Advertisement

Journal of Mathematical Biology

, Volume 66, Issue 3, pp 595–625 | Cite as

Analysis of unstable behavior in a mathematical model for erythropoiesis

  • Susana Serna
  • Jasmine A. Nirody
  • Miklós Z. Rácz
Article
  • 384 Downloads

Abstract

We consider an age-structured model that describes the regulation of erythropoiesis through the negative feedback loop between erythropoietin and hemoglobin. This model is reduced to a system of two ordinary differential equations with two constant delays for which we show existence of a unique steady state. We determine all instances at which this steady state loses stability via a Hopf bifurcation through a theoretical bifurcation analysis establishing analytical expressions for the scenarios in which they arise. We show examples of supercritical Hopf bifurcations for parameter values estimated according to physiological values for humans found in the literature and present numerical simulations in agreement with the theoretical analysis. We provide a strategy for parameter estimation to match empirical measurements and predict dynamics in experimental settings, and compare existing data on hemoglobin oscillation in rabbits with predictions of our model.

Keywords

Age-structured model Multiple delay differential equations Hopf bifurcation Erythropoiesis 

Mathematics Subject Classification

35Q92 37N25 92C50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adimy M, Crauste F (2009) Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations. Math Comp Mod 49(11–12): 2128–2137MathSciNetzbMATHCrossRefGoogle Scholar
  2. Adimy M, Crauste F, Hbid ML, Qesmi R (2010) Stability and Hopf bifurcation for a cell population model with state-dependent delay. SIAM J Appl Math 70(5): 1611–1633MathSciNetzbMATHCrossRefGoogle Scholar
  3. Adimy M, Crauste F, Marquet C (2010) Asymptotic behavior and stability switch for a mature-immature model of cell differentiation. NonLinear Anal Real World Appl 11(4): 2913–2929MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bélair J, Campbell SA (1994) Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J Appl Math 54(5): 1402–1424MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bélair J, Mackey MC, Mahaffy JM (1995) Age-structured and two-delay models for erythropoiesis. Math Biosci 128(1–2): 317–346zbMATHCrossRefGoogle Scholar
  6. Bernard S, Bélair J, Mackey MC (2003) Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J Theor Biol 223: 283–298CrossRefGoogle Scholar
  7. Burwell EL, Brickley BA, Finch CA (1953) Erythrocyte life span in small animals; comparison of two methods employing radioiron. Am J Physiol 172: 718–725Google Scholar
  8. Cornish-Bowden A (2004) Fundamentals of enzyme kinetics. Portland Press, LondonGoogle Scholar
  9. Diekmann O, van Gils SA, Verduyn Lunel SM, Walther H-O (1995) Delay equations. functional-, complex-, and nonlinear analysis. Applied mathematical sciences, vol 110. Springer, New YorkGoogle Scholar
  10. Erslev AJ (1990) Erythrokinetics. In: Hematology. McGraw-Hill, New York, pp 414–442Google Scholar
  11. Erslev AJ (1991) Erythropoietin titers in health and disease. Semin Hematol 28(Suppl. 3): 2–8Google Scholar
  12. Fishbane S, Berns JS (2007) Evidence and implications of haemoglobin cycling in anaemia management. Nephrol Dial Transplant 22: 2129–2132CrossRefGoogle Scholar
  13. Foley C, Mackey MC (2009) Dynamic hematological disease: a review. J Math Biol 58(1): 285–322MathSciNetzbMATHCrossRefGoogle Scholar
  14. Glass L, Mackey MC (1988) From clocks to chaos: the rhythms of life. Princeton University Press, PrincetonzbMATHGoogle Scholar
  15. Graber S, Krantz S (1978) Erythropoietin and the control of red cell production. Ann Rev Med 29: 51–66CrossRefGoogle Scholar
  16. Grodins FS, Gray JS, Schroeder KR, Norins AL, Jones RW (1954) Respiratory responses to CO2 inhalation: a theoretical study of a nonlinear biological regulator. J Appl Physiol 7(3): 283–308Google Scholar
  17. Guevara MR, Glass L (1982) Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J Math Biol 14(1): 1–23MathSciNetzbMATHCrossRefGoogle Scholar
  18. Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Applied mathematical sciences, vol 99. Springer, New YorkGoogle Scholar
  19. Hewitt CD, Innes DJ, Savory J, Wills MR (1989) normal biochemical and hematological values in New Zealand white rabbits. Clin Chem 35(8): 1777–1779Google Scholar
  20. Kuznetsov YA (2004) Elements of applied bifurcation theory. Springer, New YorkzbMATHGoogle Scholar
  21. Mackey MC (1978) Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis. Blood 51(5): 941–956Google Scholar
  22. Mackey MC (1979) Periodic auto-immune hemolytic anemia: an induced dynamical disease. Bull Math Biol 41(6): 829–834MathSciNetzbMATHGoogle Scholar
  23. Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197(4300): 287–289CrossRefGoogle Scholar
  24. Mackey MC, Glass L (1989) Complex dynamics and bifurcations in neurology. J Theor Biol 138: 129–147MathSciNetCrossRefGoogle Scholar
  25. Mahaffy JM, Bélair J, Mackey MC (1998) Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis. J Theor Biol 190(2): 135–146CrossRefGoogle Scholar
  26. Miller CB, Jones RJ, Piantadosi S, Abeloff MD, Spivak JL (1990) Decreased erythropoietin response in patients with the anemia of cancer. N Engl J Med 332(24): 1689–1692CrossRefGoogle Scholar
  27. Milton JG, Mackey MC (1989) Periodic haematological diseases: mystical entities or dynamical disorders?. J R Coll Phys. Lond. 23(4): 236–241Google Scholar
  28. Orr JS, Kirk J, Gray KG, Anderson JR (1968) A study of the interdependence of red cell and bone marrow stem cell populations. Br J Haematol 15(1): 23–34CrossRefGoogle Scholar
  29. Pujo-Menjouet L, Mackey MC (2004) Contribution to the study of periodic chronic myelogenous leukemia. C R Biologies 327: 235–244CrossRefGoogle Scholar
  30. Reimann HA (1951) Periodic disease. Medicine 30(3): 219–245MathSciNetCrossRefGoogle Scholar
  31. Strogatz SH (2000) Nonlinear dynamics and chaos. Westview, BoulderGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Susana Serna
    • 1
  • Jasmine A. Nirody
    • 2
  • Miklós Z. Rácz
    • 3
  1. 1.Departament de MatematiquesUniversitat Autonoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Department of Radiology and Biomedical ImagingUniversity of CaliforniaSan FranciscoUSA
  3. 3.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations