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    On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth
2013, 10(3): 803-819. doi: 10.3934/mbe.2013.10.803

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

Urszula Ledzewicz 1, , Heinz Schättler 2, , Mostafa Reisi Gahrooi 1, and  Siamak Mahmoudian Dehkordi 1,

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.
Keywords: multi-drug treatments., chemotherapy, Optimal control.
Mathematics Subject Classification: Primary: 49K15, 92C50; Secondary: 93C9.
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.

[2]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40, 2003.

[3]

H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice," (in German), 2nd ed., Deutscher Apotheker -Verlag, Stuttgart, Germany.

[4]

M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy," Lecture Notes in Biomathematics, 30, Springer Verlag, 1979.

[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), 545-554.

[6]

M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130.

[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), 557-577. doi: 10.1007/s00285-011-0424-6.

[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), 609-637. doi: 10.1023/A:1016027113579.

[9]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597.

[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), 5-13.

[11]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 129-150.

[12]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294.

[13]

U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523.

[14]

A. P. Lyss, Enzymes and random synthetics, in "Chemotherapy Source Book" (ed. M. C. Perry), Williams &Wilkins, Baltimore, (1992), 403-408.

[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), 419-428. doi: 10.1007/s00280-002-0511-x.

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964.

[17]

H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.

[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, Maui, (2012), 7691-7696.

[19]

H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design, Cancer Research, 24 (1964), 1295-1302.

[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), 297-305.

[21]

G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284. doi: 10.1016/0025-5564(90)90021-P.

[22]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172.

[23]

A. Swierniak, Some control problems for simplest models of proliferation cycle, Applied mathematics and Computer Science, 4 (1994), 223-233.

[24]

A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058.

[25]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378.

[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), 357-368.

[27]

A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139.

[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), 61-75.

[29]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386. doi: 10.1016/S0362-546X(01)00184-5.

[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), 820-821.

[31]

T. E. Wheldon, "Mathematical Models in Cancer Research," Boston-Philadelphia: Hilger Publishing, 1988.

show all references

References:
[1]

M. R. Alison and C. E. Sarraf, "Understanding Cancer-From Basic Science to Clinical Practice," Cambridge University Press, 1997.

[2]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40, 2003.

[3]

H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice," (in German), 2nd ed., Deutscher Apotheker -Verlag, Stuttgart, Germany.

[4]

M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy," Lecture Notes in Biomathematics, 30, Springer Verlag, 1979.

[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), 545-554.

[6]

M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130.

[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), 557-577. doi: 10.1007/s00285-011-0424-6.

[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), 609-637. doi: 10.1023/A:1016027113579.

[9]

U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597.

[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), 5-13.

[11]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 129-150.

[12]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294.

[13]

U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523.

[14]

A. P. Lyss, Enzymes and random synthetics, in "Chemotherapy Source Book" (ed. M. C. Perry), Williams &Wilkins, Baltimore, (1992), 403-408.

[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), 419-428. doi: 10.1007/s00280-002-0511-x.

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964.

[17]

H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.

[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, Maui, (2012), 7691-7696.

[19]

H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design, Cancer Research, 24 (1964), 1295-1302.

[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), 297-305.

[21]

G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284. doi: 10.1016/0025-5564(90)90021-P.

[22]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172.

[23]

A. Swierniak, Some control problems for simplest models of proliferation cycle, Applied mathematics and Computer Science, 4 (1994), 223-233.

[24]

A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. doi: 10.1142/S0218339095000058.

[25]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378.

[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), 357-368.

[27]

A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139.

[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), 61-75.

[29]

A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386. doi: 10.1016/S0362-546X(01)00184-5.

[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), 820-821.

[31]

T. E. Wheldon, "Mathematical Models in Cancer Research," Boston-Philadelphia: Hilger Publishing, 1988.

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