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2016, 13(6): 1223-1240. doi: 10.3934/mbe.2016040

Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity

Shuo Wang 1, and  Heinz Schättler 2,

1. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, United States
2. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

Received  December 2015 Revised  February 2016 Published  August 2016

We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.
Keywords: bang-bang controls, Cancer chemotherapy, singular controls., tumor heterogeneity, optimal control.
Mathematics Subject Classification: Primary: 49K15, 92B05; Secondary: 93C9.
Citation: Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040
References:
[1]

N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?,, Future Oncology, 7 (2011), 385.   Google Scholar

[2]

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

[3]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[4]

A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors,, Bull. of Math. Biology, 65 (2003), 407.  doi: 10.1016/S0092-8240(03)00006-5.  Google Scholar

[5]

R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy,, Cancer Research, 69 (2009), 4894.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[6]

R. A. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: 10.1038/459508a.  Google Scholar

[7]

J. H. Goldie, Drug resistance in cancer: A perspective,, Cancer and Metastasis Review, 20 (2001), 63.  doi: 10.1017/CBO9780511666544.  Google Scholar

[8]

J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents,, Mathematical Biosciences, 65 (1983), 291.   Google Scholar

[9]

R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells,, Cancer Research, 66 (2006), 1033.  doi: 10.1158/0008-5472.CAN-05-3077.  Google Scholar

[10]

J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors,, Bull. Math. Biol., 76 (2014), 627.  doi: 10.1007/s11538-014-9936-8.  Google Scholar

[11]

O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance,, Cancer Research, 73 (2013), 7168.  doi: 10.1158/0008-5472.CAN-13-1768.  Google Scholar

[12]

U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics,, in Recent Advances in Optimization and its Applications in Engineering, (2010), 267.  doi: 10.1007/978-3-642-12598-0_23.  Google Scholar

[13]

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

[14]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Contr. Optim., 46 (2007), 1052.  doi: 10.1137/060665294.  Google Scholar

[15]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine,, Control and Cybernetics, 38 (2009), 1501.   Google Scholar

[16]

J. S. Li and N. Khaneja, Ensemble control of linear systems,, Proc. of the 46th IEEE Conference on Decision and Control, (2007), 3768.   Google Scholar

[17]

J. S. Li and N. Khaneja, Ensemble control of Bloch equations,, IEEE Transactions on Automatic Control, 54 (2009), 528.  doi: 10.1109/TAC.2009.2012983.  Google Scholar

[18]

A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377.  doi: 10.1051/m2an/2012031.  Google Scholar

[19]

A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors,, Bull. Math. Biol., 77 (2015), 1.  doi: 10.1007/s11538-014-0046-4.  Google Scholar

[20]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treatment Reports, 61 (1977), 1307.   Google Scholar

[21]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited,, Cancer Treatment Reports, 70 (1986), 41.   Google Scholar

[22]

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

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Macmillan, (1964).   Google Scholar

[24]

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

[25]

H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies,, Springer, (2015).  doi: 10.1007/978-1-4939-2972-6.  Google Scholar

show all references

References:
[1]

N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?,, Future Oncology, 7 (2011), 385.   Google Scholar

[2]

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

[3]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control,, American Institute of Mathematical Sciences, (2007).   Google Scholar

[4]

A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors,, Bull. of Math. Biology, 65 (2003), 407.  doi: 10.1016/S0092-8240(03)00006-5.  Google Scholar

[5]

R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy,, Cancer Research, 69 (2009), 4894.  doi: 10.1158/0008-5472.CAN-08-3658.  Google Scholar

[6]

R. A. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: 10.1038/459508a.  Google Scholar

[7]

J. H. Goldie, Drug resistance in cancer: A perspective,, Cancer and Metastasis Review, 20 (2001), 63.  doi: 10.1017/CBO9780511666544.  Google Scholar

[8]

J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents,, Mathematical Biosciences, 65 (1983), 291.   Google Scholar

[9]

R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells,, Cancer Research, 66 (2006), 1033.  doi: 10.1158/0008-5472.CAN-05-3077.  Google Scholar

[10]

J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors,, Bull. Math. Biol., 76 (2014), 627.  doi: 10.1007/s11538-014-9936-8.  Google Scholar

[11]

O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance,, Cancer Research, 73 (2013), 7168.  doi: 10.1158/0008-5472.CAN-13-1768.  Google Scholar

[12]

U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics,, in Recent Advances in Optimization and its Applications in Engineering, (2010), 267.  doi: 10.1007/978-3-642-12598-0_23.  Google Scholar

[13]

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

[14]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Contr. Optim., 46 (2007), 1052.  doi: 10.1137/060665294.  Google Scholar

[15]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine,, Control and Cybernetics, 38 (2009), 1501.   Google Scholar

[16]

J. S. Li and N. Khaneja, Ensemble control of linear systems,, Proc. of the 46th IEEE Conference on Decision and Control, (2007), 3768.   Google Scholar

[17]

J. S. Li and N. Khaneja, Ensemble control of Bloch equations,, IEEE Transactions on Automatic Control, 54 (2009), 528.  doi: 10.1109/TAC.2009.2012983.  Google Scholar

[18]

A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377.  doi: 10.1051/m2an/2012031.  Google Scholar

[19]

A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors,, Bull. Math. Biol., 77 (2015), 1.  doi: 10.1007/s11538-014-0046-4.  Google Scholar

[20]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules,, Cancer Treatment Reports, 61 (1977), 1307.   Google Scholar

[21]

L. Norton and R. Simon, The Norton-Simon hypothesis revisited,, Cancer Treatment Reports, 70 (1986), 41.   Google Scholar

[22]

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

[23]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Macmillan, (1964).   Google Scholar

[24]

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

[25]

H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies,, Springer, (2015).  doi: 10.1007/978-1-4939-2972-6.  Google Scholar

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