Modelling of macrophages interactions in breast cancer by partial differential equations

Abstract

Tthe signalling interaction between tumor cells and macrophages will form spontaneous aggregation that causes tumor spreading. tumor cells and macrophages interchange their respective signals which results a paracrine and autocrine signalling loop. this interaction process can be represented by mathematical model in the form of chemotaxis and reaction diffusion equations. the existing models that consider paracrine signalling loop alone or with the inclusion of paracrine and autocrine signalling loop had been developed by assuming linear signals production. however, this assumption does not give a better representation on the signal dynamics where it is supposed to be in nonlinear form that saturate with increasing cell densities. therefore, in this research, two existing interaction models are improved by considering the nonlinear form of signals production. besides, another new interaction model is also developed based on the facts that tumor cells release enzyme during the signaling interaction to penetrate the surrounding tissues. the stability analysis is conducted on three separated models to investigate the condition for spontaneous aggregation. each of these conditions then are validated using numerical simulations. stability analysis shows that for all models, the formation of aggregation could be determined by the parameter that represents the secretion and degradation rates of signals together with chemotaxis rates towards signals. however, the inclusion of autocrine signalling loop in the second model increase the possibility of the aggregation. while in the third model, an additional parameter that represents the secretion and degradation rates of enzyme as well as chemotaxis rates towards them could also determine the formation of the aggregation. by numerical simulations, the results are in agreement with the stability analysis obtained for each of the interaction models. besides, cell clusters that result from the aggregation will be merged to the other cells cluster due to the “effective attraction” between them. reducing the production rates of signal or chemotaxis rates towards signals or increasing degradation rates of signal is required to prevent aggregation. the same changes towards enzymes will give the same result on preventing the aggregation. these valuable suggestions are crucial for medical experts during treatments.