Journal of Pharmacokinetics and Pharmacodynamics

, Volume 45, Issue 5, pp 693–705 | Cite as

Mathematical analysis and drug exposure evaluation of pharmacokinetic models with endogenous production and simultaneous first-order and Michaelis–Menten elimination: the case of single dose

  • Xiaotian Wu
  • Fahima NekkaEmail author
  • Jun Li
Original Paper


Drugs with an additional endogenous source often exhibit simultaneous first-order and Michaelis–Menten elimination and are becoming quite common in pharmacokinetic modeling. In this paper, we investigate the case of single dose intravenous bolus administration for the one-compartment model. Relying on a formerly introduced transcendent function, we were able to analytically express the concentration time course of this model and provide the pharmacokinetic interpretation of its components. Using the concept of the corrected concentration, the mathematical expressions for the partial and total areas under the concentration time curve (AUC) were also given. The impact on the corrected concentration and AUC is discussed as well as the relative contribution of the exogenous part in presence of endogenous production. The present findings theoretically elucidate several pharmacokinetic issues for the considered drug compounds and provide guidance for the rational estimation of their pharmacokinetic parameters.


Pharmacokinetic model X function Endogenous production Simultaneous first-order and Michaelis–Menten elimination Area under the concentration time curve (AUC) 



This work is supported by FRQNT Fellowship (X. Wu) and the NSERC-Industrial Chair in Pharmacometrics (co-funded by InVentiv Health and Pfizer) and FRQNT Projet d’équipe led by F. Nekka. X. Wu and J. Li also thank the support from NSFC (No.11501358). We thank the referees for their careful reading and valuable comments which helped improving the quality of the paper.

Supplementary material

10928_2018_9599_MOESM1_ESM.docx (14 kb)
Electronic supplementary material 1 (DOCX 14 kb)


  1. 1.
    Derendorf H, Meibohm B (1999) Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives. Pharm Res 16(2):176–185CrossRefPubMedGoogle Scholar
  2. 2.
    Li J, Nekka F (2013) A rational quantitative approach to determine the best dosing regimen for a target therapeutic effect: a unified formalism for antibiotic evaluation. J Theor Biol 319:88–95CrossRefPubMedGoogle Scholar
  3. 3.
    Craig M, Humphries AR, Nekka F, Bélair J, Li J, Mackey MC (2015) Neutrophil dynamics during concurrent chemotherapy and G-CSF administration: mathematical modelling guides dose optimisation to minimise neutropenia. J Theor Biol 385:77–89CrossRefPubMedGoogle Scholar
  4. 4.
    Câmara De Souza D, Craig M, Cassidy T, Li J, Nekka F, Bélair J, Humphries AR (2018) Transit and lifespan in neutrophil production: implications for drug intervention. J Pharmacokinet Pharmacodyn 45(1):59–77CrossRefPubMedGoogle Scholar
  5. 5.
    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 50:101S–112SCrossRefPubMedGoogle Scholar
  6. 6.
    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 9:32CrossRefPubMedPubMedCentralGoogle Scholar
  7. 7.
    Perreault S, Burzynski J (2009) Romiplostim: a novel thrombopoiesis-stimulating agent. Am J Health Syst Pharm 66:817–824CrossRefPubMedGoogle Scholar
  8. 8.
    Platanias LC, Miller CB, Mick R, Hart RD, Ozer H, McEvilly JM, Jones RJ, Ratain MJ (1991) Treatment of chemotherapy-induced anemia with recombinant human erythropoietin in cancer patients. J Clin Oncol 9(11):2021–2026CrossRefPubMedGoogle Scholar
  9. 9.
    Woo S, Krzyzanski W, Jusko WJ (2006) Pharmacokinetic and pharmacodynamic modeling of recombinant human erythropoietin after intravenous and subcutaneous administration in rats. J Pharmacol Exp Ther 319(3):1297–1306CrossRefPubMedGoogle Scholar
  10. 10.
    Gouyette A, Kerr DJ, Kaye SB, Setanoians A, Cassidy J, Bradley C, Forrest G, Soukop M (1988) Flavone acetic acid: a nonlinear pharmacokinetic model. Cancer Chemother Pharmacol 22(2):114–119CrossRefPubMedGoogle Scholar
  11. 11.
    Lee BY, Kwon KI, Kim MS, Baek IH (2016) Michaelis-Menten elimination kinetics of etanercept, rheumatoid arthritis biologics, after intravenous and subcutaneous administration in rats. Eur J Drug Metab Pharmacokinet 41:433–439CrossRefPubMedGoogle Scholar
  12. 12.
    Wagner JG, Gyves JN, Stetson PL, Walker-Andrews SC, Wollner IS, Cochran MK, Ensminger WD (1986) Steady-state nonlinear pharmacokinetics of 5-fluorouracil during hepatic arterial and intravenous infusion in cancer patients. Cancer Res 46:1499–1506PubMedGoogle Scholar
  13. 13.
    Valodia PN, Seymour MA, McFadyen ML, Miller R, Folb PI (2000) Validation of population pharmacokinetic parameters of phenytoin using the parallel Michaelis–Menten and first-order elimination model. Ther Drug Monit 22(3):313–319CrossRefPubMedGoogle Scholar
  14. 14.
    Beal SL (1982) On the solution to the Michaelis–Menten equation. J Pharmacokin Biopharm 10:109–119CrossRefGoogle Scholar
  15. 15.
    Beal SL (1983) Computation of the explicit solution to the Michaelis–Menten equation. J Pharmacokin Biopharm 11:641–657CrossRefGoogle Scholar
  16. 16.
    Schnell S, Mendoza C (1997) Closed form solution for time-dependent enzyme kinetics. J Theor Biol 187:207–212CrossRefGoogle Scholar
  17. 17.
    Tang S, Xiao Y (2007) One-compartment model with Michaelis–Menten elimination kinetics and therapeutic window: an analytical approach. J Pharmacokinet Pharmacodyn 34:807–827CrossRefPubMedGoogle Scholar
  18. 18.
    Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5:329–359CrossRefGoogle Scholar
  19. 19.
    Wu X, Li J, Nekka F (2015) Closed form solutions and dominant elimination pathways of simultaneous first-order and Michaelis–Menten kinetics. J Pharmacokinet Pharmacodyn 42:151–161CrossRefPubMedGoogle Scholar
  20. 20.
    Foley C, Mackey MC (2009) Mathematical model for G-CSF administration after chemotherapy. J Theor Biol 257:27–44CrossRefPubMedGoogle Scholar
  21. 21.
    Wright EM (1949) The linear difference-differential equation with constant coefficients. Proc R Soc Edinburgh A62:387–393Google Scholar
  22. 22.
    Asl F, Ulsoy AG (2003) Analysis of a system of linear delay differential equations. J Dyn Syst Meas Control 125(2):215–223CrossRefGoogle Scholar
  23. 23.
    Dostalek M, Gardner I, Gurbaxani BM, Rose RH, Chetty M (2013) Pharmacokinetics, pharmacodynamics and physiologically-based pharmacokinetic modelling of monoclonal antibodies. Clin Pharmacokinet 52(2):83–124CrossRefPubMedGoogle Scholar
  24. 24.
    Kozawa S, Yukawa N, Liu J, Shimamoto A, Kakizaki E, Fujimiya T (2007) Effect of chronic ethanol administration on disposition of ethanol and its metabolites in rat. Alcohol 41(2):87–93CrossRefPubMedGoogle Scholar
  25. 25.
  26. 26.
    FDA Guidance: Guidance for Industry. Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs-General Considerations (2014)
  27. 27.
    Health Canada: Guidance Document: Conduct and Analysis of Comparative Bioavailability Studies (2012)
  28. 28.
    Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, vol 41. Mathematical surveys and monographs. AMS, ProvidenceGoogle Scholar
  29. 29.
    Aston PJ, Derks G, Agoram BM, van der Graaf PH (2014) A mathematical analysis of rebound in a target-mediated drug disposition model: I. Without feedback. J Math Biol 68(6):1453–1478CrossRefPubMedGoogle Scholar
  30. 30.
    Patsatzis DG, Maris DT, Goussis DA (2016) Asymptotic analysis of a target-mediated drug disposition model: algorithmic and traditional approaches. Bull Math Biol 78(6):1121–1161CrossRefPubMedGoogle Scholar
  31. 31.
    Peletier LA, Benson N, van der Graaf PH (2009) Impact of plasma-protein binding on receptor occupancy: an analytical description. J Theor Biol 256:253–262CrossRefPubMedGoogle Scholar
  32. 32.
    Wu X, Nekka F, Li J (2016) Steady-state volume of distribution of two-compartment models with simultaneous linear and saturated elimination. J Pharmacokinet Pharmacodyn 43(4):447–459CrossRefPubMedGoogle Scholar
  33. 33.
    Quartino AL, Karlsson MO, Lindman H, Friberg LE (2014) Characterization of endogenous G-CSF and the inverse correlation to chemotherapy-induced neutropenia in patients with breast cancer using population modeling. Pharm Res 31(12):3390–3403CrossRefPubMedGoogle Scholar
  34. 34.
    Hareng L, Hartung T (2002) Induction and regulation of endogenous granulocyte colony-stimulating factor formation. Biol Chem 383(10):1501–1517CrossRefPubMedGoogle Scholar
  35. 35.
    Roberts AW (2005) G-CSF: a key regulator of neutrophil production, but that’s not all!. Growth Factors Chur Switz 23(1):33–41CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiPeople’s Republic of China
  2. 2.Faculté de pharmacieUniversité de MontréalMontréalCanada
  3. 3.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada

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