Bifurcation analysis of a free boundary problem modelling tumour growth under the action of inhibitors

Abstract

In this paper we investigate non-radial stationary solutions of a free boundary problem modelling tumour growth under the action of inhibitors. The model consists of two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the velocity of tumour cells and internal pressure. The ratio µ/γ of the proliferation rate µ and the cellto-cell adhesiveness γ plays the role of the bifurcation parameter. We prove that in certain situations there exists a positive sequence {(µ/γ ) n } n n * such that for each (µ/γ ) n (n even n * ) there exist non-radial stationary solutions bifurcating from the radial stationary solution, while in the other situations there exists at most a finite number of bifurcation points. This is a remarkable difference from the corresponding inhibitor-free model where there always exist infinitely many branches of non-radial stationary bifurcation solutions. Our analysis also indicates that inhibitor supply may lower the ability of tumour invasion, and even make the tumour unaggressive and stable.

Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors

“…As pointed out by Friedman (see page 235 of [11]), necrotic tumor model is novel in free boundary problems and many challenges arise in rigorous mathematical analysis. By taking γ as a bifurcation parameter, and using bifurcation analysis we prove that there exist infinitely many non-flat stationary solutions with the upper free boundary y = ρ s +ε cos(kx)+O(ε 2 ), the lower free boundary y = η s +εA k cos(kx)+ O(ε 2 ), and γ = γ k + O(ε) for sufficiently large k ∈ N. As shown in [12], [13], [20], [22], [23], [25] for non-necrotic tumor models, these bifurcation solutions shaped like 'fingers', are associated to the invasion of tumors into the surroundings. Hence the cell-to-cell adhesiveness γ plays an important role on tumor invasion.…”

“…Zhou et al [25] proved that there exist infinitely many non-flat non-necrotic stationary solutions bifurcating from the unique flat non-necrotic stationary solution. For similar illuminate results of non-necrotic solid tumor spheroid models which have been extensively studied, we refer to [2], [7], [8], [10], [12], [13], [14], [19], [20], [22], [23], [24] and references cited therein.…”

“…In fact, from (43) and (44), we easily show there exist infinitely many bifurcation points µ k , by taking µ as a bifurcation parameter, which is similar as shown in necrotic tumor spheroid model (see [16]) and non-necrotic tumor models (cf. [13], [23]). But for ν, if we take ν = 0, then by (65) we have KerD ρ G(γ k , ρ s ) = span{1, cos(kx)} of dimension 2, and the above bifurcation analysis based on Crandall-Rabinowitz theorem is not applicable any more.…”

A parabolic–hyperbolic system modeling the growth of a tumor

“…These models incorporate a system of partial differential equations (PDEs), where cell density, nutrients (i.e., oxygen and glucose), etc., are tracked. Modeling, mathematical analysis and numerical simulations were carried out in numerous papers, see [2,5,6,7,9,11,17,21,25,26,30,28] and the references therein. Lowengrub et al [19] provided a systematic review of tumor model studies.…”

Bifurcation analysis of amathematical model for the growth of solid tumors in the presence of external inhibitors

“…Analysis of such tumor models has been one of the most active topics of research during the past several years, and many interesting results have been obtained, cf. [1,[5][6][7][8][9][10][11][12][13][14][15][16][17], and the references cited therein. Problem (1.1) is developed from the model of Byrne and Chaplain [2], where only radial (i.e., spherically symmetric) tumors are considered.…”