CELL DYNAMICS IN SOME BLOOD DISEASES UNDER TREATMENT

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CELL DYNAMICS IN SOME BLOOD DISEASES UNDER
TREATMENT
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ANDREI HALANAY, University Politehnica of
Bucharest
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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.

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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

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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

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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

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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

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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

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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.

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Action at the stem cells level
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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

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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.

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Periodic solutions for the CML model

• The basic reference is
• M. A. Krasnoselskii, Shift operator on orbits of
  differential equations, Nauka, Moskow, 1966

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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

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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

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M. A. Krasnoselskii, Shift operator.
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Thank you !