Skip to main content
SpringerLink
Log in

Local Asymptotic Stability Conditions for the Positive Equilibrium of a System Modeling Cell Dynamics in Leukemia

  • Chapter
  • First Online:
Time Delay Systems: Methods, Applications and New Trends

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 423))

  • 2052 Accesses

Abstract

A distributed delay system with static nonlinearity has been considered in the literature to study the cell dynamics in leukemia. In this chapter local asymptotic stability conditions are derived for the positive equilibrium point of this nonlinear system. The stability conditions are expressed in terms of inequalities involving parameters of the system. These inequality conditions give guidelines for development of therapeutic actions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adimy, M., Crauste, F., El Abdllaoui, A.: Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia. J. Biological Systems 16(3), 395–424 (2008)

    Article  MATH  Google Scholar 

  2. Adimy, M., Crauste, F., El Abdllaoui, A.: Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics. C. R. Acad. Sci. Paris, Ser. I 348, 373–377 (2010)

    Article  MATH  Google Scholar 

  3. Adimy, M., Crauste, F., Ruan, S.: A Mathematical Study of the Hematopoiesis Process with Applications to Chronic Myelogenous Leukemia. SIAM J. Appl. Math. 65, 1328–1352 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonnet, D., Dick, J.E.: Human acute myeloid leukemia is organized as a hierarchy that originates from a primitive hematopoietic cell. Nature Medicine 3, 730–737 (1997)

    Article  Google Scholar 

  5. Colijn, C., Mackey, M.C.: A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. J. Theoretical Biology 237, 117–132 (2005)

    Article  MathSciNet  Google Scholar 

  6. Foley, C., Mackey, M.C.: Dynamic hematological disease: a review. J. Mathematical Biology 58, 285–322 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Huntly, B.J.P., Gilliland, D.G.: Leukemia stem cells and the evolution of cancer-stem-cell research. Nature Reviews: Cancer 5, 311–321 (2005)

    Article  Google Scholar 

  8. Kold-Andersen, L., Mackey, M.C.: Resonance in periodic chemotherapy: A case study of acute myelogenous leukemia. J. Theoretical Biology 209, 113–130 (2001)

    Article  Google Scholar 

  9. Mackey, M.C.: Unified hypothesis for the origin of aplastic anaemia and periodic hematopoiesis. Blood 51, 941–956 (1978)

    Google Scholar 

  10. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197(4300), 287–289 (1977)

    Article  Google Scholar 

  11. Mackey, M.C., Ou, C., Pujo-Menjouet, L., Wu, J.: Periodic Oscillations of Blood Cell Populations in Chronic Myelogenous Leukemia. SIAM J. Appl. Math. 38, 166–187 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Marie, J.P.: Private communication, Hôpital St. Antoine, Paris, France (July 2010)

    Google Scholar 

  13. Niculescu, S.-I., Kim, P.S., Gu, K., Lee, P.P., Levy, D.: Stability Crossing Boundaries of Delay Systems Modeling Immune Dynamics in Leukemia. Discrete and Continuous Dynamical Systems. Series B 13, 129–156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Özbay, H., Bonnet, C., Clairambault, J.: Stability Analysis of Systems with Distributed Delays and Application to Hematopoietic Cell Maturation Dynamics. In: Proc. of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 2008, pp. 2050–2055 (2008)

    Google Scholar 

  15. Özbay, H., Benjelloun, H., Bonnet, C., Clairambault, J.: Stability Conditions for a System Modeling Cell Dynamics in Leukemia. In: Preprints of IFAC Workshop on Time Delay Systems, TDS 2010, Prague, Czech Republic (June 2010)

    Google Scholar 

  16. Özbay, H., Bonnet, C., Benjelloun, H., Clairambault, J.: Global Stability Analysis of a System Modeling Cell Dynamics in AML. In: Abstracts of the 3rd Conference on Computational and Mathematical Population Dynamics (CMPD3), Bordeaux, France, June 2010, p. 186 (2010)

    Google Scholar 

  17. Özbay, H., Bonnet, C., Benjelloun, H., Clairambault, J.: Absolute Stability of a System with Distributed Delays Modeling Cell Dynamics in Leukemia. In: Proc. of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest, Hungary, July 2010, pp. 989–992 (2010)

    Google Scholar 

  18. Özbay, H., Bonnet, C., Benjelloun, H., Clairambault, J.: Stability Analysis of a Distributed Delay System Modeling Cell Dynamics in Leukemia (March 2010) (submitted for publication) (revised January 2011)

    Google Scholar 

  19. Peixoto, D., Dingli, D., Pacheco, J.M.: Modelling hematopoiesis in health and disease. Mathematical and Computer Modelling (2010), doi:10.1016/j.mcm.2010.04.013

    Google Scholar 

  20. Qu, Y., Wei, J., Ruan, S.: Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays. Physica D 239, 2011–2024 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Electrical and Electronics Eng., Bilkent Univ., Ankara, 06800, Turkey

    Hitay Özbay

  2. INRIA Saclay - Île-de-France, Parc Orsay Université, 4 rue J. Monod, 91893, Orsay Cedex, France

    Catherine Bonnet

  3. Ecole Centrale Paris, Paris, France

    Houda Benjelloun

  4. INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153, Le Chesney, Cedex, France

    Jean Clairambault

Authors
  1. Hitay Özbay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Catherine Bonnet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Houda Benjelloun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jean Clairambault
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hitay Özbay .

Editor information

Editors and Affiliations

  1. Dept. Mechanical Engineering, Northeastern University, Huntingdon Ave. 360, Boston, 02115, Massachusetts, USA

    Rifat Sipahi

  2. Technicka Street 4, Prague, 166 07, Czech Republic

    Tomáš Vyhlídal

  3. , CNRSSupélec, L2S (UMR CNRS 8506), rue Joliot Curie 3, Gif-sur-Yvette, 91192, France

    Silviu-Iulian Niculescu

  4. , Department of Electrical and Information, University of L’Aquila, Via G. Gronchi 18, L’Aquila, 67100, Italy

    Pierdomenico Pepe

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag GmbH Berlin Heidelberg

About this chapter

Cite this chapter

Özbay, H., Bonnet, C., Benjelloun, H., Clairambault, J. (2012). Local Asymptotic Stability Conditions for the Positive Equilibrium of a System Modeling Cell Dynamics in Leukemia. In: Sipahi, R., Vyhlídal, T., Niculescu, SI., Pepe, P. (eds) Time Delay Systems: Methods, Applications and New Trends. Lecture Notes in Control and Information Sciences, vol 423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25221-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-25221-1_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25220-4

  • Online ISBN: 978-3-642-25221-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation