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# A theoretical analysis of tumour containment

## 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.

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## 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.

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## 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

### 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.

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.

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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

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

• ### Spatial structure impacts adaptive therapy by shaping intra-tumoral competition

• Maximilian A. R. Strobl
• Jill Gallaher
• Alexander R. A. Anderson

Communications Medicine (2022)

• ### Spatial structure governs the mode of tumour evolution

• Robert Noble
• Dominik Burri
• Niko Beerenwinkel

Nature Ecology & Evolution (2022)

• ### The Contribution of Evolutionary Game Theory to Understanding and Treating Cancer

• Benjamin Wölfl
• Hedy te Rietmole
• Kateřina Staňková

Dynamic Games and Applications (2022)