# American Institute of Mathematical Sciences

2012, 9(2): 241-257. doi: 10.3934/mbe.2012.9.241

## Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells

 1 Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany, Germany, Germany

Received  May 2011 Revised  October 2011 Published  March 2012

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.
Citation: Maria Vittoria Barbarossa, Christina Kuttler, Jonathan Zinsl. Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells. Mathematical Biosciences & Engineering, 2012, 9 (2) : 241-257. doi: 10.3934/mbe.2012.9.241
##### References:
 [1] J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications 6 (2005), 13-33. doi: 10.1016/j.nonrwa.2004.04.002. [2] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell," 5th edition, Taylor & Francis Ltd., New York, 2007. [3] A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, New York, 2003. [4] S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theoretical Population Biology, 25 (1984), 289-311. doi: 10.1016/0040-5809(84)90011-X. [5] G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations, Journal of Differential Equations, 168 (2000), 212-237. doi: 10.1006/jdeq.2000.3885. [6] M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology, in "Tutorials in Mathematical Biosciences. III," Lecuture Notes in Math., 1872, Berlin, (2006), 131-183. [7] K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. [8] G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'' ASM Press, Washington, 1997. [9] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [10] A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Science, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571. [11] A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling, 51 (2010), 572-591. [12] J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, IN, 2000), Mathematical Biosciences, 177/178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9. [13] J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes, Mathematical Modelling of Natural Phenomena, 2 (2007), 69-100. doi: 10.1051/mmnp:2007004. [14] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0. [15] Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'' Academic Press, New York, 2003. [16] W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Mathematical Bioscience and Engineering, 4 (2007), 239-259. doi: 10.3934/mbe.2007.4.239. [17] H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed. Scientific American Books , New York, 1995. [18] N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'' Cambridge University Press, 1989. [19] R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135. doi: 10.1016/0040-5809(83)90008-4. [20] T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208. doi: 10.1137/S0036144504446291. [21] R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26-53. [22] F. R. Sharpe and A. J. Lotka, A problem in age distribution, Philosophical Magazine Series 6, 21 (1911), 435-438. doi: 10.1080/14786440408637050. [23] H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. [24] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [25] U. Veronesi and G. Quaranta, "Un Male Curabile,'' Mondadori Editore, Milano, 1986. [26] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology, 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0. [27] G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 1985.

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##### References:
 [1] J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications 6 (2005), 13-33. doi: 10.1016/j.nonrwa.2004.04.002. [2] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell," 5th edition, Taylor & Francis Ltd., New York, 2007. [3] A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, New York, 2003. [4] S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theoretical Population Biology, 25 (1984), 289-311. doi: 10.1016/0040-5809(84)90011-X. [5] G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations, Journal of Differential Equations, 168 (2000), 212-237. doi: 10.1006/jdeq.2000.3885. [6] M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology, in "Tutorials in Mathematical Biosciences. III," Lecuture Notes in Math., 1872, Berlin, (2006), 131-183. [7] K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. [8] G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'' ASM Press, Washington, 1997. [9] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032. [10] A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Science, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571. [11] A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling, 51 (2010), 572-591. [12] J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, IN, 2000), Mathematical Biosciences, 177/178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9. [13] J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes, Mathematical Modelling of Natural Phenomena, 2 (2007), 69-100. doi: 10.1051/mmnp:2007004. [14] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0. [15] Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'' Academic Press, New York, 2003. [16] W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Mathematical Bioscience and Engineering, 4 (2007), 239-259. doi: 10.3934/mbe.2007.4.239. [17] H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed. Scientific American Books , New York, 1995. [18] N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'' Cambridge University Press, 1989. [19] R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135. doi: 10.1016/0040-5809(83)90008-4. [20] T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208. doi: 10.1137/S0036144504446291. [21] R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26-53. [22] F. R. Sharpe and A. J. Lotka, A problem in age distribution, Philosophical Magazine Series 6, 21 (1911), 435-438. doi: 10.1080/14786440408637050. [23] H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. [24] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [25] U. Veronesi and G. Quaranta, "Un Male Curabile,'' Mondadori Editore, Milano, 1986. [26] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology, 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0. [27] G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 1985.
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