Global Weak Solutions in a PDE-ODE System Modeling Multiscale Cancer Cell Invasion

Abstract

We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [22] and accounts on the macroscopic level for the evolution of cell and tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble and insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior to their subcellular dynamics.The resulting system, consisting of three partial and three ordinary differential equations including a temporal delay, in particular involves chemotactic and haptotactic cross-diffusion. In order to overcome technical obstacles stemming from the corresponding highest-order interaction terms, we base our analysis on a certain functional, inter alia involving the cell and tissue densities in the diffusion and haptotaxis terms respectively, which is shown to enjoy a quasi-dissipative property. This will be used as a starting point for the derivation of a series of integral estimates finally allowing for the construction of a generalized solution as the limit of solutions to suitably regularized problems.

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

“…As mentioned above, this modelling approach follows various other recent studies on cancer cell-matrix adhesion via integrins (Stinner et al, 2014(Stinner et al, , 2015(Stinner et al, , 2016Meral et al, 2015a,b). Moreover, the coupling of PDEs for cell movement with ODEs for the dynamics of cell receptors has been considered not only for cancer cells but also for other types of cells, such as Dictyostelium cells (Höfer et al, 1995a,b).…”

“…Moreover, we assume that cell mutation changes the density of receptors (since in highly metastatic cancers, the expression of integrins is up-regulated (Kidera et al, 2010)). We follow an approach similar to (Stinner et al, 2014(Stinner et al, , 2015(Stinner et al, , 2016Meral et al, 2015a,b) and assume that the dynamics of integrins c(x,t) can be described by an ODE:…”

“…In Appendix B we present a modified mesoscale version of model (2.13)-(2.16) where we assume that the two cancer cell populations are structured by an internal variable that describes the integrin level. (Here we use the terminology of "mesoscale" and "microscale-macroscale" models in the same spirit as Stinner et al (2014); Lorenz & Surulescu (2014); Hunt & Surulescu (2017). )…”

“…Also, while previous nonlocal models have assumed constant adhesive interactions, here we will assume that these adhesive interactions depend on the level of integrins. Note that instead of assuming a kinetic (mesoscale) approach for cell-dependence on integrins as in Lorenz & Surulescu (2014); Engwer et al (2015); Perthame et al (2016); Engwer et al (2017); Hunt & Surulescu (2017), here we follow a microscale-macroscale approach similar to Stinner et al (2014Stinner et al ( , 2015Stinner et al ( , 2016; Meral et al (2015a,b) and couple the parabolic-hyperbolic model for spatial cell dynamics with an ODE for integrins dynamics. We then investigate this complex multiscale model in terms of pattern formation, with particular attention being payed to the role of mutation rate on the coexistence (or not) of cell sub-populations that form stationary aggregations or travelling wave patterns.…”

A stochastic multiscale model for acid mediated cancer invasion

“…This requires a higher dimensional and more complex modeling framework that couples the subcellular dynamics at microscopic scale with the population dynamics at the macroscopic scale. Micro-macro models of this type (in a different, but related context) were proposed and analysed e.g., in [32,31,37].…”