Optimal control for multi-input bilinear systems with an application in cancer chemotherapy.

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1 Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Volume 3, Issue 2, February 2015, PP ISSN X (Pr) & ISSN (Ole) Omal corol for mul-u blear sysems wh a alcao cacer chemoheray. Omar BALATIF Faculy of sceces Be M Sk, Dearme of Mahemacs ad Comuer Scece, LAMS, Hassa II Uversy, Casablaca, Morocco. balaf.mahs@gmal.com Issam ABDELBAKI Faculy of sceces Be M Sk, Dearme of Mahemacs ad Comuer Scece, LAMS, Hassa II Uversy, Casablaca, Morocco. Mosafa RACHIK Faculy of sceces Be M Sk, Dearme of Mahemacs ad Comuer Scece, LAMS, Hassa II Uversy, Casablaca, Morocco. m_rachk@yahoo.fr Zeb RACHIK Faculy of sceces Be M Sk, Dearme of Mahemacs ad Comuer Scece, LAMS, Hassa II Uversy, Casablaca, Morocco. Absrac: I hs aer we sudy he omal corol roblem for mul-u blear sysems. We ado a mehod based o rewrg our sysem a comarmes form, ad fdg he omal corol whch mmzes a gve cos fuco by alyg he Poryag s maxmum rcle. Also, we rese a erave rocess o fd a soluo of he omaly sysem. Keywords: Blear sysems, Omal corol, Poryag s rcle, Cacer chemoheray. 1. INTRODUCTION Blear sysems are a secal class of olear sysems, whch olear erms are cosruced by mullcao of corol vecor ad sae vecor. Through early half a ceury, hey have receved grea aeo by researchers. The morace of such sysems les he fac ha may mora rocesses, o oly egeerg [1], bu also bology [2], soco-ecoomcs [3], ad chemsry [4-5], ca be modeled by blear sysems. A overvew of he avalable corol sraeges for blear sysems ca be foud [6]-[7]. Besdes, omal corol s oe of he mos acve subjecs he corol heory. I has successful alcaos s may dscles, ecoomcs, evrome, maageme, egeerg ec. As we kow omal corol roblem for he blear sysems does o have a aalycal soluo as lear case so hs reaso movaes may researchers o ry o oba a aroxmae soluo for hs roblem. Theory ad alcao of omal corol have bee wdely used dffere felds such as arcraf sysems [8], roboc [9], bomedce [10], ec. x Ax u B x wh he al codos x(₀) = x₀. Where x Rⁿ, u = (u₁, u₂,..., u ) R, A ad B are marces. We assumed ha he rocess sars from ₀ ad eds a fxed me f 0. The ma objecve of hs aer s o develo a omal corol desg algorhm for a mulu blear sysems. We use a mehod develoed by [11] ad reseed [12] ad [13]. Ths mehod s based o he maxmum Poryg s rcle, ad a umercal algorhm s roosed o fd a soluo of he omaly sysem. ARC Page 22 (1)

2 Omal corol for mul-u blear sysems wh a alcao cacer chemoheray. The sysem (1) ca be rewre a comarmes form. x 1 a 1j u x 2 a 2j u x a j u b 1j b 2j b j (2) wh he al codos x k (₀) = x k 0, where x = (x₁, x₂,..., x ), a j = (A j ) 1,j ad b j = (B j ) 1,j. The x s he -h comoses of he sae sysem, whch ca rerese, for examle, a chemoheraeuc model, he average umber of cacer cells he -h comarme. Also, he a j ad b j ca rerese he exchages bewee hese comarmes, ad he corol u deog he drug dosage admsered. The aer s orgazed as follows. Seco 2 we rese a comarmes form for a mul-u blear sysem, ad we aalyze he omal corol roblem. I seco 3, we rese a umercal algorhm o fd a soluo of he omaly sysem. I seco 4, we rese a sudy of a omal corol for a cacer chemoheraeuc model ad he smulao corresodg resuls. Fally, he cocluso s summarzed Seco THE OPTIMAL CONTROL PROBLEM We cosder he sysem of dffereal equaos x 1 a 1j u x 2 a 2j u x a j u b 1j b 2j b j (2) wh he al codos x k = x k 0 for k {1,2,..., }. Where x = (x₁, x₂,..., x ) T, a j = (A j ) 1,j, b j = (B j ) 1,j ad u R for {1,2,..., }. We defe he objecve fucoal as f Ju x 1 f,...,x f x r 2 u 2 ds (3) where he arameers 0 ad r > 0 are he cos coeffces, hey are seleced o wegh he relave morace of x ad u. Ad ₀ ad f are he al ad fal mes. The erm, φ(x₁( f ),..., x ( f )), rereses a ye of `salvage' erm; for examle, a cacer model hs erm ca rerese a weghed average of he oal umber of cacer cells a he ed of he heray erval [₀, f ]. Our goal s o mmze hs objecve fucoal. I oher words, we seek he omal corol u = (u 1, u 2,..., u ) T such ha Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 23

3 Omar BALATIF e al. Ju mju : u U where U s he se of admssble corols defed by U u u 1,...,u : u s Lebesgue mesurable, a u b,, f, 1,..., (4) (5) Reurg o he geeral model (1), we also make he assumo ha he corol sysem s erally osve [14]:.e. For ay admssble corol u, f x (0) 0 for all = 1,...,, he x () 0for all = 1,...,, ad all mes > 0. A smle suffce codo for hs assumo o hold (for examle, see [14]) s ha all he = marces A + =1 u ()B, hey have egave dagoal eres, bu o-egave off-dagoal eres. Ths codo s aural ad wll be sasfed for ay comarmeal model whose dyamcs are gve by balace equaos where he dagoal eres corresod o he ouflows from he h comarmes ad he off-dagoal eres rerese he flows from he h o he j h comarme, j. Posve sysems lay a mora role sysems ad corol heory because may hyscal sysems he sae-varables rerese quaes ha ca ever aa egave values (e.g. oulao szes or roe coceraos) [15,16,17]. The soluo of (1) s bouded. Ideed, he soluo of (1) s, f, x x 0 Axs u sb xsds (6) where x = (x₁, x₂,..., x ) T ad x₀ = x(₀). So,, f, x x 0 x 0 x 0 A u sb xsds A u s B A c C 1 C 2 xsds B xsds xsds (7) where c = su( a, b ), C₁ = x₀ ad C₂ = A + c = =1 B. Usg Growall equaly, see [18], we oba, f, The, he boudedess of he soluo (1). x C 1 C 2 xsds C 1 ex C 2 ds C 1 exc 2 f. (8) 2.1 Exsece of a Omal Corol. The exsece of he omal corol ca be obaed usg a resul by Flemg ad Rshel [19] (see Corollary 4.1). Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 24

4 Omal corol for mul-u blear sysems wh a alcao cacer chemoheray. Theorem.1: Cosder he corol roblem wh sysem (2). There exss a omal corol u U such ha J u = m J u : u U, f he followg codos are me: (1) The se of corols ad corresodg sae varables s oemy. (2) The corol se U s covex ad closed. (3) The rgh-had sde of he sae sysem s bouded by a lear fuco he sae ad corol varables. (4) The egrad of he objecve fucoal s covex o U. (5) There exs cosas c₁, c₂ 0 ad β 1 such ha he egrad L(x₁, x₂,..., x, u) of he objecve fucoal sasfes L(x₁, x₂,..., x, u) c₁ + c₂( u₁ ² u ²) β/2. (9) To rove ha he se of corols ad corresodg sae varables s oemy, we wll use a smlfed verso of a exsece resul Boyce ad DPrma ([20], Theorem 7.1.1): Theorem.2: Le x = F (; x₁,..., x ) for {1,..., } be a sysem of dffereal equaos wh al codos x (₀) = x 0 for {1,..., }. If each of he fucos F 1,..., F ad he aral dervaves F₁/ x₁,..., F₁/ x, F₂/ x₁,..., F₂/ x,..., F / x₁,..., F / x, are couous Rⁿ+¹ sace, he here exss a uque soluo x₁,..., x ha sasfes he al codos. Proof: (Theorem.1) We use Theorem.2 o rove ha he se of corols ad corresodg sae varables s oemy. Le x 1 = F 1 (; x₁,..., x ),..., x = F (; x₁,..., x ), where he F 1,..., F form he rgh had sde of he sysem of equaos (2). Le u() = c, for some cosa, ad sce all arameers are cosas, F 1,..., F are lear. Thus, hey are couous everywhere. Addoally, he aral dervaves F₁/ x₁,..., F₁/ x, F₂/ x₁,..., F₂/ x,..., F / x₁,..., F / x are all cosas, ad so hey are also couous everywhere. Therefore, here exss a uque soluo x₁,..., x ha sasfes he al codos. Therefore, he se of corols ad corresodg sae varables s oemy, ad codo 1 s sasfed. The corol se s covex ad closed by defo. Sce he sae sysem s blear u, he rgh sde of (2) sasfes codo 3, usg he boudedess of he soluo. The egrad he objecve fucoal (4) s covex o U. I res o show ha here exss cosas c₁, c₂ 0 ad β 1 such ha he egrad L(x₁,..., x, u₁,..., u ) of he objecve fucoal sasfes Lx 1,x 2,...,x,u c 1 c 2 u u 2 /2. The sae varables beg bouded, le c₁ = 1 f(₁x₁,..., x ), c₂ = 1 f(r₁,..., r ) ad 2 2 β = 2. The follows ha : x =1 c₁ + c₂( u₁ ² u ²). =1 + r u Characerzao of he Omal Corol. We are alyg he Poryag's Maxmum Prcle [21]; he key dea s roducg he adjo fuco o aach he sysem of dffereal equaos o he objecve fucoal, resulg he formao of a fuco called he Hamloa. Ths rcle covers he roblem of fdg he corol o omze he objecve fucoal subjec o he sae dffereal equaos wh al codo, o fd he corol o omze Hamloa owse (wh resec o he corol). Now we have he Hamloa me, H x r 2 u 2 j f j x 1,...,x,u, (10) Where λ j for j = 0,1,...,, s he adjo fuco, where f j s he rgh had sde of he sysem of dffereal equaos of j h equao for j = 0,1,...,. Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 25

5 Omar BALATIF e al. Theorem.3: There exss a omal corol u ad corresodg soluos x 1, x 2,, x ha mmze J(u) over U. Furhermore, here exss adjo fucos, λ₁, λ₂,..., λ, sasfyg he equaos k k j wh he rasversaly codos a jk u b jk, k 1,2,..., (11) f x f x 1 f,...,x f, for 1,2,..., (12) Furhermore, he omal corol u s gve by u k m b;max a; 1 r k b j, k 1,2,..., (13) Proof: The adjo equaos ad rasversaly codos ca be obaed by usg Poryag's Maxmum Prcle such ha 1 H x 1, 1 f x 1 f x 1 f,...,x f H x 2, 2 f x 2 f x 1 f,...,x f H x, f x x f 1 f,...,x f The omal corol u ca be solved from he omaly codo, ha s H u k r k u k H u k 0, k 1,2,..., b j 0, k 1,2,..., By he bouds U of he corols, we oba u he form of (13). 3. Numercal algorhm. I hs seco we rese a erave mehod for he umercal soluo of he omaly sysem. The umercal algorhm reseed below s a sem-mlc fe dfferece mehod. We dscreze he erval, f a he os = + h, ( = 0,1,..., ), where h s he me se such ha = f, [22]. Nex, we defe he sae ad adjo varables x₁(), x₂(),..., x (), λ₁(), λ₂(),..., λ () ad he corol u() erms of odal os x 1,, x, λ 1,, λ ad u. Now a combao of forward ad backward dfferece aroxmao s used as follows: The mehod, develoed by [11] ad reseed [12] ad [13], s he read as: (14) (15) (16) x 1 k1 x 1 k h x k1 x k h a 11 x k1 1 a 1j x k j u k b 11 x k1 k 1 u a j x k1 k j u j2 b j k1 k d j u j b 1j j2 k (17) By usg a smlar echque, we aroxmae he me dervave of he adjo varables by her frs-order backward-dfferece ad we use he aroraed scheme as follows Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 26

6 Omal corol for mul-u blear sysems wh a alcao cacer chemoheray l1 l1 h 1 1 a 11 u l b 11 1 l1 h 1 l1 j j a j u l b j l j j2 a j1 u l b j1 The algorhm descrbg he aroxmao mehod for obag he omal corol s he followg Algorhm: Se 1: (18) x 1 0 x 1 0, x 2 0 x 2 0,...,x 0 x 0, f Se 2: for = 1,..., 1, do : x f x 1 f,...,x f 1,...,, u0 u 0. x 1 k1 x 1 k h a 1j x k k k j u b 1j j2 j2 1h a 11 u k b 11 x 2 k1 x 2 k h a 21 x k1 1 a 2j x k j u k b 21 x k1 1 k u j3 1h a 22 u k b 22 b 2j xk j j3 x k1 x k h 1 1 a j x j k u b j x j 1h a u k b (19) k1 1 k1 2 k1 k 1 h k 1 j a j1 u b j1 j2 1h a 11 u b 11 k 2 h k1 2 1 a 12 u k b 12 j j3 1h a 22 u b 22 k h 1 k1 j a j u b j 1h a u b a j2 u k b j2 T l k1 1 r k1 b l k1 j, l 1,..., u l mb;maxa;t l, l 1,..., ed for Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 27

7 Omar BALATIF e al. Se 3: for = 0,...,, wre ed for. x 1 x 1, x 2 x 2,..., x x, u u. 4. Alcao: Omal corols for a cacer chemoheraeuc model. I hs seco we formulae a geeral -comarme model for cacer chemoheray as a omal corol roblem over a fxed heray erval wh dyamcs descrbed by a blear sysem [23]. Le N = (N₁,..., N ) T deoe he sae-vecor wh N deog he umber of cacer cells he -h comarme, = 1,...,. The corol s a vecor u = (u₁,..., u m ) T wh u deog he drug dosage admsered. The corol se U s a comac m-dmesoal erval of he form [α₁, β₁] [α m, β m ] wh each erval [α, β ] [0, ). Le A ad B, = 1,..., m, be cosa marces, le r = (r₁,..., r ) be a row-vecor of osve umbers ad le s = (s₁,..., s m ) be a row-vecor of o-egave umbers. The vecors r ad s rerese subjecve weghs he objecve. We he cosder he followg omal corol roblem: Mmze he objecve Subjec o he dyamc f Ju N 1 f,...,n f N r 2 u 2 ds (20) N AN u B N, N0 N 0 (21) where he arameers 0 ad r > 0 are he cos coeffces, hey are seleced o wegh he relave morace of N ad u. ad ₀ ad f are he al ad fal mes. The erm φ(n₁( f ),..., N ( f )) rereses a weghed average of he oal umber of cacer cells a he ed of a assumed fxed heray erval, f. I oher words, we seek he omal corol u such ha Ju mju : u U (22) where U s he se of admssble corols defed by U u u 1,...,u : u s Lebesgue mesurable, a u b,, f, 1,..., We also make he assumo ha he corol sysem s erally osve [14]:.e. For ay admssble corol u, f N (0) 0 for all = 1,...,, he N () > 0 for all = 1,...,, ad all mes > 0. Before roducg a 4-comarme model for cacer chemoheray, we gve a bref bologcal backgroud o he cell cycle ad chemoheray ages[23]. Each cell asses hrough a sequece of hases from cell brh o cell dvso. Afer a al growh hase G₁, he cell eers a hase S where DNA syhess occurs. Followg a secod growh hase G₂, he cell reares for moss or hase M ha leads o cell dvso. Each of he wo daugher cells ca eher reeer hase G₁ or for some me may smly le dorma a searae hase G₀ ul reeerg G₁, hus sarg he ere rocess all over aga. Mul-comarme models combe hases of he cell cycle o clusers [24], wh he urose of effecvely modelg he dffere yes of chemoheraeuc ages used: cyooxc (kllg), cyosac (blockg) ad recrug ages. The dyamcs of hs cell cycle ad he chemoheray ages may be rereseed by he followg comarmeal model. Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 28

8 Omal corol for mul-u blear sysems wh a alcao cacer chemoheray. Fgure 1: 4-comarme model. Where he a are osve coeffces relaed o he mea ras mes of cells hrough he -h comarme. The oal umber of cacer cells a me he hases of he cell cycle G₀, G₁, S ad G₂/M, s gve by N₁, N₂, N₃ ad N₄, resecvely. The kllg age u ac he G₂/M hase whch makes sese from a bologcal sado for a coule of reasos[23]. Frs, moss M he cell becomes very h ad orous. Hece, he cell s more vulerable o a aack whle here wll be a mmal effec o he ormal cells. Secod, chemoheray durg moss wll reve he creao of daugher cells. I s assumed ha he dose rae sads drec relao o he fraco of cells whch are beg klled he G₂/M hase. Therefore oly he fraco 1 u of he ouflow of cells from he las comarme, a₄n₄, udergoes cell dvso ad reeers he frs ad secod comarme. As a resul he flow of cacer cells from he fourh o he frs ad he secod comarme, 2(a₄⁰ + a₄¹)n₂, s reduced o (1 u)2(a₄⁰ + a₄¹)n₂. However, all cells leave comarme G₂/M. The blockg age v s aled o slow he ras mes of cacer cells durg he syhess hase S. As a resul he flow of cacer cells from he hrd o he fourh comarme, a₂n₂, s reduced by a facor 1 v o (1 v)a₂n₂. The recrug age w s aled o reduce he average sejour me he quesce hase. As a resul he average ras me hrough he comarme G₀ s reduced resulg he ouflow beg creased by a facor 1 + w. The chemoheray ages ca vary bewee (o chemoheray) ad (maxmal chemoheray). (Noe: Maxmal chemoheray s esseally a sub-lehal dose, or he maxmum ha ca be gve ha wll o kll he ae). Ths model yelds he mahemacal sysem wh corols of dffereal equaos N 1 1 wa 1 N 1 1 u2a 4 N 4 N 2 1 wa 1 N 1 a 2 N 2 1 u2a 5 N 4 N 3 1 va 3 N 3 a 2 N 2 (23) N 4 1 u2a 4 a 5 N 4 1 va 3 N 3 Our goal s o reduce he umber of cacer cells hases G₀, S ad G₂/M of cell cycle ad maxmze he umber of cacer cells syhess hase S by slowg he ras mes of cacer cells durg hs hase S. Ad mmze he cos of chemoheray. Mahemacally, he roblem s o mmze he objecve fucoal Ju 4 q N f q 3 N 3 f 3 f 4 N 3 N 3 r 1 u 2 r 1 v 2 r 1 w 2 ds (24) Subjec o (23). Usg he algorhm roosed seco (3), we have he smulaos resuls reseed he grah below. These grahs, allow us o comare chages he cacer cell oulao before ad afer he roduco of he corols. The ar of daa for hs model are ake from [25], lke a₁ = 0.197, a₂ = ad a₃ = 0.107, a a 4 1 = Bu he al codos N₁ = N₃ = N₄ = 1000 ad N₂ = 9000 ad he arameer a 4 0 = 0.2 ad a 4 1 = 0.3 are arbrary academc values. 4.1 The umercal smulaos. Fgures 2 ad 5 show ha before chemoheray, G₀ ad G₂/M hases, he umber of cells crease radly. Whereas, we oce ha afer he chemoheray by usg he kllg age ad recrug age, he umber of cells decreases grealy hese hases. Also, fgure 3 shows he Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 29

9 Omar BALATIF e al. effec of he corol decreasg more radly he umber of cells durg he chemoheray rogram. I fgure 4, we ca observe ha he blockg age ca, wh success, slowg he ras mes of cacer cells durg hs hase S, so, creasg he umber of cells hs hase. Fgure 2: The umber of cells cacer wh ad whou corol G₀ hase. Fgure 3: The umber of cells cacer wh ad whou corol G₁ hase. Fgure 4: The umber of cells cacer wh ad whou corol S hase. Fgure 5: The umber of cells cacer wh ad whou corol G₂/M hase. 5. Cocluso I hs aer, we have reseed a mehod for he omal corol roblem of a mul-u blear sysem. Ths mehod based o he Poryag's maxmum rcle ad a umercal algorhm o solve he omaly sysem. A examle of cacer chemoheray has bee roosed o clarfy he mehod. Refereces [1] Mohler R. R., Blear Corol Processes, volume 106 of Mahemacs Scece ad Egeerg. Academc Press, New York, (1973). [2] Wllamso D., Observao of blear sysems wh alcao o bologcal corol, Auomaca, 13: , (1977). [3] Mohler R., Nolear sysems: Alcaos o Blear Corol, volume 2. Prece Hall, Eglewood Cls, New Jersey, (1991). [4] Esaña M., Ladau I. D., Reduced order blear models for dsllaos colums, Auomaca, 14: , (1977). [5] Bas M. V., Garca A. A., Omal flerg for blear sysem saes ad s alcao o olymerzao rocess defcao, Proceedgs of he Amerca Corol Coferece, , Dever, Colo, USA, Jue (2003). [6] R. Mohler, Naural Blear Corol Processes, New York: Academc, [7] Hofer E., Tbke B., A Ierave mehod for he fe-me blear quadrac corol roblem, J. Om. Theory Alcaos, vol. 57, , (1988). [8] Garrard W. L., Jorda J. M., Desg of olear auomac flgh corol sysems, Auomaca, vol. 13, o. 5, , Ieraoal Joural of Isrumeao ad Corol Sysems (IJICS) Vol.2, No.1, (2012). [9] We S., Zefra M., ad DeCarlo R. A., Omal corol of roboc sysem wh logcal cosras: alcao o UAV ah lag, IEEE Ieraoal Coferece o Roboc ad Auomao, Pasadea, Ca, Usa, (2008). [10] Ik M., Salamc M. U., ad Baksa S. P., Omal corol of drug heray cacer reame, Nolear Aalyss, vol. 71, , (2009). Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 30

10 Omal corol for mul-u blear sysems wh a alcao cacer chemoheray. [11] Gumel A. B., Shvakumar P. N., ad Saha B. M., A mahemacal model for he dyamcs of HIV-1 durg he ycal course of feco, Thrd world cogress of olear aalyss, 47: , (2001). [12] Haaf K., Rachk M., Saad S., Tab Y ad Yousf N., Omal Corol of Tuberculoss wh Exogeous Refeco, Aled Mahemacal Sceces, Vol. 3, o. 5, , (2009). [13] Karrakchou J., Rachk M., ad Gourar S., Omal corol ad Ifecology: Alcao o a HIV/AIDS Model, Aled Mahemacs ad Comuao,177:807818, (2006). [14] Kaczorek K., Weakly osve couous-me lear sysems, Bulle of he Polsh Academy of Sceces, 46, , (1998). [15] Berma A., Plemmos R. J., Noegave Marces he Mahemacal Sceces, SIAM, (1987). [16] Fara L., Rald S., Posve Lear Sysems: Theory ad Alcaos, Joh Wley, (2000). [17] Krasoselskj M. A., Lfshs J. A. ad Sobolev A. V., Posve Lear Sysems: he Mehod of Posve Oeraors, Helderma-Verlag, (1989). [18] Slemrod M., Ball J. ad Marsde J. E., Corollably for Dsrbued Blear sysems, SIAM J. o corol ad o, 40, , (1982). [19] Flemg W. H., Rshel R. W., Deermsc ad Sochasc Omal Corol, Srger, New York, NY, USA, (1975). [20] Boyce W. E., DPrma R. C., Elemeary Dffereal Equaos ad Boudary Value Problems, Joh Wley & Sos, New York, (2009). [21] Poryag L., Bolyask V., The Mahemacal Theory of Omal Processes, Wley, New York, (1962). [22] Gumel A. B., Padar K. C. ad Ser R. J., edors, Asymocally Cosse Nosadard Fe-Dfferece Mehods for Solvg Mahemacal Models Arsg Poulao Bology, E. Mckes ad World Scefc, Sgaore, (2005). [23] Ledzewcz U., Schaler H., Dehkord S. M. ad Res M., Omal Corol of Mul-u Sysems for Cacer Chemoheray, Laes Advaces Sysems Scece ad Comuaoal Iellgece, (2012). [24] Swerak A., Cell cycle as a objec of corol, J. of Bol. Sys., 3, , (1995). [25] Swerak A., Ledzewcz U. Schaler H., Omal corol for a class of comame models cacer chemoheray, I. J. Al. Mah. Comu. Sc. Vol. 13, No. 3, , (2003). Ieraoal Joural of Scefc ad Iovave Mahemacal Research (IJSIMR) Page 31