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May  2019, 24(5): 2315-2334. doi: 10.3934/dcdsb.2019097

Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics

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  January 2018 Revised  January 2019 Published  March 2019

We analyze the structure of optimal protocols for a mathematical model of tumor anti-angiogenic treatment. The control represents the concentration of the agent and we consider the problem to administer an a priori given total amount of agents in order to achieve a minimum tumor volume/maximum tumor reduction. In earlier work, this problem was studied with a log-kill type pharmacodynamic model for drug effects which does not account for saturation of the drug concentration. Here we study the effect of incorporating a Michaelis-Menten (MM) or $ E_{\max} $-type pharmacodynamic model, the most commonly used model in the field of pharmacometrics. We compare the formulations of both problems and the resulting solutions. The reformulated problem with $ E_{\max} $ pharmacodynamics is no longer linear in the control. This results in qualitative changes in the structure of optimal controls which, in line with an interpretation as concentrations, now are continuous while discontinuities exist if the log-kill model is used which is more in line with an interpretation of the control as dose rates. In spite of these qualitative differences, similarities in the structures of solutions can be observed. Both aspects are discussed theoretically and illustrated numerically.

Citation: Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097
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]

P. HahnfeldtD. PanigrahyJ. 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.

[4]

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

[5]

U. Ledzewicz and H. Moore, Dynamical systems properties of a mathematical model for treatment of CML, Applied Sciences, 6 (2016), p291. doi: 10.3390/app6100291.

[6]

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.

[7]

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.

[8]

M. LeszczyńskiE. RatajczykU. 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.

[9]

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.

[10]

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

[11]

A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003.

[12]

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

[13]

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

[14]

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

[15]

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.

[16]

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. doi: 10.1007/BF02459681.

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]

P. HahnfeldtD. PanigrahyJ. 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.

[4]

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

[5]

U. Ledzewicz and H. Moore, Dynamical systems properties of a mathematical model for treatment of CML, Applied Sciences, 6 (2016), p291. doi: 10.3390/app6100291.

[6]

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.

[7]

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.

[8]

M. LeszczyńskiE. RatajczykU. 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.

[9]

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.

[10]

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

[11]

A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003.

[12]

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

[13]

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

[14]

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

[15]

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.

[16]

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. doi: 10.1007/BF02459681.

Figure 1.  $ E_{\max} $ pharmacodynamic model.
Figure 2.  Geometric shape of the sets $ {\mathcal L}_0 $ and $ {\mathcal L}_m $ and definition of the associated regions $ R_i $ and $ S_i $ for ${i}{\rm{ = 1, 2, 3, 4}}{\rm{.}} $
Figure 3.  The region $ S_2 $ (inside the loop $ {\mathcal L}_m $ and above the diagonal) is not positively invariant for $ u = u_{\max} $.
Figure 4.  Graph of the value $ V(\varepsilon) $ for the one-parameter family $ {\mathcal F}_{\varepsilon} $ near its minimum value.
Figure 5.  Optimal control (left) and corresponding trajectory (right) for the initial condition $ (p_0, q_0) = (4600, 5800) $ and $ A = 12 $. The graphs are shown with $ UC_{50} = 50 $, $ U_{\max} = 1.5UC_{50} = 75 $ and $ A = 12UC_{50} = 600 $. The optimal controlled trajectory crosses the diagonal along a full dose segment and the optimal concatenation sequence is full dose $ \rightarrow $ $ \tilde{u} $ $ \rightarrow $ no dose. The initial full dose segment is shown in black, the intermediate interior control segment in blue and the final no dose segment in black; junctions are marked with an asterisk. The loop $ {\mathcal L}_m $ is shown as the dashed red curve.
Figure 6.  Optimal control (left) and corresponding trajectory (right) for the initial condition $ (p_0, q_0) = (12000, 15000) $ and $ A = 2 $. The graphs are shown with $ UC_{50} = 50 $, $ U_{\max} = 1.5UC_{50} = 75 $ and $ A = 2UC_{50} = 100 $. The optimal controlled trajectory crosses the diagonal using intermediate doses along the control $ \tilde{u} $ and the optimal concatenation sequence is of the shortened form $ \tilde{u} $ $ \rightarrow $ no dose. The optimal controlled trajectory is shown in red and as a comparison the corresponding full dose trajectory is shown in blue. Here there is a significant difference in the final tumor volumes.
Table 1.  Parameter values used in numerical computations.
variable/
coefficients
interpretation numerical value dimension
$ p $ tumor volume $ mm^{3} $
$ q $ carrying capacity of the vasculature $ mm^{3} $
$ \xi $ tumor growth parameter $ 0.192 $ per day
$ \mu $ natural loss of endothelial support $ 0.02 $ per day
$ b $ stimulation parameter, 'birth' $ 5.85 $ per day
$ d $ inhibition parameter, 'death' $ 0.00873 $ $ mm^{-2} $
per day
$ G $ anti-angiogenic killing coefficient at limiting concentration $ 10 $ per day
$ UC_{50} $ concentration with 50% effect $ 1 $
$ u_{\max} $ normalized maximum dose rate $ 1.5 $
$ A $ normalized total dose $ 2 $; $ 12 $
variable/
coefficients
interpretation numerical value dimension
$ p $ tumor volume $ mm^{3} $
$ q $ carrying capacity of the vasculature $ mm^{3} $
$ \xi $ tumor growth parameter $ 0.192 $ per day
$ \mu $ natural loss of endothelial support $ 0.02 $ per day
$ b $ stimulation parameter, 'birth' $ 5.85 $ per day
$ d $ inhibition parameter, 'death' $ 0.00873 $ $ mm^{-2} $
per day
$ G $ anti-angiogenic killing coefficient at limiting concentration $ 10 $ per day
$ UC_{50} $ concentration with 50% effect $ 1 $
$ u_{\max} $ normalized maximum dose rate $ 1.5 $
$ A $ normalized total dose $ 2 $; $ 12 $
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