Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A theoretical analysis of tumour containment

Nature Ecology & Evolution volume 5, pages 826–835 (2021)Cite this article

Subjects

Abstract

Recent studies have shown that a strategy aiming for containment, not elimination, can control tumour burden more effectively in vitro, in mouse models and in the clinic. These outcomes are consistent with the hypothesis that emergence of resistance to cancer therapy may be prevented or delayed by exploiting competitive ecological interactions between drug-sensitive and drug-resistant tumour cell subpopulations. However, although various mathematical and computational models have been proposed to explain the superiority of particular containment strategies, this evolutionary approach to cancer therapy lacks a rigorous theoretical foundation. Here we combine extensive mathematical analysis and numerical simulations to establish general conditions under which a containment strategy is expected to control tumour burden more effectively than applying the maximum tolerated dose. We show that containment may substantially outperform more aggressive treatment strategies even if resistance incurs no cellular fitness cost. We further provide formulas for predicting the clinical benefits attributable to containment strategies in a wide range of scenarios and compare the outcomes of theoretically optimal treatments with those of more practical protocols. Our results strengthen the rationale for clinical trials of evolutionarily informed cancer therapy, while also clarifying conditions under which containment might fail to outperform standard of care.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Illustration of containment and MTD treatments in Model 3.
Fig. 2: Comparison of clinical benefits of containment and MTD treatments in Model 3.
Fig. 3: Constant dose and delayed constant dose treatments in Model 3.
Fig. 4: Consequences of the costs of resistance in Model 4.

Data availability

No datasets were generated or analysed during the current study.

Code availability

Simulations were conducted in R (version 4.0.2) using the deSolve package39 (version 1.28). The code for the simulations is available at https://github.com/robjohnnoble/LogicOfContainingTumours.

References

  1. Norton, L. & Simon, R. Tumor size, sensitivity to therapy, and design of treatment schedules. Cancer Treat. Rep. 61, 1307–1317 (1977).

    CAS  PubMed  Google Scholar 

  2. Goldie, J. H. & Coldman, A. J. A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat. Rep. 63, 1727–1733 (1979).

    CAS  PubMed  Google Scholar 

  3. Gatenby, R. A. A change of strategy in the war on cancer. Nature 459, 508–509 (2009).

    Article  CAS  Google Scholar 

  4. Zhang, J., Cunningham, J. J., Brown, J. S. & Gatenby, R. A. Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer. Nat. Commun. 8, 1816 (2017).

    Article  Google Scholar 

  5. Martin, R. B., Fisher, M. E., Minchin, R. F. & Teo, K. L. Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy. Math. Biosci. 110, 201–219 (1992).

    Article  CAS  Google Scholar 

  6. Gatenby, R. A., Silva, A. S., Gillies, R. J. & Frieden, B. R. Adaptive therapy. Cancer Res. 69, 4894–4903 (2009).

    Article  CAS  Google Scholar 

  7. Gatenby, R. & Brown, J. The evolution and ecology of resistance in cancer therapy. Cold Spring Harb. Perspect. Med. 10, a040972 (2020).

    Article  CAS  Google Scholar 

  8. Bourguet, D. et al. Heterogeneity of selection and the evolution of resistance. Trends Ecol. Evol. 28, 110–118 (2013).

    Article  Google Scholar 

  9. Tabashnik, B. E., Brévault, T. & Carrière, Y. Insect resistance to Bt crops: lessons from the first billion acres. Nat. Biotechnol. 31, 510–521 (2013).

    Article  CAS  Google Scholar 

  10. Cunningham, J. J. A call for integrated metastatic management. Nat. Ecol. Evol. 3, 996–998 (2019).

    Article  Google Scholar 

  11. Bacevic, K. Spatial competition constrains resistance to targeted cancer therapy. Nat. Commun. 8, 1995 (2017).

    Article  Google Scholar 

  12. Silva, A. S. et al. Evolutionary approaches to prolong progression-free survival in breast cancer. Cancer Res. 72, 6362–6370 (2012).

    Article  CAS  Google Scholar 

  13. Enriquez-Navas, P. M. et al. Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer. Sci. Transl. Med. 8, 327ra24 (2016).

    Article  Google Scholar 

  14. Monro, H. C. & Gaffney, E. A. Modelling chemotherapy resistance in palliation and failed cure. J. Theor. Biol. 257, 292–302 (2009).

    Article  CAS  Google Scholar 

  15. Carrère, C. Optimization of an in vitro chemotherapy to avoid resistant tumours. J. Theor. Biol. 413, 24–33 (2017).

    Article  Google Scholar 

  16. Gallaher, J. A., Enriquez-Navas, P. M., Luddy, K. A., Gatenby, R. A. & Anderson, A. R. A. Spatial heterogeneity and evolutionary dynamics modulate time to recurrence in continuous and adaptive cancer therapies. Cancer Res. 78, 2127–2139 (2018).

    Article  CAS  Google Scholar 

  17. Hansen, E., Woods, R. J. & Read, A. F. How to use a chemotherapeutic agent when resistance to it threatens the patient. PLoS Biol. 15, e2001110 (2017).

    Article  Google Scholar 

  18. Cunningham, J. J., Brown, J. S., Gatenby, R. A. & Staňková, K. Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer. J. Theor. Biol. 459, 67–78 (2018).

    Article  CAS  Google Scholar 

  19. West, J., Ma, Y. & Newton, P. K. Capitalizing on competition: an evolutionary model of competitive release in metastatic castration resistant prostate cancer treatment. J. Theor. Biol. 455, 249–260 (2018).

    Article  Google Scholar 

  20. Pouchol, C., Clairambault, J., Lorz, A. & Trélat, E. Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy. J. Math. Pures Appl. 116, 268–308 (2018).

    Article  Google Scholar 

  21. Carrère, C. & Zidani, H. Stability and reachability analysis for a controlled heterogeneous population of cells. Optim. Control Appl. Methods 41, 1678–1704 (2020).

    Article  Google Scholar 

  22. Greene, J. M., Sanchez-Tapia, C. & Sontag, E. D. Mathematical details on a cancer resistance model. Front. Bioeng. Biotechnol. 8, 501 (2020).

    Article  Google Scholar 

  23. Martin, R. B., Fisher, M. E., Minchin, R. F. & Teo, K. L. Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells. Math. Biosci. 110, 221–252 (1992).

    Article  CAS  Google Scholar 

  24. Gerlee, P. The model muddle: in search of tumor growth laws. Cancer Res. 73, 2407–2411 (2013).

    Article  CAS  Google Scholar 

  25. Noble, R., Burri, D., Kather, J. N. & Beerenwinkel, N. Spatial structure governs the mode of tumour evolution. Preprint at bioRxiv https://doi.org/10.1101/586735 (2019).

  26. Hansen, E. & Read, A. F. Cancer therapy: attempt cure or manage drug resistance? Evol. Appl. 13, 1660–1672 (2020).

    Article  Google Scholar 

  27. Enriquez-Navas, P. M., Wojtkowiak, J. W. & Gatenby, R. A. Application of evolutionary principles to cancer therapy. Cancer Res. 75, 4675–4680 (2015).

    Article  CAS  Google Scholar 

  28. Gatenby, R. A. & Brown, J. S. Integrating evolutionary dynamics into cancer therapy. Nat. Rev. Clin. Oncol. 17, 675–686 (2020).

    Article  Google Scholar 

  29. Strobl, M. A. R. et al. Turnover modulates the need for a cost of resistance in adaptive therapy. Cancer Res. https://doi.org/10.1158/0008-5472.CAN-20-0806 (2020).

    Article  PubMed  Google Scholar 

  30. Bozic, I. et al. Evolutionary dynamics of cancer in response to targeted combination therapy. eLife 2, e00747 (2013).

    Article  Google Scholar 

  31. Pérez-García, V. M. et al. Universal scaling laws rule explosive growth in human cancers. Nat. Phys. 16, 1232–1237 (2020).

    Article  Google Scholar 

  32. Greene, J. M., Gevertz, J. L. & Sontag, E. S. Mathematical approach to differentiate spontaneous and induced evolution to drug resistance during cancer treatment. JCO Clin. Cancer Inform. 3, CCI.18.00087 (2019).

    PubMed Central  Google Scholar 

  33. Kuosmanen, T. et al. Drug-induced resistance evolution necessitates less aggressive treatment. Preprint at bioRxiv https://doi.org/10.1101/2020.10.07.330134 (2020).

  34. Fusco, D., Gralka, M., Kayser, J., Anderson, A. & Hallatschek, O. Excess of mutational jackpot events in expanding populations revealed by spatial Luria–Delbrück experiments. Nat. Commun. 7, 12760 (2016).

    Article  CAS  Google Scholar 

  35. Mistry, H. B. Evolutionary based adaptive dosing algorithms: beware the cost of cumulative risk. Preprint at bioRxiv https://doi.org/10.1101/2020.06.23.167056 (2020).

  36. Benzekry, S. et al. Classical mathematical models for description and prediction of experimental tumor growth. PLoS Comput. Biol. 10, e1003800 (2014).

    Article  Google Scholar 

  37. Vaghi, C. et al. Population modeling of tumor growth curves and the reduced Gompertz model improve prediction of the age of experimental tumors. PLoS Comput. Biol. 16, e1007178 (2020).

    Article  CAS  Google Scholar 

  38. Hansen, E., Karslake, J., Woods, R. J., Read, A. F. & Wood, K. B. Antibiotics can be used to contain drug-resistant bacteria by maintaining sufficiently large sensitive populations. PLoS Biol. 18, e3000713 (2020).

    Article  CAS  Google Scholar 

  39. Soetaert, K. E. R., Petzoldt, T. & Setzer, R. W. Solving differential equations in R : package deSolve. J. Stat. Softw. 33, 9 (2010).

Download references

Acknowledgements

We thank P. Lissy, S. Benzekry, F. E. Alvarez Borges and especially J. Brown for very helpful discussions and P. Ear for preliminary versions of some of the figures. This research project was initiated at the Lorentz Center Workshops Game Theory and Evolutionary Biology and Understanding Cancer Through Evolutionary Game Theory. Y.V. and R.N. acknowledge the support from the Fondation Mathématiques Jacques Hadamard Program Gaspard Monge for optimization and operation research and their interactions with data science, and from EDF, Thales, Orange and Criteo. R.N. acknowledges support from ERC Synergy grant no. 609883 and the National Cancer Institute of the National Institutes of Health (NIH) under award no. U54CA217376. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Author information

Authors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Université Paris Sciences et Lettres, Paris, France

    Yannick Viossat

  2. Department of Mathematics, City, University of London, London, UK

    Robert Noble

  3. Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland

    Robert Noble

  4. Department of Evolutionary Biology and Environmental Studies, University of Zurich, Zurich, Switzerland

    Robert Noble

Authors
  1. Yannick Viossat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Noble
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y.V. and R.N. conceived the study and wrote the manuscript. Y.V. designed and performed the mathematical analyses and conducted the literature review. R.N. designed and performed the numerical modelling.

Corresponding author

Correspondence to Yannick Viossat.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Ecology & Evolution thanks Benjamin Werner and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Ideal intermittent containment between N\({\,}_{\min }\) and N\({\,}_{\max }\).

Times to progression are shown for ideal intermittent containment between N\({\,}_{\min }\) and N0 for varied N\({\,}_{\min }\) value (solid curve), compared to ideal containment at either N0 (dashed curve) or N\({\,}_{\min }\) (dotted curve), according to a Gompertzian growth model (Model 3 in the main text). Non-varied parameter values are as in main text Table 2. The kinks in the curve for ideal intermittent containment are due to the discontinuity of the treatment when a new cycle is completed, or in mathematical terms, to the integer part that appears in the explicit formula.

Extended Data Fig. 2 Outcomes for five models with different forms of density dependence.

a, Untreated tumour growth curves for a Gompertzian growth model (black curve; Model 3 in the main text), a logistic growth model (red), a von Bertalanffy growth model (blue), an exponential model (yellow) and a superexponential model (grey). Parameter values for the Gompertzian growth model are as in Table 2 of the main text. Parameter values of the logistic and von Bertalanffy models are chosen so that their growth curves are similar to the Gompertzian model for tumour sizes between N0 and Ncrit (the lethal size), as would be the case if the models were fitted to empirical data. In the logistic growth model, K = 6.4 × 1011 and ρ = 2.4 × 10−2. In the von Bertalanffy growth model, K = 5 × 1013, ρ = 90 and γ = 1/3 (the latter value is conventional in tumour growth modelling [24, 37]). In the exponential model, ρ = 0.0175. In the superexponential model, ρ = 4.5 × 10−6 and γ = 1/3 (the latter value has been inferred from data [32]). b, Relative benefit, in terms of time to treatment failure, for ideal containment (at size Ntol) versus ideal MTD, for the five models with varied initial frequency of resistance (parameter values are the same as in panel a). Note that relative benefits for all models are independent of ρ.

Extended Data Fig. 3 Evolution of total tumour size under containment and MTD treatment in a Gompertzian growth model (Model 3 in the main text).

The initial resistant subpopulation size (R0) is varied. The maximum dose is C\({\,}_{\max }\) = 2. Fixed parameter values are as in Table 2 of the main text.

Extended Data Fig. 4 Containment at N0 and intermittent containment between N0 and 0.8 N0 in a Gompertzian growth model (Model 3 in the main text).

Dashed vertical lines indicate time to progression under containment (dashed grey) and intermittent containment (dashed black). Intermittent containment leads here to a slightly larger time to progression than containment at the upper level. However, as follows from Proposition 6 (Supplementary Material), the resistant population is larger under intermittent containment (red) than under containment (pink). After progression, tumour size quickly becomes larger under intermittent containment (solid black curve) than under containment (solid grey curve). Parameter values are as in Table 2 of the main text.

Extended Data Fig. 5 Influence of ongoing mutation in a Gompertzian growth model (Model 3 in the main text).

Outcomes are shown for a model in which mutations are neglected after the tumour reaches size N0 (solid lines) and for a model that explicitly accounts for ongoing mutation from the sensitive to the resistant phenotype at rate \(\tau_1\) (broken lines). Two different mutation rates are illustrated. The second row contains the same data as the first row of panels but with different axes so as to make visible the subtle differences between curves. Fixed parameter values are as in Table 2 of the main text.

Extended Data Fig. 6 Evolution of total tumour size under ideal and non-ideal treatments in a Gompertzian growth model (Model 3 in the main text).

The initial resistant subpopulation size (R0) and the maximum dose (C\({\,}_{\max }\)) are varied. Fixed parameter values are as in Table 2 of the main text.

Extended Data Fig. 7 Constant dose treatments in a Gompertzian growth model (Model 3 in the main text).

Tumour size for various constant dose treatments are compared to containment at the initial size (subject to C\({\,}_{\max }\) = 2), MTD (C = C\({\,}_{\max }\)) and ideal MTD. Dose 1.09 maximizes time to progression and 0.74 maximizes survival time among non-delayed constant doses (but is inferior to the optimal delayed constant dose). Parameter values are as in Table 2 of the main text.

Extended Data Fig. 8 Consequences of costs of resistance in a Gompertzian growth model (Model 4 in the main text).

Relative benefit, in terms of time to treatment failure, for ideal containment (at size Ntol) versus ideal MTD, for varied values of Kr and β. The figure is obtained from simulations, while Fig. 4a in the main text is obtained from our approximate formula. Contour lines are at powers of 2. Fixed parameter values are as in Table 2 of the main text.

Supplementary information

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Viossat, Y., Noble, R. A theoretical analysis of tumour containment. Nat Ecol Evol 5, 826–835 (2021). https://doi.org/10.1038/s41559-021-01428-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41559-021-01428-w

This article is cited by

Search

Advanced search

Quick links

Nature Briefing: Cancer

Sign up for the Nature Briefing: Cancer newsletter — what matters in cancer research, free to your inbox weekly.

Get what matters in cancer research, free to your inbox weekly. Sign up for Nature Briefing: Cancer