A multiscale model for pH-tactic invasion with time-varying carrying capacities

We propose a model for acid-mediated tumour invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumour cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behaviour of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following Bartel et al. (2012) pH-taxis: directional tumor cell migration along pH-gradients. Acta Physiol., 204, 113; Paradise et al. (2013) Directional cell migration in an extracellular pH gradient: a model study with an engineered cell line and primary microvascular endothelial cells. Exp. Cell Res., 319, 487–497; we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumour cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed to illustrate the behaviour of the model.