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Long-Time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth

  • Cecilia Cavaterra
  • Elisabetta Rocca
  • Hao WuEmail author
Article
  • 24 Downloads

Abstract

We investigate the long-time dynamics and optimal control problem of a thermodynamically consistent diffuse interface model that describes the growth of a tumor in presence of a nutrient and surrounded by host tissues. The state system consists of a Cahn–Hilliard type equation for the tumor cell fraction and a reaction–diffusion equation for the nutrient. The possible medication that serves to eliminate tumor cells is in terms of drugs and is introduced into the system through the nutrient. In this setting, the control variable acts as an external source in the nutrient equation. First, we consider the problem of “long-time treatment” under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as \(t\rightarrow +\infty \). Second, we consider the “finite-time treatment” that corresponds to an optimal control problem. Here we allow the objective cost functional to depend on a free time variable, which represents the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time. One of the main aim of the control problem is to realize in the best possible way a desired final distribution of the tumor cells, which is expressed by the target function \(\phi _\Omega \). By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we show that \(\phi _{\Omega }\) can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed.

Keywords

Tumor growth Cahn–Hilliard equation Reaction–diffusion equation Optimal control Long-time behavior Lyapunov stability 

Mathematics Subject Classification

35K61 49J20 49K20 92C50 97M60 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and helpful comments. C. Cavaterra and E. Rocca were partially supported by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). This research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese”. This research was also supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) - Dept. of Mathematics “F. Casorati”, University of Pavia. H. Wu was partially supported by NNSFC Grant No. 11631011 and the Shanghai Center for Mathematical Sciences.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  3. 3.Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNRPaviaItaly
  4. 4.School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina
  5. 5.Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University)Ministry of EducationShanghaiChina

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