Pharmacokinetic/pharmacodynamic model nonlinear models

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... 

Fig. 12.4 Pharmacokinetic-pharmacodynamic model ofthethrombopoietic effects of a thrombopoietin analogue (PEG-rHuMGDF) in healthy volunteers. The intrinsic longevity of platelets (A), nonlinear random destruction of platelets (p), and the intra-subject variability...

Bonate PL. Nonlinear Mixed Effects Models. In Bonate PL, Pharmacokinetic-Pharmacodynamic Modeling and Simulation Springer New York, 2006. 

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... 

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. 

Using the approach of Sheiner and Verotta [452], a large number of pharmacodynamic models can be considered as hierarchical models composed of a series of submodels. These submodels are linear or nonlinear, static or dynamic input-output, elementary models. Several possible combinations of such submodels have been considered, but they have systematically associated the linear with dynamic features, and the nonlinear with static ones. Is there hesitation or fear of using nonlinear dynamics in the traditional pharmacokinetic-dynamic modeling context ... 

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]. 

Biomarker models that integrate pharmacokinetics, pharmacodynamics, and biomarkers are complex because they are based on sets of differential equations, parts of the models are nonlinear, and there are multiple levels of random effects. Therefore, advanced methods from numerical analysis and applied mathematics are needed to estimate these complex models. When the model is estimated, one seeks a model that is appropriate for its intended use (see Chapter 8). 

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. 

Pharmacokinetic and pharmacodynamic modeling is not something that is taught, but something that is caught, and while this chapter is a didactic presentation of nonlinear modeling, only through practical experience and trial and error will the real nuances of how these variables interact will be understood. 

Biopharmaceutical research often involves the collection of repeated measures on experimental units (such as patients or healthy volunteers) in the form of longitudinal data and/or multilevel hierarchical data. Responses collected on the same experimental unit are typically correlated and, as a result, classical modeling methods that assume independent observations do not lead to valid inferences. Mixed effects models, which allow some or all of the parameters to vary with experimental unit through the inclusion of random effects, can flexibly account for the within-unit correlation often observed with repeated measures and provide proper inference. This chapter discusses the use of mixed effects models to analyze biopharmaceutical data, more specihcally pharmacokinetic (PK) and pharmacodynamic (PD) data. Different types of PK and PD data are considered to illustrate the use of the three most important classes of mixed effects models linear, nonlinear, and generalized linear. 

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. 

---