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access icon free Analysis of leukaemic cells dynamics with multi-stage maturation process using a new non-linear positive model with distributed time-delay

  • Author(s): Abdelhafid Zenati 1 ; Massaoud Chakir 1 ; Mohamed Tadjine 1 ; Mouloud Denai 2
    • View affiliations
  • Source: Volume 13, Issue 18, 17 December 2019, p. 3052 – 3064
    DOI:  10.1049/iet-cta.2019.0331 , Print ISSN 1751-8644, Online ISSN 1751-8652
© The Institution of Engineering and Technology
Received 26/03/2019, Accepted 02/09/2019, Revised 07/07/2019, Published 03/09/2019

This study proposes a new non-linear positive model with distributed time-delay describing the dynamics of leukaemic cells lineages and their levels of maturity. The dynamic model is first reformulated into coupled non-linear equations with distributed time-delay using an appropriate transformation of the age-structured transport partial differential equations that govern the dynamics of leukaemic cell populations. After assessing the positiveness of the proposed model, the conditions for the existence of positive steady-states are established. Furthermore, the necessary and sufficient conditions for boundedness of the solutions are derived. Finally, the convergence of the proposed model to the origin is studied as this has an important biological interpretation which is the extinction of all generations of leukaemic cells following anti-leukaemia treatments. To investigate the global asymptotic stability of this important equilibrium point, both model properties (i.e. positiveness and boundedness) are used to construct a suitable Lyapunov function for the characterisation of leukaemic cells dynamics. The results demonstrate a significant improvement for the treatment of leukaemia. Some simulations are presented to illustrate the effectiveness of the proposed model.

Inspec keywords: Lyapunov methods; delays; cancer; partial differential equations; asymptotic stability

Other keywords: nonlinear positive model; leukaemic cells lineages; dynamic model; leukaemic cell populations; age-structured transport partial differential equations; positiveness; distributed time-delay; coupled nonlinear equations; leukaemic cells dynamics; model properties; multistage maturation process

Subjects: Function theory, analysis; Stability in control theory

References

    1. 1)
      • 44. Dingli, D., Michor, F.: ‘Successful therapy must eradicate cancer stem cells’, Stem Cells, 2006, 24, (12), pp. 2603–2610.
    2. 2)
      • 23. Zenati, A., Laleg-Kirati, T.M., Chakir, M., et al: ‘Dynamical behavior of biological healthy steady state in leukemia using a new leukemic & healthy stem cells cohabitation model with distributed delay’. Accepted for presentation at 2019 American Control Conf., Philadelphia, PA, USA, 2019.
    3. 3)
      • 8. Shen, J., Lam, J.: ‘Static output-feedback stabilization with optimal L1-gain for positive linear systems’, Automatica, 2016, 63, pp. 248–253.
    4. 4)
      • 22. Dick, J.E., Lapidot, T.: ‘Biology of normal and acute myeloid leukemia stem cells’, Int. J. Hematol., 2005, 82, (5), pp. 389–396.
    5. 5)
      • 11. Zenati, A., Chakir, M., Tadjine, M.: ‘Global stability analysis and optimal control therapy of blood cell production process (hematopoiesis) in acute myeloid leukemia‘’, J. Theor. Biol., 2018, 458, pp. 15–30.
    6. 6)
      • 26. Janz, S., Potter, M., Rabkin, C.S.: ‘Lymphoma- and leukemia-associated chromosomal translocations in healthy individuals’, Genes, Chromosomes and Cancer, 2003, 36, (3), pp. 211–223.
    7. 7)
      • 14. Mazenc, F., Malisoff, M.: ‘Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach’, IEEE Trans. Autom. Control, 2016, 61, (3), pp. 771–776.
    8. 8)
      • 52. Mackey, M.C.: ‘Unified hypothesis for the origin of aplastic anaemia and periodic hematopoiesis’, Blood, 1978, 51, (5), pp. 941–956.
    9. 9)
      • 46. Ngoc, P.H.A.: ‘Stability of positive differential systems with delay’, IEEE Trans. Autom. Control, 2013, 58, (1), pp. 203–209.
    10. 10)
      • 38. Heydt, Q., Larrue, C., Saland, E., et al: ‘Oncogenic FLT3-ITD supports autophagy via ATF4 in acute myeloid leukemia’, Oncogene, 2017, 37, (6), pp. 787–797.
    11. 11)
      • 54. Khalil, H.K.: ‘Nonlinear systems’ (Prentice-Hall, USA, 2002).
    12. 12)
      • 29. Wang, S., Schättler, H.: ‘Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity’, Math. Biosci. Eng., 2016, 13, (6), pp. 1223–1240.
    13. 13)
      • 5. Rami, M.A., Tadeo, F.: ‘Controller synthesis for positive linear systems with bounded controls’, IEEE Trans. Circuits Syst. II, Express Briefs, 2007, 54, (2), pp. 151–155.
    14. 14)
      • 37. Smith, B.D.: ‘Single-agent CEP-701, a novel FLT3 inhibitor, shows biologic and clinical activity in patients with relapsed or refractory acute myeloid leukemia’, Blood, 2004, 103, (10), pp. 3669–3676.
    15. 15)
      • 28. Stiehl, T., Baran, N., Ho, A.D., et al: ‘Clonal selection and therapy resistance in acute leukaemias: mathematical modelling explains different proliferation patterns at diagnosis and relapse’, J. R. Soc. Interface, 2014, 11, (94), pp. 20140079–20140079.
    16. 16)
      • 47. De Souza, D.S, Antony, R.H.: ‘Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model’. arXiv:1712.08308v1 [math.DS], 2017, pp. 1–32.
    17. 17)
      • 27. Catlin, S.N.: ‘The kinetics of clonal dominance in myeloproliferative disorders’, Blood, 2005, 106, (8), pp. 2688–2692.
    18. 18)
      • 6. Knorn, F., Mason, O., Shorten, R.: ‘On linear co-positive Lyapunov functions for sets of linear positive systems’, Automatica, 2009, 45, (8), pp. 1943–1947.
    19. 19)
      • 18. Jan, M., Majeti, R.: ‘Clonal evolution of acute leukemia genomes’, Oncogene, 2012, 32, (2), pp. 135–140.
    20. 20)
      • 20. Fridman, E., Bonnet, C., Mazenc, F., et al: ‘Stability of the cell dynamics in acute myeloid leukemia’, Syst. Control Lett., 2016, 88, pp. 91–100.
    21. 21)
      • 36. Patnaik, M.M.: ‘The importance of FLT3 mutational analysis in acute myeloid leukemia’, Leukemia & Lymphoma, 2017, 59, (10), pp. 1–14.
    22. 22)
      • 13. Berezansky, L., Braverman, E.: ‘Oscillation properties of a logistic equation with distributed delay’, Nonlinear Anal., Real World Appl., 2003, 4, (1), pp. 1–19.
    23. 23)
      • 31. Lecca, P., Laurenzi, I., Jordan, F.: ‘Modelling in systems biology’, in Lecca, P., Laurenzi, I., Jordan, F. (Eds.): ‘Deterministic versus stochastic modelling in biochemistry and systems biology’ (Woodhead Publishing, UK, 2013), pp. 117–180.
    24. 24)
      • 50. Avila, J., Bonnet, C., Özbay, H., et al: ‘A coupled model for healthy and cancerous cells dynamics in acute myeloid leukemia’, IFAC Proc. Vol., 2014, 47, (3), pp. 7529–7534.
    25. 25)
      • 53. Bolton, W.: ‘Essential mathematics for engineering’ (Butterworth-Heinemann, UK, 1997).
    26. 26)
      • 1. Ebihara, Y., Peaucelle, D., Arzelier, D.: ‘Steady-state analysis of delay interconnected positive systems and its application to formation control’, IET Control Theory Applic., 2017, 11, (16), pp. 2783–2792.
    27. 27)
      • 35. Djema, W., Mazenc, F., Bonnet, C., et al: ‘Stability of a delay system coupled to a differential-difference system describing the coexistence of ordinary and mutated hematopoietic stem cells’. 2016 IEEE 55th Conf. on Decision and Control, Las Vegas, NV, USA, 2016.
    28. 28)
      • 2. Yang, H., Zhang, Y.: ‘Stability of positive delay systems with delayed impulses’, IET Control Theory Applic., 2018, 12, (2), pp. 194–205.
    29. 29)
      • 21. Pietras, E.M., Warr, M.R., Passegué, E.: ‘Cell cycle regulation in hematopoietic stem cells’, J. Cell Biol., 2011, 195, (5), pp. 709–720.
    30. 30)
      • 4. Li, X., Lam, H.K., Liu, F., et al: ‘Stability and stabilization analysis of positive polynomial fuzzy systems with time delay considering piecewise membership functions’, IEEE Trans. Fuzzy Syst., 2017, 25, (4), pp. 958–971.
    31. 31)
      • 41. Ledzewicz, U., Schättler, H.: ‘Anti angiogenic therapy in cancer treatment as an optimal control problem’, SIAM J. Control Optim., 2007, 46, (3), pp. 1052–1079.
    32. 32)
      • 40. Stiehl, T., Marciniak-Czochra, A.: ‘Mathematical modeling of leukemogenesis and cancer stem cell dynamics’, Math. Modell. Natural Phenom., 2012, 7, (1), pp. 166–202.
    33. 33)
      • 48. Avila, J.L., Bonnet, C., Fridman, E., et al: ‘Stability analysis of PDEs modelling cell dynamics in acute myeloid leukemia’. 53rd IEEE Conf. on Decision and Control, Los Angeles, CA, USA, 2014.
    34. 34)
      • 30. Singh, V.: ‘Modelling methodologies for systems biology’, in Singh, V., Dhar, P.K. (Eds.): ‘Systems and synthetic biology’ (Springer, Netherlands, 2014), pp. 43–62.
    35. 35)
      • 7. Zhang, J., Han, Z., Zhu, F., et al: ‘Feedback control for switched positive linear systems’, IET Control Theory Applic., 2013, 7, (3), pp. 464–469.
    36. 36)
      • 43. Jansen, G., Gatenby, R., Aktipis, C.A.: ‘Opinion: control vs. eradication: applying infectious disease treatment strategies to cancer’, Proc. Natl. Acad. Sci., 2015, 112, (4), pp. 937–938.
    37. 37)
      • 33. Clapp, G., Levy, D.: ‘A review of mathematical models for leukemia and lymphoma’, Drug Discovery Today: Disease Models, 2015, 16, pp. 1–6.
    38. 38)
      • 15. Zenati, A., Laleg-Kirati, T.M., Chakir, M., et al: ‘On instability and global asymptotic stability of age-structured distributed delay system describing pathological hematopoeisis’. Accepted for Presentation at 2019 American Control Conf., Philadelphia, PA, USA, 2019.
    39. 39)
      • 16. Stiehl, T., Lutz, C., Marciniak-Czochra, A.: ‘Emergence of heterogeneity in acute leukemias’, Biol. Direct, 2016, 11, (1), pp. 1–19.
    40. 40)
      • 3. Ebihara, Y., Peaucelle, D., Arzelier, D.: ‘Analysis and synthesis of interconnected positive systems’, IEEE Trans. Autom. Control, 2017, 62, (2), pp. 652–667.
    41. 41)
      • 34. Zenati, A., Chakir, M., Tadjine, M., et al: ‘Global stability analysis of strictly positive steady state for a surviving hematopoietic stem cells models’. American Control Conf., Milwaukee, USA, 27–29 June 2018.
    42. 42)
      • 10. Feyzmahdavian, H.R., Charalambous, T., Johansson, M.: ‘Exponential stability of homogeneous positive systems of degree one with time-varying delays’, IEEE Trans. Autom. Control, 2014, 59, (6), pp. 1594–1599.
    43. 43)
      • 9. Phat, V., Sau, N.: ‘On exponential stability of linear singular positive delayed systems’, Appl. Math. Lett., 2014, 38, pp. 67–72.
    44. 44)
      • 19. Djema, W., Mazenc, F., Bonnet, C.: ‘Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis’, Syst. Control Lett., 2017, 102, pp. 93–101.
    45. 45)
      • 25. Klco, J.M., Spencer, D.H., Miller, C.A., et al: ‘Functional heterogeneity of genetically defined subclones in acute myeloid leukemia’, Cancer Cell, 2014, 25, (3), pp. 379–392.
    46. 46)
      • 49. Zenati, A., Chakir, M., Tadjine, M.: ‘Optimal control of a coupled model for healthy and cancerous cells dynamics in acute myeloid leukemia – a therapy approach’. 2017 6th Int. Conf. on Systems and Control, Batna, Algeria, 2017.
    47. 47)
      • 12. Foley, C., Mackey, M.C.: ‘Dynamic hematological disease: a review’, J. Math. Biol., 2008, 58, (1-2), pp. 285–322.
    48. 48)
      • 42. Enderling, H.: ‘Cancer stem cells and tumor dormancy’, in Enderling, H., Almog, N., Hlatky, L. (Eds.): ‘Advances in experimental medicine and biology systems biology of tumor dormancy’ (Springer Science & Business Media, USA, 2012), pp. 55–71.
    49. 49)
      • 17. Avila, J.L., Bonnet, C., Clairambault, J., et al: ‘Analysis of a new model of cell population dynamics in acute myeloid leukemia’, in Vyhlídal, T., Lafay, J.F., Sipahi, R. (Eds.): ‘Delay systems advances in delays and dynamics’ (Springer International Publishing, Switzerland, 2014), pp. 315–328.
    50. 50)
      • 24. Hirsch, P., Zhang, Y., Tang, R., et al: ‘Genetic hierarchy and temporal variegation in the clonal history of acute myeloid leukaemia’, Nature Commun., 2016, 7, p. 12475.
    51. 51)
      • 32. Zenati, A., Chakir, M., Tadjine, M.: ‘Study of cohabitation and interconnection effects on normal and leukaemic stem cells dynamics in acute myeloid leukaemia’, IET Syst. Biol., 2018, 12, (6), pp. 279–288.
    52. 52)
      • 51. Özbay, H., Bonnet, C., Benjelloun, H., et al: ‘Stability analysis of cell dynamics in leukemia’, Math. Modell. Natural Phenom., 2012, 7, (1), pp. 203–234.
    53. 53)
      • 39. Jayachandran, D., Rundell, A.E., Hannemann, R.E., et al: ‘Optimal chemotherapy for leukemia: a model-based strategy for individualized treatment’, PLoS ONE, 2014, 9, (10), pp. 1–18.
    54. 54)
      • 45. Zenati, A., Chakir, M., Tadjine, M.: ‘Global stability analysis of healthy situation for a coupled model of healthy and cancerous cells dynamics in acute myeloid leukemia’, IFAC-PapersOnLine, 2017, 50, (1), pp. 15103–15108.
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