### Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

#### Abstract

A new optimal control model of the interactions between a growing tumour and the host immune system, along with an immunotherapy treatment strategy, is presented. The model is based on an ordinary differential equation model of interactions between the growing tumour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treatment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum allowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

**References**- Bunimovich–Mendrazitsky, S., Shochat, E. and Stone, L. Mathematical model of BCG immunotherapy in superficial bladder cancer.
*Bulletin of Mathematical Biology*, 69:1847–1870, 2007. doi:10.1007/s11538-007-9195-z - Burden, T., Ernstberger, J. and Fister, K. R. Optimal control applied to immunotherapy.
*Discrete and Continuous dynamical Systems-Series B*, 4(1):135–146, 2004. doi:10.3934/dcdsb.2004.4.135 - Cappuccio, A., Elishmereni, M. and Agur, Z. Cancer immunotherapy by interleukin-21: Potential treatment strategies evaluated in mathematical model.
*Cancer Res.*, 66(14):7293–7300, 2006. doi:10.1158/0008-5472.CAN-06-0241 - Cappuccio, A., Castiglione, F. and Piccoli, B. Determination of the optimal therapeutic protocols in cancer immunotherapy. Mathematical Biosciences, 209(1):1–13, 2007. doi:10.1016/j.mbs.2007.02.009
- Castiglione, F. and Piccoli, B. Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bulletin of Mathematical Biology, 68(2):255–274, 2006. doi:10.1007/s11538-005-9014-3
- de Pillis, L. G., Gu, W. and Radunskaya, A. E. Mixed immunotherapy and chemotherapy of tumours: modeling, applications and biological interpretations.
*Journal of Theoretical Biology*, 238:841–862, 2006. doi:10.1016/j.jtbi.2005.06.037 - de Pillis, L. G., Mallet, D. G. and Radunskaya, A. E. Spatial Tumor-Immune Modeling. Computational and Mathematical Methods in Medicine, 7(2-3):159–176, 2006. doi:10.1080/10273660600968978
- de Pillis, L. G., Radunskaya, A. E. and Wiseman, C. L. A validated mathematical model of cell-mediated immune response to tumour growth.
*Cancer Research*, 65:7950–7958, 2005. http://www.ncbi.nlm.nih.gov/pubmed/16140967 - El-Gohary, A. Chaos and optimal control of equilibrium states of tumor system with drug,
*Chaos, Soliton and Fractals*, 41:425–435, 2009. doi:10.1016/j.chaos.2008.02.003 - Ghaffari, A. and Naserifar, N. Mathematical modeling and Lyapunov-based drug administration in cancer chemotherapy.
*Iranian Journal of Electrical and Electronic Engineering*, 5(3):151–158, 2009. http://www.sid.ir/en/VEWSSID/J_pdf/106520090310.pdf - Ghaffari, A. and Naserifar, N. Optimal therapeutic protocols in cancer immunotherapy.
*Computers in Biology and Medicine*, 40:261–270, 2010. doi:10.1016/j.compbiomed.2009.12.001 - Isaeva, O. G. and Osipov, V. A. Modelling of anti-tumour immune response: Immunocorrective effect of weak centimetre electromagnetic waves.
*Computational and Mathematical Methods in Medicine*, 10:185–201, 2009. doi:10.1080/17486700802373540 - Kirschner, D. and Panetta, J. C. Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol., 37:235–252, 1998. doi:10.1007/s002850050127
- Kuznetsov, V. A., Makalkin, I. A., Taylor, M. A. and Perelson, A. S. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis.
*Bulletin of Mathematical Biology*, 56:295–321, 1994. doi:10.1007/BF02460644 - N. Larmonier, J. Fraszack, D. Lakomy, Bonnotte, B. and Katsanis, E. Killer dendritic cells and their potential for cancer immunotherapy. Cancer Immunology Immunotherapy, 59:1–11, 2010. doi:10.1007/s00262-009-0736-1
- Lenhart, S. and Workman, J. T. Optimal control applied to biological models. Chapman and Hall/CRC Mathematical and Computational Biology Series, 2007. http://www.crcpress.com/product/isbn/9781584886402
- Mallet, D. G. and de Pillis, L. G. A cellular automata model of tumour-immune system interactions.
*Journal of Theoretical Biology*, 239:334–350, 2006. doi:10.1016/j.jtbi.2005.08.002 - Moretta, A. Natural killers and dendritic cells: rendezvous in abused tissues.
*Nat. Rev. Immunol.*, 2(12):957-964, 2002. doi:10.1038/nri956 - Murray, J. M. Some optimal control problems in cancer chemotherapy with a toxicity limit.
*Mathematical Biosciences*, 100:49–67, 1990. doi:10.1016/0025-5564(90)90047-3 - Swan, G. W. Role of optimal control theory in cancer chemotherapy.
*Mathematical Biosciences*, 101:237–284, 1990. doi:10.1016/0025-5564(90)90021-P - Swierniak, A., Ledzewicz, U. and Schattler, H. Optimal control for a class of compartmental models in cancer chemotherapy.
*Int. J. Appl. Math. Comput. Sci.*, 13(3):357–368, 2003. https://www.amcs.uz.zgora.pl/?action=paper&paper=154 - Wu, Y., Xia, L., Zhang, M. and Zhao, X. Immunodominance analysis through interactions of cd\(8^+\) T cells and dcs in lymph nodes.
*Math. Biosci.*, 225(1):53–38, 2010. doi:10.1016/j.mbs.2010.01.009

DOI: http://dx.doi.org/10.21914/anziamj.v54i0.6654

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