35B40 Asymptotic behavior of solutions
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- asymptotic behavior (2)
- delay (2)
- haptotaxis (2)
- multiscale model (2)
- chemotaxis (1)
- go-or-grow (1)
- multiscale models (1)
- pH-taxis (1)
- reaction-diffusion-taxis equations (1)
- time-delayed carrying capacities (1)
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We investigate a PDE-ODE system describing cancer cell invasion in a tissue network. The model is an extension of the multiscale setting in [28,40], by considering two subpopulations of tumor cells interacting mutually and with the surrounding tissue. According to the go-or-grow hypothesis, these subpopulations consist of moving and proliferating cells, respectively. The mathematical setting also accommodates the effects of some therapy approaches. We prove the global existence of weak solutions to this model and perform numerical simulations to illustrate its behavior for different therapy strategies.
We propose a model for acid-mediated tumor 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 tumor cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behavior of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following [2,31] we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumor 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 in order to illustrate the behavior of the model.
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.