Modelling of macrophage interactions by partial differential equations

Mohd Rashid Admon, Normah Maan


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.


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

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Wilking, N., Kasteng, F., Bergh, J., Jonsson, B., Kossler, I., Martin, M., Maranon, G., Normand, C., Reed, L., and Widdershoven, G. (2009). A Review of Breast Cancer Care and Outcomes in 18 Countries in Europe, Asia, and Latin America. Retrieved from Comparator Report website:

Mayer, M., Hunis, A., Oratz, R., Glennon, C., Spicer, P., Caplan E., and Fallowfield, L. (2010). Living with metastatic breast cancer: A global patient survey, Community Oncology, 7, 406-412.

Althuis, M. D., Dozier, J. M., Anderson, W. F., Devesa, S. S., and Brinton, L. A. (2005). Global trends in breast cancer incidence and mortality 1973-1997, International Jounal of Epidemiology, 34, 405-412.

Polyak, K., (2007). Breast cancer: origins and evolution, The Journal of Clinical Investigation, 117, 3155-3163.

Koren, S. and Alj, M. B. (2015). Breast tumor heterogeneity: Source of fitness, hurdle for therapy, Molecular Cell, 60, 537-546.

Gunduz, M. and Gunduz, E. (2011). Breast cancer - Focusing tumor microenvironment, stem cell and metastasis. Croatia: InTech, 69-84.

Enderling, H., Alexander, R. A., Anderson, M. A., Chaplain, J., Munro, A. J., and Vaidya. (2006). Mathematical modelling of radiotherapy strategies for early breast cancer, Journal of Theoretical Biology, 241, 158-171.

Cameron, D. A. (1997). Mathematical modelling of the Response of Breast Cancer to Drug Therapy, Journal of Theoretical Medicine, 2, 137-151.

Zhang, X., Fang, Y., Zhao, Y., and Zheng, W. (2014). Mathematical modelling the pathway of human breast cancer, Mathematical Biosciences, 253, 25-29.

Kim, Y., and Othmer. H. G. (2013). A hybrid model of tumor-stromal interactions in a breast cancer, Bulletin of Mathematical Biology, 75, 1304-1350.

McDuffie, M. (2014). A hormone therapy model for breast cancer using linear cancer networks. Rose-Hulman Undergraduate Mathematics Journal, 15, 144-156.

Elgert, K. D., Alleva, D. G., and Mullins, D. W. (1998). Tumor –induce immune dysfunction: the macrophage connection. Journal Leukocyte Biology, 64, 275-290.

Lewis, C. E., and Pollard, J. W. (2006). Distinct role of macrophages in different tumor microenvironments, Cancer Research, 66, 605-612.

Condeelis, J. and Pollard, J. W. (2006). Macrophages: Obligate partners for tumor cell migration, invasion and metastasis, Cell, 124, 263-266.

Wynn, T. A., Chawla, A., and Pollard, J. W. (2013). Macrophage biology in development, homeostasis and disease, Nature, 496, 445-455.

Qian, B. Z. and Pollard, J. W. (2010). Macrophage diversity enhances tumor progression and metastasis, Cell, 141, 39-51.

Leek, R. D. and Harris, A. L. (2002). Tumor-associated macrophages in breast cancer, Journal of Mammary Gland Biology and Neoplasia, 7, 177-189.

van der Bij, G. J., Oosterling, S. J., Meijer, S., Beelen, R. H. J., and van Egmond, M. (2005). The role of macrophages in tumor development. Cellular Oncology, 27, 203-213.

Hao, N. B., Lu, M. H., Fan, Y. H., Cao, L. H., Zhang, Z. R., and Yang, S. M. (2012). Macrophages in tumor microenvironments and the progression of tumors, Clinical and Developmental Immunology, 2012, 1-11.

Sousa, S., Brion, R., Lintunen, M., Kronqvist, P., Sandholm, P., Monkkonen, J., Lehtinen, P. L., Lauttia, S., Tynninen, O., Joensuu, H., Heymann, D., and Maatta, J. A. (2015). Human breast cancer cells educate macrophages toward the M2 activation status, Breast Cancer Research, 17, 101-124.

Italiani, P. and Boraschi, D. (2015). New insights into tissue macrophages: from their origin to the development of memory, Immune Network, 15, 167-176.

Sica, A. and Mantovani, A. (2012). Macrophage plasticity and polarization: in vivo veritas, Journal of Clinical Investigation, 122, 787-795.

Bingle, L., Brown, N. J. and Lewis, C. E. (2002). The role of tumor associated macrophages in tumor progression: Implications for new anticancer therapies, Journal of Pathology, 196, 254-265.

Mills, C. D., Kincaid, K., Alt, J. M., Heilma, M. J., and Hill, A. M. (2000). M1/M2 Macrophages and the Th1/Th2 paradigm, The Journal of Immunology, 164, 6166-6173.

van Netten, J. P., Ashmead, B. J., Parker, R. I., Thornton, I. G., Fletcher, C., Cavers, D., Coy, P., and Brigden, M. L. (1993). Macrophage-tumor cell associations: A factor in metastasis of breast cancer?, Journal of Leukocyte Biology. 54, 360-362.

Lin, E. Y., Nguyen, A. V., Russell, R. G., and Pollard, J. W. (2001). Colony-stimulating factor 1 promotes progression of mammary tumors to malignancy, Journal Experimental Medicine, 193, 727-739.

Wyckoff, J., Wang, W., Lin, E. Y., Wang, Y., Pixley, F., Stanley, E. R., Graf, T., Pollard, J. W., Segall J., and Condeelis, J. (2004). A paracrine loop between tumor cells and macrophages is required for cell migration in mammary tumors, Cancer Research, 64, 7022-7029.

Goswami, S., Sahai, E., Wyckoff, J. B., Cammer, M., Cox,D., Pixley, F. J., Stanley, E. R., Segall, J. E., and Condeelis. J. S. (2005). Macrophages promote the invasion of breast carcinoma cells via colony stimulating factor-1/epidermal growth factor paracrine loop, Cancer Research, 65, 5278-5283.

Knutsdottir, H., Palsson, E., and Edelstein-Keshet, L. (2014). Mathematical model of macrophage-facilitated breast cancer cells invasion, Journal of Theoretical Biology, 357, 184-199.

Elitas, M. and Zeinali, S. (2016). Modeling and simulation of EGF-CSF-1 pathway to investigate glioma-macrophage interaction in brain tumors, International Journal of Cancer Study and Research, S5, 1-8.

Hillen, T. and Painter. K. J. (2009). A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58, 183-217.

Patsialou, A., Wyckoff, J., Wang, Y., Goswami, S., Stanley, E. R., and Condeelis, J. S., (2009). Invasion of human breast cancer cells in vivo requires both paracrine and autocrine loops involving the colony-stimulating factor-1 receptor, Cancer Research, 69, 9498-9506.

Keller, E. F. and Segel, L. E. (1971). Model for Chemotaxis, Journal of Theoretical Biology, 30, 225-234.

Keshet, L. E. (1976). Mathematical Models in Biology, McGraw-Hill, Inc., 393-395.

Luca, M., Ross, A. C., Keshet, L. E., and Mogilner, A. (2003). Chemotactic Signaling, Microglia, and Alzheimer’s Disease Senile Plaques: Is there a Connection? Bulletin of Mathematical Biology, 65, 693-730.

Russell., B. C. (2013). Using partial differential equations to model and analyze the treatment of a chronic wound with oxygen therapy techniques. Honors College Capstone Experience/Thesis Projects. Western Kentucky University.

Guffey, S. (2015). Application of a numerical method and optimal control theory to a partial differential equation model for a bacterial infection in a chronic wound. Masters Theses & Specialist Projects. Western Kentucky University.

Maini, P. K., Myerscough, M. R., Winters, K. H and Murray, J. D. (1991). Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bulletin of Mathematical Biology, 53, 701-719.

Myerscough, M. R., Maini, P. K., and Painter, K. J. (1998). Pattern Formation in a Generalized Chemotactic Model, Bulletin of Mathematical Biology, 60, 1-26.

Knutsdottir, H., Condeelis, J. S., and Palsson. E. (2016). 3-D individual cell based computational modelling of tumor cell-macrophage paracrine signalling mediated by EGF and CSF-1 gradients, Integrative Biology, 8, 104-119.



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