Chemotaxis Haptotaxis Model Research Articles

Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms

Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms

Numerical and statistical approach on chemotaxis-haptotaxis model for cancer cell invasion of tissue

<abstract><p>In this study, a one-dimensional chemotaxis-haptotaxis model of cancer cell invasion of tissue was numerically and statistically investigated. In the numerical part, the time dependent, nonlinear, triplet governing dimensionless equations consisting of cancer cell (CC) density, extracellular matrix (ECM) density, and urokinase plasminogen activator (uPA) density were solved by the radial basis function (RBF) collocation method both in time and space discretization. In the statistical part, mean CC density, mean ECM density, and mean uPA density were modeled by two different machine learning approaches. The datasets for modeling were originated from the numerical results. The numerical method was performed in a set of parameter combinations by parallel computing and the data in case of convergent combinations were stored. In this data, inputs consisted of selected time values up to a maximum time value and converged parameter values, and outputs were mean CC, mean ECM, and mean uPA. The whole data was divided randomly into train and test data. Trilayer neural network (TNN) and multilayer adaptive regression splines (Mars) model the train data. Then, the models were tested on test data. TNN modeling resulting in terms of mean squared error metric was better than Mars results.</p></abstract>

Some further progress for boundedness of solutions to a quasilinear higher–dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Some further progress for boundedness of solutions to a quasilinear higher–dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Qualitative behavior of solutions for a chemotaxis-haptotaxis model with flux limitation

Qualitative behavior of solutions for a chemotaxis-haptotaxis model with flux limitation

Global boundedness of weak solutions to a chemotaxis–haptotaxis model with p-Laplacian diffusion

Global boundedness of weak solutions to a chemotaxis–haptotaxis model with p-Laplacian diffusion

Optimal mass on the parabolic-elliptic-ODE minimal chemotaxis-haptotaxis in

In this paper, we study the Cauchy problem to a chemotaxis-haptotaxis model describing cancer invasion in . The main feature is to prove that 8π is the critical mass on initial data for distinguishing existence and blow-up of solutions to the model. Namely, when the initial mass is less than 8π, we prove global existence of solutions by constructing a proper free energy and using the Brezis-Merle type inequality. On the contrary, the finite time blow-up of solutions may occur if the initial mass is greater than 8π and the initial second moment is small enough. Comparing to the classical Keller-Segel model, we find that the haptotaxis has no effect on the critical mass of the model.

Global Solvability, Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

Global Solvability, Pattern Formation and Stability to a Chemotaxis-haptotaxis Model with Porous Medium Diffusion

Qualitative behavior of solutions for a chemotaxis-haptotaxis system with gradient-dependent flux-limitation

This paper deals with the coupled chemotaxis-haptotaxis model with gradient-dependent flux-limitation where is a bounded domain with smooth boundary, χ, ξ and μ are positive parameters. Under the condition that then for any and all reasonably smooth initial data , the above system possesses a globally classical solution, which is uniformly bounded in time. Under the conditions that it asserts exponential decay of w in the large time limit, whereas both u and v persist in some certain sense.

Optimal control for a chemotaxis–haptotaxis model in two space dimensions

This paper deals with a chemotaxis–haptotaxis model which described the process of cancer invasion on the macroscopic scale. We first explore the global-in-time existence and uniqueness of a strong solution. For a class of cost functionals, we prove first-order necessary optimality conditions for the corresponding optimal control problem and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.

Finite time blow-up in the higher dimensional parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system

In this work, we mainly study nonnegative classical solutions to a Neumann initial-boundary value problem for the following parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded and smooth domain Ω⊂Rn(n≥1) with χ,ξ≥0, nonnegative initial data (u0,w0) and homogeneous Neumann boundary conditions.In this setup, we first show pure haptotaxis (χ=0) cannot induce any blow-up and pattern for any n≥1, showing haptotaxis is overbalanced by diffusion on boundedness and longtime behavior. Then, in the radial setting with Ω=BR⊂Rn(n≥3), it is well-known that small mass of ‖u0‖L1(Ω) can lead to blow-up in the corresponding Keller-Segel chemotaxis-only model obtained by setting w≡0, known as generic mass blow-up phenomenon [23]. Herein, in the presence of the temporal nonlocality brought by haptotaxis, we show, in an explicit realm of the form χ‖u0‖L1(Ω)>A(n,R)ξ‖w0‖L∞(Ω), that the aforementioned generic mass finite time blow-up phenomenon preserves. This seems to be the first rigorous blow-up result in relevant chemotaxis-haptotaxis models. The blow-up result also suggests that haptotatic cross-diffusion may not be negligible compared to chemotactic aggregation in ≥3D, in contrast to the recently detected 2D negligibility of haptotaxis in a minimal chemotaxis-haptotaxis model [16].

Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model

In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller–Segel chemotaxis-only model possesses a striking feature of critical mass blowup phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model and that any global-in-time haptotaxis solution component vanishes exponentially and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, haptotaixs is neither good nor bad than chemotaxis, showing negligibility of haptotaxis effect in the underlying chemotaxis-haptotaxis model.

Global existence of classical solutions to a chemotaxis-haptotaxis model

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue. The model consists of three parabolic chemotaxis-haptotaxis PDEs describing interactions between cancer cells, matrix degrading enzymes and the host tissue. We prove the global existence and uniform boundedness in this competition system by some delicate a priori estimate techniques.

Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production

This paper deals with the quasilinear chemotaxis-haptotaxis model of cancer invasion{ut=∇⋅(D(u)∇u)+∇⋅(S1(u)∇v)+∇⋅(S2(u)∇w)+f(u,w),x∈Ω,t>0,τvt=Δv−v+g1(w)g2(u),x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded smooth domain Ω⊂RN(N≥1) with zero-flux boundary conditions, where τ∈{0,1}, the functions D(u),S1(u),S2(u)∈C2([0,∞)), f(u,w)∈C1([0,∞)2),g1(w),g2(u)∈C1([0,∞)) fulfill D(u)≥CD(u+1)−α,S1(u)≤χu(u+1)β−1,S2(u)≤ξu(u+1)γ−1, f(u,w)≤u(a−μur−1−λw),f(0,w)≥0,g1(w)≥0,0≤g2(u)≤Kuκ with CD,χ,ξ,μ,K,κ>0, λ≥0, r>1 and α,β,γ,a∈R. Under specific parameters conditions, it is shown that for any appropriately regular initial data, the associated initial-boundary value problem admits a globally bounded classical solution. Moreover, when f=u(a−μur−1−λw), g1(w)≡1 and g2(u)=uκ, the asymptotic stability of solutions is also investigated. Specifically, for some a0,μ0>0 independent of (u0,v0), the bounded classical solution (u,v,w) exponentially converges to ((aμ)1r−1,(aμ)κr−1,0) in Lp(Ω)×L∞(Ω)×W1,∞(Ω) for any p≥2 if a>a0 and μ>μ0. These results improve or extend previously known ones, and partial results are new.

Global solvability and stabilization to a cancer invasion model with remodelling of ECM * * This work is supported by NSFC (11871230) and Guangdong Basic and Applied Basic Research Foundation (2020B1515310013).

In this paper, we deal with the Chaplain–Lolas’s model of cancer invasion with tissue remodelling We consider this problem in a bounded domain (N = 2, 3) with zero-flux boundary conditions. We first establish the global existence and uniform boundedness of solutions. Subsequently, we also consider the large time behaviour of solutions, and show that the global classical solution (u, v, w) strongly converges to the semi-trivial steady state in the large time limit if δ > η; and strongly converges to if δ < η. Unfortunately, for the case δ = η, we only prove that (v, w) → (1, 0), and it is hard to obtain the large time limit of u due to lack of uniform boundedness of . It is worth noting that the large time behaviour of solutions for the chemotaxis–haptotaxis model with tissue remodelling has never been touched before, this paper is the first attempt. At last, taking advantage of the large time behaviour of solutions, we also establish the uniform boundedness of solutions in the classical sense.

A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis–haptotaxis model (with a logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis–haptotaxis system with nonlinear diffusion(0.1){ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,τvt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN(N≥1), where τ∈{0,1} and χ, ξ, and μ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfyD(u)≥CD(u+1)m−1for allu≥0andCD>0. In the present work it is shown that ifm≥2−2Nλwith0<μ<κ0,m>2−2Nλwithμ≥κ0, orm>2−2Nandμ=0 orm=2−2NandCD>CGN(1+‖u0‖L1(Ω))34(2−2N)2κ0, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in Ω×(0,∞), whereλ=κ0(κ0−μ)+ andκ0={maxs≥1⁡λ01s+1(χ+ξ‖w0‖L∞(Ω))ifτ=1,χifτ=0. Here CGN and λ0 are constants that correspond to the Gagliardo–Nirenberg inequality and the maximal Sobolev regularity, respectively. With use of new Lp-estimate techniques to obtain the a priori estimate of a solution from L1(Ω)→Lλ−ε(Ω)→Lλ(Ω)→Lλ+ε(Ω)→Lp(Ω)(for allp>1), these results significantly improve or extend previous results obtained by several authors.

Global boundedness to a chemotaxis–haptotaxis model with nonlinear diffusion

This paper deals with the following chemotaxis–haptotaxis model ut=∇⋅(D(u)∇u)−χ∇⋅(u(1+u)α∇v)−ξ∇⋅(u(1+u)β∇w)+u(a−μuk−1−λw),vt=△v−v+u,wt=−vw,under homogeneous Neumann boundary condition in a bounded domain Ω⊂R3, with χ,ξ,μ,λ>0, k>1, a∈R, and D(u)≥CD(u+1)m−1 for CD>0,m∈R. It is shown that if α>2−k and β>76−2k3, the corresponding system possesses a classical solution (u,v,w) which is globally bounded. The result improved the work proposed in Xu et al. (2019), in which, the global boundedness of solutions is established for m>0,α>0,β≥0 and k=2.

A new result for 2D boundedness of solutions to a chemotaxis–haptotaxis model with/without sub-logistic source

We consider the Neumann problem for a coupled chemotaxis–haptotaxis model of cancer invasion with/without kinetic source in a 2D bounded and smooth domain. For a large class of cell kinetic sources including zero source and sub-logistic sources, we detect an explicit condition involving the chemotactic strength, the asymptotic ‘damping’ rate, and the initial mass of cells to ensure uniform-in-time boundedness for the corresponding Neumann problem. Our finding significantly improves existing 2D global existence and boundedness in related chemotaxis–haptotaxis systems.

Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.

Global boundedness in a three-dimensional chemotaxis–haptotaxis model

This paper studies the chemotaxis–haptotaxis model of cancer invasion ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0in a bounded smooth domain Ω⊂R3 with zero-flux boundary conditions, where the parameters χ, ξ and μ are positive. It is shown that the corresponding initial–boundary problem possesses a unique classical solution which is global in time and bounded under the explicit condition μ≥(112+ξ2‖w0‖L∞(Ω)2)χ2+372+4ξ2‖w0‖L∞(Ω)2 and suitable regularity assumptions on the initial data (u0,v0,w0).

Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source \begin{document}$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), {v_t = \Delta v- v +u}, {w_t = - vw}, \quad {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, x\in \partial\Omega, t>0, {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), x\in \Omega, \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$ \end{document} in a smooth bounded domain \begin{document}$ \mathbb{R}^N(N\geq1) $\end{document} , with parameter \begin{document}$ r>1 $\end{document} . the parameters \begin{document}$ a\in \mathbb{R}, \mu>0, \chi>0 $\end{document} . It is shown that when \begin{document}$ r>2 $\end{document} , or \begin{document}$ \begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \end{array} \end{equation*} $\end{document} the considered problem possesses a global classical solution which is bounded, where \begin{document}$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $\end{document} is a positive constant which is corresponding to the maximal sobolev regularity. Here \begin{document}$ C_{\beta} $\end{document} is a positive constant which depends on \begin{document}$ \xi $\end{document} , \begin{document}$ \|u_0\|_{C(\bar{\Omega})}, \|v_0\|_{W^{1, \infty}(\Omega)} $\end{document} and \begin{document}$ \|w_0\|_{L^\infty(\Omega)} $\end{document} . This result improves or extends previous results of several authors.