# American Institute of Mathematical Sciences

May  2019, 24(5): 2039-2052. doi: 10.3934/dcdsb.2019083

## Singularity of controls in a simple model of acquired chemotherapy resistance

 1 Inter-Faculty Individual Doctoral Studies in Natural Sciences and Mathematics, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland 2 Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Sloneczna 54, 10-710 Olsztyn, Poland 3 Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

*Corresponding author

Received  December 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

Fund Project: Piotr Bajger and Mariusz Bodzioch were supported by Polish National Science Centre grant 2016/23/N/ST1/01178. Urszula Foryś was supported by Polish National Science Centre grant 2015/17/N/ST1/02564.

This study investigates how optimal control theory may be used to delay the onset of chemotherapy resistance in tumours. An optimal control problem with simple tumour dynamics and an objective functional explicitly penalising drug resistant tumour phenotype is formulated. It is shown that for biologically relevant parameters the system has a single globally attracting positive steady state. The existence of singular arc is then investigated analytically under a very general form of the resistance penalty in the objective functional. It is shown that the singular controls are of order one and that they satisfy Legendre-Clebsch condition in a subset of the domain. A gradient method for solving the proposed optimal control problem is then used to find the control minimising the objective. The optimal control is found to consist of three intervals: full dose, singular and full dose. The singular part of the control is essential in delaying the onset of drug resistance.

Citation: Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083
##### References:
 [1] P. Bajger, M. Bodzioch and U. Foryś, Role of cell competition in acquired chemotherapy resistance, Proceedings of the 16th Conference on Computational and Mathematical Methods in Science and Engineering, 1 (2016), 132-141. [2] R. H. Chisholm, T. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation, Biochim Biophys Acta, 1860 (2016), 2627-2645.  doi: 10.1016/j.bbagen.2016.06.009. [3] H. Cho and D. Levy, Modeling the chemotherapy-induced selection of drug-resistant traits during tumor growth, J Theor Biol, 436 (2018), 120-134.  doi: 10.1016/j.jtbi.2017.10.005. [4] I. Fidler and L. Ellis, Chemotherapeutic drugs – more really is not better, Nat Med, 6 (2000), 500-502.  doi: 10.1038/74969. [5] J. Foo and F. Michor, Evolution of acquired resistance to anti-cancertherapy, J Theor Biol, 355 (2014), 10-20.  doi: 10.1016/j.jtbi.2014.02.025. [6] M. Gottesman, Mechanisms of cancer drug resistance, Annu Rev of Med, 53 (2002), 615-627.  doi: 10.1146/annurev.med.53.082901.103929. [7] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term tumor burden: The logic for metronomic chemotherapeutic dosing and its antiangiogenic basis, J Theor Biol, 220 (2003), 545-554.  doi: 10.1006/jtbi.2003.3162. [8] P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res, 59 (1999), 4770-4775. [9] I. Kareva, D. Waxman and G. Klement, Metronomic chemotherapy: An attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance, Cancer Lett, 358 (2015), 100-106.  doi: 10.1016/j.canlet.2014.12.039. [10] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Res, 73 (2013), 7168-7175.  doi: 10.1158/0008-5472.CAN-13-1768. [11] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete Cont Dyn-B, 6 (2006), 129-150.  doi: 10.3934/dcdsb.2006.6.129. [12] U. Ledzewicz and H. Schättler, On optimal therapy for heterogeneous tumors, J Biol Sys, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014. [13] H. Monro and E. Gaffney, Modelling chemotherapy resistance in palliation and failed cure, Journal Theor Biol, 257 (2009), 292-302.  doi: 10.1016/j.jtbi.2008.12.006. [14] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [15] P. Savage, J. Stebbing, M. Bower and T. Crook, Why does cytotoxic chemotherapy cure only some cancers?, Nat Clin Pract Oncol, 6 (2009), 43-52.  doi: 10.1038/ncponc1260. [16] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods, Springer, 2015. doi: 10.1007/978-1-4939-2972-6. [17] H. Skipper, Prospectives in Cancer Chemotherapy: Therapeutic Design, Cancer Res, 24 (1964), 1295-1302. [18] J. Śmieja and A. Świerniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification, Int J Ap Mat Com-Pol, 13 (2003), 297-305. [19] J. Śmieja, A. Świerniak and Z. Duda, Gradient method for finding optimal scheduling in infinite dimensional models of chemotherapy, J Theor Med, 3 (2000), 25-36.  doi: 10.1080/10273660008833062. [20] G. Swan, Role of optimal control in cancer chemotherapy, Math Biosci, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P. [21] A. Świerniak, A. Polański, J. Śmieja and M. Kimmel, Modelling growth of drug resistant cancer populations as the system with positive feedback, Math Comput Model, 37 (2003), 1245-1252.  doi: 10.1016/S0895-7177(03)00134-1.

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##### References:
 [1] P. Bajger, M. Bodzioch and U. Foryś, Role of cell competition in acquired chemotherapy resistance, Proceedings of the 16th Conference on Computational and Mathematical Methods in Science and Engineering, 1 (2016), 132-141. [2] R. H. Chisholm, T. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation, Biochim Biophys Acta, 1860 (2016), 2627-2645.  doi: 10.1016/j.bbagen.2016.06.009. [3] H. Cho and D. Levy, Modeling the chemotherapy-induced selection of drug-resistant traits during tumor growth, J Theor Biol, 436 (2018), 120-134.  doi: 10.1016/j.jtbi.2017.10.005. [4] I. Fidler and L. Ellis, Chemotherapeutic drugs – more really is not better, Nat Med, 6 (2000), 500-502.  doi: 10.1038/74969. [5] J. Foo and F. Michor, Evolution of acquired resistance to anti-cancertherapy, J Theor Biol, 355 (2014), 10-20.  doi: 10.1016/j.jtbi.2014.02.025. [6] M. Gottesman, Mechanisms of cancer drug resistance, Annu Rev of Med, 53 (2002), 615-627.  doi: 10.1146/annurev.med.53.082901.103929. [7] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term tumor burden: The logic for metronomic chemotherapeutic dosing and its antiangiogenic basis, J Theor Biol, 220 (2003), 545-554.  doi: 10.1006/jtbi.2003.3162. [8] P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res, 59 (1999), 4770-4775. [9] I. Kareva, D. Waxman and G. Klement, Metronomic chemotherapy: An attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance, Cancer Lett, 358 (2015), 100-106.  doi: 10.1016/j.canlet.2014.12.039. [10] O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Res, 73 (2013), 7168-7175.  doi: 10.1158/0008-5472.CAN-13-1768. [11] U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete Cont Dyn-B, 6 (2006), 129-150.  doi: 10.3934/dcdsb.2006.6.129. [12] U. Ledzewicz and H. Schättler, On optimal therapy for heterogeneous tumors, J Biol Sys, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014. [13] H. Monro and E. Gaffney, Modelling chemotherapy resistance in palliation and failed cure, Journal Theor Biol, 257 (2009), 292-302.  doi: 10.1016/j.jtbi.2008.12.006. [14] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964. [15] P. Savage, J. Stebbing, M. Bower and T. Crook, Why does cytotoxic chemotherapy cure only some cancers?, Nat Clin Pract Oncol, 6 (2009), 43-52.  doi: 10.1038/ncponc1260. [16] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies. An Application of Geometric Methods, Springer, 2015. doi: 10.1007/978-1-4939-2972-6. [17] H. Skipper, Prospectives in Cancer Chemotherapy: Therapeutic Design, Cancer Res, 24 (1964), 1295-1302. [18] J. Śmieja and A. Świerniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification, Int J Ap Mat Com-Pol, 13 (2003), 297-305. [19] J. Śmieja, A. Świerniak and Z. Duda, Gradient method for finding optimal scheduling in infinite dimensional models of chemotherapy, J Theor Med, 3 (2000), 25-36.  doi: 10.1080/10273660008833062. [20] G. Swan, Role of optimal control in cancer chemotherapy, Math Biosci, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P. [21] A. Świerniak, A. Polański, J. Śmieja and M. Kimmel, Modelling growth of drug resistant cancer populations as the system with positive feedback, Math Comput Model, 37 (2003), 1245-1252.  doi: 10.1016/S0895-7177(03)00134-1.
Phase portraits for System (1), when (A) the positive steady state exists and is stable (the biologically realistic case), and (B) the zero steady state is stable
Typical choice for a resistance penalty: $G(z) = \tfrac{1}{2}(1 + \tanh(z))$
Singular arcs for different values c of the constant Hamiltonian: (A) c = 3, (B) c = 4, (C) c = 5 and (D) = 10
Optimal solution (A), together with the corresponding control (B), trajectory (C) and the switching function (D)
Nominal parameter values. All the parameters are non-dimensional
 Name Value Role $\gamma_1$ 0.192 Proliferation rate of sensitive cells. $\gamma_2$ 0.096 Proliferation rate of resistant cells. $\tau_1$ 0.002 Mutation rate towards the resistant phenotype. $\tau_2$ 0.001 Mutation rate towards the sensitive phenotype. $T$ 13.5 Therapy duration. $\omega_1$ 60 Weight for sensitive cell volume at the terminal point. $\omega_2$ 120 Weight for the resistant cell volume at the terminal point. $\eta_1$ 3 Weight in the overall tumour burden penalty for sensitive cells. $\eta_2$ 6 Weight in the overall tumour burden penalty for resistant cells. $\xi$ 1 Weight for the resistant phenotype penalty. $\epsilon$ 0.1 Scaling factor in the resistant phenotype penalty function $G$. $\Delta$ $10^{-6}$ Step used in finite differences gradient calculations.
 Name Value Role $\gamma_1$ 0.192 Proliferation rate of sensitive cells. $\gamma_2$ 0.096 Proliferation rate of resistant cells. $\tau_1$ 0.002 Mutation rate towards the resistant phenotype. $\tau_2$ 0.001 Mutation rate towards the sensitive phenotype. $T$ 13.5 Therapy duration. $\omega_1$ 60 Weight for sensitive cell volume at the terminal point. $\omega_2$ 120 Weight for the resistant cell volume at the terminal point. $\eta_1$ 3 Weight in the overall tumour burden penalty for sensitive cells. $\eta_2$ 6 Weight in the overall tumour burden penalty for resistant cells. $\xi$ 1 Weight for the resistant phenotype penalty. $\epsilon$ 0.1 Scaling factor in the resistant phenotype penalty function $G$. $\Delta$ $10^{-6}$ Step used in finite differences gradient calculations.
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