Free Boundary Problem Modeling Tumor Research Articles

Analysis of a radial free boundary tumor model with time-dependent absorption efficiency

We investigate an n-dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β(t) is considered. The system consists of a reaction-diffusion equation for the nutrient u(r,t) with nonlinear consumption and an integro-differential equation for the tumor radius R(t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β(t). We prove that for any sufficiently small c>0, the tumor radius R(t) will remain bounded or shrink to zero when β(t) is uniformly bounded or tends to zero, respectively; and if β(t)→β⁎, the solution (u,R) will converge to the unique steady state (u⁎,R⁎) associating with β⁎. We also give examples of R(t) blowing up as t→∞ even if β(t)→0 when c is not small.

Linear stability analysis for the free boundary problem modeling tumor growth with angiogenesis in the presence of inhibitors

In this paper, we study the linear stability of the free boundary problem for tumors. The model is a coupled system of PDE with Robin boundary, which involves concentration of nutrients, concentration of inhibitor and pressure. The presence of inhibitor in this model affects the diffusion of nutrient. We establish the existence and uniqueness of the radially-symmetric solution (us,vs,ps,Rs). We further prove that there exists a threshold value μ⁎ such that (us,vs,ps,Rs) is linearly stable under non-radially symmetric perturbations for μ∈(0,μ⁎) and linearly unstable for μ>μ⁎.

The effect of time delay in regulatory apoptosis on a tumor model with angiogenesis

In this paper, we consider a free boundary problem modeling tumor growth with angiogenesis and time delay in regulatory apoptosis. Two factors are introduced to cause tumor cell death: one is the apoptosis because of exceeding the natural lifespan, the other is the regulatory apoptosis with time delay. The existence, uniqueness and asymptotic behavior of solutions are investigated. The results show that the stabilities of stationary solutions are dramatically affected by such time delay in certain cases, and Hopf bifurcation occurs at some threshold values of time delay. Some numerical simulation results are incorporated supporting the analytical results. The impact of angiogenesis on the Hopf bifurcation is also numerically discussed. 2020 Mathematics subject classification35R35, 35B35, 35K57.

Linear stability analysis for a free boundary problem modeling tumor growth in the presence of high-molecular mass drugs

In this paper, we consider a free boundary problem modeling tumor growth in the presence of drugs with high molecular mass. This model is an improvement of a similar model proposed by Byrne and Chaplain ((1995) [1]). The novelty in our model is to better account for the fast diffusion coefficient of nutrients and the slow diffusion coefficient of drugs with high molecular mass, which is more realistic. The introduction of these terms, however, makes the analysis of linear stability more challenging in the spectral analysis of the linearized operator. A significant challenge is the lack of explicit expressions for their locations. We shall give a complete classification of parameters λσ, λβ, τ, σ‾, β‾ and σ˜ under which either (i) there exist exact one radially symmetric stationary solution (σS,βS,pS) with the boundary r=R2, or (ii) exact two solutions (σS,βS,pS) with the boundary r=R1 or r=R2 (R1<R2), or (iii) no solutions. We further show that for all γμ>0, (σS,βS,pS,R1) is linearly unstable under non-radially symmetric perturbations, while there exists a threshold value c⁎⩾0 such that (σS,βS,pS,R2) is linearly stable for γμ>c⁎ and linearly unstable for γμ∈(0,c⁎). As a biological implication, we discuss the positive and negative effects of drugs with high molecular mass on tumor growth and get a high risk region for drugs. Our results provide a strategy for tumor treatment.

Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays

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.

The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients

We study a free boundary problem modeling tumor growth with a T-periodic supply Φ(t) of external nutrients. The model contains two parameters μ and σ˜. We first show that (i) zero radially symmetric solution is globally stable if and only if σ˜⩾1T∫0TΦ(t)dt; (ii) If σ˜<1T∫0TΦ(t)dt, then there exists a unique radially symmetric positive solution σ∗(r,t),p∗(r,t),R∗(t) with period T and it is a global attractor of all positive radially symmetric solutions for all μ>0. These results are a perfect answer to open problems in Bai and Xu [Pac. J. Appl. Math. 2013(5), 217–223]. Then, considering non-radially symmetric perturbations, we prove that there exists a constant μ∗>0 such that σ∗(r,t),p∗(r,t),R∗(t) is linearly stable for μ<μ∗ and linearly unstable for μ>μ∗.

Symmetry‐breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation

In this paper, we consider bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation. In which, the vasculature supplies nutrients to the tumor at a rate α, so that holds on the boundary. For each α, we first establish the existence and uniqueness of radially symmetric stationary solutions, then prove that there exist a positive integer n∗∗ and a sequence (μ/γ )n such that symmetry‐breaking stationary solutions bifurcate from the radially symmetric one for every (μ/γ )n (even n ≥ n∗∗), where μ and γ denote the proliferation rate and the cell‐to‐cell adhesiveness, respectively. Particularly, for small α, we show that (μ/γ )2, (μ/γ )4, (μ/γ )6, (μ/γ )8, … are all bifurcation points, which includes the smallest bifurcation point that is the most significant one biologically; moreover, (μ/γ )n is monotone decreasing with respect to α for every n ≥ 2, which implies that inhibiting angiogenesis has a positive impact in limiting the ability of the invasion of tumors, at least during very early stages of angiogenesis.

A qualitative analysis of a free boundary problem modeling tumor growth with angiogenesis

In this paper we consider a free boundary problem modeling tumor growth with angiogenesis. The model is a free boundary problem of a system of partial differential equations. With Robin boundary, the model contains an ordinary differential equation describing the radius of tumor cell and two parabolic equations describing the evolution of nutrient concentration and inhibitor concentration, respectively. We study the number of the stationary solution of above problem, the existence and uniqueness of local solution and global solution, and the asymptotic behavior of the solution.

Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ=σ(x) and the inner pressure p=p(x). Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20]. In this paper we study existence of nonradial solutions by using the bifurcation method. Let {γk}k=2∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k⁎⩾2 such that in the two dimension case for any k⩾k⁎, γk is a bifurcation point, and in the three dimension case for any even k⩾k⁎, γk is also a bifurcation point.

Boundary element method for a free third boundary problem modeling tumor growth with spectral accuracy

By boundary element method, we present a numerical iterative process for solving a free third boundary problem modeling tumor growth with spectral accuracy. The piecewise quadratic curves are fitted to maintain local smoothness of the boundary at every node. The double-layer and single-layer potentials with weakly singular kernels are evaluated with spectral accuracy. The method of characteristics is employed to transform interfacial velocity PDE into discrete ODEs. The numerical integral formula for weakly singular operator with logarithmic singularity is deduced and the convergence and error are presented. The nonradially symmetric solutions of the free boundary problem on a perturbed boundary are provided to test the accuracy and effectiveness of the numerical method.

Bifurcation for a free boundary problem modeling tumor growth with ECM and MDE interactions

We study a free boundary problem modeling solid tumor growth. The simplified model contains a parameter λ. Different from previous works on bifurcation analysis, a new ingredient of the present paper is that the influence of the extracellular matrix (ECM) and matrix degrading enzymes (MDE) interactions is included in the model. We first show that for each λ>0, there exists a unique radially symmetric stationary solution with radius r=RS. Then we prove that there exist a positive integer n∗ and a sequence of λn(n>n∗) for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric one. In particular, we discover that these λn are larger than those λ˜n previously known when the effects of ECM and MDE are not considered in the model.

Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation

In this paper we study a free boundary problem modeling the growth of solid tumor spheroid. It consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor, respectively. The new feature is that nutrient concentration on the boundary is less than external supply due to a Gibbs–Thomson relation and the problem has two radial stationary solutions, which differs from widely studied tumor spheroid model with surface tension effect. We first establish local well-posedness by using a functional approach based on Fourier multiplier method and analytic semigroup theory. Then we investigate stability of each radial stationary solution. By employing a generalized principle of linearized stability, we prove that the radial stationary solution with a smaller radius is always unstable, and there exists a positive threshold value γ⁎ of cell-to-cell adhesiveness γ, such that the radial stationary solution with a larger radius is asymptotically stable for γ>γ⁎, and unstable for 0<γ<γ⁎.

Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs–Thomson relation

In this paper we study a free boundary problem modeling tumor growth. The model consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumors, respectively, and a first-order partial differential equation governing the free boundary, on which a Gibbs–Thomson relation is taken into account. We first show that the problem may have none, one or two radial stationary solutions depending on model parameters. Then by bifurcation analysis we show that there exist infinite many branches of non-radial stationary solutions bifurcating from given radial stationary solution. The result implies that cell-to-cell adhesiveness is the key parameter which plays a crucial role on tumor invasion.

Bifurcation for a free boundary problem modeling tumor growth with inhibitors

This paper deals with a free boundary problem modeling tumor growth with inhibitors. This problem has a unique radially symmetric stationary solution with radius r=Rs. The tumor aggressiveness is modeled by a positive tumor aggressiveness parameter μ. It is shown that there exist a positive integer m∗∗∈R and a sequence of μm, such that for eachμm(m>m∗∗), symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions.

A free boundary problem modeling tumor growth with different chemotactic responses and random motions for various cell types

This paper deals with a mathematical model describing the growth of an avascular tumor. Tumor cells are assumed to consist of two cell types: the proliferative cells and the quiescent non-dividing cells, which might have different chemotactic responses to gradients of extracellular nutrients (oxygen, glucose, etc.). The model is a free boundary problem for a coupled system of nonlinear convection–diffusion–reaction equations, where the diffusion rates of the two cell types might be different and the free boundary is the outer boundary of the tumor. We prove the global solvability of this model by a fixed point argument and some a priori estimate techniques. This work extends a recent result by the second author [Y. Tao, A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids, J. Differential Equations 247 (2009), 49–68] on a model with the same diffusion rate for two cell types to a new one on a model with different cell diffusion rates.

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ and the cell-to-cell adhesiveness γ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.

Bifurcation analysis for a free boundary problem modeling tumor growth

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 smooth stationary solutions of the problem which are not radially symmetric.

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.

Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation

In this paper, a free boundary problem modeling tumor growth under the direct effect of an inhibitor with time delays is studied. The delays represent the time taken for cells to undergo mitosis. Nonnegativity of solutions, the existence of the stationary solutions and their asymptomatic behavior are studied. The results show that when the inhibitor is large, and the initial tumor is not too large, the tumor will disappear. If however, the initial tumor is large enough, then it will grow. When the inhibitor is not as large, the growth of the tumor is determined by the size of the nutrients and whether the initial tumor is large or not. When the inhibitor is smaller, the tumor will grow no matter if the initial tumor is large or not.

Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays

In this paper, a free boundary problem modeling tumor growth with two discrete delays is studied. The delays respectively represents the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis. We show the influence of time delays on the Hopf bifurcation when one of delays as a bifurcation parameter.