# American Institute of Mathematical Sciences

2013, 10(3): 787-802. doi: 10.3934/mbe.2013.10.787

## On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth

 1 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States 2 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

Received  September 2012 Revised  January 2013 Published  April 2013

In this paper, a mathematical model for chemotherapy that takes tumor immune-system interactions into account is considered for a strongly targeted agent. We use a classical model originally formulated by Stepanova, but replace exponential tumor growth with a generalised logistic growth model function depending on a parameter $\nu$. This growth function interpolates between a Gompertzian model (in the limit $\nu\rightarrow0$) and an exponential model (in the limit $\nu\rightarrow\infty$). The dynamics is multi-stable and equilibria and their stability will be investigated depending on the parameter $\nu$. Except for small values of $\nu$, the system has both an asymptotically stable microscopic (benign) equilibrium point and an asymptotically stable macroscopic (malignant) equilibrium point. The corresponding regions of attraction are separated by the stable manifold of a saddle. The optimal control problem of moving an initial condition that lies in the malignant region into the benign region is formulated and the structure of optimal singular controls is determined.
Citation: Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787
##### References:
 [1] M. M. Al-Tameemi, M. A. J. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the immune system: Consequences of brief encounters, Biology Direct, 7 (2012), 31. doi: 10.1186/1745-6150-7-31. [2] N. Bellomo and N. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Reviews, 5 (2008), 183-206. [3] N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Mathematical and Computational Modelling, 32 (2000), 413-452. doi: 10.1016/S0895-7177(00)00143-6. [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40 (2003). [5] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007. [6] G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting, Annual Review of Immunology, 22 (2004), 322-360. doi: 10.1146/annurev.immunol.22.012703.104803. [7] A. Friedman, Cancer as multifaceted disease, Mathematical Modelling of Natural Phenomena, 7 (2012), 1-26. doi: 10.1051/mmnp/20127102. [8] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer Verlag, New York, 1983. [9] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: the logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. [10] T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology," W. H. Freeman, 2006. [11] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127. [12] C. M. Koebel, W. Vermi, J. B. Swann, N. Zerafa, S. J. Rodig, L. J. Old, M. J. Smyth and R. D. Schreiber, Adaptive immunity maintains occult cancer in an equilibrium state, Nature, 450 (2007), 903-905. doi: 10.1038/nature06309. [13] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295-321. [14] U. Ledzewicz, M. Faraji and H. Schättler, On optimal protocols for combinations of chemo- and immunotherapy, Proceedings of the 51st IEEE Proceedings on Decision and Control, Maui, Hawaii, (2012), 7492-7497. doi: 10.1109/CDC.2012.6427039. [15] U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem, Proceedings of the 8th AIMS Conference, Dresden, Germany, (2010), 971-980. [16] U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6. [17] U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE), 2 (2005), 561-578. doi: 10.3934/mbe.2005.2.561. [18] A. Matzavinos, M. Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour, Mathematical Medicine and Biology, 21 (2004), 1-34. [19] 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. [20] A. d'Onofrio, Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571. [21] A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS, Chaos, Solitons and Fractals, 31 (2007), 261-268. doi: 10.1016/j.chaos.2005.10.006. [22] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Mathematical and Computational Modelling, 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032. [23] A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa, Chaos, Solitons and Fractals, 41 (2009), 875-880. doi: 10.1016/j.chaos.2008.04.014. [24] A. d'Onofrio and A. Ciancio, Simple biophysical model of tumor evasion from immune system control, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.031910. [25] A. d'Onofrio, A. Gandolfi and A. Rocca, The cooperative and nonlinear dynamics of tumor-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings, Cell Proliferation, 42 (2009), 317-329. doi: 10.1111/j.1365-2184.2009.00595.x. [26] D. Pardoll, Does the immune system see tumors as foreign or self?, Annual Reviews of Immunology, 21 (2003), 807-839. [27] E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455-465. doi: 10.1038/nrclinonc.2010.82. [28] K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch'' regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. of Clinical Oncology, 23, (2005), 939-952. doi: 10.1200/JCO.2005.07.093. [29] L. G. de Pillis, A. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65 (2005), 7950-7958. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [32] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. [33] J. B. Swann and M. J. Smyth, Immune surveillance of tumors, J. of Clinical Investigations, 117 (2007), 1137-1146. doi: 10.1172/JCI31405. [34] H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy, J. of Theoretical Biology, 227 (2004), 335-348. doi: 10.1016/j.jtbi.2003.11.012. [35] S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology, J. of Clinical Oncology, 11 (1993), 820-821.

show all references

##### References:
 [1] M. M. Al-Tameemi, M. A. J. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the immune system: Consequences of brief encounters, Biology Direct, 7 (2012), 31. doi: 10.1186/1745-6150-7-31. [2] N. Bellomo and N. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Reviews, 5 (2008), 183-206. [3] N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Mathematical and Computational Modelling, 32 (2000), 413-452. doi: 10.1016/S0895-7177(00)00143-6. [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40 (2003). [5] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007. [6] G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting, Annual Review of Immunology, 22 (2004), 322-360. doi: 10.1146/annurev.immunol.22.012703.104803. [7] A. Friedman, Cancer as multifaceted disease, Mathematical Modelling of Natural Phenomena, 7 (2012), 1-26. doi: 10.1051/mmnp/20127102. [8] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer Verlag, New York, 1983. [9] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: the logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. [10] T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology," W. H. Freeman, 2006. [11] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. of Mathematical Biology, 37 (1998), 235-252. doi: 10.1007/s002850050127. [12] C. M. Koebel, W. Vermi, J. B. Swann, N. Zerafa, S. J. Rodig, L. J. Old, M. J. Smyth and R. D. Schreiber, Adaptive immunity maintains occult cancer in an equilibrium state, Nature, 450 (2007), 903-905. doi: 10.1038/nature06309. [13] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, 56 (1994), 295-321. [14] U. Ledzewicz, M. Faraji and H. Schättler, On optimal protocols for combinations of chemo- and immunotherapy, Proceedings of the 51st IEEE Proceedings on Decision and Control, Maui, Hawaii, (2012), 7492-7497. doi: 10.1109/CDC.2012.6427039. [15] U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem, Proceedings of the 8th AIMS Conference, Dresden, Germany, (2010), 971-980. [16] U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6. [17] U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE), 2 (2005), 561-578. doi: 10.3934/mbe.2005.2.561. [18] A. Matzavinos, M. Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour, Mathematical Medicine and Biology, 21 (2004), 1-34. [19] 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. [20] A. d'Onofrio, Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571. [21] A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS, Chaos, Solitons and Fractals, 31 (2007), 261-268. doi: 10.1016/j.chaos.2005.10.006. [22] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Mathematical and Computational Modelling, 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032. [23] A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa, Chaos, Solitons and Fractals, 41 (2009), 875-880. doi: 10.1016/j.chaos.2008.04.014. [24] A. d'Onofrio and A. Ciancio, Simple biophysical model of tumor evasion from immune system control, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.031910. [25] A. d'Onofrio, A. Gandolfi and A. Rocca, The cooperative and nonlinear dynamics of tumor-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings, Cell Proliferation, 42 (2009), 317-329. doi: 10.1111/j.1365-2184.2009.00595.x. [26] D. Pardoll, Does the immune system see tumors as foreign or self?, Annual Reviews of Immunology, 21 (2003), 807-839. [27] E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455-465. doi: 10.1038/nrclinonc.2010.82. [28] K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch'' regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. of Clinical Oncology, 23, (2005), 939-952. doi: 10.1200/JCO.2005.07.093. [29] L. G. de Pillis, A. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65 (2005), 7950-7958. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [32] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. [33] J. B. Swann and M. J. Smyth, Immune surveillance of tumors, J. of Clinical Investigations, 117 (2007), 1137-1146. doi: 10.1172/JCI31405. [34] H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy, J. of Theoretical Biology, 227 (2004), 335-348. doi: 10.1016/j.jtbi.2003.11.012. [35] S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology, J. of Clinical Oncology, 11 (1993), 820-821.
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