Use of distributed delay in PML
Michael Dunlavey(1), Shuhua Hu(1)
(1) Certara Corp.
Objectives: ODEs incorporating delay are called Delay Differential Equations (DDEs). They can be thought of as convolving an input function with a probability density function. In the case where the probability density function is a unit delta function at a delayed time, we call it "discrete delay". This is used to distinguish it from "distributed delay"(e.g. see [1]), in which the convolution is not with a delta function, but with a continuous probability density function. This can be used to model such things as absorption delay and delayed drug effect in a realistic manner.
Methods: A compartment-modeling statement was added to the PML language, and a previously existing function was extended, to incorporate distributed delay for the common case of a Gamma distribution. The Gamma distribution is useful in that it has a scale parameter (>0, corresponding to the mean delay time in our parameterization), and a shape parameter (>0). It models absorption and delayed drug effect well, and it has Exponential and Erlang distributions as special cases. If the optional shape parameter is not given, or if it exceeds a threshold, discrete delay is assumed.
Results: Predictions by the delay functions were compared against predictions obtained by superposition, with agreement within four or more decimal digits, over a variety of dosing histories and time scales. Performance appears to be comparable with multi-compartment models of absorption.
Conclusions: Further work will include extension to other distributions such as Weibull and log-normal distributions, and it will include processing to accomplish steady-state.
References:
[1] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011
