The Nonlinear Mixed Effects Model

For simplicity, the set of estimable regression parameters (P, 0 shall be denoted as 0. 

As a pharmacokinetic example, for the 1-compartment model after single dose intravenous administration, Y would be drug concentration, x would consist of dose (D) and sample time (t), and p would consist of clearance (CL) and volume of distribution (V) 

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 

If clearance and volume of distribution were treated as correlated random effects, then co l and toy denote the between-subject variability for clearance and volume of distribution, respectively, and 

Under all these models, the generic residuals are assumed to be independent, have zero mean, and constant variance t2. Collectively the set of all residual variance components, a2, is referred to as the residual variance matrix (X), the elements of which are not necessarily independent, i.e., the residual variance components can be correlated, which is referred to as autocorrelation. It is generally assumed that and r are independent, but this condition may be relaxed for residual error models with proportional terms (referred to as an -q — s interaction, which will be discussed later). 

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Pinheiro JC, Bates DM. Approximations to the loglikelihood function in the nonlinear mixed effects model. / Comput Graphical Stat, 1995 4 12-35. 

There are two common methods for obtaining estimates of the fixed effects (the mean) and the variability the two-stage approach and the nonlinear, mixed-effects modeling approach. The two-stage approach involves multiple measurements on each subject. The nonlinear, mixed-effects model can be used in situations where extensive measurements cannot or will not be made on all or any of the subjects. 

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. 

The models are built similar to the descriptive mechanism-based PD models. Most of them are also estimated by the nonlinear mixed effects modeling approach considering interindividual and residual variability. In addition, covariates influencing the disease progression can also be investigated. 

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]. 

Davidian, M. and Gallant, R., The nonlinear mixed effect model with a smooth random effects density, Biometrika, Vol. 80, 1993, pp. 475-488. 

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. ... 

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). 

Most of the nonlinear mixed effects modeling methods estimate the parameters by the maximum likelihood approach. The probability of the data under the model is written as a function of the model parameters, and parameter estimates are chosen to maximize this probability. This amounts to asserting that the best parameter estimates are those that render the observed data more probable than they would be under any other set of parameters. 

M. Davidian and A. R. Gallant, The nonlinear mixed effects model with a smooth random effects density. Institute of Statistics Mimeo Series No. 2206, North Carolina State University, Raleigh, NC, 1992. 

The knowledge discovery basis of PM modeling permits the generation of hypotheses from the relationship discovered during data structure analysis. These relationships can be tested in the nonlinear mixed effects modeling step. It can... 

In the subsequent sections we present the nonlinear mixed effects model approach for analyzing analgesic data and apply it to simulated analgesic study data. 

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. 

In characterizing hypnograms using the nonlinear mixed effects modeling approach, it is important to test for correlations between r values of one transition model and those from another model using individual estimates of r values. Correlations detected should be accounted for in the model. Correlations with correlation coefficient (r) > 0.75 are termed high correlations and correlations with r values between 0.5 and 0.75 are moderate correlations (25). Not accounting for such correlations may yield parameter estimates with poor precision. 

Data were analyzed using the nonlinear mixed effects model software program NONMEM (Version 5 level 1.1 double precision (32)). Nelfinavir and M8 were fitted simultaneously. The molecular weight of nelfinavir and M8 is comparable with a ratio of M8 to nelhnavir of 1.028. Therefore, the concentrations were not corrected and are expressed in nanogram per milliliter. 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

Step 4 may be redundant if the covariates were tested directly in the nonlinear mixed effects model. If the covariates were screened using some external method, e.g., regression models, then these covariates are included in the model in a forward stepwise manner. Improvement in the goodness of fit in the model is tested using either the LRT or T-test. In addition, reduction in parameter variability is expected as well. Further discussion of this topic will be made later in the chapter. 

To obtain initial estimates, an Emax model was fit to the data set in a na ive-pooled manner, which does not take into account the within-subject correlations and assumes each observation comes from a unique individual. The final estimates from this nonlinear model, 84% maximal inhibition and 0.6 ng/mL as the IC50, were used as the initial values in the nonlinear mixed effects model. The additive variance component and between-subject variability (BSV) on Emax was modeled using an additive error models with initial values equal to 10%. BSV in IC50 was modeled using an exponential error model with an initial estimate of 10%. The model minimized successfully with R-matrix singularity and an objective function value (OFV) of 648.217. The standard deviation (square root of the variance component) associated with IC50 was 6.66E-5 ng/mL and was the likely source of the... 

Steimer, J.-L., Mallet, A., Golmard, J.-L., and Boisvieux, J.-F. Alternative approachs to estimation of population pharmacokinetic parameters Comparison with the nonlinear mixed-effect model. Drug Metabolism Reviews 1984 15 265-302. 

Compliance model Compliance as an outcome modeled as a response or utilized as a covariate expressed in the nonlinear mixed effect model Indicator variable Compliance as a quantal (i.e., 0 = compliant ... 

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