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Ordinary differential equation approximation of gamma distributed delay model

  • Wojciech KrzyzanskiEmail author
Original Paper
  • 43 Downloads

Abstract

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.

Keywords

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

Notes

Supplementary material

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

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Pharmaceutical SciencesUniversity at BuffaloBuffaloUSA

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