TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE

Transcription

1 Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs and Lohar von Wolfersdorf Absrac We invesigae wo well-known basic opimal conrol problems for chemoherapeuic cancer reamen modified by inroducing a imedependen resisance facor. This facor should be responsible for he effec of he drug resisance of umor cells on he dynamical growh for he umor. Boh opimal conrol problems have common poinwise bu differen inegral consrains on he conrol. We show ha in boh models he usually pracised bang-bang conrol is opimal if he resisance is sufficienly srong. Furher, we discuss differen opimal sraegies in boh models for general resisance. MSC 2: 49 J 15; 49 K 15; 92 C 5. Keywords: Cancer chemoherapy, opimal conrol, drug resisance. 1 Inroducion Opimal conrol problems based on mahemaical models for cancer chemoherapy have a long hisory and obained a renewed ineres in he las years Acceped for publicaion on March 13, 211. Deparmen of Mahemaics, TU Darmsad, Schloßgarensr. 7, Darmsad, Germany, krabs@mahemaik.u-darmsad-de Lohar von Wolfersdorf passed away on November 3,

2 Two opimal conrol problems in cancer chemoherapy 333 (cf. [2-6, 1-26]). There are furher recen papers on mahemaical models for immunoherapy and mixed immunoherapy and chemoherapy saring wih papers by A. Kuznesov and coworkers in he nineies (cf. [19,2], for insance) bu which are no in our focus here. Insead he presen paper follows he wo pioneering papers by J.M. Murray in 199 [16,17] (see also [24]) whose basic problems are modified in he following. One difficuly in applying he considered opimal conrol problems in cancer chemoherapy is he occurance of opimal soluions which are seldom or no used in medical pracice. A desired opimal soluion by he physician is he bang-bang conrol consising of a saring inerval wih maximal dose of drug followed by an inerval of zero-herapy ill he end of reamen (considering one cycle of he chemoherapeuic reamen). In paricular, he herapy should heoreically end wih an inerval of zero-herapy o have he required minimum of he umor cells populaion also somewha laer han a he pracical end of reamen. To obain opimal soluions of his ype ofen a suiable choice of he objecive funcions is proposed (cf. [14, 17, 19, and 24]). The aim of he presen paper is o circumven his difficuly aking ino accoun he resisance of he umor cells agains drug (and furher using he size of he umor cells populaion a he final ime as he naural objecive funcion). Acquired and inrinsic resisance of he umor cells agains drug is an imporan bu very complex phenomen in umor herapy (cf. [8, 12]) and relaed deerminisic models [4, 12, 14] and sochasic ones [2, 3] in dealing wih i are developed recenly. In our highly simplified model we only consider a summarizing effec of resisance by inroducing a ime-dependen resisance facor in fron of he loss funcion of he umor cells in he deerminisic differenial equaion for he umor growh. In paricular, we do no disinguish beween drug sensiive and resisan umor cells like in [4, 12, 14]. Furher, we deal wih wo basic problems where in each problem we have wo resricions, namely he usual poinwise inequaliy for he conrol funcion (which is in he dose of he drug adminisered per uni of ime) and an inegral inequaliy for he loss funcion of he normal cells in he firs problem and for he drug dose iself in he second problem. To keep he mahemaical analysis simple oher resricions like he poinwise limi for he size of he populaion of he normal cells like in [15, 17, and 24] are no

3 334 Werner Krabs, Lohar von Wolfersdorf aken ino consideraion. There is only one dynamic equaion for he growh and suppression of he umor cells and no one for he normal cells (bu which could be easily supplemened). Boh opimal conrol problems show he desired effec ha for (properly defined) srong resisance he above-named bang-bang conrol is he unique opimal conrol (cp. wih he resuls in [3, 26], for insance). Wih respec o general resisance we have anoher picure. In he firs problem in case of weak resisance he opimal conrol is he non desired opposie bang-bang conrol wih saring inerval of zero-herapy and final inerval of maximal drug dose. On he oher hand, in he second problem for general resisance an opimal conrol similar o he desired bang-bang conrol saring and ending wih a subinerval of zero-herapy is o be expeced as an example wih Gomperzian growh show (cp. wih oher forms of opimal soluions in [14, 17], for insance). So, especially wih respec o weak resisance he second problem seems preferable o he firs problem. The plan of he paper is as follows. Afer performing he mahemaical modelling in Secion 2 we invesigae he firs opimal conrol problem in Secion 3 and he second opimal conrol problem wih he example for Gomperzian growh in Secion 4. 2 Mahemaical Models We denoe he ime-dependen number of cancer cells in he umor by a funcion T = T (), IR +, which we assume o be differeniable wih derivaive T (). The emporal developmen of he umor cells populaion T () in a given inerval [, f ] is governed by he differenial equaion T () = [f(t ()) ϕ()l(m())]t (), [, f ], (2.1) wih he iniial condiion T () = T >. (2.2) The funcion f = f(), T describes he dynamics of he umor populaion F (T ) = f(t )T if here is no adminisraion of drugs. We assume f C 1 (IR + ) wih f(t ) >, f (T ) < for all relevan T. This is fulfilled for many of he commonly used dynamics as Gomperz, logisic (Verhuls-Pearl)

4 Two opimal conrol problems in cancer chemoherapy 335 and oher growh laws in an inerval [, θ] wih maximal umor populaion θ (cf. [5, 1, 16, 22, 24]). By L = L(M), M we denoe he desrucion rae of he drug level M. We assume ha his loss funcion L C 2 (IR + ) saisfies L() = and L (M) > for all relevan M. This is fulfilled, for insance, for linear and fracional linear ( sauraed ) funcion L (cf. [1, 16, 24]). The drug level funcion M = M() obeys he linear differenial equaion and he iniial condiion Ṁ() = δm() + V (), [, f ], (2.3) M() = (2.4) wih a posiive drug decay rae δ where V () denoes he drug dose ha is adminisered per uni of ime a ime [, f ]. In he following we assume δ, hus including he mahemaical limi case δ = of no drug decay. The drug dosis V = V () per uni of ime is considered as he conrol funcion in he model. We assume i o be a bounded measurable funcion, i.e. V L (, f ), and o saisfy he poinwise condiion V () A for a.a. [, f ] (2.5) where A > is a prescribed consan, he maximum drug dosis per uni of ime. Furher, below we require addiionally an inegral condiion which we regard as responsible for he compaibiliy of he reamen. The new feaure in his model is he inroducion of he funcion ϕ = ϕ(), [, f ], which is assumed o be in C 1 [, f ] saisfying ϕ() > in [, f ) and normed by ϕ() = 1. The facor ϕ in Eq. (2.1) should - in a mos simple way - describe he oal effec of inner influences (like drug resisance) and oher ones (like accompanying herapies) on he desrucion rae of he umor cells by he drug during he reamen. Especially, he influence of he drug resisance of he umor cells will be expressed by a funcion ϕ C 1 [, f ] wih ϕ() = 1, ϕ() in [, f ] and ϕ() > in [, f ]. (We call such a funcion a resisance facor in he following). The inegral condiion in problem 1 now reads

5 336 Werner Krabs, Lohar von Wolfersdorf ϕ ()L (M())d B (2.6) wih a prescribed consan B > where L (M) is he desrucion rae of he normal cells for which we assume he same properies as for he loss funcion L(M) of he umor cells above and ϕ C 1 [, f ] wih ϕ () > in [, f ), ϕ () = 1 is a weigh funcion possessing he analogous meaning for he normal cells as ϕ for he umor cells. In problem 2 we simply require ha wih a given consan B >. V ()d B (2.7) The aim of chemoherapeuic reamen is o make he umor cells populaion T ( f ) a he end of he reamen as small as possible. In view of Eq. (2.1) his can be wrien in he usual form of he minimum condiion [f(t ()) ϕ()l(m())]t ()d min. (2.8) We furher remark ha for a given V L (, f ) he soluion of (2.3), (2.4) has he form M() = M[V ]() = e δ(s ) V (s)ds, [, f ] which implies M C[, f ]. By our assumpions on f, ϕ, L we hen have T C 1 [, f ] for he corresponding soluion T = T [V ] of Eq. (2.1). Our opimal conrol problems are now defined by he minimum condiion (2.8) for he sae equaions (2.1) - (2.4) wih he consrains (2.5), (2.6) (problem 1 ) or (2.5), (2.7) (problem 2 ). Here he inegral consrains (2.6) and (2.7) can be aken, respecively, in he form

6 Two opimal conrol problems in cancer chemoherapy 337 Q( f ) B or U( f ) B (2.9) where he addiional sae funcions Q = Q() and U = U() are given by he inegrals Q() = ϕ (s)l (M(s)ds and U() = V (s)ds, respecively, or equivalenly by he addiional sae equaions wih Q() = and wih U() =, respecively. Q() = ϕ ()L (M()), [, f ] (2.1) U() = V (), [, f ] (2.11) These opimal conrol problems always have soluions as follows by adaping he exisence proof by J.M. Murray in [16] on he basis of Theorem in [1] (aking ino accoun ha for he admissible conrol V () = in [, f ] he sae equaion (2.1) has a coninuous soluion T () in [, f ] wih finie T ( f ) and because of he finie inerval [, f ] also he parameer δ = in Eq. (2.3) is possible). Finally, we simplify he mahemaical analysis for our problems slighly by applying he usual subsiuion y = lnt for T >. Then he differenial equaion (2.1) is ransformed ino ẏ() = f(e y() ) ϕ()l(m()), [, f ] (2.12) and he iniial condiion (2.2) reads The minimum condiion (2.8) akes he form y() = y = lnt. (2.13)

7 338 Werner Krabs, Lohar von Wolfersdorf [f(e y() ) ϕ()l(m())]d Min. (2.14) The opimal conrol problems o be solved are hen given by he minimum condiion (2.14) for he sae equaions (2.12), (2.13), (2.3), (2.4), and (2.1) or (2.11), respecively, under he consrains (2.5), (2.9). 3 Soluions of he firs problem We deermine opimal soluions of problem ( ), (2.5), (2.9), (2.1) as usual wih he aid of he maximum principle [9]. The Hamilonian of he problem is given by H(, y, M, Q, V, p 1, p 2, p 3, λ ) = (f(e y ) ϕ()l(m))(p 1 λ ) (3.1) +(V δm)p 2 + ϕ ()L (M)p 3 wih he parameer λ and he adjoin sae funcions p k, k = 1, 2, 3. If (ŷ, ˆM, ˆQ, ˆV ) is an opimal quadruple here exis a number λ and hree funcions p k C 1 [, f ], k = 1, 2, 3 wih (λ, p 1, p 2, p 3 ) (,,, ) saisfying he differenial equaions ṗ 1 () = f (eŷ() )eŷ() (p 1 () λ ) (3.2) ṗ 2 () = ϕ()l ( ˆM())(p 1 () λ ) + δp 2 () ϕ ()L ( ˆM())p 3 () (3.3) ṗ 3 () = (3.4) in [, f ] and he ransversaliy condiions in f p 1 ( f ) =, p 2 ( f ) =, and p 3 ( f ), p 3 ( f )( ˆQ( f ) B) = (3.5) such ha for a.a. [, f ] he maximum condiion

8 Two opimal conrol problems in cancer chemoherapy 339 ˆV ()p 2 () = max V A [V p 2()] (3.6) is valid. From (3.4) and (3.5) i follows ha p 3 is a nonposiive consan which vanishes if ˆQ( f ) < B. We define p 1 () = p 1 () λ and g() = f (eŷ() )eŷ() >, [, f ]. (3.7) Then from (3.2) we have p 1 () = g() p 1 () which gives p 1 () = p 1 () exp( From p 1 ( f ) = we obain g(s)ds), [, f ]. (3.8) p 1 ( f ) = p 1 () exp( which shows ha p 1 () for all [, f ]. We furher pu From (3.3) we ge g()d) = λ (3.9) h() = ϕ()l ( ˆM()) p 1 () ϕ ()L ( ˆM())p 3. (3.1) which yields p 2 () = e δ H() wih ṗ 2 () = δp 2 () + h(), [, f ] (3.11) H() = p 2 () + In view of p 2 ( f ) = we have e δs h(s)ds, [, f ]. (3.12)

9 34 Werner Krabs, Lohar von Wolfersdorf which implies p 2 () = e δ h()d (3.13) p 2 () = e δ e δs h(s)ds, [, f ]. (3.14) Now we disinguish he wo cases B Q A ( f ) ϕ ()L (M A ())d (3.15) where M A () = A δ [1 e δ ] if δ > o, A if δ = is he soluion of (2.3), (2.4) for V () = A a.e. in [, f ], and B < Q A ( f ) ϕ ()L o (M A ())d. (3.16) If (3.15) is fulfilled we have he opimal soluion ˆV () = A a.e. in [, f ]. So we can assume (3.16) in he following. In his case he equaliy ˆQ( f ) ϕ ()L ( ˆM())d = B (3.17) for he opimal soluions mus hold. We prove his by conradicion. If ˆQ( f ) < B we have p 3 =. In he anormal case λ = by (3.9), (3.8) and (3.1), (3.14) his implies p 1 () = p 1 () = and p 2 () = in [, f ] which conradics he condiion (λ, p 1, p 2, p 3 ) (,,, ). In he normal case λ > by (3.8) we would have p 1 () < and hence by (3.1) also h() < in [, f ] which by (3.14) yields p 2 () > in [, f ]. The maximum

10 Two opimal conrol problems in cancer chemoherapy 341 condiion (3.6) hen would give ˆV () = A for a. a. [, f ]. This implies Q A ( f ) = ˆQ( f ) < B, a conradicion o (3.16). Therefore, (3.17) and p 3 < hold rue. We furher show because of he condiion (3.17) he abnormal case λ = canno occur. Namely, from λ = as before we obain p 1 () = p 1 () = in [, f ] implying h() = ϕ ()L ( ˆM())p 3 >, [, f ] from (3.1). By (3.14) i follows ha p 2 () < in [, f ]. Then (3.6) yields ˆV () = for a.a. [, f ] which by (2.3), (2.4) leads o ˆM() = in [, f ] and by L() = o ˆQ( f ) =, a conradicion o (3.17). Summing up, in he case (3.16) equaliy (3.17) is valid and we have λ >, p 1 () < in [, f ] and p 3 <. We inroduce he funcions q() = L ( ˆM()) L ( ˆM()), () = ϕ() p 1() ϕ ()q()p 3 (3.18) so ha by (3.1) we have h() = L ( ˆM()) () wih signh() = sign (). Now we make he assumpion ha is a sricly increasing funcion in [, f ] which is fulfilled if d d () d d [ϕ() p 1() ϕ ()q()p 3 ] > in (, f ). (3.19) We disuingish he hree cases (i) () (ii) () <, ( f ) (iii) () <, ( f ) >. In case (i) we have () > () in [, f ] implying h() > in (, f ] and p 2 () < in [, f ) by (3.14). The condiion (3.6) hen yields he soluion ˆV () = a.e. in [, f ] which is no possible. In case (ii) we have () < ( f ) in [, f ] which gives h() < in [, f ] and p 2 () > in [, f ] by (3.14) again. In view of (3.6) hen ˆV () = A a.e. in [, f ] which is also no allowed in he case of (3.16).

11 342 Werner Krabs, Lohar von Wolfersdorf I remains he case (iii). By he sric monooniciy of here exiss exacly one 1 (, f ) wih ( 1 ) =, () < in [, 1 ), and () > in ( 1, f ]. This implies he analogous inequaliies for h. Therefore he funcion H in (3.12) is sricly decreasing from H() = p 2 () o H( 1 ) < p 2 () and hen sricly increasing from H( 1 ) o H( f ) =. If now p 2 () were rue we would ge H() < and hence p 2 () < in (, f ). This would imply ˆV () = a.e. in [, f ] again. Therefore, i mus be p 2 () >. Then here exiss exacly one (, 1 ) wih H( ) =, H() > in [, ) and H() < in (, f ] which implies p 2 () > in [, ) and p 2 () < in (, f ]. The maximum condiion (3.6) yields he unique opimal soluion ˆV () = { A for a.a. [, ) for a.a. (, f ] (3.2) where (, f ) can be defined as he (unique) soluion of he equaion wih ˆM() = ϕ ()L ( ˆM())d = B (3.21) { A δ [1 ēδ ] if δ >, A if δ = for [, ] A δ [eδ 1]ē δ if δ >, A if δ = for (, f ] following from (3.17) and (2.3), (2.4) wih (3.2). We summarize he resul in THEOREM 3.1 (i) Le (3.15)be fulfilled. Then problem 1 has he unique opimal soluion ˆV () = A a.e. in [, f ]. ii) Le (3.16) be fulfilled and he funcion in (3.18) sricly increasing. Then problem 1 has he unique opimal soluion (3.2) wih (3.21). REMARKS. The monooniciy assumpion on in Theorem 3.1 is an implici condiion on ϕ (and ϕ ) since he funcions p 1 by (3.8) and q by (3.18) in general depend on he opimal soluion ˆV of he problem wih he funcion

12 Two opimal conrol problems in cancer chemoherapy 343 ϕ in (2.1) (and ϕ in (2.6)). Bu his dependence can be well derived from (3.2) wih (3.21) and Eqs. (2.1), (2.3). Moreover, in he imporan paricular case L = cl wih a consan c > and L C 1 (IR + ) (cf. [16, 17]) we have q() = c and for ϕ () = 1 in [, f ] he sufficien condiion (3.19) reduces o he simple condiion ϕ() + g()ϕ() < in (, f ) (3.22) wih he posiive funcion g = f ( ˆT ) ˆT by (3.7). Furher, in he special case of Gomperzian growh f(t ) = λln θ T (λ, θ > ) we have g = λ, a consan which is independen of he opimal soluion ˆV. Condiion (3.22) is for insance saisfied, if for some γ >. ϕ() = exp( ( g(s)ds + γ)), [, f ], In general, i remains he dependence of on he (negaive) parameer p 3 or equivalenly on he (posiive) quoien p 3 / p 1 () which are no direcly expressed by he opimal soluion ˆV. To avoid his dependence we derive a furher sufficien crierion for he opimal soluion (3.2) in he sequel. LEMMA 3.2 Under he condiions (3.16) and d d [ρ()p 2()] < in (, f ) (3.23) wih a nonnegaive funcion ρ C 1 (, f ) he opimal soluion of problem 1 is uniquely deermined and has he form (3.2). Proof. Because of (3.23) he opimal soluion canno conain singular pars in subinervals where ṗ 2 () = p 2 () = and pars of he form { a.e. in [1, τ) ˆV () = A a.e. in (τ, 2 ] wih 1 < τ < 2 f where p 2 () in ( 1, τ), p 2 (τ) =, p 2 () in (τ, 2 ) and ṗ 2 (τ). Furher, he soluions ˆV () = a.e. in [, f ] and

13 344 Werner Krabs, Lohar von Wolfersdorf ˆV () = A a.e. in [, f ] are no possible in view of (3.16) wih (3.17). This proves he lemma. In view of (3.11) he condiion (3.23) can be wrien in he form [ ρ() + δg()]p 2 () + ρ()h() < in (, f ) (3.24) wih h defined in (3.1). Taking ρ() = exp( µ(s)ds), µ C(, f ) and r() = µ() + δ C(, f ) condiion (3.24) simply wries By (3.1) and (3.14) his means where r()p 2 () + h() < in (, f ). 1 () + p 3 2 () < in (, f ) (3.25) 1 () = ϕ()l ( ˆM()) p 1 () r() e δ( s) ϕ(s)l ( ˆM(s)) p 1 (s)ds, f 2 () = r() e δ( s) ϕ (s)l ( ˆM(s))ds ϕ ()L ( ˆM()). We now choose r C(, f ) such ha 2 () = in (, f ), i.e. r() = f f e δ ϕ ()L ( ˆM()) e δs ϕ (s)l ( ˆM(s))ds Then (3.25) simplifies o he condiion 1 () < in (, f ) or defining furher he quoien q 1 () = p 1() p 1 () = exp( f ( ˆT (s)) ˆT (s)ds) > (3.26).

14 Two opimal conrol problems in cancer chemoherapy 345 by (3.7), (3.8) o he inegral inequaliy f q 1 ()ϕ() e δs ϕ (s)l ( ˆM(s))ds f > q()ϕ () e δs ϕ(s)l ( ˆM(s))q 1 (s)ds (3.27) in (, f ) where q is defined in (3.18). A sufficien condiion for (3.27) is he differenial condiion f d d [q 1()ϕ()] e δs ϕ (s)l ( ˆM(s))ds f < d d [q()ϕ ()] e δs ϕ(s)l ( ˆM(s))q 1 (s)ds in, ( f ). (3.28) Summing up we obain THEOREM 3.2 i) Le (3.16) and (3.27) wih (3.18), (3.26) be fulfilled. Then problem 1 has he unique opimal soluion (3.2) wih (3.21). ii) The inegral condiion (3.27) is saisfied if he differenial condiion (3.28) is valid. REMARKS. The condiions (3.27) and (3.28) do no conain he unknown parameers p 3 and p 1 (). In he paricular case L = cl wih ϕ () = 1 from (3.28) we ge he condiion (3.22) again. Finally, we briefly deal wih he cases where in Theorem 3.1 he funcion is sricly decreasing and he inequaliies (3.19) and (3.27) in Theorems 3.1 and 3.2, respecively, are fulfilled wih he opposie signs. In paricular, his is he case if L = cl and ϕ() = ϕ () = 1 on [, f ]. Then he above analysis shows ha he unique opimal soluion of problem 1 is ˆV () = { for a.a. [, ) A for a.a. (, f ] (3.29)

15 346 Werner Krabs, Lohar von Wolfersdorf where (, f ) is he (unique) soluion of he equaion ϕ ()L ( ˆM())d = B wih ˆM() = for [, ] and ˆM() = A δ [dδ( 1] if δ >, A( ) if δ = for [, f ] following from (3.17) and (2.3), (2.4) wih (3.29) again. In case of he condiions (3.19) or (3.27), (3.28) for a resisance facor ϕ (wih some associaed ϕ ) we say ha we have srong resisance of he umor cells agains he drug, and in case of hese condiions wih he opposie sign weak resisance. 4 Soluions of he second problem Problem ( ), (2.5), (2.9), (2.11) possesses he Hamilonian H(, y, M, U, V, p 1, p 2, p 3, λ ) = (f(e y ) ϕ()l(m))(p 1 λ ) + (V δm)p 2 + V p 3 (4.1) wih he parameer λ and he adjoin sae funcions p k, k = 1, 2, 3. If (ŷ, ˆM, Û, ˆV ) is an opimal quadruple, by he maximum principle [9], here exis a number λ and hree funcions p k C 1 [, f ], k = 1, 2, 3 wih (λ, p 1, p 2, p 3 ) (,,, ) saisfying he differenial equaions ṗ 1 () = f (eŷ() )eŷ() (p 1 () λ ) (4.2) ṗ 2 () = ϕ()l ( ˆM())(p 1 () λ ) + δp 2 () (4.3) in [, f ] and he ransversaliy condiions in f ṗ 3 () = (4.4)

16 Two opimal conrol problems in cancer chemoherapy 347 p 1 ( f ) =, p 2 ( f ) =, and p 3 ( f ), p 3 ( f )(Û( f ) B) = (4.5) such ha for a.a. [, f ] he maximum condiion ˆV ()(p 3 () + p 3 ) = max V A [V (p 2() + p 3 )] (4.6) holds. By (4.4), (4.5) p 3 is a nonposiive consan which vanishes if Û( f ) < B. We remark ha in he limi case δ = in view of (2.3), (2.4) and (2.11) he quaniies U and M coincide. Hence U, p 3 could be omied and formally p 2 () + p 3 replaced by a new p 2 (). We define he funcions p 1 and g as in problem 1 wih he relaions (3.7) - (3.9). Furher we have he relaions (3.11) - (3.14) for p 2 if we replace he funcion h in (3.1) by h () = ϕ()l ( ˆM(), p 1 (), [, f ]. (4.7) Discussing he opimal soluions of problem 2 we disinguish he wo cases B f A and B < f A. For B T f A he obvious soluion is ˆV () = A for a.a. [, f ]. For B < f A we have he equaliy Û( f ) = ˆV ()d = B (4.8) and he inequaliies p 3 <, λ >, and p 1 () < in [, f ] which can be shown as above in problem 1. By (4.7) his implies h () < in [, f ] which by (3.11) and (3.14) gives ṗ 2 () δp 2 () <, p 2 () > in [, f ). (4.9) If addiionally ϕ( f ) > hen also h ( f ) < and consequenly ṗ 2 ( f ) <. From (4.9) we obain a firs resul abou he form of he opimal soluions in he case B < f A. LEMMA 4.1

17 348 Werner Krabs, Lohar von Wolfersdorf For B < f A he opimal soluions of problem 2 do no conain a soluion par of he form ˆV () = Aa.e. for [τ, f ], τ [, f ) (4.1) and if ϕ( f ) > hey do no conain singular pars in inervals of he form [τ, f ] wih τ [, f ). Proof: The asserion (4.1) for τ = follows from (4.14). For τ > we have p 2 () + p 3 in (τ, f ] and p 2 (τ) + p 3 = implying ṗ 2 (τ), bu since p 2 (τ) = [p 2 ( f ) + p 3 ] by (4.9) i mus be ṗ 2 (τ) < δp 2 (τ) and ṗ 2 <. The proof for he singular pars is a consequence of he condiion ṗ 2 () = in [τ, f ] which leads o a conradicion o ṗ 2 ( f ) < from (4.9). Lemma 4.1 shows ha he opimal soluions of problem 2 end wih an inerval of zero-herapy if ϕ( f ) >. We now give a sufficien condiion for he opimal soluions being of he (in pracice desired) bang-bang conrol ype. LEMMA 4.2 Under he condiions B < f A and p 2 () < in (, f ) (4.11) he opimal soluion of problem 2 is uniquely deermined and has he form ˆV () = { A for a.a. [, ) for a.a. (, f ] (4.12) where = B/A (, f ). Proof. Since ṗ 2 () in (, f ) he opimal soluion does no conain singular pars. Furher, i does no conain pars of he forms ˆV () = for a.a. [, τ], τ (, f ] and ˆV () = { for a.a. (1, τ) A for a.a. (τ, 2 ) ( 1 < τ < 2 f )

18 Two opimal conrol problems in cancer chemoherapy 349 The firs one is impossible for τ = f because of (4.8) and for τ < f since we could have p 2 () + p 3 in (, τ) and p 2 (τ) + p 3 implying ṗ 2 (τ). For he second one we obain p 2 () + p 3 in ( 1, τ) and p 2 () + p 3 in (τ, 2 ) yielding ṗ 2 (τ) again. This proves he form (4.12) of he opimal soluion ˆV wih unique value following from (4.8). The proof can also be given direcly by using he fac ha (4.11) implies p 2 () > for all [, f ) and discussing he wo cases p 2 () + p 2 and p 2 () + p 3 >. By equaions (3.11) and (4.7) he condiion (4.11) is equivalen o where δp 2 () < φ() in (, f ) φ() = h () = ϕ()l ( ˆM() p 1 () > in [, f ) (4.13) wih φ( f ) and by (3.14) equivalen o he inegral inequaliy ψ() φ() δ e δ( s) φ(s)ds > in (, f ). (4.14) If (4.14) holds he opimal soluion is given by (4.12). In paricular, his is fulfilled for all posiive funcions ϕ in he limi case δ = suggesing ha (4.14) is no a oo srong condiion on ϕ for sufficienly small δ >. This can be underlined in he simple case of Gomperz growh (cf. [7, 1, 17, 23-25]) where and a linear loss funcion In his case we find ha f(t ) = λln θ, T >, (λ >, θ > ) T L(M) = km, M, (k > ). g() = f (eŷ() )eŷ() = λ, h () = kϕ() p 1 ()e λ, [, f ], and (4.14) urns ou o be equivalen wih

19 35 Werner Krabs, Lohar von Wolfersdorf ϕ()e (λ δ) δ ϕ(s)e (λ δ)s ds > for all (, f ). (4.15) If we pu ϕ() = e (λ δ), [, f ], and assume ha λ > δ, hen i follows ha ϕ C 1 [, f ], ϕ() = 1, ϕ() < and ϕ() > for all [, f ]. Furher (4.15) urns ou o be equivalen o (1 δ( f ) > for all (, f )) δ 1 <. This shows ha (4.15) can be saisfied for sufficienly small δ > and a suiable choice of ϕ. The inequaliy (4.14) is fulfilled if we have φ() < in (, f ), (4.15) since inegraion by pars of he inegral in (4.14) yields ψ() = e δ [e δ f φ( f ) e δs φ(s)ds] > in (, f ) due o φ( f ) and (4.15). Differeniaing (4.13) and using p 1 = g p 1 we furher have where φ() = p 1 ()L ( ˆM())[ ϕ() + {g() + m()}ϕ()] m() = 1 d L ( ˆM(u) d [L ( ˆM())] = L ( ˆM(u) ˆM() L ( ˆM()). (4.16) Therefore, in view of p 1 () < in [, f ] and L (M) >, condiion (4.15) is equivalen o he differenial inequaliy

20 Two opimal conrol problems in cancer chemoherapy 351 ϕ() + [g() + m()]ϕ() < in (, f ) (4.17) where m = m() is given by (4.16) and g = g( ˆT ) by (3.17), i.e. g() = f ( ˆT ()) ˆT () <, [, f ]. (4.18) Condiion (4.17) has he same form as condiion (3.22) and is like his in general an implici condiion on ϕ. Summing up, by Lemma 4.2 and (4.14) - (4.18) we obain THEOREM 4.3 (i) Under he condiions B < f A and (4.14) he opimal soluion of problem 2 is uniquely deermined and has he form (4.12). ii) Asumpion (4.14) is saisfied if he condiion (4.17) wih (4.18) and (4.16) holds rue. REMARKS. For a linear loss funcion L we have m() = in [, f ] and he condiion (4.17) reduces o (3.22). As for problem 1 we say in case of (4.14) for a resisance facor ϕ ha here is a srong resisance of he umor cells agains he drug. We conclude he paper working ou he simple case of Gomperz growh (cf. [7, 1, 17, 23-25]) f(t ) = λln θ T (λ >, θ > ) (4.19) wih a linear loss funcion L(M) = km (k > ) as an example for wha can happen for general resisance. In his case we have for y = lnt he explici expression y() = lnt e λ + λlnθ[1 e λ ] k e λ( s) ϕ(s)m(s)ds

21 352 Werner Krabs, Lohar von Wolfersdorf and he minimum condiion for y( f ) leads o he maximum condiion which can be wrien in he form e λs ϕ(s)m(s)ds max where p() = e δ p()v ()d max (4.2) e (λ δ)s ϕ(s)ds. (4.21) The maximum problem (4.2) where V L (, f ) saisfies he resricions (2.5) and (2.7) is a linear problem of he form of he Neyman-Pearson lemma and can be solved in explici form. Le be B < f A. For (4.19) he condiion (4.17) is equivalen o he inequaliy d d [eλ ϕ()] < in (, f ). If his is fulfilled he problem has he soluion (4.12). We consider furher he opposie case ha d d [eλ ϕ()] > in (, f ) (4.22) incorporaing he limi case of non-resisance ha ϕ() 1 on [, f ]. For δ > he funcion (4.21) has he derivaive ṗ() = e δ [F () C], [, f ] where C = e (λ δ) f ϕ( f ), F () = e δs d ds [eλs ϕ(s)]ds.

22 Two opimal conrol problems in cancer chemoherapy 353 Under he assumpion (4.22) he funcion F is sricly decreasing in [, f ] from he value F () = e δs d ds [eλs ϕ(s)]ds > o F ( f ) =. Hence we have wo cases (i) F () C where ṗ() < in (, f ) so ha p() is sricly decreasing in [, f ] and (ii) F () > C where here exiss a unique (, f ) such ha p() is sricly increasing in [, ] and sricly decreasing in [, f ] ill p( f ) =. In case (i) he opimal soluion is given by (4.12). In case (ii) he opimal soluion has he form ˆV () = { a.e. in [, 1 ) and ( 2, f ] A a.e. in ( 1, 2 ) (4.23) where 1, 2 wih < 1 < < 2 < f are uniquely deermined by he equaions A( 2 1 ) = B, p( 1 ) = p( 2 ). In he paricular case ϕ() = 1 on [, f ] we have (e (λ δ) f e (λ δ) )e δ if λ > δ p() = ( f )e δ if λ = δ (e (λ δ) e (λ δ) f )e δ if λ < δ and C = e (λ δ) f, F () = { λ λ δ [e(λ δ) f 1] if λ δ λ f if λ = δ. Therefore, case (i) occurs if λ f 1 for λ = δ, δe (λ δ) f λ for λ > δ, and λe (δ λ) f δ for λ < δ, and case (ii) under he opposie inequaliies. We remark ha he opimal soluion (4.23) in case (ii) sars and ends wih an inerval of zero-herapy. References [1] F.H. Clarke, Opimizaion and Nonsmooh Analysis, Wiley, New York, [2] A.J. Coldman, J.H. Goldic, A model for he resisance of umor cells o cancel chemoherapeuic agens, Mah. Biosci. 65 (1983)

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