Using Stochastic Differential Equations for PK/PD Model Development
Niels Rode Kristensen
Pharmacometrics, Experimental Medicine, Novo Nordisk A/S
Objectives: The objective of this contribution is to demonstrate the benefits of using stochastic differential equations (SDEs) instead of ordinary differential equations (ODEs) for individual PK/PD model development. Using SDEs facilitates more systematic model development by allowing information about structural model uncertainty to be extracted from data in a manner impossible when using ODEs. Allowing such information to be extracted not only provides a basis for constructing tools for performing model diagnostics, but also allows model deficiencies due to time-variations in specific parameters to be pinpointed. Furthermore, using SDEs facilitates tracking of such time-variations and in combination with nonparametric methods for feature extraction this provides a basis for intelligent model improvement.
Methods: The generic model structure used is a stochastic state space model consisting of a set of SDEs describing the dynamics of the states of the system in continuous time and a set of discrete-time measurement equations describing the relationship between the states and the observations obtained. Such models have a number of advantages compared to models based on ODEs. Most important is the inclusion of a diffusion term, the primary difference between an SDE and the corresponding ODE, which, when estimating model parameters, provides a decomposition of prediction error, which allows effects of model uncertainty, e.g. due to model structure misspecifications, to be decoupled from effects of measurement error, and also facilitates tracking of time-variations in model parameters via state filtering.
Results: Simulated as well as clinical data was used to demonstrate the performance of the proposed methodology in terms of facilitating the development of an appropiate model for the absorption kinetics of a drug. Starting from an assumption of first order absorption kinetics, simulated PK data was used for proof of concept by demonstrating that the nonlinear model used to simulate the data could be re-constructed via tracking of time-variations in the absorption rate. Using the same approach, practical applicability was demonstrated using clinical data obtained following SC administration of a long-acting insulin analogue.
Conclusion: Using SDEs for individual PK/PD model development was demonstrated to facilitate systematic model development by providing tools for performing model diagnostics and for intelligent model improvement.