Population modeling nonlinear mixed effects

Nonlinear mixed effects models consist of two components the structural model (which may or may not contain covariates) and the statistical or variance model. The structural model describes the mean response for the population. Similar to a linear mixed effects model, nonlinear mixed effects models can be developed using a hierarchical approach. Data consist of an independent sample of n-subjects with the ith subject having -observations measured at time points t i, t 2, . t n . Let Y be the vector of observations, Y = Y1 1, Yi,2,. ..Ynjl,Yn,2,. ..Yn,ni)T and let s... 

Pharmacokinetic/pharmacodynamic model using nonlinear, mixed-effects model in two compartment, best described time course of concentration strong correlation with creatinine clearance predicted concentration at the efi ect site and in reduction of heart rate during atrial fibrillation using population kinetic approach... 

Various methods are available to estimate population parameters, but today the nonlinear mixed effects modeling approach is the most common one employed. Population analyses have been performed for mAbs such as basiliximab, daclizu-mab and trastuzumab, as well as several others in development, including clenolixi-mab and sibrotuzumab. Population pharmacokinetic models comprise three submodels the structural the statistical and covariate submodels (Fig. 3.13). Their development and impact for mAbs will be discussed in the following section. 

Estimation of nonlinear mixed effects models has been implemented in a number of software packages and includes different estimation methods [12]. As NONMEM is the most commonly used software to estimate population parameters this program is base for the following description. 

Nonlinear mixed-effects modeling methods as applied to pharmacokinetic-dynamic data are operational tools able to perform population analyses [461]. In the basic formulation of the model, it is recognized that the overall variability in the measured response in a sample of individuals, which cannot be explained by the pharmacokinetic-dynamic model, reflects both interindividual dispersion in kinetics and residual variation, the latter including intraindividual variability and measurement error. The observed response of an individual within the framework of a population nonlinear mixed-effects regression model can be described as... 

Beyond pharmacokinetics and pharmacodynamics, population modeling and parameter estimation are applications of a statistical model that has general validity, the nonlinear mixed effects model. The model has wide applicability in all areas, in the biomedical science and elsewhere, where a parametric functional relationship between some input and some response is studied and where random variability across individuals is of concern [458]. 

The population pharmacokinetic aproach assesses the impact of various covariates on the pharmacokinetic of a drug. Nonlinear mixed effects modeling may be used to model the relationship between various covariates and pharmacokinetic parameters. Age or age group may be one of the covariates. This type of approach has its advantages as it involves assessment of the effect of age on the pharmacokinetics in the target population. 

The NONMEM (nonlinear mixed-effects modeling) software (Beal et al. 1992), mostly used in population pharmacokinetics, was developed at the University of California and is presently distributed by Globomax. For data management, post processing and diagnostic plots, the software S-plus (Mathsoft) is frequently used. 

The non-linear mixed effects model is the most widely used method and has proven to be very useful for continuous measures of drug effect, categorical response data, and survival-type data. The nonlinear mixed-effects modeling software (NONMEM) introduced by Sheiner and Beal is one of the most commonly used programs for population analysis. A detailed review of software for performing population PK/PD analysis is available. ... 

Steimer, J.L. Mallet, A. Golmard, J.L. Boisvieux, J.F. Alternative approaches to estimation of population pharmacokinetic parameters comparison with nonlinear mixed-effect model. Drug Metab. Rev. 1984,15 (1-2), 265-292. 

Mentre, F. Gomeni, R. A two-step iterative algorithm for estimation in nonlinear mixed-effect models with an evaluation on population pharmacokinetics. J. Biopharm. Stat. 1995, 5 (2), 141-158. 

First-Order (NONMEM) Method. The first nonlinear mixed-effects modeling program introduced for the analysis of large pharmacokinetic data was NONMEM, developed by Beal and Sheiner. In the NONMEM program, linearization of the model in the random effects is effected by using the first-order Taylor series expansion with respect to the random effect variables r], and Cy. This software is the only program in which this type of linearization is used. The jth measurement in the ith subject of the population can be obtained from a variant of Eq. (5) as follows ... 

The first attempt at estimating interindividual PK variability without neglecting the difficulties (data imbalance, sparse data, subject-specific dosing history, etc.) associated with data from patients undergoing drug therapy was made by Sheiner and co-workers (44) using the nonlinear mixed-effects model approach. The vector 6 of population characteristics is composed of aU quantities of the first two moments of the distribution of the parameters the mean values (fixed effects) and the elements of the variance-covariance matrix that characterize random effects (19, 20, 45-47). 

S. Retout and E. Mentre, Eurther developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. J Biop harm Stat 13 209-227 (2003). 

Data structure analysis is the examination of the raw data for hidden structure, outliers, or leverage observations. This is repeated during the exploratory modeling (and nonlinear mixed effects modeling) steps using case deletion diagnostics (20). This type of analysis is important since outliers or leverage observations may occur in a population PM data set. It is equally important for the reduction of the covariate vector. 

Step 1. Determine the basic PM model for the characterization of population phar-macometrics using nonlinear mixed effects modeling. 

Step 4. With the appropriate pharmacostatistical models, perform nonlinear mixed effects modeling to develop a PM model using covariates retained in step 3 with a covariate selection level of p < 0.005. Backward elimination for covariate selection should be applied to each of the 100 bootstrap samples. The covariates found to be important in explaining variability in the parameter of interest should be used to build the hnal (irreducible) population model. The choice of p < 0.005 is to indirectly take the multicomparisons that would be made into account. 

The approach involves a semimechanistic or mechanistic model that describes the joint probability of the time of remedication and the pain relief score (which is related to plasma drug concentrations). This joint probability can be written as the product of the conditional probability of the time of remedication, given the level of pain relief and the probability of the pain relief score. First, a population pharmacokinetic (PK) model is developed using the nonlinear mixed effects modeling approach (16-19) (see also Chapters 10 and 14 of this book). With this approach both population (average) and random (inter- and intraindividual) effects parameters are estimated. When the PK model is linked to an effect (pharmacodynamic (PD) model), the effect site concentration (C ) as defined by Sheiner et al. (20) can be obtained. The effect site concentration is useful in linking dose to pain relief and subsequently to the decision to remedicate. 

The aim of this chapter is to equip the pharmacometrician with sufficient theory and application to confidently approach the PK/PD-based analysis of count data and thus derive the maximum return on investment from clinical study data. Section 27.2 provides a motivating example and Section 27.3 presents relevant definitions and theory. Section 27.4 applies the theory to the example and introduces diagnostics methods. Throughout the chapter, the focus is on population approaches using nonlinear mixed effects models. Code segments of NONMEM control files are presented in the appendix. Mixed effects analysis methodology is described in detail in Chapter 4 of this text. 

This chapter endeavors to show that a population PK/PD approach to the analysis of count data can be a valuable addition to the pharmacometrician s toolkit. Nonlinear mixed effects modeling does not need to be relegated to the analysis of continuously valued variables only. The opportunity to integrate disease progression, subject level covariates, and exposure-response models in the analysis of count data provides an important foundation for understanding and quantifying drug effect. Such parametric models are invaluable as input into clinical trial and development path simulation projects. 

With the assumed population PK parameters, the PK model, and the exposure-MAE relationship, a clinical trial simulation was performed in S-Plus (Insightful Corporation, Seattle, WA). Two thousand four hundred profiles were simulated for each design and analyzed in S-Plus. Dose proportionality was estimated using the power model (26) and mixed effects modeling in S-Plus. Population PK parameter estimates were obtained using nonlinear mixed effects modeling in S-Plus. 

Nonlinear mixed effects models are similar to linear mixed effects models with the difference being that the function under consideration f(x, 0) is nonlinear in the model parameters 0. Population pharmacokinetics (PopPK) is the study of pharmacokinetics in the population of interest and instead of modeling data from each individual separately, data from all individuals are modeled simultaneously. To account for the different levels of variability (between-subject, within-subject, interoccasion, residual, etc.), nonlinear mixed effects models are used. For the remainder of the chapter, the term PopPK will be used synonymously with nonlinear mixed effects models, even though the latter covers a richer class of models and data types. Along with PopPK is population pharmacodynamics (PopPD), which is the study of a drug s effect in the population of interest. Often PopPK and PopPD are combined into a singular PopPK-PD analysis. 

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