Free Access
Issue
Math. Model. Nat. Phenom.
Volume 7, Number 6, 2012
Biological oscillations
Page(s) 1 - 22
DOI https://doi.org/10.1051/mmnp/20127601
Published online 12 December 2012
  1. J.W. Adamson. Regulation of red blood cell Production. Am. J. Med., 101 (1996), S4–S6. [CrossRef] [Google Scholar]
  2. M. Adimy, F. Crauste. Global stability of a partial differential equation with distributed delay due to cellular replication. Nonlinear Analysis, 54 (2003), 1469–1491. [CrossRef] [MathSciNet] [Google Scholar]
  3. M. Adimy, F. Crauste. Modelling and asymptotic stability of a growth factor-dependent stem cells dynamics model with distributed delay. Discrete and Continuous Dynamical Systems Series B, 8 (2007), No. 1, 19–38. [CrossRef] [MathSciNet] [Google Scholar]
  4. M. Adimy, F. Crauste. Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulation. Mathematical and Computer Modelling, 49 (2009), 2128–2137. [CrossRef] [MathSciNet] [Google Scholar]
  5. M. Adimy, F. Crauste, A. El Abdllaoui. Asymptotic Behavior of a Discrete Maturity Structured System of Hematopoietic Stem Cells Dynamics with Several Delays. Mathematical Modelling of Natural Phenomena, Vol 1 (2006), No. 2, 1–22. [CrossRef] [EDP Sciences] [Google Scholar]
  6. M. Adimy, F. Crauste, A. El Abdllaoui. Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia. J. Biol. Syst., 16 (3) (2008), 395–424. [CrossRef] [Google Scholar]
  7. M. Adimy, F. Crauste, M.L. Hbid, R. Qesmi. Stability and Hopf bifurcation for a cell population model with state-dependent delay. SIAM J. Appl. Math, 70 (5) (2010), 1611–1633. [CrossRef] [Google Scholar]
  8. M. Adimy, F. Crauste, C. Marquet. Asymptotic behavior and stability switch for a mature-immature model of cell differentiation. Nonlinear Analysis: Real World Applications, 11 (2010), 2913–2929. [CrossRef] [MathSciNet] [Google Scholar]
  9. M. Adimy, F. Crauste, S. Ruan. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM J. Appl. Math., 65 (2005), 1328–1352. [CrossRef] [MathSciNet] [Google Scholar]
  10. M. Adimy, F. Crauste, S. Ruan. Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics. Nonlinear Analysis: Real World Applications, 6 (2005), No. 4, 651–670. [CrossRef] [MathSciNet] [Google Scholar]
  11. M. Adimy, F. Crauste, S. Ruan. Periodic Oscillations in Leukopoiesis Models with Two Delays. J. Theo. Biol., 242 (2006), 288–299. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  12. M. Adimy, F. Crauste, S. Ruan. Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases. Bulletin of Mathematical Biology, 68 (8) (2006), 2321–2351. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  13. W. Aiello, H. Freedman, J. Wu. Analysis of a model representing stage-structured population growth with stage-dependent time delay. SIAM Journal of Applied Mathematics 52 (1992), 855–869. [CrossRef] [Google Scholar]
  14. U. an der Heiden. Delays in physiological systems. J. Math. Biol. 8 (1979), 345–364. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  15. T. Alarcon, M.J. Tindall. Modelling Cell Growth and its Modulation of the G1/S Transition. Bull. Math. Biol., 69 (2007), 197–214. [CrossRef] [PubMed] [Google Scholar]
  16. R. Apostu, M.C. Mackey. Understanding cyclical thrombocytopenia: a mathematical modeling approach. J. Theor. Biol., 251 (2008), 297–316. [CrossRef] [PubMed] [Google Scholar]
  17. J.J. Batzel, F. Kappel. Time delay in physiological systems: Analyzing and modeling its impact. Math. Biosciences, 234 (2011), No. 2, 61–74. [CrossRef] [Google Scholar]
  18. J. Bélair, M.C. Mackey, J.M. Mahaffy. Age-structured and two-delay models for erythropoiesis. Math. Biosci., 128 (1995), 317–346. [CrossRef] [PubMed] [Google Scholar]
  19. E. Beretta, Y. Kuang. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal., 33 (2002), 5, 1144–1165.No. [CrossRef] [MathSciNet] [Google Scholar]
  20. S. Bernard, J. Belair, M.C. Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 233–256. [CrossRef] [MathSciNet] [Google Scholar]
  21. S. Bernard, J. Bélair, M.C. Mackey. Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J. Theor. Biol., 223 (2003), 283–298. [CrossRef] [PubMed] [Google Scholar]
  22. M. Bodnar, A. Bartłomiejczyk. Stability of delay induced oscillations in gene expression of Hes1 protein model. Nonlinear Analysis: Real World Applications, 13 (2012), 2227–2239. [CrossRef] [MathSciNet] [Google Scholar]
  23. F.J. Burns, I.F. Tannock. On the existence of a G0 phase in the cell cycle. Cell Tissue Kinet., 19 (1970), 321–334. [Google Scholar]
  24. S.H. Cheshier, S. J. Morrison, X. Liao, I.L. Weissman. In vivo proliferation and cell cycle kinetics of long-term self-renewing hematopoietic stem cells. Proc. Natl. Acad. Sci. USA, 96 (1999), 3120–3125. [CrossRef] [Google Scholar]
  25. M.S. Ciupe, B.L. Bivort, D.M. Bortz, P.W. Nelson. Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. Math Biosci. 200(1) 2006, 1–27. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  26. C. Colijn, C. Foley, M.C. Mackey. G-CSF treatment of canine cyclical neutropenia: A comprehensive mathematical model. Exper. Hematol. (2007), 35, 898–907. [CrossRef] [Google Scholar]
  27. C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis – I. Periodic chronic myelogenous leukemia. J. Theor. Biol., 237 (2005), 117–132. [CrossRef] [PubMed] [Google Scholar]
  28. C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis – II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), 133–146. [CrossRef] [PubMed] [Google Scholar]
  29. L. Cooke. Stability analysis for a vector disease model. Rocky Mountain J. Math., 9 (1979), 31–42. [CrossRef] [MathSciNet] [Google Scholar]
  30. A.S. Coutts, C.J. Adams, N.B. La Thangue. p53 ubiquitination by Mdm2: a never ending tail ? DNA Repair (Amst). 8 (2009), 483–90. [CrossRef] [PubMed] [Google Scholar]
  31. F. Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Math. Bio. Eng., 3 (2006), No. 2, 325–346. [Google Scholar]
  32. F. Crauste. Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch. Mathematical Modeling of Natural Phenomena, 4 (2009), No. 2, 28–47. [Google Scholar]
  33. F. Crauste. Stability and Hopf bifurcation for a first-order linear delay differential equation with distributed delay, in Complex Time Delay Systems (Ed. F. Atay), Springer, 1st edition, 320 p., ISBN: 978-3-642-02328-6 (2010). [Google Scholar]
  34. L.A. Crews, C.H. Jamieson. Chronic myeloid leukemia stem cell biology. Curr Hematol Malig Rep., 7 (2012), No. 2, 125–132. [CrossRef] [PubMed] [Google Scholar]
  35. J.M. Cushing. Integrodifferential Equations and Delay Models in Population Dynamics. Springer-Verlag, Heidelberg, 1977. [Google Scholar]
  36. D.C. Dale, A.A. Bolyard, A. Aprikyan. Cyclic neutropenia. Semin. Hematol., 39 (2002), 89–94. [CrossRef] [PubMed] [Google Scholar]
  37. D.C. Dale, W.P. Hammond. Cyclic neutropenia: A clinical review. Blood Rev., 2 (1998), 178–185. [CrossRef] [Google Scholar]
  38. J. Dieudonné. Foundations of Modern Analysis. Academic Press, New-York, 1960. [Google Scholar]
  39. C. Foley, S. Bernard, M.C. Mackey. Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses. J. Theor. Biol. (2006), 238, 754–763. [CrossRef] [PubMed] [Google Scholar]
  40. C. Foley, M.C. Mackey. Dynamic hematological disease: a review. J. Math. Biol., 58 (2009), 285–322. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  41. A.C. Fowler, M.J. McGuinness. A delay recruitment model of the cardiovascular control system. J. Math. Biol. 51 (2005), 508–526. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  42. P. Fortin, M.C. Mackey. Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and a etiological implications. Br. J. Haematol., 104 (1999), 336–345. [CrossRef] [PubMed] [Google Scholar]
  43. A. Fowler, M.C. Mackey. Relaxation oscillations in a class of delay differential equations. SIAM J. Appl. Math., 63 (2002), 299–323. [CrossRef] [Google Scholar]
  44. H. Fuss, W. Dubitzky, S. Downes, M.J. Kurth. Mathematical models of cell cycle regulation. Brief Bioinform., 6 (2005), 163–177. [CrossRef] [PubMed] [Google Scholar]
  45. N. Geva-Zatorsky , N. Rosenfeld, S. Itzkovitz, R. Milo, A. Sigal, E. Dekel, T. Yarnitzky, Y. Liron, P. Polak, G. Lahav, U. Alon. Oscillations and variability in the p53 system. Mol Syst Biol (2006), 2.2006.0033. [Google Scholar]
  46. K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population. Dynamics, Kluwer Academic, Dordrecht, 1992. [Google Scholar]
  47. L. Glass, A. Beuter, D. Larocque. Time delays, oscillations, and chaos in physiological control systems. Mathematical Biosciences, 90 (1988), 111–125. [CrossRef] [MathSciNet] [Google Scholar]
  48. D. Guerry, D. Dale, D.C. Omine, S. Perry, S.M. Wolff. Periodic hematopoiesis in human cyclic neutropenia. J Clin Invest. 52 (1973), 3220–3230. [CrossRef] [PubMed] [Google Scholar]
  49. J. Hale, S.M. Verduyn Lunel. Introduction to functional differential equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993. [Google Scholar]
  50. Y. Haupt, R. Maya, A. Kazaz, M. Oren. Mdm2 promotes the rapid degradation of p53. Nature 387 (1997), 296–299. [CrossRef] [PubMed] [Google Scholar]
  51. C. Haurie, D.C. Dale, M.C. Mackey. Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models. Blood, 92 (1998), 2629–2640. [PubMed] [Google Scholar]
  52. C. Haurie, D.C. Dale, M.C. Mackey. Occurrence of periodic oscillations in the differential blood counts of congenital, idiopathic, and cyclical neutropenic patient before and during treatment with G-CSF. Exp. Hematol., 27 (1999), 401–409. [CrossRef] [PubMed] [Google Scholar]
  53. C. Haurie, D.C. Dale, R. Rudnicki, M.C. Mackey. Modeling complex neutrophil dynamics in the grey collie. J Theor Biol. 204 (2000), 505–519. [CrossRef] [PubMed] [Google Scholar]
  54. C. Haurie, R. Person, D.C. Dale, M.C. Mackey. Hematopoietic dynamics in grey collies. Exp. Hematol., 27 (1999), 1139–1148. [CrossRef] [PubMed] [Google Scholar]
  55. N.D. Hayes. Roots of the transcendental equation associated with a certain difference-differential equation. J. London Math. Soc., 25 (1950), 226–232. [CrossRef] [MathSciNet] [Google Scholar]
  56. T. Hearn, C. Haurie, M.C. Mackey. Cyclical neutropenia and the peripheral control of white blood cell production. J. Theor. Biol. 192 (1998), 167–181. [CrossRef] [PubMed] [Google Scholar]
  57. H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa, R. Kageyama. Oscillatory Expression of the bHLH Factor Hes1 Regulated by a Negative Feedback Loop. Science 298 (2002), 840–843. [CrossRef] [PubMed] [Google Scholar]
  58. Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press, INC., San Diego, CA (1993). [Google Scholar]
  59. L.G. Lajtha. On DNA labeling in the study of the dynamics of bone marrow cell populations, in: Stohlman, Jr., F. (Ed), The Kinetics of Cellular Proliferation, Grune and Stratton, New York (1959), 173–182. [Google Scholar]
  60. J. Lei, M.C. Mackey. Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia. J. Theor. Biol., 270 (2011), 143–153. [CrossRef] [PubMed] [Google Scholar]
  61. J. Li, Y. Kuang, C. Mason. Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays. J. Theoret. Biol., 242 (2006), 722–735. [CrossRef] [MathSciNet] [Google Scholar]
  62. G.S. Longobardo, N.S. Cherniack, A.P. Fishman. Cheyne–Stokes breathing produced by a model of the human respiratory system. J. Appl. Physiol. 21 (1966), 1839–1846. [PubMed] [Google Scholar]
  63. N. MacDonald. Time Lags in Biological Models. Springer-Verlag, Heidelberg, 1978. [Google Scholar]
  64. M.C. Mackey. Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis. Blood, 51 (1978), 941–956. [PubMed] [Google Scholar]
  65. M.C. Mackey. Periodic auto- immune hemolytic anemia: an induced dynamical disease. Bull. Math. Biol., 41 (1979), 829–834. [MathSciNet] [PubMed] [Google Scholar]
  66. M.C. Mackey. Cell kinetic status of haematopoietic stem cells. Cell Prolif., 34 (2001), 71–83. [CrossRef] [PubMed] [Google Scholar]
  67. J.M. Mahaffy, J. Bélair, M.C. Mackey. Hematopoietic model with moving boundary condition and state dependant delay. J. Theor. Biol., 190 (1998), 135–146. [CrossRef] [PubMed] [Google Scholar]
  68. J. Mallet-Paret, R.D. Nussbaum, P. Paraskevopoulos. Periodic solutions for functional differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal., 3 (1994), 101–162. [Google Scholar]
  69. J.G. Milton, M.C. Mackey. Periodic haematological diseases: mystical entities of dynamical disorders ? J.R. Coll. Phys., 23 (1989), 236–241. [Google Scholar]
  70. N.A.M. Monk. Oscillatory expression of Hes1, p53, and NF-k B driven by transcriptional time delays. Curr. Biol. 13 (2003), 1409–1413. [CrossRef] [PubMed] [Google Scholar]
  71. A. Morley. Periodic diseases, physiological rhythms and feedback control-a hypothesis. Aust. Ann. Med. 3 (1970), 244–249. [Google Scholar]
  72. A. Morley, A.G. Baikie, D.A.G. Galton. Cyclic leukocytosis as evidence for retention of normal homeostatic control in chronic granulocytic leukaemia. Lancet, 2 (1967), 1320–1322. [CrossRef] [PubMed] [Google Scholar]
  73. A. Morley, E.A. King-Smith, F. Stohlman. The oscillatory nature of hemopoiesis. In: Stohlman, F. (Ed.), Hemopoietic Cellular Proliferation. Grune & Stratton, New York, (1969), 3–14. [Google Scholar]
  74. P.W. Nelson, J.D. Murray, A.S. Perelson. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci., 163 (2000), 201–215. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  75. N. Pørksen, M. Hollingdal, C. Juhl, P. Butler, J. D. Veldhuis, O. Schmitz. Pulsatile insulin secretion: Detection, regulation, and role in diabetes. Diabetes, 51 (2002), S245–S254. [CrossRef] [PubMed] [Google Scholar]
  76. L. Pujo-Menjouet, S. Bernard, M.C. Mackey. Long period oscillations in a G0 model of hematopoietic stem cells. SIAM J. Appl. Dyn. Systems, 4 (2005), No. 2, 312–332. [CrossRef] [Google Scholar]
  77. L. Pujo-Menjouet, M.C. Mackey. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologies, 327 (2004), 235–244. [CrossRef] [PubMed] [Google Scholar]
  78. M.Z. Ratajczak, J. Ratajczak, W. Marlicz, et al. Recombinant human thrombopoietin (TPO) stimulates erythropoiesis by inhibiting erythroid progenitor cell apoptosis. Br J. Haematol., 98 (1997), 8–17. [CrossRef] [PubMed] [Google Scholar]
  79. M. Santillan, J. Bélair, J.M. Mahaffy, M.C. Mackey. Regulation of platelet production: The normal response to perturbation and cyclical platelet disease. J. Theor. Biol., 206 (2000), 585–603. [CrossRef] [PubMed] [Google Scholar]
  80. B.R. Smith. Regulation of hematopoiesis. Yale J Biol Med., 63 (1990), No. 5, 371–380. [PubMed] [Google Scholar]
  81. H.L. Smith. Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. Math. Biosc., 113 (1993), 1–23. [CrossRef] [Google Scholar]
  82. J. Sturis, K. S. Polonsky, E. Mosekilde, E. Van Cauter. Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am. J. Physiol., 260 (1991), E801–E809. [PubMed] [Google Scholar]
  83. M. Sturrock, A.J. Terry, D.P. Xirodimas, A.M. Thompson, M.A.J. Chaplain. Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways. J. Theor. Biol., 273 (2011), 15–31. [CrossRef] [PubMed] [Google Scholar]
  84. S. Tanimukai, T. Kimura, H. Sakabe et al. Recombinant human c-Mpl ligand (thrombopoietin) not only acts on megakaryocyte progenitors, but also on erythroid and multipotential progenitors in vitro. Experimental Hematology, 25 (1997), 1025–1033. [PubMed] [Google Scholar]
  85. E. Terry, J. Marvel, C. Arpin, O. Gandrillon, F. Crauste. Mathematical Model of the primary CD8 T Cell Immune Response: Stability Analysis of a Nonlinear Age-Structured System. J. Math. Biol. (to appear). [Google Scholar]
  86. I.M. Tolic, E. Mosekilde, J. Sturis. Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion. J. Theoret. Biol., 207 (2000), 361–375. [CrossRef] [Google Scholar]
  87. J.J. Tyson, B. Novak. Regulation of the Eukaryotic Cell Cycle: Molecular Antagonism, Hysteresis, and Irreversible Transitions. J. theor. Biol., 210 (2001), pp. 249–263. [Google Scholar]
  88. W. Vainchenker. Hématopoïèse et facteurs de croissance. Encycl. Med. Chir., Hematologie, 13000 (1991), M85. [Google Scholar]
  89. H.O. Walther. The solution manifold and C1-smoothness of solution operators for differential equations with state dependent delay. J. Differential Eqs., 195 (2003), 46–65. [CrossRef] [Google Scholar]
  90. G.F. Webb. Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and textbook in Pure Appl. Math., 89, Marcel Dekker, New York (1985). [Google Scholar]
  91. I.L. Weissman. Stem cells: units of development, units of regeneration, and units in evolution. Cell, 100 (2002), 157–168. [CrossRef] [PubMed] [Google Scholar]

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