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1.  Optimal Design in Population Kinetic Experiments by Set-Valued Methods 
The AAPS Journal  2011;13(4):495-507.
We propose a new method for optimal experimental design of population pharmacometric experiments based on global search methods using interval analysis; all variables and parameters are represented as intervals rather than real numbers. The evaluation of a specific design is based on multiple simulations and parameter estimations. The method requires no prior point estimates for the parameters, since the parameters can incorporate any level of uncertainty. In this respect, it is similar to robust optimal design. Representing sampling times and covariates like doses by intervals gives a direct way of optimizing with rigorous sampling and dose intervals that can be useful in clinical practice. Furthermore, the method works on underdetermined problems for which traditional methods typically fail.
Electronic supplementary material
The online version of this article (doi:10.1208/s12248-011-9291-8) contains supplementary material, which is available to authorized users.
PMCID: PMC3231860  PMID: 21761248
interval analysis; optimal experimental design; set-values methods
2.  A Fast Method for Testing Covariates in Population PK/PD Models 
The AAPS Journal  2011;13(3):464-472.
The development of covariate models within the population modeling program like NONMEM is generally a time-consuming and non-trivial task. In this study, a fast procedure to approximate the change in objective function values of covariate–parameter models is presented and evaluated. The proposed method is a first-order conditional estimation (FOCE)-based linear approximation of the influence of covariates on the model predictions. Simulated and real datasets were used to compare this method with the conventional nonlinear mixed effect model using both first-order (FO) and FOCE approximations. The methods were mainly assessed in terms of difference in objective function values (ΔOFV) between base and covariate models. The FOCE linearization was superior to the FO linearization and showed a high degree of concordance with corresponding nonlinear models in ΔOFV. The linear and nonlinear FOCE models provided similar coefficient estimates and identified the same covariate–parameter relations as statistically significant or non-significant for the real and simulated datasets. The time required to fit tesaglitazar and docetaxel datasets with 4 and 15 parameter–covariate relations using the linearization method was 5.1 and 0.5 min compared with 152 and 34 h, respectively, with the nonlinear models. The FOCE linearization method allows for a fast estimation of covariate–parameter relations models with good concordance with the nonlinear models. This allows a more efficient model building and may allow the utilization of model building techniques that would otherwise be too time-consuming.
PMCID: PMC3160160  PMID: 21725709
conditional estimation; covariate model building; NONMEM; population PK/PD
3.  Multinomial Logistic Functions in Markov Chain Models of Sleep Architecture: Internal and External Validation and Covariate Analysis 
The AAPS Journal  2011;13(3):445-463.
Mixed-effect Markov chain models have been recently proposed to characterize the time course of transition probabilities between sleep stages in insomniac patients. The most recent one, based on multinomial logistic functions, was used as a base to develop a final model combining the strengths of the existing ones. This final model was validated on placebo data applying also new diagnostic methods and then used for the inclusion of potential age, gender, and BMI effects. Internal validation was performed through simplified posterior predictive check (sPPC), visual predictive check (VPC) for categorical data, and new visual methods based on stochastic simulation and estimation and called visual estimation check (VEC). External validation mainly relied on the evaluation of the objective function value and sPPC. Covariate effects were identified through stepwise covariate modeling within NONMEM VI. New model features were introduced in the model, providing significant sPPC improvements. Outcomes from VPC, VEC, and external validation were generally very good. Age, gender, and BMI were found to be statistically significant covariates, but their inclusion did not improve substantially the model’s predictive performance. In summary, an improved model for sleep internal architecture has been developed and suitably validated in insomniac patients treated with placebo. Thereafter, covariate effects have been included into the final model.
Electronic supplementary material
The online version of this article (doi:10.1208/s12248-011-9287-4) contains supplementary material, which is available to authorized users.
PMCID: PMC3160167  PMID: 21691915
covariates; Markov; multinomial; sleep; validation
4.  Prediction-Corrected Visual Predictive Checks for Diagnosing Nonlinear Mixed-Effects Models 
The AAPS Journal  2011;13(2):143-151.
Informative diagnostic tools are vital to the development of useful mixed-effects models. The Visual Predictive Check (VPC) is a popular tool for evaluating the performance of population PK and PKPD models. Ideally, a VPC will diagnose both the fixed and random effects in a mixed-effects model. In many cases, this can be done by comparing different percentiles of the observed data to percentiles of simulated data, generally grouped together within bins of an independent variable. However, the diagnostic value of a VPC can be hampered by binning across a large variability in dose and/or influential covariates. VPCs can also be misleading if applied to data following adaptive designs such as dose adjustments. The prediction-corrected VPC (pcVPC) offers a solution to these problems while retaining the visual interpretation of the traditional VPC. In a pcVPC, the variability coming from binning across independent variables is removed by normalizing the observed and simulated dependent variable based on the typical population prediction for the median independent variable in the bin. The principal benefit with the pcVPC has been explored by application to both simulated and real examples of PK and PKPD models. The investigated examples demonstrate that pcVPCs have an enhanced ability to diagnose model misspecification especially with respect to random effects models in a range of situations. The pcVPC was in contrast to traditional VPCs shown to be readily applicable to data from studies with a priori and/or a posteriori dose adaptations.
Electronic supplementary material
The online version of this article (doi:10.1208/s12248-011-9255-z) contains supplementary material, which is available to authorized users.
PMCID: PMC3085712  PMID: 21302010
mixed-effects modeling; model diagnostics; pcVPC; prediction correction; VPC
5.  Modeling Subpopulations with the $MIXTURE Subroutine in NONMEM: Finding the Individual Probability of Belonging to a Subpopulation for the Use in Model Analysis and Improved Decision Making 
The AAPS Journal  2009;11(1):148-154.
In nonlinear mixed effects modeling using NONMEM, mixture models can be used for multimodal distributions of parameters. The fraction of individuals belonging to each of the subpopulations can be estimated, and the most probable subpopulation for each patient is output (MIXESTk). The objective function value (OFV) that is minimized is the sum of the OFVs for each patient (OFVi), which in turn is the sum across the k subpopulations (OFVi,k). The OFVi,k values can be used together with the total probability in the population of belonging to subpopulation k to calculate the individual probability of belonging to the subpopulation (IPk). Our objective was to explore the information gained by using IPk instead of or in addition to MIXESTk in the analysis of mixture models. Two real data sets described previously by mixture models as well as simulations were used to explore the use of IPk and the precision of individual parameter values based on IPk and MIXESTk. For both real data-based mixture models, a substantial fraction (11% and 26%) of the patients had IPk values not close to 0 or 1 (IPk between 0.25 and 0.75). Simulations of eight different scenarios showed that individual parameter estimates based on MIXEST were less precise than those based on IPk, as the root mean squared error was reduced for IPk in all scenarios. A probability estimate such as IPk provides more detailed information about each individual than the discrete MIXESTk. Individual parameter estimates based on IPk should be preferable whenever individual parameter estimates are to be used as study output or for simulations.
PMCID: PMC2664892  PMID: 19277871
mixture; NONMEM; pharmacometrics; population modeling; subpopulation

Results 1-5 (5)