Between-subject covariances

An alternative method to treating time as a categorical variable is to treat time as a continuous variable and to model time using a low-order polynomial, thereby reducing the number of estimable parameters in the model. In this case, rather than modeling the within-subject covariance the between-subject covariance is manipulated. Subjects are treated as random effects, as are the model parameters associated with time. The within-subject covariance matrix was treated as a simple covariance structure. In this example, time was modeled as a quadratic polynomial. Also, included in the model were the interactions associated with the quadratic term for time. 

The results are presented in Table 6.8. Again, different between-subject covariance structured resulted in different results. The AIC and AICc selected the unstructured covariance (which has seven estimable covariance parameters) as the best model, whereas the BIC selected the simple covariance (which has four estimable covariance parameters) as the best model. 

At this stage, different within-subject covariance structures, along with between-subject covariances, can... 

Here 0 is a vector of mean population pharmacokinetic parameters and Q is the variance-covariance matrix of between-subject random variability. Np represents a p-dimensional multivariate normal distribution, where p is the number of parameters. It is often more useful to consider the values of the parameters for the individual to be related to the population parameters via a covariate relationship, in which case the expression may be written as... 

In this notation, g(9, zj) is used to represent a function (g), perhaps a linear combination of CO variates, that describes the expectation of the /th subjects parameter vector 9i conditional on their demographic characteristics (z,) and population parameter values (0). The variance-covariance matrix (Q) therefore describes the random variability between subjects that is not able to be explained by covariates. 

Similar to the prior of the mean population parameter values, the prior for the between-subject variance can also be selected to have a more plausible range for PK/PD systems. If we consider the coefficient of variation of between-subject variability for most PK/PD parameters as being approximately <100%, then a choice of p for the Wishart distribution that provided a 97.5th percentile value of around this level would be biologically plausible. This is not quite as straightforward as for the precision of the population mean parameter values, since the minimum size of p is indexed to the minimum dimension of the variance-covariance matrix of between-subject effects, and p affects all variance parameters equally. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix. A series of simulations have been performed from the Wishart distribution, where the mean value of the variance of between-subject effects was set at 0.2. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix (Table 5.1). 

The covariate of with/without ritonavir may deserve more consideration. The question related to the central hypothesis test of PK similarity is Does the addition of ritonavir modify the conclusion about PK similarity From a statistical perspective, the ritonavir covariate may also deserve some special attention during model building, similar to the subject population covariate. Flowever, practically, model stability (i.e., the replication stability of the final model form) decreases as more effects are estimated. In hindsight, it may be more appropriate to prespecify that the final model include an interaction term between subject population and the ritonavir covariate, and that ritonavir will influence the clearance only. This is in part because elevation of exposure of GW433908 when given with ritonavir prompted the inclusion of ritonavir in this assessment. 

Complex pharmacokinetic/pharmacodynamic (PK/PD) simulations are usually developed in a modular manner. Each component or subsystem of the overall simulation is developed one-by-one and then each component is linked to run in a continuous manner (see Figure 33.2). Simulation of clinical trials consists of a covariate model and input-output model coupled to a trial execution model (10). The covariate model defines patient-specific characteristics (e.g., age, weight, clearance, volume of distribution). The input-output model consists of all those elements that link the known inputs into the system (e.g., dose, dosing regimen, PK model, PK/PD model, covariate-PK/PD relationships, disease progression) to the outputs of the system (e.g., exposure, PD response, outcome, or survival). In a stochastic simulation, random error is introduced into the appropriate subsystems. For example, between-subject variability may be introduced among the PK parameters, like clearance. The outputs of the system are driven by the inputs... 

We present a pediatric population PK (PPK) model development example to illustrate the impact that the model development approach to scaling parameters by size can have on pediatric PPK analyses a typical pediatric study is included. It is intuitive that patient size will affect PK parameters such as clearance, apparent volume, and intercompartmental clearance and that the range of patient size in most pediatric PPK data sets is large. Thus, it is expected that in most pediatric PPK studies subject size will affect multiple PK parameters. However, because there are complex interactions between covariates and parameters in pediatric populations, there are also intrinsic pitfalls of stepwise forward covariate inclusion. Selection of significant covariates via backward elimination has appeal in nonlinear model building however, it requires knowledge of the relationship between the covariate and model parameters (linear vs. nonlinear impact) and can encounter numerical difficulties with complex models and limited volume of data often available from pediatric studies. Thus, there is a need for PK analysis of pediatric data to treat size as a special covariate. Specifically, it is important to incorporate it into the model, in a mechanistically appropriate manner, prior to evaluations of other covariates. 

A proportional error, a constant additive error, and a combination of both error models were evaluated for the residual error model. Between-subject random effects were explored on the clearance of parent drug and metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. Interoccasion random effects were explored on the clearance of the parent drug and of the metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. The joint distribution of the between-subject random effect, the interoccasion random effects, and the residual error were assumed normal with mean 0 and variance-covariance matrices O for the between-subject and interoccasion random effects, and I, for the residual error to be estimated. The FO method was used for the estimation of the parameters. 

The effect of the following covariates was investigated on the disposition parameter of nelfinavir, M8, and ritonavir, for which a between-subject variance was estimated ... 

Both between-subject and interoccasion variances were estimated on clearance of nelfinavir, absorption rate constant, and clearance of M8. The residual error with a proportional error model was modeled for nelfinavir and M8 separately. The effect of ritonavir was found to have a statistically significant impact on the clearance of M8 but not on that of nelfinavir. The apparent clearance of M8 was 3.23 L/h it decreased to 1.87 L/h when nelfinavir was coadministered with ritonavir. After univariate selection, a large number of covariates were included in the full model. According to the acceptance criteria, none of the effect on clearance of nelfinavir on... 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

Figure 6.5 Simulated time-effect data where the intercept was normally distributed with a mean of 100 and a standard deviation of 60. The effect was linear over time within an individual with a slope of 1.0. No random error was added to the model—the only predictor in the model is time and it is an exact predictor. The top plot shows the data pooled across 50 subjects. Solid line is the predicted values under the simple linear model pooled across observations. Dashed line is the 95% confidence interval. The coefficient of determination for this model was 0.02. The 95% confidence interval for the slope was —1.0, 3.0 with a point estimate of 1.00. The bottom plot shows how the data in the top plot extended to an individual. The bottom plot shows perfect correspondence between effect and time within an individual The mixed effects coefficient of determination for this data set was 1.0, as it should be. This example was meant to illustrate how the coefficient of determination using the usual linear regression formula is invalid in the mixed effects model case because it fails to account for between-subject variability and use of such a measure results in a significant underestimation of the predictive power of a covariate.

To date, genotyping has been used solely to explain the between-subject variability in clearance with the genotype treated as any other covariate. Kvist et al. (2001) first studied the role CYP 2D6 genotype plays in the clearance of nortriptyline in 20 subjects with depression and 20 healthy volunteers. CYP 2D6 genotype can be classified into four groups based on the number of functional genes ... 

Figure 18A shows the overlaid multiplicity-edited GHSQC and 60 Hz 1,1-ADEQUATE spectra of posaconazole (47). As will be noted from an inspection of the overlaid spectra, there is an overlap of the C46 and C47 resonances of the aliphatic side chain attached to the triazolone ring that can be seen more clearly in the expansion shown in Figure 18B. In contrast, when the data are subjected to GIC processing with power = 0.5, the overlap between the C46 and C47 resonances is clearly resolved (Figure 18C). In addition, the weak correlation between the C3 and C4 resonances of the tetrahydrofuryl moiety in the structure is also observed despite the fact that this correlation was not visible in the overlaid spectrum shown in A. This feature of the spectrum can be attributed to the sensitivity enhancement inherent to the covariance processing method.50...

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