Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy
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 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 and , cancer cells can direct their movement toward regions of higher ECM densities with the haptotactic coefficient , respectively, and that uninfected cells, apart from proliferating logistically at rate , are converted into an infected state upon contact with virus particles, whereas infected cells die owing to lysis at a rate . It is assumed that the static ECM is degraded by both types of cancer cells, possibly remodeled with rate in the sense of spontaneous renewal of healthy tissue. Finally, it is also supposed that besides the random motion with the random motility coefficient, virus particles move up the gradient of ECM with the ECM-OV-taxis rate , increase at a rate β due to the release of free virus particles through infected cells, and undergo decay at the rate 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 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 . 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 in a bounded domain and obtain the asymptotic relaxation of the globally defined classical solution when in [19]. In particular, for the doubly haptotactic variant of (1.1) given by 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 .
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 is adjusted to of Beddington–deAngelis type with positive parameters ([3]) or 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 in a bounded domain with smooth boundary, where for the initial data , we suppose throughout this paper that
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 be a bounded domain with smooth boundary, , , and are positive parameters. Suppose that or . Then for any choice of fulfilling (1.5), there exists such that if , (1.4) admits a unique global classical solution , where , and are uniformly bounded for .
Remark 1.1 In line with the above discussion, the boundedness result on (1.4) with is also valid when .
Remark 1.2 When , one can see that the restriction on in Theorem 1.1 can be replaced by a certain small condition on .
A cornerstone of our analysis is to show that for the suitably small , the functional with and 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 , and , respectively (see (2.1) below). Thanks to a variant of the Gagliardo-Nirenberg inequality involving certain -type norms, the latter offers the sufficient regularity so as to allow for the -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 bounds for and a uniform bound on v. In Section 3, we are devoted to tracking the evolution of , accordingly the bounds for and c in provided by the latter will enable us to establish the -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 be a bounded domain with smooth boundary, and let and . Then there exists such that holds for all .
For the convenience in our subsequent estimation procedure, we let
Bounds for a, b and c in
This section aims to construct an Lyapunov-like functional involving the logarithmic entropy of and c, rather than that of and z, which provides some regularity information of solutions that forms the crucial step in establishing 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 in [21] can be
-bounds for a, b and c
By means of some quite straightforward testing procedures, combining Lemma 2.1 with appropriate interpolation, we can now proceed to turn the outcome of Lemma 3.4 into the -bounds for a, b and c.
Lemma 4.1 Let be the classical solution of (2.1) in . Then one can find fulfilling and as well as for all . Proof Testing the first equation in (2.1) by with , 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.
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