the growth rate of the untreated tumor regardless of the contnuous admnstraton of the drug at these concentratons. B. Smulaton of alternatve treatments of MZ We deploy the model to smulate the outcome of alternatve MZ treatment strateges on C6/lacZ tumors. We test dfferent schedule-dose combnatons usng the same amount of MZ as treatment A3 and compare the tumor volume at day 17 from mplantaton. he test treatment strateges are gven n able II. he tme progressons of the tumors under the dfferent treatments are shown n Fg. 5. Of all the test treatments, C1 and C5 show a consderable mprovement compared to treatment A3. Clearly, the man advantage of treatment C1 over A3 s the early admnstraton of the drug. At early stages, tumor cells are exposed to hgher concentratons of MZ because the attracton to the vessels domnates the response to chemorepellent(s). Moreover, a larger fracton of the tumor cells are prolferatng and therefore the tumor s more susceptble to MZ. he effect of treatment C1 s a delay on the growth of the tumor. By day 17, the tumor under treatment C1 (Fg. 6) s very smlar to the untreated tumor at day 1 (not shown). reatments C and C3 produced tumors of smlar volume than treatment A3. However, the morphology of the tumor under treatment C shows a dfferent aspect (Fg. 6). he tumor has a decreased cellular densty across the tumor rm expect at the nner sde whch s formed manly by quescent cells. However, the tumor probably would become smlar to the tumor under A3 after the suspenson of treatment. reatment C4 corresponds to the same schedule-dose than treatment A3 assumng that angogeness s completely blocked. Interestngly, blockng angogeness results n the Fg. 4. me progresson of the C6/lacZ tumors treated wth MZ. he tumor volume s normalzed wth respect to the untreated tumor. See able I for the specfcs of treatments A1, A and A3. Fg. 5. me progresson of the C6/lacZ tumors treated wth MZ. he tumor volume s normalzed wth respect to the untreated tumor. See able I for the specfcs of treatment A3 and able II for treatments C1 to C5. 97

5 largest volume of the test treatments. hs s a consequence of the more marked chemotactc gradents pontng away from the tumor core, promotng the nvasveness of the tumor. In smulatons where angogeness s allowed, newly Fg. 6. Cross sectons of C6/lacZ tumors under MZ alternatve treatments. Quescent cells, mgratng cells, pre-exstng vessels and newly formed vessels are colored n blue, green, magenta and red respectvely. Gray ndcates the locaton where necross has occurred. 973

6 formed vessels attenuate the gradents of oxygen pontng outward the core of the tumor, retardng the nvasve tumor cells. IV. DISCUSSION Complementng standard procedures (.e., surgery followed by radotherapy) wth MZ treatment resulted n a statstcally sgnfcant therapeutc beneft (an ncrease n medan survval by.5 months) [6]. However, the overall outcome s stll dsmal. he lmted effect of MZ n tumor progresson has been attrbuted n part to the development of tumor resstance to MZ [7]. It s beleved that several mechansms may be responsble for tumor resstance to MZ. One of the most accepted hypothess proposes that the resstance s assocated wth ncreased levels of the enzyme O-6-methylguann-DNA methyltransferase (MGM), whch repars the DNA lesons nduced by MZ. Patents that had an ncreased level of nactvated MGM had an mprovement on the medan survval of about 6 months [5]. Our smulaton results suggest that the tumors may become resstant to MZ by a dfferent mechansm. In our smulatons of treatments of daly admnstraton of MZ, we observed that the rato of the volume of the treated tumors to the untreated tumor stablzes around a constant value dependng on the MZ dose. hs mples that the growth rate (n terms of the volume of the tumor) of the treated tumor approaches the growth rate of the untreated tumor as tme ncreases despte the persstent admnstraton of MZ. he mechansm of the resstance observed n our smulatons depends on the response of the tumor cells to the harsh envronment created by the collapse of the vasculature. he collapse of the vasculature s accompaned by an ncrease of tumor cells dwellng under hypoxa. hese cells secrete a chemorepellent(s) that sgnals the tumor cells to flee from the hypoxc regons acceleratng the expanson of the tumor. he balance of the promotng and nhbtng factors of tumor nvason resulted n the observed progresson profles n the smulatons. he rate at whch the growth rate of the treated tumors approaches the rate of the untreated one depends on the specfc actvty of MZ aganst tumor cells. hs specfc actvty can be ncreased by treatments targetng MGM. Accordng to these smulatons, ths would dampen the acceleraton of the treated tumor but nevertheless the tumors would exhbt resstance to MZ. In attempts to ncrease the effcacy of MZ, dfferent treatment strateges are beng explored. hese studes are n general lengthy and resource consumng. herefore, t s desrable to maxmze the generaton of nsghts from the results n order to desgn new studes. Mathematcal modelng can be a powerful tool to mprove the understandng of the complex nteracton between the multple factors affectng the outcome of the studes. Here we demonstrated that the proposed model of tumor progresson s able to smulate the effect of dfferent treatment strateges n orthotopc models of gloma tumors. Analyss of the smulatons shed lght on the possble mechansms responsble for the observed expermental results and can help to desgn alternatve treatment strateges and estmate the lmtatons of the treatment. REFERENCES [1] M. Wrensch, Y. Mnn,. Chew, M. Bondy, and M. S. Berger, Neuro Oncol, vol. 4, pp , 00. [] A. Behn, K. Hoang-Xuan, A. F. Carpenter, and J. Y. Delattre, Lancet, vol. 361, pp , 003. [3] L. M. DeAngels, N Engl J Med, vol. 344, pp , 001. [4] A. D. Norden and P. Y. Wen, Neurologst, vol. 1, pp. 79-9, 006. [5] J. L. Vllano,. E. Seery, and L. R. Bressler, Cancer Chemother Pharmacol, vol. 64, pp , 009. [6] R. Stupp, et al., New England Journal of Medcne, vol. 35, pp , 005. [7] J.. Km, et al., Oncol Rep, vol. 16, pp. 33-9, 006. [8] F. B. Furnar, et al., Genes & Development, vol. 1, pp , 007. [9] M. Nakada, J. A. Nska, N. L. ran, W. S. McDonough, and M. E. Berens, Am J Pathol, vol. 167, pp , 005. [10] B. Salha, N. L. ran, M. Symons, J. A. Wnkles, J.. Rutka, and M. E. Berens, Expert Rev Mol Dagn, vol. 6, pp , 006. [11] D. B. Hoelznger,. Demuth, and M. E. Berens, Journal of the Natonal Cancer Insttute, vol. 99, pp , 007. [1] C. M. Park, et al., Cancer Research, vol. 66, pp , 006. [13] H. B. Freboes, X. Zheng, C. H. Sun, B. romberg, R. Gatenby, and V. Crstn, Cancer Research, vol. 66, pp , 006. [14] R. P. Araujo and D. L. McElwan, Bull Math Bol, vol. 66, pp , 004. [15] N. Bellomo, N. K. L, and P. K. Man, Mathematcal Models and Methods n Appled Scences, vol. 18, pp , 008. [16] H. M. Byrne, Nature Revews Cancer, vol. 10, pp. 1-30, 010. [17] J. S. Lowengrub, et al., Nonlnearty, vol. 3, pp. R1-R91, 010. [18] K. A. Rejnak and L. J. McCawley, Expermental Bology and Medcne, vol. 35, pp , 010. [19] A. R. A. Anderson and M. A. J. Chaplan, Bulletn of Mathematcal Bology, vol. 60, pp , [0] N. V. Mantzars, S. Webb, and H. G. Othmer, Journal of Mathematcal Bology, vol. 49, pp , 004. [1] A. A. Qutub, G. Lu, P. Vempat, and A. S. Popel, Pac Symp Bocomput, pp , 009. [] M. A. Chaplan, S. R. McDougall, and A. R. Anderson, Annu Rev Bomed Eng, vol. 8, pp , 006. [3] A. Swernak, M. Kmmel, and J. Smeja, European Journal of Pharmacology, vol. 65, pp , 009. [4]. S. Desboeck, L. Zhang, J. Yoon, and J. Costa, Nature Clncal Practce Oncology, vol. 6, pp. 34-4, 009. [5] H. L. P. Harpold, E. C. Alvord, and K. R. Swanson, Journal of Neuropathology and Expermental Neurology, vol. 66, pp. 1-9, 007. [6] H. Hatzkrou, A. Deutsch, C. Schaller, M. Smon, and K. Swanson, Mathematcal Models & Methods n Appled Scences, vol. 15, pp , 005. [7] K. R. Swanson, C. Brdge, J. D. Murray, and E. C. Alvord, Jr., J Neurol Sc, vol. 16, pp. 1-10, 003. [8] P. racqu, Reports on Progress n Physcs, vol. 7, 009. [9]. Demuth, et al., Clncal & Expermental Metastass, vol. 18, pp , 001. [30] C. R. Lnd, et al., Neuroscence, vol. 141, pp , 006. [31]. L. Ssknd, "Modelng the effect of temozolomde on bran tumor growth.," n Departments of Chemcal Engneerng., vol. Baccalaureate degree. State College, PA Pennsylavana State Unversty, 010. [3] X. X. an, Y. G. Zhang, J. Du, W. G. Fang, H. K. Ng, and J. Zheng, Neuropathology, vol. 6, pp , 006. [33] H. C. Ugur, et al., J Neurooncol, vol. 83, pp ,