Haptotaxis Model Research Articles

Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β<1.

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

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

Boundedness and stabilization in a haptotaxis model of oncolytic virotherapy with nonlinear sensitivity

Boundedness and stabilization in a haptotaxis model of oncolytic virotherapy with nonlinear sensitivity

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

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

Boundedness in a two-dimensional two-species cancer invasion haptotaxis model without cell proliferation

Boundedness in a two-dimensional two-species cancer invasion haptotaxis model without cell proliferation

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.

Global weak solutions in a three-dimensional two-species cancer invasion haptotaxis model without cell proliferation

This paper considers the two species cancer invasion haptotaxis model without cell proliferation in three space dimensions. The system consists of two parabolic partial differential equations (PDEs) describing the migration of differentiated cancer cells and cancer stem cells and the epithelial–mesenchymal transition between the two families of cells, a parabolic/elliptic PDE governing the evolution of matrix degrading enzymes, and an ordinary differential equation reflecting the degradation and remodeling of the extracellular matrix. We underline that the absence of a logistic source aggravates mathematical difficulties that are overcome by constructing a delicate energy-functional. For any suitably regular initial data, we establish the global existence of weak solutions to the associated initial-boundary value problem. This result affirmatively answers the open question proposed by Dai and Liu [SIAM J. Math. Anal. 54, 1–35 (2022)].

A New Result for Global Solvability of a Two Species Cancer Invasion Haptotaxis Model with Tissue Remodeling

A New Result for Global Solvability of a Two Species Cancer Invasion Haptotaxis Model with Tissue Remodeling

Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation} \begin{cases} c_{1t} = \Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{\rm EMT}c_1+\mu_1c_1(r_1-c_1^\kappa-c_2-v),\\ c_{2t} = \Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{\rm EMT}c_1+\mu_2c_2(r_2-c_1-c_2^\kappa-v),\\ \tau m_t = \Delta m+c_1+c_2-m,\\ v_t = -mv+\eta v(1-c_1-c_2-v) \end{cases}\nonumber \end{equation} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;in a bounded and smooth domain &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Omega\subset\mathbb{R}^N\;(N\geq1) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with zero-flux boundary conditions for &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ c_1,c_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ m $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, where &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \chi_i,\mu_i,r_i&amp;gt;0\;(i = 1,2) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \eta&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \kappa\geq1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ \tau\in\{0,1\} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, and &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is the epithelial-mesenchymal transition rate function such that &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \mu_{\rm EMT}\leq\mu_M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with some constant &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \mu_M&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. When &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ \kappa = 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ N = 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, by rasing the coupled a priori estimates of &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ c_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ c_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in the following way &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with any &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$ p&amp;gt;2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}$ r_i $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}$ \mu_M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. When &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}$ \kappa&amp;gt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}$ N\geq1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, by rasing the coupled a priori estimates of &lt;inline-formula&gt;&lt;tex-math id="M22"&gt;\begin{document}$ c_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M23"&gt;\begin{document}$ c_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; from &lt;inline-formula&gt;&lt;tex-math id="M24"&gt;\begin{document}$ L^1(\Omega) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; to &lt;inline-formula&gt;&lt;tex-math id="M25"&gt;\begin{document}$ L^p(\Omega) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with any &lt;inline-formula&gt;&lt;tex-math id="M26"&gt;\begin{document}$ p&amp;gt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, then to &lt;inline-formula&gt;&lt;tex-math id="M27"&gt;\begin{document}$ L^\infty(\Omega) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary &lt;inline-formula&gt;&lt;tex-math id="M28"&gt;\begin{document}$ r_i $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M29"&gt;\begin{document}$ \mu_M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The result for &lt;inline-formula&gt;&lt;tex-math id="M30"&gt;\begin{document}$ \kappa = 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; complements previously known one, and the result for &lt;inline-formula&gt;&lt;tex-math id="M31"&gt;\begin{document}$ \kappa&amp;gt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is new.&lt;/p&gt;

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.

Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models

In this paper, we introduce a weak Galerkin finite element method for 2D Keller-Segel chemotaxis models and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. The chemotaxis and haptotaxis models consist of a system of time-dependent nonlinear reaction-diffusion-taxis partial differential equations. It is well known that solutions of chemotaxis systems may blow-up in finite time in the cell density. This blow-up is a mathematical description of a cell concentration phenomenon, which mathematically results in rapidly growing solutions in small neighborhoods of concentration points or curves. Capturing such singular solutions numerically is a challenging problem. We applied the weak Galerkin finite element method (WGFEM) to approximate the spatial variables. Weak Galerkin finite element method can be considered as an extension of the standard finite element method where classical derivatives are replaced in the variational equation by the weakly defined derivatives on discontinuous weak functions. The weak derivatives of weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the WGFEM is impacted by the selection of such polynomials. We show that the proposed method produced numerical results that are nonnegative, oscillation-free and suitable for solving models that have a blow-up of the cell density. Error bounds for cell density and chemoattractant equation are obtained. The proposed method is applied to several Keller-Segel chemotaxis models and extended to the haptotaxis model for simulating the behavior of cancer cell invasion of surrounding healthy tissue. Numerical results demonstrate the accuracy and efficiency of the proposed scheme.

Collective migration during a gap closure in a two-dimensional haptotactic model

The ability of cells to respond to substrate-bound protein gradients is crucial for many physiological processes, such as immune response, neurogenesis and cancer cell migration. However, the difficulty to produce well-controlled protein gradients has long been a limitation to our understanding of collective cell migration in response to haptotaxis. Here we use a photopatterning technique to create circular, square and linear fibronectin (FN) gradients on two-dimensional (2D) culture substrates. We observed that epithelial cells spread preferentially on zones of higher FN density, creating rounded or elongated gaps within epithelial tissues over circular or linear FN gradients, respectively. Using time-lapse experiments, we demonstrated that the gap closure mechanism in a 2D haptotaxis model requires a significant increase of the leader cell area. In addition, we found that gap closures are slower on decreasing FN densities than on homogenous FN-coated substrate and that fresh closed gaps are characterized by a lower cell density. Interestingly, our results showed that cell proliferation increases in the closed gap region after maturation to restore the cell density, but that cell–cell adhesive junctions remain weaker in scarred epithelial zones. Taken together, our findings provide a better understanding of the wound healing process over protein gradients, which are reminiscent of haptotaxis.

Convergence and positivity of Finite Element methods for a haptotaxis model of tumoral invasion

In this paper, we consider a mathematical model for the invasion of host tissue by tumor cells in a d-dimensional bounded domain, d≤3. This model consists of a system of differential equations describing the evolution of cancer cell density, the extracellular matrix protein density and the matrix degrading enzyme concentration. We develop two fully discrete schemes for approximating the solutions based on a semi-implicit Euler discretization in time and Finite Element (FE) discretization on space (restricted to triangularization made up of right-angled simplices) of two equivalent systems. For the first numerical scheme, we use a splitting technique to deal with the haptotaxis term, leading to introduce an equivalent system with a new variable given by the gradient of extracellular matrix. This scheme is well-posed and preserves the non-negativity of extracellular matrix and the degrading enzyme. We analyze error estimates and convergence towards regular solutions. The second numerical scheme is based on an equivalent formulation in which the cancer cell density equation is expressed in divergence form through a suitable change of variables. This second numerical scheme preserves the non-negativity of all the discrete variables. Finally, we present some numerical simulations in agreement with the theoretical analysis.

Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system \[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\] in a smoothly bounded domain Ω ⊂ ℝ2, with parameters Dw &gt; 0, Dz &gt; 0, β &gt; 0 and ρ ⩾ 0.Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β &lt; 1, whereas whenever β &gt; 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) &gt; 1/(\beta -1)$, infinite-time blow-up occurs at least in the particular case when ρ = 0.In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β &gt; 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ &gt; 0.

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.

Global Solvability and Optimal Control to a Haptotaxis Cancer Invasion Model with Two Cancer Cell Species

This paper deals with an optimal control problem for a haptotaxis model which describes two types of cancer cell invading the surrounding healthy tissue. Cancer cells can be treated by therapeutic agents (radiotherapy or chemotherapy) considered as the control variable. Our primary goal is to characterize an optimal control which balances the therapeutic benefits with its side effects. Firstly, the global bounded weak solution of state system in spatial dimensions $$1\le N\le 5$$ are obtained by using the Banach fixed-point theorem and a new iteration criterion of $$L^p$$ estimates as well as the adapted Moser-type iteration technique of $$L^\infty $$ estimates. Subsequently, the existence of optimal pair is proved by applying the technique of minimizing sequence. Furthermore, we derive the first-order necessary optimality condition by means of the Lipschitz continuity of the control-to-state mapping and the methods of convex perturbation. Finally, we present numerically the spatio-temporal dynamics of species, optimal control and optimal regimen of therapeutic agents administrated to illustrate the concrete realization of the theoretical results obtained in this work, and to validate some clinical findings.

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.

A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy

In a two-dimensional smoothly bounded domain, we consider the oncolytic virotherapy model given by ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v,wt=DwΔw−w+uz,zt=DzΔz−z−uz+βw,with parameters Dw>0,Dz>0,β>0 and ρ≥0.According to previous findings, a corresponding Neumann initial–boundary value problem possesses a globally defined classical solution whenever the prescribed initial data are suitably regular and satisfy appropriate positivity assumptions. In the present study it is shown that if β<1, then any such solution is uniformly bounded. This complements a recent result which in the case ρ=0 has asserted the existence of unbounded solutions whenever β>1.

(In)stability of Travelling Waves in a Model of Haptotaxis

We examine the spectral stability of travelling waves of the haptotaxis model studied in Harley et al (2014a). In the process we apply Li\'enard coordinates to the linearised stability problem and use a Riccati-transform/Grassmanian spectral shooting method \'a la Harley et al (2015), Ledoux et al (2009) and Ledoux et al (2010) in order to numerically compute the Evans function and point spectrum of a linearised operator associated with a travelling wave. We numerically show the instability of non-monotone waves (type IV) and the stability of the monotone ones (types I-III) to perturbations in an appropriately weighted space.