CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT

CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT
5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

ANDREI HALANAY, University Politehnica of Bucharest

10eme Colloque Franco-Roumaine de Mathematiques Appliques
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. 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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), 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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 ), and S. Nanda, H. Moore, S. Lenhart, Optimal control of treatment in a mathematical model of chronic myelogenous leukemia, Mathematical Biosciences 210 (2007) 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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 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, 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

M.C. Mackey, A unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood 51 (1978), 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), 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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), 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: /journal. pcbi 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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. 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

Periodic solutions for the CML model
The basic reference is M. A. Krasnoselskii, Shift operator on orbits of differential equations, Nauka, Moskow, 1966 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

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 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

Treatment at the hematopoietic stem cells level
A. Halanay, Treatment induced periodic solutions in some mathematical models of tumoral cell dynamics, to appear in Mathematical Reports A. Halanay, Stability analysis for a mathematical model of chemotherapy action in hematological diseases, Bull Sci. de la Soc. des Sci. Math. de Roumanie, 53 (101) (2010), no. 1, p. 3-10 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

M. A. Krasnoselskii, Shift operator……. 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

Thank you ! 5/10/2019 10eme Colloque Franco-Roumaine de Mathematiques Appliques

CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT

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