Optimal drug regimens in cancer chemotherapy for single drugs that block progression through the cell cycle

Optimal Drug Regimens in Cancer Chemotherapy for Single Drugs that Block Progression through the Cell Cycle J. M. MURRAY School of Mathematics, University of New South Wales, PO Box 1, Kensington 2033, Australia Received 29 August 1993; reoised 21 December 1993

ABSTRACT In this paper we where the drug can tions we determine endpoint in minimal limit. We also study

1.

study the application of a single drug in cancer chemotherapy block progression through the cell cycle. Under certain condithe regimen that reduces the tumor population to a desired time while keeping a normal cell population above some lower the sensitivity of the optimal solution to various parameters.

INTRODUCTION

Mathematical models of cancer chemotherapy treatment often involve a description of the tumor cell population in terms of one or more ordinary differential equations and some constraint on the drug toxicity. The aim is a balance of killing as many tumor cells as possible while keeping toxicity to a minimum. Each of these elements can have different constructions and emphases depending on the particular problem being attacked and the approach involved. For instance the differential equations describing the tumor cells can have a variety of growth functions including exponential, Gompertz, logistic, etc., as well as a similar array of terms describing the rate at which the cytotoxic drugs kill the cancer cells. The description of the tumor cells may be even finer in that a system of differential equations may be used, one equation for each phase of the cell cycle, or one equation for each tumor subpopulation. Toxicity can be limited by fixing the total amount of drug used in the course of treatment. Alternatively, another cell population may be

MATHEMATICAL

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183 0025-5564/94/$7.00

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chosen that represents the early warning of toxic reaction to the body and a measure of its loss may be used as a quantifier of toxicity. The objective has a similar diversity. In some cases a required endpoint for the tumor population is specified and the aim is then to achieve this value by the end of treatment while minimizing the toxicity. Alternatively an allowed toxicity is specified and the aim is to minimize the tumor cell population while keeping the toxicity feasible. In some cases an attempt is made to optimize the tumor level and toxicity level at the same time. In most cases only one drug is used in the mathematical model. It is either considered to be treatment by a single drug, which is rare in practice, or it is considered to be an idealized amalgamation of several drugs, which is inaccurate. In clinical treatment of cancers, a combination of drugs is used partly to offset the development of resistance of tumor cells to individual drugs and party because different drugs can have different forms of toxicity, one attacking say while blood cells, another attacking kidney cells. If each drug still destroys tumor cells, a combined effect can be achieved with no more cumulative toxicity developing than for a single drug. Some results consider treatment with more than one drug, for instance [l, 51. But the optimization of treatment is difficult. So usually the analysis is conducted with a simplifying assumption of one idealized drug and then the hope is that these results may be helpful in indicating a suitable combined treatment. Certainly if there is no drug resistance and the toxicities are separate, this is a valid argument. An excellent reference to the literature in this area is [S]. This paper takes the approach of setting a constraint on the allowed toxicity, measured by the level of some normal cell population, and minimizing the time taken to drive the tumor population to some specified level, for instance something less than one. Only one drug is considered. We have already discussed the reasons underlying the assumption of one drug. Why have we chosen the other characteristics of the model to be of this form? It usually seems preferred by clinicians to deplete the tumor cells as much as practically possible compared to setting a desired end level for the tumor cells and minimizing the toxicity necessary to reach it. With bone marrow transplants feasible, they will first remove some of the bone marrow, undertake aggressive treatment that kills all of the remaining bone marrow and hopefully all of the cancer, and then replace the bone marrow. The preference is really on destroying the cancer. Also these other models lead to a solution where the optimal

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183

treatment is effectively to delay killing the majority of tumor cells as long as possible. The usual pattern is that the application of drug is small at the beginning of treatment becoming larger toward the end. This type of treatment is rare in practice though it can be beneficial in maintenance therapy [4]. One of the disadvantages of this approach is that it initially leaves the tumor population relatively high. Although drug resistance is not considered as part of these models, its implications are still relevant and for this approach, if development of drug resistance cells is proportional to the number of tumor cells (see for instance [11>, the treatment is more likely to fail than one that achieves the same tumor endpoint but also drives the tumor cell population down at the start of treatment. There will be less likelihood that resistant cells will develop. The development of resistance by the cancer to the cytotoxic drugs is one of the main reasons for treatment failure [3]. A novel feature of the model considered here is the inclusion of a blocking effect caused by the presence of the cytotoxic drug. Many of the drugs used not only kill tumor cells when present at some concentration, but also slow or stop the progression of cells through the cell cycle, (see the table in [7. p 2761). If no cells complete the cell cycle, then no cells will divide and the tumor does not grow. This is also true for the normal cells. In many cases the cytotoxic drug will have the following effects. For low levels there will be no apparent effect. At some concentration the drug will begin to block cells from progressing through the cell cycle, thereby slowing growth. As the drug concentration increases, cells will be killed by the drug and more cells will be blocked. At some specific concentration all cells will be blocked. This property has been used in the literature before. One proposed treatment that tried to take advantage of the blocking influence of some drugs involved using a drug to block the tumor cells in one phase of the cell cycle, say for instance, the S phase. After this was achieved, the drug concentration would be increased and the tumor cells would move into the next phase, in this case G,. The drug would then be applied to kill a proportion of cells in the G, phase. If the time taken for a tumor cell to complete one cell cycle, Tt was different to the cell cycle time of normal cells T,, then by applying the killing drug after T, time units, proportionally more tumor cells would be killed than normal cells because there would be relatively few normal cells in the G, phase since many of them were killed in the previous application. This would not be true of the tumor cells because of the different cell cycle times. Results of this type can be found in [2].

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Unfortunately the heterogeneity of tumor and normal cells and the variability of cell cycles imply that any timing of cells usually disappears within one or two cell cycles. “In many tumors the cohort of thymidinelabelled cells which pass through their first post-labelling division to give a well defined first peak on a labelled-mitoses curve have almost completely lost synchrony by the time of the second peak” [7, p. 14.51. Hence this approach is rarely used. We do not propose using blocking to improve chemotherapy regimens. This would fail for the reasons outlined above. However we do wish to incorporate this effect into our model, since it occurs so frequently. The question then arises as to how to schedule a drug for a chemotherapy model when blocking occurs. We already know the optimal policy for this model but with no blocking. It is outlined in [61. The optimal regimen is effectively one of continuous infusion. How does this change under the effects of blocking? To some extent we are searching for components of actual cancer chemotherapy that may cause the optimal drug schedule to be other than continuous infusion. Many regimens used in practice consist of a large dose of drug(s) followed by a recovery period and this is repeated several times. However most of the mathematical models predict that the optimal regime should be one of continuous infusion. These are, of course, extremely simplified in comparison to the real problem. Without moving away from the one drug case, what components can cause the are drug resistance, regime to have a periodic nature ? Possibilities toxicity, blocking, etc. In this paper we investigate the effect that blocking has on the nature of the optimal regimen. We incorporate the blocking effect of the drug into our model by multiplying the growth function by a term that has value 1 for low drug concentrations but after the drug reaches some specified level, the term decreases to 0 and then maintains this value for all higher concentrations. When the blocking function has value 1, there is full growth and as the blocking function decreases to 0, the growth function’s effect lessens until it no longer influences the rate of change of the cell population. The rate of change of the cell populations is determined by the balance between the growth function with this blocking term and the loss function associated with the killing of cells by the drug. The main results of this paper concern the derivation of optimal treatment strategies. If the blocking term stops all growth before cell loss occurs, then the optimal schedule is an essentially periodic one; at each application as much drug as possible is given, so that the normal cell population will be reduced to its lowest allowed value. Then a

CANCER

CHEMOTHERAPY

WITH

BLOCKING

resting period follows where the normal cells recover recovery level satisfies the following inequality au, < a2 -

187

to some level. This

b-Q(t).

This says that the cell populations are not allowed to grow to a level where the effective growth rate of the tumor (this equals the growth rate a, times the relative drug sensitivity parameter (Y) exceeds the growth rate of the normal cells (a, - b,x,(t)). The pattern of drug application followed by recovery is repeated until the desired tumor endpoint is achieved. We also find that under certain assumptions the amount of drug given and the recovery time is exactly the same in each cycle. At the other extreme where blocking effectively does not occur, the optimal solution entails an initial dose that is as large as possible without violating the toxicity constraint, followed by continuous infusion of the drug thereafter, keeping the normal cell population at this lowest level and driving the tumor cell population even lower. This is possible in this situation because cell growth and cell loss can occur simultaneously and since cxal < a2 - b, p the tumor cell population will decrease while the normal cells are held at their lowest level of p. Here there are ranges of the drug concentration where both the growth term and the cell loss term are nonzero. We say that cell growth and cell loss “overlap.” For the case in between these two extremes we find that the solution changes from being periodic to one with continuous infusion depending on the amount of overlap of the growth and loss terms. The structure of the paper is as follows. In the next section we formulate the mathematical model and state the results. In Section 3 we discuss the extension of results to more general problems and present some numerical solutions to these problems. Section 4 discusses our findings and their implications and the final section contains the mathematical proofs of the results of Section 2. 2.

THE

MODEL

AND

RESULTS

The model that we will use to describe cancer chemotherapy treatment with one drug uses one differential equation for the change of the tumor cells and another for the normal cells. The normal cell population is not allowed to fall below some fixed level and the aim is to minimize the time required to reduce the tumor population to some set endpoint. The mathematical representation of this is

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minimize

subject

T

(2.1)

i-l(t) = h( PW)&(-+))Ll( P(t)) i2(t) = h( Pw)qX*w) - L2( P(t)) i4t) = u(t) - &-4t)

(2.2) (2.3) (2.4) (2.5) (2.6)

to

x*(t) > p u(t) E [Gl

Xl(O)= x10,

%(O) x,(T)

Vt Vt E [0, T]

=

X20)

=

XlT

I-40)

=

PO

(2.7) (2.8)

where x,(t) is the logarithm (log,,) of the tumor cell population at time t; n,(t) is the logarithm (log,,) of the normal cell population at time t; pu(t> is the drug concentration at time t; h(p) is the blocking function; c.(x) is the growth function for cell population i = 1,2; Li( ~1 is the loss function for cell population i = 1,2; 6 is the drug decay rate; p is the lowest allowed value of the normal cell population; U is the highest rate of drug application; and xIT is the target tumor level. A measurable function u will be called a feasibk control (regimen) if (2.6) is satisfied and if the function i generated through the differential equation (2.3) using the extended control function ii(t)

=

u(t), 0

t E t>T

[OJ]

satisfies the constraint (2.5) for all t. This definition specifically requires that the constraint (2.5) must be satisfied even after the final time is reached. This is important as the practical problem must concern itself with keeping the toxicity levels feasible even after all the drugs have been administered. Since the drug level does not decay instantaneously, there is no guarantee that a regimen that is nontoxic while the drug is being administered will stay that way. Such a regimen must be considered infeasible. The blocking, growth, and loss functions can take many forms. Here we will choose some specific types, most of which can be manipulated to fit a variety of circumstances. Assumptions on h, 15;,Li, i = 1,2. (1) The blocking function h is a continuously differentiable function taking the value 1 for /L G pt, the value 0 for p 2 p2, and being monotonic in between these values. We assume 0 < p, < p7.

CANCER CHEMOTHERAPY

WITH BLOCKING;

l&iY

(2) We assume that the tumor cell population grows at an exponential rate. Hence with the log,, scaling we use for x1, we have 4(x) = ur, Vx. Here a, = log,,(2)/td, where td, is the doubling time of the tumor. (3) The growth of the normal cell population is taken to be Gompertzian so that F,(X) = a2 - b,x. We also assume x,(O) = 13, the maximum level of the normal cell population. This makes the calculations easier but if x,(O) < 0,, the same form of optimal solution will apply. We just extract a suitable piece from the optimal solution derived here. (4) The loss functions for both cell populations are general ones. However we specifically allow for an initial null effect of the drug when concentrations are small, as well as for a saturation effect for high drug levels. Therefore we assume Li( p1L)has a value 0 for 0 < p < ri, has value si for p 2 qi, and is monotonically increasing on (r,, qi>. Many drugs have this behavior. In [lo] a model of the action of vinblastine in uivo is described. Essentially the model uses a kill term for vinblastine that is discontinuous at some critical drug concentration with zero cell kill below that level and constant cell kill above it. This was in line with observations from [9] that the cells examined there lost their proliferative capacity over a relatively short range of concentrations of vinblastine at some specific level. Vinblastine is one of the drugs listed in [7, p. 2761 as having a blocking effect on the cell cycle. (5) To simplify matters further we assume L,( CL)= cLr( ~1 for some a! > 0. This implies rr = r2 = I and q1 = q2 = q.

One difficulty associated with this problem concerns the definition of a feasible control. We must ensure that any control u does not violate the toxicity constraint after T as well as before T. The optimality conditions that we will use to determine the optimal solution only consider the state of the system up until T. Hence we must find another way to ensure that any optimal solution derived by the necessary conditions satisfies our extended problem. We will achieve this by imposing two additional constraints. We will also show that these constraints do not change the problem in the sense that any optimal solution to the original problem will also be optimal for the new problem. The new constraints are

x2(T) = P

h( /-4TW*(m3)

- Ju P(T)) = 0.

(2.9) (2.10)

The motivation behind the first constraint is that if xrr is reached in minimum time, then in all practical circumstances it will be achieved by a chemotheranv strateav that gives as much drug as possible especially

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toward the end. If x,(T) > j3, then more drug could have been given and the endpoint would have been achieved earlier, violating the minimality of T. The second constraint follows from the first and our attempts to keep feasibility after T. The left-hand side of (2.10) cannot be negative, otherwise x,(t) < j3 for some interval of time after T. If it were positive, then x,(t) < p for some time interval before T. PROPOSITION 2.1

Let u be a feasible chernotheraphy regimen for the problem (2.1)-(2.8). Zf x is the state corresponding to this control and x,(T) > p, then there exists ,a control G that is also feasible and reaches the desired endpoint in time T < T. The proof of this proposition is contained in the last section. The proposition provides the basis for our claim that including the constraints (2.9) and (2.10) does not alter the problem. We will discuss several cases relevant to this model where the mix between blocking of cell progression and cell kill varies. There are two extreme cases, one where the blocking and killing do not overlap ( pZ G r), and the other case where blocking only occurs at the saturation level of the drug (q G pl>. This last situation corresponds to a drug that exhibits no blocking. The results for these cases can be obtained analytically. The remaining case that lies somewhere in between the extremes is analyzed numerically. These are perhaps the more relevant results. 2.1.

NONOVERLAPPING CASE

In this section added assumption

we consider that

the problem

(2.1)-(2.10)

but

with the

p2 =Gr. This implies that whenever the blocking function h is nonzero, the cell loss functions have no effect and vice versa as in Figure 1. Hence cell growth and cell loss do not occur at the same time. We can show (see the last section) that the optimal drug regimen for this case contains an initial does that depletes the tumor cell population as much as possible, so that the normal cell population is driven to its lowest allowed value p. After this there is periodic treatment where the cell populations are allowed to recover before giving another dose of the drug that drives the normal cell population to /3. This is repeated with exactly the same dose given each time until the tumor hits the desired level.

CANCER

CHEMOTHERAPY

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WITH BLOCKING

-

0.6

-----/ /

-

0.4

/ / /

-

0.2

/ / 0.05 FIG. 1. Plots of h-

0.15

0.1 and I,--

0.2

for p2 G r. Here

0.25

0.3

p1 = 0.05, p2 = 0.08, r = 0.1,

q = 0.2, and s = 0.6.

A typical example is shown in Figure 2. If we label a drug regimen of this type containing n cycles as u,, then the amount of drug given in each cycle will be that value that sends the tumor cell population to the specified end level in exactly IZ cycles. This dosage will vary, of course, as the number of cycles y1varies. The form of the solution has been ascertained analytically; however it is difficult to determine analytically the optimal number of cycles needed. This is a function of the values pi, pz, r, q, the relative shapes of the functions h, and L,, the value of (Y,etc..

drug

0

50

100

150

200

0

50

100

150

200

0.0

50.0100.0150.0200.0

FIG. 2. Response to treatment with 5 cycles for p = 2. Tumor cell, normal cell: and drug levels Dlotted against time (in days) after commencement of treatment.

192

J. M. MURRAY

Therefore we solve this remaining part numerically. Given the parameter and data values, etc., we determine the time needed T, for a drug regimen U, of the type described above with n cycles to send the tumor population to the desired end level. By repeating this calculation for a range of values of n we find the one that achieves the minimum T, and this is the optimal solution. The usual pattern is that as n increases the T, values will initially decrease. Eventually these values start to rise again and continue to increase. The value of n that corresponds to the turning point gives the optimal drug regimen U, with y1 cycles. Figure 3 shows a plot of T, against II for the indicated data set. The parameters used in the numerical calculations represent a fictional case. However these values reflect the characteristics of several tumor types and drugs. In the next section we present results using other data, so that we have not entirely relied on this set of values. In particular we have studied the effects of a whole range of p values. The tumor experiences exponential growth with a doubling time of 28 days (td,). The normal cells have Gompertz growth with a maximum population of 1O1’ cells (lOez> a minimum allowed level of lo2 cells (10 a>, and a doubling time of 10 days (td,), when the cell population is at this lowest level. The drug has no effect on the cells until its concentration reaches 0.1 (r), and becomes saturated at a concentration of 0.2 (4). At its saturated level the drug kills 0.6 Cs), of the cell population per day. It is equally effective on tumor cells as it is on normal cells (a = 1.0). The maximum rate at which it can be administered is 2 per day (U). The drug has a half life of 1 day (6 = ln2/1 = 0.7). We assume that no blocking occurs for drug concentration below 0.05 (pi>, and that complete blocking occurs when the concentration reaches 0.08 ( ,+I.

200 T” 196

Time

Opt&al

I

=

optimal

188.68,

number

of

cycles

=

9 x

192

x

t

x

I

180 0

I 4

I

x

x

x

x

x

x

x

x

/

I

I

I

I

I

I

I

a

12

x

x

I

I 16

FIG. 3. Plot of T_ against n for the standard data set.

x

I

x

I

I1

” 20

CANCER CHEMOTHERAPY

IYJ

WITH BLOCKING

Initially the tumor level is lOlo (x,(O) = 101, the normal cell population is at its maximum level lOlo (x,(O) = lo), and the drug concentration is 0 (p(O)). This is in line with no prior treatment. In order to eradicate the tumor, we require the final tumor population to be less than 1; to this end we have chosen the desired tumor population to be 10-l (XIT = - 1). 2.2.

COMPLETELY

OVERLAPPING

CASE

In this section we discuss the case where full growth can occur even at the effective maximum range of the drug. In terms of our parameters we have 9 =Gpl. This corresponds to the situation where the drug has no blocking effects at all, in which case the growth rate is unaffected by any level of drug as in Figure 4. For this situation the methods and results of [6] apply. The results of that paper indicate that the optimal drug schedule is one that drives the cell populations down as much as possible initially, so that x2 reaches the lower limit p. Then the drug is applied via continuous infusion so that the normal cell population stays at its lower limit whilst the tumor population is steadily decreased. This is depicted in Figure 5. 2.3.

PARTlALLY

OVERLAPPING

CASE

Here we consider the most likely case for drugs that exhibit blocking. In terms of the functions h and L, we assume p1 < r < pz < q. See Figure 6.

1

0.8

-

0.6

-

__ -

-

-

I / 0.4

/ /

0.2

/ / / 0.05

0.1

0.15

0.2

FIG. 4. Plots of h and L, for a < 4,.

0.25

0.3

J. M. MURRAY

lY4 12

12 11 Optimal

Time

normal

11

tumour

10

10

= 162.86

9

9 8 7 6 5 4 3 2 1 0

-1

0

50

100

FIG. 5.

150

200

rllllll~fl~llllll 0

50

100

150

200

Optimal treatment when there is no blocking.

Depending on the relative magnitudes of these parameters, we can imagine that either of the previous situations could occur and possibly some others as well. However it seems a reasonable supposition that the optimal solution will be one of those forms that we have discussed for the extreme cases. Which one it would be would depend on many factors. This case is too complicated and dependent on too many factors to solve analytically, so we resort to numerical computation. We do this by

0.8

-

FIG. 6.

Plots of h and L, when p1 < r < p2 < q.

CANCER

CHEMOTHERAPY

WITH BLOCKING

lY5

following the same approach as for the nonoverlapping case. We determine the time needed T,, for a drug regimen with II cycles u, to drive the tumor cell population to the specified level. The U, that gives the lowest value of T, is taken to be the optimal one. For situations that are close to being nonoverlapping we should find that the optimal T, is given by a small value of n. On the other hand, if it is close to being completely overlapping, we should find that no finite value of n attains the lowest T, and that T, is a decreasing function of IZ. Continuous infusion can be thought of as the limiting case of U, where n goes to 00. Plotting T, as a function of n we find the usual pattern is that as IZ increases the T, values will initially decrease. For the case where there is little overlap of the blocking function and the cell kill function, the T,, values follow the same pattern as for the nonoverlapping case; they decrease for the first few values of n and then start to rise again. The n that corresponds to the turning point is taken to be the optimal solution. See Figure 7. When it is close to complete overlap, the plots show that T, as a function of n is decreasing at least for the first 100 values of II. This indicates that the optimal schedule is either continuous infusion or as close to that as possible. See Figures 8 and 9. That the first 100 values of n is sufficient to indicate that a regimen close to continuous infusion is the optimal treatment can be demonstrated by Figure 10. It shows the response of the cell populations to a schedule with 55 cycles. These are almost indistinguishable from the response obtained from continuous infusion.

200 T”

195

x Optimal

Time

=

183.40,

optimal

number

of

cycles

=

13

: x

190 t

0

10

FIG. 7. Partially overlapping

20

with pz = 0.11, r = 0.1.

30

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J. M. MURRAY

165.0 T" 164.8

Optimal

164.6

Time

T, is still

decreasing

at

n=lOO

c

x

x

x

x

x

x

x

x

x

x

164.4

x

x

x

x

x

x

x

x

x

I

I

t

I

I

I

164.2

I

164.0 80.0

I

I

I

I

I

I

90.0

85.0 FIG.

I1

95.0

n

I

100.0

with p2 = 0.15, Y = 0.1

8. Partially overlapping

165.0 r 164.8

Optimal

Time

T, is still

decreasing

at

n=lOO

1 164.6

1

164.4

x

x

i

x

x

x

x

x

x

x

x

I

x

x

x

I

164.2

80.0

85 FIG.

90.0

0

95.0

100.0

9. Partially overlapping with p2 = 0.17, r = 0.1.

0.5

12

12 11

x

normal

iI

tumour

10

10

9 8

0.4

9

8

7

7

6

6

5

5

4

4

3

3

2

2

0

0

0.3 03

1 -1

:: 0

-1 50

100 FIG.

150 10.

200

0

Response

50

100

to treatment

150

200

containing

0.0

50.0100.0150.0200

55 cycles.

0

CANCER

3.

CHEMOTHERAPY

IYI

WITH BLOCKING

SENSITIVITY OF SOLUTION PROBLEM PARAMETERS

TO

In this section we investigate the sensitivity of the problem, and therefore the solution, to the data. We do this by systematically perturbing the relevant data and seeing what effect it has on the optimal time to reach the end point and for the nonoverlapping case, determining whether the number of cycles increases or decreases. Of course these are not intended to be accurate predictors but to give a guide as to what direction things will change and the order of magnitude of the change. This analysis is carried out by numerically solving each case and comparing the optimal solutions that are obtained. We only present the results for the nonoverlapping case. The indications obtained for these extend to the other cases in an appropriate manner. The standard problem and data with which we will make the comparison is the one shown in Figure 3. Consider the effect of changing the maximum drug application rate U. The standard rate is 5 = 2. If this changes to 1, there is little effect as in Figure 11. The optimal time barely increases and the number of cycles remains unchanged. Similarly if li increases to 3, the change is barely perceptible. See Figure 12. If a different drug is given that has the same specifications as the standard one except that the new one has a shorter half life, then this new drug will provide an improvement in treatment (for the cases when the optimal treatment is cyclic, but not for those where the drug level is maintained at a constant level through say continuous infusion). This can be seen in Figure 13 where the decay rate of the drug has been increased to 1.4, which implies that the half life of the drug has changed

200

T”

196

192

I

Opikval

4

Time

=

189.15,

a

Frc,. 11. Nonoverlamine

optimal

number

12

of cycles=

9

16

with B = 1 instead of 2.

20

lY8

J. M. MURRAY

200

T” 196

Optitkal

Time

=

188.54,

optimal

number

of

cycles

=

x

192

9

x x x

188

x

x

x

x

I

I

I

I

x

x

x

x

I

I

x

x

x

184

180

I

0

I

I

I

8

4

I1

I

12

FIG. 12. Nonoverlapping

” I

I

I

16

I

20

with E = 3 instead of 2.

from 1 day to l/2 day. The optimal time has dropped and the number of cycles has increased. This change is understandable in light of the fact that the drug is unproductive during the time it spends decaying (and increasing) between the areas of cell kill and cell growth (that is when p E [ p2, 1-1. Here there is neither cell growth nor cell kill and it can be considered an unproductive region). If the drug can pass between these areas faster, then the percentage of total time spent in this wasted period will decrease and so will the optimal time. Why should the number of cycles increase for this change? The fewer number of cycles used, the less time is spent traversing the region between cell growth and cell loss and as well between the upper level of drug and the lowest level of drug. This is good in one sense but it must

200 T” F 196

Optimal x

t 192

188

t

Time

=

183.69,

optimal

number

of

cycles

=

13

x

1

x I

184 1 I

180 0

I 4

FIG. 13. Nonoverlapping in half the time.

I

I

I

I

8

I

I

I 12

I1

I

I 16

I

I

I

n

I 20

with 6 = 1.4 instead of 0.7. This implies the drug decays

CANCER

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199

WITH BLOCKING

be weighted against another factor that contributes to the success of the treatment. This is that when the normal cells are kept closer to their lower limit (here /3 = 21, their growth rate is the highest so they will have the greatest advantage over the tumor cells when they are regrowing. The more cycles used the closer will the normal cell population be to this lower limit. So the optimal solution is a balance of these two competing terms. By increasing the drug decay rate we have shifted the trade off toward more cycles. A similar situation occurs when the region between the areas of cell growth and cell kill is reduced, that is when r - p2 is decreased. In Figure 14 the value of p2 equals 0.1, the same as r, the lowest level at which there is cell kill. Now there is no gap between nonzero cell growth and cell kill, but there will still be a gap between the highest level of drug used and the lowest level. We see that the optimal time has once again decreased in comparison to the standard problem, and the number of cycles has increased. The next variation that is analyzed has several implications. It considers the effects of changing the lowest allowed value of the normal cells from p = 2. If we require double the number of cells to avoid excessive toxicity so that p = 2.3 or if we require an order of magnitude greater so that p = 3.0 (we still assume that the doubling time of 10 days for the normal cells is achieved when x2 = 2.01, then the treatment time increases markedly. Studying the results displayed in Figures 15 and 16 we see that there is a measurable increase in the time needed to drive the tumor to the desired endpoint. The optimal number of cycles has increased since the normal cell population has been denied access to its highest growth rate, which was when it was at p = 2 and so compensates for this by increasing the number of cycles. It also cannot allow

196

Optixmal

1

180 0

4

Time

I

=

I

185.30,

I

I

optimal

I

8

FIG. 14. Nonoverlapping

,

number

I

I

12

of

cycles

x

x

I

I

=

x

x

I

,

11

16

with p2 = 0.1 instead of 0.08.

x I

x

I

I,

n 20

200

J. M. MURRAY

222

x

T” 221

Optimal

Time

=

217.29,

opiimal

number

of cycles

=

11

x x

219

x

x 218

x x

x

217 I

216 0

I

/

I

5

I

x

x

x

I

I

I

x

I

I

10

FIG. 15. Nonoverlapping

I

I

n

I,

I

15

I 20

with /I = 2.3 instead of 2.0.

the normal cell population to regrow to a level where the growth rate of the tumor exceeds that of the normal cells. If we use a less drastic value of p, the change in treatment time is even greater. If we take p = 8.0 with the 10 day doubling time of the normal cells occurring at this value, then the optimal pattern is described in Figure 17. Now changing p to 8.3 as in Figure 18 incurs a 33% increase in treatment time. This greater increase occurs because the growth rate of the normal cells changes from a 10 day doubling time to no growth over a smaller range of x2 values. Hence there are more cycles and there is more time when the drug level is in the unproductive region. This variation also indicates the effects of using a suboptimal regimen, as well as giving some idea as to the importance of tracking the toxicity levels of treatment if this is at all possible.

Optimal

Time

=

300.33,

optimal

number

of cycles

=16

x x

299

t

14

FIG. 16.

NOIlDVerhDDiD!?

16

with B = 3.0 instead

21 of 2-n

CANCER

‘“I

WI’I‘H BLUCKlNti

CHEMOTHERAPY

537 T” 536

E

535

-

Optimal

Time

=

532.95.

optimal

number

of

cycles

=

56

> x I

535

x

-

534

L

533

-

x

x

x x

”

532-

I

’

x

x

I

”

50

x

x

x

x

”

x

”

55

x

x

x

x

I

’

I

I

”

”

” 70

65

60

FIG. 17. Nonoverlapping

‘I

with p = 8.0.

One of the most significant influences on the optimal time, and therefore on the likelihood of success of the treatment, occurs when we vary the relative sensitivity of the two cell populations to the drug. The standard case used is (Y= 1, which means that the drug affects both cell populations exactly the same. If we change this to CY= 0.9 so that the tumor cells are more sensitive to the drug, then optimal treatment time drops dramatically from 189 to 121 as seen in Figure 19. A 10 percentage change in (Yhas caused a 36 percentage change in the optimal time. Conversely if the normal cells are more sensitive to the drug than the tumor cells as in Figure 20 where (Y= 1.1, then the optimal treatment time increases greatly. The last situation investigated looks at the effects of the relative growth rates of the two cell populations. The standard case uses a

712 T” 711

Optimal

time

=

709.60,

optimal

number

of

cycles

=

80

x t

x

x

709 t

I

708

70

I

n I

75

FIG. 18. Nonoverlapping

I

I

80

I,

,

,

I

85

with /? = 8.3 instead of 8.0.

I

I,

90

LUL

J. M. MURRAY

128

1 T.

x

Optimal

Time

=

120.73,

optimal

number

126

of

cycles x

L

5

1

I

x

t

118

I

0

I

I

I1

4

I

I

8

I

I,

I,

12

lb

I

I,

t-l

20

FIG. 19. Nonoverlapping with cy = 0.9 instead of 1 so that the normal cells are less sensitive to the drug than the tumor cells.

doubling time of 28 days for the tumor and a doubling time of 10 days for the normal cells when they are at the lowest allowed level of j3 = 2. Since we are assuming Gompertz growth for the normal cells, then as the population increases from this lowest level the doubling time will increase until it eventually exceeds that of the tumor. At that point the normal cells are losing ground to the tumor cells. This cannot be allowed in any optimal regimen. In Figure 21 we see the effects of halving the doubling time of the tumor to 14 days but keeping the same kinetics for the normal cells. There is a huge increase in the optimal treatment time and a similar increase in the number of cycles used. The number of cycles has increased for the reason specified in the previous paragraph. Since the

290 T” 284

Optimal

aime

=

264.82,

optimal

number

of

cycles

=

14

278 x x

272

x

F 266

260 0

4

8

12

lb

20

FIG. 20. Nonoverlapping with (Y= 1.1 instead of 1 so that the normal cells are more sensitive to the drug than the tumor cells.

CANCER 420

203

WITH BLOCKING

CHEMOTHERAPY

T"

418

-

416

-

414

-

412

-

I

Optimal

Time

=

412.14.

optimal

number

of

cycles

=

32

x

x

I

x x x r

x xx

x

t

"

410'1"1'1"""11'11'1"111""" 20

25

30

35

40

45

50

FIG. 21. Nonoverlapping with td, = 14 instead of 28 so that the tumor cell doubling time is closer to the doubling time of the normal cells at their lowest limit.

doubling times are closer, the normal cells only have to increase a small amount above p before the tumor cells are growing faster than the normal cells. Hence more cycles are used to keep the normal cells in that area where they outgrow the tumor cells. 4.

DISCUSSION

In this paper we have attempted to analyze the effects of cell blocking by a drug on the optimal drug regimen. We have found that the extent of the blocking determined the frequency of drug application; the more extensive the blocking (so that the growth and blocking are as depicted in Figure l), the worse is the performance and the fewer cycles used. If blocking does not arise until cell kill occurs (such as depicted in Figure 6), then the number of cycles will increase and the treatment time will decrease (see Figure 7) until we get to the stage where the optimal regimen is virtually indistinguishable from the response when there is no blocking (compare the range of treatment times in Figures 8 and 9 to the optimal time for the continuous infusion case depicted in Figure 5). Hence if two drugs are available that are identical except for the extent of blocking that they induce, then the one with the least blocking should be used. We also applied a simplistic sensitivity analysis to our numerical problem. The parameters that were most sensitive to change were the ones that bore directly on the need to have the following inequality satisfied during treatment: (Yu, < u2 - b,x,( t).

(4.1)

J. M. MURRA‘k

L”-t

If p increased, then the range of x2 values available that satisfied (4.1: was smaller and the optimal treatment time increased. Similarly if the doubling time of the tumor decreased or the relative drug sensitivity parameter CYworsened. These changes can be significant so the implication is that they should be estimated as carefully as possible. This is not an indictment on the correctness or otherwise of the model but rather a pointer to those parameters that most affect performance. In this paper we have only considered the case of exponential growth for the tumor. What can be said for the case when the tumor cell population also follows Gompertz growth? The difficulty here is that now the inequality that must be satisfied during treatment becomes

(4.2) so that there may be no guarantee that the endpoint of x,(T) = - 1 is achievable. If it is, we will need a(a, + b,) < a2 - b, p. On the supposition that it is achievable we will find that the cycles will not be identical and as xi is reduced so that the doubling time for the tumor decreases, the frequency of the cycles will increase. This is borne out by the effect of decreasing the tumor doubling time from 28 days to 14 days as shown in Figure 21. We have only considered the case of one drug. The analysis can still indicate something for the multiple drug case though, provided each drug has a different indicator of toxicity. The major determinant in that case would be the drug that incurred the most blocking since it would determine when there was regrowth. In conclusion we should say that even though the incorporation of blocking into the model would tend to push the optimal regimen to one that is periodic, it will not always be so. In fact for many cases where blocking occurs (those with partial overlap of cell kill and cell growth) the optimal treatment is close to continuous infusion. Whether the incorporation of drug resistance, multiple drugs, and other components of real cancer chemotherapy introduce effects that disfavor continuous infusion in these cases is a matter for further research. 5.

PROOFS

We will begin by showing that Proposition 2.1 holds. This proposition implies that all optimal solutions to the original problem satisfy the extra conditions (2.9) and (2.10). Proo$ For some interval of time immediately preceding T we must have the tumor cell population decreasing. If this were not the case,

CANCER CHEMOTHERAPY

205

WITH BLOCKING

then the endpoint would have been reached before T, contradicting its optimality. Let l be the largest value such that for almost all t E CT - 5, T) we have i-,(t) < 0. Because of Assumption (5) this implies that on the same interval iJt> < 0. Suppose that the set (t E (T - l,T):

u(t)

= U, or L,( p(t))

= si}

(5.1)

has measure less than <. Define for some E > 0 i(t)

u(t)7

=

t
min(u(t)+

i

E,U},

T-l
If E is suitably small, x2 will remain feasible so that Li will be a feasible control function. Then L,( jXt)) > L,( p(t)) for all t and at least on a set of positive measure we will have L,( fi(t)) > L,( p(t)). Hence i,(T) < x,(T) and we deduce that this new control can reduce the tumor level to the desired end value in shorter time and so is a better choice of control function than the original one. Now let us consider the case when the set in (5.1) has full measure. Let [ > C be the largest value such that for almost all t E (T - l’, T - l> we have u(t) = ii. Then for some r ? 0 we hav,e u(t) < U for a set S of positive measure belonging to (T - 5 - l, T - i’ 1. For some E > 0 define

i;(t)

=

min{u(t)+

E,E},

tes

otherwise.

u(t) 2

This control is also feasible for suitably small E and as before achieves lower end values for both cell populations. It therefore reaches W the desired tumor population in a shorter time. The problem (2.1)-(2.10) fits the standard problems covered in the optimal control literature. Necessary conditions for optimality for this problem imply that the following must hold. Define the Hamiltonian

+

(

P2 + I?+

1

(h(

P)F,(X,)

-

L2(

I-4)

+

P3(U - %)

Let the regimen u* be optimal and assume it generates the state described bv (x?. xZ. LL*1. Then there exists an absolutelv continuous

206

J. M. MURRAY

vector function p and a nonnegative measure v such that for almost all t E [O,Tl N(X*(t),~*(t),P*(t),u*(t))

= ~~~~~lH(x*(t):~*(t)7P*(t),u) (5.2)

A(t)

= - h( ~*(t))q.w>)P1(t)

l&(t)

= -h(

b,(t)

= p,(t)G(

(5.3)

/L*w)f-;(*:o)(

PAtI

Cc*(t)) + (P*(t)

- h’( CL*(t))

(

+ ~5w

MT))

W?)iL;(

(5.4) P*(t))

4(~TW)PlW

+ F,(n”(t)) y(t)

+jo’v(~)

= -1,

(Pz(t>

+ (Y(T)

w

support{v} c{t:xt(t)

H(x*(T),~*(T),P*(T),u*(T))

))+ sP3(t)

(5.5)

= P}

(5.6)

=I.

(5.7)

Notice that since the problem is autonomous, the Hamiltonian almost everywhere has value 1 along the optimal solution, not just at the final time. The Hamiltonian inequality (5.2) implies that the optimal control has the following form:

u*(t)

=

5,

if P3(t) > 0

undetermined,

if p3( t) = 0

0,

if p3( t) < 0.

I

(5.8)

Under our assumptions on the behavior of the growth of the tumor and normal cells, the costate equations simplify to

A(t) = 0 f%(t) = b*h( P*(t)) I-&(t) = PlL’,( p*(t))

(5.9) (P*(t)

+ (P*W

-h’(iri(t))(a,p,+(a2 + 6p,(t>.

+ @4

(5.10)

dY(4)

j+)

w+x

p*(t))

-b,x*(t))(P,(t)+~v(T)du(r)))

(5.11)

CANCER CHEMOTHERAPY 5.1.

NO GROWTH-LOSS

WITH BLOCKING

LUI

OVERLAP

Here we will assume that p2 < r so that the growth phase has no overlap with the cell kill phase. The endpoint constraint (2.10) then implies AT) The transversality

condition

p,(T)

E[

Pz7rl-

(5.12)

that follows from this is

=s0,

if I_L(T) = r

=O, >/ 0,

if p(T)

EC h,r)

if p(T)

= pz.

I

(5.13)

Since the aim is to minimize T subject to satisfying the endpoint conditions, we see that we must have p(T) = r and p(T) > r for t immediately before T. The Hamiltonian does not determine the value of the optimal control at those periods of time when p&t) = 0. The question arises as to when this is possible, that is, when can singular control occur. If p&t) = 0 on an interval of time, so that lj&t) = 0 as well, the optimality conditions give some indication as to the possible values of p. Only two cases arise that will allow singular control; all other cases violate (5.7). Singular control can occur if (1) p*.(t) E [O, pll. In this case h’( p) = L’J CL)= 0. (2) p(t) E [q, m). Here also h’( k.) = L’J p) = 0. For all other drug levels either u*(t) = U or u*(t) = 0. Notice that on these same intervals fi3(t) = Sp,(t). Hence l if p&t) < 0 when &)E[O, pl], then p3 will stay negative since u = 0, thereby keeping p in this range and violating the endpoint condition x,(T) = x17. l if p3(t) > 0 when p(t) E [q,“), then p3 will stay positive, violating the endpoint condition on p(t). so if

49 E Dd91

if l*.(t) E [qA

then p3( t) 2 0

(5.14)

then p3( t) G 0.

(5.15)

Given our comments on the feasible p3 values when p E [O, prl, the initial value of p3 must either be greater than 0 or equal to 0. We will discuss the case p,(O) = 0 later; it corresponds to the situation where x,(O) is less than the desired end value xlrr or where x, can be driven to

208

J. M. MURRAY

before x2 reaches p. Neither of these cases is of interest here. Hence the initial value of p3 is positive and the drug is applied at its maximum rate U. Proceeding from this point, k(t) E [O, pII, j&t> = 6pg(t) > 0 and p3 remains positive. When p(T) enters the interval ( pl, p2), the differential equation for p3 becomes

xlT

J+(t) = - a,p,h’( P(t)) + aP3(t) since we are assuming that x2 starts at its maximum level so that i&t> = 0. The case p1 a 0 leads to the same situation as when p3 = 0 initially. So we must have p1 < 0. If p3 = 0 for p < pz, then p3 subsequently becomes negative and the control takes the value 0. The drug level y decreases and eventually reaches the level pr. However we then have that p3 is negative, violating (5.14). Neither can we have p3 = 0 when Al.= pz since that implies that the Hamiltonian equals 0 instead of 1. Therefore the decrease in p3 is not sufficient to drive it to 0 while ,u E [ pl, pzl. When the drug level reaches the interval (r,q), the differential equation for p3 becomes

P,(I)=[pt+ Since have

p3 must

CY( p*(t) decrease

+ If$7)

dv(7)

if it is to satisfy

)] q

Pu(f>) + aP3(t).

the endpoint

(5.16)

conditions,

we

(5.17)

Notice that when p(t) 2 p2, we have p2 constant. When p3 becomes 0, the Hamiltonian indicates

that

(5.18) If p > q, this gives (5.19)

Notice that p3(t) = 0 when p(t) reaches q since it is positive before then and we have already established that p3 G 0 on [q, ~1. Also observe

CANCER CHEMOTHERAPY

LUY

WITH BLOCKlNti

that we can use singular control to keep /_L= q and this has the same effect as u = U and then u = 0 on appropriate intervals. Eventually u(t) = 0 and p3 decreases, satisfying (5.16) until the drug level reaches p2 when the differential equation becomes

(5.20)

+ 6pdt).

To satisfy the endpoint conditions we must have

PlU, + (Pdl)

+

pw w+% -b24t)) > 0.

(5.21)

Since (a, - b,+(t)) > 0 (the normal cells will regrow after depletion by we have p,(t) + /i Y(T) dv(~) > 0 which implies that pz(l) > 0. Eventually p3 becomes 0 again whence

the drug),

h(

/+))[a,p, + (p*(t) +j)W W+, - b,xdr))] =I. (5.22)

If h( p(t)) = 1, then

a,p, +

(pz(t) +@4 dv(+)(a, - bAf))

=I.

(5.23)

After this p3 becomes positive, the drug level increases, and the cycle repeats itself until the desired endpoint is reached. During this process the relations (5.18) and (5.22) will be satisfied alternately. We can draw a conclusion regarding treatment from the inequalities (5.17) and (5.21). Consider the left-hand side of (5.21) and denote it by A. When the measure Y is zero, A is differentiable and has derivative 0 when L,(p)=O. As p increases from the region of h( p) > 0 to L,(p)> 0,the system will go from satisfying (5.21) to (5.17). Since p1 + a(p,(t) + /,j Y(T) dv(~)) is constant when h( p) = 0, we have that (5.17) and (5.21) armlv at the same time (for instance when p = p2 and /1 is increasing.

210

J. M. MURRAY

This is the point at which x2 has regrown

*

(YUl

<

a2

-

b,x,(

to its highest

t) .

level). Therefore

(5.25)

Here we have used the knowledge that p2 + /y du > 0. This says that the cell populations are not allowed to grow to a level where the effective growth rate of the tumor (this equals the growth rate a, times the relative drug sensitivity parameter CY)exceeds the growth rate of the normal cells (a2 - 6,x,(t)). At the beginning of this section we made the choice ~~(0) > 0 and said that if p&O) = 0, we find that the problem corresponds to the situation when x,(O) < xlr, or when x1 is driven to _xlr before x2 reaches p. This comes about because we then have p2 + jy dv < 0. The results leading up to (5.24) still apply but now that pz + jy dv has changed sign, we get the opposite inequality to (5.25). 51.1.

Lowest Normal

Cell Values Per Cycle

The application of drug is such that there are periods of cell increase followed by cell decrease and there is some sort of repetition. In this section we will show that during each cycle enough drug is applied to drive the normal cell population to its lower limit, that is, at some point during each cycle we will have x,(t) = p. To show this we need the assumption that in each cycle, the drug decays to a level where there is no blocking, that is h( pu) = 1. Then (5.23) holds at the appropriate points in successive cycles. PROPOSITION

5.1

If during each cycle the drug decays to a level where there is no blocking, then at intervening times the normal cell population decreases to its lower limit p. ProoJ If h( p(t)) = 1 when the drug is at its lowest level during a cycle, then (5.23) applies to show that A = 1 at those times. If x,(t) # /3 in between these times, then A increases by /(YLJ pL(t))dt. This is a n contradiction. Unfortunately this proposition does not imply that the normal cells are driven to their lowest level with the first application of the drug. If we assume that the drug is of a type that has a saturation level and this level is achieved at the first instance of drug application and during

CANCER CHEMOTHERAPY

WITH BLOCKING

211

each cycle thereafter, then we obtain that the normal cell population will always be driven to its lowest limit /3. PROPOSITION 5.2

If during each cycle and at the first instance of drug application the drug reaches its saturation level (p > q), then at some intervening time the normal cell population will decrease to its lowest level p. Proof Consider an interval [T,,T~] where at the endpoints (5.19) is satisfied. Our intention is to show that in this interval x2 reaches the lowest level p. Suppose this is not the case. Then f7yr(t)dv(t) = 0 because the support of the measure v is contained in the set (t: x2(t) = p}. Since (5.19) holds at both endpoints and p1 is constant, we have that ~~(7,) = ~~(7~). Integrating the differential equation for p2, (5.4), implies that then p,(t) + /ofy(r) d&r) = 0 which violates (5.21), or that the regrowth stage is of zero length, which violates (5.22). Therefore x2 must decrease to /3 on this interval and /Qy du(t) < 0. n

51.2.

Uniform Cycles

The previous section showed that under certain assumptions, x2 is always driven to its lowest value p. The remaining question is whether the same amount of drug is applied during each cycle or whether it varies in some way from cycle to cycle. Here we will show that the former is true. The cyclical behavior of the normal cell population will be identical. We will show that the time interval for which p(t) = p1 (we use stationary control so that p(t) stays at pr rather than dropping below this level) is the same for each cycle. Since the drug increases from and decreases to this level at the same rate each cycle, we will have that the drug has zero value for the same period of time. Using the fact that x2 increases from j3 and then returns to p, we then deduce that the drug is at its maximum rate for an identical time period per cycle as well. For this result we need to assume both sets of hypotheses of the previous two propositions. PROPOSITION 5.3

During each cycle assume that the drug level reaches its saturated level ( p > q) and the nonblocking level ( t_~< p, ). Then in each cycle the period of time for which the drug is applied is the same in each cycle. This also implies that the time for which u = 0 in each cycle is also the same. Proof Consider the nth cycle and define [tin, tzn] to be the time interval in that cvcle when w.(t) = p, and p7 = 0. We have already

212

J. M. MURRAY

mentioned that we can assume that p singular control then). In each cycle p + E for some suitable, fixed E > 0 using control value 0 for the length of pz to pi. Equation (5.23) then implies

does not drop below pi (by using the value of x2 at t,, is always since x2 has increased from p time it takes p to decrease from

where 5 is the same for each cycle and each t,,. Equation (5.19) implies that when Al. reaches the value q that p2 + j’ydu = 8 for some appropriate 5 and this is the same for each cycle. However p2 + 1-ydv is constant when p increases from pz to q. For each cycle it takes the same amount of time for p to increase from pi to p2 so that when Al.is last equal to pi in each cycle the value of p2 + /y dv will be the same, say 5. Therefore on each cycle

Using the fact that that

is constant on this interval, we have

/~Y(T)~v(T)

(&(tzn) +I?’

d+))- (Pdt~n)+(‘“dT)dY(T))

We can solve this integral to obtain PAL> +

j:Y(ddV(T)

= ( Pz(tl,)

+h’lny(T)

dL’(T))~b2’“‘.‘.

Therefore t 73

-

t2,

=~ln[(p2(t2~)+~fzn~(~)~~(~))

-

which is indeoendent

(Pz(tl”> + jpd

w4)],

of n. This is the result we reouired

CANCER

CHEMOTHERAPY

WITH

BLOCKING

213

REFERENCES A. J. Coldman and J. H. Goldie, A model for the resistance of tumor cells to cancer chemotherapeutic agents, Math. Biosci. 65:291-307 (1983). 2 B. F. Dibrov, A. M. Zhabotinsky, Yu. A. Neyfakh, M. P. Orlova, and L. 1. Churikova, Mathematical model of cancer chemotherapy. Periodic schedules of phase-specific cytotoxic-agent administration increasing the selectivity of therapy, Math. Biosci. 73:1-31 (1985). 3 J. H. Goldie and A. J. Coldman, A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer Treatment Repotis 63:1727-1733 (1979). 4 R. B. Martin, M. E. Fisher, R. F. Minchin, and K. L. Teo, A mathematical model of cancer chemotherapy with an optimal selection of parameters, Math. Biosci. 99:205-230 (1990). 5 R. B. Martin, M. E. Fisher, R. F. Minchin, and K. L. Teo, Low intensity combination chemotherapy maximizes host survival time for tumors containing drug resistant cells, Math. Biosci. 110:221-252 (1992). 6 J. M. Murray, Optimal control for a cancer chemotherapy problem with general growth and loss functions, Math. Biosci. 98:273-287 (1990). I G. G. Steel, Growth Kinetics of Tumours, Oxford Univ. Press, Oxford, 1977. 8 G. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci. 101:237-284 (1990). 9 F. A. Valeriote and W. R. Bruce, An in vitro assay for growth-inhibiting activity of vinblastine, J. Nat. Cancer Inst. 35:851-856 (1965). 10 F. A. Valeriote, W. R. Bruce, and B. E. Meeker, A model for the action of vinblastine in vivo, Biophys. /. 6:145-152 (1966). 1