Pharmacokinetic-pharmacodynamic model parameters

The advantages of the one-stage analysis are that interindividual variability of the parameters can be estimated, random residual error can be estimated, covariates can be included in the model, parameters for individuals can be estimated, and pharmacokinetic-pharmacodynamic models can be... 

Dose-response models describe a cause-effect relationship. There are a wide range of mathematical models that have been used for this purpose. The complexity of a dose-response model can range from a simple one-parameter equation to complex multicompartment pharmacokinetic/pharmacodynamic models. Many dose-response models, including most cancer risk assessment models, are population models that predict the frequency of a disease in a population. Such dose-response models typically employ one or more frequency distributions as part of the equation. Dose-response may also operate at an individual level and predict the severity of a health outcome as a function of dose. Particularly complex dose-response models may model both severity of outcome and population variability, and perhaps even recognize the influence of multiple causal factors. 

With the complexity of modern pharmacokinetic-pharmacodynamic models, analytical derivation of sensitivity indexes is rarely possible because rarely can these models be expressed as an equation. More often these models are written as a matrix of derivatives and the solution to finding the sensitivity index for these models requires a software package that can do symbolic differentiation of the Jacobian matrix. Hence, the current methodology for sensitivity analysis of complex models is empirical and done by systematically varying the model parameters one at a time and observing how the model outputs change. While easy to do, this approach cannot handle the case where there are interactions between model parameters. For example, two... 

The basic problem in nonlinear least squares is finding the values of 0 that minimizes the residual sum of squares which is essentially a problem in optimization. Because the objective functions used in pharmacokinetic pharmacodynamic modeling are of a quadratic nature (notice that Eq. (3.13) is raised to the power 2), they have a convex or curved structure that can be exploited to find an estimate of 0. For example, consider the data shown in Fig. 3.1. Using a 1-compartment model with parameters 0 = (V, CL, volume of distribution (V) can be systematically varied from 100 to 200 L and clearance (CL) can be systematically varied from 2 to 60 L/h. With each parameter combination, the residual sum of squares can be calculated and plotted... 

Since dose tolerance studies usually produce adverse events, pharmacokinetic evaluation should include an assessment of the dose-concentration-toxicity relationship. In addition to regularly scheduled plasma samples obtained to calculate pharmacokinetic parameters, it is common to obtain a plasma sample at the time an adverse event is observed. Often it is possible to correlate acute toxicity with plasma concentrations as well as with dose, and occasionally even to develop a pharmacokinetic/pharmacodynamic model for acute or subchronic toxicity. 

C(t) modeled according to two-compartment model with zero-order and first-order absorption Pharmacokinetic/pharmacodynamic relationship modeled using Hill model with first-order absorption. Modeled parameters matched experimental parameters when bicompartmental model with zero-order input was used. Linear PKs, anticlockwise hysteresis loop established for all doses studied. Apomorphine and growth hormone concentration predicted with good accuracy... 

The effectiveness of drug targeting should be evaluated by taking into account not only pharmacokinetic aspects, but also the pharmacodynamic aspects. The latter include the concentration-effect relationship in the target tissue and at the sites where toxicity may occur [7,12]. The therapeutic effect of the drug and its toxic effect may be different with regard to their mechanisms, and hence their concentration-effect relationship may also be different, both qualitatively (different PD models) and quantitatively (different model parameters). 

The Optimization Module aids in the optimization (fitting) of a wide variety of model parameters including physiological, pharmacokinetic, pharmacodynamic, and formulation variables. 

In actual practice, nonlinear regression is used to fit a suitable pharmacokinetic model described by the function c (t) to time—concentration data. Then, the estimated parameters are used as constants in the pharmacodynamic model to estimate the pharmacodynamic parameters. Alternatively, simultaneous fitting of the model to the concentration-effect—time data can be performed. This is recommended as c (t) and E (t) time courses are simultaneously observed. 

Then, given a model for data from a specific drug in a sample from a population, mixed-effect modeling produces estimates for the complete statistical distribution of the pharmacokinetic-dynamic parameters in the population. Especially, the variance in the pharmacokinetic-dynamic parameter distributions is a measure of the extent of inherent interindividual variability for the particular drug in that population (adults, neonates, etc.). The distribution of residual errors in the observations, with respect to the mean pharmacokinetic or pharmacodynamic model, reflects measurement or assay error, model misspecification, and, more rarely, temporal dependence of the parameters. 

There are several approaches to population model development that have been discussed in the literature (7, 9, 15-17). The traditional approach has been to make scatterplots of weighted residuals versus covariates and look at trends in the plot to infer some sort of relationship. The covariates identified with the scatterplots are then tested against each of the parameters in a population model, one covariate at a time. Covariates identified are used to create a full model and the final irreducible, given the data, is obtained by backward elimination. The drawback of this approach is that it is only valid for covariates that act independently on the pharmacokinetic (PK) or pharmacokinetic/pharmacodynamic (PK/PD) parameters, and the understanding of the dimensionality of the covariate diata is not taken into account. 

Assumptions are included in all of the elements of any pharmacokinetic/pharma-codynamic (PK/PD) model. Some examples of common assumptions made for these models include the structure of the models for pharmacokinetics, pharmacodynamics, and their respective covariate influences, the models for the clinical effect of the drug, the parameter values of all these models, and the variance structures for model components (11). Assumptions reduce inferential certainty because if the assumptions are wrong, then the model-based conclusions are wrong. Therefore, it is the quality of the attendant assumptions, and not their existence, that is the issue with assumptions in modeling (12). 

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. 

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... 

Linear mixed effects models are primarily used in pharmacodynamic analysis or in the statistical analysis of pharmacokinetic parameters. Linear mixed effects models could also be used to analyze concentrationtime data from a 1-compartment model with bolus administration after Ln-transformation. The advantages to using mixed effects in an analysis are that observations within a subject may be correlated and that in addition to estimation of the model parameters, between- and within-subject variability may be estimated. Also, the structural model is based on the population, not on data from any one particular subject, thus allowing for sparse sampling. Most statistical packages now include linear mixed effects models as part of their analysis options, as do some pharmacokinetic software (Win-Nonlin). While linear mixed effects models are not cov-... 

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. 

In contrast to pharmacokinetic model parameters which are often modeled assuming an exponential scale, model parameters from a pharmacodynamic model are sometimes modeled on an arithmetic scale... 

When applied to real-life problems, however, there is little experience reported using the frequentist prior approach in the literature. Gastonguay et al. (1999) used prior information from adults to estimate the model parameters in a PopPK analysis in children. Simonsen et al. (2000) used prior information to estimate the pharmacokinetics and pharmacodynamics of epirubicin in rats. So, while it appears that using prior information may be useful in certain circumstances, but caveat emp-tor applies at the present time—let the buyer beware. Gisleskog, Karlsson, and Beal conclude that considerable care must be taken with the use of a frequentist prior. That would seem good advice. 

Documentation on assumptions should address those assumptions implicit in the pharmacokinetic or pharmacodynamic model and the statistical methodology chosen to evaluate the data. It should also state the assumed sensitivity of the parameters required to define the model relative to the data space being evaluated as well as any preconceived notions regarding biomarkers or surrogate markers evaluated as responses or covariates in the analysis. Hypotheses should be defined based on what was held a priori as true before the study or analysis, what was developed from preliminary or exploratory data analysis, and what would constitute a difference or equivalence in an effect or outcome, hi some instances the criteria for difference as opposed to equivalence can be defined from a statistical viewpoint independent of the actual study design. This approach does not always confer regulatory acceptance, however. 

Table 1 Biological parameters required for pyrethroid insecticide physiologically based pharmacokinetic/pharmacodynamic (PBPK/PD) models ...

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