A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia

SIAM Journal on Applied Mathematics

Volumn 65, Issue 4, 2005, Pages 1328-1352

  Adimy, Mostafa ^a   Crauste, Fabien ^a   Ruan, Shigui ^b  

^a UNIVERSITÉ DE PAU ET DES PAYS DE L'ADOUR   (France)

^b Department of Biology   (United States)

Author keywords

Asymptotic stability; Blood cells; Differential equations; Distributed delay; Hematopoiesis; Hopf bifurcation; Lyapunov functional

Indexed keywords

BLOOD CELLS; DISTRIBUTED DELAY; HEMATOPOIESIS; HOPF BIFURCATION; LYAPUNOV FUNCTIONAL;

ASYMPTOTIC STABILITY; BIFURCATION (MATHEMATICS); CELL CULTURE; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; DISEASES; LYAPUNOV METHODS; MATHEMATICAL MODELS; NONLINEAR EQUATIONS; OSCILLATIONS;

BLOOD;

EID: 24344475745     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/040604698     Document Type: Article

References (35)

1. M. ADIMY, F. CRAUSTE, AND S. RUAN, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Anal. Real World Appl., to appear.
2. R. F. V. ANDERSON, Geometric and probabilistic stability criteria for delay systems, Math. Biosci., 105 (1991), pp. 81-96.
3. R. F. V. ANDERSON, Intrinsic parameters and stability of differential-delay equations, J. Math. Anal. Appl., 163 (1992), pp. 184-199.
4. J. BÉLAIR, M. C. MACKEY, AND J. M. MAHAFFY, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), pp. 317-346.
5. S. BERNARD, J. BELAIR, 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.
6. F. G. BOESE, The stability chart for the linearized Cushing equation with a discrete delay and Gamma-distributed delays, J. Math. Anal. Appl., 140 (1989), pp. 510-536.
7. G. BRADFORD, B. WILLIAMS, R. ROSSI, AND I. BERTONCELLO, Quiescence, cycling, and turnover in the primitive haematopoietic stem cell compartment, Exper. Hematol., 25 (1997), pp. 445-453.
8. F. J. BURNS AND I. F. TANNOCK, On the existence of a Go phase in the cell cycle, Cell. Tissue Kinet., 19 (1970), pp. 321-334.
9. R. CRABB, J. LOSSON, AND M. C. MACKEY, Dependence on initial conditions in non local PDE's and hereditary dynamical systems, in Proc. Internat. Conf. Nonlinear Anal. 4, de Gruyter, Berlin, 1996, pp. 3125-3136.
10. J. DIEUDONNÉ, Foundations of Modern Analysis, Academic Press, New York, 1960.
11. 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.
12. 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.
13. J. K. HALE, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
14. 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.
15. Y. KUANG, Nonoccurrence of stability switching in systems of differential equations with distributed delays, Quart. Appl. Math., LII(3) (1994), pp. 569-578.
16. M. C. MACKEY, Unified hypothesis of the origin of aplastic anaemia and periodic hematopoïesis, Blood, 51 (1978), pp. 941-956.
17. M. C. MACKEY, Dynamic hematological disorders of stem cell origin, in Biophysical and Biochemical Information Transfer in Recognition, J. G. Vassileva-Popova and E. V. Jensen, eds., Plenum Press, New York, 1979, pp. 373-409.
18. M. C. MACKEY, Cell kinetic status of haematopoietic stem cells, Cell Prolif., 34 (2001), pp. 71-83.
19. M. C. MACKEY AND L. GLASS, From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, NJ, 1988.
20. M. C. MACKEY AND J. MILTON, Feedback, delays, and the origins of blood cell dynamics, Commun. Theor. Biol., 1 (1990), pp. 299-327.
21. M. C. MACKEY AND A. REY, Bifurcations and travelling waves in a delayed partial differential equation, Chaos, 2 (1992), pp. 231-244.
22. 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.
23. J. M. MAHAFFY, J. BÉLAIR, AND M. C. MACKEY, Hematopoietic model with moving boundary condition and state dependent delay, J. Theor. Biol., 190 (1998), pp. 135-146.
24. A. A. MORLEY, A. G. BAIKIE, AND D. A. G. GALTON, Cyclic leukocytosis as evidence for retention of normal homeostatic control in chronic granulocytic leukaemia, Lancet, 2 (1967), pp. 1320-1322.
25. G. C. NOONEY, Age distributions in dividing populations, Biophys. J., 7 (1967), pp. 69-76.
26. L. PUJO-MENJOUET, S. BERNARD, AND M. C. MACKEY, Long Period Oscillations in a Go Model of Hematopoietic Stem Cells, SIAM J. Appl. Dynam. Systems, 4 (2005), pp. 312-332.
27. L. PUJO-MENJOUET AND M. C. MACKEY, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biologies, 327 (2004), pp. 235-244.
28. S. I. RUBINOW, A maturity time representation for cell populations, Biophys. J., 8 (1968), pp. 1055-1073.
29. 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.
30. L. SACHS, The molecular control of hemopoiesis and leukemia, C. R. Acad. Sci. Paris, 316 (1993), pp. 882-891.
31. 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. Theor. Biol., 206 (2000), pp. 585-603.
32. L. F. SHAMPINE AND S. THOMPSON, Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001), 441-458; also available online at http://www.radford.edu/ thompson/webddes/.
33. E. TRUCCO, Mathematical models for cellular systems: The Von Foerster equation, Parts I and II, Bull. Math. Biophys., 27 (1965), pp. 285-304; 449-470.
34. E. TRUCCO, Some remarks on changing populations, J. Ferm. Technol., 44 (1966), pp. 218-226.
35. G. F. WEBB, Theory of Nonlinear Age-Dependent Population Dynamics, Monogr. Textbooks Pure Appl. Math. 89, Marcel Dekker, New York, 1985.