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

Singularity of controls in a simple model of acquired chemotherapy resistance

Piotr Bajger 1,, , Mariusz Bodzioch 2, and  Urszula Foryś 3,

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  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.

Keywords: Optimal control, ordinary differential equations, cancer chemotherapy, drug resistance, non-homogeneous tumour.
Mathematics Subject Classification: Primary: 49K15, 92C50; Secondary: 37N25.
Citation: Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

I. Fidler and L. Ellis, Chemotherapeutic drugs – more really is not better, Nat Med, 6 (2000), 500-502.  doi: 10.1038/74969.  Google Scholar

[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.  Google Scholar

[6]

M. Gottesman, Mechanisms of cancer drug resistance, Annu Rev of Med, 53 (2002), 615-627.  doi: 10.1146/annurev.med.53.082901.103929.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

U. Ledzewicz and H. Schättler, On optimal therapy for heterogeneous tumors, J Biol Sys, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014.  Google Scholar

[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.  Google Scholar

[14] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[17]

H. Skipper, Prospectives in Cancer Chemotherapy: Therapeutic Design, Cancer Res, 24 (1964), 1295-1302.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

G. Swan, Role of optimal control in cancer chemotherapy, Math Biosci, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

I. Fidler and L. Ellis, Chemotherapeutic drugs – more really is not better, Nat Med, 6 (2000), 500-502.  doi: 10.1038/74969.  Google Scholar

[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.  Google Scholar

[6]

M. Gottesman, Mechanisms of cancer drug resistance, Annu Rev of Med, 53 (2002), 615-627.  doi: 10.1146/annurev.med.53.082901.103929.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

U. Ledzewicz and H. Schättler, On optimal therapy for heterogeneous tumors, J Biol Sys, 22 (2014), 177-197.  doi: 10.1142/S0218339014400014.  Google Scholar

[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.  Google Scholar

[14] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, MacMillan, New York, 1964.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[17]

H. Skipper, Prospectives in Cancer Chemotherapy: Therapeutic Design, Cancer Res, 24 (1964), 1295-1302.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

G. Swan, Role of optimal control in cancer chemotherapy, Math Biosci, 101 (1990), 237-284.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[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.  Google Scholar

Figure 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
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Figure 2.  Typical choice for a resistance penalty: $G(z) = \tfrac{1}{2}(1 + \tanh(z))$
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Figure 3.  Singular arcs for different values c of the constant Hamiltonian: (A) c = 3, (B) c = 4, (C) c = 5 and (D) = 10
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Figure 4.  Optimal solution (A), together with the corresponding control (B), trajectory (C) and the switching function (D)
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Table 1.  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.
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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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