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 & 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.  Google Scholar

[2]

P. Bonate, Pharmacokinetic and Pharmacodynamic Modeling and Simulation 2nd Ed., Springer, New York, NY, 2011. Google Scholar

[3]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Series: Mathematics and Applications, Vol. 40,2003.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Derendorf and B. Meibohm, Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives, Pharm. Res., 16 (1999), 176-185.   Google Scholar

[8]

S. FaderlM. TalpazZ. EstrovS. O'BrienR. Kurzrock and H. Kantarjian, The biology of chronic myeloid leukemia, New England J. of Medicine, 341 (1999), 164-172.  doi: 10.1056/NEJM199907153410306.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Zeitschrift, 49 (1913), 333-369.   Google Scholar

[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. Google Scholar

[14]

A. Nakamura-IshizuH. 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.  Google Scholar

[15]

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

[16]

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

[17]

C. L. Sawyers, Chronic myeloid leukemia, New England J. of Medicine, 340 (1999), 1330-1340.  doi: 10.1056/NEJM199904293401706.  Google Scholar

[18]

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

[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.  Google Scholar

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.  Google Scholar

[2]

P. Bonate, Pharmacokinetic and Pharmacodynamic Modeling and Simulation 2nd Ed., Springer, New York, NY, 2011. Google Scholar

[3]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer, Series: Mathematics and Applications, Vol. 40,2003.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Derendorf and B. Meibohm, Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives, Pharm. Res., 16 (1999), 176-185.   Google Scholar

[8]

S. FaderlM. TalpazZ. EstrovS. O'BrienR. Kurzrock and H. Kantarjian, The biology of chronic myeloid leukemia, New England J. of Medicine, 341 (1999), 164-172.  doi: 10.1056/NEJM199907153410306.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Zeitschrift, 49 (1913), 333-369.   Google Scholar

[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. Google Scholar

[14]

A. Nakamura-IshizuH. 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.  Google Scholar

[15]

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

[16]

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

[17]

C. L. Sawyers, Chronic myeloid leukemia, New England J. of Medicine, 340 (1999), 1330-1340.  doi: 10.1056/NEJM199904293401706.  Google Scholar

[18]

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

[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.  Google Scholar

Figure 1.  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
Figure 2.  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
Table 1.  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 $E$ $0.7$
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