# 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). doi: 10.1186/1745-6150-7-31. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0895-7177(00)00143-6. Google Scholar [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003). Google Scholar [5] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences, (2007). Google Scholar [6] G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting,, Annual Review of Immunology, 22 (2004), 322. doi: 10.1146/annurev.immunol.22.012703.104803. Google Scholar [7] A. Friedman, Cancer as multifaceted disease,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1. doi: 10.1051/mmnp/20127102. Google Scholar [8] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer Verlag, (1983). Google Scholar [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. Google Scholar [10] T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology,", W. H. Freeman, (2006). Google Scholar [11] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. of Mathematical Biology, 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar [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. doi: 10.1038/nature06309. Google Scholar [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. Google Scholar [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, (2012), 7492. doi: 10.1109/CDC.2012.6427039. Google Scholar [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, (2010), 971. Google Scholar [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. doi: 10.1007/s00285-011-0424-6. Google Scholar [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. doi: 10.3934/mbe.2005.2.561. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.physd.2005.06.032. Google Scholar [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. doi: 10.1142/S0218202506001571. Google Scholar [21] A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS,, Chaos, 31 (2007), 261. doi: 10.1016/j.chaos.2005.10.006. Google Scholar [22] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,, Mathematical and Computational Modelling, 47 (2008), 614. doi: 10.1016/j.mcm.2007.02.032. Google Scholar [23] A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa,, Chaos, 41 (2009), 875. doi: 10.1016/j.chaos.2008.04.014. Google Scholar [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. Google Scholar [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. doi: 10.1111/j.1365-2184.2009.00595.x. Google Scholar [26] D. Pardoll, Does the immune system see tumors as foreign or self?,, Annual Reviews of Immunology, 21 (2003), 807. Google Scholar [27] E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455. doi: 10.1038/nrclinonc.2010.82. Google Scholar [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. doi: 10.1200/JCO.2005.07.093. Google Scholar [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. Google Scholar [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964). Google Scholar [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples,", Springer Verlag, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [32] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917. Google Scholar [33] J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. of Clinical Investigations, 117 (2007), 1137. doi: 10.1172/JCI31405. Google Scholar [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. doi: 10.1016/j.jtbi.2003.11.012. Google Scholar [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. Google Scholar

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). doi: 10.1186/1745-6150-7-31. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0895-7177(00)00143-6. Google Scholar [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003). Google Scholar [5] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences, (2007). Google Scholar [6] G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting,, Annual Review of Immunology, 22 (2004), 322. doi: 10.1146/annurev.immunol.22.012703.104803. Google Scholar [7] A. Friedman, Cancer as multifaceted disease,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1. doi: 10.1051/mmnp/20127102. Google Scholar [8] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer Verlag, (1983). Google Scholar [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. Google Scholar [10] T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology,", W. H. Freeman, (2006). Google Scholar [11] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. of Mathematical Biology, 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar [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. doi: 10.1038/nature06309. Google Scholar [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. Google Scholar [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, (2012), 7492. doi: 10.1109/CDC.2012.6427039. Google Scholar [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, (2010), 971. Google Scholar [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. doi: 10.1007/s00285-011-0424-6. Google Scholar [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. doi: 10.3934/mbe.2005.2.561. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.physd.2005.06.032. Google Scholar [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. doi: 10.1142/S0218202506001571. Google Scholar [21] A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS,, Chaos, 31 (2007), 261. doi: 10.1016/j.chaos.2005.10.006. Google Scholar [22] A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,, Mathematical and Computational Modelling, 47 (2008), 614. doi: 10.1016/j.mcm.2007.02.032. Google Scholar [23] A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa,, Chaos, 41 (2009), 875. doi: 10.1016/j.chaos.2008.04.014. Google Scholar [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. Google Scholar [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. doi: 10.1111/j.1365-2184.2009.00595.x. Google Scholar [26] D. Pardoll, Does the immune system see tumors as foreign or self?,, Annual Reviews of Immunology, 21 (2003), 807. Google Scholar [27] E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455. doi: 10.1038/nrclinonc.2010.82. Google Scholar [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. doi: 10.1200/JCO.2005.07.093. Google Scholar [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. Google Scholar [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964). Google Scholar [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples,", Springer Verlag, (2012). doi: 10.1007/978-1-4614-3834-2. Google Scholar [32] N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917. Google Scholar [33] J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. of Clinical Investigations, 117 (2007), 1137. doi: 10.1172/JCI31405. Google Scholar [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. doi: 10.1016/j.jtbi.2003.11.012. Google Scholar [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. Google Scholar
 [1] Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 [2] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [3] Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040 [4] Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 [5] Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347 [6] Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100 [7] Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082 [8] Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks & Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007 [9] Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925 [10] Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 [11] Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451 [12] Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415 [13] Baltazar D. Aguda, Ricardo C.H. del Rosario, Michael W.Y. Chan. Oncogene-tumor suppressor gene feedback interactions and their control. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1277-1288. doi: 10.3934/mbe.2015.12.1277 [14] Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37 [15] Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55 [16] Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006 [17] Amy H. Lin. A model of tumor and lymphocyte interactions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 241-266. doi: 10.3934/dcdsb.2004.4.241 [18] Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803 [19] Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 [20] Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066

2018 Impact Factor: 1.313