Combining interoccasion variability and mixture within a MCPEM framework
Objectives: We have developed a practical parametric implementation of the expectation-maximization (EM) methodology which accurately evaluates point estimates of population parameters from pharmacokinetic (PK)/Pharmacodynamic (PD) data without linearizing the expectation step . The basic methodology intended to estimate only Population means, variances and measurement noise. Real clinical data require often considering that the underlying population distribution is a mixture of N distributions. In addition, it is common to have some model parameters within an individual changing from occasion to occasion (inter-occasion variability). We built a new practical algorithm that allows easily handling both mixture and inter-occasion variability simultaneously. The method has been implemented within the PDx-MC-PEM program.
Methods: Suppose a data set consisting of m individuals. We assume the underlying population being a mixture of N distributions. In addition, we assume the model parameters varying from occasion to occasion (k occasions). The first step consists in duplicating the original data set N times. The expanding data set has now m.N individuals. Individual 1, m+1 ,2m+1,...(N-1)m are indeed duplicates but will be associated with distribution1,2,...N; respectively. Another expansion occurs to handle inter-occasion variability, but now with respect to the model parameters. Indeed, k random effects are added to each model parameter assumed to exhibit inter-occasion variability.
The basic MCPEM algorithm is then performed on the expanded dimensional space and results in an update of both the mixture proportions, inter-occasion variance-covariance matrix and between subject population variance-covariance matrix. The same algorithm is repeated with the new updates until no change in the population characteristics occur.
Results: 100 data sets with 100 patients each were simulated assuming the one compartment IV model. A mixture of two distributions and inter-occasion on clearance (2 occasions) were assumed. The 100 data sets were then fit automatically using the pdX-MC-PEM program. Population means and variances , inter-occasion variance and estimated proportions were tabulated for each of the 100 data sets. Summary statistics indicated no significant bias for all estimated parameters and precision of the same order of magnitude as without mixture.
Conclusions: A new practical way of combining mixture and inter-occasion has been proposed and tentatively validated by use of simulation. The idea was to expand both the data set for handling the population mixture and the parameter space to take into account inter-occasion variability. This allowed the estimation of both mixture characteristics and inter-occasion variability, using only a slight modification of the MCPEM basic algorithm.
: Robert J. Bauer and Serge Guzy. Monte Carlo Parametric Expectation Maximization (MCPEM) Method for Analyzing Population Pharmacokinetic/ Pharmacodynamic (PK/PD) Data. In: D.Z. D'Argenio, ed. Advanced Methods of Pharmacokinetic and Pharmacodynamics Systems Analysis, Vol.3. Boston: Kluwer Academic Publishers (2004),pp 135-163.