Cahn-Hilliard-Brinkman models for tumour growth: Modelling, analysis and optimal control

Phase field models recently gained a lot of interest in the context of tumour growth models. In this work we study several diffuse interface models for tumour growth in a bounded domain with sufficiently smooth boundary. The basic model consists of a Cahn–Hilliard type equation for the concentration of tumour cells coupled to a convection-reaction-diffusion-type equation for an unknown species acting as a nutrient and a Brinkman-type equation for the velocity. The system is equipped with Neumann boundary conditions for the phase field and the chemical potential, a Robin-type boundary condition for the nutrient and a “no-friction” boundary condition for the velocity which allows us to consider solution dependent source terms. We derive the model from basic thermodynamic principles, conservation laws for mass and momentum and constitutive assumptions. Using the method of formal matched asymptotics, we relate our diffuse interface model with free boundary problems for tumour growth that have been studied earlier. For the basic model, we show the existence of weak solutions under suitable assumptions on the source terms and the potential by using a Galerkin method, energy estimates and compactness arguments. If the velocity satisfies a no-slip boundary condition and is divergence free, we can establish the existence of weak solutions for degenerate mobilities and singular potentials. From a modelling point of view, it seems to be more appropriate to describe the nutrient evolution by a so-called quasi-static equation of reaction-diffusion type. For this model we establish existence of both weak and strong solutions for regular potentials and a continuous dependence result yields the uniqueness of weak solutions and thus the model is well-posed. These results build the basis to study an optimal control problem where the control acts as a cytotoxic drug. Moreover, we rigorously prove the zero viscosity limit in two and three space dimensions which allows us to relate the Cahn–Hilliard–Brinkman model with Cahn–Hilliard–Darcy models which have been studied earlier. Finally, we also analyse the model with quasi-static nutrients and classical singular potentials like the logarithmic and double-obstacle potential which enforce the phase field to stay in the physical relevant range. Under suitable assumptions on the source terms, we can establish the existence of weak solutions for these kinds of potentials

Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn-Hilliard-Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn-Hilliard inpainting model with singular potentials

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

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

Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation

In this paper, we study 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 coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable and the chemical potential, Robin boundary conditions for the nutrient and a no-friction boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the variables of the state equation during the evolution and the distribution of tumor cells at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish first-order necessary optimality conditions for the optimal control problem

On a Cahn--Hilliard--Brinkman Model for Tumor Growth and Its Singular Limits

In this work, we study a model consisting of a Cahn-Hilliard-type equation for the concentration of tumor cells coupled to a reaction-diffusion-type equation for the nutrient density and a Brinkman-type equation for the velocity. We equip the system with a Neumann boundary condition for the tumor cell variable and the chemical potential, a Robin-type boundary condition for the nutrient, and a "no-friction" boundary condition for the velocity, which allows us to consider solution-dependent source terms. Well-posedness of the model as well as existence of strong solutions will be established for a broad class of potentials. We will show that in the singular limit of vanishing viscosities we recover a Darcy-type system related to Cahn-Hilliard-Darcy-type models for tumor growth which have been studied earlier. An asymptotic limit will show that the results are also valid in the case of Dirichlet boundary conditions for the nutrient

Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn-Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn-Hilliard-Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting. (C) 2018 Elsevier Inc. All rights reserved

Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials