Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

Lu, Junfan; Hu, Bei

doi:10.1002/mma.6142

Cited by 6 publications

(4 citation statements)

References 38 publications

(106 reference statements)

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“…Because multilayered tumor cells are grown on permeable membranes which can separate FIGURE 1 Three-dimensional multilayered tumor region two reservoirs of the diffusion apparatus directly, it is an important task to study the fluidity of drug and metabolism of tumor tissue. See Lu and Hu, Mueller-Klieser, Kim et al, and Kyle et al [11][12][13][14] for the study of multilayered tumor cells.…”

Section: Introductionmentioning

confidence: 99%

The linear stability for a free boundary problem modeling multilayer tumor growth with time delay

Xing

2022

Math Methods in App Sciences

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We study a free boundary problem modeling multilayer tumor growth with a small time delay τ, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper, we establish the well‐posedness of the problem; namely, first, we prove that there exists a unique flat stationary solution false(σ∗,p∗,ρ∗,ξ∗false) for all μ>0. The stability of this stationary solution should depend on the tumor aggressiveness constant μ. It is also unrealistic to expect the perturbation to be flat. We show that, under non‐flat perturbations, there exists a threshold μ∗>0 such that false(σ∗,p∗,ρ∗,ξ∗false) is linearly stable if μ<μ∗ and linearly unstable if μ>μ∗. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.

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Section: Introductionmentioning

confidence: 99%

The linear stability for a free boundary problem modeling multilayer tumor growth with time delay

Xing

2022

Math Methods in App Sciences

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“…In the presence of inhibitors, Zhou, Wu and Cui [7] have derived the local existence and asymptotic behavior of flat stationary solutions under non-flat perturbations. For 2-dimensional tumor, Lu and Hu [8] studied the bifurcation of tumor growth with ECM and MDE interactions. Recently, for tumor model with time delay, He, Xing and Hu considered the linear stability of the positive flat stationary solution under non-flat perturbations for quasi-steady state approximation in [9] and for general case with λ = 0 in [10], respectively.…”

Section: Introductionmentioning

confidence: 99%

The existence of periodic solution and asymptotic behavior of solutions for a multi-layer tumor model with a periodic provision of external nutrients

He¹,

Xing²

2021

Preprint

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In this paper, we consider a multi-layer tumor model with a periodic provision of external nutrients. The domain occupied by tumor has a different shape (flat shape) than spherical shape which has been studied widely. The important parameters are periodic external nutrients Φ(t) and threshold concentration for proliferation σ. In this paper, we give a complete classification about Φ(t) and σ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, if 1then the zero equilibrium solution is globally stable while if 1 T T 0 Φ(t)dt > σ, then there exists a unique positive T-periodic solution and it is globally stable.

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“…Because multilayered tumor cells are grown on permeable membranes which can separate two reservoirs of the diffusion apparatus directly, it is an important task to study the fluidity of drug and metabolism of tumor tissue. See [14][15][16][17] for the study of multilayered tumor cells.…”

Section: Introductionmentioning

confidence: 99%

The linear stability for a free boundary problem modeling multi-layer tumor growth with time delay

He,

Xing,

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study a free boundary problem modeling multi-layer tumor growth with a small time delay τ , representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper we establish the well-posedness of the problem, namely, first we prove that there exists a unique flat stationary solution (σ * , p * , ρ * , ξ * ) for all µ > 0. The stability of this stationary solution should depend on the tumor aggressiveness constant µ. It is also unrealistic to expect the perturbation to be flat. We show that, under non-flat perturbations, there exists a threshold µ * > 0 such that (σ * , p * , ρ * , ξ * ) is linearly stable if µ < µ * and linearly unstable if µ > µ * . Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.

show abstract

Bifurcation for a free boundary problem modeling the growth of multilayer tumors with ECM and MDE interactions

Cited by 6 publications

References 38 publications

The linear stability for a free boundary problem modeling multilayer tumor growth with time delay

The linear stability for a free boundary problem modeling multilayer tumor growth with time delay

The existence of periodic solution and asymptotic behavior of solutions for a multi-layer tumor model with a periodic provision of external nutrients

The linear stability for a free boundary problem modeling multi-layer tumor growth with time delay

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