Geodesic
Mathematical analysis and global dynamics for a time-delayed Chronic Myeloid Leukemia model with treatment
Nawal Kherbouche1 ; Mohamed Helal1 ; Abdennasser Chekroun2 ; Abdelkader Lakmeche1
1 Biomathematics Laboratory, Univ. Sidi Bel-Abbes, P.B. 89, 22000, Algeria.
2 Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Univ. Tlemcen, Algeria.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 68

Voir la notice de l'article provenant de la source EDP Sciences

In this paper, we investigate a time-delayed model describing the dynamics of the hematopoietic stem cell population with treatment. First, we give some property results of the solutions. Second, we analyze the asymptotic behavior of the model, and study the local asymptotic stability of each equilibrium: trivial and positive ones. Next, a necessary and sufficient condition is given for the trivial steady state to be globally asymptotically stable. Moreover, the uniform persistence is obtained in the case of instability. Finally, we prove that this system can exhibits a periodic solutions around the positive equilibrium through a Hopf bifurcation.
DOI : 10.1051/mmnp/2020038

Nawal Kherbouche 1 ; Mohamed Helal 1 ; Abdennasser Chekroun 2 ; Abdelkader Lakmeche 1

1 Biomathematics Laboratory, Univ. Sidi Bel-Abbes, P.B. 89, 22000, Algeria.
2 Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Univ. Tlemcen, Algeria. 
@article{10_1051_mmnp_2020038,
     author = {Nawal Kherbouche and Mohamed Helal and Abdennasser Chekroun and Abdelkader Lakmeche},
     title = {Mathematical analysis and global dynamics for a time-delayed {Chronic} {Myeloid} {Leukemia} model with treatment},
     journal = {Mathematical modelling of natural phenomena},
     eid = {68},
     publisher = {mathdoc},
     volume = {15},
     year = {2020},
     doi = {10.1051/mmnp/2020038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020038/}
}
TY  - JOUR
AU  - Nawal Kherbouche
AU  - Mohamed Helal
AU  - Abdennasser Chekroun
AU  - Abdelkader Lakmeche
TI  - Mathematical analysis and global dynamics for a time-delayed Chronic Myeloid Leukemia model with treatment
JO  - Mathematical modelling of natural phenomena
PY  - 2020
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020038/
DO  - 10.1051/mmnp/2020038
LA  - en
ID  - 10_1051_mmnp_2020038
ER  - 
%0 Journal Article
%A Nawal Kherbouche
%A Mohamed Helal
%A Abdennasser Chekroun
%A Abdelkader Lakmeche
%T Mathematical analysis and global dynamics for a time-delayed Chronic Myeloid Leukemia model with treatment
%J Mathematical modelling of natural phenomena
%D 2020
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2020038/
%R 10.1051/mmnp/2020038
%G en
%F 10_1051_mmnp_2020038
Nawal Kherbouche; Mohamed Helal; Abdennasser Chekroun; Abdelkader Lakmeche. Mathematical analysis and global dynamics for a time-delayed Chronic Myeloid Leukemia model with treatment. Mathematical modelling of natural phenomena, Tome 15 (2020), article  no. 68. doi: 10.1051/mmnp/2020038

[1] M. Adimy, A. Chekroun, C.P. Ferreira Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase Math. Biosci. Eng 2020 1329 1354

[2] M. Adimy, A. Chekroun, T. Kuniya Coupled reaction-diffusion and difference system of cell-cycle dynamics for hematopoiesis process with Dirichlet boundary conditions J. Math. Anal. Appl 2019 1030 1068

[3] M. Adimy, F. Crauste, S. Ruan Periodic oscillations in leukopoiesis models with two delays J. Theor. Biol 2006 288 299

[4] B. Ainseba, C. Benosman Global dynamics of hematopoietic stem cells and differentiated cells in a chronic myeloid leukemia model J. Math. Biol 2010 975 997

[5] J.L. Avila, C. Bonnet, E. Fridman, F. Mazenc, J. Clairambault Stability analysis of PDEs modelling cell dynamics in acute myeloid leukemia in 53rd IEEE Conference on Decision and Control 2014

[6] J.C. Banck, D. Gorlich In-silico comparison of two induction regimens (7 + 3 vs 7 + 3 plus additional bone marrow evaluation) in acute myeloid leukemia treatment BMC Syst. Biol 2019 1 14

[7] E. Beretta, Y. Kuang Geometric stability switch criteria in delay differential systems with delay dependent parameters SIAM J. Math. Anal 2002 1144 1165

[8] A. Besse, G.D. Clapp, S. Bernard, F.E. Nicolini, D. Levy, T. Lepoutre Stability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemia Bull. Math. Biol 2017 1084 1110

[9] A. Besse, T. Lepoutre, S. Bernard Long-term treatment effects in chronic myeloid leukemia J. Math. Biol 2017 733 758

[10] M. Bouizem, B. Ainseba, A. Lakmeche Mathematical analysis of an age structured leukemia model Commun. Appl. Nonlinear Anal 2018 1 20

[11] N. Bouizem, M. Helal, N. Kherbouche, A. Lakmeche Stability analysis of leukemia mathematical model with delay Commun. Appl. Nonlinear Anal 2020 25 57

[12] F. Charif, M. Helal, A. Lakmeche Chemotherapeutic treatment models by drugs with instantaneous effects Commun. Appl. Nonlinear Anal 2017 1 24

[13] G.D. Clapp, T. Lepoutre, R.E. Cheikh, S. Bernard, J. Ruby, H. Labussiere-Wallet, F.E. Nicolini, D. Levy Implication of the autologous immune system in BCR-ABL transcript variations in chronic myelogenous leukemia patients treated with imatinib Cancer Res 2015 4053 4062

[14] C. Colijn, M. Mackey A mathematical model of hematopoiesis: II. cyclical neutropenia J. Theor. Biol 2005 133 146

[15] A.S. Corbin, A. Agarwal, M. Loriaux, J. Cortes, M.W. Deininger, B.J. Druker Human chronic myeloid leukemia stem cells are insensitive to imatinib despite inhibition of BCR-ABL activity J. Clin. Invest 2011 396 409

[16] D. Dingli, F. Michor Successful therapy must eradicate cancer stem cells Stem Cells 2006 2603 2610

[17] D. Dingli, A. Traulsen, J.M. Pacheco Stochastic dynamics of hematopoietic tumor stem cells Cell Cycle 2007 461 466

[18] J. Foo, M.W. Drummond, B. Clarkson, T. Holyoake, F. Michor Eradication of chronic Myeloid Leukemia stem cells: a novel mathematical model predicts no therapeutic benefit of adding G-CSF to Imatinib PLOS Comput. Biol 2009 e1000503

[19] H.I. Freedman, P. Moson Persistence definitions and their connections Proc. Am. Math. Soc 1990 1025 1033

[20] J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations. Springer (1993).

[21] Y. Isma, M. Helal, A. Lakmeche Existence and stability of steady states for a delay model of stem cells in leukemia under treatment Commun. Appl. Nonlinear Anal 2018 66 80

[22] C.H. Jerome Paillassa, KB / iKB Hematologie Onco-hematologie. Masson (2018).

[23] X. Jiang, Y. Zhao, C. Smith, M. Gasparetto, A. Turhan, A. Eaves, C. Eaves Chronic myeloid leukemia stem cells possess multiple unique features of resistance to BCR-ABL targeted therapies Leukemia 2007 926 935

[24] H.G. Jorgensen, E.K. Allan, N.E. Jordanides, J.C. Mountford, T.L. Holyoake Nilotinib exerts equipotent antiproliferative effects to imatinib and does not induce apoptosis in CD34+ CML cells Blood 2007 4016 4019

[25] F. Jost, E. Schalk, K. Rinke, T. Fischer, S. Sager Mathematical models for cytarabine-derived myelosuppression in acute myeloid leukaemia PLOS ONE 2019 e0204540

[26] P.S. Kim, P.P. Lee, D. Levy A PDE model for imatinib-treated chronic myelogenous leukemia Bull. Math. Biol 2008 1994 2016

[27] F. Knauer, T. Stiehl, A. Marciniak-Czochra Oscillations in a white blood cell production model with multiple differentiation stages J. Math. Biol 2019 575 600

[28] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics. Academic Press (1993).

[29] U. Ledzewicz, H. Moore Dynamical systems properties of a mathematical model for the treatment of CML Appl. Sci 2016 291

[30] T. Lorenzi, A. Marciniak-Czochra, T. Stiehl A structured population model of clonal selection in acute leukemias with multiple maturation stages J. Math. Biol 2019 1587 1621

[31] M.C. Mackey Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis Blood 1978 941 956

[32] F. Michor Quantitative approaches to analyzing imatinib-treated chronic myeloid leukemia TRENDS in Pharmacolog. Sci 2007 197 198

[33] F. Michor. Reply: The long-term response to imatinib treatment of CML Br. J. Cancer 2007 679 680

[34] F. Michor Chronic Myeloid Leukemia Blast Crisis Arises from Progenitors Stem Cells 2007 1114 1118

[35] F. Michor, T.P. Hughes, Y. Iwasa, S. Branford, N.P. Shah, C.L. Sawyers, M.A. Nowak Dynamics of chronic myeloid leukaemia Nature 2005 1267 1270

[36] S. Mustjoki, J. Richter, G. Barbany, H. Ehrencrona, T. Fioretos, T. Gedde-Dahl, B.T. Gjertsen, R. Hovland, S. Hernesniemi, D. Josefsen, P. Koskenvesa, I. Dybedal, B. Markevärn, T. Olofsson, U. Olsson-Strömberg, K. Rapakko, S. Thunberg, L. Stenke, B. Simonsson, K. Porkka, H. Hjorth-Hansen Nordic CML Study Group (NCMLSG). Impact of malignant stem cell burden on therapy outcome in newly diagnosed chronic myeloid leukemia patients Leukemia 2013 1520 1526

[37] S. Nanda, H. Moore, S. Lenhart Optimal control of treatment in a mathematical model of chronic myelogenous leukemia Math. Biosci 2007 143 156

[38] H.G. Othmer, F. Adler, M. Lewis and J. Dallon, Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology. Pearson (1997).

[39] L. Pujo-Menjouet, S. Bernard, M.C. Mackey Long period oscillations in a G0 model of hematopoietic stem cells SIAM J. Appl. Dyn. Syst 2005 312 332

[40] I.R. Radulescu, D. Cândea, A. Halanay Optimal control analysis of a leukemia model under imatinib treatment Math. Comput. Simul 2016 1 11

[41] I.R. Radulescu, D. Cândea, A. Halanay Stability and bifurcation in a model for the dynamics of stem-like cells in leukemia under treatment AIP Conf. Proc. 2012 758

[42] I. Roeder, M. Herberg, M. Horn An age structured model of hematopoietic stem cell organization with application to chronic myeloid leukemia Bull. Math. Biol. 2008 602 626

[43] I. Roeder, M. Horn, I. Glauche, A. Hochhaus, M.C. Mueller, M. Loeffler Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications Nat. Med 2006 1181 1184

[44] J.A. Sharp, A.P. Browning, T. Mapder, K. Burrage, M.J. Simpson Optimal control of acute myeloid leukaemia J. Theor. Biol 2019 30 42

[45] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics. Springer (2011).

[46] T. Stiehl, N. Baran, A.D. Ho, A. Marciniak-Czochra Clonal selection and therapy resistance in acute leukaemias: mathematical modelling explains different proliferation patterns at diagnosis and relapse J. Roy. Soc. Interface 2014 20140079

[47] T. Stiehl, C. Lutz, A. Marciniak-Czochra Emergence of heterogeneity in acute leukemias Biol. Direct 2016

[48] T. Stiehl, A. Marciniak-Czochra Mathematical modelling of leukemogenesis and cancer stem celldynamics MMNP 2012 166 202

[49] M. Tang, J. Foo, M. Gonen, J. Guilhot, F.X. Mahon, F. Michor Selection pressure exerted by imatinib therapy leads to disparate outcomes of imatinib discontinuation trials Haematologica 2012 1553 1561

Cité par Sources :