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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.
Cell migration is essential for embryogenesis, wound healing, immune surveillance, and
progression of diseases, such as cancer metastasis. For the migration to occur, cellular
structures such as actomyosin cables and cell-substrate adhesion clusters must interact.
As cell trajectories exhibit a random character, so must such interactions. Furthermore,
migration often occurs in a crowded environment, where the collision outcome is deter-
mined by altered regulation of the aforementioned structures. In this work, guided by a
few fundamental attributes of cell motility, we construct a minimal stochastic cell migration
model from ground-up. The resulting model couples a deterministic actomyosin contrac-
tility mechanism with stochastic cell-substrate adhesion kinetics, and yields a well-defined
piecewise deterministic process. The signaling pathways regulating the contractility and
adhesion are considered as well. The model is extended to include cell collectives. Numer-
ical simulations of single cell migration reproduce several experimentally observed results,
including anomalous diffusion, tactic migration, and contact guidance. The simulations
of colliding cells explain the observed outcomes in terms of contact induced modification
of contractility and adhesion dynamics. These explained outcomes include modulation
of collision response and group behavior in the presence of an external signal, as well as
invasive and dispersive migration. Moreover, from the single cell model we deduce a pop-
ulation scale formulation for the migration of non-interacting cells. In this formulation,
the relationships concerning actomyosin contractility and adhesion clusters are maintained.
Thus, we construct a multiscale description of cell migration, whereby single, collective,
and population scale formulations are deduced from the relationships on the subcellular
level in a mathematically consistent way.
We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the
extracellular matrix. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations for a two-dimensional setting.