Krzyzanski W (2011) Interpretation of transit compartments pharmacodynamic models as lifespan based indirect response models. Society for Industrial and Applied Mathematics, Philadelphia J Pharmacokinet Pharmacodyn 39(1):55–65Īscher UMPL (1998) Computer methods for ordinary differential equations and differential-algebraic equations. Koch G, Wagner T, Plater-Zyberk C, Lahu G, Schropp J (2012) Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice treated with an anti-GM-CSF antibody. CR Biol 327(11):983–994īaccam P, Beauchemin C, Macken CA, Hayden FG, Perelson AS (2006) Kinetics of influenza A virus infection in humans. J Clin Pharmacol 44(9):991–1002īachar M, Dorfmayr A (2004) HIV treatment models with time delay. Ramakrishnan R, Cheung WK, Wacholtz MC, Minton N, Jusko WJ (2004) Pharmacokinetic and pharmacodynamic modeling of recombinant human erythropoietin after single and multiple doses in healthy volunteers. J Pharmacokinet Pharmacodyn 39(1):109–123 Krzyzanski W, Perez Ruixo JJ (2012) Lifespan based indirect response models. J Pharmacokinet Pharmacodyn 41(4):291–318 Koch G, Krzyzanski W, Perez-Ruixo JJ, Schropp J (2014) Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations. For estimation of population parameters, the EM method is more stable than FOCE regardless of the DDE solver. All DDE solvers provide accurate and precise solutions with the number of significant digits controlled by the error tolerance parameters. NONMEM control streams and excerpts from datasets are provided for all discussed examples. We evaluated the accuracy of NONMEM DDE solvers, their ability to handle stiff problems, and their performance in parameter estimation using both first-order conditional estimation (FOCE) and the expectation–maximization (EM) method. The examples include previously published DDE models such as logistic growth, tumor growth inhibition, indirect response with precursor pool, rheumatoid arthritis, and erythropoiesis-stimulating agents. The purpose of this tutorial is to introduce basic concepts underlying DDE based models and to show how they can be developed using NONMEM. Two of them are based on algorithms already applied elsewhere, while others are extensions of existing ordinary differential equations (ODEs) solvers.
Several DDE solvers have been implemented in NONMEM 7.5 for the first time. Delay differential equations (DDEs) are commonly used in pharmacometric models to describe delays present in pharmacokinetic and pharmacodynamic data analysis.