2002
Paris, France
Mixture models : simulation and estimation with NONMEM
F.Hourcade-Potelleret *, C. Laveille *, M.Tod ** and R. Jochemsen *
* Institut de Recherche International Servier, 6 place des Pléiades 92415 Courbevoie cedex ; ** Hôpital Avicennes, 125 rue stalingrad 93009 Bobigny
The pharmacokinetic parameters are usually assumed to be normally or log-normally distributed in the population. If a pharmacokinetic parameter is bimodally distributed, one can use the $MIX routine in NONMEM : the $MIX record describes the probabilities of each subpopulation for a mixture population. For instance, the population is divided into two subpopulations for the catalytic activity of the cytochrome D6 : they are "fast hydroxylators" and "slow hydroxylators" (nsubpopulations = 2). In this case, the $MIX routine is able to evaluate the proportion of each subpopulation.
The purpose of the task was to test the accuracy of the $MIX routine for describing the mixture probabilities of a mixture model. Simulations followed by estimation were performed to illustrate this work.
Material and Methods
Simulation step The data set included 768 data samples from 256 subjects. The time course profile of a drug was simulated by a one-compartment model with first-order absorption. The clearance was assumed to be bimodally distributed in the population : The population total apparent clearance and the interindividual variability were 2.1 L/h and 20 % for the first subpopulation, respectively. For the second subpopulation, three cases were simulated : the population total apparent clearance were 3.5 L/h, 4.2 L/h or 6.3 L/h (ratio clearance subpopulation 1 versus subpopulation 2 equal to 1.5, 2 and 3 respectively) and the interindividual variability was 40 %. For each case, the proportion subpopulation 1 : subpopulation 2 range was 0.1 : 0.9, 0.2 : 0.8, 0.3 : 0.7, 0.4 : 0.6 and 0.5 : 0.5. As a first approach, 100 simulations were performed.
Estimation step The mixture model was implemented during the estimation step. The two estimation methods FO and FOCE with interaction were tested. The bias was calculated for the subpopulations probabilities.
Results
When using the FO estimation method, the bias was lower than ± 0.02 for the clearance ratio equal to 3 whatever the subpopulations proportion was. For the ratios equal to 1.5 or 2, the more unbalanced the subpopulations proportion was, the larger was the bias. The bias improved faster for the ratio equal to 2 compared to the ratio equal to 1.5 when equilibrating the two subpopulations probabilities ( Table 1 ).
The work is ongoing using the FOCE estimation method. This method was tested for the ratio equal to 1.5 and the subpopulations proportion 0.9 : 0.1. The subpopulations probabilities were well estimated with a bias lower than ± 0.02 ( Table 2 ).
Table 1 : FO estimation method ( N=100)
Simulation |
Subpopulation proportion (Means) ± SD Bias | ||
Ratio 1.5 |
Ratio 2 |
Ratio 3 | |
0.9 : 0.1 |
(0.78 : 0.22) ± 0.21 - 0.12 : + 0.12 |
(0.80 : 0.20) ± 0.16 - 0.10 : + 0.10 |
(0.88 : 0.12) ± 0.06 - 0.02 : + 0.02 |
0.8 : 0.2 |
(0.70 : 0.30) ± 0.21 - 0.10 : + 0.10 |
(0.74 : 0.26) ± 0.15 - 0.06 : + 0.06 |
(0.78 : 0.22) ± 0.07 - 0.02 : + 0.02 |
0.7 : 0.3 |
(0.66 : 0.34) ± 0.20 - 0.04 : + 0.04 |
(0.65 : 0.35) ± 0.14 - 0.05 : + 0.05 |
(0.69 : 0.31) ± 0.06 - 0.01 : + 0.01 |
0.6 : 0.4 |
(0.64 : 0.36) ± 0.18 + 0.04 : - 0.04 |
(0.59 : 0.41) ± 0.15 - 0.01 : + 0.01 |
(0.60 : 0.40) ± 0.08 + 0.006 : - 0.006 |
0.5 : 0.5 |
(0.57 : 0.43) ± 0.18 + 0.07 : - 0.07 |
(0.54 : 0.46) ± 0.15 + 0.04 : - 0.04 |
(0.52 : 0.48) ± 0.09 + 0.02 : - 0.02 |
Table 2 : FOCE estimation method ( N=93)*
Simulation |
Subpopulation proportion (Means) ± SD Bias |
Ratio 1.5 | |
0.9 : 0.1 |
(0.90 : 0.10) ± 0.097 - 0.003 : + 0.003 |
* : 7 non successful runs
Conclusion
Whatever the experimental conditions are, the subpopulation probabilities are likely to be well described with the FOCE estimation method. Using the FO estimation method, the accuracy of the probabilities description depends on the subpopulations proportion and the relative typical values of the bimodally distributed pharmacokinetic parameter.