Elsevier

European Journal of Cancer

Volume 37, Issue 16, November 2001, Pages 2116-2120
European Journal of Cancer

An evolutionary-game model of tumour–cell interactions: possible relevance to gene therapy

https://doi.org/10.1016/S0959-8049(01)00246-5Get rights and content

Abstract

Evolutionary games have been applied as simple mathematical models of populations where interactions between individuals control the dynamics. Recently, it has been proposed to use this type of model to describe the evolution of tumour cell populations with interactions between cells. We extent the analysis to allow for synergistic effects between cells. A mathematical model of a tumour cell population is presented in which population-level synergy is assumed to originate through the interaction of triplets of cells. A threshold of two cooperating cells is assumed to be required to produce a proliferative advantage. The mathematical behaviour of this model is explored. Even this simple synergism (minor clustering effect) is sufficient to generate qualitatively different cell-population dynamics from the models published previously. The most notable feature of the model is the existence of an unstable internal equilibrium separating two stable equilibria. Thus, cells of a malignant phenotype can exist in a stable polymorphism, but may be driven to extinction by relatively modest perturbations of their relative frequency. The proposed model has some features that may be of interest to biological interpretations of gene therapy. Two prototypical strategies for gene therapy are suggested, both of them leading to extinction of the malignant phenotype: one approach would be to reduce the relative proportion of the cooperating malignant cell type below a certain critical value. Another approach would be to increase the critical threshold value without reducing the relative frequency of cells of the malignant phenotype.

Introduction

Throughout the development of a tumour, selection of specific mutations is traditionally believed to reflect the ability of these mutations to promote growth or inhibit death by cell-autonomous modes of action 1, 2, 3, 4. An improved understanding of the genetic and biochemical mechanisms, by which mutations confer a growth advantage to cancer cells, has led to the formulation of mathematical models for tumour initiation, growth and metastatic spread 5, 6, 7, 8, 9. These models typically incorporate the effect of putative paracrine and/or autocrine growth factors, but rely nevertheless on the cell-autonomous consequences of mutations in the malignant cells. Recently, Tomlinson and Bodmer 10, 11 presented evolutionary-game mathematical models for the time evolution of a tumour cell population which incorporated non-autonomous effects of tumour cell mutations in terms of interactions among cells that have adopted individual genetic strategies. Following the work by Axelrod and Hamilton [12], evolutionary games have been studied as simple mathematical models of the evolution of cooperation in a population. The prototypical example is the Prisoner's Dilemma, where the payoff from a specific strategy depends on whether the other prisoner collaborates or defects. A popular introduction to the Prisoner's Dilemma and other evolutionary games may be found in Nowak and colleagues [13].

Tomlinson and Bodmer discussed several models, and one of these considered the production of a growth factor, exemplified by an angiogenic promoter, that was associated with a cost of production and a replication advantage. The benefit was obtained by the cell itself, as well as other cells in the population. As long as the benefit was greater than the cost of production, a stable polymorphism between producers and non-producers were predicted.

Section snippets

Materials and methods

Here, an extension of the evolutionary model by Tomlinson and Bodmer 10, 11 is proposed and its behaviour investigated. Specifically, this model incorporates a local threshold condition for deriving a benefit from collaboration between tumour cells. One concrete example could be the production of an angiogenic promoter where it is conceivable that producer cells have to collaborate in order to provide a sufficiently high local concentration of the stimulating factor. We describe the simplest

Evolutionary games and tumour population dynamics

Tomlinson and Bodmer [10] developed a mathematical model of the dynamics of a population consisting of two types of cells with strategies A+ and A−. An A+ cell produces a certain factor which conveys a proliferative advantage both to the cell itself and its neighbours. In contrast, the A− cell does not produce the factor and a population of A− cells will only have the baseline reproductive rate. The detailed mechanism of this interaction needs not to be specified, but the resulting net cost or

Discussion

The most interesting feature of the current model is the existence of an unstable internal equilibrium which forms a barrier between two stable states of the population, one with a stable polymorphism between A+ and A− cells and one where only A− cells are present. During tumorigenesis, it must be assumed that local collaboration is possible which may allow this critical threshold to be crossed locally. At a later stage, the selection pressure against the tumour cells may have increased and

Conclusion

In the light of recent advances in the description of cell–cell interactions and their biochemical signalling pathways, this kind of model is, in our view, a potentially useful tool for understanding the behaviour of populations of tumour cells. Eventually, these models could lead to more elaborate applications of spatial simulations, where local synergism and clustering effects determine the final composition of the cell population. We propose that some gene therapy approaches may benefit from

References (31)

  • S Michelson et al.

    Autocrine and paracrine growth factors in tumor growtha mathematical model

    Bull. Math. Biol.

    (1991)
  • I.P Tomlinson

    Game-theory models of interactions between tumour cells

    Eur. J. Cancer

    (1997)
  • P.M Rowe

    Starve the tumour, save the patient

    Lancet

    (1997)
  • P Armitage et al.

    A two-stage theory of carcinogenesis in relation to age distribution of human cancer

    Br. J. Cancer

    (1954)
  • P Armitage et al.

    The age distribution of cancer and a multi-stage theory of carcinogenesis

    Br. J. Cancer

    (1954)
  • J.C Fisher

    Multiple-mutation theory of carcinogenesis

    Nature

    (1958)
  • J Cairns

    Mutation selection and the natural history of cancer

    Nature

    (1975)
  • A.J Perumpanani et al.

    Biological inferences from a mathematical model for malignant invasion

    Invasion Metastasis

    (1996)
  • J.A Sherratt et al.

    Oncogenes, anti-oncogenes and the immune response to cancera mathematical model

    Proc. R. Soc. Lond. B. Biol. Sci.

    (1992)
  • I.P Tomlinson et al.

    Failure of programmed cell death and differentiation as causes of tumorssome simple mathematical models

    Proc. Natl. Acad. Sci. USA

    (1995)
  • I.P Tomlinson et al.

    The mutation rate and cancer

    Proc. Natl. Acad. Sci. USA

    (1996)
  • I.P Tomlinson et al.

    Modelling the consequences of interactions between tumour cells

    Br. J. Cancer

    (1997)
  • R Axelrod et al.

    The evolution of cooperation

    Science

    (1981)
  • M.A Nowak et al.

    The arithmetics of mutual help

    Scientific American

    (1995)
  • M.A Nowak et al.

    Evolutionary games and spatial chaos

    Nature

    (1992)
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