The Mathematical Modelling of the Motion of Biological Cells in Response to Chemical Signals

Chemotaxis is the biological process whereby a cell moves in the direction in which the concentration of a chemical in the fluid medium surrounding the cell is increasing. The chemical signal may be secreted by cells to signal other nearby cells (or clusters of cells) so that they can combine to form larger clusters. In a full simulation, spread of the chemical signal is modelled using the convection-diffusion equation which can be solved using the finite element method. This can then be coupled to a boundary integral model of the fluid motion due to the motion of the cells. However, the use of the finite element method to model the spread of the chemical is computationally expensive as the finite element approximations to some of the terms in the equations need to be recalculated at each time step as the relative velocities of the cells change. This can be avoided by adopting a simple explicit formula for the concentrations of the secreted chemical and allowing this to move with the cell that is emitting the signal. This paper will present some numerical results to show how the proposed model can be used to determine the motion of the cells with a much small computational cost when compared to the full coupled finite element method, albeit at the cost of losing some of the accuracy in the calculated concentrations of the chemical.