Concentration-response data 4-parameter fitting

Figure 5.7 Comparison of four-parameter fy-maxi mum, v-minimum. IC50, and h) and two-parameter (IC50 and h) fits of non-ideal concentration-response data. In panels A and B the data indicate a nonzero plateau at low inhibitor concentration that might reflect a low-amplitude, high-affinity second binding interaction. In panels C and D the data indicate a plateau at high inhibitor concentration that does not achieve full inhibition of the enzyme. There could be multiple causes of behavior such as that seen in panels C and D. One common cause is low compound solubility at the higher concentrations used to construct the concentration-response plot. Note that the discordance between the experimental data and the expected behavior is most immediately apparent in the plots that are fitted by the two-parameter equation.

The advantage of multiple regression is that methods are established, well described, and available in almost all statistical sofware packages, and that the fitting procedures have been well developed (Neter et al. 1996). Furthermore, the complete n + 1 dimensional concentration-response surface is fitted to the complete data set, taking into account that the parameters of the concentration-response relationships of the individual mixture components are actually predictors for the complete mixture data set. The model allows individual concentration-response curves to have their unique slopes. 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step in this process is to fit the data to an unweighted model. Table 11.7 shows a set of responses to a range of concentrations of an agonist in a functional assay. The data is fit to a three-parameter model of the form... 

FIGURE 11.13 A collection of 10 responses (ordinates) to a compound resulting from exposure of a biological preparation to 10 concentrations of the compound (abscissae, log scale). The dotted line indicates the mean total response of all of the concentrations. The sigmoidal curve indicates the best fit of a four-parameter logistic function to the data points. The data were fit to Emax = 5.2, n = 1, EC5o = 0.4 pM, and basal = 0.3. The value for F is 9.1, df=6, 10. This shows that the fit to the complex model is statistically preferred (the fit to the sigmoidal curve is indicated). 

The procedure calculates the concentrations from both curves that produce the same level of response. Where possible, one of the concentrations will be defined by real data and not the fit curve (see Figure 12.3b). The fitting parameters for both curves are shown in Table 12.3b. Some alternative fitting equations for dose-response data are shown in Figure 12.4. 

Of course, the accuracy of these determinations depends on the quality of the experimental data used to construct the concentration-response plots significant data scatter will erode the accuracy of the fitting parameter estimates. 

Y is the expected response and X is the corresponding concentration. A, B,C and D are the four parameters of the equation, where A gives an estimate of expected response at zero dose, B is the slope (e.g., response/concentration) in the middle of the calibration curve, C is the IC50 and D is the expected response at infinite dose. This curve satisfies all the conditions specified for the response-concentration relationship and also closely approximates the mass-action equations [20]. Weighting of the results is recommended for fitting dose-response data from immunoassay, in order to compensate the heterogeneity of response variances in the response-error relationship [17,18]. 

The dose-response curve after receptor alkylation is shown in Figure 12.6a (open circles). The same function is used to fit the data as employed for the control curve (for this example, Equation 12.5). The parameters of the fit dose-response curves are shown in Table 12.5b. Equiactive concentrations of oxotremorine are calculated according to the procedure given in Section 12.2.1. 

Linearity is often assessed by examining the correlation coefficient (r) [or the coefficient of determination (r )] of the least-squares regression line of the detector response versus analyte concentration. A value of r = 0.995 (r = 0.99) is generally considered evidence of acceptable fit of the data to the regression line. Although the use of r or is a practical way of evaluating linearity, these parameters, by... 

In the text which follows we shall examine in numerical detail the decision levels and detection limits for the Fenval-erate calibration data set ( set-B ) provided by D. Kurtz (17). In order to calculate said detection limits it was necessary to assign and fit models both to the variance as a function of concentration and the response (i.e., calibration curve) as a function of concentration. No simple model (2, 3 parameter) was found that was consistent with the empirical calibration curve and the replication error, so several alternative simple functions were used to illustrate the approach for calibration curve detection limits. A more appropriate treatment would require a new design including real blanks and Fenvalerate standards spanning the region from zero to a few times the detection limit. Detailed calculations are given in the Appendix and summarized in Table V. 

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