arXiv subjects

Bei Hu

Publications and source records attributed to Bei Hu.

12 recordsLinked to original sources

arXiv2022

Bifurcation analysis of a free boundary model of vascular tumor growth with a necrotic core and chemotaxis

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 (2022): 111153]. In this paper, we continue to carry out a bifurcation analysis on such a vascular tumor model with a controlled necrotic core and chemotaxis. This bifurcation analysis, to the parameter of cell proliferation, is built on the explicit formulas of radially symmetric steady-state solutions. By perturbing the tumor free boundary and establishing rigorous estimates of the free boundary system, %applying the Hanzawa transformation, we prove the existence of the bifurcation branches with Crandall-Rabinowitz theorem. The parameter of chemotaxis is found to influence the monotonicity of the bifurcation point as the mode $l$ increases both theoretically and numerically.

math.AP↗

arXiv2021

Two neural-network-based methods for solving obstacle problems

Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization of the obstacle problems. We rigorously prove the convergence of the two schemes and derive the convergence rates with the number of neurons $N$. In the simulations, we use two example problems (1-D & 2-D) to verify the convergence rate of the methods and the quality of the results.

math.NA↗

arXiv2021

The linear stability for a free boundary problem modeling multi-layer tumor growth with time delay

We study a free boundary problem modeling multi-layer tumor growth with a small time delay $\tau$, 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 $(\sigma_*, p_*, \rho_*, \xi_* )$ for all $\mu>0$. The stability of this stationary solution should depend on the tumor aggressiveness constant $\mu$. It is also unrealistic to expect the perturbation to be flat. We show that, under non-flat perturbations, there exists a threshold $\mu_*>0$ such that $(\sigma_*, p_*, \rho_*, \xi_*)$ is linearly stable if $\mu<\mu_*$ and linearly unstable if $\mu>\mu_*$. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.

math.AP↗

arXiv2020

On the first bifurcation point for a free boundary problem modeling small arterial plaque

Atherosclerosis occurs when plaque clogs the arteries. It is a leading cause of death in the United States and worldwide. In this paper, we study the bifurcation for a highly nonlinear and highly coupled PDE model of plaque formation in the early stage of atherosclerosis. The model involves LDL and HDL cholesterols, macrophage cells as well as foam cells, with the interface separating the plaque and blood flow region being a free boundary. We establish the first bifurcation point for the system corresponding to $n=1$ mode. The symmetry-breaking stationary solution studied in this paper might be helpful in understanding why there exists arterial plaque that is often accumulated more on one side of the artery than the other.

math.AP↗

arXiv2020

Convergence analysis of neural networks for solving a free boundary system

Free boundary problems deal with systems of partial differential equations, where the domain boundaries are apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problems either theoretically or numerically. In this paper, we develop a novel approach for solving a modified Hele-Shaw problem based on neural network discretization. The existence of the numerical solution with this discretization is established theoretically. We also numerically verify this approach by computing the symmetry-breaking solutions which are guided by the bifurcation analysis near the radially-symmetric branch. Moreover, we further verify the capability of this approach by computing some non-radially symmetric solutions which are not characterized by any theorems.

math.NA↗

arXiv2020

Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator \begin{equation*} -\Delta_{m}u=u^q|\nabla u|^p\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq1$, $m>1$ and $p,q\geq0$. The technique of Bernstein gradient estimates is ultilized to study the case $p<m$. Moreover, a Liouville-type theorem for supersolutions under subcritial range of exponents \begin{equation*} q(N-m)+p(N-1)<N(m-1) \end{equation*} is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type $-\Delta_m u = f(x,u,\nabla u)$, with $f$ satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved.

math.AP↗

arXiv2020

Bifurcation for a free boundary problem modeling a small arterial plaque

Atherosclerosis, hardening of the arteries, originates from small plaque in the arteries; it is a major cause of disability and premature death in the United States and worldwide. In this paper, we study the bifurcation of a highly nonlinear and highly coupled PDE model describing the growth of arterial plaque in the early stage of atherosclerosis. The model involves LDL and HDL cholesterols, macrophage cells as well as foam cells, with the interface separating the plaque and blood flow regions being a free boundary. We establish finite branches of symmetry-breaking stationary solutions which bifurcate from the radially symmetric solution. Since plaque in reality is unlikely to be strictly radially symmetric, our result would be useful to explain the asymmetric shapes of plaque.

math.AP↗

arXiv2020

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 $\mu^\ast$ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for $0<\mu<\mu^\ast$ and linearly unstable for $\mu>\mu^\ast$. 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 $0<\mu<\mu^\ast$ when neglecting translations.

math.AP↗

arXiv2019

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

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 $\sigma$ to the tumor at a rate $\beta$, then $\frac{\partial\sigma}{\partial\bf n}+\beta(\sigma-\bar\sigma)=0$ holds on the tumor boundary, where $\bf n$ is the unit outward normal to the boundary and $\bar\sigma$ is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate $\mu$. We show that for any given $\rho>0$, there exists a unique $R\in(\rho,\infty)$ such that the corresponding radially symmetric solution solves the steady-state necrotic tumor system with necrotic core boundary $r=\rho$ and outer boundary $r=R$; moreover, there exist a positive integer $n^{**}$ and a sequence of $\mu_n$, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solution for each $\mu_n$ (even $n\ge n^{**})$.

math.AP↗

arXiv2019

The Impact of Time Delay in a Tumor Model

In this paper we consider a free boundary tumor growth model with a time delay in cell proliferation and study how time delay affects the stability and the size of the tumor. The model is a coupled system of an elliptic equation, a parabolic equation and an ordinary differential equation. It incorporates the cell location under the presence of time delay, with the tumor boundary as a free boundary. A parameter $\mu$ in the model is proportional to the "aggressiveness" of the tumor. It is proved that there exists a unique classical radially symmetric stationary solution $(\sigma_*, p_*, R_*)$ which is stable for any $\mu > 0$ with respect to all radially symmetric perturbations (c.f. \cite{delay1}). However, under non-radially symmetric perturbations, we prove that there exists a critical value $\mu_*$, such that if $\mu<\mu_*$ then the stationary solution $(\sigma_*, p_*, R_*)$ is linearly stable; whereas if $\mu>\mu_*$ the stationary solution is unstable. It is actually unrealistic to expect the problem to be stable for large tumor aggressiveness parameter, therefore our result is more reasonable. Furthermore, we established that adding the time delay in the model would result in a larger stationary tumor, and if the tumor aggressiveness parameter is larger, then the time delay would have a greater impact on the size of the tumor.

math.AP↗

arXiv2018

A parabolic-hyperbolic system modeling the growth of a tumor

In this paper, we consider a model with tumor microenvironment involving nutrient density, extracellular matrix and matrix-degrading enzymes, which satisfy a coupled system of PDEs with a free boundary. For this coupled parabolic-hyperbolic free boundary problem, we prove that there is a unique radially symmetric solution globally in time. The stationary problem involves an ODE system which is transformed into a singular integro-differential equation. We establish a well-posed theorem for such general types of equations by the shooting method; the theorem is then applied to our problem for the existence of a stationary solution. In addition, for this highly nonlinear problem, we also prove the uniqueness of the stationary solution, which is a nontrivial result. In addition, numerical simulations indicate that the stationary solution is likely locally asymptotically stable for reasonable range of parameters.

math.AP↗

arXiv2010

Analysis of a Mathematical Model of Ischemic Cutaneous wounds

Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia represents a serious complicating factor in wound healing. In this paper we analyze a recently developed mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary of the open wound. The model equations involve the concentrations of oxygen, cytokines, and the densities of several types of cells. The ischemic level is represented by a parameter which appears in the boundary conditions, 0 <= gamma < 1; gamma near 1 corresponds to extreme ischemia and gamma = 0 corresponds to normal non-ischemic conditions. We establish global existence and uniqueness of the free boundary problem and study the dependence of the free boundary on gamma.

math.AP↗