On optimal chemotherapy with a strongly targeted agent for a model of tumorimmune system interactions with generalized logistic growth
Pages: 787  802,
Issue 3,
June
2013
doi:10.3934/mbe.2013.10.787 Abstract
References
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Urszula Ledzewicz  Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 620261653, United States (email)
Omeiza Olumoye  Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 620261653, United States (email)
Heinz Schättler  Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130, United States (email)
1 
M. M. AlTameemi, 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. 

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), 183206. 

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), 413452. 

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), 322360. 

7 
A. Friedman, Cancer as multifaceted disease, Mathematical Modelling of Natural Phenomena, 7 (2012), 126. 

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 longterm burden: the logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545554. 

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 tumorimmune interaction, J. of Mathematical Biology, 37 (1998), 235252. 

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), 903905. 

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), 295321. 

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), 74927497. 

15 
U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumorimmune interactions under treatment as an optimal control problem, Proceedings of the 8th AIMS Conference, Dresden, Germany, (2010), 971980. 

16 
U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumorimmune dynamics, J. of Mathematical Biology, 64 (2012), 557577. 

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), 561578. 

18 
A. Matzavinos, M. Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatiotemporal response of cytotoxic Tlymphocytes to a solid tumour, Mathematical Medicine and Biology, 21 (2004), 134. 

19 
A. d'Onofrio, A general framework for modeling tumorimmune system competition and immunotherapy: mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220235. 

20 
A. d'Onofrio, Tumorimmune system interaction: Modeling the tumorstimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Sciences, 16 (2006), 13751401. 

21 
A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS, Chaos, Solitons and Fractals, 31 (2007), 261268. 

22 
A. d'Onofrio, Metamodeling tumorimmune system interaction, tumor evasion and immunotherapy, Mathematical and Computational Modelling, 47 (2008), 614637. 

23 
A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and viceversa, Chaos, Solitons and Fractals, 41 (2009), 875880. 

24 
A. d'Onofrio and A. Ciancio, Simple biophysical model of tumor evasion from immune system control, Physical Review E, 84 (2011). 

25 
A. d'Onofrio, A. Gandolfi and A. Rocca, The cooperative and nonlinear dynamics of tumorvasculature interaction suggests lowdose, timedense antiangiogenic schedulings, Cell Proliferation, 42 (2009), 317329. 

26 
D. Pardoll, Does the immune system see tumors as foreign or self?, Annual Reviews of Immunology, 21 (2003), 807839. 

27 
E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions, Nature Reviews$$ Clinical Oncology, 7 (2010), 455465. 

28 
K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximumtolerated dose "chemoswitch'' regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. of Clinical Oncology, 23, (2005), 939952. 

29 
L. G. de Pillis, A. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth, Cancer Research, 65 (2005), 79507958. 

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. 

32 
N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917923. 

33 
J. B. Swann and M. J. Smyth, Immune surveillance of tumors, J. of Clinical Investigations, 117 (2007), 11371146. 

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), 335348. 

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), 820821. 

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