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

Harald Garcke; Kei Fong Lam; Elisabetta Rocca

doi:10.1007/s00245-017-9414-4

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

Harald Garcke
, Kei Fong Lam^*
, Elisabetta Rocca

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

66 Citations (Scopus)

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.

Original language	English
Pages (from-to)	495-544
Number of pages	50
Journal	Applied Mathematics and Optimization
Volume	78
Issue number	3
Early online date	13 Apr 2017
DOIs	https://doi.org/10.1007/s00245-017-9414-4
Publication status	Published - Dec 2018

User-Defined Keywords

Cahn–Hilliard equation
Cancer treatment
Distributed optimal control
Free terminal time
Tumor growth
Well-posedness

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10.1007/s00245-017-9414-4

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title = "Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth",

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.",

keywords = "Cahn–Hilliard equation, Cancer treatment, Distributed optimal control, Free terminal time, Tumor growth, Well-posedness",

author = "Harald Garcke and Lam, \{Kei Fong\} and Elisabetta Rocca",

note = "The financial support of the FP7-IDEAS-ERC-StG \#256872 (EntroPhase) and of the Project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d{\textquoteright}Eccellenza in biologia ed ingegneria come accelleratore di una nuona strateGia per l{\textquoteright}ATtRattivit{\`a} dell{\textquoteright}ateneo pavese” is gratefully acknowledged. Publisher Copyright: {\textcopyright} Springer Science+Business Media New York 2017",

year = "2018",

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doi = "10.1007/s00245-017-9414-4",

language = "English",

volume = "78",

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T1 - Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth

AU - Garcke, Harald

AU - Lam, Kei Fong

AU - Rocca, Elisabetta

N1 - 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. Publisher Copyright: © Springer Science+Business Media New York 2017

PY - 2018/12

Y1 - 2018/12

N2 - 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.

AB - 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.

KW - Cahn–Hilliard equation

KW - Cancer treatment

KW - Distributed optimal control

KW - Free terminal time

KW - Tumor growth

KW - Well-posedness

UR - https://www.scopus.com/pages/publications/85017409249

U2 - 10.1007/s00245-017-9414-4

DO - 10.1007/s00245-017-9414-4

M3 - Journal article

AN - SCOPUS:85017409249

SN - 0095-4616

VL - 78

SP - 495

EP - 544

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

IS - 3

ER -

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

Abstract

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