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Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth

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Abstract

We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn–Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time.

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Acknowledgements

The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and of the Project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come accelleratore di una nuona strateGia per l’ATtRattività dell’ateneo pavese” is gratefully acknowledged.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Harald Garcke & Kei Fong Lam

  2. Dipartimento di Matematica, and IMATI-C.N.R., Università degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, Italy

    Elisabetta Rocca

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  1. Harald Garcke
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  2. Kei Fong Lam
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  3. Elisabetta Rocca
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Correspondence to Kei Fong Lam.

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Garcke, H., Lam, K.F. & Rocca, E. Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth. Appl Math Optim 78, 495–544 (2018). https://doi.org/10.1007/s00245-017-9414-4

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  • DOI: https://doi.org/10.1007/s00245-017-9414-4

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