Numerical solutions for a model of tissue invasion and migration of tumour cells.

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points.

Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics.

Pseudo-spectral approximations are constructed for the model equations describing the population kinetics of human tumor cells in vitro and their responses to radiotherapy or chemotherapy. These approximations are more efficient than finite-difference approximations. The spectral accuracy of the pseudo-spectral method allows us to resolve the model with a much smaller number of spatial grid-points than required for the finite-difference method to achieve comparable accuracy.

A variant of pseudospectral method for activity-dependent dendritic branch model.

A new variant of the pseudospectral method for an activity-dependent dendritic branch model is proposed. This algorithm incorporates the Neumann boundary conditions in a more efficient way than in the algorithms proposed before for similar problems. Numerical experiments indicate that the new algorithm is more efficient than the previous algorithms discussed in the literature on the subject.