The capability of Several Population-based Approach Software to Analyze Sparse Drug Plasma Concentration Data after Intravenous Bolus Injection

Indonesian J Pharm 30(4), 2019, 293-300 | DOI: 10.14499/indonesianjpharm30iss4pp293 indonesianjpharm.farmasi.ugm.ac.id Copyright © 2019 THE AUTHOR(S). This article is distributed under a Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) The capability of Several Population-based Approach Software to Analyze Sparse Drug Plasma Concentration Data after Intravenous Bolus Injection


INTRODUCTION
In 1972, Sheiner and colleague introduced the so-called nonlinear mixed effect model, starting the era of the population-based modeling approach. This approach assumes, a particular parameter or variable, for example, the rate constant of elimination, K, is considered determined by a population or fixed effect value and inter-individual variability, resulting in a different K parameter value in each subject. This approach directly focuses on the population data, allowing analyses of the sparse-data commonly faced in clinical studies involving patients (Sheiner et al., 1972). This approach also allows the correlation of a particular parameter such as clearance (CL), distribution volume (Vd) or elimination rate constant (K) to specific covariates such as sex, age, body weight, or serum creatinine in a quantitative manner (Jonsson and Karlsson, 1998;Wählby et al., 2001). NONMEM is the first (Sheiner et al., 1972) and the gold standard (Frame, 2006;Keizer et al., 2013) software in populationbased approach, allowing quantitative analyses of the clinical data such as in a therapeutic drug 294 Volume 30 Issue 4 (2019) monitoring (TDM) (Shaker et al., 2013). NONMEM provides fast, robust, accurate, and precise computation (Plan et al., 2012). The first-order conditional estimation (FOCE) with interaction is a commonly used estimation method in NONMEM (Wang, 2007).
Several free or free-for-academic alternative tools for population modeling are available. Monolix is an example of the free-for-academic software, developed since 2003. The computation is based on SAEM (Stochastic Approximation Effect Model) algorithm using MATLAB library engine (Chan et al., 2011) developed based on a thorough statistical theory, providing a fast and efficient calculation (Lavielle and Mentré, 2007). Monolix, which is currently maintained by Lixoft company, provides an excellent graphical user interface, facilitating a more practical application (Lavielle, 2014).
Another free tool in population pharmacokinetic-pharmacodynamic analyses is PKBUGS running under WinBUGS environment (Lunn et al., 2002). WinBUGS is a general Bayesian modeling framework that can be used to analyze several different purposes and applications, using Markov chain Monte Carlo (MCMC) techniques. Bayesian inference Using Gibbs Sampling (BUGS) project has been initiated by MRC Biostatistics Unit, at University of Cambridge, in collaboration with Imperial College School of Medicine at St Mary's, London (Lunn et al., 2002;Ntzoufras, 2011).
Recently, we have reported the capability of a population-based approach to analyze limited data in the per-oral administration scenario (Nugroho et al., 2017). However, it is not yet clear to what extent of sparse-condition can be adequately described, mainly using several different administration routes. This research was aimed to analyze very sparse Cp-time data after an intravenous bolus drug administration and to estimate the minimum number of Cp-time data required for an adequate analysis.

Preparation of Data simulation
We simulated the data of Cp versus time after intravenous-bolus administration of 50mg of a hypothetical compound using Microsoft ® Excel ® 365 (running on Windows 10 machine) (Liengme, 2015) based on the one-compartment open model with intravenous bolus injection for Cp (Welling, 1986) as presented in Equation 1.

= . − .……………………………………(1)
This step resulted in the creation of the richdata of Cp, i.e., 48 data point of 6 subjects at eight time points post dosing (at 0.25; 0.5; 1; 2; 3; 4; 6 and 8h). Each individual-parameter of K and Vd was randomly selected based on the mean and standard of deviation of 0.3±0.1/h and 30±10L (Figure 1 A). Furthermore, the sparse-data were originated based on a random selection of the rich-data by choosing 2 and 1 data/datum per subject. This step resulted in 12 data points and 6 data points from 6 subjects (Figure 1 B and C).

Pi = θ. exp (ƞi) ………………………………………….(2)
Term θ is the population value, or the fixed effect parameter of P. Pi is the individual estimate value, and ηi is the inter-individual variation, assumed to be independently and normally distributed with mean zero and variance ω 2 . The interindividual variability was applied for K and Vd. The residual error is described by the additive error model (equation 3).
Term Fpij is the prediction of the j th evaluated functions (Cp). Foij is the measured value of the evaluated function (Cp), and ε represents the residual deviation of the predicted from the observed value and is assumed to be independent and distributed with mean zero and variance σ 2 . The analysis of the population parameters θ, ω 2 , and σ 2 was performed using the SAEM algorithm (Monolix), MCMC (WinBUGS-PKBUGS), and the first-order conditional estimation (FOCE) with interaction method (NONMEM).
In Monolix, analyses were performed using the structural one compartment intravenous bolus model with K and Vd parameters provided in Monolix library. No covariate was applied while the covariance implemented the default diagonal pattern.

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Data were assumed to follow a log-normal distribution. The calculation was performed using 0.3/h and 30L as initial estimates of K and Vd, respectively. The initial estimates of the random error (eta) were 0.1, and 0.1 for K and Vd, respectively. Additive error model was calculated, with the initial values of sigma of 0.001.
Modeling in NONMEM was performed using the provided one-compartment oral model (ADVAN 1 trans 1) with K and Vd parameters provided in NONMEM PK library. No covariate was applied while the covariance implemented the default diagonal pattern. The calculation was performed using 0.3/h and 30L as initial estimates of K and Vd, respectively. The initial estimates of the random error (eta) were 0.1 and 0.1 for both K and Vd. Additional residual error model was used, with the initial values of 0.1.
Analyses in WinBUGS were performed using the structural one compartment intravenous bolus model with clearance (Cl) and Vd parameters provided in PKBUGS add-in. No covariate was applied. The calculation was performed using 15 L/h and 30L as initial estimates of Cl and Vd, respectively. The initial estimates of the random error (eta) were 10% for both Cl and Vd. Table I. The comparison of the individual parameter of K and Vd of the rich-data, the sparse-data with 12 data-points (2 data per subject) and the sparse-data with 6 data-points (one datum per subject) estimated by Monolix, NONMEM, and WinBUGS-PKBUGS (data are presented as mean ± standard of deviation). 297 (Miller, 2012), the comparison of the mean values of K and Vd of the sparse-data to the rich-data values were performed based on the paired t-test method using Microsoft ® Excel ® 365.

RESULTS AND DISCUSSION
Although each population-based modeling software package has a different of complex calculation algorithms, the goodness-of-fit evaluations demonstrate that all the packages, i.e., Monolix, NONMEM, and WinBUGS-PKBUGS properly analyze the rich-data as well as the sparse-data with 12 or even 6 data-points. In all cases, a proper population fitting of the Cp data is indicated by the absence of a specific pattern or shape such as sigmoid, or shoe shapes. Moreover, the individual prediction indicates an ideal situation, where most of the DV and individual-prediction of Cp coincide on the line of identity. The analyses in Monolix, NONMEM and WinBUGS-PKBUGS, are presented in Figure 2 (the rich-data of Cp), Figure 3 (the sparse -12 data points), and Figure 4 (the sparse -6 data points). Furthermore, we analyzed the individual pharmacokinetic parameter values of K and Vd of the sparse-data to the respective reference K and Vd values used to simulate the Cp data, based on a paired t-test (Table I). In all comparisons, the K and Vd values estimated by using Monolix, NONMEM and WinBUGS-PKBUGS of the sparse-data with 12 data-points are similar to the respective reference values (p>0.05). Moreover, although the fitting analyses are adequate with all software packages, the estimated K and Vd values of the sparse-data with 6 data-points are significantly different from the reference values (p<0.05). The average percentage ratios of K values of the sparse 6 data-points to the reference values are in the range of 188% (Monolix), 184% (NONMEM) and 161% (WinBUGS-PKBUGS). Similarly, such percentage ratios of Vd values are in the range of 56% (Monolix), 69% (NONMEM), and 62% (WinBUGS-PKBUGS).
Those facts demonstrate that sparse-data obtained with a limited number of subjects (6) and a limited number of samples (2) per subject, can be appropriately fitted in all cases. In agreement with our previous report with the sparse per-oral Cp data (Nugroho et al., 2017), the conditions again highlight the power of population modeling to analyze the sparse-data. This finding is important, 299 concerning the possibility of an accurate analysis with minimal data obtained in clinical studies involving patients such as in the case of a therapeutic drug monitoring of certain drugs. In such instances, sparse-data are collected from patients (Parke and Charles, 1998;Shaker et al., 2013). The conventional approach with a two-stage approach cannot analyze such conditions as this classical method requires a rich-data situation (Nugroho et al., 2017).
We could consider that, with the case of an intravenous bolus injection of a compound following a one-compartmental open model, the minimum data-points of population modeling analyses is 12. However, we should realize that it could be related to the less complexity of the model, with only 2 model parameters (K and Vd), and without the involvement of any covariate for the analyses. In modeling cases involving more parameters, with or without any covariates, the calculation is more complicated (Mould and Upton, 2013), and the minimum number of samples might be different. It is important to find such guidance in estimating the minimum samples required for those cases, including per-oral administration of a compound following a multi-compartment open model. This information will be helpful in the preparation of an in vivo studies in patients. The appropriate scenarios can be proposed to ascertain a valid analysis of the limited data Furthermore, the fact that Monolix, NONMEM, and WinBUGS-PKBUGS demonstrated similar power in population modeling is also essential, especially for the future of the extensive application of population-modeling in Indonesia and other developing countries. It is related to the limited budget allocation support in most institutions to buy an annual and relatively expensive license cost of the commercial software package. Monolix is free for any educational use, while WinBUGS-PKBUGS is an entirely freesoftware package for everyone willing to perform a population-based approach based on Bayesian Statistics implementing an MCMC method (Lunn et al., 2002).

CONCLUSION
Population-based approach appropriately describes the sparse-data in an intravenous bolus injection scenarios. Such capability is essential to achieve the ideal therapeutic outcome based on therapeutic drug monitoring. Monolix (free-foracademic) and WinBUGS-PKBUGS (completely free) software packages have comparable performances to NONMEM, the gold standard of population based software. The possibility to use a completely free software package can facilitate a more extensive application of population modeling to facilitate an optimum therapeutic outcome in Indonesia.