Optimal control for a stochastic model of cancer chemotherapy
Andrew J. Coldman
a, J.M. Murray
b,* aBC Cancer Agency, 600 W 10th Ave, Vancouver, BC, Canada V5Z 4E6 b
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 eect 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 eect 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 eective 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. Unfor-tunately 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 asresistance[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 archi-tecture 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).
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 eect of the drug or to classes of drugs by changing proteins associated with drug in¯ux or eux. 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 chemo-therapy 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 eects and the eect of dierent 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 experi-mental systems. The eect 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 dierent determinants of chemotherapy eect 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 dierent 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 compart-ments. The eect of therapy is usually assumed instantaneous and the overall response is sum-marised 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 dierent 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
drugs),R1(resistant toT1and sensitive toT2),R2(resistant toT2and sensitive toT1) orR3(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 ratesbi tRi t, i0;1;2;3 [8]. Transitions are assumed to occur between compartments
with a constant probability per division,aik, whereiis the index of the originator state andkis 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 lnPftumor cell survivalg ÿqikdk; 1
wheredk is the dose of drug Tk;k 1;2 andqik is the parameter for drug Tk in cells of typei.
In what follows, we will assume we have available treatment times, tj; j1;. . .;N on a scale
wheret0 represents the time the tumour developed (1 cell).
The probability that the tumour is cured at timet is taken to be equivalent to the probability that there are no tumour cells alive, i.e.,
PfR0 t 0; R1 t 0; R2 t 0; R3 t 0g PfR t 0g:
IfWR t s is the probability generating function (p.g.f.) of the processR t then
PfR t 0g 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 s0. We can obtain an expression for this p.g.f. by using the well-known re-lationship that ifYi; i1;2;. . .are independent identically distributed integer valued stochastic
processes andN is another independent integer valued process, then the process
Z X
N
n1
Yn
has p.g.f. given by
WZ s 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 timet, is given by the p.g.f. prior to treatment at timetÿevaluated at a point given by
the p.g.f. of the eect of treatment on a single cell, i.e.,
WR t s WR tÿ WS s; 4
where
WS s 1ÿPD 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 t0 then we have
WR t2 s WR t1 WG t2ÿt1 s 5 for a period t1;t2when no treatments are given. The usual method for ®ndingWG(use of forward
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 per-mitting cells to undergo at most one transition in a short interval and then using Eq. (5) recur-sively to develop relationships for arbitraryt. Expressions forWGfor 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 eects on one or more normal systems. This will be summarised in a single variableXwhich 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 X1ÿ X1ÿXSeÿk1t; 6
whereX1 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 drugsTk are assumed to perturb the normal system, indicated by
DX, following a log-kill law [16] so that
DX ÿqXkdk: 7
In attempting to model clinical cancer the important outcome associated with the eect of chemotherapy on the normal tissue is the occurrence of a toxic event. The toxic event can rep-resent 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 eected 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) eect, PT, of single doses of a drug is the logistic function, i.e.,
PT dk 1ÿ 1
eb01eb0bdkÿ1; 8 where b0 and b >0 are constants. We may combine Eqs. (7) and (8) to provide a formula re-lating changes in the level of the normal system from its physiologic value to the probability of a toxic event, i.e.,
PT DX 1ÿ 1
is the size after (as given in Eq. (7)), then the probability of a toxic event associated with this dose
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
This formulation has the following desirable properties:
1. It reduces to the logistic model for single doses administered to subjects with the normal phys-iologic levels.
2. It utilises the commonly used Gompertzian growth function for normal tissue recovery. 3. It has the consistency property that two dosesd0andd00administered at the same time have the
same toxicity as a single dose d ford d0d00. 4. For givenX tÿ
j ÿX tj (reduction in size) the conditional probability of toxicity increases as
X tÿ
j declines, that is the more suppressed the normal system, the greater the toxic eect of
a ®xed dose of drug.
2.3. Optimisation
Dierent 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 uncom-plicated control,
Pfno toxicityg Pftumor is curedg 1ÿCUMPT tN PfR tN 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 ÿqX1d1 tj ÿqX2d2 tj; j1;. . .;N:
The second term in Eq. (12) is determined from Eq. (2) where the p.g.f.WRis evaluated ats0
for ttN. The p.g.f. WR, is calculated using repeated applications of Eq. (3) and combines the
individual p.g.f.s,WGof growth and mutation (Eq. (5)) between drug applications andWSat drug
WR tN 0 WG t1ÿ0 WS t1 WG t2ÿt1 WS tN 0
ÿ
ÿ
ÿ
ÿ
: 13
The assumption of the log-kill law, Eq. (1), for each of the tumor compartments and the structure of the p.g.f.sWS, determine the functional dependence ofWR tN 0on the drug regimen.
LetDdenote the vector of all drug choices for the ®xed treatment timestj; j1;. . .;N,
D 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 Dsubject to gk D60; k 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 restrictDso that
CUMPT tN6Ctox: 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 eec-tively 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 di-mensional 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 cir-cumstances. 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 PfR tN 0g. Day [8] examined the
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 proba-bility 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 settingCtox0:025 the optimal solution
(Fig. 1) achieves a probability of cure of 0.47. As expected the optimal control diers 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 eects 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 suciently 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 eects 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 chemo-therapy 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 eect of surgical cures so that those given in Table 2 represent the cured proportion amongst those with post-surgical residual disease.
Table 1
Parameter values for equivalent drugs
a0j;aj3 10ÿ5 Mutation probability to develop resistance to a drugj, or to develop double resistance from a singly resistant population
q0j;q21;q12 2 log 10 Eect of unit dose of either one drug on sensitive cells or cells resistant to other drug
qjj;q3j log 10=10 Eect of unit dose of drugjon cells resistant to drugj
bi log 2=30 Growth rate of tumour compartmenti0;1;2;3 (daysÿ1) qXj 0:78 Eect of unit dose of drugjon normal cells
X1 1010 Initial and asymptomatic normal population level
k1 ÿlog 1ÿlog10 2=2=2 Gompertzian growth rate of normal cells
XT t1ÿ 10
9 Expected initial tumor size
N 12 Number of drug application times
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.
Fig. 1. Expected cell component response and optimal drug regimen for equivalent drugs. Tumour cell compartments
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 dierence 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 tN60:01 is
pre-venting 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 ofequivalentdrugs is to give them in a symmetric manner so that equal (eective) doses of each drug are given at the same time. Also as Table 3
Parameters for breast cancer model
a01;a23 810ÿ9 Mutation probability from CMF sensitive to resistant a02;a13 310ÿ6 Mutation probability from adriamycin sensitive to resistant q0j 2 log 10 Eect of unit dose of drugjon sensitive cells
q21 log 20 Eect of unit dose of CMF on cells resistant to adriamycin
q12 log 3 Eect of unit dose of adriamycin on cells resistant to CMF qjj 0 Eect of unit dose of drugjon cells resistant to drugj
q3j 0 Eect of unit dose of drugjon doubly resistant cells
bi log 2=30 Growth rate of tumour compartmenti0;1;2;3 (daysÿ1) qXj 0:78 Eect of unit dose of drugjon normal cells
X1 1010 Initial and asymptomatic normal population level
k1 ÿlog 1ÿlog10 2=2=2 Gompertzian growth rate of normal cells
XT t1ÿ 10
9 Expected initial tumor size
N 12 Number of drug application times
tj1ÿtj 21 Days between treatments Table 2
Eects of three standard regimens
Treatment Observed probability of cure Estimated probability of cure
CMF12 0.06 0.05
A4CMF8 0.45 0.41
one might expect it is always bene®cial to give drugs as often as possible, suggesting that con-tinuous 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 eect 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 eects. 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 devel-opment 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.
Other authors using dierent growth functions, and dierent models of chemotherapeutic in-sensitivity have concluded that late intensi®cation is optimal [24]. These results are usually based on Gompertzian growth functions for the tumor and drug eects that are related to the pro-portion of cells that are progressing through the cell cycle, which is assumed to be propro-portional to the tumour growth rate. In these models, the drugs become more eective as treatment progresses since the tumor size decreases and the growth rate increases but this is oset by the increased tumour re-growth. Since, in these models, tumour re-growth depends on the post-treatment tu-mour size whereas chemotherapy eect depends on the pre-treatment tutu-mour size, there is a continuous diminishing of net therapeutic eectiveness as the regimen proceeds. Inclusion of resistance development and toxicity result in dierent optimal controls which are not solely due to diering 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 sub-clinical range [25,26]. Over the sub-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 appli-cation 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 re-sponsiveness 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 dierent 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 nb t, where B 0 1. At each birth
(transition) a signalYn t;sn is generated where sn2 0;t is the time of transition fromn ton1.
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 timet0). B andYn are assumed mutually
inde-pendent and theYn nP1 are identically distributed. Define
and
Returning to Eq. (A.2), as the integrand may be written as a product ofnÿ1 terms of common structure, each of which only depends on one of the ti, we have that
E s Y1Ynÿ1
Substituting Eq. (A.3) into the above yields
Substituting this expression into Eq. (A.1), noting that PfB t 1g 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
corresponds to exponential growth and other modes of tumour growth can be modelled by ap-propriate 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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