IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v194y2022i3d10.1007_s10957-022-02046-7.html
   My bibliography  Save this article

Optimal Control of Cancer Chemotherapy with Delays and State Constraints

Author

Listed:
  • Poh Ling Tan

    (University of Malaya)

  • Helmut Maurer

    (Westfalische Wilhelms-Universitat Münster)

  • Jeevan Kanesan

    (University of Malaya)

  • Joon Huang Chuah

    (University of Malaya)

Abstract

A mathematical model of cancer chemotherapy is considered as an optimal control problem with the objective of either minimizing a weighted sum of tumor cells and drug dosage or the terminal tumor volume. The control process is subject to three state constraints involving an upper bound on drug toxicity, a lower bound on the white blood cells (WBCs) population, and a constraint to prevent the WBCs count from staying too long below a fixed upper level. The state constraints are imposed for safeguarding the health of patients during treatment. The dynamics of the WBCs population involves delays due to the delay chain of granulocyte development. The control problem is based on a similar model presented in Iliadis and Barbolosi (Comput. Biomed. Res. 33(3):211–226, 2000). However, the authors do not treat necessary conditions which only recently have been presented in the literature. We introduce two variants of the control problem imposing different state constraints. For a basic control problem, we give a thorough discussion of the necessary optimality conditions. Discretization and nonlinear programming methods are employed to determine extremal solutions that precisely satisfy the necessary conditions. It is surprising that in both control problems the time-delayed solution agrees with the non-delayed solution except for the WBCs which are obtained as backward time shifts of the non-delayed WBCs. Since the control variable appears linearly in the Hamiltonian, we find that the control is a combination of bang-bang arcs and boundary arcs of the state constraints. The direct optimization of the switching times and junction times with the boundary defines a finite-dimensional optimization problem for which we can verify second-order sufficient conditions (SSC). To obtain a more practical protocol, we construct an approximative control with fewer control sub-arcs resulting in only a marginal increase in the cost function.

Suggested Citation

  • Poh Ling Tan & Helmut Maurer & Jeevan Kanesan & Joon Huang Chuah, 2022. "Optimal Control of Cancer Chemotherapy with Delays and State Constraints," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 749-770, September.
  • Handle: RePEc:spr:joptap:v:194:y:2022:i:3:d:10.1007_s10957-022-02046-7
    DOI: 10.1007/s10957-022-02046-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-022-02046-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10957-022-02046-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. G. Vossen, 2010. "Switching Time Optimization for Bang-Bang and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 409-429, February.
    2. Jinghua Shi & Oguzhan Alagoz & Fatih Erenay & Qiang Su, 2014. "A survey of optimization models on cancer chemotherapy treatment planning," Annals of Operations Research, Springer, vol. 221(1), pages 331-356, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Najmeddine Dhieb & Ismail Abdulrashid & Hakim Ghazzai & Yehia Massoud, 2023. "Optimized drug regimen and chemotherapy scheduling for cancer treatment using swarm intelligence," Annals of Operations Research, Springer, vol. 320(2), pages 757-770, January.
    2. M. Soledad Aronna & J. Frédéric Bonnans & Pierre Martinon, 2013. "A Shooting Algorithm for Optimal Control Problems with Singular Arcs," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 419-459, August.
    3. Laura Poggiolini & Gianna Stefani, 2020. "Strong Local Optimality for a Bang–Bang–Singular Extremal: General Constraints," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 24-49, July.
    4. Ursula Felgenhauer, 2016. "Discretization of semilinear bang-singular-bang control problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 295-326, May.
    5. Kai He & Lisa M. Maillart & Oleg A. Prokopyev, 2019. "Optimal sequencing of heterogeneous, non-instantaneous interventions," Annals of Operations Research, Springer, vol. 276(1), pages 109-135, May.
    6. Itziar Irurzun-Arana & Alvaro Janda & Sergio Ardanza-Trevijano & Iñaki F Trocóniz, 2018. "Optimal dynamic control approach in a multi-objective therapeutic scenario: Application to drug delivery in the treatment of prostate cancer," PLOS Computational Biology, Public Library of Science, vol. 14(4), pages 1-16, April.
    7. Nazila Bazrafshan & M. M. Lotfi, 2020. "A finite-horizon Markov decision process model for cancer chemotherapy treatment planning: an application to sequential treatment decision making in clinical trials," Annals of Operations Research, Springer, vol. 295(1), pages 483-502, December.
    8. Temitayo Ajayi & Seyedmohammadhossein Hosseinian & Andrew J. Schaefer & Clifton D. Fuller, 2024. "Combination Chemotherapy Optimization with Discrete Dosing," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 434-455, March.
    9. U. Felgenhauer, 2012. "Structural Stability Investigation of Bang-Singular-Bang Optimal Controls," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 605-631, March.
    10. K. H. Wong & H. W. J. Lee & C. K. Chan & C. Myburgh, 2013. "Control Parametrization and Finite Element Method for Controlling Multi-species Reactive Transport in an Underground Channel," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 168-187, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:194:y:2022:i:3:d:10.1007_s10957-022-02046-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.