Mathematical model of macrophage-facilitated breast cancer cells invasion.

Mortality from breast cancer stems from its tendency to invade into surrounding tissues and organs. Experiments have shown that this metastatic process is facilitated by macrophages in a short-ranged chemical signalling loop. Macrophages secrete epidermal growth factor, EGF, and respond to the colony stimulating factor 1, CSF-1. Tumor cells secrete CSF-1 and respond to EGF. In this way, the cells coordinate aggregation and cooperative migration. Here we investigate this process in a model for in vitro interactions using two distinct but related mathematical approaches. In the first, we analyze and simulate a set of partial differential equations to determine conditions for aggregation. In the second, we use a cell-based discrete 3D simulation to follow the fates and motion of individual cells during aggregation. Linear stability analysis of the PDE model reveals that decreasing the chemical secretion, chemotaxis coefficients or density of cells or increasing the chemical degradation in the model could eliminate the spontaneous aggregation of cells. Simulations with the discrete model show that the ratio between tumor cells and macrophages in aggregates increases when the EGF secretion parameter is increased. The results also show how CSF-1/CSF-1R autocrine signalling in tumor cells affects the ratio between the two cell types. Comparing the continuum results with simulations of a discrete cell-based model, we find good qualitative agreement.