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

Song, Huijuan; Hu, Bei; Wang, Zejia

doi:10.3934/dcdsb.2020084

Cited by 11 publications

(9 citation statements)

References 32 publications

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“…In order to prove

μ = μ_{1}

is a bifurcation point, it is required to verify the four conditions of the Crandall‐Rabinowitz theorem at this point. First of all, the differentiability of the map

ℱ

follows the same argument as in the papers, 10,23,25,29,30 and the condition (1) in the Crandall‐Rabinowitz theorem is naturally satisfied. For the remaining conditions, we need to prove

([], ℱ_{\overset{R}{true}} false(0, μ_{1} false)) \cos m θ \neq 0, for \forall m \neq 1 .

As is mentioned, it has been established in Zhao and Hu 10 that there exists a bound E 2 > 0, when 0 < ϵ < E 2 , we have () to be true for each m ≥ 2.…”

Section: Proof Of Theorem 11mentioning

confidence: 91%

“…In order to prove 𝜇 = 𝜇 1 is a bifurcation point, it is required to verify the four conditions of the Crandall-Rabinowitz theorem at this point. First of all, the differentiability of the map ℱ follows the same argument as in the papers, 10,23,25,29,30 and the condition (1) in the Crandall-Rabinowitz theorem is naturally satisfied. For the remaining conditions, we need to prove…”

Section: Proof Of Theorem 11mentioning

confidence: 92%

“…(1.16) In recent years, there are considerable research projects on the bifurcation analysis which are based upon the Crandall-Rabinowitz theorem (see literature [17][18][19][20][21][22][23][24][25][26][27][28][29][30] ). In these papers, the solutions on the n = 1 bifurcation branch are just 𝜖-translations of the origin of the radially symmetric solution; after transformation of coordinates, this kind of solutions are still radially symmetric.…”

Section: Figure 1 the Cross Section Of An Arterymentioning

confidence: 99%

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On the first bifurcation point for a free boundary problem modeling a small arterial plaque

Zhao

2022

Math Methods in App Sciences

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When plaques block the arteries, atherosclerosis occurs. Severe atherosclerosis would result in fatal cardiovascular diseases such as stroke, heart attack, coronary artery disease, or peripheral artery disease; these are leading causes of death in the world. In the paper, we investigate a free boundary PDE model to describe the formation of an arterial plaque in the early stage of atherosclerosis.The bifurcation analysis is carried out for the model. In particular, we establish the first bifurcation point for the system corresponding to the n = 1 mode. The symmetry-breaking stationary solution studied in this paper can be helpful in understanding why there exists arterial plaque that is often accumulated more on one side of the artery than the other. KEYWORDS atherosclerosis, bifurcations in context of PDEs, free boundary problems for PDEs MSC CLASSIFICATION 35R35; 35B32

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“…In order to prove

μ = μ_{1}

is a bifurcation point, it is required to verify the four conditions of the Crandall‐Rabinowitz theorem at this point. First of all, the differentiability of the map

ℱ

follows the same argument as in the papers, 10,23,25,29,30 and the condition (1) in the Crandall‐Rabinowitz theorem is naturally satisfied. For the remaining conditions, we need to prove

([], ℱ_{\overset{R}{true}} false(0, μ_{1} false)) \cos m θ \neq 0, for \forall m \neq 1 .

As is mentioned, it has been established in Zhao and Hu 10 that there exists a bound E 2 > 0, when 0 < ϵ < E 2 , we have () to be true for each m ≥ 2.…”

Section: Proof Of Theorem 11mentioning

confidence: 91%

Section: Proof Of Theorem 11mentioning

confidence: 92%

Section: Figure 1 the Cross Section Of An Arterymentioning

confidence: 99%

See 1 more Smart Citation

On the first bifurcation point for a free boundary problem modeling a small arterial plaque

Zhao

2022

Math Methods in App Sciences

Self Cite

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“…Very recently, Wu-Xu [22] analyzed the model (1.1)-(1.5) with (1.3) replaced by the boundary condition (1.10), and complete existence, uniqueness and stability results were given. Finally, for other related study, we refer the reader to [2,3,6,7,10,14,16,19,24] and the references cited therein.…”

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

Cited by 11 publications

References 32 publications

On the first bifurcation point for a free boundary problem modeling a small arterial plaque

On the first bifurcation point for a free boundary problem modeling a small arterial plaque

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

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

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