Analysis of a diffuse interface model of multispecies tumor growth

Abstract

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins-Daruud et al. in [25]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction ϕ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

Optimal distributed control of a diffuse interface model of tumor growth

“…The first contributions in this direction dealt with the case where the nutrient is neglected, which then leads to the so-called Cahn-Hilliard-Hele-Shaw system (see [3,31,34,41,42]). Only very recently, in the paper [17], the authors proved existence of weak solutions and some rigorous sharp interface limit for a model introduced in [6] (cf. also [15,16,18,33,43]), where both velocities (satisfying a Darcy law with Korteweg term) and multispecies tumor fractions, as well as the nutrient evolutions are taken into account.…”

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

“…Note in particular that only µ T is needed to drive the evolution of ϕ i , 2 ≤ i ≤ L. However, the mathematical treatment of these types of models is difficult due to the fact that the equation for ϕ i is now a transport equation with a high order source term div (M (ϕ i )∇µ T ), and the natural energy identity of the model does not appear to yield useful a priori estimates for ϕ i . In the case that the mobility M is a constant, the existence of a weak solution for the model of [9] has been studied by Dai et al in [15]. The specific forms of the source terms U (ϕ, σ) and S(ϕ, σ) will depend on the specific situation we want to model.…”

Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation

“…The basis of phase-field tumor models, i.e., a Cahn-Hilliard equation for the tumor volume fraction φ T and a reaction-diffusion equation for the nutrient concentration φ σ , has been proposed in [31] and has been extended to general multiphase models in [26]. The existence analysis for this model is provided by Garcke et al [10,24] and additionally, several flow models for the velocity field of the mixture have been proposed and analyzed, e.g., flow models by Darcy [11,23,25,27,33], Brinkman [16,17], Darcy-Forchheimer-Brinkman [22] and Navier-Stokes [36].…”