A Unified Parametric/Nonparametric Approach to PK/PD Population Modeling
R. H. Leary(1), R. Jelliffe(2), A. Schumitzky(2), and M. Van Guilder(2)
(1) UCSD, San Diego, CA and (2) Laboratory for Applied Pharmacokinetics, USC School of Medicine, Los Angeles, CA
Currently the most popular computational methods for PK/PD population modeling are based on a parametric maximum likelihood (ML) approach that assumes normality or lognormality for the underlying population distribution. In order to reduce computational requirements to reasonable levels, these direct parametric methods make approximations in the computation of the likelihood function that significantly compromise statistical consistency. We have often observed, for example, that such methods introduce artificial correlations between population parameters.
Nonparametric ML methods, such as the NPEM and NPAG programs from our laboratory, have obvious advantages for situations where unimodal parametric distributions may be unrealistic, such as multimodal populations of fast and slow metabolizers. Here it is shown that nonparametric ML methods can also be advantageously applied to the unimodal parametric case to obtain consistent parametric estimators that avoid the difficulties caused by likelihood approximations.
Nonparametric ML methods produce distribution estimators that can be interpreted as a set of direct observations of the PK/PD parameters for a finite set of virtual subjects, even though there are no such direct observations in the available data. If the nonparametric ML problem is solved exactly, i.e. a global maximum to the likelihood function is found, then the means and covariances of these virtual direct observations are in fact consistent estimators of the means and covariances of an assumed normal or lognormal parametric distribution.
We have recently combined the use of adaptive grids with a primal-dual interior-point algorithm to obtain such globally optimal nonparametric solutions with no introduced approximations. In many cases this can now be done on a single-processor PC, whereas previous nonparametric methods based on the EM algorithm often required a supercomputer to obtain the necessary accuracy. By the simple extension mentioned above, a computationally efficient and statistically consistent common method for both the parametric and nonparametric PK/PD population problems is obtained.
Examples using both simulated and real data are given to illustrate the method.