Elsevier

Journal of Differential Equations

Volume 270, 5 January 2021, Pages 94-113
Journal of Differential Equations

Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy

https://doi.org/10.1016/j.jde.2020.07.032Get rights and content

Abstract

This paper is concerned with a haptotactic cross-diffusion system as the model for oncolytic virotherapy, which describes the influence of the extracellular matrix (ECM) taxis over the tumor-oncolytic virus interaction in the form of haptotaxis of both cancer cells and oncolytic virus toward higher ECM densities. The main results assert the global boundedness of solutions to an associated spatially two-dimensional initial-boundary value problems under suitably assumptions on the system parameters.

Introduction

Oncolytic viruses (OV), sometimes referred to as ‘smart bomb’ cancer viruses, are replicating viruses that have been designed to selectively bind to receptors on the tumor cell surface, but not to the surface of normal healthy cells, and accordingly the oncolytic virotherapy has been emerging as a promising novel cancer treatment modality that use replication-competent viruses to attaches to a cancer cell, gain entry and proliferate exponentially, eventually causing cell death (lysis). At lysis, the virus particles inside them are liberated and become available to infect adjacent cancer cells, thereby eventually leading to a reduced overall degrading impact on the surrounding healthy tissue fibers (extracellular matrix, ECM) ([5], [11], [8], [9], [14]).

It is observed that oncolytic virotherapy currently has some limitations in the oncolytic efficacy, which might be the result of virus clearance and the physical barriers inside tumours such as the interstitial fluid pressure, ECM deposits and tight inter-cellular junctions. To better understand the physical barriers that limit virus spread, the authors of [1] recently proposed the PDE–ODE system of the form{ut=DuΔuξu(uv)+μuu(1u)ρuuz,wt=DwΔwξw(wv)δww+ρwuz,vt=(αuu+αww)v+μvv(1v),zt=DzΔzξz(zv)δzzρzuz+βw, to describe the coupled dynamics of uninfected cancer cells u, OV-infected cancer cells w, ECM v and oncolytic viruses z. Herein, the underlying modeling hypotheses are that in addition to random diffusion with the respective motility coefficient Du and Dw, cancer cells can direct their movement toward regions of higher ECM densities with the haptotactic coefficient ξu,ξw, respectively, and that uninfected cells, apart from proliferating logistically at rate μu, are converted into an infected state upon contact with virus particles, whereas infected cells die owing to lysis at a rate δw. It is assumed that the static ECM is degraded by both types of cancer cells, possibly remodeled with rate μv in the sense of spontaneous renewal of healthy tissue. Finally, it is also supposed that besides the random motion with Dz the random motility coefficient, virus particles move up the gradient of ECM with the ECM-OV-taxis rate ξz, increase at a rate β due to the release of free virus particles through infected cells, and undergo decay at the rate δz accounting for the natural virions' death as well as the trapping of these virus particles into the cancer cells.

From a mathematical perspective, model (1.1) on the one hand involves three simultaneous haptotaxis processes, but on the other hand contains the production term ρuz in w-equation which distinguishes (1.1) from the most of the previous haptotaxis ([6], [12], [10], [16], [23]) and chemotaxis–haptotaxis models ([13], [15], [18]). In fact, the haptotactic migration of u,z toward higher densities v simultaneously, in which no smoothing action on the spatial regularity of v can be expected, renders us unable to apply smoothing estimates for the Neumann heat semigroup to gain a priori boundedness information on u and z beyond the norm in L1(Ω). Accordingly this superlinear production term ρuz in (1.1) seems likely to increase the destabilizing potential in the sense of enhancing the tendency towards blow-up of solutions, and thus becomes the key contributor to mathematical challenges already even in the derivation of global solvability theory of (1.1), which is also indicated in the qualitative analysis of chemotaxis-May-Nowak model ([3], [2], [7], [22]).

Though the methodological limitations seem to widely restrict the theoretical understanding of the full model (1.1), some analytical works on simplifications of the latter have recently been achieved in [21], [20], [19]. Indeed, upon neglecting haptotactic migration processes of infected tumor cells and oncolytic viruses, renewal of ECM as well as proliferation of infected tumor cells, Tao and Winkler consider{ut=Δu(uv)ρuz,wt=DwΔww+uz,vt=(u+w)v,zt=DzΔzzuz+βw in a bounded domain ΩR2 and obtain the asymptotic relaxation of the globally defined classical solution when β<1 in [19]. In particular, for the doubly haptotactic variant of (1.1) given by{ut=DuΔuξu(uv)+μuu(1u)ρuuz,wt=DwΔwξw(wv)δww+ρwuz,vt=(αuu+αww)v+μvv(1v),zt=DzΔzδzzρzuz+βw, authors of [21] assert the global classical solvability of the corresponding no-flux initial-boundary problem for any suitably regular initial data and the possibility of μu=0.

The purpose of the present work is to a more comprehensive understanding of model (1.1) in the biologically most relevant constellation in which the haptotactic motion of virus particles is taken into account particularly, and either the production term uz or proliferating term μuu(1u) is adjusted to uzku+θu of Beddington–deAngelis type with positive parameters ku,θ ([3]) or μuu(1ur) of superquadratic type, respectively. Specially, on closing the system by convenient boundary conditions and initial data, we shall be concerned with the PDE-ODE system given by{ut=DuΔuξu(uv)+μuu(1ur)ρuzku+θu,xΩ,t>0,wt=DwΔwξw(wv)δww+ρuzku+θu,xΩ,t>0,vt=(αuu+αww)v+μvv(1v),xΩ,t>0,zt=DzΔzξz(zv)δzzρuzku+θu+βw,xΩ,t>0,(Duuξuuv)ν=(Dwwξwwv)ν=(Dwzξzzv)ν=0,xΩ,t>0,u(x,0)=u0(x),w(x,0)=w0(x),v(x,0)=v0(x),z(x,0)=z0(x),xΩ in a bounded domain ΩR2 with smooth boundary, where for the initial data (u0,w0,v0,z0), we suppose throughout this paper that{u0,w0,z0andv0are nonnegative functions fromC2+ϑ(Ω¯)for someϑ(0,1),withu00,w00,z00,v00andw0ν=0onΩ.

Beyond the global classical solvability, in the present work we focus on the global boundedness of classical solutions to (1.4)–(1.5) stated as follows, which can be regarded as a first step toward the qualitative comprehension of (1.4).

Theorem 1.1

Let ΩR2 be a bounded domain with smooth boundary, Du,Dw, Dz, ξu,ξw,ξz,μu,μv,ρ,ku,αu,αw,β,δw and δz are positive parameters. Suppose that r=1,θ>0 or r>1,θ0. Then for any choice of (u0,w0,v0,z0) fulfilling (1.5), there exists C>0 such that if ξwαw<C, (1.4) admits a unique global classical solution (u,w,v,z), where u(,t)L(Ω), w(,t)L(Ω) and z(,t)L(Ω) are uniformly bounded for t(0,).

Remark 1.1

In line with the above discussion, the boundedness result on (1.4) with r=1,θ=0 is also valid when ξz=0.

Remark 1.2

When μv=0, one can see that the restriction on ξwαw in Theorem 1.1 can be replaced by a certain small condition on v0.

A cornerstone of our analysis is to show that for the suitably small ξwαw, the functionalF(t):=AΩeχwv(,t)b(,t)lnb(,t)+Ωeχuv(,t)a(,t)lna(,t)+Ωeχzv(,t)c(,t)lnc(,t) with a=ueχuv,b=weχwv and c=zeχzv enjoys a certain quasi-dissipative property under appropriate choice of the positive constant A (of (3.22)). As the first step in this direction, we perform the variable change used in several precedents, by which the crucial haptotactic contributions to the equations in (1.4) are reduced to zero-order terms χua(αuu+αww)vχuμvav(1v), χwb(αuu+αww)vχwμvbv(1v) and χzc(αuu+αww)vχzμvcv(1v), respectively (see (2.1) below). Thanks to a variant of the Gagliardo-Nirenberg inequality involving certain LlogL-type norms, the latter offers the sufficient regularity so as to allow for the L-bounds of solutions in present two-dimensional setting.

The structure of this paper is as follows: besides the local well-posedness of (1.4) with a convenient criterion for extensibility of local solutions, Section 2 provides both the L1 bounds for u,w,z and a uniform L bound on v. In Section 3, we are devoted to tracking the evolution of F, accordingly the bounds for a,b and c in LlogL provided by the latter will enable us to establish the L-bounds for u, w and z in Section 4.

Section snippets

Preliminaries

The following lemma is a special case of Lemma A.5 in [17] and can be regarded as a variant of the Gagliardo–Nirenberg inequality originally derived in [4], which will be of importance in the later analysis.

Lemma 2.1

Let ΩR2 be a bounded domain with smooth boundary, and let p(1,) and ε>0. Then there exists K(p,ε)>0 such thatφL2(p+1)p(Ω)2(p+1)pεφL2(Ω)2Ω|φ|2p|ln|φ||+K(p,ε)(φL2p(Ω)2(p+1)p+1) holds for all φW1,2(Ω).

For the convenience in our subsequent estimation procedure, we letχu:=ξuDu,χw:=

Bounds for a, b and c in LlogL

This section aims to construct an Lyapunov-like functional involving the logarithmic entropy of a,b and c, rather than that of u,w and z, which provides some regularity information of solutions that forms the crucial step in establishing L bounds for u, w and z in the present spatially two-dimensional setting. It should be mentioned that upon the special structure of (1.3), inter alia neglecting haptotactic migration processes of oncolytic viruses z, the energy-like functional F in [21] can be

L-bounds for a, b and c

By means of some quite straightforward Lp testing procedures, combining Lemma 2.1 with appropriate interpolation, we can now proceed to turn the outcome of Lemma 3.4 into the L-bounds for a, b and c.

Lemma 4.1

Let (a,b,v,c) be the classical solution of (2.1) in Ω×[0,Tmax). Then one can find C>0 fulfillinga(,t)L(Ω)C andb(,t)L(Ω)C as well asc(,t)L(Ω)C for all t(0,Tmax).

Proof

Testing the first equation in (2.1) by eξuvap1 with p>4, integrating by parts and using to the Young inequality, we can

Acknowledgments

This work is partially supported by NSFC (No. 11571363), the Research Grant Funds of Minzu University of China and the Beijing Key Laboratory on MCAACI.

References (23)

  • N. Bellomo et al.

    Stabilization in a chemotaxis model for virus infection

    Discrete Contin. Dyn. Syst., Ser. S

    (2020)
  • Cited by (16)

    View all citing articles on Scopus
    View full text