Ordinary differential equation approximation of gamma distributed delay model

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 [Formula: see text] where [Formula: see text] is the number of differential equations and [Formula: see text] 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, [Formula: see text] has resulted in an accurate approximation, if [Formula: see text]. However, if [Formula: see text] the error of approximation decreases slowly with increasing [Formula: see text], and [Formula: see text] 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; Chemotherapy-induced myelosuppression; Convolution; Gamma distribution; Pharmacodynamics; Transit compartments model.