Bifurcation for a free boundary problem modeling tumor growth with ECM and MDE interactions

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate normalλ, which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider “flat stationary solution” in this paper. We first show that there exists a unique flat stationary solution for any normalλ>0. Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points normalλ=normalλkfalse(ρ*false) ( k≥1).

“…By taking

normalλ

as a bifurcation parameter and considering the systems to as a bifurcation problem, we obtain that the sequences

normalλ = {normalλ}_{k} false(ρ_{*} false)

(

k \geq 1

) are bifurcation points of nonflat stationary solutions. These results are different from those in spherical tumors model, while Pan and Xing proved that there exist a positive integer

n^{*}

and a sequence of

{normalλ}_{n}

(

n > n^{*}

) for which branches of symmetry‐breaking stationary solutions bifurcate from the radially symmetric one, we find positive bifurcation points starting from

normalλ = {normalλ}_{1} false(ρ_{*} false)

, which is quite significant in biology, as this might indicate the stability and instability of the nonflat stationary solutions (see previous studies).…”

Section: Conclusion: Biological Interpretationcontrasting

confidence: 92%

“…We denote the ECM density by

E

and the concentration of MDEs by

M

, respectively. MDEs are released by tumor cells, diffuse throughout the tissue and undergo some form of decay (see previous studies). The equation governing the evolution of MDEs is given by

M_{t} = D_{M} normalΔ M + {normalλ}_{p 1}^{M} false(1 - M M_{0}^{- 1} false) + {normalλ}_{p 2}^{M} σ - {normalλ}_{d}^{M} M in normalΩ false(t false),

with the boundary conditions

\frac{\partial M}{\partial n} = 0 on {normalΓ}_{ρ}, {normalΓ}_{0} .

…”

Section: Introducionmentioning

confidence: 95%

“…Now, the problem is reduced to finding three unknown functions

σ

M

, and

p

, together with the tumor region

{normalΩ}_{ρ}

. Pan and Xing considered this model in three‐dimensional space. They showed that for each parameter

normalλ > 0

, there exists a unique radially symmetric stationary solution with radius

r = R_{s}

.…”

Section: Introducionmentioning

confidence: 99%

“…Moreover, they also established the existence of symmetry‐breaking bifurcations for this model. To simplify this model, we normalize as in Pan and Xing by setting

L = \sqrt{\frac{λ_{B} + λ_{d}^{σ}}{D_{σ}}}, B = \frac{σ_{B} λ_{B}}{λ_{B} + λ_{d}^{σ}}, C = \frac{λ_{p 1}^{M} + B λ_{p 2}^{M}}{λ_{p 1}^{M} M_{0}^{- 1} + λ_{d}^{M}}, G = \frac{χ}{μ γ L} \frac{E_{0}}{λ_{p}^{E}},

α = \frac{λ_{p 1}^{M} M_{0}^{- 1} + λ_{d}^{M}}{D_{M} L^{2}}, β = \frac{G λ_{d}^{E} λ_{p 2}^{M} (σ_{\infty} - B)}{D_{M} L^{2}}, λ = \frac{λ_{p}^{σ} (σ_{\infty} - B)}{μ γ L^{3}}, \tilde{σ} =

…”

Section: Introducionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

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

2020

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

Zhao

2022

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

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

Xing

2022

Self Cite

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.