A Combined Boundary Element and Finite Element Model of Cell Motion due to Chemotaxis

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. In some cases of chemotaxis, cells secrete the chemical in order to create a concentration gradient that will attract other nearby cells to form clusters. When the cell secreting the chemical is stationary the linear diffusion equation can be used to model the concentration of the chemical as it spreads out into the surrounding fluid medium. However, if the cell is moving then its motion and the resulting motion of the surrounding fluid need to be taken into account in any model of how the chemical spreads out. In the case of a single, circular cell it is possible to express the fluid velocity which results from the motion of the cell in terms of a dipole located at the center of the cell. However if the cell is not circular and/or there is more than one cell in the fluid, a more sophisticated method of determining the fluid velocity is needed. This paper presents a mathematical model for simulating the concentrations of chemical secreted into the surrounding fluid medium from a moving cell. The boundary integral method is used to determine the velocity of the fluid due to the motion of the cell. The concentration of the chemical in the fluid is modelled by the convection–diffusion equation where the fluid velocity term is that given by the boundary integral equation. The resulting differential equation is then solved using the finite element method. The method is illustrated with a number of typical examples.