Long period oscillations in a G 0 model of hematopoietic stem cells

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 2, 2005, Pages 312-332

  Pujo Menjouet, Laurent ^a   Bernard, Samuel ^b   Mackey, Michael C ^a  

^a MCGILL UNIVERSITY   (Canada)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Cell proliferation; Delay differential equations; G 0 stem cell model; Hill function; Long period oscillations; Periodic chronic myelogenous leukemia

Indexed keywords

CELL PROLIFERATION; DELAY DIFFERENTIAL EQUATIONS; G 0 STEM CELL MODEL; HILL FUNCTION; LONG PERIOD OSCILLATIONS; PERIODIC CHRONIC MYELOGENOUS LEUKEMIA;

BLOOD; DISEASES; MATHEMATICAL MODELS; NUMERICAL METHODS; OSCILLATIONS;

CELLS;

EID: 24344441267     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/030600473     Document Type: Article

References (66)

1. M. ADIMY AND L. PUJO-MENJOUET, A singular transport model describing cellular division., C. R. Acad. Sci. Paris Sér I. Math., 332 (2001), pp. 1071-1076.
2. M. ADIMY AND L. PUJO-MENJOUET, Asymptotic behavior of a singular transport equation modelling cell division, Discrete Contin. Dyn. Syst. Sci. B, 3 (2003), pp. 439-456.
3. U. AN DER. HEIDEN, M. C. MACKEY, AND H. O. WALTHER, Complex oscillations in a simple deterministic neuronal network, in Mathematical Aspects of Physiology, F. Hoppensteadt, ed., AMS, Providence, RI, 1981, pp. 355-360.
4. U. AN DER HEIDEN AND H. O. WALTHER, Existence and chaos in control systems with delayed feedback, J. Differential Equations, 47 (1983), pp. 273-297.
5. J. BÉLAIR, M. C. MACKEY, AND J. M. MAHAFFY, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), pp. 317-346.
6. E. BERETTA AND Y. KUANG, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), pp. 1144-1165.
7. S. BERNARD, J. BÉLAIR, AND M. C. MACKEY, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), pp. 233-256.
8. S. BERNARD, J. BÉLAIR., AND M. C. MACKEY, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J. Theoret. Biol., 223 (2003), pp. 283-298.
9. S. BERNARD, J. BÉLAIR, AND M. C. MACKEY, Bifurcations in a white blood cell production model, C. R. Biologies, 327 (2004), pp. 201-210.
10. S. BERNARD, B. ČAJAVEC, L. PUJO-MENJOUET, M. C. MACKEY, AND H. HERZEL, Modeling transcriptional feedback loops: The role of Gro/TLE1 in Hesl oscillations, Proc. Roy. Soc., submitted.
11. H. BOLOURI AND E. H. DAVIDSON, Transcriptional regulatory cascades in development: Initial rates, not steady state, determine network kinetics, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 9371-9376.
12. A.-M. BUCKLE, R. MOTTRAM, A. PIERCE, G. S. LUCAS, N. RUSSEL, J. A. MIYAN, AND A. D. WHETTON, The effects of Bcr-Abl protein tyrosine kinase on maturation and proliferation of primitive haematopoietic cells, Molecular Medicine, 6 (2000), pp. 892-902.
13. W. S. BULLOUGH, Mitotic control in adult mammalian tissues, Biol. Rev., 50 (1975), pp. 99-127.
14. 0 phase in the cell cycle, Cell Tissue Kinet., 3 (1970), pp. 321-334.
15. R. CRABB, M. C. MACKEY, AND A. REY, Propagating fronts, chaos and multistability in a cell replication model, Chaos, 3 (1996), pp. 477-492.
16. G. DALEY, R. VANETTEN, AND D. BALTIMORE, Induction of chronic myelogenous leukemia in mice by the p210 bcr-abl gene of the Philadelphia-chromosome, Science, 247 (1990), pp. 824-830.
17. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, A singular transport equation modelling a proliferating maturity structured cell population, Canad. Appl. Math. Quart., 4 (1996), pp. 65-95.
18. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, Hypercyclicity of solutions of a transport equation with delays, Nonlinear Anal., 29 (1997), pp. 1343-1351.
19. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, An age and maturity structured model of cell population dynamics, in Mathematical Models in Medical and Health Science, G. W. M. Horn and G. Simonnett, eds., Vanderbilt University Press, Nashville, TN, 1998, pp. 99-116.
20. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, A nonlinear age and maturity structured model of population dynamics. I. Basic theory, J. Math. Anal. Appl., 242 (2000), pp. 93-104.
21. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, A nonlinear age and maturity structured model of population dynamics. II. Chaos, J. Math. Anal. Appl., 242 (2000), pp. 255-270.
22. J. DYSON, R. VILLELLA-BRESSAN, AND G. F. WEBB, A singular transport equation with delays, Int. J. Math. Math. Sci., 32 (2003), pp. 2011-2026.
23. C. J. EAVES AND A. C. EAVES, Stem cell kinetics, Baillieres Clinical Haematology, 10 (1997), pp. 233-257.
24. A. ELEFANTY, I. HARIHARAN, AND S. CORY, Bcr-Abl, the hallmark of chronic myeloid-leukemia in man, induces multiple hematopoietic neoplasms in mice, EMBO J., 9 (1990), pp. 1069-1078.
25. J. J. FERRELL, Tripping the switch fantastic: How protein kinase cascade convert graded into switch-like outputs, TIBS, 21 (1996), pp. 460-466.
26. P. FORTIN AND M. C. MACKEY, Periodic chronic myelogenous leukemia: Spectral analysis of blood cell counts and etiological implications, Brit. J. Haematol., 104 (1999), pp. 336-345.
27. A. C. FOWLER AND M. C. MACKEY, Relaxation oscillations in a class of delay differential equations, SIAM J. Appl. Math., 63 (2002), pp. 299-323.
28. M. GISHIZKY, J. JOHNSONWHITE, AND O. WITTE, Efficient transplantation of bcr-abl-induced chronic myelogenous leukemia-like syndrome in mice, Proc. Natl. Acad. Sci. USA, 90 (1993), pp. 3755-3759.
29. T. HAFERLACH, M. WINKEMANN, C. NICKENIG, M. MEEDER, L. RAMMPETERSEN, R. SCHOCH, M. NICKELSEN, K. WEBERMATTHIESEN, B. SCHLEGELBERGER, C. SCHOCH, W. GASSMAN, AND H. LOFFLER, Which compartments are involved in Philadelphia- chromosome positive chronic myeloid leukaemia? An answer at the single cell level by combining May-Grunwald-Giemsa staining and fluorescence in situ hybridization techniques, Brit. J. Haematol., 97 (1997), pp. 99-106.
30. C. HAURIE, D. C. DALE, AND M. C. MACKEY, Cyclical neutropenia and other periodic hematological diseases: A review of mechanisms and mathematical models, Blood, 92 (1998), pp. 2629-2640.
31. N. D. HAYES, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc., 25 (1950), pp. 226-232.
32. T. HEARN, C. HAURIE, AND M. C. MACKEY, Cyclical neutropenia and the peripheral control of white blood cell production, J. Theoret. Biol., 192 (1998), pp. 167-181.
33. K. H. HOWARD, Size and Maturity Structured Models of Cell Population Dynamics Exhibiting Chaotic Behavior, Ph.D. thesis, Vanderbilt University, Nashville, TN, 2000.
34. Y. IIZUKA, A. HORIKOSHI, S. SEKIYA, U. SAWADA, T. OHSHIMA, AND I. AMAKI, Periodic fluctuation of leukocytes, platelets and reticulocytes in a case of chronic myelocytic leukemia: The relation between leukocyte counts, CFU-C colony formation, CSA and CIA, Acta Haematologica Japanica, 47 (1984), pp. 71-79.
35. X. JIANG, C. J. EAVES, AND A. C. EAVES, IL-3 and G-CSF gene expression in primitive PH+CD34(+) cells from patients with chronic myeloid leukemia (CML), Blood, 90 (1997), pp. 1745-1745.
36. M. KELLIHER, J. MCLOUGHLIN, O. WITTE, AND N. ROSENBERG, Induction of a chronic myelogenous leukemia-like syndrome in mice with v-Abl and Bcr-Abl, Proc. Natl. Acad. Sci. USA, 87 (1990), pp. 6649-6653.
37. J. KEVORKIAN AND J. D. COLE, Perturbation Methods in Applied Mathematics, Springer-Verlag, Berlin, 1981.
38. L. KOLD-ANDERSEN AND M. C. MACKEY, Resonance in periodic chemotheray: A case study of acute myelogenous leukemia, J. Theoret. Biol., 209 (2001), pp. 113-130.
39. J. KONOPKA, S. WATANABE, AND O. WITTE, An alteration of the human c-Abl protein in K562 leukemiacells unmasks associated tyrosine kinase-activity, Cell, 37 (1984), pp. 1035-1042.
40. L. G. LAJTHA, On DNA labeling in the study of the dynamics of bone marrow cell populations, in The Kinetics of Cellular Proliferation, F. Stohlman, Jr., ed., Grune & Stratton, New York, 1959, pp. 173-182.
41. A. LASOTA, M. C. MACKEY, AND M. WAZEWSKA-CZYZEWSKA, Minimizing therapeutically induced anemia, J. Math. Biol., 13 (1981), pp. 149-158.
42. I. LEWIS, L. MCDIARMID, L. SAMUEL, L. To, AND T. HUGUES, Establishment of a reproducible model of chronic-phase chronic myelogenous leukemia in NOD/SCID mice using blood-derived mononuclear or CD34(+) cells, Blood, 91 (1998), pp. 630-640.
43. T. LUGO, A. PENDERGAST, A. MULLER, AND O. WITTE, Tyrosine kinase-activity and transformation potency of Bcr-Abl oncogene products, Science, 247 (1990), pp. 1079-1082.
44. M. C. MACKEY, Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), pp. 941-956.
45. M. C. MACKEY, Dynamic haematological disorders of stem cell origin, in Biophysical and Biochemical Information Transfer in Recognition, J. G. Vassileva-Popova and E. V. Jensen, eds., Plenum Publishing, New York, 1979, pp. 373-409.
46. M. C. MACKEY, Periodic auto-immune hemolytic anemia: An induced dynamical, disease, Bull. Math. Biol., 41 (1979), pp. 829-834.
47. M. C. MACKEY, Mathematical models of hematopoietic cell replication and control, in The Art of Mathematical Modeling: Case Studies in Ecology, Physiology and Biofluids, Prentice-Hall, Upper Saddle River, NJ, 1997, pp. 149-178.
48. M. C. MACKEY, Cell kinetic status of hematopoietic stem cells, Cell Prolif., 34 (2001), pp. 71-83.
49. M. C. MACKEY AND U. AN DER HEIDEN, The dynamics of production and destruction: Analytic insight into complex behavior, J. Math. Biol., 16 (1982), pp. 75-101.
50. M. C. MACKEY AND P. DÖRMER, Continuous maturation of proliferating erythroid precursors, Cell Tissue Kinet., 15 (1982), pp. 381-392.
51. M. C. MACKEY AND R. RUDNICKI, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), pp. 89-109.
52. M. C. MACKEY AND R. RUDNICKI, A new criterion for the global stability of simultaneous cell replication and maturation processes, J. Math. Biol., 38 (1999), pp. 195-219.
53. H. MOORE AND N. K. LIB, A mathematical model for chronic myelogenous leukemia (cml) and t cell interaction, J. Theoret. Biol., 227 (2004), pp. 513-523.
54. H. G. OTHMER, F. R. ADLER, M. A. LEWIS, AND J. C. DALTON, eds., The Art of Mathematical Modeling: Case Studies in Ecology, Physiology and Biofluids, Prentice-Hall, Upper Saddle River, NJ, 1997.
55. H. PETERS, Comportment chaotique d'une équation retardée, C. R. Acad. Sci. Paris Sér 1. Math., 290 (1980), pp. 1119-1122.
56. L. PUJO-MENJOUET, Contribution à l'étude d'une équation de transport à retards décrivant une dynamique de population cellulaire, Ph.D. thesis, Université de Pau et des Pays de l'Adour, Pau, France, 2001.
57. L. PUJO-MENJOUET AND M. C. MACKEY, Contribution to the study of periodic chronic myelogenous leukemia, Comptes Rendus de l'Académie des Sciences Paris Biologie, 327 (2004), pp. 235-244.
58. S. I. RUBINOW AND J. L. LEBOWITZ, A mathematical model of neutrophil production and control in normal man, J. Math. Biol., 1 (1975), pp. 187-225.
59. M. SANTILLAN, J. BÉLAIR, J. M. MAHAFFY, AND M. C. MACKEY, Regulation of platelet production: The normal response to perturbation and cyclical platelet disease, J. Theoret. Biol., 206 (2000), pp. 585-903.
60. L. F. SHAMPINE AND S. THOMPSON, Solving DDEs, Appl. Numer. Math., 37 (2001), pp. 441-445.
61. J. A. SMITH AND L. MARTIN, Do cells cycle?, Proc. Natl. Acad. Sci. USA, 70 (1973), pp. 1263-1267.
62. J. SWINBURNE AND M. C. MACKEY, Cyclical thrombocytopenia: Characterization by spectral analysis and a review, J. Theoret. Med., 2 (2000), pp. 81-91.
63. N. TAKAHASHI, I. MIURA, K. SAITOH, AND A. B. MIURA, Lineage involvment of stem cells bearing the Philadelphia chromosome in chronic myeloid leukemia in the chronic phase as shown by a combination of fluorescence-activated cell sorting and fluorescence in situ hybridization, Blood, 92 (1998), pp. 4758-4763.
64. H. O. WALTHER, Homoclinic solution and chaos in x(t) = -μx(t) + f(x(t-1)), Nonlinear Anal., 5 (1981), pp. 775-788.
65. G. F. WEBB, Theory of Nonlinear Age-Dependent Population Dynamics, Monogr. Textbooks Pure Appl. Math. 89, Dekker, New York, 1985.
66. X. W. ZHANG AND R. B. REN, Bcr-Abl efficiently induces a myeloproliferative disease and production of excess interleukin-3 and granulocyte-macrophage colony-stimulating factor in mice: A novel model for chronic myelogenous leukemia, Blood, 92 (1998), pp. 3829-3840.