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

Wu, Junde

doi:10.3934/dcds.2019140

Cited by 13 publications

(5 citation statements)

References 21 publications

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“…Moreover, tumor will finally converge to the necrotic dormant state forσ > σ * , or converge to the nonnecrotic dormant state for σ <σ ≤ σ * , or finally disappear forσ ≤σ. For the existence of stationary solutions of a similar necrotic multilayered tumor model, we refer to [16].…”

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confidence: 99%

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Wu¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

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confidence: 99%

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Wu¹,

Xu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…Under the assumption ν=0, Hao et al [35] derived the first bifurcation result for the tumor model with a necrotic core, in which they studied the two-dimensional case by taking µ as a bifurcation parameter with the aid of numerical calculations. Very recently, Wu [47] rigorously analyzed the necrotic multilayered tumor model, and obtained the existence of bifurcation branches of non-flat stationary solutions for γ k (k ≥ K). Asymptotic stability of stationary solutions to the above two types of necrotic tumor models was also studied, cf.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [35,47], we shall perform rigorous mathematical analysis of the stationary state of the problem (1.2)-(1.7) in the three-dimensional case, under the assumption that 0 < σ < σ < σ = 1, β(t) ≡ β, ν = 0, where β is a positive constant. That is, ∆σ = σI Ω\D in Ω, (1.9)…”

Section: Introductionmentioning

confidence: 99%

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

Song¹,

Hu²,

Wang³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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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 σ to the tumor at a rate β, then ∂σ ∂n + β(σ −σ) = 0 holds on the tumor boundary, where n is the unit outward normal to the boundary andσ is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate µ. We show that for any given ρ > 0, there exists a unique R ∈ (ρ, ∞) such that the corresponding radially symmetric solution solves the steady-state necrotic tumor system with necrotic core boundary r = ρ and outer boundary r = R; moreover, there exist a positive integer n * * and a sequence of µ n , symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solution for each µ n (even n ≥ n * * ).

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“…Following the early biomechanical models of avascular tumor growth proposed by Greenspan [14], the bifurcation analysis [15,16,17,18,19,20,21,22], numerical simulations, and computational modeling [23,24,25,26,27,28,29] have contributed significantly to the tumor modeling area. Recently, tumor growth models with a necrotic core have also been developed and analyzed via the bifurcation theory [30,31,32,33,34,35,36].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis

Lu,

Hao,

Liu

et al. 2021

Preprint

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In this paper, we develop a sharp interface tumor growth model to study the effect of both the intratumoral structure using a controlled necrotic core and the extratumoral nutrient supply from vasculature on tumor morphology. We first show that our model extends the benchmark results in the literature using linear stability analysis. Then we solve this generalized model numerically using a spectrally accurate boundary integral method in an evolving annular domain, not only with a Robin boundary condition on the outer boundary for the nutrient field which models tumor vasculature, but also with a static boundary condition on the inner boundary for pressure field which models the control of tumor necrosis. The discretized linear systems for both pressure and nutrient fields are shown to be well-conditioned through tracing GMRES iteration numbers. Our nonlinear simulations reveal the stabilizing effects of angiogenesis and the destabilizing ones of chemotaxis and necrosis in the development of tumor morphological instabilities if the necrotic core is fixed in a circular shape. When the necrotic core is controlled in a noncircular shape, the stabilizing effects of proliferation and the destabilizing ones of apoptosis are observed. Finally, the values of the nutrient concentration with its fluxes and the pressure level with its normal derivatives, which are solved accurately at the boundaries, help us to characterize the corresponding tumor morphology and the level of the biophysical quantities on interfaces required in keeping various shapes of the necrotic region of the tumor. Interestingly, we notice that when the necrotic region is fixed in a 3-fold non-circular shape, even if the initial shape of the tumor is circular, the tumor will evolve into a shape corresponding to the 3-fold symmetry of the shape of the fixed necrotic region.

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

Cited by 13 publications

References 21 publications

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

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

Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis

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