Free Boundary Problem Modeling Research Articles

Analysis and simulations of a free boundary problem modelling phototrophic granular biofilms

Analysis and simulations of a free boundary problem modelling phototrophic granular biofilms

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Symmetry-breaking combined latitude-longitude bifurcations for a free boundary problem modeling small plaques

Bifurcation analysis of a free boundary problem modeling multilayered tumor growth with time delay

In this paper, we consider a free boundary problem modeling 3-dimensional multilayered tumor growth with time delay. While the time delay and the spatial displacement are biologically very reasonable, their introduction forces the system to be non-local, rendering a system without any possibility of explicit expressions for pressure and the displacement function. These are the challenges for bifurcation analysis. In comparison to the 2-dimensional case, the kernel of the linearized operator is no longer one-dimensional for our 3-dimensional multilayered tumor model. In particular, some monotonicity properties enjoyed in Friedman and Hu (2008) [6] and Zhao and Hu (2020) [26] are no longer valid here, which adds complexity to the kernel of the linearized operator. The novelty of this paper is to explore periodicity and symmetry in order to develop an effective method for the bifurcation in this case. We not only give bifurcation results for all eigenvalues μj, but also show the existence of a plethora of bifurcation solutions.

Qualitative properties of solutions to a mass-conserving free boundary problem modeling cell polarization

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. We have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at t = 0. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.

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.

Analysis of a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients

In this paper, we study a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients. The mathematical model is a free boundary problem since the external radius of the tumor denoted by R(t) changes with time. The characteristic of this model is the consideration of both angiogenesis and periodic external nutrient supply. The cells inside the tumor absorb nutrient u(r,t) through blood vessels and attracts blood vessels at a rate proportional to α. Thus on the boundary, we have ur(r,t)+α(u(r,t)−ψ(t))=0,r=R(t),t>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} u_{r}(r,t)+\\alpha \\bigl(u(r,t)-\\psi (t)\\bigr)=0,\\quad r=R(t),t>0, \\end{aligned}$$ \\end{document} where psi (t) is the nutrient concentration provided externally. Considering that the nutrient provided externally to the tumor are generally provided periodically, in this paper, we assume that psi (t) is a periodic function. Sufficient conditions for a tumor to disappear are given. We investigate the existence, uniqueness, and stability of solutions. The results show that when the nutrient concentration exceeds a certain value and c is sufficiently small, the solutions of the model can be arbitrarily close to the unique periodic function as trightarrowinfty.

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⁎.

Periodic solution for a free boundary problem modeling small plaques

Periodic solution for a free boundary problem modeling small plaques

Symmetry-breaking longitude bifurcations for a free boundary problem modeling small plaques in three dimensions.

Atherosclerosis, one of the leading causes of death in USA and worldwide, begins with a lesion in the intima of the arterial wall, allowing LDL to penetrate into the intima where they are oxidized. The immune system considers these oxidized LDL as a dangerous substance and tasks the macrophages to attack them; incapacitated macrophages become foam cells and leads to the formation of a plaque. As the plaque continues to grow, it progressively restricts the blood flow, possibly triggering heart attack or stroke. Because the blood vessels tend to be circular, two-space dimensional cross section model is a good approximation, and the two-space dimensional models are studied in Friedman et al. (J Differ Equ 259(4):1227-1255, 2015) and Zhao and Hu (J Differ Equ 288:250-287, 2021). It is interesting to see whether a true three-space dimensional stationary solution can be developed. We shall establish a three-space dimensional stationary solution for the mathematical model of the initiation and development of atherosclerosis which involves LDL and HDL cholesterols, macrophages and foam cells. The model is a highly nonlinear and coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions which bifurcate from the annular stationary solution in the longitude direction.

Bifurcation for a free-boundary problem modeling small plaques with reverse cholesterol transport

In this paper, we rigorously analyze a free boundary model of plaque growth under the influence of reverse cholesterol transport (RCT). We will analyze a model with a highly nonlinear and mutually coupled PDE system of plaque growth dynamics, including LDL and HDL cholesterol, M1 inflammatory and M2 anti-inflammatory macrophages, and foam cells. This free boundary problem is associated with two different types of surfaces: a vascular surface with the boundary set to r=1 and the interface Γ(t) between the plaque and blood flow. For this model, the explicit stationary solutions are nonexistent due to the highly nonlinear nature of the problem. We shall first prove the existence and uniqueness of the radially symmetric stationary solution. Then, we shall combine parameters into a single bifurcation parameter μ, and establish the existence of a sequence μn, such that for each μn(n≥2), the symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions for sufficiently small plaques.

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.

The linear stability for a free boundary problem modeling multilayer tumor growth with time delay

We study a free boundary problem modeling multilayer tumor growth with a small time delay , 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 for all . The stability of this stationary solution should depend on the tumor aggressiveness constant . It is also unrealistic to expect the perturbation to be flat. We show that, under non‐flat perturbations, there exists a threshold such that is linearly stable if and linearly unstable if . Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.

Dynamics of a free boundary problem modelling species invasion with impulsive harvesting

<p style='text-indent:20px;'>To understand the role of impulsive harvesting in dynamics of the invasive species, we explore an impulsive logistic equation with free boundaries. The criteria whether the species spreads or vanishes are given, and some sufficient conditions based on threshold values are established. We then discuss the spreading speeds of moving fronts when the species spreads. Our numerical simulations reveal that impulsive harvesting can reduce the spreading speed of the species, and a large impulsive harvesting is unfavorable for persistence of the species. Moreover, when impulsive harvesting is moderate, the species occurs spreading or vanishing depending on its expanding capability or initial number, that is, the species will die out with a small expanding capability or small initial number and spread with a large expanding capability. <br/><br/>Note: KYCX22_3446 is added in Acknowledgments.</p>

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.

Error analysis of finite difference/collocation method for the nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery

The main target of this paper is to present a new and efficient method to solve a nonlinear free boundary mathematical model of atherosclerosis. This model consists of three parabolic, one elliptic and one ordinary differential equations that are coupled together and describes the growth of a plaque in the artery. We start our discussion by using the front fixing method to fix the free domain and simplify the model by changing the mixed boundary condition to a Neumann one by applying suitable changes of variables. Then, after employing a finite difference using the second-order backward difference formula (BDF2) and the collocation method on this model, we prove the stability and convergence of methods. Finally, some numerical results are considered to show the efficiency of the method.

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.

Asymptotic stability for a free boundary tumor model with angiogenesis

In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value μ⁎ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for 0<μ<μ⁎ and linearly unstable for μ>μ⁎. In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for 0<μ<μ⁎ when neglecting translations.

Bifurcation for a free-boundary tumor model with extracellular matrix and matrix degrading enzymes

We study a free boundary problem modeling solid tumor growth with ECM and MDE interactions. The production rate of MDE by tumor cells is a nonlinear function depending on nutrients and MDE. This nonlinear term is more complicated. For this model, we can not give the explicit expressions or integral expressions for MDE terms while we only get the existence and uniqueness. At first, we show the existence and the uniqueness of radially symmetric stationary solutions. Then the existence of symmetry-breaking solutions bifurcating from the radially symmetric stationary solutions is obtained.