We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.
In this paper, we study a tumor growth model with nutrients. The contact inhibition for the tumor cells, presented in the model, results in the evolution of a congested tumor patch. We study the regularity of the tumor patch as the nutrients' diffusion strength $D$ diminishes. In particular, we show that for small $D>0$ the boundary of the tumor patch stays in a small neighborhood of the smooth tumor patch boundary obtained with $D=0$, uniformly with respect to the Hausdorff distance.
In this contribution, the non-local, integro-partial differential system of equations proposed by Hillen et al. (Bull. Math. Biol. 75, 2013, no.1, 161-184) to account for the the tumor growth paradox (or the observation that some incomplete cancer treatments may enhance tumor growth) is reviewed. It is shown that when cancer cells' diffusion effects and Neumann boundary conditions are taken into consideration, the same paradoxical tumor growth emerges.
In this paper we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the contact inhibition among the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics. In particular, we provide contraction estimates, exponential rate of asymptotic convergence, and boundary regularity of the tumor patch. These results are in sharp contrast to the models either with nutrient diffusion or with death rate in tumor cells.
We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a…
We study a model introduced by Perthame and Vauchelet that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.
We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. The model features two moving interfaces separating the tumor, the normal tissue, and the exterior vacuum. We prove local-in-time existence and uniqueness of strong solutions for their evolution starting from a nearly radial initial configuration. It is assumed that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fing…
We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration. We prove that, under physically motivated assumptions on parameters and data, the corresponding initial-boundary value problem generates a dissipative dynamical system that admits the global attractor in a proper phase space.
In this paper we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter $ε$, representing the interface thickness between the tumorous and non tumorous cells, tends to zero. More in particular, we analyze here a gradient-flow type model arising from a modification of the recently introduced model for tumor growth dynamics in [A. Hawkins-Daruud, K. G. van der Zee, J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3--24.] (cf. a…
The present paper deals with a free boundary problem modeling the growth process of necrotic multi-layer tumors. We prove the existence of flat stationary solutions and determine the linearization of our model at such an equilibrium. Finally, we compute the solutions of the stationary linearized problem and comment on bifurcation.
We introduce here a new diffuse interface thermodynamically consistent non-isothermal model for tumor growth in presence of a nutrient in a domain $Ω\subset \mathbb{R}^3$. In particular our system describes the growth of a tumor surrounded by healthy tissues, taking into account changes of temperature, proliferation of cells, nutrient consumption and apoptosis. Our aim consists in proving an existence result for weak entropy solutions to our model.
In this paper we study a mathematical model for the growth of nonnecrotic solid tumor. The tumor is assumed to be radially symmetric and its radius R(t) is an unknown function of time t as tumor growth, and the model is in the form of a free boundary problem. The feature of the model is that a Gibbs-Thomson relation is taken into account, which resulting an interesting phenomenon that there exist two stationary solutions (depending on the model parameters). The global existence and uniqueness of solution are established. By denoting c the ratio of the diffusion time scale to the tumor doubling…
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G. van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction coupled to a reaction-diffusion equation for a variable representing the nutrient-rich extracellular water volume fraction. The distributed control monitors as a right-hand side the reaction-diffusion e…
In this paper we deal with a free boundary problem modeling the growth of nonnecrotic tumors.The tumor is treated as an incompressible fluid, the tissue elasticity is neglected and no chemical inhibitor species are present. We re-express the mathematical model as an operator equation and by using a bifurcation argument we prove that there exist stationary solutions of the problem which are not radially symmetric.
This paper is concerned with a multi-dimensional free boundary problem modeling the growth of a tumor with two species of cells: proliferating cells and quiescent cells. This free boundary problem has a unique radial stationary solution. By using the Fourier expansion of functions on unit sphere via spherical harmonics, we establish some decay estimates for the solution of the linearized system of this tumor model at the radial stationary solution, so that proving that the radial stationary solution is linearly asymptotically stable when neglecting translations.
In this paper, we conduct a thorough mathematical analysis of a tumor growth model with treatments. The model is a system describing the evolution of metastatic tumors and the number of cells present in a primary tumor. The former evolution is described by a linear transport equation and the latter by an ordinary differential equation of Gompertzian type. The two dynamics are coupled through a nonlocal boundary condition that takes into account the tumor colonization rate. We prove an existence result where the main difficulty is to deal with the coupling and to take into account the time disc…
We analyze a system of cross-diffusion equations that models the growth of an avascular-tumor spheroid. The model incorporates two nonlinear diffusion effects, degeneracy type and super diffusion. We prove the global existence of weak solutions and justify the convergence towards the free boundary problem of the Hele-Shaw type when the pressure gets stiff. We also investigate the convergence rate of the solutions in $L^1-$Lebesgue spaces.
We consider the asymptotic limit of a diffuse interface model for tumor-growth when a parameter $\varepsilon$ proportional to the thickness of the diffuse interface goes to zero. An approximate solution which shows explicitly the behavior of the true solution for small $\varepsilon$ will be constructed by using matched expansion method. Based on the energy method, a spectral condition in particular, we establish a smallness estimate of the difference between the approximate solution and the true solution.
We discuss the sharp interface limit of a diffuse interface model for a coupled Cahn-Hilliard--Darcy system that models tumor growth when a certain parameter $\varepsilon>0$, related to the interface thickness, tends to zero. In particular, we prove that weak solutions to the related initial boundary value problem tend to varifold solutions of a corresponding sharp interface model when $\varepsilon$ goes to zero.
In this paper we study a model describing the growth of necrotic tumors in different regimes of vascularisation. The tumor consists of a necrotic core of death cells and a surrounding nonnecrotic shell. The corresponding mathematical formulation is a moving boundary problem where both boundaries delimiting the nonnecrotic shell are allowed to evolve in time.We determine all radially symmetric stationary solutions of the problem and reduce the moving boundary problem into a nonlinear evolution. Parabolic theory provides us the perfect context in order to show local well-posed of the problem for…
Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions.
In this paper we study a free boundary problem for the growth of multi-layer tumors in necrotic phase. The tumor region is strip-like and divided into necrotic region and proliferating region with two free boundaries. The upper free boundary is tumor surface and governed by a Stefan condition. The lower free boundary is the interface separating necrotic region from proliferating region, its evolution is implicit and intrinsically governed by an obstacle problem. We prove that the problem has a unique flat stationary solution, and there exists a positive constant $γ_*$, such that the flat stati…
A two-dimensional free boundary model for the growth of multi-layer tumors has been proposed in [S. Cui, J. Escher: ARMA 191 (2009) 173-193] where the authors derive well-posedness in a functional analytic setting, the stationary solutions and their asymptotic stability. In this note, we consider once again the cancer growth model of [S. Cui, J. Escher: ARMA 191 (2009) 173-193] but coping without periodic boundary conditions for the $x$ variable which is reasonable from the point of view of modeling. The paper points out that the generalized model allows for the same stationary solutions as co…
This paper concerns a free boundary problem modeling tumor growth with angiogenesis and two time delays. The two delays represent the time taken for cells to undergo mitosis and modify the rate of cell loss because of apoptosis, respectively. We study the stability of stationary solutions and find that Hopf bifurcation occurs under some conditions, which extends the results of Xu. Furthermore, numerical simulations are performed to investigate the relationship among the rate of angiogenesis, two time delays and Hopf bifurcation.
We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hoelder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties o…
A considerable number of research works has been devoted to the study of tumor models. Several biophysical factors, such as cell proliferation, apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have an impact on the complicated biological system of tumors. An indicator of the aggressiveness of tumor development is the instability of the shape of the tumor boundary. Complex patterns of tumor morphology have been explored by Lu, Min-Jhe et al. [Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis, Journal of Computational Physics 459 (…
In this paper, we present a rigorous mathematical analysis of a free boundary problem modeling the growth of a vascular solid tumor with a necrotic core. If the vascular system supplies the nutrient concentration $σ$ to the tumor at a rate $β$, then $\frac{\partialσ}{\partial\bf n}+β(σ-\barσ)=0$ holds on the tumor boundary, where $\bf n$ is the unit outward normal to the boundary and $\barσ$ is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate $μ$. We show that for any given $ρ>0$, there exists a unique $R\in(ρ,\infty)$ such that the…
A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observatio…
We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $γ_*>0$ for the surface tension coefficient $γ$, such that in the case $γ>γ_*$ this radia…
We consider weak solutions to a problem modeling tumor growth. Under certain conditions on the initial data, solutions can be obtained by passing to the stiff (incompressible) limit in a porous medium type problem with a Lotka-Volterra source term describing the evolution of the number density of cancerous cells. We prove that such limit solutions solve a free boundary problem of Hele-Shaw type. We also obtain regularity properties, both for the solution and for its free boundary. The main new difficulty arises from the competition between the growth due to the source, which keeps the initial …
We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associat…
We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the "tumor" is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the "waste" and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media…
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.
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 co…
In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn-Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in the preprint arXiv:1401.5943, letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interact…
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 Darcy's equation. The domain occupied by the tumor in this setting is a growing conti…
In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describ…
Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the 'stiff pressure law' limit. Even though vis…
In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in {\em Bull. Math. Biol.} (2001). By using a functional approach and the $C_0$ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman ({\em Trans. Amer. Math. Soc.}, 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear …
Mathematical models that describe the tumor growth process have been formulated by several authors in order to understand how cancer develops and to develop new treatment approaches. In this study, it is aimed to investigate the long-time behavior of the two-phase diffuse-interface model, which was proposed in [11] to model a tumor tissue as a mixture of cancerous and healthy cells. Up to now, studies on the long-time behavior of this model have neglected chemotaxis and active transport, which have a significant effect on tumor growth. In this research, our main aim is to study this model with…
In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn-Hilliard type equation for the phase variable with a reaction-diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn-Hilliar…
In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth o…
We study a free boundary problem modeling multi-layer tumor growth with a small time delay $τ$, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper we establish the well-posedness of the problem, namely, first we prove that there exists a unique flat stationary solution $…
The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness 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.
We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asym…
We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New diffi…
