Modelling chemotherapy resistance in palliation and failed cure

https://doi.org/10.1016/j.jtbi.2008.12.006Get rights and content

Abstract

The goal of palliative cancer chemotherapy treatment is to prolong survival and improve quality of life when tumour eradication is not feasible. Chemotherapy protocol design is considered in this context using a simple, robust, model of advanced tumour growth with Gompertzian dynamics, taking into account the effects of drug resistance. It is predicted that reduced chemotherapy protocols can readily lead to improved survival times due to the effects of competition between resistant and sensitive tumour cells. Very early palliation is also predicted to quickly yield near total tumour resistance and thus decrease survival duration. Finally, our simulations indicate that failed curative attempts using dose densification, a common protocol escalation strategy, can reduce survival times.

Introduction

The interest in modelling chemotherapy protocols to support clinical insight is driven by efforts to improve outcomes by adjusting drug scheduling (Browder et al., 2000, Citron et al., 2002, Fallik et al., 2003, Midgley and Kerr, 2005) and is illustrated by the seminal work of Norton and Simon, 1976, Norton and Simon, 1986. This led to their initial hypothesis, namely that chemotherapy results in a rate of regression in tumour volume that is proportional to the rate of growth for an unperturbed tumour of the same size. Further work resulted in the influential concepts of dose intensification and especially dose densification. These constitute protocol escalation via either reducing rest phases, i.e. densification, or increasing drug dosage, i.e. intensification, and were predicted to increase the probability of tumour eradication (Norton, 1998, Norton, 1997, Norton, 2001). In particular the dose densification strategy was tested in a series of clinical trials by the Cancer and Leukaemia Group B (CALGB) and the American Breast Intergroup (Citron et al., 2002, Citron et al., 2003). The results supported the theory that dose dense treatment would lead to a significant improvement in clinical outcomes, additionally exemplifying how a mathematical model can make successful predictions for improved chemotherapy protocols (Piccart-Gebhart, 2003). This is further highlighted by recent phase I trials with a novel Capecitabine schedule for metastatic breast cancer (Tiffany et al., 2008). However, one should also note that, in general, regimens utilising protocol escalation, either by dose densing or intensification, yield variable results (Piccart et al., 2000).

The focus of Norton and Simon's modelling strategies was to maximise cell kill; other objectives have been considered in the literature. For example, whereas Norton and Simon neglect genetic mutation to resistance, Goldie and Coldman (Goldie et al., 1982) modelled chemotherapy scheduling with the goal of minimising the development of drug resistance. Resistance was assumed to be a result of random genetic mutation to a resistant state (Goldie and Coldman, 1998). This assumption was based on the Nobel Prize winning work of Luria and Delbruck (1943), who showed that bacterial cultures developed resistance to bacteriophages at random. Spontaneous mutation has also been found to arise in in vitro cancer cell systems (Jaffrezou et al., 1994, Summer and Handshumacher, 1973), and Law (1952) later found that the same applied to methotrexate resistance in vivo with leukaemic L1210 cells.

The Goldie–Coldman model focuses on maximising the probability that no tumour cells will have mutated sufficiently to become resistant to all drugs present. When more than one non-cross-resistant chemotherapeutic is used, it was predicted that the drugs should be alternated as quickly as possible in order to reduce the occurrence of resistant cells, thus maximising the probability of cure (Goldie et al., 1982). Evidence supporting this hypothesis has been sought in numerous clinical trials (De Placido et al., 1995, Sieber et al., 2002, Siodlak et al., 1990), though it is typically refuted rather than validated. Gaffney, 2004, Gaffney, 2005 extended the Goldie and Coldman model to consider cell cycle phase specific drugs and the effects of drug delivery. It was shown that the Goldie and Coldman's alternation hypothesis often breaks down both due to the effects of pharmacokinetics and due to resonances between the application time of a cell cycle phase specific drug and the tumour cell cycle time.

However, such models focus on cure, as do virtually all the models in the literature. In contrast, we will investigate the consequences of protocols when tumour eradication does not occur, either in the context of palliation or failed cure, and whether this reveals the need to consider different protocol strategies.

In particular, models have traditionally predicted that an increased dose will yield an increased response, at least within any implicit or explicit toxicity constraints. However, lower doses of chemotherapy may indeed lead to longer survival outcomes in the absence of cure. The failure of an aggressive chemotherapy schedule could occur due to the preferential removal of sensitive cells, leaving behind a population more resistant as a whole. At extremes, a non-curative but escalated chemotherapeutic regimen could be expected to leave behind an uncontrollable mass of cells insensitive to any further therapy. In contrast, too low a dose will allow even cells that are sensitive to the drug to grow out of control. Therefore we immediately have the question of whether one can reasonably anticipate that an intermediate level of chemotherapy will restrict tumour growth and increase survival duration in the palliative setting.

Protocol escalation is in conflict with the above reasoning, as escalation requires either a higher dosage of chemotherapy, a shorter administration time, or both. The source of this disagreement arises from the different objectives. The aim of protocol escalation is to improve the chances of eradicating the tumour; our above reasoning is based upon prolonging survival when tumour eradication is not possible. However, it cannot always be certain whether a tumour is curable or not. This emphasises that the effect of an attempted, but unsuccessful, curative protocol escalation on survival time also needs to considered in modelling chemotherapy scheduling and we will investigate this in detail.

We note that there is experimental evidence supporting the concept that lower chemotherapy doses could be preferable in the presence of resistance. In particular an investigation by Aabo et al. (1994) showed that low-dose chemotherapy delayed the relapse of a dominated and resistant sub-population in a human small-cell lung cancer (SCLC) xenograft in mice. The xenograft consisted of an artificially mixed BCNU-sensitive, dominating sub-population and a BCNU-resistant, undetectable (dominated) sub-population, where BCNU is an anti-cancer drug. At the time of tumour regrowth after low-dose treatment, most of the tumours continued to be dominated by the sensitive population, and thus remain susceptible to chemical control. At the time of regrowth after the response to high-dose treatment, the resistant cell line was the predominant population.

Similar concepts to those presented in this paper have been detailed in the modelling study of Hahnfeldt et al. (2003) with particular emphasis on the anti-angiogenic effects of metronomic chemotherapy, which is a low-dose drug scheduling regime. Such regimens are observed to differentially target the endothelial cells of the growing blood vessels found in tumours (Bello et al., 2001, Bertolini et al., 2003, Browder et al., 2000, Kerbel and Kamen, 2004, Man et al., 2002), therefore inhibiting angiogenesis. The model of Hahnfeldt et al. (2003) focuses on how heterogeneity in resensitisation rates between tumour and endothelial cells can provide a logical framework explaining such observations. Our model has an analogous mathematical framework though the central focus here is tumour cell heterogeneity per se rather than the difference between endothelial and tumour cells. This variation is primarily one of model interpretation; the core difference is that Hahnfeldt et al.'s model considers exponential cell growth. In contrast, we consider Gompertzian tumour growth; this is a constant, exponential, retardation of the growth rate which has been found to provide a good empirical description of the decelerating growth curves exhibited by more advanced tumours (Simpson-Herren and Lloyd, 1970, Sullivan and Salmon, 1972). The Gompertz (1825) model was first applied in actuarial statistics, and subsequently in the study of growth by Winsor (1932), with Laird (1964) further illustrating that the growth for a variety of primary and transplanted tumours in the mouse, rat and rabbit satisfied the Gompertzian relation.

Gompertzian growth is critical in this paper as it provides the theoretical basis of protocol escalation (Norton, 1997) in addition to being highly relevant for advanced tumours (Simpson-Herren and Lloyd, 1970, Sullivan and Salmon, 1972). For example, reducing a protocol's rest phase entails that a Gompertzian tumour is growing at faster and faster rates at each drug application since each application yields a smaller tumour. By the eponymous Norton–Simon hypothesis this compounds a greater and greater cell kill effect. This compounding entails that protocol outcomes are anticipated to be very sensitive to the dose densing effect. Such dynamics thus need to be explicitly considered when investigating protocols involving advanced tumours and when examining the consequences of dose densing, or more generally protocol escalation, and is central to our model.

In Section 2, the development of the model for palliative chemotherapy applied to advanced tumour growth will be outlined. Two main sections ofresults and discussion will follow. In Section 3.1 we consider continuous chemotherapy for our initial investigation into the effects of protocol escalation in the palliative setting. In Section 3.2, we proceed to consider protocol escalation for cycles of drug administration interspersed with rest phases. This study is subsequently extended to the case where the tumour is only just incurable to allow us to investigate the relationship between survival time and protocol escalation for a failed curative attempt. Finally, we discuss our results and observations in Section 4.

Section snippets

The model

The assumptions of the model are:

  • Tumour growth is represented by a continuous, Gompertzian, model.

  • The effect of the chemotherapeutic on the sensitive cells is to induce a regression rate proportional to the unperturbed growth rate, in accordance with the Norton–Simon hypothesis (Norton and Simon, 1976).

  • The chemotherapeutic induces sensitive tumour cell kill in proportion to the intensity of the administration.

  • Mutation to resistance is Darwinian (Law, 1952, Luria and Delbruck, 1943) at a rate

Indefinite continuous protocol

Simulations of indefinite continuous chemotherapy are performed for different, fixed, values of cell kill λC(t)C0 with survival times depicted in Fig. 1. It clearly shows a local maximum, illustrating that an intermediate level of chemotherapy is predicted to prolong survival time the most.

Plots A, B, C of Fig. 2, respectively, show the cell population dynamics for the three values of cell kill, C0, depicted by A, B, C in Fig. 1. Note that in the optimal scenario, B, both sensitive and

Indefinite continuous protocols

The results from indefinite continuous chemotherapy, as illustrated in Fig. 1, support the hypothesis that intermediate palliative doses of chemotherapy yield higher survival times compared to higher doses.

A more detailed examination of the cell population dynamics for drug doses around this optimum level reveals the reason for this behaviour; see Fig. 2. At lower doses, the chemotherapy cannot control the sensitive cell growth, resulting in the total cell number exceeding the fatal level

Acknowledgements

This work was supported in part by the Engineering and Physical Sciences Research Council, UK (EPSRC GR/S72023/01). We are also grateful to Prof. D.J. Kerr for useful and stimulating discussions. We are grateful to an anonymous referee for helpful comments.

References (47)

  • T. Browder et al.

    Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer

    Cancer Res.

    (2000)
  • N. Brünner et al.

    Effect on growth and cell cycle kinetics of Estradiol and Tamoxifen on MCF-7 human breast cancer cells grown in vitro and in nude mice

    Cancer Res.

    (1989)
  • M. Citron et al.

    Superiority of dose-dense (DD) over conventional scheduling (CS) and equivalence of sequential (SC) vs combination adjuvant chemotherapy (CC) for node-positive breast cancer (CALGB INT C9741)

    Breast Cancer Res. Treat.

    (2002)
  • M.L. Citron et al.

    Randomized trial of dose-dense versus conventionally scheduled and sequential versus concurrent combination chemotherapy as postoperative adjuvant treatment of node-positive primary breast cancer: first report of Intergroup trial C9741/Cancer and Leukemia Group B trial 9741

    J. Clin. Oncol.

    (2003)
  • S. De Placido et al.

    CMF vs alternating CMF/EV in the adjuvant treatment of operable breast cancer. A single centre randomised clinical trial (Naples GUN-3 study)

    Br. J. Cancer

    (1995)
  • E.A. Gaffney

    The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling

    J. Math. Biol.

    (2004)
  • J.H. Goldie et al.

    Drug Resistance in Cancer

    (1998)
  • J.H. Goldie et al.

    Rationale for the use of alternating noncross resistant chemotherapy

    Cancer Treat. Rep.

    (1982)
  • B. Gompertz

    On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies

    Philos. Trans. R. Soc. London

    (1825)
  • W. Gorczyca et al.

    The cell cycle related differences in susceptibility of HL-60 cells to apoptosis induced by various antitumor agents

    Cancer Res.

    (1993)
  • E.J. Hall et al.

    Radiobiology for the Radiologist

    (2006)
  • J.P. Jaffrezou et al.

    Mutation rates and mechanisms of resistance to etoposide determined from fluctuation analysis

    J. Natl. Cancer Inst.

    (1994)
  • R.S. Kerbel et al.

    The anti-angiogenic basis of metronomic chemotherapy

    Nat. Rev. Cancer

    (2004)
  • Cited by (0)

    View full text