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CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT

This presentation focuses on mathematical models for treating blood diseases, specifically chronic myelogenous leukemia (CML). Topics include models for action on leukocytes in CML and action at the stem cells level. Other models and periodic solutions are also discussed. The presentation will be held at the 10th Franco-Romanian Colloquium on Applied Mathematics.

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CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT

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  1. CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  2. ANDREI HALANAY, University Politehnica of Bucharest 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  3. Topics of Discussion • Description of the models • Periodic solutions for the model for the treatment of CML involving both stem and mature cells • Periodic solutions for the model where only hematopoietic stem cells are considered. 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  4. Models for therapy in blood diseases • Two types of treatment will be considered : 1. Action on leukocytes in CML 2. Action at the stem cells level 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  5. Action on leukocytes in CML • The model is derived from the one given in the paper of C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis I- Periodic chronic myelogenous leukemia, Journal of Theoretical Biology 237 (2005), 117-132 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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  14. Other models • Another type of model is investigated in the papers H. Moore, N. K. Li, A mathematical model for chronic myelogenous leukemia ( CML) and T-cell interaction, J. Theor. Biol. 227 92004), 235-244 and S. Nanda, H. Moore, S. Lenhart, Optimal control of treatment in a mathematical model of chronic myelogenous leukemia, Mathematical Biosciences 210 (2007) 143-156 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  15. Other models • In these papers the models that are studied take into consideration the action of the immune system during a complex therapy that acts on the mature cells but has also some toxic effects on the cells of the immune system. • The models use ordinary differential equations 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  16. Action at the stem cells level • At the stem cells level two stages will be retained in the complex process of renewing and differentiation. One variable will account for non-proliferating (quiescent) stem cells and another one for the proliferating stem cells. The starting point of this model is in 1977-1978 with some papers promoting what has since been called the Mackey-Glass equations. M.C. Mackey, L. Glass, Oscillation and Chaos in Physiological Control Systems, Science, New Series, vol 197 (1977), no 4300, 287-289 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  17. Action at the stem cells level • M.C. Mackey, A unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood 51 (1978), 941-956 • Periodic solutions are obtained in M. C. Mackey, C. Ou, L. Pujo-Menjouet, J. Wu, Periodic Oscillations of blood cell population in chronic myelogenous leukemia, SIAM J. Math.Anal.38 (2006), 166-187. 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  18. Action at the stem cells level 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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  20. Other models at the stem cells level Other models are also intensively studied .One class has its origins in F. Michor, T. Hughes, Y. Iwasa, S. Branford, N. P.Shah, C. Sawyers, M. Novak, Dynamics of chronic myeloid leukemia, Nature 435 (2005), 1267-1270. A recent more elaborated model of this type is investigated in P. Kim, P. Lee, D. Levy, Dynamics and Potential Impact of the Immune Response to Chronic Myelogenous Leukemia, PLOS Comput. Biology 4 (6) (2008), doi: 10.1371/journal. pcbi. 1000095 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  21. Other models at the stem cells level • These models use also ordinary differential equations to describe the four stage stem cells’ evolution. • In the second paper, a delay equation is added to account for the immune system action. 10eme Colloque Franco-Roumaine de Mathematiques Appliques

  22. Periodic solutions for the CML model • The basic reference is M. A. Krasnoselskii, Shift operator on orbits of differential equations, Nauka, Moskow, 1966 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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  45. Stability of the periodic solution • Theorem of stability by the first approximation : J. Hale, Theory of Functional Differential equations, Springer, 1977, Theorem 18.3 V. Kolmanovskii, A. Mishkis, Applied Theory of Functional Differential equations, Kluwer, 1992, Theorem 1.9. Also it will be used a criterion of stability in Aristide Halanay, Differential Equations: stability, oscillations, time lag, Academic Press, 1966, Theorem 4.18 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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