Generalized electro-biothermo-fluidic and dynamical modeling of cancer growth: state-feedback controlled cesium therapy approach

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1 J. Bomdcal Scnc and Engnrng, 0, 4, JBSE do:0.436/jbs Publsd Onln Sptmbr 0 (ttp:// Gnralzd lctro-botrmo-fludc and dynamcal modlng of cancr growt: stat-fdback controlld csum trapy approac Murad l-sbl Collg Rqurmnt Unt, bu Dab Polytcnc, Insttut of ppld Tcnology, bu Dab, Untd rab Emrats. Emal: murad.alsbl@at.ac.a Rcvd Dcmbr 00; rvsd 3 Fbruary 0; accptd 5 July 0. BSTRCT Ts papr dvlops a gnralzd dynamcal modl to dscrb t ntractv dynamcs btwn normal clls, tumor clls, mmun clls, drug trapy, lctromagntc fld of t uman clls, xtracllular at and flud transfr, and ntrcllular fractonal mass of Oxygn, cll acdty and Pancratn nzym. T ovrall dynamcs stablty, controllablty and obsrvablty av bn nvstgatd. Morovr, Csum trapy s consdrd as a control nput to t -dmnsonal dynamcs usng stat-fdback controlld systm and pol placmnt tcnqu. Ts approac s found to b ffctv n drvng t dsrd rat of tumor cll kll and convrgng t systm to alty qulbrum stat. Furtrmor, t rangs of t systm dynamcs paramtrs wc lad to nstablty and growt of tumor clls av bn dntfd. Fnally, smulaton rsults ar dmonstratd to vrfy t ffctvnss of t appld approac wc can b mplmntd succssfully to cancr patnts. Kywords: Cancr; Tumor Growt; Tumor Dynamcs and Modlng; Immun Systm; Csum Trapy; Stat-Fdback Control; Pol Placmnt. INTRODUCTION Tr ar ovr 00 dffrnt typs of cancr tat affct vrtually vry organ n t body. Ty can sm bwldrngly dffrnt but all cancrs sar crtan faturs as outlnd by Douglas anaan and Robrt Wnbrg []. Sx ssntal altratons n cll pysology tat collctvly dctat malgnant growt: slf-suffcncy n growt sgnals, nsnstvty to antgrowt sgnals, vason of programmd cll dat (apoptoss), lmtlss rplcatv potntal, sustand angognss, and tssu nvason and mtastas. T tcncal rport [] prsnts wat s known today concrnng non-onzng lctromagntc fld ntracton wt t uman body. Many sgnfcant and ntrstng ffcts ar dntfd. s an xampl, bot tors and obsrvatons lnk non-onzng lctromagntc fld to cancr n umans, n at last tr dffrnt ways: as a caus, as a mans of dtcton and as an ffctv tratmnt. pas-spac analyss of a matmatcal modl of tumor growt wt an mmun rspons and cmotrapy s ntroducd n [3]. It s provd tat all orbts and trajctors ar boundd and convrg to on of svral possbl qulbrum ponts. T addton of a drug to t systm can mov t soluton trajctory nto a dsrabl basn of attracton. Stat Dpndnt Rccat Equaton (SDRE) basd optmal control tcnqu to a nonlnar tumor growt modl s appld n [4]. T modl conssts of tr bologcal clls wc ar normal tssu, tumor and mmun clls. T ffct of cmotrapy tratmnt s also ncludd n t modl. Cmotrapy admnstraton s consdrd as a control nput to t nonlnar cancr dynamcs and t amount of admnstrd drug s dtrmnd by usng SDRE optmal control. T optmal control s appld to t modl n ordr not only to drv t tumor clls to t alty qulbrum stat but also to mnmz t amount of t drug usd. basc matmatcal modl of t mmun rspons wn cancr clls ar rcognzd s proposd. T modl conssts of sx ordnary dffrntal quatons [5]. It s xtndd by takng nto account two typs of mmunotrapy: actv mmunotrapy and adoptv mmunotrapy. n analyss of t corrspondng modls s mad to answr t quston wc of t prsntd mtods of mmunotrapy s bttr. Rvw [6] xplans wy matmatcs s a powrful tool for ntrprtng suc data by prsntng cas studs tat llustrat t typs of nsgt tat ralstc tortcal Publsd Onln Sptmbr 0 n ScRs. ttp://

2 570 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) modls of sold tumour growt may yld. Ts rang from dscrmnatng btwn comptng ypotss for t formaton of collagnous capsuls assocatd wt bngn tumours to prdctng t most lkly stmulus for protas producton n arly brast cancr. Undrstandng t dynamcs of uman osts and tumors s of crtcal mportanc [7]. matmatcal modl was dvlopd tat xplord t mmun rspons to tumors tat was usd to study a spcal typ of tratmnt. Ts tratmnt approac uss lmnts of t ost to boost ts mmun rspons n t ops tat t ost can clar t tumor. Two modls of optmzng tumor growt ar prsntd n [8] In t frst modl optmal controls mnmz t tumor volum for a gvn amount of angognc nbtors to b admnstrd wl t scond formulaton trs to acv a balanc btwn tumor rducton and total amount of angognc nbtors gvn. For bot modls a full syntss of optmal solutons dtrmnd by portons of bang and sngular controls wt rst prods s prsntd. T dffrncs n t two solutons ar dscussd. nw matmatcal modl s dvlopd for t dynamcs btwn tumor clls, normal clls, mmun clls, cmotrapy drug concntraton and drug toxcty [9]. Tn, t torm of Lyapunov stablty s appld to dsgn tratmnt stratgs for drug protocols tat nsur a dsrd rat of tumor cll kll and pus t systm to t ara wt smallr tumor clls. Clls, wtr cancrous or normal can only lv and rproduc (undrgo mtoss) n a p rang of btwn 6.5 and 7.5. alty cll as a p of 7.35 wl a cancr cll s mor acdc. Wn t p of a cancr cll gos abov 7.5 t ds and f t gos abov 8.0 t wll d n a mattr of ours. Evry cll n t body works as mll-volt battry. To succssfully brng noursmnt n, and tak posons out, t as to b fully cargd. In a cancrous cll, t cll voltag drops from 90 mllvolts to lss tan 40 mllvolts. Wn t cll voltag gts to t vry bottom, only 5 substancs can pass n or out of t cll. Ty ar watr, sugar, potassum, csum and rubdum. Oxygn cannot ntr nto a cancr cll. Evn f tr s a lot of oxygn n t blood, t won t gt nto t cll. Potassum ons ar rsponsbl for t ablty of glucos to ntr t cll. Potassum ntrs cancr clls n a normal mannr so glucos stll ntrs t cancr cll. Cancr clls av only % of t calcum contnt found n normal alty clls. T calcum, magnsum and sodum ons, wc ar rsponsbl for t ntak of oxygn nto t cll, cannot ntr t cancr cll but t potassum on stll ntrs ts clls. Tus w av cancr clls contanng glucos but no oxygn. alty ndvdual as a blood oxygn lvl of btwn 98 and 00 as masurd by a puls oxmtr. Nobl Prz Laurat, Dr. Otto Warburg, dscovrd tat wn lowrd t oxygn lvls of tssus by 35 % for 48 ours normal clls wr convrtd nto rrvrsbl cancr clls [0]. Cancr patnts av low lvls of oxygn n tr blood usually around 60 compard to normal valus of about 00 by puls oxmtry. T common traps usd to trat cancr (cmotrapy and radaton) bot caus drastc falls n t body's oxygn lvls. Tssus tat ar acdotc contan low lvls of oxygn wras tssus tat ar alkalotc av g lvls of oxygn. Wn oxygn fals to ntr t cll t cll's ablty to control, ts p s lost and t cll bcoms qut acdc. Ts s causd by t apparanc of abnormal mtabolsm (anrobc glycolyss) n wc glucos s convrtd (frmntaton) nto two partcls of lactc acd. Ts producton of lactc acd promptly lowrs t p wtn t cll to 6.5 or lowr. T lactc acd damags t tmplat for propr DN formaton. Mssngr RN s also cangd so t ablty of t cll to control ts growt s lackng. Rapd and uncontrolld cancr cll growt and dvson occurs. Vtamn C and znc ar abl to nanc t uptak of csum, rubdum, and potassum nto cancr clls. Cancr clls dvlop a protn coatng 3 tms tckr tan normal clls. Ts maks t dffcult for t mmun systm to attack tm. By ngstng g doss of pancratn, you can actually dssolv cancr clls nsd t body []. In t natural cours of on s lftm, mllons of cancr clls dvlop, and ar armlssly dgstd by t mmun systm. T body uss pancratn, a scrton from t pancras to dssolv t cancr clls. s w ag, t pancras s lss and lss abl to mak ts mportant substanc. By takng pancratn orally, t s possbl to ncras t lvls of ts actv ngrdnts n t blood, trby lpng t body brak down t cancr clls and rmov tm from crculaton. T actv ngrdnts n pancratn wc av sown to av tumor dssolvng ablts ar trypsn and cymotrypsn. Ts ngrdnts wr takn out of vrtually all t pancratn supplmnts avalabl to consumrs yars ago. Ts actv ngrdnts ar bng bougt n massv quantts by t swrag ndustrs to dgst t swrag nto lss noxous forms. ltoug many mprovmnts and matmatcal modlng av bn ntroducd n t tratmnt of cancr, but majorty as bn lmtd to of modlng of normal clls, tumor clls, mmun clls and cancr trapy and toxcty ffcts. Morovr, dvlopmnt of tratmnt stratgs rqurs many clncal xprmnts frst on anmals and tn on umans n ordr to fgur Copyrgt 0 ScRs. JBSE

3 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) out a convnnt way for t admnstraton of t trapy. Ts xprmnts, n gnral, tak a long tm and most mportantly may rsult n many dats durng t dvlopmnt prod. Clncal xprmnts also rval t fact tat tr s a strong rlatonsp btwn t cancr stat (t numbr and/or volum of tumor clls, tumor typ), t mmun systm of t patnt and t tratmnt stratgy. nc, undrstandng t dynamcal bavor of cancr as rcvd a grat ntrst. Ts papr dvlops a gnralzd dynamcal modl to dscrb t ntractv dynamcs btwn normal clls, tumor clls, mmun clls, drug trapy, lctromagntc fld of t uman cll, xtracllular at and flud transfr, and ntrcllular fractonal mass of oxygn, cll acdty and pancratn. T ovrall dynamcs stablty and controllablty as bn nvstgatd. Morovr, Csum trapy s consdrd as a control nput to t -dmnsonal dynamcs usng stat-fdback controlld systm and pol placmnt tcnqu. Ts approac s found to b ffctv n drvng t dsrd rat of tumor cll kll and convrgng t systm to alty qulbrum stat. Furtrmor, t rangs of t systm dynamcs paramtrs wc lad to nstablty and growt of tumor clls av bn dntfd. Fnally, smulaton rsults ar dmonstratd to vrfy t ffctvnss of t appld approac wc can b mplmntd for ac ndvdual cas. Ts papr s organzd as follows. Scton ntroducs modlng of t lctromagntc fld of a lv cll, cllular at and flud transfr ar prsntd n Sctons 3 and 4, rspctvly. Cllular factonal composton s outlnd n Scton 5, gnralzd cllular-tumor dynamcs s dtald n Scton 6. Controllablty and stat-fdback control dsgn of tumor dynamcs ar prsntd n Scton 7. Scton 8 dmonstrats smulaton wc followd fnally conclusons.. ELECTROMGNETIC CELLULR MODELING grat varty of tors av bn dvlopd to dscrb t lctro-magntc fld of t uman body. Som tors rgardd t uman body as a wol as sngl prolat sprod wt a sngl st of lctromagntc constants: prmttvty, prmablty, and conductvty []. In tat sns, t body s a smpl antnna or prob capabl o f ntrcptng a crtan amount of lctromagntc nrgy, wc s convrtd ntrly n to at. t t otr xtrm, t body may b rgardd as a collcton of countlss lctronc mcrocrcuts, ac on corrspondng to an ndvdual cll or partally. Elctromagntc nrgy somow fnds ts way to ndvdual mcrocrcuts and nfluncs t lctronc functons tr. Ts functons nclud varous communcaton and control procsss ssntal to lf and ts actvts. Efforts to undrstand ts av focusd attnton on t mcroscopc componnts of t tssus suc clls, mmbrans, fluds, molculs n soluton ratr tan t tssu takn as a wol. Som tors av bn dvlopd concrnng ndvdual clls. n quvalnt crcut as bn dvlopd as sown blow. Gvn an ncdnt currnt or currnt dnsty actually passng across t cll mmbran troug and troug t cll can b calculatd. If t currnt s suffcntly noug, dffrnt rsponss ar possbl. For xampl a currnt dnsty of m/cm s about t amount assocatd wt t acton potntal of nrv and muscl clls. Praps a pulsd lctromagntc fld could smulat ts acton potntals and confus t body by gnratng fals sgnals. Of t clls part, t mmbran probably as attractd most attnton at last wt rspct top lctromagntc ffcts. Many mmbran proprts av bn quantfd. Typcal tcknss 45 ng, typcal capactanc mcrofarad/cm, typcal lakag conductanc - 0 mos/cm, typcal rstng potntal 00 mllvolts, dlctrc constant 5, Elctrc fld Mllon V/m, Surfac carg dnsty C/cm = 6. carg/cm. Of ts praps t lctrc fld s t most rmarkabl on snc t s almost gratr tan any otr found n natur. For xampl t lctrc fld of t Eart at ts surfac s only about 00 V/m. So far t mmbran as bn tratd as a omognous substanc and caractrzd by a capactanc and nonlnar conductanc. In fact t mmbran s not omognous. T basc structur s a doubl layr of molculs calld lpds. Lpds ar ydropobc, tat s, ty rpl watr. It s t rpulson tat olds t mmbran togtr. Protns ar cans of mno-acds. Protns ar so mportant bcaus ty can conduct cargs wl lpds ar good nsulators. Tus protns ar t conductanc n t lctronc crcut of t mmbran. Elctromagntc fld can ntract wt protns ovr a wd rang orquncs btwn - 0 Mz. It dpnds on ts sz and mass protn can b modld as sngl dpol wc rotats n rspons to an oscllatng fld. Protns fgur prdomnantly n at last on of t tors of cancr advancd by Nobl Laurat lbrt Sznt-Gyorgy wc currntly bng rsarcd [,0]. ccordng to ts tory, protns conduct lctron out of t cll ntror. Oxygn molculs at t cll xtror accpt t lctrons and carry tm away. Ts fr lctrons ar t products of som cmcal procss nsd t cll tat nbts rproducton. If t lctrons ar not conductd away, tn t procss stops, and t cll dvds at uncontrollabl rat. Evntually, tr ar Copyrgt 0 ScRs. JBSE

4 57 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) noug clls to form a tumor wc caractrzd by poorly crculatd systm. So, lttl or no oxygn-carryng blood racs t clls. T quvalnt mpdanc (rsstanc) Z q of t quvalnt lctrc crcut of t uman cll can now b xprssd n t Laplac doman as: Z q R R C s R R R C s R C R C R C s w w w w wr R s t ntracllular mdum rsstanc, R s t xtracllular mdum rsstanc, C w s t capactanc of t cll wall, C s t capactanc of xtracllular mdum. Now t s possbl to xprss t voltag across t cll V() s usng Om s law suc tat () V() s Zq Is () () wr t Is () s t passng cllular currnt. T voltag-currnt tm cang can b formulatd by takng t Laplac nvrs of t formr quaton as follows (assumng zro ntal condtons) R R C V ( t) R C R C R C V ( t) V ( t) w w w R R C I( t) R I( t) w (3) T rgt and sd of t last Eq.3 rprsnts t nput sourc to t cll. Snc R rprsnts t xtracllular rsstanc tn t trm RI ( t) V( t) s t xtrnal voltag sourc to t cll and t trm R I( t) Vx ( t) t rat of cang of t xtrnal voltag sourc. T quaton now can b modfd to R R C V ( t) R C R C R C V ( t) V ( t) w w w R C V ( t) V ( t) u ( t) w x x (4) Fgur. Passag of lctrc currnt I n t uman cll and ts /quvalnt lctrc crcut. R s t xtracllular rsstanc, R s t ntracllular rsstanc, C w s t capactanc of t wall of t cll and C s t xtracllular capactanc. Eq.4 It rprsnts an nput-output rlatonsp of t lctrc bavor of t cll n a scond ordr ordnary lnar dffrntal quaton form. Nxt scton wll ntroduc t dynamcs of xtracllular at transfr as t s so mportant to mantan stady-stat at from and nto t uman clls. 3. EXTRCELLULR ET TRNSFER Trmal systm nvolvs transfr of at from on cll to anotr. It may b analyzd I trms of trmal rsstanc and capactanc. To smplfy t analyss t s assumd tat t cllular trmal systm can b rprsntd by a lumpd-paramtr modl, tat substancs tat ar caractrzd by rsstanc to at flow av nglgbl at capactanc, and tat substancs tat ar caractrzd by at capactanc av nglgbl rsstanc to at flow. Conducton and convcton at flow s only consdrd. For conducton or convcton at transfr, qk (5) wr q s t at flow rat (kcal/sc), s t tmpratur dffrnc ( C) and K s t at coffcnt (kcal/sc. C). T coffcnt K s gvn by K k / x for conducton and K for convcton, wr k s trmal conductvty kcal/m.sc. C, s t ara normal to at flow m, x s t tcknss of t conductor (m) and s t convcton coffcnt ( kcal/m.sc. C). Trmal rsstanc for at transfr btwn two clls may b dfnd as R Cang n Tpmratur Dffrnc C Cang n at Flow Rat kcal / sc T trmal rsstanc for conducton or convcton at transfr s gvn by d( ) R (6) dq K T trmal capactanc C Cang n at Stord kcal Cang n Tmpratur C m c wr m s mass of t cll consdrd (kg), and c s t spcfc at of t cll kcal / kg. C. It s dsrd now to consdr t tmpratur cang of a uman cll suc tat flud nsd t cll s prfctly mxd suc tat t cll tmpratur s omognous. sngl tmpratur s consdrd t ntracllular tmpratur and t outflowng flud. T dffrntal quaton for t systm s now d C qn qout (7) dt wr Copyrgt 0 ScRs. JBSE

5 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) ( ) R (8) q out out Combng t last two Eqs.7-8 ylds R C R q (9) n out It now possbl now to xprss t gnratd cllular q ntropy S by S. Eq.9 sows tat t tmpratur of t cll s cangng as frst ordr dffrntal dynamcs gvn a at nput sourc and at losss to t surroundng. Nxt scton wll consdr ow watr flow n t cll cangs. 3. INTRCELLULR FLUID MODELING Consdr now t flow btwn two adjacnt clls troug tr walls. T flow rsstanc for lqud flow troug t cllular walls s dfnd as t cang n t lvl dffrnc ncssary to caus a unt cang n flow rat, tat s R f Cang n Lvl Dffrnc m 3 Cang n Flow Rat m / sc It s assumd tat t flow s lamnar. T rlatonsp btwn t stady-stat flow and stady-stat ad s dfnd as Q K (0) 3 wr Q s t stady-stat flow rat( m / sc ), K s flow coffcnt ( m / sc ), and s t stady-stat ad ( m ). T capactanc C of a tank s dfnd to b t cang n quantty of stord flud ncssary to caus a unt cang n t potntal ad. C f 3 Cang n Lqud Stord m Cang n ad m Snc t nflow mnus t outflow durng a small tm ntrval dt s qual to t addtonal amount stord n t cll, tn C d ( Q Q ) dt () f n out From t dfnton of t flow rsstanc, t rlatonsp btwn Q and s gvn by out Q out / R () T dffrntal quaton for t fludc systm now bcoms Rf C f Rf Qn (3) T formr quaton s a frst ordr dffrntal dynamcal systm sow ow t watr contnt lvl s cangng nsd t cll gvn a rgular watr nflow. Suc a dynamcs can b usful n modlng t fractonal mass of substancs nsd t uman cll. Oxygn, and ydrogn ons and Pancratn ar consdrd n ts papr for tr major rol n kpng alty clls and kllng t tumor clls or at last stoppng tr growt. 4. CELLULR COMPOSITION DYNMICS To dtrmn t proprts of a mxtur, w nd to know t composton of t mxtur as wll as t proprts of t ndvdual componnts. Tr ar two ways to dscrb t composton of a mxtur: tr by spcfyng t numbr of mols of ac componnt, calld molar analyss, or by spcfyng t mass of ac componnt, calld gravmtrc analyss. Consdr a flud mxtur composd of n componnts. T mass of t mxtur m T s t sum of t masss of t ndvdual componnts, and t mol numbr of t mxtur N T s t sum of t mol numbrs of t ndvdual componnts m T n m and N T n N (4) T rato of t mass of a componnt to t mass of t mxtur s calld t mass fracton x, and t rato of t mol numbr of a componnt to t mol numbr of t mxtur s calld t mol fracton y : x m and mt y m (5) mt W can asly sow tat t sum of t mass fractons or mol fractons for a mxtur s qual to n x and n y (6) It s assumd tat t ntrcllular mxtur s omognous suc tat t s possbl to xprss t rato of t mass of ac substanc wt rspct to t watr volum as follows (substanc-watr rato): m m v V (7) wr v s t substanc mass raton n a unt volum of watr xsts n t cll, mass m of a substanc, V s t watr volum, s t avrag cross sctonal ara of t cll, and s t watr ad ndcator. T mass m of a substanc s rlatd to t numbr of mols N troug t rlaton m NM, wr M s t molar mass. T formr Eq.7 can now b modfd to m (8) T rat of cang of ac ndvdual substanc mass n t cllular flud can b xprssd as Copyrgt 0 ScRs. JBSE

6 574 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) m v v v (9) Nxt scton wll outln ow ts quaton can b usd to rprsnt t contnts of oxygn, ydrogn ons (as a masur of P) and Pancratn nzym as a kllr for t tumor clls. 5. GENERLIZED ELECTRO-BIO-TERML TUMOR DYNMICS Matmatcal modls for cancr dynamcs av bn studd by many scntsts usng dffrnt matmatcal mtods. Som of ts modls consdr t growt of tumor clls as populaton dynamcs problms wc nclud t ntracton of tumor clls wt otr clls (.g. normal clls and mmun clls). In ordr to dvlop tratmnt stratgs, t ffcts of trapy ar also ncludd n t modls as control nputs. In ts study, w analyz t modl orgnally dscussd n [4]. T modl dos not am to concntrat on a spcfc knd of cancr and uss normalzd paramtrs. It ncluds tr dffrnt cll populatons and cmotrapy drug concntraton. Intracton of t tumor clls wt normal and mmun cll populaton n t absnc of any tratmnt s gvn by t systm x r x ( b x ) c x x (0) x r x ( b x ) c x x c x x () 3 3 xx x s c x x d x x () wr x () t, x () t and x () 3 t dnot t numbr of normal clls, t numbr of tumor clls and t numbr of mmun clls at tm t, rspctvly. T frst trm n t normal cll populaton s t logstc growt of normal cll populaton wt growt rat r and maxmum carryng capacty (/ b ). T scond trm s t loss of normal clls du to comptton among tumor-normal clls for local rsourcs. In a tumor cll populaton, t frst trm dnots t logstc growt of tumor clls n t absnc of mmun survllanc wt t growt rat r and maxmum tumor carryng capacty (/ b ). T scond and t trd trms n (4) ar dat trms for tumor clls du to t ntracton btwn mmun and normal clls, rspctvly. Immun clls av a constant sourc s 0 wc can b suppld from bon marrow or lymp nods. In t prsnc of tumor clls, mmun clls ar stmulatd by tumor clls wt a Mcals-Mntn typ saturaton functon wt t postv paramtrs and. Immun clls ar dactvatd by tumor clls at t rat c 4 and ty also d at t natural dat rat d. Tr ar dffrnt ways to nclud t ffct of cmotrapy n t tumor growt modl. W assum tat cmotrapy klls all cll populatons wt dffrnt ratos usng mass acton trm. T ffct of drug trapy n t modl s sown wt an addtonal stat x () 4 t and control nput ut () wc dnot drug concntraton n t blood stram, and xtrnal drug njcton rspctvly. T nonlnar systm (0)-() wt t ffct of drug trapy s: 4 x r x ( b x ) c x x a x x (3) x r x ( b x ) c x x c x x a x x (4) xx x s c x x d x a x x x (5) x4 dx4 u() t (6) r, a, a, and a 3 ar t dffrnt kllng ffcts of cmotrapy on t cll populatons. Cmotrapy drug dcay rat n t blood stram s dnotd by d. T systm paramtrs wc ar normalzd to t maxmum carryng capacty of t normal clls ar gvn n Tabl. nalyss of t modl for possbl qulbrum ponts n t absnc of trapy s gvn n t nxt sub-scton. In ordr now to gnralz t dynamcs of t cll n trms of lctromagntc modl, at transfr modl, flud transfr modl, fractonal mass modl, cllular growt, lt us ntroduc t stat spac dynamc varabls as follows. Dfn t lctrc carg passng from or nto t call Vt () as x 5. Snc t lctrc currnt flow s t rat of cang of carg passng troug an nductor I q tn x5 x6 V () t (7) x6 V () t (8) Tn t quvalnt lctrc crcut Eq.4 can b rxprssd as R R C x ( t) R C R C R C x ( t) x ( t) w 6 w w 6 5 R C V ( t) V ( t) u ( t) w x x T lctrc dynamcs (9) can b modfd as R C R C R C w w 6 x6 R RCw x R R C w x u () t 5 (9) (30) Furtrmor, rprsntng t at modl (9) n t stat spac form assumng x7 ylds x7 x7 Rqn out (3) Copyrgt 0 ScRs. JBSE

7 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) Tabl. Ovrall dynamcs paramtrs. Paramtr Valu r r.5 b b c c 0.5 c 3 c 4 a a 0.5 a d 0. d s R 5 0 R 5 0 C 6 W 0 C 6 0 R / 0.06 C 700 R 6 f 0 C f O O Smlarly, for t cllular flud transfr (3) assumng x wll lad to 8 x x Q (3) 8 8 For t sgnfcant of t oxygn, p and Pancratn nzym for t survval of t normal cll, tr fractonal mass rato wll b consdrd. Dnot tr corrspondng masss, rspctvly, as x m, x0 m and x mpancratn (33) 9 oxygn Rfrrng to (8) and (9), last Eq.33 can b rformulatd as n m m m m (34) v v By t vrtu of t flud dynamcs quaton t formr quaton t fractonal mass of t oxygn, p and Pancratn nzym can b dscrbd, rspctvly, by O 9 8 O 9 O n x x x Q (35) x x x Q (36) x x x Q (37) 8 n wr v O, v and v ar t fractonal ratos of oxygn, ydrogn as a masur of P and Pancratn, rspctvly. Fractonal rat of cang of t tr substancs wt rspct to tr ntal valus ar dfnd, vo v v pn rspctvly, as O, and v v v O along wt t rat of cang of t cll surfac ara wt rspct to ts ntal valu as. Slcton of suc substancs ar basd on tr sgnfcant rol n t survval of uman cll and kllng or at last slowng down t tumor clls. Now t ovrall gnralzd modl s 4 n x r x ( b x ) c x x a x x (38) x r x ( b x ) c x x c x x a x x (39) xx x s c x x d x a x x x x (40) x4 dx4 u() t (4) x x (4) 5 6 R C R C R C w w 6 x6 R RCw R R C w 7 7 x u () t 5 (43) x x R q (44) 8 8 n out x x Q (45) x x x Q (46) O 9 8 O 9 O n n Copyrgt 0 ScRs. JBSE

8 576 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) v, 0 x x x Q (47) x x x Q (48) 8 v, n T ovrall gnralzd dynamcs rprsnts t botrmal lctro-fludc dynamcs wt dmnson. Nxt scton wll nvstgat t stablty of t ntr dynamcal systm at qulbrum ponts. 5.. Equlbrum Stat It s ncssary to nvstgat t qulbrum stat of t cll snc t ndcats ts stablty and t non-growt of t tumor clls. s a start, lt us assum a fr trapy and fnd out t corrspondng qulbrum ponts for t dynamcs (38)-(4). s for t normal clls, qulbrum s govrnd by x r x ( b x ) c x x 0 ndcats tat x 0, x r ( b x ) / c, and x wn x / b Tumor clls qulbrum xsts wn x r x ( b x ) c x x c x x 0 (49) at ponts satsfy 3 3 x 0 and x (/ b ) ( c / r b ) x ( c / r b ) x 3 3 n (50) On t otr and, t mmun systm wll xprnc qulbrum wn at stat xx x s c x x d x x s0( x) x3 d ( x ) c x ( x ) x 4 (5) (5) Provdd tat d( x ) c4x ( x ) x Fnally wt fr trapy x4 0 at x4 0 wt zro control nput ut (). It can b sn asly tat t systm as tr dffrnt typs of qulbra: Tumor-fr (no tumor clls), Dad (no normal tssu clls), and Coxstng (bot normal and tumor clls xst) qulbrum ponts [4]. In t contxt of dvlopng trapy stratgy, Tumor-fr or Coxstng typ qulbrum ponts sould b racd, snc n ts typs of stats, t normal cll populaton s clos to ts alty stat. In ts study, our am s to dtrmn t trapy dosag to brng t systm to t tumor-fr qulbrum pont. T tumor-fr qulbrum pont of t systm s obtand as (/ b,0, s0 / d,0) wc gvs us a alty normal cll populaton of (/ b ) and mmun cll populaton of ( s0 / d) wt zro tumor lvl and fr drug njcton. Consdrng now t lctromagntc qulbrum gvn tat x5 x6 0, along wt R C R C R C x x x u ( t) 0 w w R RCw R RCw (53) wn stats x6 x5 u ( t) 0 wc s not fasbl snc t lv cll sould kp a cllular voltag n t rang of mllvolts. Wt otr qulbrum stat R R C x x u() t w 6 5 RCw RCw RC RCw RCw RC (54) T uman cll wll av a stady-stat at transfr wn x7 x7 Rqn out 0 (55) at x R C q R C (56) 7 n out stady-stat cllular flud flow occurs wn x x Q 0 (57) 8 8 x n R C Q. (58) 8 n Tus, t flud stady-stat occurs wn x8 Rf C f Qn. Fractonal mass balanc xst n t uman cll wn t oxygn flow, P and Pancratn satsfy, rspctvly, vo x9 x8 O x9 vo Q 0 n (59) f f 0 8 v, 0 x x x Q 0 (60) x x x Q (6) 8 v, n 0 wt oxygn qulbrum stat gvn by vo v O x9 x8 Q O O and P qulbrum stat dfnd by v x0 x8 Q, v v, Fnally Pancratn stablty wn v x x Q 8 v v,, n n n n (6) (63) Snc t normal cll-tumor-mmun dynamcs s nonlnar, t s ntrstng to lnarzd t systm at Copyrgt 0 ScRs. JBSE

9 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) qulbrum ponts for t purposd of control dsgn. 5.. utors and fflatons It s dsrd at t momnt to sk a lnarzd modl of t clls growt at t qulbrum ponts wt zro control nput n t gvn form x x (64) wr s t Jacoban Matrx. To fulfll ts goal, lt x f ( x, x, x, x ) (65) 3 4 x f ( x, x, x, x ) (66) 3 4 x f ( x, x, x, x ) (67) x f ( x, x, x, x ) (68) T Jacoban matrx for t dynamcs (38)-(4) can b vrfd as r r b x c x c x 0 a x cx r r b x c x c3 x3 a x4 c3x ax x3 ( x ) x x3 x 0 c 4 x a x4 d a3x 3 ( x ) x d Substtutng now t qulbrum pont (/ b,0, s0 / d,0) ylds to t followng lnarzd Jacoba Matrx r r b / b c / b 0 a / b 0 r c / b c3s0 / d 0 0 s0 0 d a3s0 / d d d (69) (70) T followng tabls sow t gnralzd systm paramtrs valus usd n t modlng and lnarzd stat-fdback control. Ts lnarzd ovrall dynamcal modl wll b t ntrst of nxt scton to nvstgat controllablty, obsrvablty and dsgnng a stat-fdback controllr usng pol placmnt approac. Latr analyss wll sow ts paramtrs ar so mportant n dtrmnng t stablty of t uman cll, ts survval and n t growt of t tumor clls and dcrasng t mmun systm. Som paramtrs affct t normal clls growt, tumor growt and mmun systm strngt, cllular voltag, at and flud flow, oxygn rat, acdty and pancratn nzym. r r b / b c / b 0 a / b r c / b c3s0 / d s0 0 d a3s0 / d d d RCw RCw RC R RCw R RC w vo O 0 0 v , 0 v v v, T nput vctor s gvn by (7) T B u( t) u ( t) Rqn out Qn vo Q n v Q n Q n (7) Copyrgt 0 ScRs. JBSE

10 578 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) CONTROLBILITY ND OF TUMOR DYNMICS ND CONTROLLER DESIGN systm s sad to b controllabl at tm t 0 f t s possbl by mans of unconstrand control vctor to transfr t systm from an ntal stat xt ( 0) to any otr stat n a fnt ntrval tm. T concpts of controllablty and obsrvablty wr ntroducd by Kalman. Ty play an mportant rol n t dsgn of control systms n stat spac. In fact, t condtons of controllablty and obsrvablty may govrn t xstnc of a complt soluton to t control systm dsgn. ltoug most pyscal systms ar controllabl and obsrvabl, corrspondng matmatcal modls may not possss t proprty of controllablty and obsrvablty. In wat follows, w sall drv t condton for complt stat controllablty. Fgurs and 3 sow t opn-loop cancr controlld systms and closd-loop cancr controlld systm, rspctvly. Consdr t contnuous-tm systm x x B w (73) y Cx D w (74) wr, x s a stat vctor y s m -output vctor w s a control sgnal s n n matrx B s n matrx C s n m matrx T systm dscrbd n Eq.73 s sad to b stat controllabl at t t0 f t s possbl to construct an unconstrand control sgnal tat wll transfr an ntal stat to any fnal stat n a fnt tm ntrval t0 t t. If vry stat s controllabl, tn t systm s sad to b compltly stat controllabl []. T systm s sad to b controllabl f and only f t followng n n matrx s full rank n n B B B B (75) Ts matrx s calld t controllablty matrx. systm s sad to b obsrvabl at tm t 0, f wt t systm n stat xt ( 0), t s possbl to dtrmn ts stat from t obsrvaton of t output ovr a fnt tm ntrval. T concpt of obsrvablty s vry mportant bcaus, n practc, t dffculty s ncountrd wt stat fdback control s tat som of t stat varabls ar not accssbl for drct masurmnt, wt t rsult tat t bcoms ncssary to stmat t unmasurabl stat varabls n ordr to construct t control sgnals. T systm s sad to b obsrvabl f and only f t followng n nm matrx s oull rank n T T T T T T n T C C C C (76) Matrx (76) s commonly calld obsrvablty matrx. Ts followng analyss prsnts a dsgn mtod commonly calld t pol-placmnt tcnqu. W assum tat all stat varabls ar masurabl and ar avalabl for fdback. It s sown tat f t systm consdrd s compltly stat controllabl, tn pols of t closd-loop systm may b placd at any dsrd locatons by mans of stat fdback troug an approprat stat fdback gan matrx. Lt us assum t dsrd closd-pols ar to b at s, s,, sn n. W sall coos t control sgnal to b w Kx (77) Ts mans tat t control sgnal s dtrmnd by an nstantanous stat. Suc a scm s calld stat fdback. T n matrx K s calld t stat fdback gan matrx. Substtutng (77) nto Eq.73 gvs x( t) BK x ( t) (78) T soluton of ts quaton s gv by t BK x( t) x (0) (79) wr s t ntal stat causd by xtrnal dsturbancs. T stablty and transnt rspons caractrstcs ar dtrmnd by t gnvalus of matrx BK. If matrx K s cosn proprly, t matrx BK can b mad asymptotcally stabl matrx. Dfn a transformaton matrx T by T MW (80) Fgur. Opn-loop cancr controlld systm. Fgur 3. Closd-loop cancr controlld systm. Copyrgt 0 ScRs. JBSE

11 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) wr M s t controllablty matrx and n B B B B (8) an an a an an 3 0 W (8) a wr t as ar t coffcnts of t caractrstc polynomal n n s s as an s an I (83) Lt us coos a st dsrd gnvalus as s, s,, sn n. Tn t dsrd caractrstc quaton bcoms ( s )( s ) ( s ) s s s n n n n n (84) T suffcnt condton tat t systm to b compltly controllabl wt all gnvalus arbtrarly placd by coosng t gan matrx a a a a K n n n n (85) T formr outlnd control dsgn wll b valdatd n t followng scton by dmonstratng t smulaton rsults. Smulaton Rsults Evry cll n t uman body functons as a mcro battry. To succssfully brng noursmnt n and tak posons out, t as to b fully cargd. In a cancrous cll, t cll voltag drops from 90 mllvolts to lss tan 40 mllvolts. Wn t cll voltag gts to t vry bottom, only 5 substancs can pass n or out of t cll. Ty ar watr, sugar, potassum, csum and rubdum []. Oxygn cannot ntr nto a cancr cll. Evn f tr s a lot of oxygn n t blood, t cannot gt nto t cll. Csum, bcaus of ts lctrcal proprts can stll ntr t cancrous cll. Wn t dos so, bcaus of ts xtrm alkalnty, t cll ds. Luckly, alty clls ar not affctd by csum bcaus tr cll voltag allows tm to balanc tmslvs. Ts uptak can b nancd by Vtamns and C as wll as salts of znc and slnum. T quantty of csum takn up was suffcnt to ras t cll to t 8 p rang. Wr cll mtoss cass and t lf of t cll s sort. T objctv of t Csum trapy s to kll t tumor clls, mnmz t amount of drug applcaton and to kp cllular voltag, trmo-flud transfr and cllular ngrdnts at fxd rats. In t smulatd gnralzd cancr modl, t alty qulbrum pont wt t paramtr st gvn n Tabl s (, 0,.65, 0, 0., 0, ,.6 0, 0, 0, 0) s locally stabl. It mans tat t growt of cancr s controllabl f suffcnt drug survllanc s guarantd. In t absnc of suffcnt mmun control, t tumor clls grow n numbr and kll t alty tssu clls and rac t lmt capacty, wc s rfrrd to as dad qulbrum pont. T ntal stats,.., t condtons wn t cmotrapy tratmnt s startd, ar assumd to b: N(0) =, T(0) = 0.0, I(0) =, M(0) = ). T rspons of tratmnt-fr cancr growt wt rspct to normalzd tm scal s gvn n Fgurs 4-6. Smulaton analyss sows t followng snstvty of paramtrs tat affct t cll stablty. T gnvalus of matrx av som postv valus, zro, and magnary wt ngatv ral valus. Tus t orgnal systm s unstabl. Ts gnvalus ar lstd as follows: , , Snc t orgnal dynamcs s unstabl, t s now dsrd to dsgn a stat-fdback controllr wt t followng dsrd gnvalus: 0., 0., 0., 0., 0.3, 0.3, 0.4, 0.4, 0.5, 0., 0.. s t can b sn, all dsrd gnvalus ar ngatv and clos to t orgn to guarant a stabl and fast rspons. Bfor procdng n t controllr dsgn, t controllablty condton dfnd by Eq.75 must b satsfd. Consdrng coffcnt matrx dfnd (7), t nput vctor gvn by (7), and paramtrs valus lstd n Tabl, t s found out tat t controllablty matrx s oull rank. Smulaton rsults n Fgurs 4-6 sows tat t csum trapy s so ffctv n brnng t normal and mmun systm to ts qulbrum stat and forcng t tumor cll kll. s t can b sn from t arlr analays, avng no closd-loop controlld tumor trapy, t ovrall bo-trmo-fludc-dynamcal systm was unstabl. Enforcng a stat-fdback pol placmnt clord-csum tumor trapy, a complt trapy of dsas s acvd as sown n Fgur 4. Normal and mmun clls av bn brougt to an qulbrum stat Copyrgt 0 ScRs. JBSE

12 580 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) Fgur 4. Normal clls, tumor clls, mmun clls and trapy nput rspons. Fgur 5. Cllular voltag and ts rat of cang, cllular at-flud transfr. Copyrgt 0 ScRs. JBSE

13 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) Fgur 6. Fractonal mass of oxygn, ydrogn on and Pancratn Enzym. by nforcng fr kll clls. Smlarly, t cllular voltag s rgulatd at t rqurd valu. Transfr of at and flud from and nto t cll s mantand fxd as wll. Snc oxygn s so sgnfcant to mantan a alty cll and rduc or kll t tumor clls ts control was so ffcnt to kp t normal and mmun clls as dsrd. cdty s controlld by t amount of ydrogn ons n t cll as wll. Fnally, Pancratn rato (wc as an mportant rol n kllng tumor clls) s kpt wtn t normal rang as wll. 9 9 If R 0 or R 0 (Indpndntly), t systm s unstabl. 7 If bot R R 0, t systm s unstabl. Incrasng R, mmunty ncrass. Dcrasng R, mmunty ncrass sgnfcantly (0 tms manfold). Dcrasng wall capactanc C W ncrass t normal clls, tumor clls and mmun clls and all otrs. Incrasng of ara rato to.8, oscllatons occurs; and at.9 t rsults n unstabl clls. Incrasng oxygn rato O to 3 wll gnrat oscllatons; and to 3.8 producs unstabl clls. Dcrasng ydrogn rato to 0.0 and oxygn rat O at 0.0, gnrats unstabl clls. Incrasng Pancratn rato to 6 producs unstabl clls. 7. CONCLUSIONS Ts papr prsnts a mor gnralzd dynamcal modl dscrbng t normal-tumor-mmun growt consdrng t lctro-trmo-fludc-cmcal caractrstcs of a uman cll. Equlbrum and stablty of suc a modl s valdatd va stat-fdback controllr dsgn. Csum trapy s found to b so ffctv n controllng t wol systm. T snstvty and t rang of t dynamc paramtrs tat nflunc t normalty and mmunty of t lv cll av bn dntfd. It s fgurd out tat t cllular voltag s so mportant to rgulat t cll normal opraton. Tr cllular componnts av bn nvstgatd: oxygn (to kp t cll alv and prvnt cancr growt, ydrogn (acdty: P valu) and Pancratn nzym (to kll t tumor clls). Smulaton of t controlld trapy sows ow s approac s so ffcnt and can b mplmntd clncally. Copyrgt 0 ScRs. JBSE

14 58 M. l-sbl / J. Bomdcal Scnc and Engnrng 4 (0) REFERENCES [] anaan, D. and Wnbrg, R.. (000) T allmarks of cancr. Cll Journal, 00, do:0.06/s (00) [] Rans, J.K. (98) Elctromagntc fld ntractons wt t uman body: Obsrvd ffcts and tors. Tcncal Rport, NS, Goddard Spac Flgt Cntr, Wasngton. [3] D Plls, L.G. and Radunskaya,.E. (003.) T dynamcs of an optmally controlld tumor modl: cas study. Matmatcal and Computr Modlng, 37, do:0.06/s (03)0033-x [4] Itk M., Salamc, M.U. and Banks, S.P. (00) SDRE optmal control of drug admnstraton n cancr tratmnt. Turks Journal of Elctrcal Engnrng and Computr Scncs, 8, [5] Szymanska, Z. (003) nalyss of mmunotrapy modls n t contxt of cancr dynamcs. Intrnatonal Journal of ppld Matmatcs and Computr Scnc, 3, , [6] Byrn,.M., larcon, T., Own, R., Wbb, S. D. and Mak, P.K. (006) Modllng aspcts of cancr dynamcs: rvw. Plosopcal Transacton of t Royal Socty, 364, [7] Krscnr, D. (009) On t global dynamcs of a modl for tumor mmunotrapy. Matmatcal Boscncs and Engnrng, 6, [8] U. Ldzwcz, (005) Optmal control for a systm modllng tumor ant-angognss. CSE 005 Confrnc, Caro, 9- Dcmbr 005. [9] Gaffar,. and Nassrfar, N. (009) Matmatcal modlng and lyapunov-basd drug admnstraton n cancr cmotrapy. Journal of Elctrcal & Elctronc Engnrng, 5, [0] Warburg, O.. (969) T prm caus and prvnton of cancr. nd Edton, Konrad Trltsc, Würzburg. [] Brwr.K. (984) T g p trapy for cancr, tsts on mc and umans. Parmacology Bocmstry & Bavor,, -5. [] nold J. (003) Clan out your artrs at om, wtout a ndl, and at a fracton of t cost, alt Scncs Insttut Mmbrs lrt, -4. Copyrgt 0 ScRs. JBSE