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January  2021, 26(1): 483-499. doi: 10.3934/dcdsb.2020213

On the role of pharmacometrics in mathematical models for cancer treatments

Urszula Ledzewicz 1,2,, and  Heinz Schättler 3,

1. 

Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland
2. 

Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il, 62026-1653, USA
3. 

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

* Corresponding author: Urszula Ledzewicz

Received  April 2020 Revised  April 2020 Published  January 2021 Early access  July 2020

We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as pharmacodynamic (PD) models are varied. The results will be illustrated with a sample of models for cancer treatment.

Keywords: optimal control, pharmacokinetics, pharmacodynamics, cancer treatment, singular controls.
Mathematics Subject Classification: primary: 92C50, 49K15, secondary: 93C15.
Citation: Urszula Ledzewicz, Heinz Schättler. On the role of pharmacometrics in mathematical models for cancer treatments. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 483-499. doi: 10.3934/dcdsb.2020213
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]

C. S. Chou and A. Friedman, Introduction to Mathematical Biology - Modeling, Analysis and Simulation, Springer Verlag, 2016. doi: 10.1007/978-3-319-29638-8.

[4]

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

[5]

L. A. Fernández and C. Pola, Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy, Discrete and Continuous Dynamical Systems, Series B, 24 (2019), 2577-2612.  doi: 10.3934/dcdsb.2018266.

[6]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. 

[7]

A. Källén, Computational Pharmacokinetics, Chapman and Hall, CRC, London, 2007.

[8]

H. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River, NJ, USA, 2002.

[9]

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

[10]

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, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267–276. doi: 10.1007/978-3-642-12598-0_23.

[11]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.  doi: 10.3934/mbe.2011.8.307.

[12]

U. Ledzewicz and H. Moore, Dynamical systems properties of a mathematical model for the treatment of CML, Applied Sciences, 6 (2016), 291. doi: 10.3390/app6100291.

[13]

U. Ledzewicz and H. Moore, Optimal control applied to a generalized Michaelis-Menten model of CML therapy, Dicrete and Continuous Dynamical Systems, Series B, 23 (2018), 331-346.  doi: 10.3934/dcdsb.2018022.

[14]

U. Ledzewicz and H. Schättler, Controlling a model for bone marrow dynamics in cancer chemotherapy, Mathematical Biosciences and Engineering, 1 (2004), 95-110.  doi: 10.3934/mbe.2004.1.95.

[15]

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.

[16]

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.

[17]

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.

[18]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312.  doi: 10.1016/j.jtbi.2008.02.014.

[19]

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

[20]

M. Leszczyński, E. Ratajczyk, U. Ledzewicz and H. Schättler, Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics, Opuscula Mathematica, 37 (2017), 403-419.  doi: 10.7494/OpMath.2017.37.3.403.

[21]

M. Leszczyński, U. Ledzewicz and H. Schättler, Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics, Discrete and Continuous Dynamical Systems, Series B, 24 (2019), 2315-2334.  doi: 10.3934/dcdsb.2019097.

[22]

M. Leszczyński, U. Ledzewicz and H. Schättler, Optimal control for a mathematical model for chemotherapy with pharmacometrics, Mathematical Modelling of Natural Phenomena, 2020. doi: 10.1051/mmnp/2020008.

[23]

H. Moore, U. Ledzewicz and L. Strauss, Optimization of combination therapy for chronic myeloid leukemia with dosing constraints, J. of Mathematical Biology, 77 (2018), 1533-1561.  doi: 10.1007/s00285-018-1262-6.

[24]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222, (2009), 13–26. doi: 10.1016/j.mbs.2009.08.004.

[25]

P. Macheras and A. Iliadin, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, Interdisciplinary Applied Mathematics, Vol. 30, 2nd ed., Springer, New York, 2016. doi: 10.1007/978-3-319-27598-7.

[26] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific Press, ingapore, 1994.  doi: 10.1142/2048.
[27]

L. G. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J. of Theoretical Medicine, 3 (2001), 79-100. 

[28]

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

[29]

M. Rowland and T.N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia, 1995.

[30]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2.

[31]

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.

[32]

H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong locak optimality in optimal control problems with $L_2$-type objectives and control constraints, Dicrete and Continuous Dynamical Systems, Series B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.

[33]

S. Shimoda, K. Nishida, M. Sakakida, Y. Konno, K. Ichinose, M. Uehara, T. Nowak and M. Shichiri, Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas, Frontiers of Medical and Biological Engineering, 8 (1997), 197-211. 

[34]

H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. 

[35]

G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984.

[36]

G. W. Swan, General applications of optimal control theory in cancer chemotherapy, IMA J. of Mathematical Applications in Medicine and Biology, 5 (1988), 303-316.  doi: 10.1093/imammb/5.4.303.

[37]

G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101, (1990), 237–284.

[38]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Biomedical Systems Modelling and Simulation (Paris, 1988), 51–53, IMACS Ann. Comput. Appl. Math., 5, IMACS Trans. Sci. Comput. '88, Baltzer, Basel, 1989.

[39]

A. Swierniak, Cell cycle as an object of control,, Journal of Biological Systems, 3 (1995), 9-54.  doi: 10.1007/978-3-319-28095-0_2.

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]

C. S. Chou and A. Friedman, Introduction to Mathematical Biology - Modeling, Analysis and Simulation, Springer Verlag, 2016. doi: 10.1007/978-3-319-29638-8.

[4]

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

[5]

L. A. Fernández and C. Pola, Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy, Discrete and Continuous Dynamical Systems, Series B, 24 (2019), 2577-2612.  doi: 10.3934/dcdsb.2018266.

[6]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. 

[7]

A. Källén, Computational Pharmacokinetics, Chapman and Hall, CRC, London, 2007.

[8]

H. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, Upper Saddle River, NJ, USA, 2002.

[9]

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

[10]

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, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267–276. doi: 10.1007/978-3-642-12598-0_23.

[11]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323.  doi: 10.3934/mbe.2011.8.307.

[12]

U. Ledzewicz and H. Moore, Dynamical systems properties of a mathematical model for the treatment of CML, Applied Sciences, 6 (2016), 291. doi: 10.3390/app6100291.

[13]

U. Ledzewicz and H. Moore, Optimal control applied to a generalized Michaelis-Menten model of CML therapy, Dicrete and Continuous Dynamical Systems, Series B, 23 (2018), 331-346.  doi: 10.3934/dcdsb.2018022.

[14]

U. Ledzewicz and H. Schättler, Controlling a model for bone marrow dynamics in cancer chemotherapy, Mathematical Biosciences and Engineering, 1 (2004), 95-110.  doi: 10.3934/mbe.2004.1.95.

[15]

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.

[16]

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.

[17]

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.

[18]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312.  doi: 10.1016/j.jtbi.2008.02.014.

[19]

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

[20]

M. Leszczyński, E. Ratajczyk, U. Ledzewicz and H. Schättler, Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics, Opuscula Mathematica, 37 (2017), 403-419.  doi: 10.7494/OpMath.2017.37.3.403.

[21]

M. Leszczyński, U. Ledzewicz and H. Schättler, Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics, Discrete and Continuous Dynamical Systems, Series B, 24 (2019), 2315-2334.  doi: 10.3934/dcdsb.2019097.

[22]

M. Leszczyński, U. Ledzewicz and H. Schättler, Optimal control for a mathematical model for chemotherapy with pharmacometrics, Mathematical Modelling of Natural Phenomena, 2020. doi: 10.1051/mmnp/2020008.

[23]

H. Moore, U. Ledzewicz and L. Strauss, Optimization of combination therapy for chronic myeloid leukemia with dosing constraints, J. of Mathematical Biology, 77 (2018), 1533-1561.  doi: 10.1007/s00285-018-1262-6.

[24]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222, (2009), 13–26. doi: 10.1016/j.mbs.2009.08.004.

[25]

P. Macheras and A. Iliadin, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, Interdisciplinary Applied Mathematics, Vol. 30, 2nd ed., Springer, New York, 2016. doi: 10.1007/978-3-319-27598-7.

[26] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific Press, ingapore, 1994.  doi: 10.1142/2048.
[27]

L. G. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: An optimal control approach, J. of Theoretical Medicine, 3 (2001), 79-100. 

[28]

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

[29]

M. Rowland and T.N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia, 1995.

[30]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2.

[31]

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.

[32]

H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong locak optimality in optimal control problems with $L_2$-type objectives and control constraints, Dicrete and Continuous Dynamical Systems, Series B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.

[33]

S. Shimoda, K. Nishida, M. Sakakida, Y. Konno, K. Ichinose, M. Uehara, T. Nowak and M. Shichiri, Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas, Frontiers of Medical and Biological Engineering, 8 (1997), 197-211. 

[34]

H. E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology, 48 (1986), 253-278. 

[35]

G. W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York, 1984.

[36]

G. W. Swan, General applications of optimal control theory in cancer chemotherapy, IMA J. of Mathematical Applications in Medicine and Biology, 5 (1988), 303-316.  doi: 10.1093/imammb/5.4.303.

[37]

G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101, (1990), 237–284.

[38]

A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Biomedical Systems Modelling and Simulation (Paris, 1988), 51–53, IMACS Ann. Comput. Appl. Math., 5, IMACS Trans. Sci. Comput. '88, Baltzer, Basel, 1989.

[39]

A. Swierniak, Cell cycle as an object of control,, Journal of Biological Systems, 3 (1995), 9-54.  doi: 10.1007/978-3-319-28095-0_2.

Figure 1.  A schematic representation of the interactions described by pharmacometric models
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Figure 2.  $ E_{max} $ model for the PD of a drug
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Figure 3.  Sigmoidal model for the PD of a drug
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Figure 4.  Formulation [CC0]: Example of an optimal control of the type $ \mathbf{u}_{\max}\mathbf{s0} $ as function of time (left) and corresponding controlled trajectory (right) shown with the tumor volume $ p $ on the vertical axis and the vasculature $ q $ on the horizontal axis for initial data $ (p_{0}, q_{0}, A) = (12000, 15000, 300) $. The full dose segment for the control and its corresponding trajectory are shown in red, the singular pieces in blue and the no dose segments in green. While the control is singular, the system follows the singular arc $ S $. The dynamics is very fast horizontally along the full and no-dose segments which generates long segments of the trajectory for small time intervals, but it is much slower vertically along the singular control. This segment is the dominant piece in time and the only segment where a significant reduction of the vasculature is achieved
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Figure 5.  Comparison of the solution to the restricted optimization problem where controls are kept constant for 3 months for year 3 (top row; controls shown on the left and the corresponding states are given on the right) with the solutions to the optimal control problem for year 3 (shown in the bottom row)
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