2023 Volume 13 Issue 4
Article Contents

Lorand Gabriel Parajdi, Flavius Pătrulescu, Radu Precup, Ioan Ştefan Haplea. TWO NUMERICAL METHODS FOR SOLVING A NONLINEAR SYSTEM OF INTEGRAL EQUATIONS OF MIXED VOLTERRA-FREDHOLM TYPE ARISING FROM A CONTROL PROBLEM RELATED TO LEUKEMIA[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1797-1812. doi: 10.11948/20220197
Citation: Lorand Gabriel Parajdi, Flavius Pătrulescu, Radu Precup, Ioan Ştefan Haplea. TWO NUMERICAL METHODS FOR SOLVING A NONLINEAR SYSTEM OF INTEGRAL EQUATIONS OF MIXED VOLTERRA-FREDHOLM TYPE ARISING FROM A CONTROL PROBLEM RELATED TO LEUKEMIA[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1797-1812. doi: 10.11948/20220197

TWO NUMERICAL METHODS FOR SOLVING A NONLINEAR SYSTEM OF INTEGRAL EQUATIONS OF MIXED VOLTERRA-FREDHOLM TYPE ARISING FROM A CONTROL PROBLEM RELATED TO LEUKEMIA

  • The aim of this paper is to present two algorithms for numerical solving of a fixed final state control problem in connection with the leukemia treatment strategy. In the absence of the controllability condition, our model leads to a nonlinear integral system of Volterra type to whom explicit iterative techniques apply and converge. Once using the controllability condition, the control variable is expressed in terms of the state variables and the integral system changes to a mixed Volterra-Fredholm type one making direct iterative techniques inoperative. However, two paths can be followed. One consists in an iterative procedure where at each step the control variable is calculated using the approximate values of the state variables from the previous step. The other looks for the numerical value of the control variable by using the bisection method. Numerical simulations, error analysis and biological interpretation are given.

    MSC: 45G15, 34H05, 65R20, 92C50, 93B05
  • 加载中
  • [1] B. Ainseba and C. Benosman, Optimal control for resistance and suboptimal response in CML, Math. Biosci., 2010, 227(2), 81–93. doi: 10.1016/j.mbs.2010.06.005

    CrossRef Google Scholar

    [2] C. J. S. Alves, P. M. Pardalos and L. N. Vicente, Optimization in Medicine, Springer Science & Business Media, New York, 2008.

    Google Scholar

    [3] V. Barbu, Mathematical Methods in Optimization of Differential Systems, Springer Science & Business Media, Dordrecht, 1994.

    Google Scholar

    [4] D. Bufnea, V. Niculescu, G. Silaghi and A. Sterca, Babeş-Bolyai University's high performance computing center, Stud. Univ. Babeş-Bolyai Inf., 2016, 61(2), 54–69.

    Google Scholar

    [5] G. Cedersund, O. Samuelsson, G. Ball, J. Tegnér and D. Gomez-Cabrero, Optimization in biology parameter estimation and the associated optimization problem. In: Uncertainty in Biology, A Computational Modeling Approach, Springer, Cham, 2016, 177–197.

    Google Scholar

    [6] Y. Cherruault, Global optimization in biology and medicine, Mathl. Comput. Modelling, 1994, 20(6), 119–132. doi: 10.1016/0895-7177(94)90027-2

    CrossRef Google Scholar

    [7] A. Cucuianu and R. Precup, A hypothetical-mathematical model of acute myeloid leukaemia pathogenesis, Comput. Math. Methods Med., 2010, 11(1), 49–65. doi: 10.1080/17486700902973751

    CrossRef Google Scholar

    [8] M. W. Deininger, N. P. Shah, et al., Chronic myeloid leukemia, Version 2.2021, NCCN Clinical practice guidelines in oncology, J. Natl. Compr. Canc. Netw., 2020, 18(10), 1385–1415. doi: 10.6004/jnccn.2020.0047

    CrossRef Google Scholar

    [9] D. Dingli and F. Michor, Successful therapy must eradicate cancer stem cells, Stem Cells, 2006, 24(12), 2603–2610. doi: 10.1634/stemcells.2006-0136

    CrossRef Google Scholar

    [10] S. Faderl, M. Talpaz, Z. Estrov, S. O'Brien, R. Kurzrock and H. M. Kantarjian, The biology of chronic myeloid leukemia, N. Engl. J. Med., 1999, 341(3), 164–172. doi: 10.1056/NEJM199907153410306

    CrossRef Google Scholar

    [11] J. Gong, B. Wu, T. Guo, S. Zhou, B. He and X. Peng, Hyperleukocytosis: A report of five cases and review of the literature, Oncol. Lett., 2014, 8(4), 1825–1827. doi: 10.3892/ol.2014.2326

    CrossRef Google Scholar

    [12] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn., Springer-Verlag, Berlin, Heidelberg, 1993.

    Google Scholar

    [13] B. Hanfstein, M. C. Müller, et al., Early molecular and cytogenetic response is predictive for long-term progression-free and overall survival in chronic myeloid leukemia (CML), Leukemia, 2012, 26(9), 2096–2102. doi: 10.1038/leu.2012.85

    CrossRef Google Scholar

    [14] I. Ş. Haplea, L. G. Parajdi and R. Precup, On the controllability of a system modeling cell dynamics related to leukemia, Symmetry, 2021, 13(10), 1867. doi: 10.3390/sym13101867

    CrossRef Google Scholar

    [15] Q. He, J. Zhu, D. Dingli, J. Foo and K. Z. Leder, Optimized treatment schedules for chronic myeloid leukemia, PLoS Comput. Biol., 2016, 12(10), e1005129. doi: 10.1371/journal.pcbi.1005129

    CrossRef Google Scholar

    [16] A. Hochhaus, R. A. Larson, et al., Long-term outcomes of Imatinib treatment for chronic myeloid leukemia, N. Engl. J. Med., 2017, 376(10), 917–927. doi: 10.1056/NEJMoa1609324

    CrossRef Google Scholar

    [17] A. Hochhaus, S. Saussele, et al., Chronic myeloid leukaemia: ESMO Clinical Practice Guidelines for diagnosis, treatment and follow-up, Ann. Oncol., 2017, 28, iv41-iv51.

    Google Scholar

    [18] A. Hochhaus, M. Baccarani, et al., European LeukemiaNet 2020 recommendations for treating chronic myeloid leukemia, Leukemia, 2020, 34(4), 966–984. doi: 10.1038/s41375-020-0776-2

    CrossRef Google Scholar

    [19] T. Hughes, M. Deininger, et al., Monitoring CML patients responding to treatment with Tyrosine Kinase Inhibitors: review and recommendations for harmonizing current methodology for detecting BCR-ABL transcripts and kinase domain mutations and for expressing results, Blood, 2006, 108(1), 28–37. doi: 10.1182/blood-2006-01-0092

    CrossRef Google Scholar

    [20] E. Jabbour and H. Kantarjian, Chronic myeloid leukemia: 2020 update on diagnosis, therapy and monitoring, Am. J. Hematol., 2020, 95(6), 691–709. doi: 10.1002/ajh.25792

    CrossRef Google Scholar

    [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 1977, 197(4300), 287–289. doi: 10.1126/science.267326

    CrossRef Google Scholar

    [22] S. B. Mendrazitsky and B. Shklyar, Optimization of combined leukemia therapy by finite-dimensional optimal control modeling, J. Optim. Theory. Appl., 2017, 175(1), 218–235. doi: 10.1007/s10957-017-1161-9

    CrossRef Google Scholar

    [23] S. B. Mendrazitsky, N. Kronik and V. Vainstein, Optimization of interferon-alpha and imatinib combination therapy for chronic myeloid leukemia: a modeling approach, Adv. Theory. Simul., 2019, 2(1), 1800081–8. doi: 10.1002/adts.201800081

    CrossRef Google Scholar

    [24] F. Michor, T. P. Hughes, Y. Iwasa, S. Branford, N. P. Shah, C. L. Sawyers and M. A. Nowak, Dynamics of chronic myeloid leukaemia, Nature, 2005, 435(7046), 1267–1270. doi: 10.1038/nature03669

    CrossRef Google Scholar

    [25] M. C. Müller, N. Gattermann, et al., Dynamics of BCR-ABL mRNA expression in first-line therapy of chronic myelogenous leukemia patients with Imatinib or Interferon Alpha/Ara-C, Leukemia, 2003, 17(12), 2392–2400. doi: 10.1038/sj.leu.2403157

    CrossRef Google Scholar

    [26] S. Nanda, H. Moore and S. Lenhart, Optimal control of treatment in a mathematical model of chronic myelogenous leukemia, Math. Biosci., 2007, 210(1), 143–156. doi: 10.1016/j.mbs.2007.05.003

    CrossRef Google Scholar

    [27] S. G. O'Brien, F. Guilhot, et al., Imatinib compared with interferon and low-dose cytarabine for newly diagnosed chronic-phase chronic myeloid leukemia, N. Engl. J. Med., 2003, 348(11), 994–1004. doi: 10.1056/NEJMoa022457

    CrossRef Google Scholar

    [28] L. G. Parajdi, R. Precup, E. A. Bonci and C. Tomuleasa, A mathematical model of the transition from normal hematopoiesis to the chronic and accelerated-acute stages in myeloid leukemia, Mathematics, 2020, 8(3), 376. doi: 10.3390/math8030376

    CrossRef Google Scholar

    [29] L. G. Parajdi, Analysis of Some Mathematical Models of Cell Dynamics in Hematology, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2021.

    Google Scholar

    [30] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic, Dordrecht, 2002.

    Google Scholar

    [31] Y. Wang, X. Zhang and L. Chen, Optimization meets systems biology, BMC Syst. Biol., 2010, 4(2), 1–4.

    Google Scholar

Figures(8)

Article Metrics

Article views(1586) PDF downloads(274) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint