Shangbin Cui

Analysis of a mathematical model of the effect of inhibitors on the growth of tumors

In this paper, we study a model of tumor growth in the presence of inhibitors. The tumor is assumed to be spherically symmetric and its boundary is an unknown function r=R(t). Within the tumor the concentration of nutrient and the concentration of inhibitor (drug) satisfy a system of reaction-diffusion equations. The important parameters are Lambda(0) (which depends on the tumors parameters when no inhibitors are present), gamma which depends only on the specific properties of the inhibitor, and beta; which is the (normalized) external concentration of the inhibitor. In this paper, we give precise conditions under which there exist one dormant tumor, two dormant tumors, or none. We then prove that in the first case, the dormant tumor is globally asymptotically stable, and in the second case, if the radii of the dormant tumors are denoted by R(s)(-),R(s)(+) with R(s)(-)infinity)R(t)=R(s)(-), provided the initial radius R(0) is smaller than R(s)(+); if however R(0)R(s)(+) then the initial tumor in general grows unboundedly in time. The above analysis suggests an effective strategy for treatment of tumors.

A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

First published in Transactions of the American Mathematical Society in volume 355, issue 9, published by the American Mathematical Society.

Asymptotic Behaviour of Solutions of a Multidimensional Moving Boundary Problem Modeling Tumor Growth

We study a moving boundary problem modeling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumors body, respectively. The driving mechanism of the evolution of the tumor surface is governed by Darcys law. Finally surface tension effects on the moving boundary are taken into account which are considered to counterbalance the internal pressure. To put our analysis on a solid basis, we first state a local well-posedness result for general initial data. However, the main purpose of our study is the investigation of the asymptotic behaviour of solutions as time goes to infinity. As a result of a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in the C m+μ-norm with m ≥ 3 and μ ∈ (0,1), then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly shifted) ball, provided the surface tension coefficient γ is larger than a positive threshold value γ*. In the case 0 < γ < γ* the radially symmetric equilibrium is unstable.

A Hyperbolic Free Boundary Problem Modeling Tumor Growth

In this paper we study a free boundary problem modeling the growth of tumors with three cell populations: proliferating cells, quiescent cells and dead cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, with tumor surface as a free boundary. The nutrient concentration satisfies a diffusion equation, and the free boundary r = R(t) satisfies an integro-differential equation. We consider the radially symmetric case of this free boundary problem, and prove that it has a unique global solution for all the three cases 0 0, while limt!1 R(t) = 1 in the case KR = 0.

BIFURCATION ANALYSIS OF AN ELLIPTIC FREE BOUNDARY PROBLEM MODELLING THE GROWTH OF AVASCULAR TUMORS

We study bifurcations from radially symmetric solutions of a free boundary problem modelling the dormant state of nonnecrotic avascular tumors. This problem consists of two semilinear elliptic equations with a Dirichlet and a Neumann boundary condition, respectively, and a third boundary condition coupling surface tension effects on the free interface to the internal pressure. By reducing the full problem to an abstract bifurcation equation in terms of the free boundary only and by characterizing the linearization as a Fourier multiplication operator, we carry out a precise analysis of local bifurcations of this problem.

A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior

First published in Transactions of the American Mathematical Society in volume 357, issue 12, published by the American Mathematical Society.