A mathematical model of the effects of drug resistance in cancer chemotherapy

A Mathematical Model of the Effects of Drug Resistance in Cancer Chemotherapy B. G. BIRKHEAD AND W. M. GREGORY Clinical Operational Research Unit, Department of Statistical Science, University College London, Gower Street, London WC1 E 6BT, England Received 12 December

1983; revised 23 May I984

ABSTRACT A mathematical model is proposed relating tumor response under repeated doses of a single cytotoxic agent to the presence and accumulation of phenotypic drug resistance. The latter is assumed to comprise two elements: that present at diagnosis and that acquired in response to and during treatment. New analytic expressions are presented for quantities such as the fractional tumor reduction effected by each dose, the minimum tumor size achieved under therapy, the changing composition of the tumor, and the number of doses before apparent clinical resistance (the nadir) is observed. A similar model accommodates the sequential delivery of different drugs between which there is some degree of cross-resistance, and it is shown how competing treatment strategies can be simulated and compared.

INTRODUCTION Although cytotoxic chemotherapy is now the principal form of treatment for many tumors [l], the design of effective regimens and strategies has proved to be a difficult and complex problem. While certain agreed treatment principles seem to have emerged from empirical experimentation, for some diseases these have failed to produce any significant improvements in the rate of clinical response, survival, or cure, and for others, initial progress has not been consolidated. There is a need, in such cases, to provide insights to help the clinician understand the reasons for failure and to help him make a rational choice of his next strategy. The aim of a model is to illuminate by judiciously simplifying. Indeed, most clinical experimenters will invoke their own unformalized models when designing new treatment trials. Such models may have evolved over several years and will have been molded primarily by the literature and experience. They are valuable and necessary, but often lack the facility to quantify the relative importance of included factors and the means to investigate the

MATHEMATICAL

BIOSCIENCES

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12159-69 (1984)

OElsevier Science Publishing Co., Inc., 1984 52 Vanderbilt Ave., New York, NY 10017

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B. G. BIRKHEAD

AND W. M. GREGORY

“shape” of their effects. In reality, of course, each of these factors subsumes more complex processes which would themselves require complex models, and many detailed mathematical models [2-51 exist which address a wide range of pharmacokinetic and cell-kinetic considerations relevant to successful regimen design. But for some diseases the main problem is still one of strategy rather than detail, and clinically oriented models of the kind developed successfully by Skipper [6] and his associates may be more appropriate. It is an extension of such a model which is presented here. Starting from the simple exponential tumor growth model summarized by Lloyd [7] and incorporating the established critical effects [8,9] of drug resistance, a model of tumor regression and reprogression under the repeated administration of a single drug or the sequential delivery of different drugs is formulated. New closed expressions for measures of treatment efficacy are derived for use in investigating broad treatment strategies. THE MODEL Consider a single cytotoxic drug administered repeatedly in the same dose at constant intervals of T days to a tumor which on presentation has size (volume or number of cells) X0, of which proportions R, and 1 - R, are respectively resistant and sensitive to the chosen agent. R, (the primmy resistance) will be patient-specific and a function of X0 and the spontaneous mutation rate towards resistance during the pretreatment growth of the tumor [9]. Generally, let X,-i, R, m1 denote the tumor size and resistant proportion just before the i th course. Suppose that each dose kills a constant proportion k,, of the sensitive cells in the tumor, generally reducing X, _ 1 to x 1 say, and let k, 1 denote the fractional tumor reduction of the i th dose (k, _ 1 = 1 - y 1/X, 1). Let the tumor regrow exponentially (parameter a) between doses, and suppose that when regrowth in the i th interdose interval is complete, a proportion E, of the sensitive cells surviving the i th course have acquired (secondary) resistance-as a consequence, for example, of host-defence reactions [lO,ll], spontaneous mutation [9,12], and other effects. Figure 1 illustrates the size and composition of the tumor during the period covering the first two doses. THE FRACTIONAL

TUMOR

REDUCTION

OF EACH

DOSE

(k,)

Under the assumption of unperturbed exponential growth between doses, all the information about the efficacy of a drug delivered in this fashion will be contained in the sequence of fractional tumor reductions { k, }. The i th dose reduces X,_ 1 to y_ 1 given by Yt-1 =

xi-, - kd-

R,-,1X,-,

[ = cells before dose - (sensitive)

cells killed] ,

DRUG

RESISTANCE DOSE

IN CANCER

1

OOSE

FIG. 1.

and x:i

61

CHEMOTHERAPY

Diagram

2

of the model.

regrows to X, during the i th interdose interval.

So,

The sensitive cells X,’ of X, consist of that fraction 1 - e, of cells derived from the sensitive cells of y_ 1 which have not defected (through secondary resistance) to the resistant population [ = (1- t,)(lk.J(lR,_,) X,_leaT]. Thus, X, has sensitive proportion

1_

R

=

(l-c>(l-kd-R,-1)

I

Using k,_, = k,(lrelation

l-(1-

and k, = k,(l-

R,_,)

k

=

(1)

R,_l)k,

R,), (1) leads to the recurrence

k,-,(I- kd(l-

I

c,>

l-k,_,

where k, = k&l - R,). It follows, if E, is constrained { k, } is a nonincreasing sequence for k k i-1

I=

(l-k&l-c,) 1-k,+

k,R,pl

(4

’

<1

to be nonnegative,

that

62

B. G. BIRKHEAD

AND W. M. GREGORY

That is, successive doses of the drug result in lower fractional tumor shrinkage, owing to a higher proportion of the tumor being resistant. From this point, to avoid undue complexity in the initial formulation, we will adopt the simplifying assumption that e, = E [constant (net 2 0) for all i]; so that given constant T, the potency of the mechanisms of secondary resistance is assumed independent of the number of existing sensitive cells and the time since the start of therapy. Solving (2) with this simplification leads to an explicit formula for each k,: u’(l-a)k,

k, =

(34

l-a-(l-a’)kO where u=(I-k,)(l-E),

k, = k,(l-

R,),

and so

l-k,=

l-u-(l-a’f’)k,

(3b)

l-a-(l-a’)k,,

0.2

12

4 Nvmbrr

FIG. 2.

Sequences

Of

Co”r*rr

of fractional

lb

I I)

tumor reductions

( k, )

20

DRUG

RESISTANCE

IN CANCER

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CHEMOTHERAPY

The general (laterally inverted S) shape of the sequence can be seen in Figure 2, which shows the predicted effect of intensifying therapy (increasing k,) for fixed values of the primary (R,) and secondary (c) resistance factors, and demonstrates the effect of resistance in markedly decreasing the efficacy of successive doses. For example, for regimens A and B it can be seen that treatment beyond the fifth and eighth dose, respectively, would be superfluous, since only negligible tumor volume reduction is subsequently achieved. The choice of values for R, is from the range expected on the basis of spontaneous mutation [9], and the range of k, chosen spans weak to moderate values observed in animal model experiments. The value of E, however, about which little is known, has been chosen arbitrarily for illustration. MONITORING

TUMOR

SIZES

A clinical response to therapy is partly determined by the reduction of tumor size below some prespecified level. This is often monitored by taking measurements (generally using radiographic techniques) immediately before each dose, { Xi }, and/or immediately after, { y }. Given { k, }, predictions of these quantities are readily computable: r-1

X, =F_lear=

(l-

k,_l)X,_lear=

XoPT n

(l-

k,),

j=O

FIG. 3.

Predicted

tumor sizes (A and B as in Figure 2).

B. G. BIRKHEAD

64

which, using (3b) and after cross-canceling, x,= I

l-a-(l-a’)k, l--U

AND W. M. GREGORY

gives

XaP’T.

The sequence {x } [ = ((1 - k,)X,}] follows. Figure 3 shows { X, }, { y }, and the predicted tumor size during regrowth plotted for the parameter values of curves A and B in Figure 2, assuming a tumor doubling time of approximately 35 days, and interdose interval of 30 days, and an initial tumor burden of 1012 cells. The absolute number of resistant cells throughout treatment is also traced, the initial curvature in these tracks being attributable to a nonzero value of the secondary resistance fraction E, whose effect is visually more significant in the early phases of treatment while the number of sensitive cells is still relatively large. CLINICAL

MEASURES

OF TREATMENT

EFFICACY

From the basic expressions (l)-(4) explicit formulae for a range of measures used to assess treatment efficacy can be derived, and a selection of these is presented with explanation in the Appendix. They can be used to explore directly the effects on treatment success of different strategies underlying both regimen design factors (e.g. drug intensity and timing) and patient-specific factors (e.g. the rate of growth of an individual tumor and its composition at diagnosis). For treatments with curative intent, for example, the minimum tumor size achieved during treatment will be critical. A suite of interactive computer programs has been written to mechanize sensitivity analysis with respect to all model variables for a selection of outcome measures. It is possible, given a reliable method of tumor size measurement, that the model may also be helpful in summarizing prognostic information about individual patients. Since by using serial tumor sizes { X, } taken in the early stages of treatment to estimate the model’s parameters, predictions about response, duration of response and possibility of cure, etc. under the continued use of the same treatment would follow. NON-CROSS-RESISTANCE In an attempt to overcome the limiting effects of drug resistance on the success of single-agent therapy, multiple agents have been tried simultaneously in combination [l]. This effectively enlarges the sensitive-cell population at each course through the independent cell-killing activity of the constituent drugs. Even a compound agent, however, if used repeatedly, eventually selects to a (perhaps smaller) resistant core. More recently, sequences of so-called non-cross-resistant drugs (see below) or combinations

DRUG

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CHEMOTHERAPY

have been experimented with [12-141. Such regimens, however, are proving difficult to evaluate. The total non-cross-resistance between two drugs A and B has been defined [12] to mean that “cells resistant to A have the same mutuation rate to resistance to B as do purely sensitive cells, and the therapeutic kill of A against cells resistant to B is the same as for the purely sensitive cells and vice versa.” The limiting factor on exclusive use of two such agents is the presence of doubly resistant cells (viz. resistant to both A and B). This can be quantified through the above model in the following way: Suppose in an initial tumor X0, proportions R, and S, of cells are resistant to A and B respectively, and that of those cells resistant to A a proportion p are also resistant to B (doubly resistant). Let k, and I, be the sensitive-cell kills of the two drugs, and to minimize complexity, suppose secondary resistance is negligible. The clinical experimenter will be interested in the question (and its more general counterparts): “Under what circumstances would it be better to give drug A followed by drug B (A --) B) rather than two doses of A (A 4 A)?” Under the regime A + A, k, and k, given by (3) are the consecutive factional tumor reductions. For A + B, k, and I, will be the corresponding terms, and I, will be derived below. The tumor can be considered to comprise four kinds of cells: those resistant to A alone (+A), those resistant to B alone ($B), those resistant to A and B (+AAB),and those sensitive to both A and B (+,,). (See Figure 4.) The

BEFORE

A

AFTER

FIG. 4.

A

Multiple

BEFORE

drug resistance.

B

AFTER

B

66

B. G. BIRKHEAD TABLE

AND W. M. GREGORY

1

II I

After A ( = before B)

Before A @A

(l-p)&

9e

% - PRO

@Aas @O

PRO l+pR,-R,-

III After B (1-

(1-P)R, Cl- k,)(% - pRoI S,,

Cl- k,)U+

PRO pR, - R, ~ So)

1,)(1-

p)R,

Cl-k,K+pR,,) (I-

PRO I,)(1 ~ k.)(l+ pR,

- R, - S,,,

changes in the ratios of cells in each category under the A + B regimen are shown in Table 1. As before,

which, in this case =I-

which simplifies

sum of terms in column III sum of terms in column II ’

to [,=

~*{(1-~0)(1-k*)+k*~o(l-~)} R, +(1-

k*)(l-

R,)

It follows from [3] and [S] that the A + B strategy will be superior I, > k,) when

k*(l- R,)(l- k*) I*’

(l-S,,)(l-k.)+k*R,(l-p).

For complete non-cross-resistance shown to be of the order of [12]

(i.e.

(6)

the expected value of p, say pO, has been

pa = aln(&Xo), where LYis the rate of spontaneous mutation towards resistance to B. For realistic X0 ( - 10i2) and OL( - 10-5), therefore, p0 will be small. An effect of cross-resistance will be to increase p, either through an increased mutation rate from singly resistant cells to doubly resistant cells or through resistance to A carrying with it resistance to B and vice versa. For p = 1 (all cells resistant to A are also resistant to B) the A + B strategy wins

DRUG RESISTANCE IN CANCER CHEMOTHERAPY

67

when 1,(1- S,,) > k*(l - R,). Generally, the higher p is, the higher I, must be to make the switch to the second drug worthwhile, and similarly as R, decreases (other parameters remaining fixed). The dependence on k,, however, is less clear and will depend critically on the relative values of p, R,, and S,. In situations when k, is sufficiently high to eliminate nearly all A-sensitive cells, then B will be preferable to A as the second agent for relatively smaller values of I*, but where large numbers of A-sensitive cells survive the first course, this will no longer necessarily be the case.

DISCUSSION The selection and outgrowth of resistant cells during cancer chemotherapy is now being recognized as one of the major obstacles to its success, and new strategies to overcome the problem are evolving. An attempt has been made to construct a framework which clarifies some of the mechanisms of failure and which at the same time provides a means of comparing the new approaches. The model rests on assumptions supported by experimental studies in animal model systems, such as the logarithmic kill effect of cytotoxic drugs on sensitive cells [15], exponential tumor growth (a good approximation over a wide range of population sizes), and the -existence of a number of cells resistant to any chosen agent at the beginning of treatment as the result of spontaneous mutation [9]. A further assumption about the acquisition of resistance during therapy is included to make the formulation more general. With a reduction in algebraic appeal the same approach can be adopted when alternative tumor growth laws are chosen, for nonconstant intervals between doses and for variable secondary-resistance fractions. There is scope to extend the basic model (and so increase its complexity) in many ways. For example, dose-response formulae [16] could appear explicitly in the sensitive-cell kill parameter, the primary resistance R, could be replaced by a function of spontaneous mutation rates and tumor size [9], and stochastic variation could be incorporated. Multiple drug resistance limits the efficacy of combination and sequential chemotherapy, and an attempt has been made to describe this effect using the structure of the model. Levels of multiple resistance will depend largely on the degree of cross-resistance between the chosen agents, and the formulae derived may provide some initial quantitative insights into this difficult concept. In the absence of suitable techniques for measuring tumor size or levels of cross-resistance, and with inadequate knowledge about the value in human neoplasms of tumor burden at diagnosis, tumor-cell kill per dose, and repopulation rates, validation of the model rests largely on the level of confirmation given to the underlying assumptions, and appropriate caution should be exercised in interpreting results in the light of the true biological

68

B. ti. BIRKHEAD

AND W. M. GREGORY

situation. It is hoped, however, that by clarifying some of the relationships between the main variables accepted as being important to the success or failure of chemotherapy, a more rational design of treatment strategies will be encouraged. APPENDIX.

CLINICAL

(a) The number of doses tumor apparently ceases to gresses. It occurs after n X, > X,- 1. It is interpreted usefully be continued. From

MEASURES

OF TREATMENT

EFFICACY

to the nadir. The nadir is the point at which a regress under therapy and subsequently reprodoses, where n is the minimum i such that clinically in terms of how long treatment can the formulae for k, it follows that (1-

e-“‘)(l-

a - k,)

k,,( emaT - a) where (x) is the smallest integer greater than or equal to x. (b) The minimum tumor size attained during repeated single-drug therapy is predictive of whether a clinical response will be achieved and whether cure will be possible. If the nadir occurs after n courses (n > 2) the minimum tumor size will be T_, (follows from X, > X,_,). For n = 1, 5 will be the minimum. So Ymin= X, 1eFaT, with n defined as above and X,_ 1 defined in the text. (c) The mix of sensitive and resistant cells during treatment can be measured by the ratio R, : (l- Ri). Its value at the nadir, or at the end of a first period of treatment, will determine the likelihood of subsequent response to the same drugs. It can easily be shown that R,

_ k,(l-a-k,)-a’k,(l-a-k.)

l-R,

a’(l-a)k,

In the absence of secondary resistance (e = 0) this simplifies, as expected, to R l-R,

2

=&(l-

k.)-‘.

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in the chemotherapy

of acute