Nonlinear Analysis: Real World Applications 2 (2001) 455 – 465 www.elsevier.com/locate/na

Nonlinear mathematical model of pulsed-therapy of heterogeneous tumors Abdelkader Lakmechea; b; ∗ , Ovide Arinob a Laboratoire

ADEL, D epartement de Math ematiques, Facult e des Sciences, Universit e Djillali LIABES, BP. 89, Sidi Bel Abbes 22000, Algeria b Laboratoire de Math ematiques Appliqu ees, I.P.R.A., Avenue de l’Universit e, Universit e de Pau, 64000 Pau, France

Received 4 November 1999; received in revised form 4 February 2000; accepted 15 September 2000

Abstract In this paper, we deal with a nonlinear impulsive di0erential equations modelling the chemotherapy of a heterogeneous tumor. We consider the case of several drugs with instantaneous e0ects. We take into account the interactions between sensitive cells and drug resistant cells. We are interested in the stability of the disease. We also study the loss of stability and the bifurcation c 2001 Elsevier Science Ltd. All rights reserved. of nontrivial solutions.  Keywords: Impulsive di0erential equation; Nonlinear mathematical model; Chemotherapeutic treatment; Heterogeneous tumor

1. Introduction One of the principal causes of the failure of chemotherapeutic treatment of cancer is its resistant development.There are, in general, two types of resistance, acquired resistance, which comes from cellular mutations, and induced resistance coming from the chemotherapeutic use. The cancer might increase strongly if a noncross-resistant drug is not available. In order to counter the resistance of the cancer, we must understand the di0erent kinetics of the cancer, which might give us information on the e0ects of the resistance. The two types of resistant tumor cells are physically completely di0erent, and hence di0erently modelized. In this work, we are interested in induced resistance. We examine a nonlinear mathematical model, describing the dynamics of ∗ Corresponding author. Laboratoire ADEL, D9 epartement de Math9ematiques, Facult9e des Sciences, Universit9e Djillali LIABES, BP. 89, Sidi Bel Abbes 22000, Algeria. Fax: +213-756-1642. E-mail addresses: [email protected] (A. Lakmeche), [email protected] (O. Arino).

c 2001 Elsevier Science Ltd. All rights reserved. 1468-1218/01/$ - see front matter
PII: S 1 4 6 8 - 1 2 1 8 ( 0 1 ) 0 0 0 0 3 - 7

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a heterogeneous tumor, constituted by two compartments, the sensitive cells and the drug resistant cells. Many mathematical models on heterogeneous tumors exist in the literature (see [3–7,11,12]). In these models the authors distinguish between compartments of sensitive and resistant cells. Birkhead et al. [6] consider a linear model of heterogeneous tumor with four types of cells. This model consists of four linear ordinary di0erential equations, describing the mass of the compartments of sensitive, resistant, proliferating and nonproliferating cells. In particular, they are interested in determining qualitative strategies for the treatment. Other models are presented in Coldman and Goldie [10], and Rosen [23]. Coldman and Goldie [10] consider a probabilistic model, describing the resistance to drugs and to combination of drugs and its e0ects on the resistant cells. Rosen is concerned with the acquired resistance, which is independent of the drug dose, and considers the interactions between di0erent types of cells. Other authors also consider this model; for instance, Michelson and Leith [17–19], Birkhead et al. [5,6], Gyori et al. [13], Gregory et al. [12], Panetta [21,22]. It was Panetta [22], who recently presented a model on the heterogeneous tumor, di0erent from the previous ones. This model consists of a continuous part (linear ordinary di0erential equations) describing the dynamic of the cancer, and a discrete part (algebraic equations) describing the e0ects of the drug on the di0erent types of cancerous cells, which is called pulsed-therapy (see [22]). Panetta justiHes such a model by the fact that some drugs have an instantaneous e0ect since the reduction of the mass of the tumor follows immediately drug dose shot. Panetta’s model is discrete: this is justiHed by the clinical data being discretely collected, we model it discretely. Panetta [22] compares the results of his model and those of a linear model. He considers di0erent combinations of two drugs; the best one is used periodically to eradicate the disease. In earlier work, Panetta [21] considered a nonlinear impulsive model of chemotherapeutic treatment of a population, consisting of normal and tumor cells, taking into account the interaction between the di0erent types of cells. He considered the nonlinearities only in the di0erential equations, while the impulsions were linear. This model was considered in [15] with nonlinearities in both the di0erential and impulse equations. The impulsive models arise, generally, in the description of phenomena subjected to abrupt external changes, where the time of the change can be neglected, and the change can be modelled as jump in the phenomena under study. A rich literature on the theory of impulsive di0erential equations can be found in [1,2,16]. This new branch of di0erential equations developed quickly for the past 30 years (see [20]). In particular, important contributions have been made by Bainov and Simeonov [2], Lakshmikantham et al. [16], and their colleagues. In this work, we deal with a nonlinear impulsive di0erential equations modelling the chemotherapy of a heterogeneous tumor. We consider the case of several drugs with instantaneous e0ects described by impulses. We take into account the interactions between sensitive cells and drug resistant cells, which justiHes the nonlinearities in the di0erential equations with impulsive e0ects. We are interested in the stability of the disease. We also study the loss of stability and eventually the bifurcation of nontrivial solutions. In the next section, we present the model, while in Section 3 we reduce the problem to a Hxed point problem, and then we examine the stability of the trivial

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solution, the loss of stability and the bifurcation of nontrivial solutions. Section 4 is devoted to a brief conclusion and the appendix is the subject of the last section. 2. The model We consider the following heterogeneous tumor model x˙ = r1 (x; y)x;

(1)

y˙ = r2 (x; y)y;

(2)

x(tn+ ) = n (D; x(tn ); y(tn ));

(3)

y(tn+ ) = n (D; x(tn ); y(tn ));

(4)

where D is the drug dose administered, x and y are, respectively, the biomass of sensitive cells and drug resistant cells. The values r1 (x; y); r2 (x; y) are, respectively, the growth rates of sensitive and drug resistant cells. The values n (D; x(tn ); y(tn )) and n (D; x(tn ); y(tn )) are, respectively, the biomass of sensitive and resistant cells which survive after the nth drug dose D administered at the moment tn , the sequence (ti ) is strictly increasing. We consider that In := (n ; n ) is positive and that the positive quadrant is invariant with respect to the Low associated to (1) – (2), denoted by = ( 1 ; 2 ). The functions r1 ; r2 ; n and n , for n ∈ N ∗ , are considered smooth enough. Our model is more general than Panetta’s [22], both with regards to di0erential equations and impulsive equations. In Panetta [22], r1 and r2 are constant and positive, and i (D; x; y) = A(D)x and n (D; x; y) = B(D)x + C(D)y, where A; B and C are positive functions depending on the drug dose D administered, a combination of drugs is considered and administered periodically. Our aim is to study the case of several drugs administered one at a time, in sequence, with some period T : drug 1 is administered at time t1 , drug 2 at time t2 , and so on until the last drug, drug n, then the sequence of shots is regarded: drug 1 at time, drug 2 at time tn + t2 ; : : : ; T = tn . In this work, without any loss of generality, we consider only two drugs. In this case, the period is T = t2 . To simplify, we denote by A the Hrst drug and by B the second one. The case of several drugs can be studied in the same way. The tumor is constituted by sensitive and resistant cells, the evolution of their biomass is governed by system (1), (2). At time t = t1 , the tumor biomass is equal to the biomass of sensitive cells x(t1 ) and the biomass of resistant cells y(t1 ). When a dose D of the drug A is administered at time t1 the tumor biomass becomes x(t1+ ) + y(t1+ ) = 1 (D; x(t1 ); y(t1 )) + 1 (D; x(t1 ); y(t1 )). The drug eliminates only a small fraction of the resistant cell biomass. To reduce a signiHcant resistant cell biomass, we administer a dose D of the drug B at the moment t = t2 , t2 ¿ t1 . We start again with drug A at time t2 + t1 , etc : : : . This will be used periodically until eradication of the tumor.

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We assume that the drugs have no e0ect in case of absence of the tumor, i.e. n (D; 0; y) ≡ 0;

∀y;

n (D; 0; 0) = 0;

∀D ∈ R:

D∈R

(5)

and (6)

Then the zero function veriHes system (1) – (4), it represents an equilibrium of system (8:1)–(8:4), it will be called the trivial solution. We say that Z =(x; y); x =x(t); y =y(t), is a solution of (1) – (4), if it is t2 -periodic, and it veriHes, respectively, (1), (2) on the interval (0; t1 ) ∪ (t1 ; t2 ), and (3), (4) at t1 . 3. Stability and critical cases If Z = (x; y) is a solution of problem (1) – (4) with the initial condition Z(0) = Z0 , then it veriHes Z(t2+ ) = I2 (D; (t2 ; I1 (D; (t1 ; Z0 )))) = Z0 ;

(7)

moreover, Z(t) = (t; Z0 ), for 0 6 t 6 t1 and Z(t) = (t; I1 (D; (t1 ; Z0 ))), for t1 ¡ t 6 t2 . Let  = (1 ; 2 ) be deHned by (D; Z) := I2 (D; (t2 ; I1 (D; (t1 ; Z)))):

(8)

Hence a t2 -periodic solution of problem (1) – (4) is a Hxed point of the map (D; :). From Iooss [14], Z0 is stable if the spectral radius, (@=@Z(D; Z0 )), of the derivative of (D; :), at Z0 , is less than 1 (see Appendix A). Theorem 1. The trivial solution is stable if


@2
@1

¡ exp(−r1 (0; 0)t2 ) (D; 0) (D; 0)
@x
@x and





@ 2 @ 1
¡ exp(−r2 (0; 0)t2 ):
(D; 0) (D; 0)

@y @y

(9)

(10)

The last two inequalities mean that the rates of destruction of the sensitive and resistant cells are suMciently large. In the case when the product @ 2 =@y(D; 0)@ 1 =@y(D; 0) exp(r2 (0; 0)t2 ) is close to 1, the resistance is very strong. In the following we examine the case where the above term is equal to 1. Assume that, for some given dose D0 ¿ 0 of the administered drugs A and B, we have @ 2 =@y(D0 ; 0)@ 1 =@y(D0 ; 0) exp(r2 (0; 0)t2 ) = 1. Take N (; Z) := Z − (D0 + ; Z): Then N (; Z) = 0 is equivalent to Z = (D0 + ; Z).

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We have N (0; 0)=0 and DZ N (0; 0) is singular, that is, properties necessary conditions for bifurcation are satisHed (see [9]). Put DX N (0; 0) = E. Then dim ker(E) = 1 = co dim R(E). Let P and Q be the projections onto ker(E) and R(E), respectively, such that P+Q=IdR2 ; PR2 =span{Y0 }=ker E, with Y0 = (−c=a; 1), and QR2 = span{X0 } = R(E), oNu X0 = (1; 0); a = 1 − @=@x1 (D0 ; 0) and c = −@=@x2 (D0 ; 0). Then (I − P)R2 = span{(1; 0)}, and (I − Q)R2 = span{(0; 1)}. The equation N (; Z) = 0 is equivalent to f1 (; (; )) = N1 (; )X0 + (Y0 ) = 0; f2 (; (; )) = N2 (; )X0 + (Y0 ) = 0;

(11)

where ; ); ( ∈ R and N = (N1 ; N2 ). From the Hrst equation of (11) we have @ @N1 (0; 0) @x f1 (0; 0; 0) = @) @x @) @2 @1 =1− (D0 ; 0) (D0 ; 0) exp(r1 t2 ) = 0: @x @x From the implicit function theorem (see [8]), we Hnd a constant * ¿ 0 and a unique continuous function )(; () deHned in a neighbourhood of zero, such that )(0; 0) = 0 and ∀ || ¡ *; |(| ¡ *, f1 (; (; )(; ()) = N1 (; )(; ()X0 + (Y0 ) = 0: To solve problem (1) – (4), it remains to study the following equation: f(; () = N2 (; )(; ()X0 + (Y0 ) = 0: We have f(; () = A( + B + o(|(| + ||)

(12)

where @f=@(0; 0) = A and @f=@((0; 0) = B. If AB = 0, we have (= = −B=A; for AB ¡ 0, we have supercritical bifurcation and for AB ¿ 0, we have subcritical bifurcation. The number of nontrivial solutions is equal to the number of the nontrivial solutions of Eq. (12). If AB = 0, it will be necessary to expand f to the higher order. Finally, we have the following conclusion: Theorem 2. If AB = 0; we have a bifurcation of nontrivial solutions. We have a subcritical case if AB ¿ 0; and if AB ¡ 0 we have a supercritical case. If BC = 0; we have an undetermined case. 4. Conclusion In this work, we have investigated a nonlinear impulsive model of chemotherapeutic treatment of heterogeneous tumor. We have studied the case of delivery of two drugs.

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We have studied the stability of the trivial solution, which is equivalent to the eradication of the tumor; we have shown that stability holds if the two inequalities (9), (10) are satisHed. After that, we have considered the case of strong resistance, that is when the rate of resistant cell destruction is equal to 1. In this case stability is lost and we have a bifurcation phenomenon if AB = 0, in this situation, the tumor is not eradicated, it may be very small but still is viable. The study of this model allows us to choose the best drug strategy, to be used periodically in order to have conditions (9), (10), and consequently to eradicate the tumor. In our model we have assumed that impulses are nonlinear. It would be interesting to consider the case where the right-hand side of Eqs. (1) – (4) depends functionally on the biomass of the cancer, for example, when the nonlinearities depend on the average, the minimum or the maximum of the cancer biomass, during the treatment. Appendix A.1. First partial derivatives of We have 

@ 1 (t; 0) d   @x1 dt  @ 2 (t; 0) @x1

i.e.



@ 1 (t; 0)  d  @x1 dt  0

  @F1 (0) @ 1 (t; 0)  @x1 @x2  = @ 2 (t; 0)   @F2 (0) @x1 @x2 



@F1 (0)   @x1 = @ 2 (t; 0)   0 @x2 0

 @ 1 (t; 0) @F1 (0)  @x1 @x2   @F2 (0)   @ 2 (t; 0) @x2 @x1 

@ 1 (t; 0)   @x1  @F2 (0)   0 @x2 0

 @ 1 (t; 0) @x2  ; @ 2 (t; 0)  @x2 0



 : @ 2 (t; 0)  @x2

Then @ 1 (t; 0)=@x1 =@ 1 (0; 0)=@x1 exp(tr1 (0; 0)); @ 2 (t; 0)=@x2 =@ 2 (0; 0)=@x2 exp(tr2 (0; 0)), @ 1 (t; 0)=@y ≡ 0 and @ 2 (t; 0)=@x ≡ 0. A.2. First partial derivatives of )(:; :) We have @=@(f1 (; (; )(; ()) = 0; ∀|| ¡ *; |(| ¡ *, then @)(0; 0) c =− @( a and @)(0; 0) = @

@,2 (0; 0) @,2 (0; 0) @ 1 (t2 ; 0) @,1 (0; 0) + @ @x @x @  −1 @,2 (0; 0) @ 1 (t2 ; 0) @,1 (0; 0) @ 1 (t1 ; 0) × 1− @x @x @ @x



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A.3. First partial derivatives of f @ @f(; () = (( − -2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))))) @( @( @-2 =1− (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 )))) @x

@ 1 × (t2 ; I2 (; (t1 ; )(; ()X0 + (Y0 ))) @x  @,1 (; (t1 ; )(; ()X0 + (Y0 )) × @x  @ 1 @ 1 @) @( (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) × @x @( @y @( @,1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @ 2 @) @( @ 2 (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) × @x @( @y @(

+

@ 1 (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))) @y  @-1 × (; (t1 ; )(; ()X0 + (Y0 )) @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @( @y @( +

@-1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @ 2 @) @( @ 2 (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) × @x @( @y @(

+

@-2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 )))) @y 

@,1 @ 2 × (t2 ; I1 (; (t1 ; )(; ()X0 +(Y0 ))) (; (t1 ; )(; ()X0 + (Y0 )) @x @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @( @y @( −

@,1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @) @( @ 2 @ 2 × (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @( @y @(

+

461

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 @-1 @ 2 (; (t1 ; )(; ()X0 + (Y0)) (t2 ; I1 (; (t1 ; )(; ()X0 +(Y0 ))) @x @y  @ 1 @( @) @ 1 (t1 ; )(; ()X0 + (Y0 ) × (t1 ; )(; ()X0 +(Y0 ) (; ()+ @y @( @x @(

+

@-1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @) @( @ 2 @ 2 ; × (t1 ; )(; ()X0 +(Y0) (; ()+ (t1 ; )(; ()X0 + (Y0 ) @x @( @y @(

@f @ 1 @-2 @,1 @ 1 (0; 0) (t1 ; 0) (0; 0) = 1 − (t2 ; 0) (0; 0) @x @x @( @x @x 

@ 1 @-2 @-1 @) @ 2 (t1 ; 0) (0; 0) (0; 0) − (0; 0) (t2 ; 0) @x @( @y @y @x @-1 @ 2 × + ; (0; 0) (t1 ; 0) @y @y

@f @,1 @-2 2 (0; 0) = 1 − (0; 0) r1 (0; 0) (0; 0) @( @x @x 

@-1 @) @-2 @-1 − (0; 0) r2 (0; 0) (0; 0)r1 (0; 0) (0; 0) + (0; 0)r2 (0; 0) @y @x @( @y +

and @f(; () @ = (( − -2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))))) @ @ =−

@-2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 )))) @

@-2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 )))) @x

@ 1 × (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))) @x  @,1 @,1 × (; (t1 ; )(; ()X0 + (Y0 )) + (; (t1 ; )(; ()X0 + (Y0 )) @ @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @ −

@,1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @ 2 @) @( @ 2 (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) × @x @ @y @ +

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463

@ 1 (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))) @y  @-1 @-1 (; (t1 ; )(; ()X0 + (Y0 )) + (; (t1 ; )(; ()X0 + (Y0 )) × @ @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @

+

@-1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @ 2 @) @( @ 2 (t1 ; )(; ()X0 +(Y0 ) (; ()+ (t1 ; )(; ()X0 +(Y0 ) × @x @ @y @ +

@-2 (; (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 )))) @y

@ 2 (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))) × @x  @,1 @,1 × (; (t1 ; )(; ()X0 + (Y0 )) + (; (t1 ; )(; ()X0 + (Y0 )) @ @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @ −

@,1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @) @( @ 2 @ 2 × (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @ +

@ 2 (t2 ; I1 (; (t1 ; )(; ()X0 + (Y0 ))) @y  @-1 @-1 × (; (t1 ; )(; ()X0 + (Y0 )) + (; (t1 ; )(; ()X0 + (Y0 )) @ @x  @ 1 @) @( @ 1 × (t1 ; )(; ()X0 + (Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @

+

@-1 (; (t1 ; )(; ()X0 + (Y0 )) @y  @) @( @ 2 @ 2 ; × (t1 ; )(; ()X0 +(Y0 ) (; () + (t1 ; )(; ()X0 + (Y0 ) @x @ @y @ +

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@-2 @f @-2 @ 1 (t2 ; 0) (0; 0) = (0; 0) − (0; 0) @ @ @x @x  @,1 @ 1 (t1 ; 0) @) @,1 (0; 0) + (0; 0) (0; 0) × @ @x @x @  @-2 @-1 @ 1 (t1 ; 0) @) @ 2 (t2 ; 0) @-1 − (0; 0) + (0; 0) (0; 0) ; (0; 0) @y @y @ @x @x @  @f @-2 @,1 @) @,1 @-2 (0; 0) = (0; 0) − (0; 0)r1 (0; 0) (0; 0) + (0; 0)r1 (0; 0) (0; 0) @ @x @ @ @ @x  @-1 @) @-2 @-1 − (0; 0)r2 (0; 0) (0; 0) + (0; 0)r1 (0; 0) (0; 0) = B: @y @ @x @ References [1] D. Bainov, P. Simeonov, Systems with Impulsive E0ect: Stability, Theory and Applications, Ellis Horwood, Chichester, 1989. [2] D. Bainov, P. Simeonov, Theory of Impulsive Di0erential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. [3] N. Bellomo, G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Modelling 20 (1994) 107–122. [4] M.C. Berenbaum, Dose-response curves for agents that impair cell reproductive integrity, Br. J. Cancer 23 (1969) 434–445. [5] B.G. Birkhead, W.M. Gregory, A mathematical model of the e0ects of drug resistance in cancer chemotherapy, Math. Biosci. 72 (1) (1984) 59–69. [6] B.G. Birkhead, E.M. Rakin, S. Gallivan, L. Dones, R.D. Rubens, A mathematical model of the development of drug resistance to cancer chemotherapy, Eur. J. Cancer Clin. Oncol. 23 (9) (1987) 1421–1427. [7] R.A. Burger, E.A. Grosen, G.R. Ioli, M.E. Van Eden, H.D. Brightbill, M. Gatanaga, P.J. DiSaia, G.A. Granger, T. Gatanaga, Host–tumor interaction in ovarian cancer spontaneous release of tumor necrosis factor and interleukin-1 inhibitors by puriHed cell populations from human ovarian carcinoma in vitro, Gynecol. Oncol. 55 (1994) 294–303. [8] H. Cartan, Cours de Calcul Di09erentiel, Edition, Herman, Paris, 1977. [9] S.N. Chow, J. Hale, Methods of Bifurcation Theory, Springer, Berlin, 1982. [10] A.J. Coldman, J.H. Goldie, Role of mathematical modeling in protocol formulation in cancer chemotherapy, Cancer Treat. Rep. 69 (10) (1985) 1041–1045. [11] G.P. Dotto, A. Weinberg, A. Ariza, Maligant transformation of mouse primary keratinocytes by Harvey sarcoma virus and its modulation by surrounding normal cells, Proc. Acad. Sci. USA 85 (1988) 6389–6393. [12] W.M. Gregory, B.G. Birkhead, R.L. Souhami, A mathematical model of drug resistance applied to treatment for small-cell lung cancer, J. Clin. Oncol. 6 (3) (1988) 457–461. [13] I. Gyori, S. Michelson, J. Leith, Time-dependent subpopulation induction in heterogeneous tumors, Bull. Math. Biol. 50 (6) (1988) 681–696. [14] G. Iooss, Bifurcation of maps and applications, Study of Mathematics, North-Holland, Amsterdam, 1979. [15] A. Lakmeche, O. Arino, Bifurcation of nontrivial periodic solutions of impulsive di0erential equations arising from chemotherapeutic treatment, Dynamics Control Discrete Impl. Systems 7 (2) (2000) 265–288. [16] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Di0erential Equations, World ScientiHc, Singapore, 1989.

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