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2013, 10(3): 803-819. doi: 10.3934/mbe.2013.10.803

## On the MTD paradigm and optimal control for multi-drug cancer chemotherapy

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

Received  October 2012 Revised  December 2012 Published  April 2013

In standard chemotherapy protocols, drugs are given at maximum tolerated doses (MTD) with rest periods in between. In this paper, we briefly discuss the rationale behind this therapy approach and, using as example multi-drug cancer chemotherapy with a cytotoxic and cytostatic agent, show that these types of protocols are optimal in the sense of minimizing a weighted average of the number of tumor cells (taken both at the end of therapy and at intermediate times) and the total dose given if it is assumed that the tumor consists of a homogeneous population of chemotherapeutically sensitive cells. A $2$-compartment linear model is used to model the pharmacokinetic equations for the drugs.
Citation: 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
##### References:
 [1] M. R. Alison and C. E. Sarraf, "Understanding Cancer-From Basic Science to Clinical Practice,", Cambridge University Press, (1997). Google Scholar [2] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003). Google Scholar [3] H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice,", (in German), (). Google Scholar [4] M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy,", Lecture Notes in Biomathematics, 30 (1979). Google Scholar [5] 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 [6] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120. Google Scholar [7] 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 [8] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609. doi: 10.1023/A:1016027113579. Google Scholar [9] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183. doi: 10.1142/S0218339002000597. Google Scholar [10] U. Ledzewicz, H. Schättler and A. Swierniak, Finite dimensional models of drug resistant and phase specific cancer chemotherapy,, J. of Medical Information Technology, 8 (2004), 5. Google Scholar [11] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems, 6 (2006), 129. Google Scholar [12] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar [13] U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine,, Control and Cybernetics, 38 (2009), 1501. Google Scholar [14] A. P. Lyss, Enzymes and random synthetics,, in, (1992), 403. Google Scholar [15] J. C. Panetta, Y. Yanishevski, C. H. Pui, J. T. Sandlund, J. Rubnitz, G. K. Rivera, W. E. Evans and M. V. Relling, A mathematical model of in vivo methotrexate accumulation in acute lymphoblastic leukemia,, Cancer Chemotherapy and Pharmacology, 50 (2002), 419. doi: 10.1007/s00280-002-0511-x. Google Scholar [16] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964). Google Scholar [17] 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 [18] H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy,, Proc. of the 51st IEEE Conference on Decision and Control, (2012), 7691. Google Scholar [19] H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design,, Cancer Research, 24 (1964), 1295. Google Scholar [20] J. Smieja and A. Swierniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification,, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 297. Google Scholar [21] G. W. Swan, Role of optimal control in cancer chemotherapy,, Mathematical Biosciences, 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar [22] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, Proceedings of the 12th IMACS World Congress, 4 (1988), 170. Google Scholar [23] A. Swierniak, Some control problems for simplest models of proliferation cycle,, Applied mathematics and Computer Science, 4 (1994), 223. Google Scholar [24] A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41. doi: 10.1142/S0218339095000058. Google Scholar [25] A. Swierniak, Direct and indirect control of cancer populations,, Bulletin of the Polish Academy of Sciences, 56 (2008), 367. Google Scholar [26] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357. Google Scholar [27] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell Proliferation, 29 (1996), 117. Google Scholar [28] A. Swierniak, A. Polanski, M. Kimmel, A. Bobrowski and J. Smieja, Qualitative analysis of controlled drug resistance model - inverse Laplace and semigroup approach,, Control and Cybernetics, 28 (1999), 61. Google Scholar [29] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2000), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar [30] 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 [31] T. E. Wheldon, "Mathematical Models in Cancer Research,", Boston-Philadelphia: Hilger Publishing, (1988). Google Scholar

show all references

##### References:
 [1] M. R. Alison and C. E. Sarraf, "Understanding Cancer-From Basic Science to Clinical Practice,", Cambridge University Press, (1997). Google Scholar [2] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003). Google Scholar [3] H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice,", (in German), (). Google Scholar [4] M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy,", Lecture Notes in Biomathematics, 30 (1979). Google Scholar [5] 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 [6] M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy,, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120. Google Scholar [7] 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 [8] U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy,, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609. doi: 10.1023/A:1016027113579. Google Scholar [9] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy,, J. of Biological Systems, 10 (2002), 183. doi: 10.1142/S0218339002000597. Google Scholar [10] U. Ledzewicz, H. Schättler and A. Swierniak, Finite dimensional models of drug resistant and phase specific cancer chemotherapy,, J. of Medical Information Technology, 8 (2004), 5. Google Scholar [11] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems, 6 (2006), 129. Google Scholar [12] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. on Control and Optimization, 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar [13] U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine,, Control and Cybernetics, 38 (2009), 1501. Google Scholar [14] A. P. Lyss, Enzymes and random synthetics,, in, (1992), 403. Google Scholar [15] J. C. Panetta, Y. Yanishevski, C. H. Pui, J. T. Sandlund, J. Rubnitz, G. K. Rivera, W. E. Evans and M. V. Relling, A mathematical model of in vivo methotrexate accumulation in acute lymphoblastic leukemia,, Cancer Chemotherapy and Pharmacology, 50 (2002), 419. doi: 10.1007/s00280-002-0511-x. Google Scholar [16] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964). Google Scholar [17] 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 [18] H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy,, Proc. of the 51st IEEE Conference on Decision and Control, (2012), 7691. Google Scholar [19] H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design,, Cancer Research, 24 (1964), 1295. Google Scholar [20] J. Smieja and A. Swierniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification,, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 297. Google Scholar [21] G. W. Swan, Role of optimal control in cancer chemotherapy,, Mathematical Biosciences, 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar [22] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle,, Proceedings of the 12th IMACS World Congress, 4 (1988), 170. Google Scholar [23] A. Swierniak, Some control problems for simplest models of proliferation cycle,, Applied mathematics and Computer Science, 4 (1994), 223. Google Scholar [24] A. Swierniak, Cell cycle as an object of control,, J. of Biological Systems, 3 (1995), 41. doi: 10.1142/S0218339095000058. Google Scholar [25] A. Swierniak, Direct and indirect control of cancer populations,, Bulletin of the Polish Academy of Sciences, 56 (2008), 367. Google Scholar [26] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy,, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357. Google Scholar [27] A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy,, Cell Proliferation, 29 (1996), 117. Google Scholar [28] A. Swierniak, A. Polanski, M. Kimmel, A. Bobrowski and J. Smieja, Qualitative analysis of controlled drug resistance model - inverse Laplace and semigroup approach,, Control and Cybernetics, 28 (1999), 61. Google Scholar [29] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance,, Nonlinear Analysis, 47 (2000), 375. doi: 10.1016/S0362-546X(01)00184-5. Google Scholar [30] 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 [31] T. E. Wheldon, "Mathematical Models in Cancer Research,", Boston-Philadelphia: Hilger Publishing, (1988). Google Scholar
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