# Control by Viability in a Chemotherapy Cancer Model

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## Abstract

The aim of this study is to provide a feedback control, called the Chemotherapy Protocol Law, with the purpose to keep the density of tumor cells that are treated by chemotherapy below a “tolerance level” \(L_c\), while retaining the density of normal cells above a “healthy level” \(N_c\). The mathematical model is a controlled dynamical system involving three nonlinear differential equations, based on a Gompertzian law of cell growth. By evoking viability and set-valued theories, we derive sufficient conditions for the existence of a Chemotherapy Protocol Law. Thereafter, on a suitable viability domain, we build a multifunction whose selections are the required Chemotherapy Protocol Laws. Finally, we propose a design of selection that generates a Chemotherapy Protocol Law.

## Keywords

Cancer model Feedback control Viability Set-valued analysis## Mathematics Subject Classification

34H05 34A60 49 J52 49K15 92C50 9C60## Notes

### Acknowledgements

We are very grateful to the editor in chief of the journal “Acta Biotheoretica”, and the anonymous reviewers for their useful remarks and constructive suggestions that improve substantially the manuscript.

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