On the long time behavior of a tumor growth model (original) (raw)

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

Applied Mathematics & Optimization, 2019

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.

On a diffuse interface model of tumour growth

European Journal of Applied Mathematics, 2015

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruudet al.((2013)J. Math. Biol.671457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potentialFandpsatisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided thatpsatisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and tha...

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

Mathematical Methods in the Applied Sciences, 2017

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dyna...

On a Cahn-Hilliard type phase field system related to tumor growth

Discrete and Continuous Dynamical Systems, 2014

The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove a well-posedness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.

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

Mathematical Models and Methods in Applied Sciences, 2014

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.

Analysis of a mixture model of tumor growth

European Journal of Applied Mathematics, 2013

We study an initial-boundary value problem (IBVP) for a coupled Cahn-Hilliard-Hele-Shaw system that models tumor growth. For large initial data with finite energy, we prove global (local resp.) existence, uniqueness, higher order spatial regularity and Gevrey spatial regularity of strong solutions to the IBVP in 2D (3D resp.). Asymptotically in time, we show that the solution converges to a constant state exponentially fast as time tends to infinity under certain assumptions.

Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials

arXiv (Cornell University), 2022

We analyze a phase field model for tumor growth consisting of a Cahn-Hilliard-Brinkman system, ruling the evolution of the tumor mass, coupled with an advectionreaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn-Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

A MODEL FOR THE GROWTH OF TUMORS: STATIONARY SOLUTIONS

We prove the existence of stationary solutions for a nonlinear problem modelling the growth of a nonnecrotic spheroid tumor in absence of inhibitor agents. We assume that the rate of consumption of nutrients by the cells is greater than the rate of transference of nutrients from the vasculature and that this balance is an increasing function of the nutrient concentration. The proliferation rate considered depends on the nutrient concentration and is given by either an increasing function or one that assumes only a negative minimum value. Some bounds for the stabilizing radius also are presented.

Tumor growth dynamics with nutrient limitation and cell proliferation time delay

Discrete and Continuous Dynamical Systems - Series B

It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity

2006

In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix). Two crucial components of tissue invasion are (i) cancer cell proliferation, and (ii) over-expression and secretion of proteolytic enzymes by the cancer cells. The proteolytic enzymes are responsible for the degradation of the tissue, enabling the proliferating cancer cells to actively invade and migrate into the degraded tissue. Our model focuses on the role of nonlocal kinetic terms modelling competition for space and degradation. The model consists of a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. We first of all prove results concerning the local existence, uniqueness and regularity of solutions of our system of nonlinear PDEs. We then extend these results to prove global existence, uniqueness and regularity of the solutions. Using Green's functions, we transform our original nonlocal equations into a coupled system of parabolic and elliptic equations and we undertake a numerical analysis of this equivalent system, presenting computational simulation results from our model showing travelling waves of cancer cells, degrading, invading and replacing the tissue. Finally, concluding 1 2 Szymańska et al.