Analysis of necrotic core formation in angiogenic tumor growth

Xu, Shihe; Su, Dan

doi:10.1016/j.nonrwa.2019.103016

Cited by 8 publications

(3 citation statements)

References 22 publications

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“…Since nutrients enter the sphere by the vascular system, the Robin boundary condition (1.3) is assumed on the tumor boundary (see [12]). For the quasi-stationary version of the model (1.1)-(1.6), Xu and Su [13] analyzed the well-posedness and asymptotic behavior of solutions. By more delicate calculations, Song et al [14] and Xu et al [15] extended the results to the nonlinear case, respectively.…”

Section: Introductionmentioning

confidence: 99%

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

Song,

Wang,

2024

Math Methods in App Sciences

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In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem.

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Section: Introductionmentioning

confidence: 99%

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

Song,

Wang,

2024

Math Methods in App Sciences

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“…is prescribed on the tumor boundary, the model was discussed by Xu-Su [23]. Recently, Wu-Wang [21] and Song-Hu-Wang [20] extended the results to the cases of general nonlinear functions f and g, respectively.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a nonlinear necrotic tumor model with angiogenesis and a periodic supply of external nutrients

Song,

Huang,

Wang

2021

Preprint

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In this paper, we consider a free boundary problem modeling the growth of spherically symmetric necrotic tumors with angiogenesis and a ω-periodic supply φ(t) of external nutrients. In the model, the consumption rate of the nutrient and the proliferation rate of tumor cells S(σ) are both general nonlinear functions. The well-posedness and asymptotic behavior of solutions are studied. We show that if the average of S(φ(t)) is nonpositive, then all evolutionary tumors will finally vanish; the converse is also ture. If instead the average of S(φ(t)) is positive, then there exists a unique positive periodic solution and all other evolutionary tumors will converge to this periodic state.

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“…For the analysis of other related tumor models, see [1,6,12] for example. When replacing the Dirichlet boundary condition (1.6) with the Robin condition (1.3), the model (1.1)-(1.5) with linear functions (1.7) was recently studied by Xu and Su [11]. As for nonlinear functions f and g, the nonnecrotic case was analyzed by Zhuang and Cui [13].…”

Section: Introductionmentioning

confidence: 99%

Analysis of a nonlinear free-boundary tumor model with angiogenesis and a connection between the nonnecrotic and necrotic phases

Song

Wang

2020

Preprint

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This paper is concerned with a nonlinear free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, set with a Robin boundary condition. In which, both nonnecrotic tumors and necrotic tumors are taken into consideration. The well-posedness and asymptotic behavior of solutions are studied. It is shown that there exist two thresholds, denoted by σ and σ * , on the surrounding nutrient concentration σ. If σ ≤ σ, then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if σ > σ, then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if σ < σ ≤ σ * and necrotic if σ > σ * . The connection and mutual transition between the nonnecrotic and necrotic phases are also given.

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Analysis of necrotic core formation in angiogenic tumor growth

Cited by 8 publications

References 22 publications

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

Analysis of a nonlinear necrotic tumor model with angiogenesis and a periodic supply of external nutrients

Analysis of a nonlinear free-boundary tumor model with angiogenesis and a connection between the nonnecrotic and necrotic phases

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