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

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B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003).   Google Scholar

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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

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A. Friedman, Cancer as multifaceted disease,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1.  doi: 10.1051/mmnp/20127102.  Google Scholar

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J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer Verlag, (1983).   Google Scholar

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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

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T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology,", W. H. Freeman, (2006).   Google Scholar

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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

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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

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