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

2017 ◽  
Vol 78 (3) ◽  
pp. 495-544 ◽  
Author(s):  
Harald Garcke ◽  
Kei Fong Lam ◽  
Elisabetta Rocca
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Treatment Time ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

scholarly journals Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

2020 ◽  
Vol 26 ◽  
pp. 71 ◽  
Author(s):  
Matthias Ebenbeck ◽  
Patrik Knopf
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Chemical Potential ◽  
Type Equation ◽  
Field Variable ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

scholarly journals Optimal distributed control of a diffuse interface model of tumor growth

Nonlinearity ◽  
2017 ◽  
Vol 30 (6) ◽  
pp. 2518-2546 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Elisabetta Rocca ◽  
Jürgen Sprekels
Keyword(s):  
Tumor Growth ◽  
Distributed Control ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Optimal Distributed Control

scholarly journals Formal asymptotic limit of a diffuse-interface tumor-growth model

2015 ◽  
Vol 25 (06) ◽  
pp. 1011-1043 ◽  
Author(s):  
Danielle Hilhorst ◽  
Johannes Kampmann ◽  
Thanh Nam Nguyen ◽  
Kristoffer George Van Der Zee
Keyword(s):  
Tumor Growth ◽  
Growth Model ◽  
Singular Limit ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Sharp Interface Limit ◽  
Field System ◽  
Tumor Growth Model ◽  
Phase Field System

We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit.

scholarly journals Optimal control of time discrete two-phase flow governed by a diffuse interface model

PAMM ◽  
2016 ◽  
Vol 16 (1) ◽  
pp. 785-786 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Two Phase Flow ◽  
Diffuse Interface ◽  
Phase Flow ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase

scholarly journals Analysis of a diffuse interface model of multispecies tumor growth

Nonlinearity ◽  
2017 ◽  
Vol 30 (4) ◽  
pp. 1639-1658 ◽  
Author(s):  
Mimi Dai ◽  
Eduard Feireisl ◽  
Elisabetta Rocca ◽  
Giulio Schimperna ◽  
Maria E Schonbek
Keyword(s):  
Tumor Growth ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

scholarly journals Optimal control of time-discrete two-phase flow driven by a diffuse-interface model

2019 ◽  
Vol 25 ◽  
pp. 13 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Time Discretization ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Fully Discrete

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in Abelset al.[M3AS22(2012) 1150013]. On the basis of the stable time discretization proposed in Garckeet al.[Appl. Numer. Math.99(2016) 151] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

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