Non-Linear Mixed Effects Models with Stochastic Differential Equations – Implementation of an Estimation Algorithm

R. V. Overgaard(1,2), E. N. Jonsson(3), C. W. Tornøe(4), H. Madsen(2)

(1) Pharmacometrics, Experimental Medicine, Novo Nordisk A/S, Denmark; (2) Informatics and Mathematical Modelling, Technical University of Denmark; (3) Department of Pharmaceutical Biosciences Uppsala University; (4) Experimental Medicine, Ferring Pharmaceuticals

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Objectives: Stochastic Differential Equations (SDEs) could potentially aid some parts of PK/PD modelling by better parameter estimates, as a diagnostic tool, to pinpoint model deficiencies, by incorporating true variations in the parameters, etc. These models offer a general intra-individual error structure, where the residuals are decomposed into dynamical noise from the SDEs and uncorrelated measurement noise. The focus of the present study is on two fundamental issues concerning the implementation of SDEs in non-linear mixed effects models. The first is how the likelihood function of non-linear mixed-effects models with SDEs can be approximated to facilitate estimation in these models. The second focus concerns identifiability: Can the inter-individual variability, the measurement- and the dynamical noise be separated?

Methods: The likelihood function was approximated by combining the First Order Conditional Estimation (FOCE) method used in non-linear mixed-effects models, with the Extended Kalman Filter (EKF) [1] used for SDEs. This aproximation was implemented in MATLAB for a non-linear mixed-effects model with SDEs corresponding to a one-compartment model. Several simulations of- and successive estimations with this model have been used to test the estimates produced by the proposed approximation of the likelihood function.

Results: Simulations confirm that the estimated dynamical noise is small when none is used in the simulations, such that only few type I errors are likely to occur. Other simulation studies demonstrate that the algorithm is able to detect a significant amount of dynamical noise when it is present in data (no type II errors), and that higher levels of dynamical noise do not produce higher estimates of the measurement noise or the inter-individual variability.

Conclusion: A novel approximation of the likelihood function was presented for non-linear mixed-effects models based on SDEs. It was confirmed that inter-individual variability, measurement- and dynamical noise can be separated, such that these models can be treated meaningfully.

Reference
[1] A.H Jazwinski. Stochastic Processes and Filtering Theory. Academic Press, New York (1970).

Reference: PAGE 13 () Abstr 543 [www.page-meeting.org/?abstract=543]

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