Stochastic differential equations in NONMEM
C. W. Tornøe (1,2,3), H. Agersø (1), R. V. Overgaard (2), H. A. Nielsen (2), H. Madsen (2), E. N. Jonsson (3)
(1) Experimental Medicine, Ferring Pharmaceuticals, (2) Informatics and Mathematical Modelling, Technical University of Denmark, (3) Division of Pharmacokinetics and Drug Therapy, Uppsala University
Objectives: The objective of the present analysis is to explore the use of stochastic differential equations (SDEs) in nonlinear mixed-effects modelling. The purpose of using SDEs in population PK/PD modelling is to account for correlated residual errors due to possible model misspecification. SDEs can furthermore be used as a diagnostic tool for model appropriateness and provide a framework for pinpointing model deficiencies. The focus of the presentation will be on the implementation and application of SDEs in population PK/PD modelling using NONMEM.
Methods:The intra-individual variability in SDE models is decomposed into two types of noise, i.e. a measurement noise term representing uncorrelated error due to e.g. assay error and a system noise term accounting for model misspecification or true random physiological fluctuations. Parameter estimation in SDE models involves the use of state filtering methods such as the Extended Kalman Filter (EKF)  which has been implemented in NONMEM. In the case of no system noise, the SDE model reduces to an ordinary differential equation (ODE) model traditionally used in NONMEM.
Results:Clinical data of GnRH antagonist degarelix was used to explore the use of SDEs in population PK/PD modelling. The PK/PD of IV administered degarelix was analyzed using a three-compartment PK model and a turnover PD model with a pool compartment. The estimated system noise in the PK model was significant but small and likely due to random physiological fluctuations. For the PD model, the estimated system noise was large indicating possible model misspecifications which were attempted pinpointed.
Conclusion:: The SDE algorithm was successfully implemented in NONMEM VI and applied to clinical PK/PD data of GnRH antagonist degarelix. The obtained results illustrated that it is possible to decompose the residual error into measurement and system noise in nonlinear mixed-effects models. Identified advantages of using SDEs compared to ODEs in population PK/PD modelling are: 1) More realistic description of the observed variations and thereby improved simulation properties, 2) can be used as a diagnostic tool for model appropriateness, and 3) provide a framework for pinpointing model deficiencies.
 AH Jazwinski. Stochastic Processes and Filtering Theory. Academic Press, New York (1970).