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 [1]. 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 [4]. 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 [58]. In 2004, the FDA released a report entitled: “Innovation or Stagnation, Challenge and Opportunity on the Critical Path to New Medical Products” [9]. 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 followup 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 [12][1315].
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 [16]; clearance is calculated from dose and AUC; Cmax and Tmax are calculated from concentrations and their associated time points; terminal halflife is usually calculated from the last two to four sampling timepoints directly; and etc. All these parameters cannot be used directly in a compartment model, and their transformation to compartment model PK parameters is essential.
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 = (AUC, T_{max,} T_{1/2}), are necessary to recover the onecompartment model parameters, β = (k_{a,} k_{e,} V).
(1)
(2)
where, F is an assumed known bioavailability, and dose denotes the oral dose. If however, only oral clearance, CL_{po} is reported, instead of AUC, then CL _{po} = V × K_{e}. On the other hand, when dosing is through IV, only w = (AUC, T_{1/2}), are necessary to recover the one compartment (3), with β = (k_{e,} V). The transformation formulas are defined in (4).
(3)
(4)
Similarly, if CL_{iv} is reported, instead of AUC, then CL_{IV} = V × k_{e}. These onecompartmentmodel and noncompartment model parameters and transformation were defined and discussed in great detail by [16].
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 = (Vd, AUC, T_{max,} CL_{iv,} T_{1/2,slow,} T_{1/2,fast}), are necessary to recover the twocompartment model parameters, β = (k_{a,} k_{e,} V_{1,} k_{12,} k_{21}). Their transformations are defined in (6).
(5)
(6)
If a drug’s pharmacokinetics follows a twocompartment model with IV dose (7), the following noncompartment model PK parameters, w = (Vd, AUC, CL_{iv,} T_{1/2, slow,} T_{1/2, fast}), are necessary to recover the twocompartment model parameters, β = (k_{e,}V_{1,} k_{12,} k_{21}). Their transformations are defined in (8).
(7)
(8)
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 w_{jk} as the observed j th noncompartment PK parameter (j=1,…, J_{k}) from study k (k=1,..,K). Please note that not every study published all of the noncompartment parameters, hence J_{k} varies from study to study. β_{k} is the study level compartmentmodel PK parameter vector, and g_{j }(β_{k}) represents the transformation function. Because noncompartment model PK parameter,w_{jk}, is usually published in the form of a sample mean, model (9) shows that its variance is , where is the within study variance (assumed to be homogeneous across studies), and n_{k} is study k sample size
(9)
Model (9) also shows that the observed noncompartment model parameters, , are independent. This is a multivariate nonlinear regression model.
Study level compartment model parameter β_{k} is assumed to follow multivariate normal distribution (10), in which µ is the population PK parameter vector and Ω_{k} is its general covariance matrix.
. (10)
The joint likelihood of population/subject parameters and their covariance is shown in equation (11).
(11)
whereis a J×1 () observed noncompartment 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 compartmentmodel PK parameter vector; is a diagonal J×J covariance matrix for W, and ; is a Kp×p design matrix relating studyspecific parameter β to population parameter µ, and I_{k} is an identity matrix; and is a Kp×Kp covariance matrix for studyspecific parameter β.
This multivariate nonlinear mixed model (11) is different from the conventional univariate nonlinear mixed model [17] structurally in the additional design matrix X in front of the nonlinear function ( i.e. transformation function g(.)). Model (11) is a metaanalysis approach, in which sample mean noncompartment model PK parameters are formulated. Among the existing nonlinear mixed model metaanalysis literatures, some dealt with the subjectlevel data from multiple studies [18,19]; the others dealt with sample mean drug concentration data [20,21]; and none of them discussed the metaanalysis on summarized PK parameters through the noncompartment 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 [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 [24]. We revise their derivation based on our special metaanalysis multivariate nonlinear mixed model (11). This twostep estimation scheme is described as following.
Step 1: given the current estimate of variance componentand, minimize the following objective function, L_{1}, with respect to (β, µ).
. (12)
Computationally, minimizing L_{1} on (β, µ) is an iterative process. Within each iteration, a linearization is applied to Xg(β) with respect to β, and a linear mixed model (13) is fitted [24].
(13)
Parameters (μ, b, β)’s estimates and their covariance are
(14)
(15)
Step 2: given the current estimate,, minimize the following objective function, L_{2} , with respect to θ, which is the variance component parameter vector in (Ω, Σ), and it is of dimension q.
(16)
This L_{2} likelihood function is the restricted maximum likelihood for variance component estimates. The scores and the elements of information matrix for θ are defined in (17).
(17)
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 [2527].
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 1. (C_{max,} AUC, T_{1/2,slow}) were reported with high frequencies, i.e. 8 to 10 out 10 publications. T_{1/2,fast} was published only twice. Both V_{d} and CL_{iv} were published 5 to 6 times.
Table 1. Summary of Published NonCompartment Model Midazolam Pharmacokinetics Parameters
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 (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 noncompartment model parameters were logtransformed. They were assumed to have the same within study variance in logscale (i.e. same coefficient of variance in the raw scale). All of the compartment model PK parameters were also logtransformed, and their between study standard deviations can be interpreted as coefficient of variance in raw scale.
Figure 1 displays the convergence plots for all five compartment PK parameters (V_{1,} k_{a,} k_{12,} k_{21,} k_{e}). The xaxes are these PK parameters’ domain, and the yaxes are the likelihood function (13). It appears that all these PK parameter estimates reach maximum likelihood, and we don’t observe any nonidentifiable parameters.
Figure 1. 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 2 reported the PK parameter estimates. V_{1} = 33 L, k_{a} = 0.68 1/h, k_{12} = 0.33 1/h, k_{21} = 0.27 1/h, and k_{e} = 0.67 (1/h). Please notice that V_{1} has very small between study variances, CV= 10%; k_{a} has high between study variance, CV = 84%; and k_{e}’s variation is moderate, CV = 23%. On the other hand, the withinstudy variation of reported noncompartment PK parameters is moderate, CV = 27%.
Table 2. Midazolam Compartment Model Pharmacokinetics Parameter Estimates
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 (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 noncompartment model PK parameters were simulated based on the twocompartment model transformation relationship (5) and (6), their metaanalysis multivariate nonlinear mixed model (9) and (10), and MDZ PK parameter estimates and variances from Table 2.
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 3 reported the simulation results. Among fixed effects, all of the relative biases are less than 1%. (V1, k_{a,} k _{e}) had lower 95% coverage probabilities than (K_{12,} K_{21}) did, because (V_{1,} k_{a,} k _{e}) were assumed to have between study variances, but (K_{12,} K_{21}) didn’t have. Therefore, the low 95% coverage probability was probably due to the under estimated standard error. On the other hand, the biases of between study variance estimates were between 5% and 12.8%, though their 95% CP were all around 95%.
Table 3. Simulation Results with No Missing Data
Simulation 2 (With Missing Data)
Table 4 reported the simulation results. Among fixed effects, all of the relative biases can be as high as 2.84% (i.e. k_{e}). All of their 95% coverage probabilities were outside of the normal range, (92.93%, 97.07%). The low coverage of (V_{1,} k_{a,} k _{e}) was probably due to their between subject variations; and the low coverage probability for k_{12} and high coverage for k_{21} were probably due to the missing data. As in the MDZ example, T_{1/2,fast} had only 2 out of 10 papers published. On the other hand, the biases of between study variance estimates were between 3.6% and 9.8%, though their 95% CP were within (92.93%, 97.07%).
Table 4. Simulation Results with Missing Data
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.
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/17520509/4?issue=S1.
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