On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation

Abstract

We propose a multiscale model for tumor cell migration in a tissue network. The system of equations involves a structured population model for the tumor cell density, which besides time and position depends on a further variable characterizing the cellular state with respect to the amount of receptors bound to soluble and insoluble ligands. Moreover, this equation features pH-taxis and adhesion, along with an integral term describing proliferation conditioned by receptor binding. The interaction of tumor cells with their surroundings calls for two more equations for the evolution of tissue fibers and acidity (expressed via concentration of extracellular protons), respectively. The resulting ODE-PDE system is highly nonlinear. We prove the global existence of a solution and perform numerical simulations to illustrate its behavior, paying particular attention to the influence of the supplementary structure and of the adhesion.

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

“…If we substitute this state into (B.1) we recover model (2.22), with the exception of equation (2.22d) which is now a steady-state equation. We note here that while different microscale-macroscale models (Stinner et al, 2014(Stinner et al, , 2015(Stinner et al, , 2016Meral et al, 2015a,b) and mesoscale models (Lorenz & Surulescu, 2014;Engwer et al, 2015;Perthame et al, 2016;Engwer et al, 2017;Hunt & Surulescu, 2017) for tumour-ECM interactions via integrins have been developed over the last two decades, to our knowledge there are not many studies that compare and contrast the outcomes of different ways the integrins are incorporated into these different models.…”

“…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). However, the past 10-15 years have seen the development of structured cell population models (described by mesoscale models) where the macroscopic cell dynamics is dependent on a microscopic variable that characterises the cells at molecular level (Erban & Othmer, 2004Xue et al, 2011;Lorenz & Surulescu, 2014;Engwer et al, 2015;Perthame et al, 2016;Engwer et al, 2017;Hunt & Surulescu, 2017). 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.…”

“…In particular, Domschke et al (2014) also assumed mutation between different cell populations. A different class of nonlocal models for cancer invasion has been developed in the context of the velocity-jump framework introduced in Othmer et al (1988) and the kinetic models for active particles framework described, for example, in Bellomo et al (2010); see the transport models in Kelkel & Surulescu (2012); Lorenz & Surulescu (2014); Engwer et al (2015Engwer et al ( , 2017; 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.…”

Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation

“…Several nonlocal mathematical models of adhesion (meaning models in which events or cell concentrations at a point x in the tumor domain depend on events at points distinct from x but within a finite neighborhood of x) have been proposed in the literature. For an example of such cell-to-cell adhesion models, see Armstrong et al [3], Chaplain et al [1,9], Engwer et al [18], and Stinner et al [55], the latter two references addressing the effects of adhesion on tumor-cell invasion.…”