Division of Biostatistics, Department of Medicine, School of Medicine, Indiana University, Indianapolis, IN, 46032, USA
Department of Biostatistics, School of Public Health, University of Michigan, Ann Arbor, MI, 48109, USA
Abstract
Background
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 noncompartment model pharmacokinetics (PK) parameters into compartment model PK parameters. This metaanalysis was performed with a multivariate nonlinear mixed model. A conditional firstorder linearization approach was developed for statistical estimation and inference.
Results
Using MDZ as an example, we showed that this approach successfully transformed 6 noncompartment 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 noncompartment PK parameters were reported in every study. If there missing noncompartment 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%.
Conclusions
This noncompartment model PK parameter transformation into compartment model metaanalysis approach possesses valid statistical inference. It can be routinely used for model based drug development.
Background
In recent decades, a new drug requires an average of 15 years and approaching a billion dollars in research and development
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 noncompartment model PK parameters. For example, area under the concentration curve (AUC) is calculated from drug plasma concentration data based on trapezoidrule
Methods
NonCompartment Model to OneCompartment Model Transformation
When a drug follows a onecompartment model of oral dose (1), the following noncompartment model PK parameters, w = (
where,
Similarly, if
NonCompartment Model to TwoCompartment Model Transformation
If a drug’s pharmacokinetics follows a twocompartment model with oral dose (5), the following noncompartment model PK parameters, w = (
If a drug’s pharmacokinetics follows a twocompartment model with IV dose (7), the following noncompartment model PK parameters, w = (
A Multivariate Nonlinear Mixed Effect Model (Model Specification)
Based on the multiple transformation equations between noncompartment model PK parameters and one or two compartment models, a multivariate nonlinear mixed effect model is established to estimate the population level PK parameters and their between study variances. Denote
Model (9) also shows that the observed noncompartment model parameters,
Study level compartment model parameter β
The joint likelihood of population/subject parameters and their covariance is shown in equation (11).
where
This multivariate nonlinear mixed model (11) is different from the conventional univariate nonlinear mixed model
A Multivariate Nonlinear Mixed Effect Model (Estimation and Inference)
As a conditional first order linearization approach provides the least biased estimate in estimating the PK parameter with comparable efficiency
Step 1: given the current estimate of variance component
Computationally, minimizing
Parameters (μ, b, β)’s estimates and their covariance are
Step 2: given the current estimate,
This
Hence, θ can be estimated through an iterative Fisher algorithm. An alternative derivation of this twostep first order linearization is through a second order Laplace’s approximation
Results
Midazolam NonCompartment Model Parameters to Compartment Model Parameters Transformation Data Analysis
After extensive literature search, 10 midazolam pharmacokinetics studies were identified, and their published noncompartment PK parameters are reported in Table
Summary of Published NonCompartment Model Midazolam Pharmacokinetics Parameters
NonCompartment PK Parameters
Reported
Missed
C_{max}
9
1
AUC
10
0
T_{max}
7
3
T_{1/2,fast}
2
8
T_{1/2,slow}
8
2
V_{d}
5
5
CL_{iv}
4
6
There are totally 10 studies available from publications. This table shows the number of reported and missed records for the sample means of noncompartment PK parameters among those 10 studies.
A multivariate nonlinear mixed effect model is fitted to these published noncompartment PK parameters to estimate their compartment model PK parameters. The NONMEM code is reported in Appendix I. In this metaanalysis, between study variances are assumed for (
Figure
Convergence plots for five twocompartment midazolam pharmacokinetics parameters. The xaxes are logtransformed PK parameters, and yaxes are the loglikelihood functions. The dots on the top represent the maximum likelihood estimates.
Convergence plots for five twocompartment midazolam pharmacokinetics parameters. The xaxes are logtransformed PK parameters, and yaxes are the loglikelihood functions. The dots on the top represent the maximum likelihood estimates.
Table
Midazolam Compartment Model Pharmacokinetics Parameter Estimates
Compartment Model PK Parameters
NonCompartment Model to Compartment Model Transformation
FixedEffect
logscale
rawscale
Between Study CV*
3.5
33.11
10%
0.68
1.97
84%
1.1
0.33

1.32
0.27

0.403
0.67
23%
WithinStudy CV**
27%
These compartment model PK parameters are estimated from reported noncompartment model PK parameters. *These are betweenstudy variances for compartment model PK parameters. **This is the withinstudy variance for all noncompartment PK parameters.
Simulation Studies
Simulation Schemes
The primary concern of this noncompartment 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 noncompartment 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 (
Simulation Evaluation Criteria
Both fixed effect and variance components were evaluated in the simulation studies. The bias was calculated as the relative bias: abs(trueest)/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 halfwidth 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 1 (All Reported and No Missing Data)
Table
Simulation Results with No Missing Data
Estimate
TRUE(logscale)
Mean
SE
RelativeBias (%)
95% CP
3.5
3.505
0.110
0.14
0.89
0.68
0.680
0.115
0.02
0.87
FixedEffect
0.403
0.397
0.112
0.75
0.89
1.1
1.097
0.088
0.24
0.93
1.32
1.322
0.053
0.15
0.99
0.09
0.083
0.045
8.05
0.97
Between Study Variance
0.09
0.078
0.053
12.8
0.93
0.09
0.085
0.045
5.33
0.96
Sigma^{2}
0.01
0.01
0.003
4.43
0.95
Simulation 2 (With Missing Data)
Table
Simulation Results with Missing Data
Estimate
TRUE(logscale)
Mean
SD
RelativeBias (%)
95% CP
V_{1}
3.5
3.494
0.129
0.17
0.92
k_{a}
0.68
0.672
0.159
1.13
0.87
FixedEffect
k_{e}
0.403
0.389
0.141
2.84
0.90
k_{12}
1.1
1.09
0.172
0.59
0.84
k_{21}
1.32
1.323
0.070
0.19
0.99
V1
0.09
0.081
0.052
9.82
0.98
Between Study Variance
ka
0.09
0.082
0.066
9.43
0.95
ke
0.09
0.087
0.055
3.60
0.97
Sigma^{2}
0.01
0.01
0.003
13.9
0.96
Conclusions
This paper proposed an important approach to transform published noncompartment model pharmacokinetics parameters into compartment model PK parameters. This metaanalysis was performed with a multivariate nonlinear mixed model. A conditional firstorder 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 noncompartment 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 followup simulation studies, we have shown that our metaanalysis multivariate nonlinear mixed model had little relative bias (<1%) in estimating compartment model PK parameters if all noncompartment PK parameters were reported in every study. If there existed missing noncompartment 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 noncompartment model PK parameters to twocompartment model with oral dose PK parameters, we think it is probably the most complicated case among published drug PK studies. One compartment models and twocompartment model with IV dose have simpler transformation function and less computational expense.
Sometimes, not all of the required noncompartment 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 noncompartment 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.
List of Abbreviations
AUC: area under the concentration curve; MDZ: Midazolam; PK: Pharmacokinetics.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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 noncompartment model transformation formulas; LL initialized the idea, and developed the model transformation schemes, confirmed the statistical theory, and wrote the paper.
Authors’ information
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.
Acknowledgements
Dr. Lang Li is supported by NIH grants, R01 GM74217. Dr. Seongho Kim is partially supported by DOE grants, DEEM0000197, and an Intramural Research Incentive Grant from the University of Louisville.
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