On a nonlinear model for tumour growth with drug application

Abstract

We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Brinkman's equation. The domain occupied by the tumor in this setting is a growing continuum Ω with boundary ∂Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data. This article is part of a research program whose aim is the investigation of the effect of drug application in tumor growth.

A convergent explicit finite difference scheme for a mechanical model for tumor growth

“…But in contrast to the present nonlinear system, the transport equation for the evolution of cancerous cells in [10,9] has a source term which is linear with respect to cell density.…”

On the vanishing viscosity approximation of a nonlinear model for tumor growth

“…Proof. We present here the main ingredients of the proof of the Theorem 3.1 presented in [4] (in order to simplify the notations we drop the index µ).…”

“…This form of boundary penalty approximation appeared by Courant in [2], in the context of slip conditions for stationary incompressible fluids by Stokes and Carrey in [14], and more recently in a series of articles (cf. [3], [5], [4], [6], [7]). More specifically, the boundary condition (1.13) is treated as a weakly enforced constraint, in the sense that the variational (weak) formulation of the Brinkman equation is supplemented by a singular forcing term • Keeping ε and ω fixed, we solve the modified problem in a (bounded) reference domain B ⊂ R 3 chosen in such way that Ω τ ⊂ B for any τ ≥ 0 with the aid of a Faedo-Galerkin approximation.…”

“…More specifically, the boundary condition (1.13) is treated as a weakly enforced constraint, in the sense that the variational (weak) formulation of the Brinkman equation is supplemented by a singular forcing term • Keeping ε and ω fixed, we solve the modified problem in a (bounded) reference domain B ⊂ R 3 chosen in such way that Ω τ ⊂ B for any τ ≥ 0 with the aid of a Faedo-Galerkin approximation. We refer the reader to [4] for the details. The solution {P ω,ε , Q ω,ε , D ω,ε , v ω,ε } constructed satisfy the following uniform bounds: 6) this entails that for any p ≥ 1…”

On a Nonlinear Model for Tumor Growth in a Cellular Medium

“…In [12], the same authors treat a related nonlinear model and discuss the effect of drug application on tumor growth. The main contribution of the present article to the existing theory can be characterized as follows:…”