Nonlinear calibration curve

Accuracy When spectral and chemical interferences are minimized, accuracies of 0.5-5% are routinely possible. With nonlinear calibration curves, higher accuracy is obtained by using a pair of standards whose absorbances closely bracket the sample s absorbance and assuming that the change in absorbance is linear over the limited concentration range. Determinate errors for electrothermal atomization are frequently greater than that obtained with flame atomization due to more serious matrix interferences. 

Nonlinear calibration curves are not forbidden, but they do complicate things quite a bit more calibration points are necessary, and interpolation from signal to concentration is often tedious. It would be improper to apply... 

In the absence of analyte, many CL systems show a low emission background level. Hence, in flow systems, as the CL intensity is proportional to the analyte concentration, the emission appears as a sharp peak superimposed on a low constant blank signal, which is measured when the mixture of analyte and CL reagents passes through the detector cell. Because only a small portion of CL emission is measured from this time profile, nonlinear calibration curves may be obtained for reactions with complex kinetics [1],... 

In Section III.C various detectors were mentioned that respond selectively to nitrogen-containing compounds some applications to amines follow. A nonlinear calibration curve... 

In some instances, calibration test data may need to be subjected to some kind of mathematical transformation, prior to the regression analysis, in order to obtain linear calibration plots. In some cases, however, such as in immunochemical assays, linearity cannot be demonstrated even after any transformation. The use of nonlinear calibration curves for analysis has been discussed (27). 

The main limitation of this model [6,14] is that it assumes that the measured response at a given sensor is due entirely to the constituents considered in the calibration step, whose spectra are included in the matrix of sensitivities, S. Hence, in the prediction step, the response of the unknown sample is decomposed only in the contributions that are found in S. If the response of the unknown contains some contributions from constituents that have not been included in S (in addition to background problems and baseline effects), biased predicted concentrations may be obtained, since the system will try to assign this signal to the components in S. For this reason, this model can only be used for systems of known qualitative composition (e.g. gas-phase spectroscopy, some process monitoring or pharmaceutical samples), in which the signal of all the pure constituents giving rise to a response can be known. For the same reason, CLS is not useful for mixtures where interaction between constituents or deviations from the Lambert-Beer law (nonlinear calibration curves) occur. 

Nonlinear calibration curve. Following the procedure in Box 4-2, find how many micrograms (pg) of protein are contained in a sample with a corrected absorbance of 0.350 in Figure 4-11. 

Fic. 2. Distortion of normal results distribution using linear and nonlinear calibration curves as described by Picart et al. (1978). Copyright Elsevier Science Publishers B.V., Amsterdam. 

The approach described by Garland and collaborators (Garland and Powell, 1981 Min et al., 1978) is generally applicable and is different from the previous ones in that they use nonlinear calibration curves. By dividing both numerator and denominator by p, Eq. (1) was rewritten as... 

Calibration is performed by measuring Rm for a number of standard mixtures with a known concentration of analyte x. Depending on the type of ion interference, linear regression analysis is applied if both C1 and C2 (type 1 ion interference, Section 3.1) or only C2 (type 2) are negligible. In all other cases (types 3 and 4), a NONLIN computer program is used to calculate the nonlinear calibration curve after experimental determination of Clt C2, and C3 on the pure components. Apart from the fact that the exact form of the NONLIN data regression was not specified, the experimental determination of ion interferences and the assumption that qj = pt are limiting factors to accuracy (Jonckheere, 1982). 

Because pressure transducers from different manufacturers can vary significantly, it is important to understand their performances such as accuracy. An ideal device would have a direct linear relationship between pressure and output voltage. In reality, there will always be some deviations this is referred to as nonlinearly. The best straight line is fitted to the nonlinear curve. The deviation is quoted in their specifications and expressed as a percent of full scale. The nonlinear calibration curve is determined in ascending direction from zero to full rating. This pressure will be slightly different from the pressure measured in descending mode. This difference is termed hysteresis it can be reduced via electrical circuits. 

Calibration is necessary for in-situ spectrometry in TLC. Either the peak height or the peak area data are measured, and used for calculation. Although the nonlinear calibration curve with an external standard method is used, however, it shows only a small deviation from linearity at small concentrations [94.95 and fulfils the requirement of routine pharmaceutical analysis 96,97J. One problem may be the saturation function of the calibration curve. Several linearisation equations have been constructed, which serve to calculate the point of determination on the basis of the calibration line and these linearisation equations are used in the software of some scanners. A more general problem is the saturation function of the calibration curve. It is a characteristic of a wide variety of adsorption-type phenomena, such as the Langmuir and the Michaelis-Menten law for enzyme kinetics as detailed in the literature [98. Saturation is also evident for the hyperbolic shape of the Kubelka-Munk equation that has to be taken into consideration when a large load is applied and has to be determined. 

With reagents prepared in advance, the Lowry assay requires 1 h. A 400-pL sample is required, containing 2-100-pg protein (5-250 pg/mL). Nonlinear calibration curves are obtained, due to decomposition of the Folin reagent at alkaline pH following addition to the sample that results in incomplete reaction. Interferences include agents that acidify the solution, chelate copper, or cause reduction of cop-per(II). 

When an assay presents a nonlinear calibration curve (Fig. 16.4), the data can be linearized using standard functions.4 The log-logit function transforms a sigmoid curve with a single point of inflection into a straight line, and is used extensively with data from competitive immunoassays. 

A linear calibration curve is preferred because of its mathematical simplicity and because it makes detecting an abnormal response easy. With linear calibration curves, fewer standards and a linear regression procedure can be used. Nonlinear calibration curves can be employed, but more standards are required to establish the calibration function. A large linear dynamic range is desirable because a wide range of concentrations can be determined without dilution. In some determinations, such as the determination of sodium in blood serum, only a small dynamic range is required because variations of the sodium level in humans are quite small. 

Some concentration nonlinearities can be treated by a simple extension of the trilinear model. If there are concentration nonlinearities, then the model estimates apparent concentrations in C. Making a simple nonlinear calibration curve between the values in C and the actual concentrations in the standards gives the solution since the apparent concentration of the analyte in the mixture can then be transformed to the actual concentration domain [Booksh et al. 1994],... 

Chapter 5 discusses in depth the statistical considerations related to LB A development and validation. In addition to the most appropriate algorithms for describing the nonlinear calibration curves typically found in LBAs, the authors also provide further insight into the performance characteristics to be evaluated during assay validation, including the concepts of total error in prestudy validation and the use of the 4-6-X rule. The decision rules at the prestudy validation and routine assay implementation stages are also discussed in some detail in Chapter 5. 

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 ... 

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