Sigmoidal 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). 

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. 

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. 

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. 

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

Sigmoid Emax Model Jonkers and colleagues [80] studied the pharmacodynamics of racemic metoprolol, a cardioselective beta-blocker, and the active S-isomer in extensive metabolizers (EMs) and poor metabolizers (PMs). The drug effect studied was the antagonism by metoprolol of terbutaline-induced hypokalemia (abnormally low potassium concentration in the blood). The pharmacodynamic interaction was described by a sigmoidal function for competitive antagonism based on the earlier work of Holford and Sheiner [81] ... 

The corresponding inhibitory sigmoid Emax model is functionally described as follows ... 

The PK-PD relationship for G-CSF following IV and SC administration was well characterized in healthy volunteers (53). The PK model was a two-compartment PK model with bisegmental absorption from the site of SC administration, parallel first-order and saturable elimination pathways, and an indirect effect PD model describing the time course of neutrophils. A sigmoidal Emax model was applied for the stimulation of the neutrophil input rate. In addition, a time-variant scahng factor for absolute neutrophil count (ANC) observations was introduced to account for the early transient depression of ANC. 

What this equation shows is that the F-test and AIC are not independent and that given one of them the other can be determined. These equations also show that sometimes the two criteria can lead to different conclusions. Suppose a modeler fit an Emax model with two estimable parameters and a sigmoid Emax model with three estimable parameters to a data set with 14 observations, such as might be the case when fitting a pharmacodynamic model to individual data. In this case, an F-test greater than 3.84 is required to declare the sigmoid Emax model the superior model at the 0.05 level, which is equivalent to a AAIC of —2.19. An F-test value less than 3.84 is considered to be not statistically significant at the 0.05 level and the reduced model is chosen as the superior model. However, any AAIC less than 0, even values between 0 and —2.19, is still considered to be indicative that the full model is the superior model. Hence, the possibility exists for the two criteria to reach different conclusions. 

The next model examined was a sigmoid Emax model... 

Figure 9.3 Goodness of fit plots for the zifrosilone data under the sigmoid Emax model with an effect compartment. Final values are reported in Table 9.2. Solid line in upper left plot is the line of unity. Dashed line in upper right plot is the LOESS smooth to the data with a 0.3 sampling proportion.

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