Stephan Benay and Athanassios Iliadis
Dpt. of Pharmacokinetics, Aix-Marseille University, Inserm, CRO2 U 911, 13385 Marseille, France.
Objectives: Even for the simplest pharmacokinetic model, the state variable is a nonlinear function with respect to the model parameters. For this reason, given observations, parameter values cannot be directly computed, but must be estimated by using iterative optimization algorithms. Our proposal is to obtain linear expressions of the state variables for the common pharmacokinetic models, allowing computation of parameter values without using iterative optimization algorithms.
Methods: The one- and two-compartment models with infusion have been transformed from their initial form in continuous time (set of first-order linear differential equations) to a new form in discrete-time (linear transfer function between inputs and states). The transform results in a polynomial form of the transfer function called ARX (Auto-Regression with eXtra inputs) involving a set of new parameters obtained from the parameters involved in the compartmental configuration. In the discrete-time ARX model, the state variables are linear with respect to the parameters. Therefore, the parameter estimation has been performed straightforwardly without requiring an iterative algorithm. The ARX model parameters have then been estimated using a least squares estimator [1], either on the whole set of data, or recursively by incorporating new data to the set as it becomes available. The ARX and the compartmental parameters are linked by simple algebraic relationships.
Results: The method was applied on real pharmacokinetic data and allowed to successfully estimate parameters of one compartment (fotemustine) and two-compartment (mitoxantrone) models without using any optimization algorithm. Additionally, data was simulated using a one-compartment model with Michaelis-Menten elimination. Using the ARX recursive least squares estimator, the model with linear elimination revealed able to track the time-varying elimination, and correlate it with the drug concentration to discover the Michaelis-Menten nonlinear relationship.
Conclusions: The approach allows estimation of the system parameters through direct, non-iterative calculation. It also has the potential to track apparent time-varying parameters and detect nonlinearities in the process. These kinds of discrete-time, linear-in-parameters models should be proposed to be applied in the population approaches for mixed-effects modeling or in the optimization of experimental designs.
References:
[1] Ljung L, System Identification: Theory for the User, 2nd edition, p. 609. 1999. Prentice Hall, New Jersey.
Reference: PAGE 24 (2015) Abstr 3594 [www.page-meeting.org/?abstract=3594]
Poster: Methodology - Estimation Methods