Sigmoidal models

In graded response one can resort to using surrogates, and the classic Hill model (or sigmoid model as described in Equation 18.16 above) is used to correlate observed effect (E(t)) with the concentration modified by an exponent that is called Hill coefficient in classic sigmoidal model, the Hill coefficient h) would be equal to 1 ... 

For simplicity, a linear relationship between concentration and effect is often assumed, reducing the problem of PK/PD to the pharmacokinetics. However, the concentration-effect relationship of any drug tends towards a plateau, and a sigmoidal model (sigmoid E ax model or Hill equation) is more appropriate [21-24] ... 

Brisbin et al. (30) described the development of the Richards sigmoidal model and presented it as an equation for contaminant uptake based on the amount of contaminant in an organism or compartment. For our data, the equation was reparameterized to use a partition coefficient rather than a concentration. It takes the form... 

Acclimation conditions, susceptibilities of organisms to AS toxicity, 547 Accumulation of contaminants, sigmoidal model, 565-566... 

Some of the isothermal relative rate [= (dfli d/V(da/d/LJ - time plots corresponding to the rate equations in Table 3.3. (a) sigmoid models (b) geometrical and reaction order (RO) models. (NOTE (i) the relative rate - time plots for the third-order rate equation and the difhision models are too deceleratory for usehil comparison and (ii) calculation on an arbitrary basis, e.g. that a = 0.98 at / = 100 min., results in plots of relative rates against a, in place of time, having similar shapes). 

Figures 5.2-5.5 show that the non-electrostatic model completely fails for alumina (one example of relatively good agreement between the calculated and experimental charging curve in Fig, 5.5 is probably a fortuitous coincidence). On the other hand the sigmoidal model curves roughly reflect the charging behavior of silica, especially at low ionic strengths. For silica, the number of adjustable parameters in the model can be reduced to one by fixing the K or N, . Figures 5.11-5.15 show the model curves calculated for log K (reaction 5.25) = 8 (fixed value). The best-fit values are summarized in Table 5.6.

FIGURE 31.8 Effect site drug concentration-response curve. The thin line reflects the observed effect associated with the theoretical effect site concentration. The thick line reflects the sigmoidal model fit to the observed data. 

Calculation of C-50. The value concentration that provide CL-SI = 50 % was termed C-50. C-50 was calculated by the data to the sigmoid model ... 

Figure 19.9 Sigmoidal model of gradient of interface. Source Reprodnced with permission from Koberstein JT, Morra B, Stein RS. J Appl CrystaUogr 1980 13 34 [27]. Copyright 1980 lUCr (International Union of Crystallography) (http //dx.doi.org/10.1107/S0021889880011478).

The exit effect results make it clear that a simple sigmoidal model for the density profile is not sufficient to describe the physical reality. Rather, there are complex geometric effects which are also important. 

Obviously, the fit now is almost perfect with an adjusted equal to 0.9987. Let us have a look at further analysis regarding the general sigmoid model goodness. Click the Analysis button as shown in Fig. 5.15. Figure 5.29 shows the Analysis window showing the 95% confidence interval for the... 

Figure 9.32 Pasture yield vs. growth time with sigmoidal model.

Figure 4 (a) Volume expansion ratio for microgels in squalane obtained from swelling experiments modeled with an exponential-sigmoid equation with the temperature in an absolute scale. X represents observed data, — represents an empirical sigmoid model with the following eoefficients E 3.21, Eq = 1.36, Ti = 351.2, and m = 70.95. (b) Particle volume fraction in a 5 wt% microgel suspension. 

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