Bulletin of Mathematical Biology

, Volume 73, Issue 12, pp 2983–3007 | Cite as

A Model of Oscillatory Blood Cell Counts in Chronic Myelogenous Leukaemia

Original Article


In certain blood diseases, oscillations are found in blood cell counts. Particularly, such oscillations are sometimes found in chronic myelogenous leukaemia, and then occur in all the derived blood cell types: red blood cells, white blood cells, and platelets. It has been suggested that such oscillations arise because of an instability in the pluri-potential stem cell population, associated with its regulatory control system. In this paper, we consider how such oscillations can arise in a model of competition between normal (S) and genetically altered abnormal (A) stem cells, as the latter population grows at the expense of the former. We use an analytic model of long period oscillations to describe regions of oscillatory behaviour in the SA phase plane, and give parametric criteria to describe when such oscillations will occur. We also describe a mechanism which can explain dynamically how the transformation from chronic phase to acute phase and blast crisis can occur.


Chronic myelogenous leukaemia CML Chronic phase Oscillations Delay equations Blast crisis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adimy, M., Crauste, F., & Ruan, S. (2005a). Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics. Nonlinear Analysis: Real World Applications, 6, 651–670. MathSciNetMATHCrossRefGoogle Scholar
  2. Adimy, M., Crauste, F., & Ruan, S. (2005b). A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM Journal on Applied Mathematics, 65, 1328–1352. MathSciNetMATHCrossRefGoogle Scholar
  3. Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., & Watson, J. (1989). Molecular biology of the cell. New York: Garland Publishing. Google Scholar
  4. Bélair, J., & Mackey, M. C. (1987). A model for the regulation of mammalian platelet productiona. Annals of the New York Academy of Sciences, 504, 280–282 (Perspectives in Biological Dynamics and Theoretical Medicine). CrossRefGoogle Scholar
  5. Bedi, A., Zehnbauer, B. A., Barber, J. P., Sharkis, S. J., & Jones, R. J. (1994). Inhibition of apoptosis by BCR–ABL in chronic myeloid leukemia. Blood, 83(8), 2038–2044. Google Scholar
  6. Bennett, M., & Grunwald, A. J. (2001). Hydroxyurea and periodicity in myeloproliferative disease. European Journal of Haematology, 66(5), 317–323. CrossRefGoogle Scholar
  7. Bernard, S., Bélair, J., & Mackey, M. C. (2001). Sufficient conditions for stability of linear differential equations with distributed delay. Discrete and Continuous Dynamical Systems. Series B, 1, 233–256. MathSciNetMATHCrossRefGoogle Scholar
  8. Bessonov, N., Pujo-Menjouet, L., & Volpert, V. (2006). Cell modelling of hematopoiesis. Mathematical Modelling of Natural Phenomena, 1, 81–103. MathSciNetCrossRefGoogle Scholar
  9. Buckle, A.-M., Mottram, R., Pierce, A., Lucas, G. S., Russell, N., Miyan, J. A., & Whetton, A. D. (2000). The effect of bcr-abl protein tyrosine kinase on maturation and proliferation of primitive haematopoietic cells. Molecular Medicine, 6(10), 892–902. Google Scholar
  10. Colijn, C., Fowler, A. C., & Mackey, M. C. (2006). High frequency spikes in long period blood cell oscillations. Journal of Mathematical Biology, 53, 499–519. MathSciNetMATHCrossRefGoogle Scholar
  11. Colijn, C., & Mackey, M. C. (2005). A mathematical model of hematopoiesis—I. Periodic chronic myelogenous leukemia. Journal of Theoretical Biology, 237, 117–132. MathSciNetCrossRefGoogle Scholar
  12. Cortes, J., Talpaz, M., & Kantarjian, H. (1996). Chronic myelogenous leukaemia: a review. The American Journal of Medicine, 100, 555–570. CrossRefGoogle Scholar
  13. De Klein, A., Geurts van Kessel, A., & Grosveld, G. (1982). A cellular oncogene is translocated to the Philadelphia chromosome in chronic myelocytic leukemia. Nature, 300, 765–767. CrossRefGoogle Scholar
  14. Druker, B. J., Ford, J. M., Sawyers, C. L., Capdeville, R., Baccarani, M., & Goldman, J. M. (2001). Chronic myelogenous leukemia. In American society of hematology education program book (pp. 87–112), Orlando, Florida. Google Scholar
  15. Eaves, C., Cashman, J., & Eaves, A. (1998). Defective regulation of leukemic hematopoiesis in chronic myeloid leukemia. Leukemia Research, 22, 1085–1096. CrossRefGoogle Scholar
  16. Faderl, S., Kantarjian, H. M., & Talpaz, M. (1999). Chronic myelogenous leukemia: update on biology and treatment. Oncology, 13(2), 169–184. Google Scholar
  17. Fortin, P., & Mackey, M. C. (1999). Periodic chronic myelogenous leukaemia. British Journal of Haematology, 104, 336–345. CrossRefGoogle Scholar
  18. Fox, S. I. (1996). Human physiology (5th edn.). Dubuque: Brown. Google Scholar
  19. Fowler, A. C., & Mackey, M. C. (2002). Relaxation oscillations in a class of delay differential equations. SIAM Journal on Applied Mathematics, 63(1), 299–323. MathSciNetMATHCrossRefGoogle Scholar
  20. Frassoni, F., Podsta, M., & Piaggio, G. (1999). Normal and leukaemic haematopoiesis in bone marrow and peripheral blood of patients with chronic myeloid leukaemia. Baillieres Clinical Haematology, 12(1/2), 199–208. Google Scholar
  21. Goldman, J. (1997). ABC of clinical haematology: chronic myeloid leukaemia. British Medical Journal, 314(7081), 657–665. CrossRefGoogle Scholar
  22. Gordon, M. Y., Dowding, C. R., Riley, G. P., Goldman, J. M., & Greaves, M. F. (1987). Altered adhesive interactions with marrow stroma of haematopoietic progenitor cells in chronic myeloid leukaemia. Nature, 328, 342–344. CrossRefGoogle Scholar
  23. Gordon, M. Y., & Blackett, N. M. (1998). Reconstruction of the hematopoietic system after stem cell transplantation. Cell Transplantation, 7(4), 339–344. CrossRefGoogle Scholar
  24. Gordon, M. Y., Dazzi, F., Marley, S. B., Lewis, J. L., Nguyen, D., Grand, F. H., Davidson, R. J., & Goldman, J. M. (1999). Cell biology of CML cells. Leukemia, 13, S65–S71. CrossRefGoogle Scholar
  25. Guerry, D., Dale, D. C., Omine, M., Perry, S., & Wolff, S. M. (1973). Periodic hematopoiesis in human cyclic neutropenia. The Journal of Clinical Investigation, 52, 3220–3230. CrossRefGoogle Scholar
  26. Haurie, C., Dale, D. C., & Mackey, M. C. (1998). Cyclical neutropenia and other periodic hematological disorders: a review of mechanisms and mathematical models. Blood, 92(8), 2629–2640. Google Scholar
  27. Haurie, C., Dale, D. C., Rudnicki, R., & Mackey, M. C. (2000). Modeling complex neutrophil dynamics in the grey collie. Journal of Theoretical Biology, 204, 505–519. CrossRefGoogle Scholar
  28. Hill, J. M., & Meehan, K. R. (1999). Chronic myelogenous leukemia: Curable with early diagnosis and treatment. Postgraduate Medicine, 106(3), 149–159. CrossRefGoogle Scholar
  29. Hoffbrand, A. V., & Pettit, J. E. (1993). Essential haematology (3rd edn.). Oxford: Blackwell Science. Google Scholar
  30. Hughes-Jones, N. C., & Wickramasinghe, S. N. (1997). Lecture notes on haematology (6th edn.). Oxford: Blackwell Science. Google Scholar
  31. Iizuka, Y., Horikoshi, A., Sekiya, S., Sawada, U., Ohshima, T., & Amaki, I. (1984). Periodic fluctuation of leukocytes, platelets and reticulocytes in a case of chronic myelocytic leukemia: the relation between leukocyte counts, CFU–C colony formation, CSA and CIA. Acta Haematol. Jpn., 47(1), 71–79. Google Scholar
  32. Jorgensen, H. G., & Holyoake, T. L. (2001). A comparison of normal and leukemic stem cell biology in chronic myeloid leukemia. Hematological Oncology, 19, 89–106. CrossRefGoogle Scholar
  33. Kamada, N., & Uchino, H. (1978). Chronologic sequence in appearance of clinical and laboratory findings characteristic of chronic myelocytic leukemia. Blood, 51(5), 843–850. Google Scholar
  34. Kummermehr, J., & Trott, K.-R. (1997). Tumour stem cells. In Stem cells (pp. 401–419). London: Academic Press. Google Scholar
  35. Lebowitz, J. L., & Rubinow, S. I. (1969). Grain count distributions in labelled cell populations. Journal of Theoretical Biology, 23, 99–123. CrossRefGoogle Scholar
  36. Lodish, H., Baltimore, D., Berk, A., Zipursky, S. L., Matsudaira, P., & Darnell, G. (1995). Molecular Cell Biology (3rd edn.). New York: Scientific American Books. Google Scholar
  37. Mackey, M. C. (1978). A unified hypothesis for the origin of aplastic anemia and periodic haematopoiesis. Blood, 51, 941–956. Google Scholar
  38. Mackey, M. C. (1979). Dynamic haematological disorders of stem cell origin. In J. G. Vassileva-Popova & E. V. Jensen (Eds.), Biophysical and biochemical information transfer in recognition (pp. 373–409). New York: Plenum. Google Scholar
  39. Mackey, M. C. (1981). Some models in hemopoiesis: predictions and problems. In M. Rotenberg (Ed.), Biomathematics and cell kinetics (pp. 23–28). North Holland: Elsevier. Google Scholar
  40. Mackey, M. C. (1997). Mathematical models of hematopoietic cell replication and control. In H. G. Othmer, F. R. Adler, M. A. Lewis, & J. C. Dallon (Eds.), The art of mathematical modelling: case studies in ecology, physiology and biofluids (pp. 149–178). New Jersey: Prentice-Hall. Google Scholar
  41. Mackey, M. C., & Rudnicki, R. (1994). Global stability in a delayed partial differential equation describing cellular replication. Journal of Mathematical Biology, 33, 89–109. MathSciNetMATHCrossRefGoogle Scholar
  42. Mahaffy, J. M., Bélair, J., & Mackey, M. C. (1998). Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis. Journal of Theoretical Biology, 190, 135–146. CrossRefGoogle Scholar
  43. 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, 1267–1270. CrossRefGoogle Scholar
  44. Moore, H., & Li, N. K. (2004). A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. Journal of Theoretical Biology, 227, 513–523. MathSciNetCrossRefGoogle Scholar
  45. Neiman, B. (2000). A mathematical model of chronic myelogenous leukemia. M.Sc. Dissertation, Oxford University. Google Scholar
  46. Nowell, P. C., & Hungerford, D. A. (1960). A minute chromosome in human chronic granulocytic leukemia. Science, 132, 1497–1501. Google Scholar
  47. Potten, C. S. (1997). Stem cells. New York: Academic Press. Google Scholar
  48. Pujo-Menjouet, L., Bernard, S., & Mackey, M. C. (2005). Long period oscillations in a G 0 model of hematopoietic stem cells. SIAM Journal on Applied Dynamical Systems, 4, 312–332. MathSciNetMATHCrossRefGoogle Scholar
  49. Pujo-Menjouet, L., & Mackey, M. C. (2004). Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologies, 327, 235–244. CrossRefGoogle Scholar
  50. Rubinow, S. I., & Lebowitz, J. L. (1975). A mathematical model of neutrophil production and control in normal Man. Journal of Mathematical Biology, 1, 187–225. MathSciNetMATHCrossRefGoogle Scholar
  51. Rubinow, S. I., & Lebowitz, J. L. (1976). A mathematical model of the acute myeloblastic leukemic state in Man. Biophysical Journal, 16, 897–910. CrossRefGoogle Scholar
  52. Schwarzenberger, P., Kolls, J. K., & La Russa, V. (2002). Hematopoietic stem cells. Cancer Investigation, 20(1), 124–138. CrossRefGoogle Scholar
  53. Strife, A., & Clarkson, B. (1988). Biology of chronic myelogenous leukemia: Is discordant maturation the primary defect? Seminars in Hematology, 25(1), 1–19. Google Scholar
  54. Strife, A., Lambek, C., Wisniewski, D., Wachter, M., Gulati, S. C., & Clarkson, B. D. (1988). Discordant maturation as the primary biological defect in chronic myelogenous leukemia. Cancer Research, 48, 1035–1041. Google Scholar
  55. Whittaker, J. A. (1987). Leukaemia. Oxford: Blackwell Scientific Publications. Google Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  1. 1.Centre for Medical Image Computing, Department of Computer ScienceUniversity College LondonLondonUK
  2. 2.MACSIUniversity of LimerickLimerickIreland

Personalised recommendations