- Email: [email protected]
Conflicting objectives in chemotherapy with drug resistance
BuIldn
of Mathematical
Biology, Vol. 59, No. 4, pp. 707-724, Elsevkr
Science
8 1997 Society for Mathematical 0092-8240/97
1997 Inc.
Biology
$17.00 + 0.00
SOWZ-8240@7)00013-X
CONFLICTING OBJECTIVES WITH DRUG RESISTANCE
IN CHEMOTHERAPY
?? M.
I. S. COSTA? and J. L. BOLDRINIt * Laborathio National de Computa@o Cientifica-CNPq, Rua Lam-o Miiller, 455, 22290-160, Rio de Janeiro, RJ, Brazil (Email:
[email protected])
Wniversidade Estadual de Campinas, Instituto de Matematica Estatistica e Cihcia CP 6065, 13081-970, Campinas, SP, Brazil (Email:
da Computa@o,
[email protected])
A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Spontaneously acquired drug resistance is accounted for, as well as the evolution in time of normal cells. In addition, optimization of conflicting objectives forms the aim of the chemotherapeutic treatment. For general cell growth, some results are given, whereas for the special case of Malthusian (exponential) growth of tumor cells and rather general growth rate for normal cells, the optimal strategy is worked out. The latter, from the clinical standpoint, corresponds to maximum drug concentration throughout the treatment. 0 1997 Society for Mathematical Biology
1. Introduction. Optimal use of drugs in cancer chemotherapy is hindered, amongst others, by several factors such as lack of detailed knowledge about the kill rates of the drugs, drug resistance, cell growth models and appropriate criteria for measuring toxicity. In the context of spontaneous acquisition of drug resistance, some optimal chemotherapeutic treatments were devised in Costa et al. (1992) and Costa et al. (1994). Maximum drug concentration throughout appeared as an optimal strategy and rest periods were not allowed. However, in clinical protocols, variable (or constant) levels of drug concentration interspersed with rest periods seem to be a clinical practice. Nonetheless, it should be stressed that clinical protocols used in practice are not necessarily optimal, but that they support the belief that maximum drug concentration throughout cannot be optimal in a satisfactory sense. In turn, this suggests that the protocol formed by the highest allowable level of drug concentration throughout should not be the sole optimal solution of the optimization of realistic theoretical models of cancer chemotherapy. 707
708
M. I. S. COSTA AND J. L. BOLDRINI
By utilizing normal cells as an indicator of toxicity (with the requirement that a given minimum level of normal cells should not be violated), Murray (1990a) and Murray (1990b) and Zietz and Nicolini (1979) were able to find optimal treatments that could include rest periods. From the point of view of modelling, unlike the ones used in Costa et al. (1992) and Costa et al. (1994), this toxicity criterion is noncumulative. However, the models in Murray (1990a) and Murray (199Ob) and Zietz and Nicolini (1979) did not include drug resistance. In Costa et al. (1995b) we attempted to devise optimal treatments when this feature was taken into account and toxicity, like in Murray (1990a) and Murray (199Ob) was modelled via a minimum level of normal cells that should not be violated along the treatment. Unfortunately, as soon as drug resistance was included in the model, the sole optimal strategy for general growth and drug kill rates was the maximum drug concentration throughout (again no rest periods were allowed). This issue may point to some features that are not present in the cited models, since in clinical practice there are impediments that preclude clinicians from resorting to protocols formed by the highest allowable level of drug concentration throughout. So in the light of the results that evidence maximum drug concentration to be almost always the optimal strategy in presence of drug resistance [although, rest periods might feature as candidates for optimal treatment when drug kinetics was also accounted for, but in a nonconclusive way, see Costa et al. (1995a)], we decided to take another approach. In Costa and Boldrini (1997) we set up a rather theoretical frame for a phenomenological notion of patient’s recuperation. In that setting it was proved that strategies with alternation of maximum drug concentration and rest periods could be optimal even in presence of drug resistance. This outcome tends to point towards a likely relationship between rest periods in continuous chemotherapy and mechanisms of recuperation. However, to make completely clear the necessity of inclusion of some sort of recuperation mechanism in order to obtain the possibility of rest periods in optimal therapies for models with drug resistance, it is still necessary to test another possibility: the reason for the fact that maximum drug injection throughout the treatment was in general the optimal strategy in the previous models might actually be due to imperfect choices of the performance criteria. As a matter of fact, there is an intrinsic conflicting situation in chemotherapy: from clinical experience it is evidenced that drugs not only kill malignant cells but normal cells as well. Therefore, to keep the patient up to a safe health state, one should, while minimizing the level of tumoral cells during the treatment, maximize the level of normal cells-a situation of conflicting objectives. In this case, it is not easy to reach a realistic compromise and devise a suitable unique performance
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
709
criterion involving level of tumoral cells and toxicity [like the one employed in Costa et al. (1992) and Costa et al. (199411. As we said above, this raises the question whether the cause of the absence of rest periods in optimal strategies was not due to the nonadequate mathematical choice of the functional. In an attempt to answer this question, instead of trying to build an adequate unique performance criterion, in this work we decided to attack directly the problem of optimal chemotherapy with conflicting functionals. The modelling of this multiple-objective situation can be accomplished in several ways (Swan, 1990); we will employ one that was also utilized by Zietz and Nicolini (1979), i.e., the minimization of tumoral cells and maximization of normal cells at the end of the treatment. The difference is in that our setting takes into account the dynamics of drug-resistant cells, increasing then the order of the system of differential equations (three equations, two of which are coupled), while in Zietz and Nicolini (1979) the two equations are uncoupled. In fact, as it will be shown, this renders the mathematical analysis of determining optimal strategies quite cumbersome. In the next section we set up the mathematical problem of finding optimal treatments which deal with conflicting objectives. We seek solutions to this problem resorting to Pareto optimization techniques (Leitmann, 1974) and some results are derived for general growth and drug kill rates. However, due to the difficult intrinsic non-linearities of the problem, these partial results do not lead to any conclusive form of the optimal strategies. On the other hand, in Section 3 we will be able to describe the optimal treatment for the special case of exponential growth of tumor cells and linear drug kill rate, but with rather general nonlinear growth rate for normal cells. The result, which is relegated to be discussed in Section 4, is that rest periods are still not allowed.
2. A Pareto Optimal@ Criterion: Preparatory Results. In order to carry out the analysis of tumor growth submitted to chemotherapy, some assumptions are taken: (a) The tumor will be viewed as a cell population undergoing homogeneous growth, that is, it does not depend on the cell position within the tumor. (b) The tumor will consist also of drug-resistant cells whose growth rate depends not only on the size of its own population, but on the size of the sensitive cells as well. This latter dependence is due to a randomly spontaneous mutation during mitosis towards drug resistance, which will occur according to a constant probability. In this way no sensitive cell becomes drug resistant during its life time; only their daughter cells may acquire drug resistance by spontaneous mutation during mitosis. A
710
(4
M. I. S. COSTA AND J. L. BOLDRINI
biological validation of this kind of drug resistance was performed by in vitro experiments with T-cell lymphoblastic cell line CCRF-CEM. A description of these experiments can be found in Vendite (1988). [The importance of drug resistance in designing chemotherapeutic protocols is also emphasized in Skipper (1983)]. The kill rate of the drug (number of cells killed/unit drug concentration) will be considered as function of the size of the sensitive cells population. As for the normal cells the kill rate will also be a function of their own population size.
The following system is a model for the behaviour of tumoral and normal cells submitted to chemotherapy when the assumptions mentioned above are taken into account:
z dY
=AjYy)
+
cqf(y)(y 4,
=yf(y)
-
uWg(y 4,
(2.1)
dn
!x(O)
=x0,
y(O) =y,,
n(O)= no.
Here, t > 0 represents the elapsed time; y(t) E R stands for the total number of tumor cells at time t, while x(t) E R stands for the number of drug-resistant cells within the tumor and n(t) is the number of normal cells. Clearly, any initial condition (x0, yo> is such that x0 < y,; f(y) and fr(n> are the specific growth rates for the tumor and normal cells, respectively; 0 < (Y< 1 is the fraction per unit of time of the drug sensitive cells that mutates into drug resistant cells; 0 < u(t) < u, is the instantaneous drug concentration at the tumor site, assumed to be limited, i.e., u, < +a (this limit could be set by the maximum allowable level of drug injection, which, in turn, may be restricted within dosage bounds); g and g, give the kill rate of the drug per unit of drug concentration as function of the drug-sensitive cells or as a function of the normal cells. The functions f, fi, g, and g, in (2.1) will be described later on. As a criterion to choose an optimal strategy, we would like to minimize the total number of tumor cells at the end of the treatment:
Concomitantly, one possible way to take account of the toxicity is to choose the optimal strategy in such a way so as to maximize the number of normal cells at final time, n, which for mathematical convenience we restate as
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
711
to minimize
J*(u,tf) = -n(+). This corresponds to an optimal control problem with a vector performance criterion. Moreover, we remark that the above functionals Jr and J2 are conflicting. In fact, the problem of minimizing J1 subject to (2.1) and the restriction 0 Q u $ u, in the control variable is a particular case of the problem analyzed in Costa et al. (1992). There the optimal strategy that minimized J, was u = u, throughout the treatment. But this strategy obviously decreases n (this can immediately be seen from the third equation in (2.1), and this is the same as to increase J,, as compared with the strategy of giving no drug at all (u = 0)). On the other hand, with the same kind of arguments, we see that to minimize J2 we should choose u = 0, whereas this last strategy clearly increases J,. Thus, J, and J2 are conflicting, and to try to find a compromising solution, in this work we will use Pareto optimization techniques. Now, according to Leitmann (1974, p. 8), a problem with a vector of performance criteria can be converted into a problem of finding optimal solutions for the family of convex combinations of each of the individual performance criterion. More precisely, we must analyze the following family of Free End-Time Optimal Control Problem associated with (2.1). For each pair of real numbers (p, 4) satisfying 0 (p, q < 1 and p + q = 1, find a time 0 G tf* < +a and a BV[O, tf*]-function u: [O,t;] + IF4satisfying 0 G u*(t) d u, in [0, t,*], and in such a way that it will be the optimal drug concentration in the sense that
J,,(u*(-), t;) = minimum(J,,(u,
tf), u Ed(tf),
Vt,> O}.
(2.2)
Here BV[O, tf] indicates the class of bounded variation functions defined in [0, t,], tit,> = (u E BV[O, t,], 0 Q u(t) < u, Vt E [0, t,]). The functional. JP4 is defined by JP4 =pJI + qJ,, that is,
(2.3) This functional is a convex combination of the number of tumoral cells and minus the number of normal cells at the end of the treatment; p and q may serve as penalization factors. This structure stems from the following fact: the problem deals with conflicting objectives, that is to say, the minimization of tumoral cells and maximization of normal cells at the end of the treatment.
712
M. I. S. COSTA AND J. L. BOLDRINI
As to the functions f, fi and g, g, that appear in (2.1), we will consider the following natural assumptions: f,fi,g,gi
are C’functions;
g(s), g,(s) > 0 and g’(s),g$s)
g(0) =g,(O) = 0, > 0 when s > 0.
(2.4)
and mere
exist y,,, > 0 and n, > 0 such that f( y,) = 0, fi( %) = 0 andf(y),f,(n)>OforO
(2.5)
or
f(y),f,(n)
> 0 f or y 2 0, n 2 0; f, fi and g, g, are globally Lipschitz. (2.6)
In (2.4) the two first expressions indicate that the drug effect is strictly related to the existence of sensitive (normal) cells and the third one states that the drug effect increases as the level of sensitive (normal) cells increases. In (2.5) it is stated that the tumor (normal cells) exhibits a density dependent growth, where y,(n,> is the maximum attainable level of tumor (normal) cells. In (2.6) it is assumed that there is no maximum attainable level of tumor (normal) cells and that the relative increment of kill rate per unit concentration is bounded. The behaviour of system (2.1) without any drug concentration, that is, u(t) = 0 for all t > 0, corresponds to the dynamics obtained in Goldie and Coldman (1979, p. 1732) describing the evolution of resistant cells in relation to the number of tumoral cells. Now we define an open set 0 in R2 as (i)
~={(x,y)ElR2:O
(ii)
~={(~,y)ElR~:O
(2.7)
if assumptions (2.4) and (2.5) hold. Before proceeding to the analysis of the optimal control problem (2.2), we enunciate a lemma that relates the trajectories of system (2.1) with the open set n and gives a small amount of information on n(t). This lemma can be proved in a similar way as Lemma 2 in Costa et al. (1992): it is
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
713
enough to observe that the first two equations of (2.1) are the same as the ones in Costa et al. (19921, that the third equation (in 2.1) is a scalar equation decoupled from the other two, with n = 0 as an equilibrium point, and that conditions (2.4) and either (2.6) or (2.5) hold for it. LEMMA 2.1. Consider u(t) 2 0 a function of bounded variation. The corresponding solution [x(t), y(t),n(t)] of (2.1) with initial conditions (xO,yO,no> satisfying (xO,yO) E n is such that its projection on the x, y-plane, that is, [x(t), y(t)], never touches the boundary of KI in finite time. Moreover, if 0 < n(O), then 0 < n(t) < 00for all finite time t.
This lemma implies in particular that 0 = J,(u, tf> = y(t,). Then, as we remarked above, we are in a special case of Costa et al. (1992), and the optimal strategy is u = u, throughout the treatment. When p = 0, then 4 = 1 and Jol(u, tf> =J&u, tf> = -n(tf). Again as we remarked above, in this trivial case the optimal strategy is u = 0. Thus, hereafter we will consider only the following situation: p+q=l.
O
Also, a relationship among u, and the functions fi, g,, and n must be required so that the present problem should not fall exactly in the same frame as that tackled in Costa et al. (1992). In fact, if dn/dt = nfi(n> u,g,(n) > 0 with n(0) > 0, then n(t) is always increasing for any treatment such that 0 Q u(t) G u,. In this case, no harm is impinged on the normal cells and the problem reduces to the one studied in Costa et al. (1992). To rule out this possibility, in this analysis, we will require nfi(n)
- u,g,(n)
< 0,
c2.8)
either for all n E (0, n,) in case of (2.5) or for all n > 0 in case of (2.6). To analyze the above optimal control problem (2.2), we will resort to the Pontryagin Minimum Principle (Kirk, 1970). For this, we start by observing that the Hamiltonian associated with the optimal control problem formed by (2.0, (2.2), and (2.3) is given by:
Hpq(x,y,n, 4, A,, A,,u)
= h,[ICf(y) +
+ A,[yf(y)
cyf(y)(y
--x)1
- ug(y
-x)1+
&f,(n)
- ugAn)I,
c2.9)
714
M. I. S. COSTA AND J. L. BOLDRINI
where A,, A,, A, are the costate (adjoint) variables whose dynamics are controlled by the following differential equations (Kirk, 1970): /
iI = - [ A,f(y)(l
- a) + A,ug'(y
i, = - [A,(xf'(y)
+af'(y)(y
+&(yfl(y)
(
+f(y)
-x,]
-xl + af(y))
- ug’ty
-x>)]
(2.10)
i, = - A3[fib> + nf;h) - z&(n)] \A,($) = 0; A,($) =p; A&) = -4. We remark that throughout this work a “s” over a variable will mean the same as d/dt applied to it. We will use both of these notations. Also relevant to our analysis is the fact that along the solution corresponding to any optimal strategy u(t), the Hamiltonian is identically zero, since we have a Free End-Time Optimal Control Problem. That is, for all t E [0, t,], with u(t) being an optimal strategy and x(t), y(t), n(t), A,(t), A,(t), and A,(t) the corresponding solutions of (2.1) and (2.10), there holds
HJt)
=H,,(x(t>,y(t),nW,
A,(t), A,(t), A&),df))
= A,(t)[x(t)f(y(t))
+ af(y(f))(Y(t) --x(t))]
+ A,(t)[y(t)f(y(t)) + A&)[
-u(t)&(t)
--x(t))]
n(t>fi(n(t)> - u(t>g,(n(d)]
= 0.
(2.11)
Since the Hamiltonian is linear in the control variable u, by using Pontryagin Minimum Principle (Kirk, 1970), we easily obtain the Optimal Control Law as follows: if
A,g(y -x> + A,g,(n)
< 0,
if
A,g(y -x)
> 0,
if
A,g(y -x> + A,g,(n) = 0.
+ A,g,(n)
(2.12)
We remark that if the third case of the above control law occurs, the corresponding control is usually called a singulur control. We will use this terminology throughout this work. Since A, and A, appear in (2.12), some preparatory results regarding their behaviour will be derived in order to yield more specific information on the optimal strategy. These results will be used in the subsequent section.
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
715
We start with two results concerning the sign of A, and the possible sign of A,. LEMMA 2.2.
h,(t)
< 0 for all t E [0, tf]
irrespective of u.
Proof It follows from the special form of the equation for A, in (2.10) and the final condition A,(tf) = -4. In fact, it is easily seen that A&t) = -9 exp[ - j,‘~f, 0. LEMMA 2.3.
A,(t)
>
0 when either u is singular or u = u,.
Proof It follows immediately from the optimal control law (2.12), the signs of g and g, given in hypotheses (2.4), and the signs of II and A, given ?? in Lemmas 2.1 and 2.2, respectively. Before we proceed, let us recall certain facts concerning equations (2.1) and (2.10). Since u E BV[O, t,], it can be discontinuous, but for each t E (0, t,), there always exists the lateral limits u(t’> and u(t-1, together with u(O+) and u. Moreover, even though the derivatives with respect to time of the functions x(t), y(t), n(t), A,(t), A,(t), and A,(t) can also be discontinuous, since they depend directly on u in equations (2.1) and (2.10), their lateral limits exist and the functions x(t), y(t), n(t), A,(t), A,(t), and A,(t) themselves are continuous. This remark is used in the next lemma that describes the behaviour of A, in a special case. LEMMA 2.4.
Suppose thatfor any t in an interval [tl, t2), with 0 < t, < t, < t,, the optimal control satisfies u(t) > 0, and, moreover, Ai 2 0; then A,(t) > 0 in the interval [t 1, t2), and it is also strictly decreasing. The same result holds if n(t) is not increasing in [tl, t,) and Al(t2) > 0. Proof Since u(t) > 0 for t E [tl, t2], by the optimal control law (2.12), at each time t E [tl, t2], u(t) must be singular or u,, and we can apply the previous lemma to conclude that A,(t) > 0 for t E [t,, t2]. On the other hand, the final conditions AI = 0 and A2(t2) > 0 applied to the differential equation for A, in (2.10) imply that i,(t;)
= -A2(t2h(t2)g’(y(t,)
-dt,))
< 0.
Thus, A,(t) > 0 for t in a left neighborhood of t,. Let us show that actually A,(t) > 0 for t E [tl, t2) by contradiction. Suppose then that it is not so, and call 7 the largest time in .[tl, t2) for which A, is zero. Since Al(r) = 0 and A(t) > 0 for t E (T, t,> then A1(7+) 2 0. By using again the differential equation for A, in (2.10), but now computed at T+, we
716
M. I. S. COSTA AND J. L. BOLDRINI
obtain
0 G &CT+) = -A1(7)f(y(7))(1=
-&(dU(T+)g'(y(T)
a) -
h*WU(T’)g’(y(T)
--x(T))
--x(T))
< 0,
since by Lemma 2.3, A.&T) is strictly positive, and from (2.41, g’ is also strictly positive. This contradiction proves that A,(t) > 0 for t E [tl, t2). Looking back to the first equation in (2.10), we see that A,(?+) < 0 for any t in the same interval. Therefore, we conclude that h,(t) is strictly decreasing in the interval. For the last part of the lemma, we observe that since n(t) is not increasing in the interval of time being considered, we must have u(t) > 0. Otherwise, from the equation for A, in (2.Q we would have ri = nf, > 0. Thus, we can apply the first part of the lemma, and conclude that the statement is true. ?? The next lemma determines the behaviour of A,(t)n(t), which appears in the control law (2.12). Provided that the growth rate fr(n) for the normal cells satisfies a rather natural condition (which is satisfied in the case of logistic growth, for instance) and the corresponding kill rate gl(n> is given with a particular structure, the following result holds: 2.5. Suppose that (2.4) and (2.5) hold, with the additionalhypothesis that the growth rate of the normal cells satisfiesf;(n) < 0 for 0 < n < nm and that the corresponding kill rate is g,(n) = Fin, with F, > 0. Then A&(t) is strictlydecreasing in time, irrespectiveof u(t). When f,(n) = rl is constant, A,(t)n(t) is constant in time. LEMMA
Proof
It is enough to compute the derivative of A,(t)n(t) with respect to
t:
d( A&
-
dt
= -A3( f&n) + nf;(n> - F,u)n + A,(nf,(n) - Flun) = - A,n*f;h).
(2.13)
?? Since, by Lemma 2.2 A, is negative, we obtain the above stated results. It is possible to prove some other results with partial information for the general case of growth conditions and kill rates. Since they are not used in the subsequent section of this work, we will describe them in the following remarks.
Remark 1. If we knew that the optimal energy strategy u(t) had only a finite number of switching points (or we were working with a class of
OE3JEClWEiS IN CHEMOTHERAPY
WITH DRUG RESISTANCE
717
admissible controls having already this property), we could prove that there would be a time T E (0, tf] such that A,(t) is strictly positive and strictly decreasing in [0, T) and it is identically zero in the interval [T, tf] (this could reduce just to the point tf>. In fact, suppose that it is so. Then, starting from the final time tf, it is enough to reason by going backwards in time. Call 0 Q t, < t, < --- < t, < tf the switching times between non-zero (that is, u, or singular) and identically zero controls. Observe that from the continuity of the state and adjoint variables, by using the optimal control law (2.12), at these switching times we must have A2(ti)g(y(ti) -x(Q) + h,(t,)g,(n(t,)) = 0 for i = 1,. . . , n. Since g > 0 and by Lemma 2.2 we know that A3(ti) < 0, we conclude that h,(ti) > 0. Suppose first that the final subinterval [t,, tf> of the optimal control is identically zero. By using the final condition AI = 0 and the first equation in (2.10), it is easy to verify that A,(t) is also identically zero there. So, at t, we have A,(&) = 0. Now, in the subinterval [t,_ 1, fn> we have u(t) > 0 and we are exactly in the situation of Lemma 2.4, and we conclude that in [t, _ 1, t,>, h,(t) is strictly positive and decreasing. In particular A,(t, _ 1) > 0. In the subinterval [tn_2, t,_ I> the control is u(t) = 0, and the first equation of (2.10) simplifies. Since the final value of A,(t, _ 1) is positive, it is easy to see that A,(t) is also strictly positive and strictly decreasing in [L*, t,-1). In the subinterval [tn_3, fn_2), we repeat the above argument with the help of Lemma 2.4, and so on. After II steps in the above reasoning we obtain the stated result. In the case where in the final subinterval [t,, tf> the optimal control is u(t) > 0, we repeat the same kind of argument above by invoking the use of Lemma 2.4. The rest of the reasoning follows as before. Remark 2. With the notation of the above remark, the optimal control is zero in the last subinterval [t,, t,), only if at the switching time t, the corresponding optimal trajectory at that time reaches a certain surface in the state space. The equation for this surface is obtained as follows: Suppose that for t E [t,, tf> there holds u(t) = 0. Then, A,(t) = 0 in the same subinterval, and from the fact that the Hamiltonian is identically zero, we must have A,(t,)y(t,)f(y(t,)) + A,(f,)n(t,)f,(n(t,)) = 0. Also, as above, at the switching time t, we must have from the optimal control law A2(fn)g(y(tn) -x(Q) + A3(tJgl(tn) = 0. This yields two homogeneous linear equations in the variables A2(fn) and A&&). Since A&&) < 0, the only possibility is that the determinant of the matrix of coefficients of the linear system be zero. Thus, we conclude that we must have y(t,)f(y(t,))gl(t,) n(t,>fi(n(t,>)g(y(t,) -x(Q) = 0, which is the required equation for the
718
M. I. S. COSTA AND J. L. BOLDRINI
surface. Maybe this information could be used to conclude that in most cases u(t) > 0 in the neighborhood of the end of the treatment, as expected. The next section deals with a special case in which it is possible to find the corresponding optimal strategy.
3. Optimal Strategy in the Case of Malthusian Tumor Growth. We proceed to the search of optimal treatments in the special case when the tumor cells obey a Malthusian (exponential) growth, but normal cells may possess a rather general nonlinear growth rate (which includes the usual logistic type growth rate, for instance). We will also assume that the kill rates for normal and sensitive tumor cells are respectively linearly proportional to their populations. This model assumes that the specific growth rate of the tumor is constant [f(y) = r]. Its importance is centered on the introduction of the concept of doubling time and, although it does not have a strong physiological basis, it starts with a reasonable assumption Vaidya (1982) [this model is used for modelling cell growth in Eisen (1978)]. An account of the assessment of the exponential model in human tumors can also be found in Swan (1987) and references therein. In this section, the system corresponding to (2.1) has the following form:
- = dt dY z
=
Ix +
ar(y
ry -Fu(t)(y
4, -4,
(3.1)
dn
-
dt
$0)
=nfi(n) =x0,
-FF,u(t)n,
y(O) =y,, 40) = no,
where the tumor cells have specific growth rate r. As the drug may act differently on each kind of cell, different constants of proportionality F and F1 for tumor and normal cells, respectively, were assigned to the kill rates. For the main result of this section to hold, it will be necessary to assume the following conditions on the growth rate for normal cells: f&n) is a C’ function satisfying fi(0) =f,(n,)
= 0, f,(n) > 0
andf;(n)
(3.2)
OBJECTIVES IN CHEMOTHERAPY
WITH DRUG RESISTANCE
719
Moreover, with the above hypothesis, condition (2.8) reduces to
f&O) -&urn < 0.
(3.3)
This will be assumed throughout this section, thus guaranteeing that the problem does not fall in the same frame of Costa et al. (1992). The basic result of this section states that if an optimal treatment exists, then it should consist basically of maximum drug concentration (u,) throughout at the tumor site. In order to prove this, we will have to derive further results, specific to this section. We observe that the Hamiltonian is reduced to HP4 = A,(m + ar(y -x>) + h&y - Fu(y -xl)
+ A&f,(n)
- Qn).
(3.4)
The adjoint equations simplify to
‘i, = -(l (
i,=
- /_+A, -Fu(t)A,,
-cx~Ai - (r-Fu(t))A,, (3.51
h, = (Flu -f, - nf;(n))A,, A&)
The corresponding
= 0,
A&)
=p,
A&)
= -4.
Optimal Control Law is reduced to
0
if
A,F(y --xl + A,F,n < 0,
% Undetermined
if
A,F(y -xl
if
A,F(y --xl + A,F,n = 0.
+ A,F,n > 0,
(3.6)
As before, the last case of (3.6) is referred to as singular control. In what follows a sequence of lemmas is proved so as to form the underlying basis of the optimal strategy. LEMMA 3.1. In an optimal strategy with time interval [O,tf], there is i E (0, tf] such that A,(t) > 0 for any t E [O,i) and A,(t) = 0 for all t E fi, t,].
ProoJ: From the first equation in (3.5) and the final condition we obtain A,(t) =Fexp[-(1
- cY)rt]l’~A,(s)u(s)e~[(l t
- a)rs]ds.
AI
= 0,
720
M. I. S. COSTAANDJ. L. BOLDRINI
Now, the optimal control law (3.6), Lemma 2.3 and the fact that u(t) 2 0 imply that h,(t)u(t) 2 0 for all t E [0, tf]. Thus, the above expression implies that h,(t) > 0, and, moreover, that, if for some t, > 0, h2(t,)u(t,) > 0, then A,(t) > 0 for all t E [0, tl>. This implies the stated result. ?? LEMMA3.2. If (3.2) is true, there is not singular control in optimal strategies. Proof By contradiction, suppose there singular on an interval [Cl,t2] C [0, tr]. Then, the switching function in (3.61, F1A&t>n(t), must be identically zero on the ing the derivative with respect to time of obtains
-Far(A,(t)
+ A,(t)(y(t)
-n(t))
is a strategy
such that
u is
which is A,(t)F(y(t) -x(t)) + interval [t,, t2], and, by computthis function at t E [cl, tz], one
+F,
d( A,(th(t)) dt
= 0.
Now, Lemma 2.5 says that the second term of the above expression is nonpositive; Lemma 2.3 guarantees that A,(t) > 0, Lemma 3.1 furnishes that A,(t) 2 0, and, by Lemma 2.1, y(t) -x(t) > 0. Thus, the above terms cannot add up to zero. This contradiction proves the lemma. ?? Therefore we are left with two possibilities for an optimal strategy at any time t: either u(t) = u, or u(t) = 0. We proceed by proving the following technical lemma that will be important in the analysis of the possibilities of switchings in optimal strategies. LEMMA3.3. In an optimal strategy, if u(t) = u, on a certain interval of time [tl, tzl, then A&)(y(t) - x(t)> is strictly decreasing in that same interval. Prooj By calculating the derivative of A,(t)(y(t) one obtains
-$A,(t)(y(t)
--x(t)))
= --ar(A1(t)
-x(t)> with respect to t,
+ A,(t))(y(t)
--x(t)).
Now, for any t E [Cl, tz], there hold Al(t) z 0 (by Lemma 3.1), A&f) > 0 (by Lemma 2.3) and y(t) -x(t) > 0 (by Lemma 2.1). Thus, for t E [Cl, f2],
-&A,(t)(y(t) and we have the result.
-x(t)))
<0 ??
OBJECI'IVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
Now we proceed strategies.
to rule out a certain
721
kind of switching in optimal
LEMMA 3.4.In the case of (3.2), there cannot be an optimal strategy with switch& from u = 0 to u = u ,,,.
Proof Suppose by contradiction that on a certain interval of time on an interval [tz, t3) the optimal Then, from the optimal control
that there is such kind of switching and the optimal strategy be u = 0, and strategy be u = u,. law (3.61, the switching function G(t) = h,(t)F(y(t) -x(t)) + F,A,(t)n(t) must pass from negative values on [tl, t2) to +(tJ = 0 to positive values on (t2, t3). But, from Lemma 3.3, $(t) must be strictly decreasing on [t2, tJ, and, since $(t2) = 0, we conclude that @(t> < 0 for t E (f2, tJ, in contradiction with the previous conclusion. Thus, it is impossible to have switchings from zero to u, in optimal strategies. ?? Finally, the above lemmas immediately imply the following characterization of the optimal strategies. [tl, t2)
THEOREM 3.1. If (3.2) holds, any optimal treatment consists of maximum drug concentration throughout, that is, either u(t) = u, for all t E [0, tf] or there may be 0 < i < t f such that u(t) = u, for t E [0, i) and u(t) = 0 for t E Tt,tJ. Remark 3. The results of the above lemmas and theorem hold true from standpoint of considering the case of fi(n) as a constant.
the mathematical
Remark 4. In this section we have completely
characterized the optimal strategy to be used when the growth rate for tumor cells is constant and the growth rate for normal cells is rather general. Unfortunately, we were not able to derive such a specific result for the most general case, even in the logistic situation, which corresponds to the equations (2.1) with f(y) = r(1 - ky), f, = r,(l - k,n) and r, k, rl, k, positive constants. In this general case we were only able to obtain partial information concerning optimal strategies. For instance, one result that could be of some interest is the following: although we were not able to rule out the possibility of singular control in optimal strategies, we could prove that in such strategies there is no switching from maximum drug concentration to a concentration corresponding to singular control while the state of the system is in a relevant defmed region in the x, y plane (according to the related parameters of the problem). Yet this sort of information was not enough to yield conclusive results for the optimal control.
122
M. I. S. COSTA AND J. L. BOLDRINI
4. Discussion. In this work we attempted to devise optimal chemotherapeutic treatments in continuous chemotherapy, when drug resistance is taken into account and the objectives to be attained, namely, the minimization of tumoral cells and the maximization of normal cells at the end of the treatment, are conflicting. In the special case of constant growth rate for tumor cells (Malthusian growth), but rather general growth rate for normal cells, the optimal strategy was worked out, and, from the practical point of view, maximum drug concentration throughout the treatment proved to be the optimum, except in the trivial case p = 0. Although we were not able to work out optimal strategies for the general growth rates for both tumor and normal cells, we conjecture that in this case treatments other than maximum drug concentration throughout may be confined to restricted sets of initial levels of tumoral and normal cells and/or to relations among parameters that do not necessarily express a commonly biological evidence. We point out that under the set-up of qualitative modeling of spontaneously acquired drug resistance presented in this work as well as in our previous ones, maximum drug concentration featured as a candidate that almost always presented itself as an optimal treatment (let alone the cases when it is the sole optimal treatment). This prevalence of maximum drug concentration as an optimal treatment is, in part, corroborated by probabilistic models that take drug resistance into account [see Coldman and Goldie (1983) and Coldman and Goldie (1986), Harnevo and Agur (1992)], though their setting concerns discrete drug administration. According to these models rapid depletion of sensitive cells is essential and if low dose therapies are continued over long periods of time, they are unlikely to be successful. Indeed it is an intriguing issue that rest periods almost do not feature as part of any optimal strategy in continuous chemotherapy when drug resistance is accounted for [at least as modelled as in Costa et al. (19921, Costa et al. (1994), and Costa et al. (1995b) and also in this work]. It seems that spontaneously acquired drug resistance modelled by an ever-increasing population induces in the model a built-in inclination for the suppression of rest periods. In fact, rest periods only proved to be clearly part of optimal strategies in the setting where the patient’s recuperation was considered [see Costa and Boldrini (1997)]. In the course of our previous works without patient’s recuperation, alternation between maximum drug concentrations and rest periods as optimal treatment was ruled out in all cases except for the setup including drug kinetics [see Costa et al. (1995a)l. Thus, it is still open the question whether the inclusion of a pharmacokinetic equation may engender optimal solutions with rest periods, since the drug decay rate plays an important
OBJECTIVES
IN CHEMOTHERAPY
WITH DRUG RESISTANCE
123
role in the determination of optimal treatments. To reinforce this issue we point out that in a recent work Murray (1995) modelled drug resistance via an effectiveness term that decreases the tumoral and normal cell loss as the accumulation of the drug (which is expressed by a drug kinetics equation) grows. There rest periods belong to the class of optimal treatments. In a sense, due to the fact that his work deals with resistance induced by drug, his result supports our conjecture about the relation between rest periods and drug kinetics in the context of spontaneously acquired drug resistance. Finally we would like to mention that the absence of rest periods in the protocols proposed by the models of theoretical chemotherapy with drug resistance analyzed in our previous works as well as in this one, may indicate that they are not entirely appropriate to design drug regimens (especially when the model yields maximum drug concentration throughout as the sole optimal strategy since impediments in clinical protocols prevent its current use). Besides, this may serve as guide to devise models that are more qualitatively realistic, that is, models that contain relevant aspects and features for deriving suggestions of clinically accepted protocols. At this stage, considering deterministic settings, our analysis points at two possible directions: inclusion of drug kinetics and mechanisms of patient’s recuperation in the models. The authors acknowledge the helpful comments and suggestions made by the referees on an earlier version of this work. This research was supported in part by CNPq grant No. 300153/95-3.
REFERENCES Coldman, A. J. and J. H. Goldie. 1983. A model for the resistance of tumor cells to cancer chemotherapeutic agents. Math. Biosci. 65, 291-307. Coldman, A. J. and J. H. Goldie. 1986. A stochastic model for the origin and treatment of tumors containing drug-resistant cells. Bull. Math. Biol. 48, 279-292. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1992. Optimal chemical control of populations developing drug resistance. ZMA J. Math. Appl. Med. Biol. 9, 215-226. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1994. Optimal chemotherapy: a case study with drug resistance, saturation effect and toxicity. Ii&f J. Math. Appl. Med. Biol. 11, 45-59.
Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995a. Drug kinetics and drug resistance in optimal chemotherapy. Math. Biosc. 125, 191-209. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995b. Chemotherapeutic treatments involving drug resistance and level of normal cells as a criterion of toxicity. Math. Biosci. 125, 211-228. Costa, M. I. S. and J. L. Boldrini. 1997. Chemotherapeutic treatments: a study of the interplay among drug resistance, toxicity and recuperation from side effects. Unpublished manuscript. Eisen, M. 1978. Mathematical Models in Cell Biology and Cancer Chemotheram. Lecture Notes in Biomathematics, Vol. 30. Heidelberg: Springer-Verlag. Goldie, J. H. and A. J. Coldman. 1979. A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat. Rep. 63,1727-1733.
724
M. I. S. COSTA AND J. L. BOLDRINI
Hamevo, L. and Z. Agur. 1992. Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency. Cancer Chemother. PhatmacoL 30,469~476. Kirk, D. 1970. Optimal Control Theory. Englewood Cliffs, NJ: Prentice-Hall. Leitmann, G. 1974. Cooperativeand Non-CooperatiueMany Player DifferentialGames, International Centre for Mechanical Sciences Course Lectures, No. 190. Udine, Italy: Springer-Verlag. Murray, J. M. 1990a. Optimal control for a cancer chemotherapy problem with general growth and loss functions. Math. Biosci 98, 273-287. Murray, J. M. 1990b. Some optimal control problems in cancer chemotherapy with a toxicity limit. Math. Biosci. 100, 49-67. Murray, J. M. 1995. An example of the effects of drug resistance on the optimal schedule for a single drug in cancer chemotherapy. ZUA J. Math. AppL Med BioL 12, 55-71. Skipper, H. E. 1983. The forty year old mutation theory of Luria and Delbruck and its pertinence to cancer chemotherapy. Aa!u. Cancer Research 40,331-363. Swan, G. W. and T. L. Vincent. 1977. Optimal control analysis in the chemotherapy of ZBG multiple myeloma. Bull. Math. BioL 39, 317-337. Swan, G. W. 1987. Tumor growth models and cancer chemotherapy. In Cancer Modeling, J. R. Thompson and B. W. Brown (Eds), pp. 91-179 New York: Marcel Dekker. Swan, G. W. 1990. Role of optimal control theory in cancer chemotherapy. Math. Biosc. 101, 237-284.
Vaidya, V. G. and F. J. Alexandro, Jr. 1982. Evaluation of some mathematical models for tumor growth. Znt.J. Bio-Med. Comp. 13, 19-35. Vendite, L. L. 1988. Modelagem matem&ca para o crescimento tumoral e o problema de resist&cia celular aos fhrmacos anti-bl&ticos. Ph.D. thesis, Faculdade de Engenharia El&rica, Universidade Estadual de Campinas, SP, Brazil. Zietz, S. and C. Nicolini. 1979. Mathematical approaches to optimization of cancer chemotherapy. Bull. Math. BioL 41,305-324.
Received 28 August 1995 Revised version accepted 6 June 1996
of Mathematical
Biology, Vol. 59, No. 4, pp. 707-724, Elsevkr
Science
8 1997 Society for Mathematical 0092-8240/97
1997 Inc.
Biology
$17.00 + 0.00
SOWZ-8240@7)00013-X
CONFLICTING OBJECTIVES WITH DRUG RESISTANCE
IN CHEMOTHERAPY
?? M.
I. S. COSTA? and J. L. BOLDRINIt * Laborathio National de Computa@o Cientifica-CNPq, Rua Lam-o Miiller, 455, 22290-160, Rio de Janeiro, RJ, Brazil (Email:
[email protected])
Wniversidade Estadual de Campinas, Instituto de Matematica Estatistica e Cihcia CP 6065, 13081-970, Campinas, SP, Brazil (Email:
da Computa@o,
[email protected])
A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Spontaneously acquired drug resistance is accounted for, as well as the evolution in time of normal cells. In addition, optimization of conflicting objectives forms the aim of the chemotherapeutic treatment. For general cell growth, some results are given, whereas for the special case of Malthusian (exponential) growth of tumor cells and rather general growth rate for normal cells, the optimal strategy is worked out. The latter, from the clinical standpoint, corresponds to maximum drug concentration throughout the treatment. 0 1997 Society for Mathematical Biology
1. Introduction. Optimal use of drugs in cancer chemotherapy is hindered, amongst others, by several factors such as lack of detailed knowledge about the kill rates of the drugs, drug resistance, cell growth models and appropriate criteria for measuring toxicity. In the context of spontaneous acquisition of drug resistance, some optimal chemotherapeutic treatments were devised in Costa et al. (1992) and Costa et al. (1994). Maximum drug concentration throughout appeared as an optimal strategy and rest periods were not allowed. However, in clinical protocols, variable (or constant) levels of drug concentration interspersed with rest periods seem to be a clinical practice. Nonetheless, it should be stressed that clinical protocols used in practice are not necessarily optimal, but that they support the belief that maximum drug concentration throughout cannot be optimal in a satisfactory sense. In turn, this suggests that the protocol formed by the highest allowable level of drug concentration throughout should not be the sole optimal solution of the optimization of realistic theoretical models of cancer chemotherapy. 707
708
M. I. S. COSTA AND J. L. BOLDRINI
By utilizing normal cells as an indicator of toxicity (with the requirement that a given minimum level of normal cells should not be violated), Murray (1990a) and Murray (1990b) and Zietz and Nicolini (1979) were able to find optimal treatments that could include rest periods. From the point of view of modelling, unlike the ones used in Costa et al. (1992) and Costa et al. (1994), this toxicity criterion is noncumulative. However, the models in Murray (1990a) and Murray (199Ob) and Zietz and Nicolini (1979) did not include drug resistance. In Costa et al. (1995b) we attempted to devise optimal treatments when this feature was taken into account and toxicity, like in Murray (1990a) and Murray (199Ob) was modelled via a minimum level of normal cells that should not be violated along the treatment. Unfortunately, as soon as drug resistance was included in the model, the sole optimal strategy for general growth and drug kill rates was the maximum drug concentration throughout (again no rest periods were allowed). This issue may point to some features that are not present in the cited models, since in clinical practice there are impediments that preclude clinicians from resorting to protocols formed by the highest allowable level of drug concentration throughout. So in the light of the results that evidence maximum drug concentration to be almost always the optimal strategy in presence of drug resistance [although, rest periods might feature as candidates for optimal treatment when drug kinetics was also accounted for, but in a nonconclusive way, see Costa et al. (1995a)], we decided to take another approach. In Costa and Boldrini (1997) we set up a rather theoretical frame for a phenomenological notion of patient’s recuperation. In that setting it was proved that strategies with alternation of maximum drug concentration and rest periods could be optimal even in presence of drug resistance. This outcome tends to point towards a likely relationship between rest periods in continuous chemotherapy and mechanisms of recuperation. However, to make completely clear the necessity of inclusion of some sort of recuperation mechanism in order to obtain the possibility of rest periods in optimal therapies for models with drug resistance, it is still necessary to test another possibility: the reason for the fact that maximum drug injection throughout the treatment was in general the optimal strategy in the previous models might actually be due to imperfect choices of the performance criteria. As a matter of fact, there is an intrinsic conflicting situation in chemotherapy: from clinical experience it is evidenced that drugs not only kill malignant cells but normal cells as well. Therefore, to keep the patient up to a safe health state, one should, while minimizing the level of tumoral cells during the treatment, maximize the level of normal cells-a situation of conflicting objectives. In this case, it is not easy to reach a realistic compromise and devise a suitable unique performance
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
709
criterion involving level of tumoral cells and toxicity [like the one employed in Costa et al. (1992) and Costa et al. (199411. As we said above, this raises the question whether the cause of the absence of rest periods in optimal strategies was not due to the nonadequate mathematical choice of the functional. In an attempt to answer this question, instead of trying to build an adequate unique performance criterion, in this work we decided to attack directly the problem of optimal chemotherapy with conflicting functionals. The modelling of this multiple-objective situation can be accomplished in several ways (Swan, 1990); we will employ one that was also utilized by Zietz and Nicolini (1979), i.e., the minimization of tumoral cells and maximization of normal cells at the end of the treatment. The difference is in that our setting takes into account the dynamics of drug-resistant cells, increasing then the order of the system of differential equations (three equations, two of which are coupled), while in Zietz and Nicolini (1979) the two equations are uncoupled. In fact, as it will be shown, this renders the mathematical analysis of determining optimal strategies quite cumbersome. In the next section we set up the mathematical problem of finding optimal treatments which deal with conflicting objectives. We seek solutions to this problem resorting to Pareto optimization techniques (Leitmann, 1974) and some results are derived for general growth and drug kill rates. However, due to the difficult intrinsic non-linearities of the problem, these partial results do not lead to any conclusive form of the optimal strategies. On the other hand, in Section 3 we will be able to describe the optimal treatment for the special case of exponential growth of tumor cells and linear drug kill rate, but with rather general nonlinear growth rate for normal cells. The result, which is relegated to be discussed in Section 4, is that rest periods are still not allowed.
2. A Pareto Optimal@ Criterion: Preparatory Results. In order to carry out the analysis of tumor growth submitted to chemotherapy, some assumptions are taken: (a) The tumor will be viewed as a cell population undergoing homogeneous growth, that is, it does not depend on the cell position within the tumor. (b) The tumor will consist also of drug-resistant cells whose growth rate depends not only on the size of its own population, but on the size of the sensitive cells as well. This latter dependence is due to a randomly spontaneous mutation during mitosis towards drug resistance, which will occur according to a constant probability. In this way no sensitive cell becomes drug resistant during its life time; only their daughter cells may acquire drug resistance by spontaneous mutation during mitosis. A
710
(4
M. I. S. COSTA AND J. L. BOLDRINI
biological validation of this kind of drug resistance was performed by in vitro experiments with T-cell lymphoblastic cell line CCRF-CEM. A description of these experiments can be found in Vendite (1988). [The importance of drug resistance in designing chemotherapeutic protocols is also emphasized in Skipper (1983)]. The kill rate of the drug (number of cells killed/unit drug concentration) will be considered as function of the size of the sensitive cells population. As for the normal cells the kill rate will also be a function of their own population size.
The following system is a model for the behaviour of tumoral and normal cells submitted to chemotherapy when the assumptions mentioned above are taken into account:
z dY
=AjYy)
+
cqf(y)(y 4,
=yf(y)
-
uWg(y 4,
(2.1)
dn
!x(O)
=x0,
y(O) =y,,
n(O)= no.
Here, t > 0 represents the elapsed time; y(t) E R stands for the total number of tumor cells at time t, while x(t) E R stands for the number of drug-resistant cells within the tumor and n(t) is the number of normal cells. Clearly, any initial condition (x0, yo> is such that x0 < y,; f(y) and fr(n> are the specific growth rates for the tumor and normal cells, respectively; 0 < (Y< 1 is the fraction per unit of time of the drug sensitive cells that mutates into drug resistant cells; 0 < u(t) < u, is the instantaneous drug concentration at the tumor site, assumed to be limited, i.e., u, < +a (this limit could be set by the maximum allowable level of drug injection, which, in turn, may be restricted within dosage bounds); g and g, give the kill rate of the drug per unit of drug concentration as function of the drug-sensitive cells or as a function of the normal cells. The functions f, fi, g, and g, in (2.1) will be described later on. As a criterion to choose an optimal strategy, we would like to minimize the total number of tumor cells at the end of the treatment:
Concomitantly, one possible way to take account of the toxicity is to choose the optimal strategy in such a way so as to maximize the number of normal cells at final time, n
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
711
to minimize
J*(u,tf) = -n(+). This corresponds to an optimal control problem with a vector performance criterion. Moreover, we remark that the above functionals Jr and J2 are conflicting. In fact, the problem of minimizing J1 subject to (2.1) and the restriction 0 Q u $ u, in the control variable is a particular case of the problem analyzed in Costa et al. (1992). There the optimal strategy that minimized J, was u = u, throughout the treatment. But this strategy obviously decreases n
J,,(u*(-), t;) = minimum(J,,(u,
tf), u Ed(tf),
Vt,> O}.
(2.2)
Here BV[O, tf] indicates the class of bounded variation functions defined in [0, t,], tit,> = (u E BV[O, t,], 0 Q u(t) < u, Vt E [0, t,]). The functional. JP4 is defined by JP4 =pJI + qJ,, that is,
(2.3) This functional is a convex combination of the number of tumoral cells and minus the number of normal cells at the end of the treatment; p and q may serve as penalization factors. This structure stems from the following fact: the problem deals with conflicting objectives, that is to say, the minimization of tumoral cells and maximization of normal cells at the end of the treatment.
712
M. I. S. COSTA AND J. L. BOLDRINI
As to the functions f, fi and g, g, that appear in (2.1), we will consider the following natural assumptions: f,fi,g,gi
are C’functions;
g(s), g,(s) > 0 and g’(s),g$s)
g(0) =g,(O) = 0, > 0 when s > 0.
(2.4)
and mere
exist y,,, > 0 and n, > 0 such that f( y,) = 0, fi( %) = 0 andf(y),f,(n)>OforO
(2.5)
or
f(y),f,(n)
> 0 f or y 2 0, n 2 0; f, fi and g, g, are globally Lipschitz. (2.6)
In (2.4) the two first expressions indicate that the drug effect is strictly related to the existence of sensitive (normal) cells and the third one states that the drug effect increases as the level of sensitive (normal) cells increases. In (2.5) it is stated that the tumor (normal cells) exhibits a density dependent growth, where y,(n,> is the maximum attainable level of tumor (normal) cells. In (2.6) it is assumed that there is no maximum attainable level of tumor (normal) cells and that the relative increment of kill rate per unit concentration is bounded. The behaviour of system (2.1) without any drug concentration, that is, u(t) = 0 for all t > 0, corresponds to the dynamics obtained in Goldie and Coldman (1979, p. 1732) describing the evolution of resistant cells in relation to the number of tumoral cells. Now we define an open set 0 in R2 as (i)
~={(x,y)ElR2:O
(ii)
~={(~,y)ElR~:O
(2.7)
if assumptions (2.4) and (2.5) hold. Before proceeding to the analysis of the optimal control problem (2.2), we enunciate a lemma that relates the trajectories of system (2.1) with the open set n and gives a small amount of information on n(t). This lemma can be proved in a similar way as Lemma 2 in Costa et al. (1992): it is
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
713
enough to observe that the first two equations of (2.1) are the same as the ones in Costa et al. (19921, that the third equation (in 2.1) is a scalar equation decoupled from the other two, with n = 0 as an equilibrium point, and that conditions (2.4) and either (2.6) or (2.5) hold for it. LEMMA 2.1. Consider u(t) 2 0 a function of bounded variation. The corresponding solution [x(t), y(t),n(t)] of (2.1) with initial conditions (xO,yO,no> satisfying (xO,yO) E n is such that its projection on the x, y-plane, that is, [x(t), y(t)], never touches the boundary of KI in finite time. Moreover, if 0 < n(O), then 0 < n(t) < 00for all finite time t.
This lemma implies in particular that 0
O
Also, a relationship among u, and the functions fi, g,, and n must be required so that the present problem should not fall exactly in the same frame as that tackled in Costa et al. (1992). In fact, if dn/dt = nfi(n> u,g,(n) > 0 with n(0) > 0, then n(t) is always increasing for any treatment such that 0 Q u(t) G u,. In this case, no harm is impinged on the normal cells and the problem reduces to the one studied in Costa et al. (1992). To rule out this possibility, in this analysis, we will require nfi(n)
- u,g,(n)
< 0,
c2.8)
either for all n E (0, n,) in case of (2.5) or for all n > 0 in case of (2.6). To analyze the above optimal control problem (2.2), we will resort to the Pontryagin Minimum Principle (Kirk, 1970). For this, we start by observing that the Hamiltonian associated with the optimal control problem formed by (2.0, (2.2), and (2.3) is given by:
Hpq(x,y,n, 4, A,, A,,u)
= h,[ICf(y) +
+ A,[yf(y)
cyf(y)(y
--x)1
- ug(y
-x)1+
&f,(n)
- ugAn)I,
c2.9)
714
M. I. S. COSTA AND J. L. BOLDRINI
where A,, A,, A, are the costate (adjoint) variables whose dynamics are controlled by the following differential equations (Kirk, 1970): /
iI = - [ A,f(y)(l
- a) + A,ug'(y
i, = - [A,(xf'(y)
+af'(y)(y
+&(yfl(y)
(
+f(y)
-x,]
-xl + af(y))
- ug’ty
-x>)]
(2.10)
i, = - A3[fib> + nf;h) - z&(n)] \A,($) = 0; A,($) =p; A&) = -4. We remark that throughout this work a “s” over a variable will mean the same as d/dt applied to it. We will use both of these notations. Also relevant to our analysis is the fact that along the solution corresponding to any optimal strategy u(t), the Hamiltonian is identically zero, since we have a Free End-Time Optimal Control Problem. That is, for all t E [0, t,], with u(t) being an optimal strategy and x(t), y(t), n(t), A,(t), A,(t), and A,(t) the corresponding solutions of (2.1) and (2.10), there holds
HJt)
=H,,(x(t>,y(t),nW,
A,(t), A,(t), A&),df))
= A,(t)[x(t)f(y(t))
+ af(y(f))(Y(t) --x(t))]
+ A,(t)[y(t)f(y(t)) + A&)[
-u(t)&(t)
--x(t))]
n(t>fi(n(t)> - u(t>g,(n(d)]
= 0.
(2.11)
Since the Hamiltonian is linear in the control variable u, by using Pontryagin Minimum Principle (Kirk, 1970), we easily obtain the Optimal Control Law as follows: if
A,g(y -x> + A,g,(n)
< 0,
if
A,g(y -x)
> 0,
if
A,g(y -x> + A,g,(n) = 0.
+ A,g,(n)
(2.12)
We remark that if the third case of the above control law occurs, the corresponding control is usually called a singulur control. We will use this terminology throughout this work. Since A, and A, appear in (2.12), some preparatory results regarding their behaviour will be derived in order to yield more specific information on the optimal strategy. These results will be used in the subsequent section.
OBJECTIVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
715
We start with two results concerning the sign of A, and the possible sign of A,. LEMMA 2.2.
h,(t)
< 0 for all t E [0, tf]
irrespective of u.
Proof It follows from the special form of the equation for A, in (2.10) and the final condition A,(tf) = -4. In fact, it is easily seen that A&t) = -9 exp[ - j,‘~f,
A,(t)
>
0 when either u is singular or u = u,.
Proof It follows immediately from the optimal control law (2.12), the signs of g and g, given in hypotheses (2.4), and the signs of II and A, given ?? in Lemmas 2.1 and 2.2, respectively. Before we proceed, let us recall certain facts concerning equations (2.1) and (2.10). Since u E BV[O, t,], it can be discontinuous, but for each t E (0, t,), there always exists the lateral limits u(t’> and u(t-1, together with u(O+) and u
Suppose thatfor any t in an interval [tl, t2), with 0 < t, < t, < t,, the optimal control satisfies u(t) > 0, and, moreover, Ai 2 0; then A,(t) > 0 in the interval [t 1, t2), and it is also strictly decreasing. The same result holds if n(t) is not increasing in [tl, t,) and Al(t2) > 0. Proof Since u(t) > 0 for t E [tl, t2], by the optimal control law (2.12), at each time t E [tl, t2], u(t) must be singular or u,, and we can apply the previous lemma to conclude that A,(t) > 0 for t E [t,, t2]. On the other hand, the final conditions AI = 0 and A2(t2) > 0 applied to the differential equation for A, in (2.10) imply that i,(t;)
= -A2(t2h(t2)g’(y(t,)
-dt,))
< 0.
Thus, A,(t) > 0 for t in a left neighborhood of t,. Let us show that actually A,(t) > 0 for t E [tl, t2) by contradiction. Suppose then that it is not so, and call 7 the largest time in .[tl, t2) for which A, is zero. Since Al(r) = 0 and A(t) > 0 for t E (T, t,> then A1(7+) 2 0. By using again the differential equation for A, in (2.10), but now computed at T+, we
716
M. I. S. COSTA AND J. L. BOLDRINI
obtain
0 G &CT+) = -A1(7)f(y(7))(1=
-&(dU(T+)g'(y(T)
a) -
h*WU(T’)g’(y(T)
--x(T))
--x(T))
< 0,
since by Lemma 2.3, A.&T) is strictly positive, and from (2.41, g’ is also strictly positive. This contradiction proves that A,(t) > 0 for t E [tl, t2). Looking back to the first equation in (2.10), we see that A,(?+) < 0 for any t in the same interval. Therefore, we conclude that h,(t) is strictly decreasing in the interval. For the last part of the lemma, we observe that since n(t) is not increasing in the interval of time being considered, we must have u(t) > 0. Otherwise, from the equation for A, in (2.Q we would have ri = nf,
Proof
It is enough to compute the derivative of A,(t)n(t) with respect to
t:
d( A&
-
dt
= -A3( f&n) + nf;(n> - F,u)n + A,(nf,(n) - Flun) = - A,n*f;h).
(2.13)
?? Since, by Lemma 2.2 A, is negative, we obtain the above stated results. It is possible to prove some other results with partial information for the general case of growth conditions and kill rates. Since they are not used in the subsequent section of this work, we will describe them in the following remarks.
Remark 1. If we knew that the optimal energy strategy u(t) had only a finite number of switching points (or we were working with a class of
OE3JEClWEiS IN CHEMOTHERAPY
WITH DRUG RESISTANCE
717
admissible controls having already this property), we could prove that there would be a time T E (0, tf] such that A,(t) is strictly positive and strictly decreasing in [0, T) and it is identically zero in the interval [T, tf] (this could reduce just to the point tf>. In fact, suppose that it is so. Then, starting from the final time tf, it is enough to reason by going backwards in time. Call 0 Q t, < t, < --- < t, < tf the switching times between non-zero (that is, u, or singular) and identically zero controls. Observe that from the continuity of the state and adjoint variables, by using the optimal control law (2.12), at these switching times we must have A2(ti)g(y(ti) -x(Q) + h,(t,)g,(n(t,)) = 0 for i = 1,. . . , n. Since g > 0 and by Lemma 2.2 we know that A3(ti) < 0, we conclude that h,(ti) > 0. Suppose first that the final subinterval [t,, tf> of the optimal control is identically zero. By using the final condition AI = 0 and the first equation in (2.10), it is easy to verify that A,(t) is also identically zero there. So, at t, we have A,(&) = 0. Now, in the subinterval [t,_ 1, fn> we have u(t) > 0 and we are exactly in the situation of Lemma 2.4, and we conclude that in [t, _ 1, t,>, h,(t) is strictly positive and decreasing. In particular A,(t, _ 1) > 0. In the subinterval [tn_2, t,_ I> the control is u(t) = 0, and the first equation of (2.10) simplifies. Since the final value of A,(t, _ 1) is positive, it is easy to see that A,(t) is also strictly positive and strictly decreasing in [L*, t,-1). In the subinterval [tn_3, fn_2), we repeat the above argument with the help of Lemma 2.4, and so on. After II steps in the above reasoning we obtain the stated result. In the case where in the final subinterval [t,, tf> the optimal control is u(t) > 0, we repeat the same kind of argument above by invoking the use of Lemma 2.4. The rest of the reasoning follows as before. Remark 2. With the notation of the above remark, the optimal control is zero in the last subinterval [t,, t,), only if at the switching time t, the corresponding optimal trajectory at that time reaches a certain surface in the state space. The equation for this surface is obtained as follows: Suppose that for t E [t,, tf> there holds u(t) = 0. Then, A,(t) = 0 in the same subinterval, and from the fact that the Hamiltonian is identically zero, we must have A,(t,)y(t,)f(y(t,)) + A,(f,)n(t,)f,(n(t,)) = 0. Also, as above, at the switching time t, we must have from the optimal control law A2(fn)g(y(tn) -x(Q) + A3(tJgl(tn) = 0. This yields two homogeneous linear equations in the variables A2(fn) and A&&). Since A&&) < 0, the only possibility is that the determinant of the matrix of coefficients of the linear system be zero. Thus, we conclude that we must have y(t,)f(y(t,))gl(t,) n(t,>fi(n(t,>)g(y(t,) -x(Q) = 0, which is the required equation for the
718
M. I. S. COSTA AND J. L. BOLDRINI
surface. Maybe this information could be used to conclude that in most cases u(t) > 0 in the neighborhood of the end of the treatment, as expected. The next section deals with a special case in which it is possible to find the corresponding optimal strategy.
3. Optimal Strategy in the Case of Malthusian Tumor Growth. We proceed to the search of optimal treatments in the special case when the tumor cells obey a Malthusian (exponential) growth, but normal cells may possess a rather general nonlinear growth rate (which includes the usual logistic type growth rate, for instance). We will also assume that the kill rates for normal and sensitive tumor cells are respectively linearly proportional to their populations. This model assumes that the specific growth rate of the tumor is constant [f(y) = r]. Its importance is centered on the introduction of the concept of doubling time and, although it does not have a strong physiological basis, it starts with a reasonable assumption Vaidya (1982) [this model is used for modelling cell growth in Eisen (1978)]. An account of the assessment of the exponential model in human tumors can also be found in Swan (1987) and references therein. In this section, the system corresponding to (2.1) has the following form:
- = dt dY z
=
Ix +
ar(y
ry -Fu(t)(y
4, -4,
(3.1)
dn
-
dt
$0)
=nfi(n) =x0,
-FF,u(t)n,
y(O) =y,, 40) = no,
where the tumor cells have specific growth rate r. As the drug may act differently on each kind of cell, different constants of proportionality F and F1 for tumor and normal cells, respectively, were assigned to the kill rates. For the main result of this section to hold, it will be necessary to assume the following conditions on the growth rate for normal cells: f&n) is a C’ function satisfying fi(0) =f,(n,)
= 0, f,(n) > 0
andf;(n)
(3.2)
OBJECTIVES IN CHEMOTHERAPY
WITH DRUG RESISTANCE
719
Moreover, with the above hypothesis, condition (2.8) reduces to
f&O) -&urn < 0.
(3.3)
This will be assumed throughout this section, thus guaranteeing that the problem does not fall in the same frame of Costa et al. (1992). The basic result of this section states that if an optimal treatment exists, then it should consist basically of maximum drug concentration (u,) throughout at the tumor site. In order to prove this, we will have to derive further results, specific to this section. We observe that the Hamiltonian is reduced to HP4 = A,(m + ar(y -x>) + h&y - Fu(y -xl)
+ A&f,(n)
- Qn).
(3.4)
The adjoint equations simplify to
‘i, = -(l (
i,=
- /_+A, -Fu(t)A,,
-cx~Ai - (r-Fu(t))A,, (3.51
h, = (Flu -f,
The corresponding
= 0,
A&)
=p,
A&)
= -4.
Optimal Control Law is reduced to
0
if
A,F(y --xl + A,F,n < 0,
% Undetermined
if
A,F(y -xl
if
A,F(y --xl + A,F,n = 0.
+ A,F,n > 0,
(3.6)
As before, the last case of (3.6) is referred to as singular control. In what follows a sequence of lemmas is proved so as to form the underlying basis of the optimal strategy. LEMMA 3.1. In an optimal strategy with time interval [O,tf], there is i E (0, tf] such that A,(t) > 0 for any t E [O,i) and A,(t) = 0 for all t E fi, t,].
ProoJ: From the first equation in (3.5) and the final condition we obtain A,(t) =Fexp[-(1
- cY)rt]l’~A,(s)u(s)e~[(l t
- a)rs]ds.
AI
= 0,
720
M. I. S. COSTAANDJ. L. BOLDRINI
Now, the optimal control law (3.6), Lemma 2.3 and the fact that u(t) 2 0 imply that h,(t)u(t) 2 0 for all t E [0, tf]. Thus, the above expression implies that h,(t) > 0, and, moreover, that, if for some t, > 0, h2(t,)u(t,) > 0, then A,(t) > 0 for all t E [0, tl>. This implies the stated result. ?? LEMMA3.2. If (3.2) is true, there is not singular control in optimal strategies. Proof By contradiction, suppose there singular on an interval [Cl,t2] C [0, tr]. Then, the switching function in (3.61, F1A&t>n(t), must be identically zero on the ing the derivative with respect to time of obtains
-Far(A,(t)
+ A,(t)(y(t)
-n(t))
is a strategy
such that
u is
which is A,(t)F(y(t) -x(t)) + interval [t,, t2], and, by computthis function at t E [cl, tz], one
+F,
d( A,(th(t)) dt
= 0.
Now, Lemma 2.5 says that the second term of the above expression is nonpositive; Lemma 2.3 guarantees that A,(t) > 0, Lemma 3.1 furnishes that A,(t) 2 0, and, by Lemma 2.1, y(t) -x(t) > 0. Thus, the above terms cannot add up to zero. This contradiction proves the lemma. ?? Therefore we are left with two possibilities for an optimal strategy at any time t: either u(t) = u, or u(t) = 0. We proceed by proving the following technical lemma that will be important in the analysis of the possibilities of switchings in optimal strategies. LEMMA3.3. In an optimal strategy, if u(t) = u, on a certain interval of time [tl, tzl, then A&)(y(t) - x(t)> is strictly decreasing in that same interval. Prooj By calculating the derivative of A,(t)(y(t) one obtains
-$A,(t)(y(t)
--x(t)))
= --ar(A1(t)
-x(t)> with respect to t,
+ A,(t))(y(t)
--x(t)).
Now, for any t E [Cl, tz], there hold Al(t) z 0 (by Lemma 3.1), A&f) > 0 (by Lemma 2.3) and y(t) -x(t) > 0 (by Lemma 2.1). Thus, for t E [Cl, f2],
-&A,(t)(y(t) and we have the result.
-x(t)))
<0 ??
OBJECI'IVES IN CHEMOTHERAPY WITH DRUG RESISTANCE
Now we proceed strategies.
to rule out a certain
721
kind of switching in optimal
LEMMA 3.4.In the case of (3.2), there cannot be an optimal strategy with switch& from u = 0 to u = u ,,,.
Proof Suppose by contradiction that on a certain interval of time on an interval [tz, t3) the optimal Then, from the optimal control
that there is such kind of switching and the optimal strategy be u = 0, and strategy be u = u,. law (3.61, the switching function G(t) = h,(t)F(y(t) -x(t)) + F,A,(t)n(t) must pass from negative values on [tl, t2) to +(tJ = 0 to positive values on (t2, t3). But, from Lemma 3.3, $(t) must be strictly decreasing on [t2, tJ, and, since $(t2) = 0, we conclude that @(t> < 0 for t E (f2, tJ, in contradiction with the previous conclusion. Thus, it is impossible to have switchings from zero to u, in optimal strategies. ?? Finally, the above lemmas immediately imply the following characterization of the optimal strategies. [tl, t2)
THEOREM 3.1. If (3.2) holds, any optimal treatment consists of maximum drug concentration throughout, that is, either u(t) = u, for all t E [0, tf] or there may be 0 < i < t f such that u(t) = u, for t E [0, i) and u(t) = 0 for t E Tt,tJ. Remark 3. The results of the above lemmas and theorem hold true from standpoint of considering the case of fi(n) as a constant.
the mathematical
Remark 4. In this section we have completely
characterized the optimal strategy to be used when the growth rate for tumor cells is constant and the growth rate for normal cells is rather general. Unfortunately, we were not able to derive such a specific result for the most general case, even in the logistic situation, which corresponds to the equations (2.1) with f(y) = r(1 - ky), f,
122
M. I. S. COSTA AND J. L. BOLDRINI
4. Discussion. In this work we attempted to devise optimal chemotherapeutic treatments in continuous chemotherapy, when drug resistance is taken into account and the objectives to be attained, namely, the minimization of tumoral cells and the maximization of normal cells at the end of the treatment, are conflicting. In the special case of constant growth rate for tumor cells (Malthusian growth), but rather general growth rate for normal cells, the optimal strategy was worked out, and, from the practical point of view, maximum drug concentration throughout the treatment proved to be the optimum, except in the trivial case p = 0. Although we were not able to work out optimal strategies for the general growth rates for both tumor and normal cells, we conjecture that in this case treatments other than maximum drug concentration throughout may be confined to restricted sets of initial levels of tumoral and normal cells and/or to relations among parameters that do not necessarily express a commonly biological evidence. We point out that under the set-up of qualitative modeling of spontaneously acquired drug resistance presented in this work as well as in our previous ones, maximum drug concentration featured as a candidate that almost always presented itself as an optimal treatment (let alone the cases when it is the sole optimal treatment). This prevalence of maximum drug concentration as an optimal treatment is, in part, corroborated by probabilistic models that take drug resistance into account [see Coldman and Goldie (1983) and Coldman and Goldie (1986), Harnevo and Agur (1992)], though their setting concerns discrete drug administration. According to these models rapid depletion of sensitive cells is essential and if low dose therapies are continued over long periods of time, they are unlikely to be successful. Indeed it is an intriguing issue that rest periods almost do not feature as part of any optimal strategy in continuous chemotherapy when drug resistance is accounted for [at least as modelled as in Costa et al. (19921, Costa et al. (1994), and Costa et al. (1995b) and also in this work]. It seems that spontaneously acquired drug resistance modelled by an ever-increasing population induces in the model a built-in inclination for the suppression of rest periods. In fact, rest periods only proved to be clearly part of optimal strategies in the setting where the patient’s recuperation was considered [see Costa and Boldrini (1997)]. In the course of our previous works without patient’s recuperation, alternation between maximum drug concentrations and rest periods as optimal treatment was ruled out in all cases except for the setup including drug kinetics [see Costa et al. (1995a)l. Thus, it is still open the question whether the inclusion of a pharmacokinetic equation may engender optimal solutions with rest periods, since the drug decay rate plays an important
OBJECTIVES
IN CHEMOTHERAPY
WITH DRUG RESISTANCE
123
role in the determination of optimal treatments. To reinforce this issue we point out that in a recent work Murray (1995) modelled drug resistance via an effectiveness term that decreases the tumoral and normal cell loss as the accumulation of the drug (which is expressed by a drug kinetics equation) grows. There rest periods belong to the class of optimal treatments. In a sense, due to the fact that his work deals with resistance induced by drug, his result supports our conjecture about the relation between rest periods and drug kinetics in the context of spontaneously acquired drug resistance. Finally we would like to mention that the absence of rest periods in the protocols proposed by the models of theoretical chemotherapy with drug resistance analyzed in our previous works as well as in this one, may indicate that they are not entirely appropriate to design drug regimens (especially when the model yields maximum drug concentration throughout as the sole optimal strategy since impediments in clinical protocols prevent its current use). Besides, this may serve as guide to devise models that are more qualitatively realistic, that is, models that contain relevant aspects and features for deriving suggestions of clinically accepted protocols. At this stage, considering deterministic settings, our analysis points at two possible directions: inclusion of drug kinetics and mechanisms of patient’s recuperation in the models. The authors acknowledge the helpful comments and suggestions made by the referees on an earlier version of this work. This research was supported in part by CNPq grant No. 300153/95-3.
REFERENCES Coldman, A. J. and J. H. Goldie. 1983. A model for the resistance of tumor cells to cancer chemotherapeutic agents. Math. Biosci. 65, 291-307. Coldman, A. J. and J. H. Goldie. 1986. A stochastic model for the origin and treatment of tumors containing drug-resistant cells. Bull. Math. Biol. 48, 279-292. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1992. Optimal chemical control of populations developing drug resistance. ZMA J. Math. Appl. Med. Biol. 9, 215-226. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1994. Optimal chemotherapy: a case study with drug resistance, saturation effect and toxicity. Ii&f J. Math. Appl. Med. Biol. 11, 45-59.
Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995a. Drug kinetics and drug resistance in optimal chemotherapy. Math. Biosc. 125, 191-209. Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995b. Chemotherapeutic treatments involving drug resistance and level of normal cells as a criterion of toxicity. Math. Biosci. 125, 211-228. Costa, M. I. S. and J. L. Boldrini. 1997. Chemotherapeutic treatments: a study of the interplay among drug resistance, toxicity and recuperation from side effects. Unpublished manuscript. Eisen, M. 1978. Mathematical Models in Cell Biology and Cancer Chemotheram. Lecture Notes in Biomathematics, Vol. 30. Heidelberg: Springer-Verlag. Goldie, J. H. and A. J. Coldman. 1979. A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat. Rep. 63,1727-1733.
724
M. I. S. COSTA AND J. L. BOLDRINI
Hamevo, L. and Z. Agur. 1992. Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency. Cancer Chemother. PhatmacoL 30,469~476. Kirk, D. 1970. Optimal Control Theory. Englewood Cliffs, NJ: Prentice-Hall. Leitmann, G. 1974. Cooperativeand Non-CooperatiueMany Player DifferentialGames, International Centre for Mechanical Sciences Course Lectures, No. 190. Udine, Italy: Springer-Verlag. Murray, J. M. 1990a. Optimal control for a cancer chemotherapy problem with general growth and loss functions. Math. Biosci 98, 273-287. Murray, J. M. 1990b. Some optimal control problems in cancer chemotherapy with a toxicity limit. Math. Biosci. 100, 49-67. Murray, J. M. 1995. An example of the effects of drug resistance on the optimal schedule for a single drug in cancer chemotherapy. ZUA J. Math. AppL Med BioL 12, 55-71. Skipper, H. E. 1983. The forty year old mutation theory of Luria and Delbruck and its pertinence to cancer chemotherapy. Aa!u. Cancer Research 40,331-363. Swan, G. W. and T. L. Vincent. 1977. Optimal control analysis in the chemotherapy of ZBG multiple myeloma. Bull. Math. BioL 39, 317-337. Swan, G. W. 1987. Tumor growth models and cancer chemotherapy. In Cancer Modeling, J. R. Thompson and B. W. Brown (Eds), pp. 91-179 New York: Marcel Dekker. Swan, G. W. 1990. Role of optimal control theory in cancer chemotherapy. Math. Biosc. 101, 237-284.
Vaidya, V. G. and F. J. Alexandro, Jr. 1982. Evaluation of some mathematical models for tumor growth. Znt.J. Bio-Med. Comp. 13, 19-35. Vendite, L. L. 1988. Modelagem matem&ca para o crescimento tumoral e o problema de resist&cia celular aos fhrmacos anti-bl&ticos. Ph.D. thesis, Faculdade de Engenharia El&rica, Universidade Estadual de Campinas, SP, Brazil. Zietz, S. and C. Nicolini. 1979. Mathematical approaches to optimization of cancer chemotherapy. Bull. Math. BioL 41,305-324.
Received 28 August 1995 Revised version accepted 6 June 1996