Empirical Pharmacodynamic Models

Assuming relatively rapid drug-receptor equilibrium with respect to c (t) variations, then c c(t) (assumption H2), so the previous equation becomes 

With 7=1, (10.6) has been used extensively in pharmacology to describe the effect of chemicals on tissues in the modified form  

Combined pharmacokinetic-dynamic studies seek to characterize the time course of drug effects through the application of mathematical modeling to dose-effecttime data. This definition places particular emphasis on the time course of drug 

The most common function used to relate drug concentration c with effect is the Amax model  

Other simpler empirical models have also been used since the early days of pharmacodynamics [412,413] to describe the drug concentration-effect relationship. The linear model relies on a linear relationship between E and c  

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An important aspect of PD models is the link between the pharmacokinetics and the pharmacodynamic model, i.e. which concentration drives the drug effect. For empirical models there are many different PK/PD link approaches described and the theory is presented in several review papers [33-35]. The two most popular approaches are described in the following and are also illustrated in Fig. 17.7. The direct link approach assumes that a change in the measured concentration is directly reflected in a change in the measured P D. This is most often observed if the site of PK measurement and the site of PD measurement are identical (e.g. PK measurement in plasma and clotting time as PD measurement). 

Although the maximum-effect pharmacodynamic models are empirically based/ they do incorporate the concept of a maximum effect predicted by the drug-receptor interactions described earlier. The Hill equation/ which takes the same form as the equation describing drug effect as a function of receptor occupancy/ relates a continuous drug effect to the drug concentration at the effect site as shown ... 

The result is that a much more empirical approach is used to describe the relationship between concentration and effect in clinical pharmacology studies. After a pharmacodynamic experiment has been conducted, concentration-effect plots are generated. The shape of the concentration-effect curve is used to determine which pharmacodynamic model will be used to describe the data. Because of this, the pharmacodynamic models used in a clinical pharmacology study are deterministic in the same way that the shape of the serum-concentration-versus-time curve determines which pharmacokinetic model is used in clinical pharmacokinetic studies. 

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... 

In clinical trials only a limited amount of data can be obtained. Thus, if parameters of models describing these data should be estimated often empirical models are completely sufficient. These models allow a thorough investigation of the influence of extrinsic and intrinsic factors on the observed pharmacokinetic or pharmacodynamic data. 

To obtain the empirical estimates of a, Kowalski and Hutmacher (33) simulated 300 chnical trials for each combination of sample size and p, where the proportional reduction in CUP (0) was fixed to zero. Covariate and base models were fitted to each of the trials and the likelihood ratio tests were performed at the 5% level of significance. The percentage of trials where a statistically significant difference in CUP was observed provided an empirical estimate of a (i.e, PIoi = 0 is rejected when i/o is true). The data were analyzed with the NONMEM population phar-macokinetics/pharmacodynamics analysis software. The results suggested that an approximate nine-point change in the objective function should be used to assess statistical significance at the 5% level rather than the commonly used critical value of 3.84 for one degree of freedom. 

W. Krzyzanski and W. J. Jusko, Note Caution in the use of empirical equations for pharmacodynamic indirect response models. J Pharmacokinet Biopharm 26 735-741 (1998). 

Pharmacodynamic investigations in the IE animal model allow quantitation of bacterial densities within vegetations over time as a function of antibiotic concentration. These models empirically confirm many of the observed IE treatment principles. The antibiotic concentration in serum that is needed to kill bacteria within vegetations may be many times the minimal bactericidal concentration... 

Exposure-response modeling can be an important component of a totality of evidence assessment of the risk of QTc prolongation. It can be evaluated in early-phase studies and as part of the conventiontil study of QTc prolongation, and may help inform further evaluation. There are many different types of models for the analysis of concentration-response data, including descriptive pharmacodynamic (PD) models and empirical models that link pharmacokinetic (PK) models (dose-concentration-response) with PD models. 

Lunn, D.J., Best, N., Thomas, A., Wakefield, J., and SpiegeUialter, D. 2002. Bayesian analysis of population PK/PD models general concepts and software. /. Pharmacokinet. Pharmacodynam. 29 271-307. Magni, R, BeUazzi, R., Nauti, A., Patrini, C., and Rindi, G. 2001. Compartmental model identification based on an empirical Bayesian approach the case of thiamine kinetics in rats. Med. Biol Eng. Comput. 39 700-706. 

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