Four parameter logistic model

The empirical models are based on mathematical functions that mimic the distribution of the standards measured in the assay. They can be based on point to point (interpolation) methods or regression methods. The most widely used empirical models that have been applied to MIP-ILAs include the log-logit model and the four-parameter logistic model. 

For calibration, a seven-point standard curve ranging from 2-32 ng/mL in rat or dog EDTA plasma was used. The four-parameter logistic model was used to describe the relationship between the OD readings and nominal concentration (CONC) of the analyte (DeLean et al. 1978) ... 

The most frequently used nonlinear calibration curve models [18] are the four- and five-parameter logistic models (4PL and 5PL). For example, the four-parameter logistic model is expressed mathematically as follows ... 

O Connell, M.A., Belanger, B.A., and Haaland, P.D. (1993) Calibration and assay development using the four parameter logistic model. Chemometrics and Intelligent Laboratory Systems, 20, 97 114. 

Finke. M. D., DeFoliart, G. R., and Benevenga, N. J. (1987a). Use of simultaneous curve fitting and a four-parameter logistic model to evaluate the nutritional quality of protein sources at growth rates of rats from maintenance to maximum gain. J. Nutr. 117,1681-1688. 

The responses are values of y, and n is the number of responses. A calculated SSq value will have associated with it a value for the degrees of freedom. If there are no fitting parameters involved in applying the model, the number of degrees of freedom will be n. For the data in Figure 11.13, dfs = 10. A more complex model for these data is a four-parameter logistic function of the form... 

This value is identified in F tables for the corresponding dfc and dfs. For example, for the data in Figure 11.13, F = 7.26 for df=6, 10. To be significant at the 95% level of confidence (5% chance that this F actually is not significant), the value of F for df = 6, 10 needs to be > 4.06. In this case, since F is greater than this value there is statistical validation for usage of the most complex model. The data should then be fit to a four-parameter logistic function to yield a dose-response curve. 

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). 

A commonly accepted fitting approach is the four-parameter logistic equation (4PL) used to fit the mean concentration-response relationship, and the power-of-the-mean equation to fit the response-error relationship [15,18]. In some applications a fifth parameter describing the asymmetry of the curve might be incorporated, ensuring a higher versatility of the model [18]. 

This relationship is established by measurement of samples with known amounts of analyte (calibrators). One may distinguish between solutions of pure chemical standards and samples with known amounts of analyte present in the typical matrix that is to be measured (e.g., human serum). The first situation applies typically to a reference measurement procedure, which is not influenced by matrix effects, and the second case corresponds typically to a field method that often is influenced by matrix components and so preferably is calibrated using the relevant matrix. Calibration functions may be linear or curved, and in the case of immunoassays often of a special form (e.g., modeled by the four-parameter logistic curve) This model (logistic in log x) has been used for both radioimmunoassay and enzyme immunoassay techniques and can be written in several forms as shown (Table 14-1). Nonlinear regression analysis is applied to estimate the relationship, or a logit transforma-... 

Another method of detecting a dose-response relationship is to fit the data to various models for dose-response curves. This method statistically determines whether or not a dose-response model (such as a Logistic function) fits the data points more accurately than simply the mean of the values this method is described fully in Chapter 12. The most simple model would be to assume no dose-response relationship and calculate the mean of the ordinate data as the response for each concentration of ligand (horizontal straight line parallel to the abscissal axis). A more complex model would be to fit the data to a sigmoidal dose-response function (Equation 11.2). A sum of squares can be calculated for the simple model (response — mean of all response) and then for a fit of the data set refit to the four parameter Logistic shown... 

A more complex model for these data is a four-parameter logistic function of the form... 

Selection of an appropriate model to fit the calibration curve is critical for the accuracy of the method. The four-parameter logistic (4PL) model is applied... 

The topic of nonlinear calibration for LBAs, such as immunoassays, has been reviewed in detail in a number of publications [4,8,9,15 17]. Typically, immunoassay calibration curves are inherently nonlinear [9]. Because the response error relationship is a nonconstant function of the mean response, weighting is needed to account for the heterogeneity in response variances. The four- or five-parameter logistic models are accepted widely as the standard models for fitting nonlinear sigmoidal calibration data [3 5,8,9,16,17], This model can be described... 

A calibration (standard) curve describes the concentration-response curve typically including more than eight calibrators and additional ones serving as anchor points thus facilitating curve fitting. All calibrators are prepared in duplicates in the matrix analyzed. The concentration-response relationship is most often fitted with a four- to five-parameter logistic model. 

Physico-chemical measurements using chromatographic methods produce responses that are linear to the concentrations. As IA measures the resulting signals of a reaction, however, the response is a nonlinear function of the analyte concentration. Often, the regression model used to describe this relationship is a four- or five-parameter logistic function, as shown in the sigmoid shape standard curve in Fig. 6.4. 

Computers incorporated into instruments or connected through stand-alone PCs have expanded the capabilities of instrument systems to include those previously available only by connection to the mainframes of LISs. These capabilities include summaries of quality control (production of Shewhart plots), fitting of immunoassay calibration curves using various models (e.g., four parameter log-logistic), and linkage of patient information with specimen identification. 

The model (1) has four parameters R, K, A, and B. As usual, there are various ways to nondimensionalize the system. For example, both A and K have the same dimension as A, and so either N/A or N/K could serve as a dimensionless population level. It often takes some trial and error to find the best choice. In this case, our heuristic will be to scale the equation so that all the dimensionless groups are pushed into the logistic part of the dynamics, with none in the predation part. This turns out to ease the graphical analysis of the fixed points. 

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