Non-compartment model to compartment model pharmacokinetics transformation meta-analysis – a multivariate nonlinear mixed model
© Li et al. 2010
Published: 28 May 2010
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© Li et al. 2010
Published: 28 May 2010
To fulfill the model based drug development, the very first step is usually a model establishment from published literatures. Pharmacokinetics model is the central piece of model based drug development. This paper proposed an important approach to transform published non-compartment model pharmacokinetics (PK) parameters into compartment model PK parameters. This meta-analysis was performed with a multivariate nonlinear mixed model. A conditional first-order linearization approach was developed for statistical estimation and inference.
Using MDZ as an example, we showed that this approach successfully transformed 6 non-compartment model PK parameters from 10 publications into 5 compartment model PK parameters. In simulation studies, we showed that this multivariate nonlinear mixed model had little relative bias (<1%) in estimating compartment model PK parameters if all non-compartment PK parameters were reported in every study. If there missing non-compartment PK parameters existed in some published literatures, the relative bias of compartment model PK parameter was still small (<3%). The 95% coverage probabilities of these PK parameter estimates were above 85%.
This non-compartment model PK parameter transformation into compartment model meta-analysis approach possesses valid statistical inference. It can be routinely used for model based drug development.
In recent decades, a new drug requires an average of 15 years and approaching a billion dollars in research and development . Unfortunately, only one in 10 drugs that enter clinical testing receives eventual FDA approval [2, 3]. Scientists have become increasingly mechanistic in their approach to drug development . The recent ability to integrate genetic mutations and altered protein expression to pharmacokinetics (PK) and pharmacodynamic (PD) models allow a deeper understanding of the mechanisms of disease and therapies that are genuinely targeted [5–8]. In 2004, the FDA released a report entitled: “Innovation or Stagnation, Challenge and Opportunity on the Critical Path to New Medical Products” . Among its six general topic areas, three of them emphasized the importance of computational modeling and bioinformatics in biomarker development and streamlining clinical trials [10, 11]. In multiple follow-up papers, clinical researchers, experimental biologists, computational biologists, and biostatisticians from both academia and industry all supported the FDA leadership in this critical path, and pointed out the challenges and opportunities of the PK/PD model based approach in drug development [13–15].
Pharmacokinetics model is the central piece of model based drug development. Almost all of the published PK data were summarized without fitting a compartment model. They are usually called non-compartment model PK parameters. For example, area under the concentration curve (AUC) is calculated from drug plasma concentration data based on trapezoid-rule ; clearance is calculated from dose and AUC; Cmax and Tmax are calculated from concentrations and their associated time points; terminal half-life is usually calculated from the last two to four sampling time-points directly; and etc. All these parameters cannot be used directly in a compartment model, and their transformation to compartment model PK parameters is essential.
Similarly, if CL iv is reported, instead of AUC, then CL IV = V × k e . These one-compartment-model and non-compartment model parameters and transformation were defined and discussed in great detail by .
Model (9) also shows that the observed non-compartment model parameters, , are independent. This is a multivariate nonlinear regression model.
where is a J×1 ( ) observed non-compartment model PK parameter vector; is a J×p indicator matrix, and X k is a J k ×p matrix indicating the corresponding transformation function; g(.) is a p×1 transformation function vector; is a study level compartment-model PK parameter vector; is a diagonal J×J covariance matrix for W, and ; is a Kp×p design matrix relating study-specific parameter β to population parameter µ, and I k is an identity matrix; and is a Kp×Kp covariance matrix for study-specific parameter β.
This multivariate nonlinear mixed model (11) is different from the conventional univariate nonlinear mixed model  structurally in the additional design matrix X in front of the nonlinear function ( i.e. transformation function g(.)). Model (11) is a meta-analysis approach, in which sample mean non-compartment model PK parameters are formulated. Among the existing nonlinear mixed model meta-analysis literatures, some dealt with the subject-level data from multiple studies [18, 19]; the others dealt with sample mean drug concentration data [20, 21]; and none of them discussed the meta-analysis on summarized PK parameters through the non-compartment model.
As a conditional first order linearization approach provides the least biased estimate in estimating the PK parameter with comparable efficiency [22, 23]), it is chosen as the estimation approach for this multivariate nonlinear mixed model. This conditional first order linearization approach was firstly introduced by Lindstrom and Bates . We revise their derivation based on our special meta-analysis multivariate nonlinear mixed model (11). This two-step estimation scheme is described as following.
Summary of Published Non-Compartment Model Midazolam Pharmacokinetics Parameters
Non-Compartment PK Parameters
A multivariate nonlinear mixed effect model is fitted to these published non-compartment PK parameters to estimate their compartment model PK parameters. The NONMEM code is reported in Appendix I. In this meta-analysis, between study variances are assumed for (V 1, k a, k e ). (k 12, k 21 ) were assumed to be the fixed effects across different studies without random effects, because only two papers published the MDZ distribution information, i.e. T 1/2,fast . All of the non-compartment model parameters were log-transformed. They were assumed to have the same within study variance in log-scale (i.e. same coefficient of variance in the raw scale). All of the compartment model PK parameters were also log-transformed, and their between study standard deviations can be interpreted as coefficient of variance in raw scale.
Midazolam Compartment Model Pharmacokinetics Parameter Estimates
Compartment Model PK Parameters
Non-Compartment Model to Compartment Model Transformation
Between Study CV*
The primary concern of this non-compartment PK parameter transformation to compartment model PK parameter is the bias of PK parameter estimates. Two simulation studies were designed to investigate this problem. In the first simulation, every non-compartment PK parameter was observed for each study. In the second simulation, the same amount of missing data as our MDZ example was assumed to be present.
In each simulation, 1000 simulated data sets were generated. Each data set had 10 studies, and each study reported either all (C max, AUC, T 1/2,slow , T 1/2,fast , V d , CL iv ) in simulation 1, or a partial amount of (C max, AUC, T 1/2,slow , T 1/2,fast , V d , CL iv ) in simulation 2. These non-compartment model PK parameters were simulated based on the two-compartment model transformation relationship (5) and (6), their meta-analysis multivariate nonlinear mixed model (9) and (10), and MDZ PK parameter estimates and variances from Table 2.
Both fixed effect and variance components were evaluated in the simulation studies. The bias was calculated as the relative bias: abs(true-est)/est; and their 95% coverage probabilities were also reported based on model based 95% confidence interval. Coverage probabilities outside of (92.93, 97.07) were highlighted. The half-width of this interval is three times the binomial stand error, which is [(95%)(5%)/1000]1/2=0.6892%. Standard error was also reported based on 1000 simulation results.
Simulation Results with No Missing Data
Between Study Variance
Simulation Results with Missing Data
Between Study Variance
This paper proposed an important approach to transform published non-compartment model pharmacokinetics parameters into compartment model PK parameters. This meta-analysis was performed with a multivariate nonlinear mixed model. A conditional first-order linearization approach was developed for statistical estimation and inference, and it was implemented in R. Using MDZ as an example, we have shown that this approach transformed 6 non-compartment model PK parameters from 10 publications into 5 compartment model PK parameters, and the conditional first order linearization approach converged to the maximum likelihood. In the follow-up simulation studies, we have shown that our meta-analysis multivariate nonlinear mixed model had little relative bias (<1%) in estimating compartment model PK parameters if all non-compartment PK parameters were reported in every study. If there existed missing non-compartment PK parameters, the relative bias of compartment model PK parameter was still small (<3%). The 95% coverage probabilities of these PK parameter estimates were usually above 85% or more. Therefore, this approach possesses adequately valid inference.
Although this paper only showed the transformation performance of non-compartment model PK parameters to two-compartment model with oral dose PK parameters, we think it is probably the most complicated case among published drug PK studies. One compartment models and two-compartment model with IV dose have simpler transformation function and less computational expense.
Sometimes, not all of the required non-compartment model PK parameters are available in the literature. Whether it is feasible to transform these data into compartment model is an interesting and important question. In this paper, MDZ was chosen as an example. Because MDZ has been a well studied probe drug, its published non-compartment model PK parameters were expected to be rich. Other rarely studied drugs may not have all these published information, and their compartment model developments from literature need further investigations.
ZW developed the theory of multivariate nonlinear mixed effect model, and run the implementation; SK developed the theory of multivariate nonlinear mixed effect model; SKQ provided the MDZ example background; JZ integrated the compartment model non-compartment model transformation formulas; LL initialized the idea, and developed the model transformation schemes, confirmed the statistical theory, and wrote the paper.
ZW is currently a Ph.D. Computer Science student in the Indiana University; SK is an assistant professor in the University of Louisville; SKQ is an assistant professor in the Indiana University; JZ is a PhD student in the University of Michigan; and LL is an association professor in the Indiana University.
area under the concentration curve
Dr. Lang Li is supported by NIH grants, R01 GM74217. Dr. Seongho Kim is partially supported by DOE grants, DE-EM0000197, and an Intramural Research Incentive Grant from the University of Louisville.
This article has been published as part of BMC Systems Biology Volume 4 Supplement 1, 2010: Proceedings of the ISIBM International Joint Conferences on Bioinformatics, Systems Biology and Intelligent Computing (IJCBS). The full contents of the supplement are available online at http://www.biomedcentral.com/1752-0509/4?issue=S1.
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.