# American Institute of Mathematical Sciences

January  2018, 23(1): 331-346. doi: 10.3934/dcdsb.2018022

## Optimal control applied to a generalized Michaelis-Menten model of CML therapy

 1 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA 2 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland 3 Bristol-Myers Squibb, Quantitative Clinical Pharmacology, Princeton, NJ 08543, USA

* Corresponding author

Received  January 2017 Published  January 2018

We generalize a previously-studied model for chronic myeloid leu-kemia (CML) [13,10] by incorporating a differential equation which has a Michaelis-Menten model as the steady-state solution to the dynamics. We use this more general non-steady-state formulation to represent the effects of various therapies on patients with CML and apply optimal control to compute regimens with the best outcomes. The advantage of using this more general differential equation formulation is to reduce nonlinearities in the model, which enables an analysis of the optimal control problem using Lie-algebraic computations. We show both the theoretical analysis for the problem and give graphs that represent numerically-computed optimal combination regimens for treating the disease.

Citation: 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
##### References:
 [1] E. K. Afenya and C. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells, Bull. Math. Biol., 62 (2000), 527-542.  doi: 10.1006/bulm.1999.0165. [2] P. Bonate, Pharmacokinetic and Pharmacodynamic Modeling and Simulation 2nd Ed., Springer, New York, NY, 2011. [3] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Series: Mathematics and Applications, Vol. 40,2003. [4] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [5] G. E. Briggs and J. B. Haldane, A note on the kinetics of enzyme action, Biochem. J., 19 (1925), 338-339.  doi: 10.1042/bj0190338. [6] B. Chereda and J. V. Melo, Natural course and biology of CML, Ann. Hematol., 94 (2015), 107-121.  doi: 10.1007/s00277-015-2325-z. [7] H. Derendorf and B. Meibohm, Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives, Pharm. Res., 16 (1999), 176-185. [8] S. Faderl, M. Talpaz, Z. Estrov, S. O'Brien, R. Kurzrock and H. Kantarjian, The biology of chronic myeloid leukemia, New England J. of Medicine, 341 (1999), 164-172.  doi: 10.1056/NEJM199907153410306. [9] A. Källén, Computational Pharmacokinetics, Chapman and Hall, CRC, London, 2007. [10] 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. [11] 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. [12] L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Zeitschrift, 49 (1913), 333-369. [13] H. Moore, L. Strauss and U. Ledzewicz, Mathematical optimization of combination therapy for chronic myeloid leukemia (CML), A poster presented at the 6th American Conference on Pharmacometrics, Crystal City, VA, October 4-7,2015. [14] A. Nakamura-Ishizu, H. Takizawa and T. Suda, The analysis, roles and regulation of quiescence in hematopoietic stem cells, Development, 141 (2014), 4656-4666.  doi: 10.1242/dev.106575. [15] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [16] M. Rowland and T. N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia, 1995. [17] C. L. Sawyers, Chronic myeloid leukemia, New England J. of Medicine, 340 (1999), 1330-1340.  doi: 10.1056/NEJM199904293401706. [18] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [19] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies Springer Verlag, 2015. doi: 10.1007/978-1-4939-2972-6.

show all references

##### References:
 [1] E. K. Afenya and C. Calderón, Diverse ideas on the growth kinetics of disseminated cancer cells, Bull. Math. Biol., 62 (2000), 527-542.  doi: 10.1006/bulm.1999.0165. [2] P. Bonate, Pharmacokinetic and Pharmacodynamic Modeling and Simulation 2nd Ed., Springer, New York, NY, 2011. [3] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Series: Mathematics and Applications, Vol. 40,2003. [4] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. [5] G. E. Briggs and J. B. Haldane, A note on the kinetics of enzyme action, Biochem. J., 19 (1925), 338-339.  doi: 10.1042/bj0190338. [6] B. Chereda and J. V. Melo, Natural course and biology of CML, Ann. Hematol., 94 (2015), 107-121.  doi: 10.1007/s00277-015-2325-z. [7] H. Derendorf and B. Meibohm, Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives, Pharm. Res., 16 (1999), 176-185. [8] S. Faderl, M. Talpaz, Z. Estrov, S. O'Brien, R. Kurzrock and H. Kantarjian, The biology of chronic myeloid leukemia, New England J. of Medicine, 341 (1999), 164-172.  doi: 10.1056/NEJM199907153410306. [9] A. Källén, Computational Pharmacokinetics, Chapman and Hall, CRC, London, 2007. [10] 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. [11] 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. [12] L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Zeitschrift, 49 (1913), 333-369. [13] H. Moore, L. Strauss and U. Ledzewicz, Mathematical optimization of combination therapy for chronic myeloid leukemia (CML), A poster presented at the 6th American Conference on Pharmacometrics, Crystal City, VA, October 4-7,2015. [14] A. Nakamura-Ishizu, H. Takizawa and T. Suda, The analysis, roles and regulation of quiescence in hematopoietic stem cells, Development, 141 (2014), 4656-4666.  doi: 10.1242/dev.106575. [15] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [16] M. Rowland and T. N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia, 1995. [17] C. L. Sawyers, Chronic myeloid leukemia, New England J. of Medicine, 340 (1999), 1330-1340.  doi: 10.1056/NEJM199904293401706. [18] H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2. [19] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies Springer Verlag, 2015. doi: 10.1007/978-1-4939-2972-6.
Diagram of the dynamical system. The shaded circular regions represent the "populations" or "states" included in the model. Solid arrows extending from or to the populations represent changes in numbers, with inward-pointing arrows signifying increases in numbers and outward-pointing arrows decreases in numbers. Dashed arrows indicate indirect effects leading to increases or decreases. Colored bars represent inhibition of a production or of an indirect effect, due to the represented treatment. Colored arrows represent amplification of a rate or of an indirect effect
Comparison of locally optimal controls (left column) and corresponding time-evolution of the states (right column). The top row shows the solutions for the optimal control problem [OC1] with dynamics (29)-(31) while the middle and bottom row show solutions for problem [OC2] with dynamics (32)-(37). For the solution shown in the middle row we chose $(b_1,b_2,b_3)=(5,5,5)$ while $(b_1,b_2,b_3)=(0.05,0.02,0.010)$ for the solution shown in the bottom row
Cell populations, dose levels, and parameters used in calculations
 Symbol Interpretation Units values $Q$ concentration of quiescent leukemic cells cells/$\mu$L $Q_\text{init}$ initial value of Q cells/$\mu$L $0.100$ $P$ concentration of proliferating leukemic cells cells/$\mu$L $P_\text{init}$ initial value of P cells/$\mu$L $1670$ $P_\text{ss}$ steady-state of P cells/$\mu$L $3330$ $PC_{50}$ value of $P$ with half the maximum effect cells/$\mu$L $1110$ $E$ effector T cells cells/$\mu$L $E_\text{init}$ initial value of E cells/$\mu$L $1700$ $E_\text{ss}$ carrying capacity of effector T cells cells/$\mu$L $3500$ $EC_{50}$ value of $E$ with half the maximum effect cells/$\mu$L $2500$ $r_{Q}$ replication rate constant of quiescent cells 1/month $0.011$ $\delta_{Q}$ natural death rate constant of quiescent cells 1/month $0.00225$ $k_{P}$ rate constant of $Q$ cells differentiating into $P$ 1/month $1.6$ $r_{P}$ replication rate constant 1/month $0.03$ of proliferating leukemic cells $\delta_{P}$ natural death rate constant 1/month $0.002$ of proliferating leukemic cells $s_{E}$ growth rate constant for effector T cells 1/month $0.01$ $\delta_{E}$ natural death rate constant of effector T cells 1/month $0.02$ $P_{\max,1}$ maximum stimulation effect of $P$ on $E$ $0.8$ $P_{\max,2}$ maximum inhibition effect of $P$ on $E$ $0.5$ $E_{\max,1}$ maximum effect of $E$ on $Q$ $2$ $E_{\max,2}$ maximum effect of $E$ on $P$ $2$ $u_{1}$ dose level of a general BCR-ABL1 inhibitor mg (e.g., imatinib) $u_{1}^{\max}$ maximum dose level of $u_1$ mg $800$ $U1C_{50}$ level of $u_{1}$ that gives half the maximum effect mg $300$ $U1_{\max,1}$ maximum possible effect of $u_{1}$ $0.8$ on new $P$ from $Q$ and growth of $P$ $U1_{\max,2}$ maximum effect of $u_{1}$ on death of $P$ $10$ $u_{2}$ dose level of a BCR-ABL1 inhibitor that also has immunomodulatory effects (e.g., dasatinib) mg $u_{2}^{\max}$ maximum dose level of $u_2$ mg $140$ $U2C_{50}$ dose level of $u_{2}$ that gives half the maximum effect mg $40$ $U2_{\max,1}$ maximum effect of $u_{2}$ on death of $Q$ or $P$ $2$ $U2_{\max,2}$ maximum effect of $u_{2}$ on new $P$ from $Q$ and growth of $P$ $0.6$ $U2_{\max,3}$ maximum effect of $u_{2}$ on death of $P$ $10$ $U2_{\max,4}$ maximum effect of $u_{2}$ on stimulating proliferation of $E$ $10$ $U2_{\max,5}$ maximum effect of $u_{2}$ on prevention of the death of $E$ $0.4$ $u_{3}$ dose level of an immunomodulatory agent mg (e.g., nivolumab) $u_{3}^{\max}$ maximum dose level of $u_3$ mg $240$ $U3C_{50}$ dose level of $u_{3}$ that gives half the maximum effect mg $80$ $U3_{\max,1}$ maximum effect of $u_{3}$ on death of $Q$ or $P$ $1$ $U3_{\max,2}$ maximum effect of $u_{3}$ on stimulating proliferation of $E$ $1$ $U3_{\max,3}$ maximum effect of $u_{3}$ on prevention of the death of $E$ $0.7$
 Symbol Interpretation Units values $Q$ concentration of quiescent leukemic cells cells/$\mu$L $Q_\text{init}$ initial value of Q cells/$\mu$L $0.100$ $P$ concentration of proliferating leukemic cells cells/$\mu$L $P_\text{init}$ initial value of P cells/$\mu$L $1670$ $P_\text{ss}$ steady-state of P cells/$\mu$L $3330$ $PC_{50}$ value of $P$ with half the maximum effect cells/$\mu$L $1110$ $E$ effector T cells cells/$\mu$L $E_\text{init}$ initial value of E cells/$\mu$L $1700$ $E_\text{ss}$ carrying capacity of effector T cells cells/$\mu$L $3500$ $EC_{50}$ value of $E$ with half the maximum effect cells/$\mu$L $2500$ $r_{Q}$ replication rate constant of quiescent cells 1/month $0.011$ $\delta_{Q}$ natural death rate constant of quiescent cells 1/month $0.00225$ $k_{P}$ rate constant of $Q$ cells differentiating into $P$ 1/month $1.6$ $r_{P}$ replication rate constant 1/month $0.03$ of proliferating leukemic cells $\delta_{P}$ natural death rate constant 1/month $0.002$ of proliferating leukemic cells $s_{E}$ growth rate constant for effector T cells 1/month $0.01$ $\delta_{E}$ natural death rate constant of effector T cells 1/month $0.02$ $P_{\max,1}$ maximum stimulation effect of $P$ on $E$ $0.8$ $P_{\max,2}$ maximum inhibition effect of $P$ on $E$ $0.5$ $E_{\max,1}$ maximum effect of $E$ on $Q$ $2$ $E_{\max,2}$ maximum effect of $E$ on $P$ $2$ $u_{1}$ dose level of a general BCR-ABL1 inhibitor mg (e.g., imatinib) $u_{1}^{\max}$ maximum dose level of $u_1$ mg $800$ $U1C_{50}$ level of $u_{1}$ that gives half the maximum effect mg $300$ $U1_{\max,1}$ maximum possible effect of $u_{1}$ $0.8$ on new $P$ from $Q$ and growth of $P$ $U1_{\max,2}$ maximum effect of $u_{1}$ on death of $P$ $10$ $u_{2}$ dose level of a BCR-ABL1 inhibitor that also has immunomodulatory effects (e.g., dasatinib) mg $u_{2}^{\max}$ maximum dose level of $u_2$ mg $140$ $U2C_{50}$ dose level of $u_{2}$ that gives half the maximum effect mg $40$ $U2_{\max,1}$ maximum effect of $u_{2}$ on death of $Q$ or $P$ $2$ $U2_{\max,2}$ maximum effect of $u_{2}$ on new $P$ from $Q$ and growth of $P$ $0.6$ $U2_{\max,3}$ maximum effect of $u_{2}$ on death of $P$ $10$ $U2_{\max,4}$ maximum effect of $u_{2}$ on stimulating proliferation of $E$ $10$ $U2_{\max,5}$ maximum effect of $u_{2}$ on prevention of the death of $E$ $0.4$ $u_{3}$ dose level of an immunomodulatory agent mg (e.g., nivolumab) $u_{3}^{\max}$ maximum dose level of $u_3$ mg $240$ $U3C_{50}$ dose level of $u_{3}$ that gives half the maximum effect mg $80$ $U3_{\max,1}$ maximum effect of $u_{3}$ on death of $Q$ or $P$ $1$ $U3_{\max,2}$ maximum effect of $u_{3}$ on stimulating proliferation of $E$ $1$ $U3_{\max,3}$ maximum effect of $u_{3}$ on prevention of the death of $E$ $0.7$
 [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] 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 [3] 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 [4] 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 [5] Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 195-216. doi: 10.3934/mbe.2017013 [6] Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100 [7] Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037 [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] Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030 [10] Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021 [11] Enrique Fernández-Cara, Juan Límaco, Laurent Prouvée. Optimal control of a two-equation model of radiotherapy. Mathematical Control and Related Fields, 2018, 8 (1) : 117-133. doi: 10.3934/mcrf.2018005 [12] Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219 [13] Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control and Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015 [14] IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial and Management Optimization, 2018, 14 (2) : 427-446. doi: 10.3934/jimo.2017054 [15] Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456 [16] Qingkai Kong, Zhipeng Qiu, Zi Sang, Yun Zou. Optimal control of a vector-host epidemics model. Mathematical Control and Related Fields, 2011, 1 (4) : 493-508. doi: 10.3934/mcrf.2011.1.493 [17] Erin N. Bodine, Louis J. Gross, Suzanne Lenhart. Optimal control applied to a model for species augmentation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 669-680. doi: 10.3934/mbe.2008.5.669 [18] Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229 [19] Sophia R-J Jang, Hsiu-Chuan Wei. On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3261-3295. doi: 10.3934/dcdsb.2021184 [20] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639

2020 Impact Factor: 1.327