# American Institute of Mathematical Sciences

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February  2004, 4(1): 81-98. doi: 10.3934/dcdsb.2004.4.81

## Macrophage-tumour interactions: In vivo dynamics

 1 Centre for Mathematical Medicine, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom, United Kingdom, United Kingdom

Received  December 2002 Revised  April 2003 Published  November 2003

Experimentalists are developing new therapies that exploit the tendency of macrophages, a type of white blood cell, to localise within solid tumours. The therapy studied here involves engineering macrophages to produce chemicals that kill tumour cells. Accordingly, a simple mathematical model is developed that describes interactions between normal cells, tumour cells and infiltrating macrophages. Numerical and analytical techniques show how the ability of the engineered macrophages to eliminate the tumour changes as model parameters vary. The key parameters are $m^*$, the concentration of engineered macrophages injected into the vasculature, and $k_1$, the rate at which they lyse tumour cells. As $k_1$ or $m^*$ increases, the average tumour burden decreases although the tumour is never completely eliminated by the macrophages. Also, the stable solutions are oscillatory when $k_1$ and $m^*$ increase through well-defined bifurcation values. The physical implications of our results and directions for future research are also discussed.
Citation: H.M. Byrne, S.M. Cox, C.E. Kelly. Macrophage-tumour interactions: In vivo dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 81-98. doi: 10.3934/dcdsb.2004.4.81
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