Christoffer W. Tornøe(1), Henrik Agersø(1), E. Niclas Jonsson(2), Henrik Madsen(3), and Henrik A. Nielsen(3)
(1)Clinical Pharmacology and Experimental Medicine, Ferring Pharmaceuticals; (2)Department of Pharmaceutical Biosciences, Uppsala University; (3)Informatics and Mathematical Modelling, Technical University of Denmark
Objectives: The aim of the present analysis was to explore the possibility of implementing differential equations in the non-linear mixed-effects library NLME using an ordinary differential equation (ODE) solver with simultaneous sensitivity analysis.
Methods: The odesolve package [1] which can handle stiff and non-stiff systems of first-order ODE’s was used in combination with NLME for parameter estimation in non-linear mixed-effects models. The gradient matrix was calculated by simultaneous solution of the system of ODE’s in Eq.(1) and the corresponding first-order parametric sensitivity equations in Eq.(2).
ÂÂÂÂÂ Eq.(1) Â dy/dt = f(y,t,p)
 Eq.(2)  dS/dt = JAC · S + df/dp
where y and t are the dependent and independent variable, respectively, f is the structural model and p is a vector of fixed-effects parameters. S is the gradient matrix dy/dp, JAC is the jacobian matrix df/dy, and df/dp is a matrix of partial derivatives. The sensitivity equations were included in order to investigate whether they increase the numerical stability and the rate of convergence of the algorithm compared with numerical calculation of the gradient matrix.
Results: The pharmacokinetic (PK) data of the anti-asthmatic drug Theophylline was used to validate the proposed method. These data were reported and analyzed in Boeckmann et al. [2] and Pinheiro et al. [3] using a one-compartment pharmacokinetic model with first-order absorption and elimination. The proposed algorithm with and without sensitivity equations was numerical stable and the parameter estimates and predictive performance were accurate and comparable with results obtained from NONMEM and the SSfol function distributed with NLME.
Conclusion: The implementation of ODE’s in the non-linear mixed-effects library NLME makes it a promising tool for population PK/PD analysis of complicated systems which cannot be solved analytically. The proposed algorithm can easily be extended to include other PK models as well as indirect response models for analysis of PD data.
References:
[1] Setzer, R.W. (2003). The odesolve package. http://cran.us.r-project.org.
[2] Boeckmann, A.J., Sheiner, L.B., Beal, S.L. (1994). NONMEM Users Guide – Part V, NONMEM Project Group, University of California, San Francisco.
[3] Pinheiro, J., Bates, D. (2000). Mixed-effects in S and S-PLUS, Springer-Verlag.
Reference: PAGE 12 (2003) Abstr 399 [www.page-meeting.org/?abstract=399]
Poster: poster