Ordinary differential equation approximation of gamma distributed delay model

  • Wojciech KrzyzanskiEmail author
Original Paper


In many models of pharmacodynamic systems with delays, a delay of an input is introduced by means of the convolution with the gamma distribution. An approximation of the convolution integral of bound functions based on a system of ordinary differential equations that utilizes properties of the binomial series has been introduced. The approximation converges uniformly on every compact time interval and an estimate of the approximation error has been found \(O\left( {\frac{1}{{N^{\nu } }}} \right)\) where \(N\) is the number of differential equations and \(\nu\) is the shape parameter of the gamma distribution. The accuracy of approximation has been tested on a set of input functions for which the convolution is known explicitly. For tested functions, \(N \ge 20\) has resulted in an accurate approximation, if \(\nu \ge 1\). However, if \(\nu < 1\) the error of approximation decreases slowly with increasing \(N\), and \(N > 100\) might be necessary to achieve acceptable accuracy. Finally, the approximation was applied to estimate parameters for the distributed delay model of chemotherapy-induced myelosuppression from previously published WBC count data in rats treated with 5-fluorouracil.


Binomial series Convolution Pharmacodynamics Gamma distribution Transit compartments model Chemotherapy-induced myelosuppression 


Supplementary material

10928_2018_9618_MOESM1_ESM.docx (257 kb)
Supplementary material 1 (DOCX 253 kb)


  1. 1.
    Smith H (2010) An introduction to delay differential equations with application to the life sciences. Springer, New YorkGoogle Scholar
  2. 2.
    Davis PJ (1972) Gamma function and related function. In: Abramowitz M, Stegun IA (eds) Handbook of mathematical functions. Dover Publications, New YorkGoogle Scholar
  3. 3.
    Sun YN, Jusko WJ (1998) Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics. J Pharm Sci 87:732–737CrossRefGoogle Scholar
  4. 4.
    Krzyzanski W, Perez-Ruixo JJ, Vermeulen A (2008) Basic pharmacodynamic models for agents that alter the lifespan distribution of natural cells. J Pharmacokinet Pharmacodyn 35:349–377CrossRefPubMedCentralGoogle Scholar
  5. 5.
    Savic RM, Jonker DM, Kerbusch T, Karlsson MO (2007) Implementation of a transit compartment model for describing drug absorption in pharmacokinetic studies. J Pharmacokinet Pharmacodyn 34:711–726CrossRefGoogle Scholar
  6. 6.
    De Suza DC, Craig M, Cassidy T, Li J, Nekka F, Belair J (2017) Humphries AR (2017) Transit and lifespan in neutrophil production: implication for drug intervension. J Pharmacokinet Pharmacodyn 45(1):59–77Google Scholar
  7. 7.
    Mager DE, Wyska E, Jusko WJ (2003) Diversity of mechanism based pharmacodynamic models. Drug Metab Dispos 31:510–519CrossRefGoogle Scholar
  8. 8.
    Koch G, Schropp J (2015) Distributed transit compartments for arbitrary lifespan distributions in aging populations. J Theor Biol 380:550–558CrossRefGoogle Scholar
  9. 9.
    Friberg LE, Freijs A, Sandstro M, Karlsson MO (2000) Semiphysiological model for the time course of leukocytes after varying schedules of 5-fluorouracil in rats. J Pharmacol Exp Ther 295:734–740Google Scholar
  10. 10.
    Friberg LE, Henningsson A, Maas H, Nguyen L, Karlsson MO (2002) Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J Clin Oncol 20:4713–4721CrossRefGoogle Scholar
  11. 11.
    Krzyzanski W, Hu S, Dunlavey M (2018) Evaluation of performance of distributed delay model for chemotherapy-induced myelosuppression. J Pharmacokinet Pharmacodyn 45:329–337CrossRefGoogle Scholar
  12. 12.
    Krantz SG, Parks HR (2002) A primer of real analytic functions, 2nd edn. Birkhauser, BostonCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Pharmaceutical SciencesUniversity at BuffaloBuffaloUSA

Personalised recommendations