Nonlinear calibration functions

Finally, a comparison between real and calculated signal intensities can demonstrate the quality of calibration, even if a nonlinear calibration function correlates better with the observed response (Table 4). 

Table 6.2. Suitable nonlinear calibration functions, their normal equations and sensitivity functions... 

One of the most widely used methods is AAS. GFAAS provides low detection limits for the PGM. FAAS provides less satisfactory detection limits. It is thus less suitable for PGM determinations (Table 4). In the presence of strongly interfering matrixes, procedures for selective enrichment and matrix elimination should be applied (see Sec. 4.1.3) [23,24,27]. The determination of osmium leads to nonlinear calibration functions and is therefore difficult to perform. 

Figure 4.36. Cross validation between two HPLCs A stock solution containing two compounds in a fixed ratio was diluted to three different concentrations (1 10 20) and injected using both the 10 and the 20 /xl loop on both instruments. The steps observed at Amount = 100 (gray ellipses) can be explained with effective loop volumes of 9.3 and 20 pi (model 1) and 14.3 and 20 pi (model 2) instead of nominally 10 and 20 pi. This is irrelevant as both a sample and the calibration solution will be run using the same equipment configuration. The curved portion of the model 2 calibration function was fitted using Y = A /x this demonstrates the nonlinearity of the response at these high concentrations. The angle between the full and the dotted line indicates the bias that would obtain if a one-point calibration scheme were used.

Consequently, the proof of calibration should never be limited to the presentation of a calibration graph and confirmed by the calculation of the correlation coefficient. When raw calibration data are not presented in such a situation, most often a validation study cannot be evaluated. Once again it should be noted that nonlinearity is not a problem. It is not necessary to work within the linear range only. Any other calibration function can be accepted if it is a continuous function. 

In general, from nonlinear calibrations result variable sensitivities expressed by sensitivity functions S(x) ... 

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. 

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. 

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-... 

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. 

For nonlinear detectors the calibration curve may be determined by mathematical curve fitting. For complex calibration functions with more than one unknown parameter, additional information (e.g. about the physical theory of the measuring method) or pulse experiments with different concentrations are necessary. In any case, it is advised to check the low and high concentration range by separate experiments with different feed concentrations. There should be no detector overflow at maximum concentration. 

Belanger, B.A., Davidian, M., and Giltinan, D.M. The effect of variance function estimation on nonlinear calibration inference in immunoassay data. Biometrics 1996 52 158-175. 

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... 

Calibration refers to the procedures used for correlating test method ontpnt or response to an amount of analyte (concentration or other quantity). The characteristics of a calibration fnnction and justification for a selected calibration model should be demonstrated dnring SLV and ILS stndies. The performance of a calibration technique and the choice of calibration model (e.g., first-order linear, cnrvifinear, or nonlinear mathematical function) are critical for minimizing method bias and optimizing precision. The parameters of the model are nsnally estimated from the responses of known, pnre materials. Calibration errors can result from failure to identify the best calibration model inaccnrate estimates of the parameters of the model errors in the composition of calibration materials or inadeqnately smdied, systematic effects from matrix components. This section focnses on the critical issne of the traceability and supply of materials used for calibration of marine biotoxin methods. 

Equation [8.89] is linear with respect to parameters 4> and 0, but nonlinear with respect to q, and therefore the data must be fitted to this calibration function using nonlinear least-squares regression (Section 8.3.8) it is emphasized that it is very important to ensure that the initial estimates for the unkown parameters should be reasonably close to the final best estimates (see the text box dealing with nonlinear regression). In the present example (Equation [8.89]) excellent initial estimates can be obtained experimentally (see below) but if this is not possible tricks can be employed to obtain reasonable first estimates. One way is to plot the experimental data for Ra VRsis s Qa VQsis" nd draw an approximate curve though the points by hand. Experimental data expected to be well represented by Equation [8.89] should extrapolate to a value of (Ra /Rsis ) = 0 as (Qa"/Qsis ) zero, and to (Ra VRsis") = as (Qa"/Qsis") becomes... 

While, from a general measurement point of view a two point calibration is probably acceptable, there are situations where this might not be so. At the nuclear level, energies are very precisely defined (see Chapter 1, Section 1.6.4), and studies of these levels might require a much greater degree of accuracy. A multi-point nonlinear calibration would then be needed of a mathematical form chosen to reflect the shape of the non-linearity. The most obvious first choice might be a quadratic function ... 

Sensitivity is given by the slope of the calibration function applied for the desired operating point. Because we are also going to consider nonlinear calibrations, with their sensitivity being dependent on the operating point, we focus on the middle of the working range. This will enable sensitivity data to be compared numerically and visually for any calibration function that may apply. 

The calibration function is defined exclusively within the working range given by the experimental calibration. GC-MS systems achieve very low LODs so that, at a correspondingly dense collection of calibration points near the blank value, a nonlinear area is described. This area can be caused by unavoidable active sites (residual activities) in the system and swallows up a small but constant quantity of substance. Such a calibration function tends to approach the a -axis before reaching the origin. 

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