Emax model

Pharmacokinetics. Figure 2 Sigmoid Emax model of pharmacodynamics with Hill coefficient (H), concentration producing half-maximum effect (CE50), threshold concentration (CE05), and ceiling concentration (CE95). 

There was a direct relationship between the effect and the plasma concentration in the rat pharmacodynamic data and it was well described by a simple Emax model. Based on preclinical models for efficacy, a 90% effect was considered as the target for therapeutic effect. Finally the human C90 (human concentration corresponding to 90% effect, C90 man) was estimated by accounting for the different affinities and unbound fractions of each compound for the rat and human receptors as follows ... 

It follows that the effect is at half maximum when [D] = ATd. In pharmacology, Eq. (8) or the so-called -Emax model is conventionally written as Eq. (9) ... 

Fig. 3. Concentration-effect relationship for the sigmoid Emax model with 5 = 0.5, 1, 3 and 5, respectively, (a) Linear concentration scale, (b) logarithmic concentration scale.

Fig. 4. Change in heart rate produced by apomorphine in the rat. Slowing of heart rate predominates at low dmg concentrations, while tachycardia is most prominent at high steady-state concentration. Two sigmoid Emax models have been combined for the PK-PD analysis. Cp(50) corresponds to Cso% (From Paalzow LK, Paalzow GHM, Tfelt-Hansen P Variability in bioavailability concentration versus effect. In Rowland M, Sheiner LB, Steimer J-L, editors. Variability in dmg therapy description, estimation, and control. New York Raven Press 1985.)...

Dutta S, Matsumoto Y, Ebling WF. Is it possible to estimate the parameters of the sigmoid Emax model with truncated data typical of clinical studies J Pharm Sci 1996 85 232-8. 

In the simplest case, drug effects are directly related to plasma concentrations, but this does not necessarily mean that effects simply parallel the time course of concentrations. Because the relationship between drug concentration and effect is not linear (recall the Emax model described in Chapter 2 Drug Receptors Pharmacodynamics), the effect will not usually be linearly proportional to the concentration. 

Estimation of plasma EC50 by an inhibitory effect Emax model at equilibrium. 

Fig. 15.10 Concentration-effect relationship of pegfilgrastim based on a simple Emax model. The average concentrations of pegfilgrastim from patients with breast cancer at doses of 30, 60, and 100 pg/kg are indicated ( ).

Fig. 17.6 Emax model with a given Emax = 100, EC50 = 10 and ten different n values, ranging from 0.5 up to 5.0.

As is implicit from all the above, the measured concentration in plasma is directly linked to the observed effect for these simple mechanistic, pharmacokinetic-dynamic models. Accordingly, these models are called direct-link models since the concentrations in plasma can be used directly in (10.6) and (10.7) for the description of the observed effects. Under the assumptions of the direct-fink model, plasma concentration and effect maxima will occur at the same time, that is, no temporal dissociation between the time courses of concentration and effect is observed. An example of this can be seen in the direct-fink sigmoid Emax model of Racine-Poon et al. [418], which relates the serum concentration of the anti-immunglobulin E antibody CGP 51901, used in patients for the treatment of seasonal allergic rhinitis, with the reduction of free anti-immunglobulin E. 

Under the assumptions of the direct-link model, neither a counterclockwise (Figure 10.2) nor a clockwise hysteresis loop (Figure 10.4) will be recorded in an effect vs. concentration plot. In principle, the shape of the effect vs. concentration plot for an ideal direct-link model will be a curve identical to the specific pharmacodynamic model, relating effect with concentration, e.g., linear for a linear pharmacodynamic model, sigmoid for the sigmoid Emax model (cf. Table 10.1 and following paragraphs and sections), etc. 

First, mention can be made of cases in which the measured effect instead of being proportional to the activated receptors, follows a more general function E = T (v). This model is called receptor-transducer and was introduced by Black and Leff [409]. The function T is called a transducer function and its most common form is yet again the Emax function, which when replaced in (10.5) results in an Emax model but with different shape parameters called an operational model [441]. 

In a direct-response model, the output of a linear dynamic model (the link model) with input c (t) drives a nonlinear static model (usually the Emax model) to produce the observed response. 

The Emax model is defined as E=(Emax. Cj/ECso + C, where Emax is maximum effect, C the concentration, and EC50 the drug concentration that results in 50 % of the maximum response. 

This model is defined as E = (Emax CY) / (CY+ EC50y), where Emax = E when concentration, C, approaches infinity. EC50 is drag concentration that results in 50 % of the maximum response. This model is characterized by a sigmoidal shape and y is the degree of sigmoidicity in effect versus concentration. When y = 1, the model reduces itself to the Emax model. The values of y may vary depending upon receptor/affector attributes. 

Population pharmacodynamic data, i.e., observed 24-hour efficacy scores were modeled as a function of individual predicted 24-hour steady state AUCs. Various pharmacodynamic models were explored including linear, Emax, and sigmoidal Emax models. Fixed and random-effect parameters were used to describe the PK/PD relationship. The results of the model development are presented in Table 7. 

Emax is the maximal fractional reduction in seizure frequency and ED50 is the dose that produces a 50% decrease in seizure frequency from maximum. PLAC is the fractional change in seizure frequency from baseline after placebo treatment. Drug treatment was modeled as an Emax model (see Chapter 18) and placebo treatment was modeled as a constant. This model describes a dose-related reduction in seizure frequency with a maximum decrease in seizure frequency of 38%. Half that reduction (ED50) was achieved with a dose of 48.7 mg/day. However, the ED50 was not well estimated, since the symmetrical 95% confidence interval included zero. After placebo treatment the average increase in seizure frequency was 10% of baseline. 

The Emax model is a simpler form of the sigmoid Emax model, with a slope factor n = 1, so that... 

In Figure 18.16, the Emax model is used to quantify the relationship between theophylline serum level and improvement in pulmonary function as measured by the increase in forced expiratory volume in 1 second (FEVi) in six patients who were treated with placebo and three incremental doses of theophylline (14). 

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