Abstract
In this paper, results about the structure of cancer treatment protocols that can be inferred from an analysis of mathematical models with the methods and tools of optimal control are reviewed. For homogeneous tumor populations of chemotherapeutically sensitive cells, optimal controls are bang-bang corresponding to the medical paradigm of maximum tolerated doses (MTD). But as more aspects of the tumor microenvironment are taken into account, such as heterogeneity of the tumor cell population, tumor angiogenesis and tumor-immune system interactions, singular controls which administer agents at specific time-varying reduced dose rates become optimal and give an indication of what might be the biologically optimal dose (BOD).
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References
N. André, L. Padovani, E. Pasquier, Metronomic scheduling of anticancer treatment: the next generation of multitarget therapy?. Fut. Oncol. 7(3), 385–394 (2011)
T. Boehm, J. Folkman, T. Browder, M.S. O’Reilly, Antiangiogenic therapy of experimental cancer does not induce acquired drug resistance. Nature 390, 404–407 (1997)
B. Bonnard,, M. Chyba, Singular trajectories and their role in control theory. Mathématiques & Applications, vol. 40 (Springer, Paris 2003)
A. Bressan, A. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences (2007)
S. Davis, G.D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis. Cur. Top. Microbio. Immun. 237, 173–185 (1999)
M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lecture Notes in Biomathematics, vol. 30 (Springer, NewYork 1979)
A. Ergun, K. Camphausen, L.M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. Math. Biol. 65, 407–424 (2003)
J. Folkman, Tumor angiogenesis: therapeutic implications. New Engl. J. Med. 295, 1182–1196 (1971)
J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Ann. Surg. 175, 409–416 (1972)
J. Folkman, M. Klagsburn, Angiogenic factors. Science 235, 442–447 (1987)
U. Forys, Y. Keifetz, Y. Kogan, Critical-point analysis for three-variable cancer angiogenesis models. Math. Biosci. Eng. 2, 511–525 (2005)
R.A. Gatenby, A.S. Silva, R.J. Gillies, B.R. Frieden, Adaptive therapy. Canc. Res. 69, 4894–4903 (2009)
J.H. Goldie, Drug resistance in cancer: a perspective. Canc. Meta. Rev. 20, 63–68 (2001)
J.H. Goldie, A. Coldman, Drug Resistance in Cancer (Cambridge University Press, Cambridge 1998)
P. Hahnfeldt, L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations. Radiat. Res. 150, 681–687 (1998)
P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Can. Res. 59, 4770–4775 (1999)
P. Hahnfeldt, J. Folkman, L. Hlatky, Minimizing long-term burden: the logic for metronomic chemotherapy dosing and its angiogenic basis. J. Theo. Biol. 220, 545–554 (2003)
D. Hanahan, G. Bergers, E. Bergsland, Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice. J. Clin. Invest. 105, 1045–1047 (2000)
R.K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med., 7, 987–989 (2001)
R.K. Jain, L.L. Munn, Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents, Princ. Pract. Oncol. 21, 1–7 (2007)
B. Kamen, E. Rubin, J. Aisner, E. Glatstein, High-time chemotherapy or high time for low dose? J. Clin. Oncol. 18, Editorial, 2935–2937 (2000)
R.S. Kerbel, Tumor angiogenesis: past, present and near future, Carcinogensis, 21, 505–515 (2000)
T.J. Kindt, B.A. Osborne, R.A. Goldsby, Kuby Immunology (W.H. Freeman, New York 2006)
M. Kimmel, A. Swierniak, Control theory approach to cancer chemotherapy: benefiting from phase dependence and overcoming drug resistance, in Tutorials in Mathematical Biosciences III: Cell Cycle, Proliferation, and Cancer. Lecture Notes in Mathematics, vol. 1872 (Springer, Newyork, 2006), pp. 185–221
M. Klagsburn, S. Soker, VEGF/VPF: the angiogenesis factor found?. Curr. Biol. 3, 699–702, (1993)
G. Klement, S. Baruchel,, Rak, J., Man, S., Clark, K., Hicklin, D.J., Bohlen, P., Kerbel, R.S.: Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity, J. Clin. Invest. 105, R15–R24 (2000)
V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bul. Math. Biol. 56, 295–321 (1994)
U. Ledzewicz, K. Bratton, H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations. Acta Appl. Matem. (2014) doi: 10.1007/s10440-014-9952-6
U. Ledzewicz, M.S. Faraji Mosalman, H. Schättler, Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost, Discr. Cont. Dyn. Syst. Ser. B 18, 1031–1051 (2013)
U. Ledzewicz, A. d’Onofrio, H. Schättler, Tumor development under combination treatments with anti-angiogenic therapies. in Mathematical Methods and Models in Biomedicine (Springer, NewYork, 2012), pp. 311–337
U. Ledzewicz, J. Marriott, H. Maurer, H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for novel cancer treatments, Math. Med. Biol. 27, 157–179, (2010).
U. Ledzewicz, M. Naghnaeian, H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J. Math. Biol. 64, 557–577 (2012)
U. Ledzewicz, H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Th. Appl. 114, 609–637 (2002)
U. Ledzewicz, H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy. J. Biol. Syst. 10, 183–206 (2002)
U. Ledzewicz, H. Schättler, Optimal control for a bilinear model with recruiting agent in cancer chemotherapy, Proc. of the 42nd IEEE Conference on Decision and Control (CDC), Maui, Hawaii, 2762–2767 (2003)
U. Ledzewicz, H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Math. Biosci. Engr. 2, 561–578 (2005)
U. Ledzewicz, H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discr. Cont. Dyn. Syst. Ser. B, 6, 129–150 (2006)
U. Ledzewicz, H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Contr. Optim. 46, 1052–1079 (2007)
U. Ledzewicz, H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. of Theo. Biol. 252, 295–312, (2008)
U. Ledzewicz, H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. of Optim. Th. Appl. 153, 195–224 (2012)
U. Ledzewicz, H. Schättler, M. Reisi Gahrooi, S. Mahmoudian Dehkordi, On the MTD paradigm and optimal control for combination cancer chemotherapy. Math. Biosci. Engr. 10, 803–819 (2013)
L.A. Loeb, A mutator phenotype in cancer. Canc. Res. 61, 3230–3239 (2001)
R. Martin, K.L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy (World Scientific Publishers, Singapore 1994)
L. Norton, R. Simon, The Norton-Simon hypothesis revisited. Canc. Treat. Rep. 70, 163–169 (1986)
A. d’Onofrio, A general framework for modelling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedial inferences. Phys. D 208, 202–235, (2005)
A. d’Onofrio, Rapidly acting antitumoral antiangiogenic therapies. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76, 031920 (2007)
A. d’Onofrio, A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al., Math. Biosci. 191, 159–184 (2004)
A. d’Onofrio, A. Gandolfi, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation. Appl. Math. and Comp. 181, 1155–1162 (2006)
A. d’Onofrio, A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy. Math. Med. Biol., 26, 63–95 (2009)
A. d’Onofrio, A. Gandolfi, Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular “pruning”. J. Theo. Biol. 264, 253–265, (2010)
A. d’Onofrio, A. Gandolfi, A. Rocca, The dynamics of tumour-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings. Cell Prolif., 42, 317–329, (2009)
A. d’Onofrio, U. Ledzewicz, H. Maurer, H. Schättler, On optimal delivery of combination therapy for tumors. Math. Biosci., 222, 13–26 (2009)
A. d’Onofrio, U. Ledzewicz, H. Schättler, On the dynamics of tumor immune system interactions and combined chemo- and immunotherapy, in: New Challenges for Cancer Systems Biomedicine eds. by A. d’Onofrio, P. Cerrai, A Gandolfi, vol. 1 (SIMAI Springer series, 2012). pp. 249–266
D. Pardoll, Does the immune system see tumors as foreign or self? Ann. Rev. Immun. 21, 807–839 (2003)
E. Pasquier, U. Ledzewicz, Perspective on “More is not necessarily better”: Metronomic Chemotherapy. Newslet. Soc. Math. Biol. 26(2), 9–10, (2013)
E. Pasquier, M. Kavallaris, N. André, Metronomic chemotherapy: new rationale for new directions. Nat. Rev. | Clin. Onc. 7, 455–465 (2010)
K. Pietras, D. Hanahan, A multi-targeted, metronomic and maximum tolerated dose “chemo-switch” regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer. J. Clin. Onc. 23, 939–952 (2005)
J. Poleszczuk, U. Forys, Derivation of the Hahnfeldt et al. model (1999) revisited, Proceedings of the 16th Nat. Conf. on Applications of Mathematics in Biology and Medicine, Krynica, Poland 87–92 (2010)
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes (MacMillan, New York 1964)
H. Schättler, U. Ledzewicz, B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis. Math. Biosci. Engr. 8, 355–369 (2011)
H. Schättler, U. Ledzewicz: Geometric Optimal Control (Springer, NewYork 2012)
H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi, M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, Proc. of the 51st IEEE Conf. on Decision and Control, Maui, Hawaii, 7691–7696, (2012)
N.V. Stepanova, Course of the immune reaction during the development of a malignant tumour. Biophys. 24, 917–923 (1980)
G.W. Swan, Role of optimal control in cancer chemotherapy. Math. Biosci., 101, 237–284 (1990)
J.B. Swann, M.J. Smyth, Immune surveillance of tumors. J. Clin. Invest. 117 1137–1146, (2007)
A. Swierniak, Optimal treatment protocols in leukemia-modelling the proliferation cycle, Proc. of the 12th IMACS World Congress, Paris, vol. 4, 170–172 (1988)
A. Swierniak, Cell cycle as an object of control, J. Biol. Syst. 3, 41–54 (1995)
A. Swierniak, Direct and indirect control of cancer populations. Bul. Pol. Acad. Sci. Techn. Sci. 56, 367–378 (2008)
A. Swierniak, U. Ledzewicz, H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy. Int. J. Appl. Math. Comp. Sci. 13, 357–368 (2003)
A. Swierniak, A. d’Onofrio, A. Gandolfi, Optimal control problems related to tumor angiogenesis. Proc. IEEE-IECON’2006, 667–681 (2006)
A. Swierniak, J. Smieja, Cancer chemotherapy optimization under evolving drug resistance. Nonlin. Ana. 47, 375–386 (2000)
H.P. de Vladar, J.A. González, Dynamic response of cancer under the influence of immunological activity and therapy. J. Theo. Biol. 227, 335–348 (2004)
Acknowledgements
This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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Ledzewicz, U., Schättler, H. (2014). Tumor Microenvironment and Anticancer Therapies: An Optimal Control Approach. In: d'Onofrio, A., Gandolfi, A. (eds) Mathematical Oncology 2013. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0458-7_10
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