
- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
Applications of stochastic semigroups to cell cycle models
Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics
1. | Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, 76010, USA |
2. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA |
We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.
References:
[1] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. |
[2] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. |
[3] |
A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. |
[4] |
U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, in: 50 Years of Optimal Control, A. Ioffe, K. Malanowski, F. Tröltzsch, Eds., Control and Cybernetics, 38 (2009), 1305–1325. |
[5] |
R. A. Gatenby,
A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.
doi: 10.1038/459508a. |
[6] |
J. H. Goldie,
Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.
|
[7] |
J. H. Goldie and A. Coldman,
A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307.
|
[8] |
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), 1033-1039.
doi: 10.1158/0008-5472.CAN-05-3077. |
[9] |
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., 74 (2014), 627-653.
doi: 10.1007/s11538-014-9936-8. |
[10] |
P. Hahnfeldt and L. Hlatky,
Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687.
doi: 10.2307/3579891. |
[11] |
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.
doi: 10.1006/jtbi.2003.3162. |
[12] |
O. Lavi, J. Greene, D. Levy and M. Gottesman,
The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.
doi: 10.1158/0008-5472.CAN-13-1768. |
[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), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[14] |
U. Ledzewicz, K. Bratton and H. Schättler,
A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207.
doi: 10.1007/s10440-014-9952-6. |
[15] |
U. Ledzewicz and H. Schättler,
On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197.
doi: 10.1142/S0218339014400014. |
[16] |
M. Leszczy\'nski, E. Ratajczyk, U. Ledzewicz and H. Schättler,
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics, Opuscula Math., 37 (2017), 403-419.
doi: 10.7494/OpMath.2017.37.3.403. |
[17] |
J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768–3773. |
[18] |
J. S. Li and N. Khaneja,
Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
[19] |
D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, 2012.
![]() ![]() |
[20] |
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), 377-399.
doi: 10.1051/m2an/2012031. |
[21] |
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), 1-22.
doi: 10.1007/s11538-014-0046-4. |
[22] |
L. Norton and R. Simon,
Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317.
|
[23] |
L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41–61. |
[24] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. |
[25] |
H. Schättler, On classical envelopes in optimal control theory, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta, USA, (2010), 1879–1884. |
[26] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[27] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, vol. 42, Springer, 2015.
doi: 10.1007/978-1-4939-2972-6. |
[28] |
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. |
[29] |
S. Wang and J. S. Li,
Fixed-endpoint optimal control of bilinear ensemble systems, SIAM J. Control and Optimization (SICON), 55 (2017), 3039-3065.
doi: 10.1137/15M1044151. |
[30] |
S. Wang and H. Schättler,
Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Mathematical Biosciences and Engineering - MBE, 13 (2016), 1223-1240.
doi: 10.3934/mbe.2016040. |
show all references
References:
[1] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003. |
[2] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. |
[3] |
A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. |
[4] |
U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, in: 50 Years of Optimal Control, A. Ioffe, K. Malanowski, F. Tröltzsch, Eds., Control and Cybernetics, 38 (2009), 1305–1325. |
[5] |
R. A. Gatenby,
A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.
doi: 10.1038/459508a. |
[6] |
J. H. Goldie,
Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.
|
[7] |
J. H. Goldie and A. Coldman,
A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307.
|
[8] |
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), 1033-1039.
doi: 10.1158/0008-5472.CAN-05-3077. |
[9] |
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., 74 (2014), 627-653.
doi: 10.1007/s11538-014-9936-8. |
[10] |
P. Hahnfeldt and L. Hlatky,
Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687.
doi: 10.2307/3579891. |
[11] |
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.
doi: 10.1006/jtbi.2003.3162. |
[12] |
O. Lavi, J. Greene, D. Levy and M. Gottesman,
The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.
doi: 10.1158/0008-5472.CAN-13-1768. |
[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), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[14] |
U. Ledzewicz, K. Bratton and H. Schättler,
A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207.
doi: 10.1007/s10440-014-9952-6. |
[15] |
U. Ledzewicz and H. Schättler,
On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197.
doi: 10.1142/S0218339014400014. |
[16] |
M. Leszczy\'nski, E. Ratajczyk, U. Ledzewicz and H. Schättler,
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics, Opuscula Math., 37 (2017), 403-419.
doi: 10.7494/OpMath.2017.37.3.403. |
[17] |
J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768–3773. |
[18] |
J. S. Li and N. Khaneja,
Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
[19] |
D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, 2012.
![]() ![]() |
[20] |
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), 377-399.
doi: 10.1051/m2an/2012031. |
[21] |
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), 1-22.
doi: 10.1007/s11538-014-0046-4. |
[22] |
L. Norton and R. Simon,
Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317.
|
[23] |
L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41–61. |
[24] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. |
[25] |
H. Schättler, On classical envelopes in optimal control theory, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta, USA, (2010), 1879–1884. |
[26] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[27] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, vol. 42, Springer, 2015.
doi: 10.1007/978-1-4939-2972-6. |
[28] |
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. |
[29] |
S. Wang and J. S. Li,
Fixed-endpoint optimal control of bilinear ensemble systems, SIAM J. Control and Optimization (SICON), 55 (2017), 3039-3065.
doi: 10.1137/15M1044151. |
[30] |
S. Wang and H. Schättler,
Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Mathematical Biosciences and Engineering - MBE, 13 (2016), 1223-1240.
doi: 10.3934/mbe.2016040. |






Require: |
Ensure: optimal control start with an initial guess |
while solve |
end while |
Require: |
Ensure: optimal control start with an initial guess |
while solve |
end while |
Require: |
Ensure: initial guess Start with an initial trajectory |
while Solve Update |
end while |
Require: |
Ensure: initial guess Start with an initial trajectory |
while Solve Update |
end while |
[1] |
Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 |
[2] |
Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022 |
[3] |
Karl Peter Hadeler. Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1541-1560. doi: 10.3934/mbe.2013.10.1541 |
[4] |
Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173 |
[5] |
Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 |
[6] |
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 |
[7] |
Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks and Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007 |
[8] |
Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086 |
[9] |
Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 |
[10] |
Yangjin Kim, Khalid Boushaba. An enzyme kinetics model of tumor dormancy, regulation of secondary metastases. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1465-1498. doi: 10.3934/dcdss.2011.4.1465 |
[11] |
Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082 |
[12] |
Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Optimal control of an Allen-Cahn model for tumor growth through supply of cytotoxic drugs. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022003 |
[13] |
Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control and Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002 |
[14] |
Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 |
[15] |
Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 |
[16] |
Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 |
[17] |
Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial and Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63 |
[18] |
Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107 |
[19] |
Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149 |
[20] |
Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]