Tutorial for Modeling Delays in Biological Systems in the <scp>NONMEM</scp> Software

Delays in biological systems are a common phenomenon. The models for delays require specialized mathematical and numerical techniques such as transit compartments, delay differential equations (DDEs), and distributed DDEs (DDDEs). Because of mathematical complexity, DDEs and particularly DDDEs are infrequently used for modeling. DDEs are supported by most pharmacometric programs. Recently, DDDEs have been implemented in NONMEM that greatly improve the applicability of this technique in pharmacokinetic and pharmacodynamic (PKPD) modeling. The objective of this tutorial is to provide examples of PKPD models with delays and demonstrate how to implement them in NONMEM. All examples provide a brief description of the biology and pharmacology underlying model equations, explain how they are coded in the NONMEM control stream, and discuss results of data analysis models were used for. NONMEM codes for all models are presented in supporting information (Data S1). The tutorial concludes with a discussion of the pros and cons of presented delay modeling techniques with guidelines for which one might be preferred given the nature of the delay, available data, and the task to be performed.