2003 A 3D host-immune-tumor system

Christophe LETELLIER
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Recent advances in oncology lead to promote the role of the tumor micro-environment in tumor growth [1] . In fact, mathematical cancer model taking into account normal (healthy) cells - thus the micro-environment - interacting with immune and tumor cells are not numerous. There is one proposed by Owen and Sherratt [2] which remains mainly focused on tumor–macrophage interactions, the normal cells being only considered for their ability to colonize the site studied. One of the most interesting models seems to be the model proposed by Lisette De Pillis and Ami Radunskaya [3] which incorporates host,immune and tumor cells to reproduce certain qualitative aspects as oscillations in tumor size(Jeff’s phenomenon) [4] or tumor dormancy [5]. This model is rather generic in the sense that it is not specific to a given type of cancer. It is indeed based on quite common interactions between host, immune and tumor cells.The model is

      \displaystyle \dot{x}=\rho_{1}x(1-x)-\alpha_{13}xz \\[0.1 cm]
      \displaystyle \dot{y}=\frac{\rho_{2}yz}{1+z}-\alpha_{23}yz
                             -\delta_{2}y \\[0.3 cm]
      \displaystyle \dot{z}=\rho_{3}z(1-z)-\alpha_{31}zx-\alpha_{32}zy 

where x designates the normalized population of host cells, y being the population of effector immune cells and z the population of tumor cells. Tumor cells compete for resources with the two other populations of cells. From that point of view, they are the ``generalist’’ competitors while the two others are ``specific competitors’’.

This model produces a chaotic attractor for the set of parameter values as follows [6]

- \rho_{1}=0.518 host cell growth rate ;
- \alpha_{13}=1.5 : host cell killing rate by tumor cells ;
- \rho_{2}=4.5 : effector immune cell growth rate ;
- \alpha_{23}=0.2 : effector immune cell inhibition rate by tumor cells ;
- \delta_{2}=0.5 : effector immune cell natural death rate ;
- \rho_{3}=1 : tumor growth rate ;
- \alpha_{31}=1 : tumor killing rate by host cells ;
- \alpha_{32}=2.5 : tumor cell killing rate by effector immune cells.

The chaotic attractor (Fig. 1) was investigated in details in [7].

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Fig. 1. Unimodal chaotic attractor produced by the cancer model.

[1] M. J. Bissell & W. C. Hines, Why don’t we get more cancer ? A proposed role of the microenvironment in restraining cancer progression, Nature Medicine, 17 (3), 320-329, 2011,

[2] M. R. Owen & J. A. Sherratt, Modelling the macrophage invasion of tumours : effects on growth and composition, IMA Journal of Mathematics Applied in Medicine and Biology, 15, 165-185, 1998.

[3] L. G. De Pillis & A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy : an optimal control approach, Journal of Theoretical Medicine, 3, 79-100, 2001. [[L. G. De Pillis & A. Radunskaya, The dynamics of an optimally controlled tumor model : a case study, Mathematical and Computer Modelling, 37, 1221-1244, 2003.

[4] R. Tholimson, Measurement and management of carcinoma of the breast, Clinical Radiobiology, 33 (5), 481-493, 1982.

[5] J. Farrar, K. Katz, J. Windsor, G. Trush, R. Scheuermann, J. Uhr & N. Street, Cancer dormancy vii. A regulatory role for CD8+ T cells and IFN-gamma in establishing and maintaining the tumor-dormant state, Journal of Immunology, 162 (5), 2842-2849, 1999.

[6] M. Itik & S. P. Banks, Chaos in three-dimensional cancer model, International Journal of Bifuraction & Chaos, 20, 71-73, 2010.

[7] C. Letellier, F. Denis & L. A. Aguirre, What can be learned from a chaotic cancer model ?, Journal of Theoretical Biology, 322, 7-16, 2013.

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