Chapter 21
Considerations in Immunoassay Calibration Thomas L. Fare, Robert G. Sandberg, and DavidP.Herzog
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The appeal of immunoassays has been their ability to provide precise and accurate quantitative results for a specific analyte at a low cost per test. This quantitative capability has been the motivation for analytical chemists to evaluate the technique for use with environmental samples. While some regulators have initially viewed immunoassays as a qualitative method, more recently they have expressed interest in extending the application of immunoassay beyond screening results to quantitative analyses. Although many of the principles underlying qualitative and quantitative techniques are similar, qualitative evaluation of data, particularly when applied to the detection of complex mixtures, is sufficiently complex to deserve separate treatment. This paper will cover calibration of small molecule immunoassays for quantitative analyses. A general approach to important considerations will be given along with a set of recommendations for the analyst using immunochemical methods.
The goal of immunoassay calibration is to estimate the concentration of an analyte as accurately as possible while understanding the practical limitations of this estimation. To calibrate an immunoassay requires obtaining the assay response as a function of known concentrations (or calibrators). Besides the contribution to calibration from analytical sources (e.g., pipette or calibrator accuracy), practical considerations that also affect the accurate estimate of concentration include 1) the confidence level required for the given application and 2) the economics of the analysis. Many problems associated with immunoassay calibration could be reduced by an increased number of more closely spaced calibrators, each analyzed with greater replication. Unfortunately, costs may preclude this approach and compromises are made at the expense of higher certainty.
0097-6156/96/0646-0240$15.00/0 © 1996 American Chemical Society
In Environmental Immunochemical Methods; Van Emon, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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Immunoassay calibration does not relieve the analyst of responsibility for reliability of his measurements; it provides statistically sound results and a means to assess reliability. Several elements go into the development and support of a sound and reliable immunoassay: an understanding of the chemistry, careful validation of the method, and a comprehensive quality control program. Faithful execution of the protocol developed from such a program should yield high quality results; however, the analyst is ultimately responsible for rational application of the calibration method. Internal, institutional, or government regulations (e.g., GLPs) should be reviewed for practical effects on the quality of these elements.
Characteristics of Immunoassay Calibration Curves There are a few characteristics that are constant for all immunoassay calibration curves, regardless of the immunoassay technique being employed. First, the measured assay response has a nonlinear relationship to analyte concentration. A simple, straight-line analysis cannot be applied over the working range of the assay (typically 2 to 3 orders of magnitude). If an immunoassay working range were limited to a "linear" portion of the calibration curve, it would result in the loss of large amounts of valuable analytical data. Data can be transformed to yield a linear relationship (e.g., logarithmic, logistic); however, the analysis is essentially nonlinear. Since the calibration curve is nonlinear and a limited number of calibrators are used, there are many curves that could pass through a given set of calibrator points. A choice of fit must be made, which introduces a risk of bias. Under some measurement conditions, assay errors may be large relative to the analyte levels being measured. When measuring the calibrators, these errors may contribute a significant uncertainty in determining the relative position of a calibration line, even when its general shape is defined. Errors are not constant in every region of the assay's working range and, as a result, there is less confidence in the calibration curve in some parts of the concentration range than others. Since calibration is not constant in every batch, a new curve may need to be determined for every run. For these reasons, method developers should specify recommended procedures and operating conditions to minimize the potential for error, including the use of calibrator replicates and curve-fitting methods for the calibration curve.
Curve Fitting Methods Numerous mathematical methods to adjust the calibration curve have been proposed and are well characterized (1-3). Some examples are given in Table I. These methods can be divided into three major groups; first, the empirical methods, so named because their use is based on practical success, not on some physicochemical model for the assay process. In several of these methods (e.g., point-to-point, spline functions, polygonal interpolations), the calibration curve will closely fit the experimental data, regardless of how unlikely the data are on chemical grounds. The position of each segment of the calibration curve is largely independent of the rest of
In Environmental Immunochemical Methods; Van Emon, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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the curve and it is possible that some segments will be accurate and others not. Consider the polynomial interpolation, for which an n-th order polynomial is made to fit a given set of n+1 data points. The fit can result in highly erratic oscillations between data points. Such oscillations are typical of higher order polynomial fits, and, consequently, a cubic fit is the most commonly used with immunoassay data. In general, the order selected should be much lower than the available number of points to be fitted.
Table I. Curve Fitting Methods Semi-empirical methods Empirical methods Log-Log Manual curve fitting Reciprocals - 1/B, T/B, F/B, Bo/B Point-to-point linear Logistic - two, three, four, five, six Polygonal interpolation parameter Spline function interpolation Polynomials - straight line, parabolic, cubic, quartic, adjustable order Model-based methods Rectangular hyperbola Scatchard Two, three, four, five, N-parameter Exponential function of concentration Log concentration Log response
The second group of methods can be referred to as semi-empirical because there are theoretical justifications, under very rigid, simplifying assumptions, that predict the calibration curve. The most common of these is the logistic function, first named by Berkson in the 1920's and used in population studies, tumor growth, and economic models. This method was first introduced for immunoassay calibration by Rodbard in the late 1960's (4). The simplest form of logistic function is the popular log-logit. The log-logit model produces two parameters: the slope and the intercept of the linear regression fit to the transformed data. The final group of methods is based on equations derived from the Law of Mass Action applied to antibody-antigen binding systems at equilibrium. This approach is attractive because it is based on sound chemical theory and is therefore likely to be more reliable than any arbitrary model. In practice, however, these models are partly empirical because the actual mechanism of the reaction is more complex than the assumptions. Log-linear Curve Fitting. A plot of a typical immunoassay calibration curve is shown on linear axes in Figure 1. Since data from immunoassays may form a straight line when plotted as the log concentration versus response, some investigators have referred to it as "linear." Clearly the relationship is not linear in the same sense that absorbance and concentration are linearly related by Beer's Law. One drawback to the log-linear transform is that unphysical response values are predicted at extreme concentrations. At very low concentrations, the transform will result in responses
In Environmental Immunochemical Methods; Van Emon, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
Downloaded by UNIV MASSACHUSETTS AMHERST on October 8, 2012 | http://pubs.acs.org Publication Date: October 23, 1996 | doi: 10.1021/bk-1996-0646.ch021
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approaching infinity; at high concentrations, negative responses would be obtained. As a result, the analyst must be careful to use this type of transform across a limited range of concentrations. At higher concentrations, the responsiveness (the ability to quantify small changes in concentrations, see also section on Precision Profiles) of an immunoassay decreases, so the upper limit imposed by the transform itself will not be unnecessarily restrictive. Limiting the calibration curve at low concentrations, however, may result in the loss of useful information where the assay might still provide accurate and precise results. In general, kit developers should recommend appropriate lower limits for their protocols and discourage extrapolating concentrations beyond the standards. Log-logit Curve Fitting. In practice, the shape of the immunoassay calibration curve is sigmoidal (Figure 2). Unlike a log-linear relationship, the actual calibration curve of an immunoassay has a maximum and minimum that are approached asymptotically at extremes of concentration. The maximum response is referred to as B and the calibration curves are typically given in terms of B/B , where Β is the assay response at a given concentration, c. The value of B/B has a maximum of 1 and can approach 0 at its minimum. In general, a sigmoidal formula for competitive immunoassays can be written as 0
0
0
Β B ~l 0
1 + (c/c )
(1)
b
0
where c is the concentration at which B/B is 0.5 and b is a fitting parameter for the model (where typically 0 < b
