# 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 & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022
##### References:

show all references

##### References:
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] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [2] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [3] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [4] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [5] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [6] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 [7] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [8] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [9] Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 [10] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [11] Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020343 [12] José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271 [13] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [14] Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302 [15] Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114 [16] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [17] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [18] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [19] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [20] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

2019 Impact Factor: 1.27