Abstract
We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181–1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728–744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells.
An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbott, L.H., Michor, F., 2006. Mathematical models of targeted cancer therapy. Br. J. Cancer. 95(9), 1136–141.
Adimy, M., Pujo-Menjouet, L., 2003. A mathematical model describing cellular division with a proliferating phase duration depending on the maturity of cells. Electron. J. Differ. Equ. 2003(107), 1–4.
Campbell, J.D., Cook, G., Holyoake, T.L., 2001. Evolution of bone marrow transplantation—the original immunotherapy. Trends Immunol. 22(2), 88–2.
Colijn, C., Mackey, M.C., 2005. A mathematical model of hematopoiesis–I. Periodic chronic myelogenous leukemia. J. Theor. Biol. 237(2), 117–32.
DeConde, R., Kim, P.S., Levy, D., Lee, P.P., 2005. Post-transplantation dynamics of the immune response to chronic myelogenous leukemia. J. Theor. Biol. 236(1), 39–9.
Dingli, D., Michor, F., 2006. Successful therapy must eradicate cancer stem cells. Stem Cells 24(12), 2603–610.
Druker, B.J., Lydon, N.B., 2000. Lessons learned from the development of an ABL tyrosine kinase inhibitor for chronic myelogenous leukemia. J. Clin. Invest. 105(1), 3–.
Fokas, A.S., Keller, J.B., Clarkson, B.D., 1991. Mathematical model of granulocytopoiesis and chronic myelogenous leukemia. Cancer Res. 51(8), 2084–091.
Kim, P.S., Lee, P.P., Levy, D., 2007. Mini-Transplants for Chronic Myelogenous Leukemia: A Modeling Perspective. In: Queinnec et al. (Eds.), Biology and Control Theory: Current Challenges, Lecture Notes in Control and Information Sciences, vol. 357, pp. 3–0.
Kim, P.S., Lee, P.P., Levy, D., 2008. Modeling imatinib-treated chronic myelogenous leukemia: reducing the complexity of agent-based models. Bull. Math. Biol. 70(3), 728–44.
Komarova, N.L., Wodarz, D., 2005. Drug resistance in cancer: Principles of emergence and prevention. Proc. Natl. Acad. Sci. USA 102(27), 9714–719.
Michor, F., Hughes, T.P., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C.L., Nowak, M.A., 2005. Dynamics of chronic myeloid leukaemia. Nature 435, 1267–270.
Moore, H., Li, N.K., 2004. A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol. 225(4), 513–23.
Neiman, B., 2002. A mathematical model of chronic myelogenous leukemia. Master’s thesis, University College, Oxford University, UK.
Pujo-Menjouet, L., Mackey, M.C., 2004. Contribution to the study of periodic chronic myelogenous leukemia. C. R. Biol. 327, 235–44.
Roeder, I., 2003. Dynamical modeling of hematopoietic stem cell organization—Design and validation of the new concept of within-tissue plasticity. PhD thesis, University of Leipzig, Germany.
Roeder, I., Glauche, I., 2008. Pathogenesis, treatment effects, and resistance dynamics in chronic myeloid leukemia—insights from mathematical model analyses. J. Mol. Med. 85(1), 17–7.
Roeder, I., Horn, M., Glauche, I., Hochhaus, A., Mueller, M.C., Loeffler, M., 2006. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat. Med. 12(10), 1181–184.
Thijsen, S.F.T., Schuurhuis, G.J., van Oostveen, J.W., Ossenkoppele, G.J., 1999. Chronic mlyeloid leukemia from basics to bedside. Leukemia 13(11), 1646–674.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, P.S., Lee, P.P. & Levy, D. A PDE Model for Imatinib-Treated Chronic Myelogenous Leukemia. Bull. Math. Biol. 70, 1994–2016 (2008). https://doi.org/10.1007/s11538-008-9336-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-008-9336-z