# American Institute of Mathematical Sciences

May  2019, 24(5): 2383-2405. doi: 10.3934/dcdsb.2019100

## Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics

 1 Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, 76010, USA 2 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA

* Corresponding author: Heinz Schättler

Received  January 2018 Revised  January 2019 Published  May 2019 Early access  March 2019

We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.

Citation: 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
##### 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] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. [4] U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, in: 50 Years of Optimal Control, A. Ioffe, K. Malanowski, F. Tröltzsch, Eds., Control and Cybernetics, 38 (2009), 1305–1325. [5] R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.  doi: 10.1038/459508a. [6] J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68. [7] J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. [8] R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039.  doi: 10.1158/0008-5472.CAN-05-3077. [9] J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 74 (2014), 627-653.  doi: 10.1007/s11538-014-9936-8. [10] P. Hahnfeldt and L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687.  doi: 10.2307/3579891. [11] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554.  doi: 10.1006/jtbi.2003.3162. [12] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.  doi: 10.1158/0008-5472.CAN-13-1768. [13] 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. [14] U. Ledzewicz, K. Bratton and H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207.  doi: 10.1007/s10440-014-9952-6. [15] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014. [16] M. Leszczy\'nski, E. Ratajczyk, U. 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. [17] J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768–3773. [18] J. S. Li and N. Khaneja, Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.  doi: 10.1109/TAC.2009.2012983. [19] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, 2012. [20] A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399.  doi: 10.1051/m2an/2012031. [21] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22.  doi: 10.1007/s11538-014-0046-4. [22] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. [23] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41–61. [24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. [25] H. Schättler, On classical envelopes in optimal control theory, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta, USA, (2010), 1879–1884. [26] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2. [27] 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. [28] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386.  doi: 10.1016/S0362-546X(01)00184-5. [29] S. Wang and J. S. Li, Fixed-endpoint optimal control of bilinear ensemble systems, SIAM J. Control and Optimization (SICON), 55 (2017), 3039-3065.  doi: 10.1137/15M1044151. [30] S. Wang and H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Mathematical Biosciences and Engineering - MBE, 13 (2016), 1223-1240.  doi: 10.3934/mbe.2016040.

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##### 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] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. [4] U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, in: 50 Years of Optimal Control, A. Ioffe, K. Malanowski, F. Tröltzsch, Eds., Control and Cybernetics, 38 (2009), 1305–1325. [5] R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.  doi: 10.1038/459508a. [6] J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68. [7] J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. [8] R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039.  doi: 10.1158/0008-5472.CAN-05-3077. [9] J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 74 (2014), 627-653.  doi: 10.1007/s11538-014-9936-8. [10] P. Hahnfeldt and L. Hlatky, Cell resensitization during protracted dosing of heterogeneous cell populations, Radiation Research, 150 (1998), 681-687.  doi: 10.2307/3579891. [11] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554.  doi: 10.1006/jtbi.2003.3162. [12] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.  doi: 10.1158/0008-5472.CAN-13-1768. [13] 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. [14] U. Ledzewicz, K. Bratton and H. Schättler, A 3-compartment model for chemotherapy of heterogeneous tumor populations, Acta Applicandae Matematicae, 135 (2015), 191-207.  doi: 10.1007/s10440-014-9952-6. [15] U. Ledzewicz and H. Schättler, On optimal chemotherapy for heterogeneous tumors, J. of Biological Systems, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014. [16] M. Leszczy\'nski, E. Ratajczyk, U. 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. [17] J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768–3773. [18] J. S. Li and N. Khaneja, Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.  doi: 10.1109/TAC.2009.2012983. [19] D. Liberzon, Calculus of Variations and Optimal Control Theory, Princeton University Press, 2012. [20] A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399.  doi: 10.1051/m2an/2012031. [21] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22.  doi: 10.1007/s11538-014-0046-4. [22] L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. [23] L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41–61. [24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. [25] H. Schättler, On classical envelopes in optimal control theory, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta, USA, (2010), 1879–1884. [26] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2. [27] 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. [28] A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386.  doi: 10.1016/S0362-546X(01)00184-5. [29] S. Wang and J. S. Li, Fixed-endpoint optimal control of bilinear ensemble systems, SIAM J. Control and Optimization (SICON), 55 (2017), 3039-3065.  doi: 10.1137/15M1044151. [30] S. Wang and H. Schättler, Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity, Mathematical Biosciences and Engineering - MBE, 13 (2016), 1223-1240.  doi: 10.3934/mbe.2016040.
A numerically computed extremal control (top row, left) for the effectiveness function $\varphi(x) = \frac{2}{1+x^2}$ (top row, right), weights $\alpha = (1, \ldots, 1)$, $\beta = 500\alpha$ and $\gamma = 5000$, and therapy horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) remain finite over the full interval $[0, T]$
A numerically computed extremal control (top row, left) for the effectiveness function $\varphi_1$ and $\varphi_2$ given in (22), (top row, right), $\alpha = (1, \ldots, 1)$, $\beta = 50 \alpha$, $\gamma = (600, 1400)$ and time horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) remain finite over the full interval $[0, T]$
A numerically computed extremal control (top row, left) for the effectiveness function $\varphi_1$ and $\varphi_2$ given in (23), (top row, right), $\alpha = (1, \ldots, 1)$, $\beta = 50 \alpha$, $\gamma = (600, 1400)$ and time horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is not a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) blow up in the interval
Examples of the system response, for the same parameters as in Fig. 3, for the constant controls $u_1 = 1$ and $u_2 = 2.5$
Illustration of the functions $\Upsilon$ (top, left), $\eta$ (top, right) and the full Hamiltonian $H$ (bottom) as function of the controls. These graphs depict the actual values for the solution shown in Fig. 6 below at the time $t = 19$
A numerically computed locally optimal control (top row, left) for $\alpha = (1, \ldots, 1)$, $\beta = 15 \alpha$, $\gamma = (90, 30)$ with therapy horizon $T = 20$ and effectiveness functions $\varphi_1 = \frac{2}{1+x^2}$ and $\varphi_2 = \frac{2}{1+3x^6}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(20, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) remain finite in the full interval $[0, 20]$
A numerically computed locally optimal control (top row, left) for α = (1, ...., 1), β = 15α, γ = (90, 30) with therapy horizon T = 20 and effectiveness functions ${\varphi _1}{\rm{ = }}\frac{2}{{1 + 5{x^2}}}$ and ${\varphi _2}{\rm{ = }}\frac{2}{{1 + 60{{\left( {1 - x} \right)}^6}}}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density n(0, x) ≡ $\frac{{200}}{{21}}$ and the terminal density n(20, x) shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution S to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) blow up in the interval.
A numerically computed extremal control (top row, left) for α = (1, ..., 1), β = 15α, γ = (90, 30) with shortened therapy horizon T = 5 and effectiveness function ${\varphi _1}{\rm{ = }}\frac{2}{{1 + 5{x^2}}}$ and ${\varphi _2}{\rm{ = }}\frac{2}{{1 + 60{{\left( {1 - x} \right)}^6}}}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density n(0, x) ≡ $\frac{{200}}{{21}}$ and the terminal density n(5, x) shown as red curve is given in the bottom row on the left. This control still is a strong local minimum as the entries of the solution S to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) remain finite in the full interval [0, T].
Shooting Method
 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: optimal control   start with an initial guess $\lambda(0)$ and the stepsize $s=0.01$. while $\|\lambda(T)-\alpha\|>\epsilon$ do  solve $N, \; \lambda$.   $\lambda(0)\Leftarrow \lambda(0)-s\Big( \frac{\delta \lambda(T)}{\delta \lambda(0)}\Big) ^{-1} (\lambda(T)-\alpha)$ end while
 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: optimal control   start with an initial guess $\lambda(0)$ and the stepsize $s=0.01$. while $\|\lambda(T)-\alpha\|>\epsilon$ do  solve $N, \; \lambda$.   $\lambda(0)\Leftarrow \lambda(0)-s\Big( \frac{\delta \lambda(T)}{\delta \lambda(0)}\Big) ^{-1} (\lambda(T)-\alpha)$ end while
Iterative Method
 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: initial guess   Start with an initial trajectory $N^{(0)}$ and $k=1$. while $\|N^{(k)}-N^{(k-1)}\|>\epsilon$ do   $A^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   $O^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   Solve $S^{(k)}, \; \nu^{(k)}$ backward in time by assuming $\lambda^{(k)} = S^{(k)}N^{(k)}+\nu^{(k)}$.  Update $N^{(k)}$ forward in time:  $\qquad \dot{N}^{(k)} =[A^{(k-1)}+O^{(k-1)}S^{(k)}]N^{(k)}+O^{(k-1)}\nu^{(k)}$)  $u^{(k)} \Leftarrow$ function of $N^{(k)}$, $\lambda^{(k)}$  $k+1 \Leftarrow k$ end while
 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: initial guess   Start with an initial trajectory $N^{(0)}$ and $k=1$. while $\|N^{(k)}-N^{(k-1)}\|>\epsilon$ do   $A^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   $O^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   Solve $S^{(k)}, \; \nu^{(k)}$ backward in time by assuming $\lambda^{(k)} = S^{(k)}N^{(k)}+\nu^{(k)}$.  Update $N^{(k)}$ forward in time:  $\qquad \dot{N}^{(k)} =[A^{(k-1)}+O^{(k-1)}S^{(k)}]N^{(k)}+O^{(k-1)}\nu^{(k)}$)  $u^{(k)} \Leftarrow$ function of $N^{(k)}$, $\lambda^{(k)}$  $k+1 \Leftarrow k$ end while
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