Modelling of macrophage interactions by partial differential equations


  • Mohd Rashid Admon Universiti Teknologi Malaysia
  • Normah Maan Universiti Teknologi Malaysia



Partial differential equations, macrophages, breast cancer, saturating functions, Stability region


The recruitment of macrophages at the tumor sites is the earliest immune response takes place during tumor progression. In breast cancer, experimental studies reveals that the tumor cells are capable of taking advantage on the plasticity of macrophages. Tumor cells respond to epidermal growth factor, EGF that released by macrophages while macrophages respond to colony stimulating factor 1, CSF-1 that released by tumor cells. This chains continues and results a paracrine signalling loop. Consequently, tumor cells and macrophages will aggregate and invade to other tissues or organ. Tumor cells also receive their own signals, adding a new feature of interaction called autocrine signalling loop. By considering in vitro interactions, a system of partial differential equations that incorporate the saturating functions for secretion terms was developed. This functions describes the production of chemical signals saturates with increasing cell density. Stability analysis are then performed to investigate the conditions for aggregation. For a given average of cells density, the homogeneous steady state is non-trivial and the concentration of CSF-1 and EGF are produced in the saturated production. Stability results show that regions for instability are reduced, compared to previous model which assumes the production rates are linear with increasing cell density. Besides, the inclusion of autocrine signalling loop increase the occurrence of aggregation. Decreasing the production rates and chemotaxis sensitivity, together with increasing the decay rates are required to impede the aggregation from initiated. This results should provide valuable clinical suggestions in guiding medical experts during drug designs.

Author Biographies

Mohd Rashid Admon, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Faculty of Science, UTM Skudai

Normah Maan, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Faculty of Science, UTM Skudai


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