Analytical Solution and Exposure Analysis of a Pharmacokinetic Model with Simultaneous Elimination Pathways and Endogenous Production: The Case of Multiple Dosing Administration

  • Xiaotian Wu
  • Fahima Nekka
  • Jun LiEmail author
Special Issue: Mathematics to Support Drug Discovery and Development


In this paper, a typical pharmacokinetic (PK) model is studied for the case of multiple intravenous bolus-dose administration. This model, of one-compartment structure, not only exhibits simultaneous first-order and Michaelis–Menten elimination, but also involves a constant endogenous production. For the PK characterization of the model, we have established the closed-form solution of concentrations over time, the existence and local stability of the steady state. Using analytical approaches and the concept of corrected concentration, we have shown that the area under the curve (\(\hbox {AUC}^{corr}_{ss,\tau }\)) at steady state is higher compared to that at the single dose (\(\hbox {AUC}^{corr}_{0-\infty }\)). Moreover, by splitting the dose and dosing interval into halves, we have revealed that it can result in a significant decrease in the steady-state average concentration. These model-based findings, which contrast with the current knowledge for linear PK, confirm the necessity to revisit drugs exhibiting nonlinear PK and to suggest a rational way of using mathematical analysis for the dosing regimen design.


Pharmacokinetic model Simultaneous first-order and Michaelis–Menten elimination Endogenous production Area under the curve at steady state Average concentration at steady state 

Mathematics Subject Classification

92C45 92C50 34N05 37N25 



This research is supported by NSERC-Industrial Chair in Pharmacometrics—Novartis, Pfizer and Inventiv Health Clinical and FRQNT Projet d’équipe led by F. Nekka as well as NSERC and FRQNT (F.N. and J.L.). FRQNT Fellowship and NSFC (No. 11501358) hold by X.W. are also acknowledged.


  1. Agarwal Ravil P (2000) Difference equations and inequalities. Theory, methods, and applications, 2nd edn., rev. and expanded. New YorkGoogle Scholar
  2. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert \(W\) function. Adv Comput Math 5:329–359MathSciNetzbMATHGoogle Scholar
  3. Craig M, Humphries AR, Nekka F, Bélair J, Mackey MC LJ (2015) Neutrophil dynamics during concurrent chemotherapy and G-CSF administration: mathematical modelling guides dose optimisation to minimise neutropenia. J Theor Biol 385:77–89zbMATHGoogle Scholar
  4. Craig M, Humphries AR, Mackey MC (2016) A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internalization. Bull Math Biol 78(12):2304–2357MathSciNetzbMATHGoogle Scholar
  5. Dirks NL, Meibohm B (2010) Population pharmacokinetics of therapeutic monoclonal antibodies. Clin Pharmacokinet 49(10):633–659Google Scholar
  6. FDA Guidance (2014) Guidance for industry. Bioavailability and bioequivalence studies submitted in NDAs or INDs—general considerations. Accessed 11 Aug 2019
  7. Foley C, Mackey MC (2009) Mathematical model for G-CSF administration after chemotherapy. J Theor Biol 257:27–44MathSciNetzbMATHGoogle Scholar
  8. Frymoyer A, Juul SE, Massaro AN, Bammler TK, Wu YW (2017) High-dose erythropoietin population pharmacokinetics in neonates with hypoxic-ischemic encephalopathy receiving hypothermia. Pediatr Res 81(6):865–872Google Scholar
  9. Gibaldi M, Perrier D (2007) Pharmacokinetics. Informa Healthcare USA Inc, New YorkGoogle Scholar
  10. Jin F, Krzyzanski W (2004) Pharmacokinetic model of target-mediated disposition of thrombopoietin. AAPS Pharm Sci 6(1):E9Google Scholar
  11. Keizer RJ, Huitema AD, Schellens JH, Beijnen JH (2010) Clinical pharmacokinetics of therapeutic monoclonal antibodies. Clin Pharmacokinet 49(8):493–507Google Scholar
  12. Klitgaard T, Nielsen JN, Skettrup MP, Harper A, Lange M (2009) Population pharmacokinetic model for human growth hormone in adult patients in chronic dialysis compared with healthy subjects. Growth Horm IGF Res 19(6):463–470Google Scholar
  13. Kloft C, Graefe EU, Tanswell P, Scott AM, Hofheinz R, Amelsberg A, Karlsson MO (2004) Population pharmacokinetics of sibrotuzumab, a novel therapeutic monoclonal antibody, in cancer patients. Invest New Drugs 22(1):39–52Google Scholar
  14. Kuester K, Kovar A, Lüpfert C, Brockhaus B, Kloft C (2008) Population pharmacokinetic data analysis of three phase I studies of matuzumab, a humanised anti-EGFR monoclonal antibody in clinical cancer development. Br J Cancer 98(5):900–906Google Scholar
  15. Leader B, Baca QJ, Golan DE (2008) Protein therapeutics: a summary and pharmacological classification. Nat Rev Drug Discov 7:21–39Google Scholar
  16. Li J, Nekka F (2007) A pharmacokinetic formalism explicitly integrating the patient drug compliance. J Pharmacokinet Pharmacodyn 34(1):115–139Google Scholar
  17. Mager DE, Jusko WJ (2001) General pharmacokinetic model for drugs exhibiting target-mediated drug disposition. J Pharmacokinet Pharmacodyn 28(6):507–532Google Scholar
  18. Mager DE (2006) Target-mediated drug disposition and dynamics. Biochem Pharmacol 72(1):1–10Google Scholar
  19. Mehdi B (2015) Pharmacokinetics and toxicokinetics. CRC Press, Boca RatonGoogle Scholar
  20. 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):3390–3403Google Scholar
  21. Schnell S, Mendoza C (1997) Closed form solution for time dependent enzyme kinetics. J Theor Biol 187:207–212Google Scholar
  22. Shi S (2014) Biologics: an update and challenge of their pharmacokinetics. Curr Drug Metab 15(3):271–290Google Scholar
  23. Tang S, Xiao Y (2007) One-compartment model with Michaelis–Menten elimination kinetics and therapeutic window: an analytical approach. J Pharmacokinet Pharmacodyn 34:807–827Google Scholar
  24. van der Graaf PH, Benson N, Peletier LA (2016) Topics in mathematical pharmacology. J Dyn Diff Equ 28:1337–1356MathSciNetzbMATHGoogle Scholar
  25. Wong H, Chow TW (2017) Physiologically based pharmacokinetic modeling of therapeutic proteins. J Pharm Sci 106(9):2270–2275Google Scholar
  26. Woo S, Krzyzanski W, Jusko WJ (2007) Target-mediated pharmacokinetic and pharmacodynamic model of recombinant human erythropoietin (rHuEPO). J Pharmacokinet Pharmacodyn 34(6):849–868Google Scholar
  27. 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 42:151–161Google Scholar
  28. 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):447–459Google Scholar
  29. Wu X, Nekka F, Li J (2018) 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. J Pharmacokinet Pharmacodyn 45(5):693–705Google Scholar
  30. Yu RH, Cao YX (2017) A method to determine pharmacokinetic parameters based on andante constant-rate intravenous infusion. Sci Rep 7(1):13279Google Scholar
  31. Zhao L, Shang EY, Sahajwalla CG (2012) Application of pharmacokinetics–pharmacodynamics/clinical response modeling and simulation for biologics drug development. J Pharm Sci 101(12):4367–4382Google Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiPeople’s Republic of China
  2. 2.Faculté de pharmacieUniversité de MontréalMontrealCanada
  3. 3.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada

Personalised recommendations