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

Abstract

We investigate the long-time dynamics and optimal control problem of a 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 source and prove the convergence of any global solution to a single equilibrium as t → +∞. Then we consider the "finite-time treatment" of a tumor, which corresponds to an optimal control problem. Here we also 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 φ Ω . By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we see that φ Ω can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed.

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

“…Moreover, the investigation has to be restricted to regular potentials with polynomial growth, so that the double-obstacle one is not allowed. In that regards, we also point out the recent [4], where the authors extend the analysis of [9] employing a time-dependent cost functional which, in addition, penalizes the long-time treatments and the large mass of the tumor at the end of the medication. Next, we point out [34], where the author, adding the two relaxation terms α∂ t µ, β∂ t ϕ generalizes the optimal control problem [9] extending the analysis to the case of singular and regular double-well potentials.…”

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

“…Moreover, we want to mention the papers [23] and [8] where optimal control problems of treatment time are studied. In [23] the control enters the phase field equation in the same way as ours whereas in [8] it enters the nutrient equation. Although the nutrient equation in both papers is non-stationary, some of the major difficulties do not occur since the velocity is assumed to be negligible (v = 0).…”

A Distributed Control Problem for a Fractional Tumor Growth Model

“…The other class of models, to which the model considered here belongs, neglects the velocity. Typical contributions in this direction were given in [4,6,[8][9][10]26], to name just a few.…”

“…We remark at this place that it would be desirable to minimize the duration, i.e., the time T > 0, of the medical treatment as well, in order to prevent that the tumor cells develop a resistance against the drug. However, such an approach, which was possible (see, e.g., [4]) in the special case when A 2ρ = B 2σ = C 2τ = −∆, becomes very complicated in the situation considered here and was therefore not included.…”