Bifurcation Analysis of an Elliptic Free Boundary Problem Modelling the Growth of Avascular Tumors

Cui, Shangbin; Escher, Joachim

doi:10.1137/060657509

Cited by 68 publications

(82 citation statements)

References 10 publications

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“…The simpler model, as presented in [4,8,11], has been studied extensively by different authors, see, e.g. [3,6,7,9] and the references therein. In particular it is shown in these papers that if the parameters belong to an appropriate range, then the mathematical formulation possesses a unique radially symmetric solution.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Radially symmetric growth of nonnecrotic tumors

Escher

Matioc

2009

Nonlinear Differ. Equ. Appl.

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Abstract. The growth of tumors is an important subject in recent research. We present here a mathematical model for the growth of nonnecrotic tumors in all the three regimes of vascularisation. This leads to a freeboundary problem which we treat by means ODE techniques. We prove the existence of a unique radially symmetric stationary solution. It is also shown that, if the initial tumor is radially symmetric, there exists a unique radially symmetric solution of the evolution equation, which exists for all times. The asymptotic behaviour of this solution will be discussed in relation to the parameters characterizing cell proliferation and cell death.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…During the last four decades an increasing number of mathematical models have been proposed to describe the growth of solid tumors (see [2,[6][7][8] and the literature therein). There is a three level approach in modeling the complex phenomena influencing and describing the processes inside a tumor.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Radially symmetric growth of nonnecrotic tumors

Escher

Matioc

2009

Nonlinear Differ. Equ. Appl.

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“…While in [8,20] the equations coincide with those remaining in (2.1), the authors in [6] consider the problem of existence of non-radial stationary solution for the model when the first and the third equation of (2.1) are replaced with the more general relations…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that this model is an extension (taxis is taken into account) of previous single phase models [1,4], where the driving mechanisms were the pressure and cell proliferation. The moving boundary problems associated to the models presented in [1,4] have been investigated by many authors (see [5][6][7][8][9]11,12,[19][20][21] and the references therein). Unlike the models studied in [10,13,23,24] where existence of a necrotic core consisting of death cells is taken into account, the model derived in [3] and studied herein does not possess this feature.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, let us observe that if the nutrient concentration σ H is constant in Ω H (t), the taxis is absent, C = 0, and we drop the sixth equation of (2.1) to arrive at the one-phase tumor model analyzed in [6,8,20]. While in [8,20] the equations coincide with those remaining in (2.1), the authors in [6] consider the problem of existence of non-radial stationary solution for the model when the first and the third equation of (2.1) are replaced with the more general relations…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a two-phase model describing the growth of solid tumors

Escher¹,

Matioc²

2012

Eur. J. Appl. Math

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In this paper we consider a two-phase model describing the growth of avascular solid tumors when taking into account the effects of cell-to-cell adhesion and taxis due to nutrient. The tumor is surrounded by healthy tissue which is the source of nutrient for tumor cells. In a three-dimensional context, we prove that the mathematical formulation corresponds to a well-posed problem, and find radially symmetric steady-state solutions of the problem. They appear in the regime where the rate of cell apoptosis to cell proliferation is less than the far field nutrient concentration. Furthermore, we study the stability properties of those radially symmetric equilibria and find, depending on the biophysical parameters involved in the problem, both stable and unstable regimes for tumor growth.

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

Cui

2014

Math Methods in App Sciences

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We study bifurcations from radial solution of a free boundary problem modeling the dormant state of nonnecrotic solid tumors in the presence of external inhibitors. This problem consists in three linear elliptic equations with two Dirichlet and one Neumann boundary conditions and a fourth boundary condition coupling surface tension effects on free boundary. In this paper, surface tension coefficient γ plays the role of bifurcation parameter. We prove that in certain situations there exists a positive null point sequence for γ where bifurcation occurs from radial solution, while in the other situations, either bifurcation occurs at only finite many points of γ or even it does not occur for any γ > 0. Copyright © 2014 John Wiley & Sons, Ltd.

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Bifurcation Analysis of an Elliptic Free Boundary Problem Modelling the Growth of Avascular Tumors

Cited by 68 publications

References 10 publications

Radially symmetric growth of nonnecrotic tumors

Radially symmetric growth of nonnecrotic tumors

Analysis of a two-phase model describing the growth of solid tumors

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

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