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Mathematical Biosciences 168 (2000) 187±200
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Optimal control for a stochastic model of cancer chemotherapy
Andrew J. Coldman a, J.M. Murray b,*
a

b

BC Cancer Agency, 600 W 10th Ave, Vancouver, BC, Canada V5Z 4E6
School of Mathematics, University of New South Wales, UNSW, Sydney, NSW 2052, Australia
Received 10 May 1999; received in revised form 22 August 2000; accepted 25 August 2000

Abstract
Chemotherapy is useful in a number of cancers to reduce or eliminate residual disease. When used in this
way the objective is to maximise the likelihood that the cancer will be eliminated. In this article, we extend a
stochastic model of chemotherapy for cancer to incorporate its concomitant e€ect on the normal system and
derive overall measures of outcome. The model includes the development of drug resistance and is suciently ¯exible to include a variety of tumour and normal system growth functions. The model is then applied
to situations previously examined in the literature and it is shown that early intensi®cation is a common
feature of successful regimens in situations where drug resistance is likely. The model is also applied to data
collected from clinical trials analysing the e€ect of adriamycin, and cyclophosphamide, methotrexate and 5¯ourouracil (CMF) therapy in the treatment of operable breast cancer. The model is able to mimic the data

and provides a description of the optimal regimen. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords: Optimal control; Stochastic; Chemotherapy; Breast cancer

1. Introduction
The addition of chemotherapy as an e€ective modality for the treatment of cancer over the past
30 years has had a signi®cant impact upon the mortality and morbidity of the disease. Unfortunately not all cancer types are sensitive to one or more of the available anticancer drugs and
those types which are, show variation between individuals and over time. This lack of sensitivity is
commonly referred to as resistance [1] and the characterisation and causes of resistance has been
an active area of oncologic research.
Various diverse mechanisms of resistance have been identi®ed which may relate to the architecture and location of the tumour, the pharmacokinetic characteristics of the host and the ability
*

Corresponding author. Tel.: +61-2 9385 7042; fax: +61-2 9385 7123.
E-mail addresses: acoldman@bccancer.bc.ca (A.J. Coldman), j.murray@unsw.edu.au (J.M. Murray).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 4 5 - 6

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A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

of the individual tumour cells to absorb, metabolise and excrete drugs [2]. Research has shown
that genetic changes may confer resistance to speci®c drugs by altering protein products necessary
for the activation or e€ect of the drug or to classes of drugs by changing proteins associated with
drug in¯ux or e‚ux. Experiments of the Luria±Delbruck [3] type have consistently implicated
random mutational changes as being the origin of these genetic changes [4] although this may not
be the exclusive mechanism [5]. It is believed that the observed high frequency of these alterations
is associated with genomic instability that is a characteristic of cancer and is a result of the nature
of the steps that lead to cancer [6].
Several authors [7±9] have attempted to quantitatively model the response of cancer to chemotherapy assuming that resistance can be present and that it is produced by random mutations.
Typically they have used a birth and death model for tumour growth and assumed that resistant cells
arise at a constant frequency in proportion to the division rate of the tumour cells. Treatments are
incorporated as instantaneous e€ects and the e€ect of di€erent strategies are summarised in the
probability of cure [7], which is the long term likelihood of the tumour being eliminated. The resulting
models provide predictions that agree with results from a variety, but certainly not all, of experimental systems. The e€ect of chemotherapy on the normal system has only been acknowledged in
these approaches, by restricting the dose and combination of drugs, and was not quantitatively
included. Although these models are reasonably ¯exible it has only been possible to identify optimal
controls [10], the optimal application of chemotherapy, in a number of restricted situations.
Other authors have used di€erent determinants of chemotherapy e€ect to produce models for

response to chemotherapy. Mostly other approaches are of two types: those which incorporate
models of the cell cycle speci®c activity of some anticancer drugs [11] and those utilising di€erent
kinetic models of tumour growth and chemotherapy response. Kinetic models of tumour growth,
typically exponential or Gompertz [12] have also included resistant cells as one of the compartments. The e€ect of therapy is usually assumed instantaneous and the overall response is summarised in the total tumour size. Explicit models of the behaviour of the normal system have been
included and optimal controls found under a variety of di€erent constraints. These models have
been deterministic in nature and it is not easy to use them to model human cancer where response
is discrete (alive or dead) and variability in response is the rule rather than the exception.
In this research we combine a dynamical model for the re-population of the normal system with
stochastic models of tumour growth, the development of resistance and the likelihood of life
threatening toxicity. We develop a single stochastic framework which incorporates many of the
recognised determinants of outcome in the chemotherapeutic treatment of human cancer and
provides a description of the resulting optimal controls. We use the model to re-analyse previous
theoretical calculations that have been made and simulate clinical data from the treatment of
breast cancer.

2. Setting the problem
2.1. The malignant population
We consider a situation where two active drugs are available. Tumour cells may be in one of
four mutually exclusive states de®ned by sensitivity to the drugs T1 and T2 : R0 (sensitive to both

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

189

drugs), R1 (resistant to T1 and sensitive to T2 ), R2 (resistant to T2 and sensitive to T1 ) or R3 (resistant
to both drugs) [7]. Let Ri …t† be the number of cells in the ith compartment at time t. Each
compartment is assumed to grow with the kinetics of a pure birth process with compartment
speci®c rates bi …t†Ri …t†, i ˆ 0; 1; 2; 3 [8]. Transitions are assumed to occur between compartments
with a constant probability per division, aik , where i is the index of the originator state and k is the
destination state. Each tumour cell is assumed to obey the log-kill law in its response to drugs in
which the log of the probability of cell survival, PD , is proportional to the drug dose, i.e.,
ln…PD …dk †† ˆ ln‰P ftumor cell survivalgŠ ˆ ÿqik dk ;

…1†

where dk is the dose of drug Tk ; k ˆ 1; 2 and qik is the parameter for drug Tk in cells of type i.
In what follows, we will assume we have available treatment times, tj ; j ˆ 1; . . . ; N on a scale
where t ˆ 0 represents the time the tumour developed (1 cell).
The probability that the tumour is cured at time t is taken to be equivalent to the probability
that there are no tumour cells alive, i.e.,

P fR0 …t† ˆ 0; R1 …t† ˆ 0; R2 …t† ˆ 0; R3 …t† ˆ 0g ˆ P fR…t† ˆ 0g:
If WR…t† …s† is the probability generating function (p.g.f.) of the process R…t† then
P fR…t† ˆ 0g ˆ WR…t† …0†:

…2†

Thus if we can calculate the p.g.f. we can obtain the required probability by evaluating it at a
particular point s ˆ 0. We can obtain an expression for this p.g.f. by using the well-known relationship that if Yi ; i ˆ 1; 2; . . . are independent identically distributed integer valued stochastic
processes and N is another independent integer valued process, then the process
Zˆ

N
X

Yn

nˆ1

has p.g.f. given by
WZ …s† ˆ WN …WY …s††;

…3†

where WN …s† and WY …s† are the p.g.f.s of N and Y. In particular we have that the p.g.f. after
treatment at time t, is given by the p.g.f. prior to treatment at time tÿ evaluated at a point given by
the p.g.f. of the e€ect of treatment on a single cell, i.e.,
WR…t† …s† ˆ WR…tÿ † …WS …s††;

…4†

where
WS …s† ˆ 1 ÿ PD …d† ‡ sPD …d†;
and PD …d† is given by Eq. (1). Similarly if we use G…t† to designate the process of growth and
transformation (into resistant types) of a single cell present at t ˆ 0 then we have
WR…t2 † …s† ˆ WR…t1 † …WG…t2 ÿt1 † …s††

…5†

for a period …t1 ; t2 Š when no treatments are given. The usual method for ®nding WG (use of forward
and backward equations) results in a series of di€erential equations which have no obvious

general solution for the process described although useful special solutions have been found [7].
The complexity in ®nding general solutions results from the possibility that individual cells may

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A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

undergo multiple transitions (e.g., R0 ! R1 ! R3 ) and growth. Because transition rates are low
such events are rare in short intervals. Approximate solutions have been developed [8] by permitting cells to undergo at most one transition in a short interval and then using Eq. (5) recursively to develop relationships for arbitrary t. Expressions for WG for small intervals are calculated
using results for ®ltered Poisson processes [13], which provide simple integral expressions for the
p.g.f.s required. We have extended these results to apply to a larger class of birth processes that
include ®ltered Poisson processes as a particular case (see Appendix A). In particular it is possible
to evaluate results for which growth is assumed to follow a stochastic Gompertzian process [14].
2.2. The normal population
Both drugs are assumed to have unwanted dose dependent toxic e€ects on one or more normal
systems. This will be summarised in a single variable X which is equal to the logarithm of the size
of the critical normal population which is assumed to re-populate following a Gompertzian form
of growth [15], i.e.,
X …t† ˆ X1 ÿ …X1 ÿ XS †eÿk1 t ;

…6†

where X1 is the asymptotic size, k1 a growth parameter and t is the elapsed time from when the
system was of size XS . If the normal system is perturbed, then its re-growth is described by the
same equation. The anticancer drugs Tk are assumed to perturb the normal system, indicated by
DX , following a log-kill law [16] so that
DX ˆ ÿqXk dk :

…7†

In attempting to model clinical cancer the important outcome associated with the e€ect of
chemotherapy on the normal tissue is the occurrence of a toxic event. The toxic event can represent a variety of situations. As well as the most drastic, death of the patient, it can also typify a
medical outcome such as kidney failure or neurological damage, that the therapist is trying to
avoid. On a more basic level it denotes any outcome that causes the cessation of treatment. This
variety of meanings can be e€ected by appropriate choices of the parameters in Eqs. (8) and (9).
A commonly used model for the probability of a speci®c binary toxic [17] (or therapeutic) e€ect,
PT , of single doses of a drug is the logistic function, i.e.,


ÿ1

PT …dk † ˆ 1 ÿ 1 ‡ eb0 1 ‡ eb0 ‡bdk ;
…8†
where b0 and b…> 0† are constants. We may combine Eqs. (7) and (8) to provide a formula relating changes in the level of the normal system from its physiologic value to the probability of a
toxic event, i.e.,


ÿ1
…9†
PT …DX † ˆ 1 ÿ 1 ‡ eb0 1 ‡ eb0 ÿbDX :

We assume in Eq. (9) that the determinant of the likelihood of toxicity is only in¯uenced by the
net kill on the normal system of the drug and not by which drug is used. One of the characteristics
of cancer chemotherapy is that normal systems are being repeatedly perturbed by the ongoing
sequential application of therapy and may not return to their physiologic values during the course
of therapy. We may utilise the parameterisation of Eq. (9) to model this situation as follows. If
X …tÿ † is the size of the normal system prior to the administration of a drug dose at time t and X …t†

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

191

is the size after (as given in Eq. (7)), then the probability of a toxic event associated with this dose
is given by
PT …t† ˆ

PT …X …t† ÿ X1 † ÿ PT … X …tÿ † ÿ X1 †
1 ‡ exp ‰b0 ‡ b…X1 ÿ X …tÿ ††Š
:
ˆ
1
ÿ
1 ÿ PT …X …tÿ † ÿ X1 †
1 ‡ exp ‰b0 ‡ b…X1 ÿ X …t††Š

…10†

Eq. (10) provides an expression for the probability of toxicity conditional on no preceding toxic
event. Using Bayes theorem we may simply calculate the cumulative probability of a toxic event,
CUMPT …t†, from
i

h
ÿ
ÿ
…11†
CUMPT …tj † ÿ CUMPT …tj † ˆ PT …t† 1 ÿ CUMPT …tj † ;
with the condition

CUMPT …t1ÿ † ˆ 0:
1.
2.
3.
4.

This formulation has the following desirable properties:
It reduces to the logistic model for single doses administered to subjects with the normal physiologic levels.
It utilises the commonly used Gompertzian growth function for normal tissue recovery.
It has the consistency property that two doses d 0 and d 00 administered at the same time have the
same toxicity as a single dose d for d ˆ d 0 ‡ d 00 .
For given X …tjÿ † ÿ X …tj † (reduction in size) the conditional probability of toxicity increases as
X …tjÿ † declines, that is the more suppressed the normal system, the greater the toxic e€ect of
a ®xed dose of drug.

2.3. Optimisation
Di€erent objectives can be used that each combines the features of maximising probability of
cure while minimising toxicity. Here we achieve this by maximising the probability of uncomplicated control,
P fno toxicityg  P ftumor is curedg ˆ …1 ÿ CUMPT …tN ††  P fR…tN † ˆ 0g:

…12†

A patient can fail treatment for two reasons. The treatment may not be able to eliminate the
tumour. The treatment itself may violate toxicity for a patient, to the point where the regimen
cannot be completed. Either of these results leads to treatment failure. The function in Eq. (12)
expresses the probability that neither of these circumstances occur. It determines the probability
that a patient can complete treatment (it will not be prematurely stopped due to toxicity) and have
the tumour eliminated, for a given regimen.
The ®rst term in Eq. (12) is determined from Eq. (11) with the change in the normal population
DX being given by
DX …tj † ˆ ÿqX 1 d1 …tj † ÿ qX 2 d2 …tj †;

j ˆ 1; . . . ; N :

The second term in Eq. (12) is determined from Eq. (2) where the p.g.f. WR is evaluated at s ˆ 0
for t ˆ tN . The p.g.f. WR , is calculated using repeated applications of Eq. (3) and combines the
individual p.g.f.s, WG of growth and mutation (Eq. (5)) between drug applications and WS at drug
application, i.e.,

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A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

ÿ
ÿ
ÿ ÿ





WR…tN † …0† ˆ WG…t1 ÿ0† WS…t1 † WG…t2 ÿt1 †


WS…tN † …0†



:

…13†

The assumption of the log-kill law, Eq. (1), for each of the tumor compartments and the
structure of the p.g.f.s WS , determine the functional dependence of WR…tN † …0† on the drug regimen.
Let D denote the vector of all drug choices for the ®xed treatment times tj ; j ˆ 1; . . . ; N ,
D ˆ …d1 …t1 †; . . . ; d1 …tN †; d2 …t1 †; . . . ; d2 …tN ††:
Choosing the drug regimen that achieves the best response in terms of our objective, Eq. (12), is
equivalent to the optimisation problem
…P†

maximise

f0 …D†subject to gk …D† 6 0;

k ˆ 1; . . . ; K;

where f0 is the function in Eq. (12) and the gk describe any explicit constraints on the drug
regimen. For our problem we will restrict D so that
CUMPT …tN † 6 Ctox :

…14†

Ctox denotes any explicit limits on toxicity, apart from the implicit ones within the objective
function Eq. (12), that the clinician may wish to impose. A large value for this parameter e€ectively removes this constraint and the objective becomes the maximisation of uncomplicated
control as described earlier. On the other hand the best regimen determined by maximisation of
Eq. (12) may prescribe a level of toxicity that is unacceptable for certain practical reasons. In this
case a value of Ctox can be set that will ensure any regimen returned by the maximisation of Eq.
(12) in conjunction with Eq. (14) will satisfy any additional constraints. P describes a ®nite dimensional constrained optimisation problem and as such can be solved by an array of numerical
methods. We choose a sequential quadratic programming method implemented within Matlab.

3. Case studies
3.1. Equivalent drugs
The study of the optimal scheduling of two drugs has been considered before in several circumstances. Coldman and Goldie [7] determined the optimal sequence of separate applications for
equivalent drugs, a situation in which all drugs are assumed to have the same values for analogous
parameters. They were able to prove that, for equivalent drugs which were not permitted to be
given simultaneously, sequential alternation maximised P fR…tN † ˆ 0g. Day [8] examined the behaviour of P fR…t† ˆ 0g under a wide range of parameter values for 16 patterns of drug sequencing
chosen to represent a range of di€erent amounts of each drug and di€erent sequences of administration. He was able to infer heuristics for the optimal pattern of drug utilisation. Neither of
these authors included the behaviour of the normal system under cancer chemotherapy and were
therefore unable to appropriately examine whether dose alteration provided the opportunity to
signi®cantly improve therapeutic outcomes. Here we re-examine the use of equivalent drugs but
now allow both drugs to be given simultaneously. The parameters used in this example are displayed in Table 1. The mutational probabilities are taken from the literature estimates [18]. The
tumour growth parameters were chosen to duplicate the e€ects for the equivalent drug case
analysed in [7,8]. The normal cell parameters were chosen to achieve acceptable toxicity, 2.5%, for

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

193

Table 1
Parameter values for equivalent drugs
a0j ; aj3

10ÿ5

q0j ; q21 ; q12

2 log…10†

qjj ; q3j
bi
qXj
X1
k1
XT …t1ÿ †
N
tj‡1 ÿ tj

log…10†=10
log…2†=30
0:78
1010
ÿ log…1 ÿ log10 …2†=2†=2
109
12
30

Mutation probability to develop resistance to a drug j, or to
develop double resistance from a singly resistant population
E€ect of unit dose of either one drug on sensitive cells or cells
resistant to other drug
E€ect of unit dose of drug j on cells resistant to drug j
Growth rate of tumour compartment i ˆ 0; 1; 2; 3 (daysÿ1 )
E€ect of unit dose of drug j on normal cells
Initial and asymptomatic normal population level
Gompertzian growth rate of normal cells
Expected initial tumor size
Number of drug application times
Days between treatments

the standard alternating and sequential regimens. As in [8] we restrict treatment protocols to 12,
monthly applications and assume treatment commences after 30 doublings of the tumour.
The alternating schedule, previously identi®ed as optimal amongst ®xed dose non-overlapping
treatments, has a probability of cure 0.39 for the parameter values in Table 1 where the probability of toxicity was 0.025, being compatible with that seen in chemotherapy protocols, where
cure is possible. Imposing this as a limit on toxicity, so setting Ctox ˆ 0:025 the optimal solution
(Fig. 1) achieves a probability of cure of 0.47. As expected the optimal control di€ers from the
alternating control in two distinct ways: (1) Both drugs are given concurrently at the same levels
and (2) dosage is front end loaded to diminish the singly resistant compartments as quickly as
possible. As the regimen proceeds the dose of each drug is reduced since the negative e€ects on
normal tissue of constant high dose chemotherapy outweigh the marginal bene®ts these have on
controlling the tumour. Notice, however, that there always must be a suciently high probability
of cure to generate this pattern of optimal control. Where tumour eradication is unlikely the
framework developed here will have little application. In such cases the optimal control is
governed more by the e€ects on the normal system and exhibits minimal front end loading.
This would represent a palliative treatment situation.

4. Adjuvant treatment of breast cancer
As a clinical application of our methods we determine the optimal regimen for the adjuvant
treatment of breast cancer with two groups of drugs [19,20]. In this case women receive chemotherapy post-surgically to reduce the probability that unrecognised metastatic disease will cause
subsequent treatment failure. Estimates of post-surgical tumour burden and growth rates are
taken from the earlier publication [21] as analysed by Skipper [22]. The tumour burden was set at
1010 cells and tumour growth was assumed to be exponential with a doubling time of 30 days.
Observed cure rates in chemotherapy groups were then adjusted to account for the e€ect of
surgical cures so that those given in Table 2 represent the cured proportion amongst those with
post-surgical residual disease.

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A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

Fig. 1. Expected cell component response and optimal drug regimen for equivalent drugs. Tumour cell compartments
R0 , R1 , R2 , R3 represented by 0, 1, 2, 3 (left-hand axis); normal damage by n (right-hand axis).

The ®rst group of drugs consists of cyclophosphamide, methotrexate, and 5-¯uorouracil
(CMF). The second group contains the single drug adriamycin. The standard dosage of each of
these drugs was cyclophosphamide 600 mg=m2 , methotrexate 40 mg=m2 , 5-¯uorouracil
600 mg=m2 , and adriamycin 75 mg=m2 . For convenience these were rescaled so that the standard
CMF single drug approximation had a level of 1, and the adriamycin standard dose was also 1.
The time between drug applications was 21 days. There were 12 treatment times. The toxicity
parameters were ®tted to be such that they gave an overall probability of toxicity of 0.01 for the
three regimens in Table 2.
Other parameter values were estimated by ®tting the model previously described to the data
[19,20]. The algorithm of Nelder and Mead [23] was used as it is generally quite robust for sparse
data. The best ®t parameters are displayed in Table 3 although a range of parameter values were
compatible with the data when account is taken of the statistical accuracy with which each data
point was estimated. In particular, treatment outcome is very sensitive to some parameter values
…q12 ; a23 †, and insensitive to many others as long as they are less than some value. For ease of
convenience we label CMF as drug 1 and adriamycin as drug 2.

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

195

Table 2
E€ects of three standard regimens
Treatment

Observed probability of cure

Estimated probability of cure

CMF  12
A  4 ‡ CMF  8
fCMF; CMF; Ag  4

0.06
0.45
0.13

0.05
0.41
0.11

Table 3
Parameters for breast cancer model
a01 ; a23
a02 ; a13
q0j
q21
q12
qjj
q3j
bi
qXj
X1
k1
XT …t1ÿ †
N
tj‡1 ÿ tj

8  10ÿ9
3  10ÿ6
2 log…10†
log…20†
log…3†
0
0
log…2†=30
0:78
1010
ÿ log…1 ÿ log10 …2†=2†=2
109
12
21

Mutation probability from CMF sensitive to resistant
Mutation probability from adriamycin sensitive to resistant
E€ect of unit dose of drug j on sensitive cells
E€ect of unit dose of CMF on cells resistant to adriamycin
E€ect of unit dose of adriamycin on cells resistant to CMF
E€ect of unit dose of drug j on cells resistant to drug j
E€ect of unit dose of drug j on doubly resistant cells
Growth rate of tumour compartment i ˆ 0; 1; 2; 3 (daysÿ1 )
E€ect of unit dose of drug j on normal cells
Initial and asymptomatic normal population level
Gompertzian growth rate of normal cells
Expected initial tumor size
Number of drug application times
Days between treatments

This choice of parameters produced estimated probabilities of cure of 0.05, 0.41 and 0.11,
respectively, for the three regimens in Table 2 each with a 0.01 probability of toxicity. Imposing
the additional constraint that the optimal regimen could be no more toxic than the three standard
regimens, we obtain the schedule and the expected response for each of the cell populations
displayed in Fig. 2. Each of the drug applications is allowed to vary freely so that they can be
greater than a unit dose.
The optimal regimen is approximately sequential and is rather similar to the best of the three
regimens that the investigators used (four cycles of drug 2 followed by eight cycles of drug 1). The
di€erence relates to a somewhat more intensive (increased ®rst cycle) and extensive (six versus four
cycles) use of adriamycin. Examination of Fig. 2 shows that at no time is the expected number of
cells less than unity. This would seem to suggest that the constraint CUMPT …tN † 6 0:01 is preventing the consideration of controls which have higher net overall survival rates. Removal of this
constraint results in an optimal control with an increased cure rate but also an increased rate of
toxicity. This control, however, improves the probability of uncomplicated control.

5. Discussion
As anticipated, the optimal regimen for the administration of equivalent drugs is to give them in
a symmetric manner so that equal (e€ective) doses of each drug are given at the same time. Also as

196

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

Fig. 2. Expected response of optimal breast cancer regimen.

one might expect it is always bene®cial to give drugs as often as possible, suggesting that continuous infusions would prove to be the optimal control in a general setting. As an example of
this, the equivalent drug case with applications every day over a 12 month period, rather than
every month as in the simulation, can produce a 20% improvement in cure with no additional
toxicity.
In order to e€ect cures it is necessary to reduce and eventually eliminate the singly resistant
compartments: to do this each drug must be used to control the cells that are resistant to the other
drug. The pattern of administration of the two drugs is determined by the balance between toxic
and therapeutic e€ects. If the likelihood of cure is low, then the optimal control is primarily in¯uenced by its toxicity to normal tissue. In this case, drug doses are quite uniform across the
regimen. If the potential of cure is appreciable, then the nature of the mechanism of the development of resistance implies that early control of the resistant sub-compartments is important.
For this situation the pattern of optimal control consists of elevated initial doses, especially the
®rst, and then once the singly resistant compartments have been substantially reduced, the level of
individual doses declines gradually over the rest of the regimen.

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

197

Other authors using di€erent growth functions, and di€erent models of chemotherapeutic insensitivity have concluded that late intensi®cation is optimal [24]. These results are usually based
on Gompertzian growth functions for the tumor and drug e€ects that are related to the proportion of cells that are progressing through the cell cycle, which is assumed to be proportional to
the tumour growth rate. In these models, the drugs become more e€ective as treatment progresses
since the tumor size decreases and the growth rate increases but this is o€set by the increased
tumour re-growth. Since, in these models, tumour re-growth depends on the post-treatment tumour size whereas chemotherapy e€ect depends on the pre-treatment tumour size, there is a
continuous diminishing of net therapeutic e€ectiveness as the regimen proceeds. Inclusion of
resistance development and toxicity result in di€erent optimal controls which are not solely due to
di€ering kinetic assumptions.
Breast tumor growth is also frequently modelled by Gompertzian growth in the observable
range although some authors have used exponential functions to describe growth in the subclinical range [25,26]. Over the sub-clinical range, for realistic Gompertzian parameters, the
doubling time varies less than two-fold. For the numerical simulations we describe here we have
used an exponential growth model, as an approximation, since the initial chemotherapy application reduces the tumor to the sub-clinical range. However, it is recognised that the choice of
growth function can have considerable impact [12], especially in situations in which tumor responsiveness is also linked to tumor growth rate [24].
Although the equivalent drug scenario demonstrates the bene®ts of giving both drugs when
possible and the early intensi®cation of treatment, the breast cancer scenario seems to imply some
di€erent guideline. The lesson to be learnt from this is that although simple scenarios produce
some clean results the basic problem is a complicated dynamical process.

Appendix A
Theorem. Let B…t† be a birth process with rates bn …t† ˆ nb…t†, where B…0† ˆ 1. At each birth
(transition) a signal Yn …t; sn † is generated where sn 2 ‰0; tŠ is the time of transition from n to n ‡ 1.
Also a signal Y0 …t† is generated regardless of the number of transitions (which may be viewed as
associated with a `transition' from 0 to 1 at time t ˆ 0). B…
† and Yn …
† are assumed mutually independent and the Yn …
† …n P 1† are identically distributed. Define
Z…t† ˆ

B…t†
X

Ynÿ1 …t; snÿ1 †

nˆ1

then the p.g.f. of Z…t† is given by


 Y …t† 
p…t†
0
;
WZ…t† …s† ˆ E s
1 ÿ I…s; t†
where

 Z t

p…t† ˆ exp ÿ
b…u† du
0

198

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

and
I…s; t† ˆ

Z

t



Y …t;v†

E s

0





b…v† exp



ÿ

Z

t

b…u† du

v



dv:

Proof. By de®nition WZ…t† …s† ˆ E‰sZ…t† Š and conditioning on B…t† we have
1
X



WZ…t† …s† ˆ
E sY0 ‡


‡Ynÿ1 B…t† ˆ n P fB…t† ˆ ng

…A:0†

nˆ1

"
#
1
X
Y
 Y ‡


‡Y 

nÿ1 
ˆ E s 0 P fB…t† ˆ 1g ‡
E s1
B…t† ˆ n P fB…t† ˆ ng :

…A:1†

nˆ2

If we let fnÿ1 …t1 ; . . . ; tnÿ1 † be the density of the ordered transition times then
Z t Z tnÿ1
Z t2




 Y ‡


‡Y 

1
nÿ1 
E s
B…t† ˆ n ˆ
E sY1 …t1 †


E sYnÿ1 …tnÿ1 † fnÿ1 …t1 ; . . . ; tnÿ1 † dt1


dtnÿ1 :




0

0

0

…A:2†

Now
fnÿ1 …t1 ; . . . ; tnÿ1 †
h R
i
h R
i
 Rt

t
t
exp ÿ 01 b1 …u† du b1 …t1 † exp ÿ t12 b2 …u† du b2 …t2 †


bnÿ1 …tnÿ1 † exp ÿ tnÿ1 bn …u† du
:
ˆ
P fB…t† ˆ ng
Re-ordering gives
fnÿ1 …t1 ; . . . ; tnÿ1 † ˆ

b1 …t1 † exp

 R t1
0


Rt

…b2 …u† ÿ b1 …u††du


bnÿ1 …tnÿ1 † exp 0nÿ1 …bn …u† ÿ bnÿ1 …u††du
Rt

:
exp 0 bn …u†du P fB…t† ˆ ng

In general this is somewhat intractable but is considerably simpli®ed if bj …u† ÿ bjÿ1 …u† ˆ bi …u† ÿ
biÿ1 …u† for all i; j [In the case of filtered Poisson processes bi …u† ˆ bj …u†]. In particular for bn …t† ˆ
nb…t† we have
 Rt
 nÿ1
Rt

…n ÿ 1†! exp ÿ 0 b…u† du Y
b…ti † exp 0i b…u† du
Rt
 :
fnÿ1 …t1 ; . . . ; tnÿ1 † ˆ
…A:3†
P fB…t† ˆ ng
exp 0 b…u† du
iˆ1

Returning to Eq. (A.2), as the integrand may be written as a product of n ÿ 1 terms of common
structure, each of which only depends on one of the ti , we have that



E sY1 ‡


‡Ynÿ1 B…t† ˆ n
Z tZ t
Z t




1
ˆ
E sY1 …t1 †


E sYnÿ1 …tnÿ1 † fnÿ1 …t1 ; . . . ; tnÿ1 † dt1


dtnÿ1 :




…n ÿ 1†! 0 0
0
Substituting Eq. (A.3) into the above yields



p…t†
nÿ1
E sY1 ‡


‡Ynÿ1 B…t† ˆ n ˆ
‰ I…s; t†Š :
P fB…t† ˆ ng

A.J. Coldman, J.M. Murray / Mathematical Biosciences 168 (2000) 187±200

199

Substituting this expression into Eq. (A.1), noting that P fB…t† ˆ 1g ˆ p…t†, and summing the
resulting series provides the desired result. 
We may use the above theorem to calculate the p.g.f. for growth, WG , for a cell of type j by
setting


E sY0 …t† ˆ sj

and by putting


Y …t;v†

E s



ˆ sj 1 ÿ

X
k6ˆj

ajk

!

‡

X
k6ˆj

ajk

sk pk …t ÿ v†
;
1 ÿ sk ‡ sk pk …t ÿ v†

where pk …t† is the same as in the theorem but with the parameter bk …t† for cells of type k. bk …t† ˆ bk
corresponds to exponential growth and other modes of tumour growth can be modelled by appropriate speci®cation. The resulting integral (Eq. (A.0)) will not, in general, have a closed form
solution, however, it will typically permit simple numerical evaluation.

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