Bulletin of Mathematical Biology

, Volume 78, Issue 12, pp 2304–2357 | Cite as

A Mathematical Model of Granulopoiesis Incorporating the Negative Feedback Dynamics and Kinetics of G-CSF/Neutrophil Binding and Internalization

  • M. CraigEmail author
  • A. R. Humphries
  • M. C. Mackey
Original Article


We develop a physiological model of granulopoiesis which includes explicit modelling of the kinetics of the cytokine granulocyte colony-stimulating factor (G-CSF) incorporating both the freely circulating concentration and the concentration of the cytokine bound to mature neutrophils. G-CSF concentrations are used to directly regulate neutrophil production, with the rate of differentiation of stem cells to neutrophil precursors, the effective proliferation rate in mitosis, the maturation time, and the release rate from the mature marrow reservoir into circulation all dependent on the level of G-CSF in the system. The dependence of the maturation time on the cytokine concentration introduces a state-dependent delay into our differential equation model, and we show how this is derived from an age-structured partial differential equation model of the mitosis and maturation and also detail the derivation of the rest of our model. The model and its estimated parameters are shown to successfully predict the neutrophil and G-CSF responses to a variety of treatment scenarios, including the combined administration of chemotherapy and exogenous G-CSF. This concomitant treatment was reproduced without any additional fitting to characterize drug–drug interactions.


Granulopoiesis Mathematical modelling State-dependent delay differential equations Physiologically constructed pharmacokinetics G-CSF Bound and unbound drug concentrations 



A.R.H. and M.C.M. are grateful to the National Science and Engineering Research Council (NSERC), Canada, for funding through the Discovery Grant program. M.C. wishes to thank NSERC for funding from the PGS-D program. We are grateful to Fahima Nekka, Jun Li, Jacques Bélair, and David Dale for their insight and support. We would also like to thank both anonymous reviewers for their helpful and insightful comments.


  1. Baiocci G, Scambia G, Benedetti P, Menichella G, Testa U, Pierelli L, Martucci R, Foddai ML, Bizzi B, Mancuso S, Peschle C (1993) Autologous stem cell transplantation: sequential production of hematopoietic cytokines underlying granulocyte recovery. Cancer Res 53:1297–1303Google Scholar
  2. Barreda DR, Hanington PC, Belosevic M (2004) Regulation of myeloid development and function by colony stimulating factors. Dev Comp Immunol 28(5):509–554CrossRefGoogle Scholar
  3. Basu S, Hodgson G, Katz M, Dunn AR (2002) Evaluation of role of G-CSF in the production, survival, and release of neutrophils from bone marrow into circulation. Blood 100:854–861CrossRefGoogle Scholar
  4. Bernard S, Bélair J, Mackey MC (2003) Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J Theor Biol 223:283–298MathSciNetCrossRefGoogle Scholar
  5. Brooks G, Langlois GP, Lei J, Mackey MC (2012) Neutrophil dynamics after chemotherapy and G-CSF: the role of pharmacokinetics in shaping the response. J Theor Biol 315:97–109MathSciNetCrossRefGoogle Scholar
  6. Bugl S, Wirths S, Müller MR, Radsak MP (2012) Current insights into neutrophil homeostasis. Ann N Y Acad Sci 1266:171–178CrossRefGoogle Scholar
  7. Cairo MS, Suen Y, Sender L, Gillan ER, Ho W, Plunkett JM, van de Ven C (1992) Circulating granulocyte colony-stimulating factor (G-CSF) levels after allogenic and autologous bone marrow transplantation: endogenous G-CSF production correlates with myeloid engraftment. Blood 79:1869–1873Google Scholar
  8. Colijn C, Mackey MC (2005a) A mathematical model of hematopoiesis: I. Periodic chronic mylogenous leukemia. J Theor Biol 237:117–132MathSciNetCrossRefGoogle Scholar
  9. Colijn C, Mackey MC (2005b) A mathematical model of hematopoiesis: II. Cyclical neutropenia. J Theor Biol 237:133–146MathSciNetCrossRefGoogle Scholar
  10. Craig M, González-Sales M, Li J, Nekka F (2016) Impact of pharmacokinetic variability on a mechanistic physiological pharmacokinetic/pharmacodynamic model: a case study of neutrophil development, PM00104, and filgrastim. In: Toni B (ed) Mathematical sciences with multidisciplinary applications, Springer Proceedings in Mathematics and Statistics. Springer Science + Business Media, New York, ISBN 978-3-319-31323-8Google Scholar
  11. Craig M, Humphries AR, Bélair J, Li J, Nekka F, Mackey MC (2015) Neutrophil dynamics during concurrent chemotherapy and G-CSF administration: mathematical modelling guides dose optimisation to minimise neutropenia. J Theor Biol 385:77–89CrossRefzbMATHGoogle Scholar
  12. Dale DC, Mackey MC (2015) Understanding, treating and avoiding hematological disease: better medicine through mathematics? Bull Math Biol 77:739–757MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dale DC, Welte K (2011) Hematopoietic growth factors in oncology. Springer, HeidelbergGoogle Scholar
  14. Dancey JT, Deubelbeiss KA, Harker LA, Finch CA (1976) Neutrophil kinetics in man. J Clin Investig 58:705–715CrossRefGoogle Scholar
  15. DiPiro JT, Spruill WJ, Wade WE, Blouin RA, Pruemer JM (eds) (2010) Concepts in clinical pharmacokinetics, vol 5. American Society of Health-System Pharmacists, BethesdaGoogle Scholar
  16. Durand C, Charbord P (2010) Stem cell biology and regenerative medicine, vol 3. River Publishers, AalborgGoogle Scholar
  17. Endele M, Etzrodt M, Schroeder T (2014) Instruction of hematopoietic lineage choice by cytokine signaling. Exp Cell Res 329:207–213CrossRefGoogle Scholar
  18. Foley C, Bernard S, Mackey MC (2006) Cost-effective G-CSF therapy strategies for cyclical neutropenia: mathematical modelling based hypotheses. J Theor Biol 238:756–763MathSciNetCrossRefGoogle Scholar
  19. Foley C, Mackey MC (2009a) Dynamic hematological disease: review. J Math Biol 58:285–322MathSciNetCrossRefzbMATHGoogle Scholar
  20. Foley C, Mackey MC (2009b) Mathematical model for G-CSF administration after chemotherapy. J Theor Biol 257:27–44MathSciNetCrossRefGoogle Scholar
  21. Friberg LE, Karlsson MO (2003) Mechanistic models for myelosuppression. Invest New Drugs 21:183–194CrossRefGoogle Scholar
  22. González-Sales M, Valenzuela B, Pérez-Ruixo C, Fernández Teruel C, Miguel-Lillo B, Soto Matos A et al (2012) Population pharmacokinetic-pharmacodynamic analysis of neutropenia in cancer patients receiving PM00104 (Zalypsis\(\textregistered \) ). Clin Pharmacokinet 51:751–764CrossRefGoogle Scholar
  23. Greeenbaum AM, Link DC (2011) Mechanisms of G-CSF-mediated hematopoietic stem and progenitor mobilization. Leukemia 25:211–217CrossRefGoogle Scholar
  24. Hammond WP, Csiba E, Canin A, Hockman H, Souza LM, Layton JE, Dale DC (1991) Chronic neutropenia. A new canine model induced by human granulocyte colony-stimulating factor. J Clin Investig 87(2):704–710CrossRefGoogle Scholar
  25. Hearn T, Haurie C, Mackey MC (1998) Cyclical neutropenia and the peripherial control of white blood cell production. J Theor Biol 192:167–181CrossRefGoogle Scholar
  26. Kawakami M, Tsutsumi H, Kumakawa T, Abe H, Hirai M, Kurosawa S, Mori M, Fukushima M (1990) Levels of serum granulocyte colony-stimulating factor in patients with infections. Blood 76(10):1962–1964Google Scholar
  27. Kazarinoff ND, van den Driessche P (1979) Control of oscillations in hematopoiesis. Science 203:1348–1350MathSciNetCrossRefzbMATHGoogle Scholar
  28. King-Smith EA, Morley A (1970) Computer simulation of granulopoiesis: normal and impaired granulopoiesis. Blood 36:254–262Google Scholar
  29. Krinner A, Roeder I, Loeffler M, Scholz M (2013) Merging concepts—coupling an agent-based model of hematopoietic stem cells with an ODE model of granulopoiesis. BMC Syst Biol 7:117CrossRefGoogle Scholar
  30. Krzyzanski W, Wiczling P, Lowe P, Pigeolet E, Fink M, Berghout A, Balser S (2010) Population modeling of filgrastim PK-PD in healthy adults following intravenous and subcutaneous administrations. J Clin Pharmacol 9(Suppl):101S–112SCrossRefGoogle Scholar
  31. Kuwabara T, Kato Y, Kobayashi S, Suzuki H, Sugiyama Y (1994) Nonlinear pharmacokinetics of a recombinant human granulocyte colony-stimulating factor derivative (Nartograstim): species differences among rats, monkeys and humans. J Pharmacol Exp Ther 271:1535–1543Google Scholar
  32. Layton JE, Hall NE (2006) The interaction of G-CSF with its receptor. Front Biosci 31:177–199Google Scholar
  33. Lei J, Mackey MC (2011) Multistability in an age-structured model of hematopoiesis: cyclical neutropenia. J Theor Biol 270:143–153MathSciNetCrossRefzbMATHGoogle Scholar
  34. Lui G, Yang H, Wang X, Chu Y (2013) Modulation of neutrophil development and homeostasis. Curr Mol Med 13:1270–1283CrossRefGoogle Scholar
  35. Mackey MC (2001) Cell kinetic status of hematopoietic stem cells. Cell Prolif 34:71–83CrossRefGoogle Scholar
  36. Mackey MC, Aprikyan AAG, Dale DC (2003) The rate of apoptosis in post mitotic neutrophil precursors of normal and neutropenic humans. Cell Prolif 36:27–34CrossRefGoogle Scholar
  37. Maholtra V, Perry MC (2003) Models of anti-cancer therapy. Classical chemotherapy: mechanism, toxicities, and the therapeutic window. Cancer Biol Ther 2:S2–S4CrossRefGoogle Scholar
  38. Mathworks (2015) MATLAB 2015a. Mathworks, NatickGoogle Scholar
  39. Molineux G (2011) Granulocyte colony-stimulating factors. In: Lyman GH, Dale DC (eds) Hematopoietic growth factors in oncology. Springer Science + Business Media, New YorkGoogle Scholar
  40. Molineux G, Arvedson T, Foote M (2012) Twenty years of G-CSF clinical and nonclinical discoveries. Springer Basel AG, BaselCrossRefGoogle Scholar
  41. Pérez-Ruixo C, Valenzuela B, Fernández Teruel C, González-Sales M, Miguel-Lillo B, Soto-Matos A et al (2012) Population pharmacokinetics of PM00104 (Zalypsis\(\textregistered \)) in cancer patients. Cancer Chemother Pharmacol 69:15–24CrossRefGoogle Scholar
  42. Pfreundschuh M, Trümper L, Kloess M, Schmits R, Feller AC, Rudolph C et al (2004a) Two-weekly or 3-weekly chop chemotherapy with our without etoposide for the treatment of elderly patients with aggressive lymphomas: results of the NHL-B2 trial of the DSHNHL. Blood 104:634–641CrossRefGoogle Scholar
  43. Pfreundschuh M, Trümper L, Kloess M, Schmits R, Feller AC, Rudolph C et al (2004b) Two-weekly or 3-weekly chop chemotherapy with our without etoposide for the treatment of young patients with good prognosis (normal LDH) aggressive lymphomas: results of the NHL-B1 trial of the DSHNHL. Blood 104:626–633CrossRefGoogle Scholar
  44. Price TH, Chatta GS, Dale DC (1996) Effect of recombinant granulocyte colony-stimulating factor on neutrophil kinetics in normal young and elderly humans. Blood 88:335–340Google Scholar
  45. Pujo-Menjouet L, Bernard S, Mackey MC (2005) Long period oscillations in a G\(_0\) model of hematopoietic stem cells. SIAM J Appl Dyn Syst 4:312–332MathSciNetCrossRefzbMATHGoogle Scholar
  46. Quartino AL, Friberg LE, Karlsson MO (2012) A simultaneous analysis of the time-course of leukocytes and neutrophils following docetaxel administration using a semi-mechanisitic myelosuppression model. Invest New Drugs 30:833–845CrossRefGoogle Scholar
  47. Rankin SM (2010) The bone marrow: a site of neutrophil clearance. J Leukoc Biol 88:241–251CrossRefGoogle Scholar
  48. Riether C, Schürch CM, Ochsenbein AF (2015) Regulation of hematopoietic and leukemic stem cells by the immune system. Cell Death Differ 22:187–198CrossRefGoogle Scholar
  49. Rubinow S, Lebowitz J (1975) A mathematical model of neutrophil production and control in normal man. J Math Biol 1:187–225MathSciNetCrossRefzbMATHGoogle Scholar
  50. Ryan DH (2016) Examination of blood cells. In: Kaushansky K, Lichtman MA, Prchal JT, Levi MM, Press OW, Burns LJ, Caligiuri M (eds) Williams Hematology, vol 9. McGraw-Hill Companies Inc., New YorkGoogle Scholar
  51. Santillán M (2008) On the use of the Hill functions in mathematical models of gene regulatory networks. Math Model Nat Phenom 3:85–97MathSciNetCrossRefzbMATHGoogle Scholar
  52. Sarkar CA, Lowenhaupt K, Wang PJ, Horan T, Lauffenburger DA (2003) Parsing the effects of binding, signaling, and trafficking on the mitogenic potencies of granulocyte colony-stimulating factor analogues. Biotechnol Prog 19:955–964CrossRefGoogle Scholar
  53. Schirm S, Engel C, Loeffler M, Scholz M (1996) Modelling chemotherapy effects on granulopoiesis. Br J Haematol 95:616–625CrossRefGoogle Scholar
  54. Schmitz S (1988) Ein mathematisches Modell der zyklischen Haemopoese. PhD thesis, Universitat KolnGoogle Scholar
  55. Schmitz S, Franke H, Loeffler M, Wichmann HE, Diehl V (2014) Model analysis of the contrasting effects of GM-CSF and G-CSF treatment on peripheral blood neutrophils observed in three patients with childhood-onset cyclic neutropenia. BMC Syst Biol 8:1–18CrossRefGoogle Scholar
  56. Schmitz S, Loeffler M, Jones JB, Lange RD, Wichmann HE (1990) Synchrony of bone marrow proliferation and maturation as the origin of cyclic haemopoiesis. Cell Tissue Kinet 23:425–441Google Scholar
  57. Scholz M, Engel C, Loeffler M (2005) Modelling human granulopoiesis under polychemotherapy with G-CSF support. J Math Biol 50:397–439MathSciNetCrossRefzbMATHGoogle Scholar
  58. Scholz M, Schirm S, Wetzler M, Engel C, Loeffler M (2012) Pharmacokinetic and -dynamic modelling of G-CSF derivatives in humans. Theor Biol Med Model 9:1497–1502CrossRefGoogle Scholar
  59. Semerad CL, Liu F, Gregory AD, Stumpf K, Link DC (2002) G-CSF is an essential regulator of neutrophil trafficking from the bone marrow to the blood. Immunity 17:413–423CrossRefGoogle Scholar
  60. Shochat E, Rom-Kedar V, Segel LA (2007) G-CSF control of neutrophils dynamics in the blood. Bull Math Biol 69:2299–2338MathSciNetCrossRefzbMATHGoogle Scholar
  61. Shvitra D, Laugalys R, Kolesov YS (1983) Mathematical modeling of the production of white blood cells. In: Marchuk G, Belykh LN (eds) Mathematical modeling in immunology and medicine. North-Holland, Amsterdam, pp 211–223Google Scholar
  62. Smith CW (2016) Production, distribution, and fate of neutrophils. In: Kaushansky K, Lichtman MA, Prchal JT, Levi MM, Press OW, Burns LJ, Caligiuri M (eds) Williams hematology, vol 9. McGraw-Hill Companies Inc., New YorkGoogle Scholar
  63. Spiekermann K, Roesler J, Emmendoerffer A, Elsner J, Welte K (1997) Functional features of neutrophils induced by G-CSF and GM-CSF treatment: differential effects and clinical implications. Leukemia 11:466–478CrossRefGoogle Scholar
  64. Terashi K, Oka M, Ohdo S, Furukubo T, Ikeda C, Fukuda M, Soda H, Higuchi S, Kohno S (1999) Close association between clearance of recombinant human granulocyte colony-stimulating factor (G-CSF) and G-CSF receptor on neutrophils in cancer patients. Antimicrob Agents Chemother 43:21–24CrossRefGoogle Scholar
  65. Vainas O, Ariad S, Amir O, Mermershtain W, Vainstein V, Kleiman M, Inbar O, Ben-Av R, Mukherjee A, Chan S, Agur Z (2012) Personalising docetaxel and G-CSF schedules in cancer patients by a clinically validated computational model. Br J Cancer 107:814–822CrossRefGoogle Scholar
  66. Vainstein V, Ginosar Y, Shoham M, Ranmar DO, Ianovski A, Agur Z (2005) The complex effect of granulocyte colony-stimulating factor on human granulopoiesis analyzed by a new physiologically-based mathematical model. J Theor Biol 235:311–327MathSciNetCrossRefGoogle Scholar
  67. van der Graaf P, Benson N (2011) Systems pharmacology: bridging systems biology and pharmacokinetics-pharmacodynamics (PKPD) in drug discovery and development. Pharm Res 28:1460–1464CrossRefGoogle Scholar
  68. von Schulthess GK, Mazer NA (1982) Cyclic neutropenia (CN): a clue to the control of granulopoiesis. Blood 59:27–37Google Scholar
  69. Wang B, Ludden TM, Cheung EN, Schwab GG, Roskos LK (2001) Population pharmacokinetic-pharmacodynamic modeling of filgrastim (r-metHuG-CSF) in healthy volunteers. J Pharmacokinet Pharmacodyn 28:321–342CrossRefGoogle Scholar
  70. Ward AC, Monkhouse JL, Csar XF, Touw IP, Bello PB (1998) The Src-like tyrosine kinase Hck is activated by granulocyte colony-stimulating factor (G-CSF) and docks to the activated G-CSF receptor. Biochem Biophys Res Commun 251(1):117–123CrossRefGoogle Scholar
  71. Watari K, Asano S, Shirafuji N, Kodo H, Ozawa K, Takaku F, Kamachi S (1989) Serum granulocyte colony-stimulating factor levels in healthy volunteers and patients with various disorders as estimated by enzyme immunoassay. Blood 73(1):117–122Google Scholar
  72. Wichmann HE, Loffler M (1988) Mathematical modeling of cell proliferation: stem cell regulation in hemopoiesis. CRC Press, Boca RatonGoogle Scholar
  73. Wu X, Nekka F, Li J Steady-state volume of distribution of two compartmental models with simultaneous linear and saturated eliminations. Under reviewGoogle Scholar

Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  1. 1.Faculté de PharmacieUniversité de MontréalMontréalCanada
  2. 2.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA
  3. 3.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada
  4. 4.Departments of Mathematics, Physics and PhysiologyMcGill UniversityMontréalCanada

Personalised recommendations