323© 2013 David G. Wild. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/B978-0-08-097037-0.0002...
324 The Immunoassay Handbook
log of the standard concentrations. The logit transform
yields unreliable results at the low ...
325CHAPTER 3.6 Calibration Curve Fitting
These variance regressions, which are obtained from
the normal behavior of random...
326 The Immunoassay Handbook
the same error as the squared residual error observed
with the fitted regression models, assum...
327CHAPTER 3.6 Calibration Curve Fitting
under consideration. This χ2 probability has been called a
fit probability.
, (6)
...
328 The Immunoassay Handbook
number of parameters that any general asymmetric sig-
moidal function can possess is five: one...
329CHAPTER 3.6 Calibration Curve Fitting
approach the left and right asymptotes at different rates
unless g = 1.
Considera...
330 The Immunoassay Handbook
G The number of replicates of the calibrator points and
the unknown sample,
G The amount of e...
331CHAPTER 3.6 Calibration Curve Fitting
utility when developing an assay, when troubleshooting
the effects of changed con...
332 The Immunoassay Handbook
Outliers
Identifying outliers is a requirement for all immunoassay
data. Outliers are not to ...
333CHAPTER 3.6 Calibration Curve Fitting
Residual outliers are isolated points on the curve that do
not match the behavior...
334 The Immunoassay Handbook
be achieved by running at least four adjusters in duplicate,
but this would negate the potent...
335CHAPTER 3.6 Calibration Curve Fitting
Fitting process
The provision of many extra data points has another ben-
efit. It ...
336 The Immunoassay Handbook
calibration would comprise of a periodic determination of
analyzer bias from the master analy...
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The immuassay handbook parte37

  1. 1. 323© 2013 David G. Wild. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/B978-0-08-097037-0.00022-1 Quantitative immunoassay relies on a calibration curve to indirectly determine the analyte concentration of a sample from its response. The calibration, or standard, curve is a plot of known concentrations against their respective sig- nal responses. These calibration curves usually span two or more orders of magnitude on the concentration axis and one or two orders of magnitude on the response axis. If an infinite number of concentration dilutions, each with an infinite number of replicates, could be assayed, the true curve of each assay could be easily determined. However, since only a limited number of dilutions can be run, the true curve must be estimated from a relatively sparse num- ber of noisy points. Since there cannot be a standard at every concentration, a means of interpolating between standards is necessary to estimate the dose–response rela- tionship between standard dilutions. While the curve can be drawn by hand with a pencil, flexicurve, and sheet of graph paper, the process is normally done by using a math- ematical function, or regression, to approximate the shape of the true curve. This approximating function is called a curve model. The curve model, which typically describes a family of curves using two or more parameters, is then fitted to the assay data by adjusting the curve model’s parameters to obtain the one curve from the family of curves that best describes the assay data. The computation of these curve models is typically performed by software programs of varying sophistication. In any regression, regardless of the curve model used, there are two sources of the total error of that curve. Ran- dom variation in the assay data is one source and can be reduced by increasing the number of replicates and/or dilutions of the standard. The second type is caused when the curve model cannot fit the shape of the true curve accurately. Nothing can reduce this lack-of-fit error except using a more appropriate curve model. For exam- ple, a straight line cannot fit a sigmoidal curve shape with- out error no matter how many dilutions are run. There has been a divergence between the highly auto- mated clinical diagnostics sector and other fields, includ- ingpharmaceutical,lifescienceresearch,andenvironmental monitoring. For areas where immunoassays are being applied for research, the general trend is toward greater accuracy and precision and the terminology reflects that. The analyte is typically prepared from a standard solution by a series of dilutions, hence the term standard curve is commonly used. Samples are tested in batches, or assays, with their own standard curve. Curve-fit algorithms may use error data derived from multiple assays. This is where advanced, flexible curve-fit software programs from spe- cialized vendors are useful tools. However, in the clinical diagnostics sector, for mainstream analytes, the trend is toward random access using singleton measurements and stored calibration curves. The term “calibration curve” is historically derived from immunoassay kits, which included prepared calibrator sets with the reagents. In the health care sector, the primary objectives are cost reduction and the ability to test individual samples on demand. The achieve- ment of these additional requirements reduces the options available to developers to achieve accurate and precise results, although the high level of instrument automation and reagent manufacturing control reduce some of the sources of error. Although the science of curve fitting is much the same, the applications have followed different paths. Response Transformations The two fundamental variables involved in curve fitting are the analyte concentration (often termed dose) and sig- nal level, or response, of a sample. There are many differ- ent types of signal, such as radioactive counts in radioimmunoassay, light intensity in luminescence immu- noassays, and color in ELISA immunoassays. Immunoas- say concentration gradients are roughly logarithmic in relation to the linear signal responses. When concentra- tions are plotted on a logarithmic scale and the responses plotted on a linear scale, the curves are sigmoidal in shape. Sometimes responses are transformed from their origi- nal signal, but this should be done with caution. For exam- ple, subtracting the average blank response from samples results in low standard responses less than or very close to zero, which in turn distorts the regression fit. Expressing the response as percent bound ratio changes the distribu- tion of that expression at low percent bound ratios from normal to binomial and requires an arc sine transform to convert its distribution back to normal. The logit transform is an attempt to transform a sym- metric S-shaped curve with a single point of inflection into a straight line. In competitive assays, the logarithm of the concentration is plotted against the logit of the signal, cor- rected for the binding at zero concentration (B0) and the non-specific binding (NSB). See Fig. 1. , (1) The B0 is obtained from the zero concentration calibrator in a competitive binding assay or the highest calibrator in an immunometric assay. The NSB may be estimated from zero calibrator tubes into which no antibody is added in a competitive binding assay or the zero calibrator standard from an immunometric assay. A linear regression is then computed from the logit-transformed responses and the Calibration Curve Fitting John Dunn (jdunn@brendan.com) David Wild (david@davidwild.net) C H A P T E R 3.6
  2. 2. 324 The Immunoassay Handbook log of the standard concentrations. The logit transform yields unreliable results at the low and high concentration regions, is not effective with asymmetrical immunometric assays, and is not used much anymore. Sometimes a log function has been used in an attempt to reduce asymmetry in sigmoidal dose–response curves when computing with a four-parameter logistic (4PL). However, the log transform has the opposite effect and increases asymmetry and worsens the 4PL fit, when the transition point is above the midpoint of the curve, which is typical for most ELISA and immunometric assays. In practice, unless the errors introduced by the transfor- mation are correctly handled in the regression fitting, these transformations will introduce more error in addi- tion to the random error and the lack-of-fit error already present in the regression. Determining the Response– Error Relationship It is necessary to determine the random error associated with each sample response in order to obtain what is called the maximum likelihood estimate of the true curve in regression theory. In immunoassays, the random error, expressed as the variance of the response, is caused by three factors: G Magnitude of the response G Kinetics of the antigen–antibody reaction at each concentration G Signal error from the detector. It is common for the variances at the high and low responses to differ by three or four orders of magnitude. This is not surprising since the responses themselves can differ by two or more orders of magnitude. Most detectors produce signal noise with a standard deviation that is proportional to the magnitude of the response. An exception is the error from isotopic and luminescent detectors. These detectors measure discrete “counts” of light photons, and their error is Poisson, i.e., the square root of the number of counts. The error from the kinetics associated with antibody binding is nonlinear, with the result that the kinetic varia- tions in the reaction change disproportionately as the ratio of analyte to primary binder and tracer binding changes. It is this second factor that is responsible for most of the dis- similarity in variance patterns between test methods, which is not surprising since the reaction kinetics between differ- ent antibodies vary so much. As a result, immunoassay responses are always heteroscedastic and do not have a constant variance. Because of the substantial error from the kinetics, these variances cannot be made homoscedas- tic, or constant, using, for example, a log transformation or a simple 1/Y or 1/Y² formula and must be determined individually for each test method. This response–error relationship is usually expressed as a regression function itself. The variance of the standard dilutions can usually be approximated by a power function of the response, (2) where A is a function of the magnitude of the responses and the average noise level, and B falls in a range of 1.0– 2.2. In some cases, adding a constant minimum variance parameter (C) will improve the variance fit of very low responses, (3) Since it is impractical to run enough replicates to get a reliable estimate of the true variance function from a sin- gle assay, responses from a pool of randomly selected assays of that test method are required. This has the added benefit of incorporating the differences in intra-assay vari- ation observed between assays. A one-way analysis of vari- ance (ANOVA) is performed separately on each dilution, using the replicates from each assay. A power linear regres- sion is then computed using the log of the mean response versus the log of the error mean sum of squares of each dilution to generate the variance regression. These expected variances are then used to weight the dose–response regressions using the inverse of the variance at each point. See Fig. 2. FIGURE 1 Logit–log method. FIGURE 2 Only weighting equations from pooled assays model the actual variances of a dose response curve. (The color version of this figure may be viewed at www.immunoassayhandbook.com)
  3. 3. 325CHAPTER 3.6 Calibration Curve Fitting These variance regressions, which are obtained from the normal behavior of randomly-selected assays from a test method, can also be used to evaluate the precision of replicates from assays of that test method. These variance regressions are also required to compute the error profiles that will be discussed below, as well as determining the limits of quantitation, the limits of detection, parallelism curves in potency assays, and other computations. Curve-Fitting Methods A good curve model should possess three properties. First, the curve model must do a good job of approximating the shape of the true curve. If the curve model does not do this, there is no way to compensate for this lack-of-fit com- ponent of the total error. Second, a good curve model must be able to average out as much of the random varia- tion as practical to produce concentration estimates with low error. Third, a good curve model must be able to pre- dict accurate concentration estimates for points between the anchor points of the standard dilutions. EMPIRICAL Empirical, or interpolatory, methods such as point-to- point and cubic splines have been used because they are simple to run. These functions pass exactly through the mean data points. Because these empirical methods pass through the data points, there is no averaging of the data to reduce random variation. Since the random error of these points shifts the data from their true value, these empirical curves are guaranteed not to be good estimates of the true curve. Point-to-point curves do not attempt to approximate the area between the data points (see Fig. 3), rendering concentration estimates in these regions unreliable. Cubic splines are not always monotonic and can oscil- late up and down because of the random variation in every node point, instead of producing a continuous, smooth functional form (see Fig. 4). Because of these and other weaknesses with empirical methods, the concentration estimates contain a greater amount of error than curve regressions with fewer parameters that are able to average out the random variation. REGRESSION Regression methods fit a given functional form or model to the data, so that errors in the calibrator points are par- tially corrected for, making the calibration curve more robust. These regression models are chosen for their abil- ity, with parameterization, to assume a shape that matches the dose–response shape of the standard curve. Use of a good regression model is especially important if the cali- brators are run as singletons. The statistical technique most commonly used to esti- mate the parameters in any regression method is least- squares fitting. In least-squares fitting, the vertical response distance, or residual, of each point from the curve is calculated and squared. The sum of these squared resid- uals is called the sum of squares error (SSE). The least- squares procedure selects the curve that gives the smallest SSE. An illustration of this is shown in Fig. 5. It should be obvious that there is a problem if the SSE is simply the sum of the individual squared residuals. If the individual errors between points are proportional to each other, the squared difference between the observed and the curve responses will be very, very high at the high- response end and very, very low at the low-response end. For example, if the error of all points is 5%, at the high end an observed response of 11,500−11,000 from the com- puted curve is a squared residual of 250,000. Conversely, a 5% difference at the low end is a residual of 105−100, observed minus computed, or a squared residual of 25. Clearly this means that the regression algorithms would be fitting the curve using, essentially, only the high end. The low end would contribute virtually nothing to the SSE that would not be overwhelmed by the high end no matter how bad the fit was at the low end. The random error estimate generated from the repli- cate variances of the pooled assays described above is FIGURE 3 Point-to-point linear interpolation. FIGURE 4 Spline function between nodes.
  4. 4. 326 The Immunoassay Handbook the same error as the squared residual error observed with the fitted regression models, assuming no model lack-of-fit error. Therefore, the expected variance from the variance regression will be the same as the squared residual error at each response, when averaged from pooled assays. When each squared residual is divided by its expected variance from the variance regression, i.e., weighting the responses, all points contribute equally to the fitted curve. This means that those concentra- tions that have the least relative error have a greater impact on the curve than those points with more pro- portional error. Now, the best fitting curve, i.e., the curve with the lowest SSE, will be the optimal maximum likelihood estimate of the true curve, as predicted from regression theory. That explains why sample concentra- tions computed from unweighted regression fits can dif- fer from properly weighted curves by hundreds of percent. (4) The weighted SSE is sometimes referred to as wSSE (weighted sum of squares error), or RSSE for residual sum of squares error. Statistical theory adds the stipulation that the error at each point is normally distributed. In practice, this works fine so long as each dilution point displays a central ten- dency in its distribution. This behavior is generally true for all immunoassays. Data points should not be used when the signal values from the detector are beyond the linear range of their capabilities. All detectors have a finite range where their signal is proportionately linear to the amount of label material producing the signal. When assay ranges are extended below or above this linear range, the mean signal itself is wrong, and the distribution of that signal becomes distorted and is no longer as predicted. CURVE-FIT METRICS It is necessary to have a means of assessing the quality of a curve fit. There are no metrics that can assess empirical methods. When measuring unweighted nonlinear regres- sions, there are no statistically-appropriate metrics that are meaningful, so the goodness-of-fit is usually assessed with the r2. This metric is a measure of the proportion of the responses that fit the regression model, not the residual amount that did not fit. The r2 is primarily a measure of whether there is a causal relationship between the concen- tration and its associated response and is not well suited for nonlinear regressions. This is why even obviously bad curves usually have good r2 values. With weighted regressions, the SSE itself is a metric that is a direct measure of how well the curve model fits the data. The SSE is often expressed as a residual variance by dividing the SSE by the degrees of freedom of the curve (number of points minus number of parameters) to normalize the metric between different curve models and data point num- bers. However, neither of these metrics provides any infor- mation about how good or bad the fit is relative to all of the values that could be obtained from good assay curves. It is a statistical property of the weighted SSE that, if the responses are normally distributed (a requirement satisfied with the central tendency of this data), the SSE is a χ2- distributed value at number of points minus number of parameters degrees of freedom. (5) This property allows a χ2 probability to be determined. The p value of the SSE can be viewed as the fraction of an infinite number of assays that, if performed under exactly the same conditions, would be expected to have a worse curve fit, i.e., a larger SSE, than the curve fit of the assay FIGURE 5 Illustration of wSSE from individual squared residuals.
  5. 5. 327CHAPTER 3.6 Calibration Curve Fitting under consideration. This χ2 probability has been called a fit probability. , (6) Since this metric is a probability, a fit probability of 0.01 or above can be considered acceptable. WEIGHTED LEAST-SQUARES METHODS There are three weighted least-squares regression meth- ods in general use for immunoassay dose–response curves. These are linear regression, the 4PL and the five-parame- ter logistic (5PL). All three of these methods use weighting for the individual points, and all of them find the lowest SSE for the solution, a process called minimization. All least-square regression curves require at least one more data point than there are parameters in the model. These extra degrees of freedom are what allow the averaging of the error in these models. Linear Regression Model Linear regression is the simplest of these methods because it is a closed form function that can be solved alge- braically. This means that there will be an exact solution for the regression parameters. This makes the computa- tion simple enough to perform on a handheld calculator, or simple software programs, and all will get the same solution. The formula: where Y is the response and X is the concentration, gener- ates a straight line having a slope of b and a Y intercept of a. Concentrations are determined by inverting the formula to The problem, of course, is that all but the shortest immu- noassay curves are nonlinear. Various methods have been used to “linearize” this curve, the most popular being the logit transform discussed above. These linearization schemes were necessitated by the poor or nonexistent com- puting resources that were available at the time. But attempt- ing to linearize a nonlinear curve is a poor solution, and for many years, these transformation attempts have been replaced by nonlinear curve models like the 4PL and 5PL. The linear logit–log model is sometimes considered to be related to the 4PL model (a 4PL curve transforms to a straight line in logit–log space). These in turn have been shown to have certain approximations to the mass action model, the only model where the parameters are measures of physical properties. But the complexities of ascertaining these physical properties, and modeling them in a kinetic regression formula, have prevented any practical applica- tion of such a model from appearing. Nonlinear Curve Models Nonlinear curve modeling is much more difficult than linear regression. Finding the solutions to these models requires using numeric processes to find a solution. Numeric processes are iterative processes that incrementally reparam- eterize the coefficients to find better solutions (i.e., lower SSEs). There are two major steps involved, and each of them is critical to the process. The first is finding the initial starting estimates of the parameters, and the second is find- ing the best solution in the region the first step placed the fitting algorithms. The region identified in the starting esti- mates is very important because, in the four- or five-dimen- sional geometric space of the 4PL or 5PL, respectively, there is one, and only one, set of coefficients that is the global minimum, i.e., the best fit. But there are many local minima that can fool the fitting algorithms into settling on a solution that is not the best fit, and sometimes a local set of minima gives a terrible fit. Sometimes the fitting algo- rithm cannot find a solution at all. Marquardt–Levenberg and Gauss–Newton are popu- lar minimization algorithms used in many immunoassay software programs. They are generally adequate for find- ing solutions to 4PL models. But these minimization algo- rithms have well-documented problems finding solutions in 5PL space. Software programs that use more powerful numeric algorithms are more successful with the 5PL. Although they require more computational speed and memory, modern PCs are generally powerful enough to run these programs. It is important to note that these nonlinear curve models are mathematical shape functions only. Their parameters do not correlate with any physical properties of the immu- noassay reaction. Sometimes the coefficients between sim- ilar appearing curves can be widely different, especially the b, c, and g coefficients of the 5PL, with very little differ- ence in the shapes. This is to be expected and does not matter so long as the SSE is low. The 4PL and 5PL models The 4PL model is widely used, in large part because the model is easier to fit computationally than the 5PL model. Widespread use of the 5PL has only become possible with the more powerful fitting algorithms available in recent years. The 4PL model works well when the dose–response curve is symmetrical. The formula for the 4PL curve is: (7) where a and d are the asymptotic ends, b controls the tran- sition between the two asymptotes, and c is the transition point midway between the two asymptotes where the curve changes direction. Concentration estimates for the 4PL curve can be obtained from the formula: (8) The 4PL model can be extended by adding a fifth param- eter, g, which controls the degree of asymmetry of the curve. With the extra flexibility afforded by the asymmetry parameter, the 5PL model is able to eliminate the lack- of-fit error that occurs when the 4PL is fitted to asym- metric dose–response data. The 5PL model provides an excellent compromise between over-parameterized mod- els that can fit data closely at the cost of a large variance in the predictions and under-parameterized models that suf- fer from large lack-of-fit errors. This is because the fewest
  6. 6. 328 The Immunoassay Handbook number of parameters that any general asymmetric sig- moidal function can possess is five: one for the upper asymptote, one for the lower asymptote, one for the over- all length of the function’s transition region, one for the location of the transition region, and one for the degree of asymmetry. It is unlikely that any function with less than five parameters will have the flexibility necessary to pro- duce a high-quality fit to asymmetric sigmoidal dose– response data. The formula for the 5PL curve is: (9) where a and d are the asymptotic ends, b controls the tran- sition between the two asymptotes, c is the transition point where the curve changes inflection, and g, with b, controls the rate of approach to the lower asymptote. Note that in the 5PL, the transition point c is not in the center between the two asymptotes except when g=1, in which case the 5PL reduces down to a 4PL. Concentration estimates for the 5PL curve can be obtained from the formula:     (10) The families of curve shapes that the 5PL can assume by varying one parameter at a time are shown in Fig. 6. Figure 6 makes several characteristics of the 5PL function apparent. The function approaches a horizon- tal asymptote as the dose approaches zero, and it approaches a horizontal asymptote as the dose approaches infinity. Between the asymptotic regions of the curve is a transition region which contains a single inflection point. On either side of the inflection point the curve will FIGURE 6 Effects of varying the parameters of the 5PL function.
  7. 7. 329CHAPTER 3.6 Calibration Curve Fitting approach the left and right asymptotes at different rates unless g = 1. Consideration of Fig. 6 gives some insight into how the five parameters of the 5PL function affect the resulting curves. Parameters a and d control the position of the curve’s horizontal asymptotes. Examining the behavior of the curve in its asymptotic regions provides additional insights, especially into the roles of b and g. When approaching the “a” asymptote, only parameter b controls the rate of approach to the asymptote. However, when approaching the “d” asymptote, the rate of approach is controlled by the product bg. This coupling of parameters is one of the reasons the 5PL is harder to model. The table below summarizes the effect of the parame- ters a, b, and d on the slope of the logistic function. Because the 4PL function is point symmetric on semi- log axes about its midpoint (g=1), the literature has adopted two conventions for eliminating this redundancy in the parameterization of the 4PL function: either a>d is fixed and the sign of b determines the slope of the logistic or b>0 is fixed and the ordering of a and d determines the slope of the logistic. In the 4PL case, neither of these con- ventions restricts the range of functions that can be produced. In contrast, the 5PL function has no symmetry. There- fore, the cases of Table 1 all yield distinct functional forms. Cases 1 and 4 are both suitable for modeling decreasing dose–response data, while cases 2 and 3 are both suitable for modeling increasing dose–response data. The a>d form can model sharp transitions at the high end of the curve (with shallow transitions at the low end), and the a<d form can model sharp transitions at the low end of the curve (with shallow transitions at the high end), but neither can model the reverse. These cases can arise with some bioassay dose–response curves but are not typically observed with cell-free immunoassay data. For immunoas- say curves, either 5PL form can model the curves accurately. Figures 7 and 8 show an immunoassay curve fit with a weighted 4PL and the same data fitted with a weighted 5PL. The squared residuals plotted below the graphs illus- trate the greater curve error with the 4PL in the curve-fit metrics observed compared to the magnitude of the squared residuals of the 5PL. Few immunoassay curves are completely symmetric, with immunometric or sandwich assays such as ELISAs being particularly asymmetric. Also, improvements in signal-to-noise ratios have tended to increase the amount of asymmetry observed in these dose–response curves. The 5PL will have a lower SSE because of its greater flexibility in shape and is usually the better choice. But if the curve is fairly symmetrical, the SSEs from the 4PL and 5PL will be similar. This in turn may result in a slightly higher fit prob- ability for the 4PL than the 5PL due to the extra degree of freedom of the 4PL curve. In this case, either curve model would be justified. If the curve does not reach the inflection point, there is only one end region and the 4PL will be sufficient. ESTABLISHING STANDARD DILUTION CONCENTRATIONS Standard dilution concentrations should be chosen that generate an even gradient of responses. It is not necessary that the concentrations be evenly distributed on the log scale. What is necessary is that the response gradient should be evenly separated when plotted on a linear scale, and none of the responses should be clumped together at the same response value. The plot of the squared residuals of the assay below the assay curve in Fig. 9 shows how the distribution should appear. Note that there is an even distribution of individual responses for the entire response range. In Fig. 10, the squared residuals of another assay shows the effect of an uneven gradient of responses on the mag- nitude of the squared residuals in the region with the clumped responses, and hence on the SSE of the weighted regression fit. This clumping of responses relative to the rest of the responses results in uneven fitting throughout the curve, as shown by the squared residual plot. Concentrations should be chosen that evenly span the range of dose-dependent changes in response. When the dose no longer has an effect on the response (the plateau regions), there should not be more than one dilution. More than one dilution in the plateau regions makes it likely that these points will be nonmonotonic. Even if the variation of these nonmonotonic points is within their variance ranges, they will decrease the acceptability of the curve fit. Since these points only contribute noise, they only shift the shape from the more accurate estima- tion of the true curve derived from the more sensitive points. This is especially true when hooks are not excluded. Error Profiles The purpose of immunoassay tests is to determine the concentrations of unknown specimens. These concentra- tion estimates are only valid if the error of their estimates is not so high that the result is meaningless. There are five factors that make up the error of the concentration estimates: G The response variation at that point, G The response/dose slope at that point, G The closeness of adjacent standard calibrators to that point, TABLE 1 The Relationship between the Order of a and d, the Sign of b, and the Slope of the Monotonic 5PL Function Case No. Order of a and d Sign of b Slope 1 a>d b>0 Down 2 a>d b<0 Up 3 a<d b>0 Up 4 a<d b<0 Down Note: For g=1 (4PL curve), case no.1 generates the same functional form as case no. 4, and case no. 2 generates the same functional form as case no. 3. For 5PL, all four cases produce distinct functional forms.
  8. 8. 330 The Immunoassay Handbook G The number of replicates of the calibrator points and the unknown sample, G The amount of error in the regression. Each of these factors can be addressed during assay development to reduce their contribution, but this is only practical when the amount of error can be measured. The response variation, discussed earlier, can be reduced by further optimizing incubation conditions and reagents. The response/dose slope can be made steeper in diagnosti- cally important regions by varying assay conditions. Cali- brator concentrations, number of dilutions, and replicate numbers can be adjusted. More appropriate curve models can be selected. But what is needed to measure the effect of any changes to the assay is an accurate determination of the error profile throughout the dose range. The mathematical determination of this error is very com- plex for the nonlinear curves produced by immunoassays. Earlier methods from the literature simplified the error determinations by only including the error from the first two bullet points above. But these methods can miss a third or more of the actual error and give unreliable error estimates at the low and high ends. These earlier methods were usually termed precision profiles, and the error was expressed as a %CV to approximate the empirical values of the variation obtained from repeated measurements of known specimen concentrations. Recently, estimations using Monte Carlo methods have enabled accurate determinations of the error profile of an individual assay to be made that include all of the error factors. In these more accurate error profiles, the error is typically expressed as a percent error of the concentration. Individual sample concentration error from both individ- ual replicates and dilution averages is then derived from these error profiles. These error profiles are of inestimable FIGURE 8 The SSE and fit probability of the 5PL fit are 1.386 and 0.350, respectively. FIGURE 7 The SSE and fit probability of the 4PL fit are 29.931 and 0.0003, respectively.
  9. 9. 331CHAPTER 3.6 Calibration Curve Fitting utility when developing an assay, when troubleshooting the effects of changed conditions on assay results, and for establishing the limits of quantitation that will determine the reportable range of an assay. An error profile plot from a standard curve plots the concentration on the x-axis and the %Error on the y-axis. Determinations of the limits of quantitation can be easily made by setting the amount of error allowed for report- able results and calculating the lowest and highest concen- trations that do not exceed that amount of error. Three sets of immunoassay data were computed using the pro- gram StatLIA® from Brendan Technologies, Inc., in the examples below using a weighted 5PL curve model, as described in Gottschalk and Dunn (2005a). All of the curves had fit probabilities above 0.3. One assay curve was normal (Fig. 11), one assay curve matched an incubation temperature that was too low (Fig. 12), and one assay curve increased the number of dilution points at each end and reduced the number of dilutions in the center of the curve (Fig. 13). Error profiles at a 95% significance level were computed for each curve, and the results are dis- played below. Assuming a 50% acceptable %Error for the limits of quantitation, the minimum and maximum acceptable concentrations (MinAC and MaxAC, respectively) can be easily seen from the error profiles below their respective dose–response curves. Note the reduced reportable range and increased %Errors in Fig. 12, which had suboptimal incubation conditions. In Fig. 13, note how increasing the number of dilutions at each end increased the reportable range half an order of magnitude at the low end and 30% at the high end when compared to the limits in Fig. 11. Also note the increased %Error in the regions where there are no close standard points to anchor the regression. FIGURE 9 The concentrations are all 1:2 serial dilutions and evenly spaced apart, and the responses are also evenly spaced apart. The squared residuals are similar in magnitude for all doses. FIGURE 10 The concentrations are all 1:2 serial dilutions and evenly spaced apart, but the responses are not evenly spaced. The squared residuals are all high where the responses are clumped and small where they are sparsely distributed.
  10. 10. 332 The Immunoassay Handbook Outliers Identifying outliers is a requirement for all immunoassay data. Outliers are not to be confused with failed samples. Outliers are responses whose values are so far removed from the normal distribution of that sample that the probability that it is a member of that population is prohibitively low, generally less than 0.0001. A failed sample response, how- ever, is statistically likely still to be a member of that popula- tion but has behavior that does not pass acceptability, such as a probability less than 0.01 but above 0.0001. The differ- ence between the two is that an outlier response can, and should, be ignored and the assay recomputed; but a failed sample response should not be ignored even if the assay fails. There are two types of outlier responses: precision outliers and residual outliers. Precision outliers are rep- licate samples that have too much difference between their response values. Often this variation is expressed as %CV (the standard deviation of the responses as a percentage of their mean). Acceptability is determined by an arbitrarily selected %CV applied to all replicate sets. When three or more replicates are assayed and the %CV is unacceptable, attempts are sometimes made to identify the replicate response farthest from the other samples as an outlier. Occasionally, Dixon or similar outlier tests are used to fil- ter these outlier replicates. The problem with methods like Dixon is that the sample size is too small for a reliable esti- mation of the distribution of that sample. A more sound method for evaluating replicate behavior is to use the variance regression discussed above. Using the ratio of the observed replicate variance divided by the expected variance of that response generates an F statistic, and an F probability can then be obtained using the appro- priate degrees of freedom. These would be the number of replicates—one for the numerator, and the sum of the individual ANOVA error degrees of freedom—two for the denominator. These F probabilities (precision probabili- ties) are tailored for that test method because the compari- son is to the normal behavior at that response level in that test method, as determined by its historical behavior. If there are three or more replicates, a Grubbs threshold can be determined using the degrees of freedom of the vari- ance regression to determine if a replicate can be ignored. 0.5 1 5 10 50 Concentration 25 50 75 100 125 150 175 % Error MinAC MaxAC 0.5 1 5 10 50 100 Concentration 500 1000 1500 2000 2500 3000 Response FIGURE 11 Normal curve. 0.5 1 5 10 50 Concentration 25 50 75 100 125 150 175 MinAC MaxAC %Error 0.5 1 5 10 50 100 Concentration 500 1000 1500 2000 2500 3000 Response FIGURE 12 Depressed curve. Concentration 0.5 1 5 10 50 100 25 50 75 100 125 150 175 % Error MinAC1 MaxAC1 MinAC2 MaxAC2 0.5 1 5 10 50 100 Concentration 500 1000 1500 2000 2500 3000 Response FIGURE 13 Dilution point effects.
  11. 11. 333CHAPTER 3.6 Calibration Curve Fitting Residual outliers are isolated points on the curve that do not match the behavior of the other points and make it impossible to get an acceptable curve fit. These residual out- liers can be the mean dilution response from replicates with acceptable precision, meaning that the dilution itself was improperly prepared, or from a single outlier replicate response. Residual dilution outliers are identified by comput- ing an F probability using the squared residual of the point as the F statistic, with the number of points minus number of parameters as the numerator degrees of freedom, and the variance regression degrees of freedom for the denominator. Stored Calibration Curves, Factory Master Curves, and Adjusters In life sciences and pharmaceutical research, and many other fields where immunoassays are applied, the samples are usually run in batches with freshly-prepared standard curves made by diluting a master standard solution. Assay kits usually contain a set of calibrators supplied with the reagents. But in the clinical diagnostic field, to improve the turnaround time, individual samples can be loaded onto a random-access analyzer at any time. This requires that calibration curves are stable over a period of time. Most immunoassay systems on the market have calibration curves that are stable for at least 2 weeks and across reagent packs within one lot, although controls should still be run regularly. This reflects the level of stability and consis- tency now achievable with reagents and equipment. Recently, this stability has also led to a reduction in the number of calibrators required for user calibrations, through the provision of a master calibration curve from the manufacturer. A reduced number of calibrators are run by the user to adjust the master curve to take account of bias due to the user’s analyzer. Hence the calibrators, in this context, are really adjusters. In order to achieve random-access operation, the num- ber of requirements that must be fulfilled by the curve-fit and adjustment algorithms, and by the assay reagents and instrumentation, is significantly greater than for a batch assay. That being said, manufacturers have been able to provide remarkably precise sample measurements consid- ering the many potential sources of error, through rigor- ous attention to detail and extensive validation. It is beyond the scope of this chapter to comment on all the possible methods for using stored and reduced calibration curves, however, there are a few general principles. There are two different stages to the process: establish- ing the initial calibration curve—sometimes referred to as the “master curve”—at the manufacturer’s laboratory and the user adjustment in the local analyzer. This corrects for bias due to the analyzer or changes in the reagents since they were manufactured. MASTER CALIBRATION CURVE A master calibration curve is established by the manufac- turer for each reagent pack lot created. Its details are usu- ally encoded in some way on the packs, by a conventional barcode, a two-dimensional barcode, a magnetic card, or a smart card. The data may also be made available from a web site on the Internet or communicated to analyzers via e-mail. For each reagent pack lot, around 6–10 calibrator concentrations are used with high order replication. The responses are obtained from at least two different analyz- ers. Generally, at least 20 “replicates” for each calibrator are used to establish the master calibration curve. The cali- bration curve-fitting method chosen is usually 4PL or 5PL. Due to the closed nature of commercial systems, the curve-fitting process can be specially tailored to the par- ticular assay or family of assays. There are two major advantages of master curves, other than their obvious convenience for the user. First, the number of replicates is much greater than in conventional user calibrations. This is economically viable because the master calibration is only run once for all of the users of each lot. More concentration levels may be used, and many replicates run at each level, without excessive cost. The second advantage is that one set of master calibrator sets may be used to provide calibration curves to all the ana- lyzers, possibly for many years, removing a potential source of bias (due to calibration error) when conventional selling calibrators are calibrated from the master set. How- ever, for the purist, there is no substitute for running freshly made standards, unknown samples, and controls together in a batch format, because it removes many potential sources of error. But this is impractical for many clinical situations. ADJUSTERS No matter how many analyzers are used by the manufac- turer to produce a master curve, any bias due to the user’s analyzer will produce a bias if the master calibration curve is used. For this reason, some local calibration activity is necessary for each analyzer. The user calibration consists of running two or three adjuster calibrators with known concentrations and an algorithm to move the master calibration curve based on the signal levels for each adjuster. For example, if the mas- ter calibration curve gives a signal level of 1000 signal units at 100 concentration units and the adjuster, which has a concentration of 100 units, gives a signal level of 950 signal units, the algorithm may lead to a shift in the master cali- bration curve of 5% at this point. Using multiple adjusters, the entire master calibration curve is moved to allow for bias in the user analyzer. However, this highlights the weakness of the adjustment stage. In this example, was the difference of 5% between the signal levels due to genuine bias of the user analyzer from the analyzers used for the master calibration or was it just due to normal assay varia- tion? The error in this type of calibration system thus derives largely from the assay imprecision when the adjuster calibrators are run at the user laboratory. Any deterioration of the calibrators during shipment from the manufacturer, or due to insufficient refrigeration on stor- age, can cause the entire calibration curve to be biased. For the immunoassay system designer, the questions are: how many adjusters should be used, where should they be placed, and how should the user’s analyzer process the information they produce? Clearly the best results would
  12. 12. 334 The Immunoassay Handbook be achieved by running at least four adjusters in duplicate, but this would negate the potential economic advantages of the master curve. The compromise solution is that two or three adjusters are usually run, sometimes in singleton and sometimes in duplicate. Little has appeared in the literature about the theory of curve fitting associated with factory calibration with user adjustment, but we can offer some guidelines in respect of the problems involved. One key principle applies through- out: the total number of adjuster replicates must at least equal the number of parameters in the model that may change from the master curve determination. Linear Master Curves It is helpful to distinguish between different types of lin- earity. There is the direct linear form where there are only two parameters: slope (m) and intercept (c), that need to be determined. These parameters could well appear as the natural parameters after suitable transformations of response and dose. The other situation is where there is a pseudo-linear curve shape, where the response and dose metameters are linear in the direct sense, but there are other unknown parameters needed to specify the transfor- mation. An example is the 4PL, where if it is assumed that the NSB and B0 values are known, the logit–log transfor- mation produces a linear plot. These two situations need to be differentiated. Direct linear form There are two cases to consider: fixed and non-fixed slope. If the master curve has a fixed slope, then there is only one parameter to determine, namely the intercept. The mini- mum number of adjusters is therefore 1. For a homosce- dastic assay, if the slope of the line is m, the error in the response metameter for the adjuster σA, and the error in the response metameter for the unknowns σU, then the error in the interpolated dose is: (11) σU and σA can be reduced by replication. So if the maxi- mum error in prediction is specified, this will not only put constraints on m and σU, but also on σA, a result that could mean extra adjusters being needed in the form of replicates. If the master curve has a non-fixed slope, then the mini- mum number of adjusters needed is two as there are the two unknown parameters: intercept (c) and slope (m). Standard statistical theory indicates that the placement of the adjusters should be such that they span the linear range, thus avoiding the increased loss of precision due to extrapolation. Pseudo-linear form As mentioned earlier, the use of only two adjusters could be challenged on theoretical grounds, for most immunoas- says, since the linear relationship might be a consequence of a transformation from a mathematical form that had more than two parameters. For example, the logit–log transformation might well linearize a plot, but there are fundamentally four parameters to describe the response, the slope and the intercept in the logit–log domain, and the NSB and B0. If the NSB and B0 are “known” then the logit– log plot is a truly linear one with the slope and intercept defining its properties. If NSB and B0 are “unknown,” then extra information must be introduced into the process to infer their values. One possible approach is now described. Suppose conventional calibration curves are established in a number of assay runs at the factory, then sometimes a plot of percent bound against dose will produce a profile that is very stable across assays, even though the signal lev- els vary considerably between assays. This constant feature can be exploited to reduce the number of calibrators/ adjusters needed. Suppose two adjusters have the fixed per- centage bound B1 and B2, respectively, and for a particular assay run, they have responses R1 and R2, respectively. Constant percent bound means can be written as: , (12) Here B1, B2, R1, and R2 are known and so NSB and B0 can be determined. This is all that is required, together with the master curve to run the assay. The statistical problem that has to be resolved by the manufacturer is to determine the optimum adjuster concentrations, taking account of the impact on the interpolated dose. Also, with any calibration curve with asymptotes (NSB and B0), what procedure is adopted by the software when responses fall outside of the range of the calibration? Integration of the Model, the Master Curve and the Adjustment Process Most manufacturers currently provide adjusters so that the curve shape can be adjusted empirically, by pulling it up and/or down, without taking account of the underlying model. This approach is validated by measuring precision on many assays, over the full reagent shelf lives. But the method is very prone to single-point error and outliers. A better approach may be to record the key model parame- ters for the master curve, and then use the adjuster data to change the particular parameters in the model that have been shown, experimentally, to change from analyzer to analyzer and across the reagent shelf life. The number of adjusters would then be determined by the number of parameters that can change. Master curve model We can start the process of integration with the master curve model. The increased number of calibrators and replicates run in the factory laboratory allows for more complex modeling. For example a 4PL fit could be applied with additional constants included for variables unique to the chemistry and system, such as a reduction of horserad- ish peroxidase efficiency in the presence of high substrate levels, at very low concentrations in an immunometric assay. With as many as 200 replicates for the master cali- bration, there would be a more than adequate supply of data. So the master curve model could have many con- stants, reflecting the true variables in the assay. This would allow very precise, but constrained, modeling of the mas- ter calibration curve.
  13. 13. 335CHAPTER 3.6 Calibration Curve Fitting Fitting process The provision of many extra data points has another ben- efit. It is possible to systematically determine the optimum values of the parameters in the curve-fit model in a logical fashion. Conventional curve fitting may involve the deter- mination and fixing of parameters in a stepwise fashion to obtain best fit (least sum of squares) leaving one last param- eter to be optimized. This may have to take on an extreme value to make the model fit, due to errors in the determi- nation of the values of the other parameters. However, using the larger amount of data available in a factory cali- bration, it may be possible to determine each parameter independently, before finally making small adjustments to the values to obtain the best fit. Previous theoretical knowledge about the parameters that can be affected by reagent lot-to-lot variation can be used to restrain param- eter changes from going outside previously established ranges. Parameters affected by analyzer-to-analyzer variation Analyzer variation needs to be investigated to identify which parameters in the curve-fit model are affected. For example, there could be a variation in absolute signal lev- els, variation in low signal sensitivity, or differences in high-signal saturation characteristics. These should be incorporated in the model using as few parameters as pos- sible. They may be assay specific. For example, analyzer variation in the background noise level at low concentra- tions may only be a significant factor in a sensitive TSH assay. Assay-specific parameters During development of new kits, the model could be explored further. Is the assay competitive or immunomet- ric? How linear is the dose response? Are the zero refer- ence calibrators, used for master curve generation, truly zero? Is the assay likely to involve very low levels of enzyme at the signal generation stage? The aim of this work would be to derive the basic model for the assay, perhaps chosen from a family of options for the system. Any parameters in the system model not relevant to the assay could be set to a fixed number or removed from the model. Stability During transport and stability studies, the model would be used to determine which parameters change in value with time. It may be that more than one parameter changes with time, but that two parameters vary according to a fixed ratio. Knowledge of this can be used to determine the number of adjusters and their concentrations. Number of adjusters The number of adjusters can be determined from knowl- edge about analyzer–analyzer and stability effects. The minimum number depends on the number of parameters (or linked pairs of parameters) that can change. Position of adjuster concentrations Knowledge of the changes that can occur may be applied to the choice of adjuster concentration. For example, if background signal at zero concentration does not vary, the lowest adjuster does not have to be at zero. Replication of adjusters This is a matter of trade-off between convenience and avoidance of adjustment bias. However, the replication is strongly influenced by the number of adjusters, the preci- sion of the system in the user’s laboratory, and the desired precision for the assay. As a rule-of-thumb, for assays with less than four replicates overall (e.g., four adjusters in sin- gleton or two adjusters in duplicate), adjustment bias will be a significant source of overall assay imprecision. Method used to adjust master curve using adjuster signal levels Using the accumulated information about the assay, obtained during development, it should be possible to use the adjuster data to modify the relevant parameters in the model, while retaining the parameter values that are not expected to change. In this way, the model is less likely to be forced into bias simply due to the error in the adjuster signal determination. However, the problem still remains that adjuster signal determination error can unduly influ- ence the parameter(s) allowed to change and distort the curve. It is important that the system has some error checks to warn of signal changes that are outside of expected limits. MODELING CALIBRATION CURVE CHANGES OVER SHELF LIFE For reagents that are very stable, periodic recalibration within the shelf life may not be necessary. However, a con- cept that does not seem to have attracted much attention in the literature is that of modeling the time dependence of parameter values. As an example, suppose an appropriate calibration curve-fit model for a particular assay is the 4PL. Also suppose that a time series plot of the four parameter values reveals a profile that can be accurately modeled over a period of time. If this turned out to be the case, then no adjusters would be required, since all that would be neces- sary is the date of use of the reagents. A periodic calibra- tion of the analyzer may be required, but this may not need to be assay specific. We are aware of this technique being used on one system, but it places great demands on the manufacturer to produce materials that have consistent and predictable changes during the shelf life. It is not likely to be robust to temperature fluctuation during transporta- tion or storage. USE OF ELECTRONIC DATA TRANSMISSION As explained earlier, use of a master curve determined using a large number of replicates from a set of secondary reference standards at the manufacturer’s laboratory has several advantages that are offset by the additional error due to use of very few adjuster concentrations and repli- cates in the user laboratory. In the future, we may see increasing use of master curve updating via the Internet, using modems in immunoassay analyzers. User analyzer
  14. 14. 336 The Immunoassay Handbook calibration would comprise of a periodic determination of analyzer bias from the master analyzers at the manufac- turer, which may not be assay specific. SUMMARY The key issues about using stored calibration curves are stability and analyzer-to-analyzer variation. If, in a con- ventional assay, the underlying dose–response relationship requires k parameters to determine its form, then there should be at least k adjusters used, unless it has been shown experimentally that fewer parameters can change over time or between analyzers. It would be interesting to see if manufacturers could explain the fundamental theory of their stored calibration systems, as well as providing data to support stability and precision claims. There is no doubt that these “black box” proprietary calibration systems should be approached with caution, with great attention paid to quality assurance and control schemes, to check that calibration integrity has been maintained. Suitable Calibration Curve-Fit Software STATLIA Quantum®, by Brendan Technologies (www. brendan.com), offers a complete data reduction program for quantitative immunoassays, potency bioassays, and qualitative screening tests. The program is a fully inte- grated enterprise system networking multiple detectors, liquid handlers, workstations, users, and LIM systems to a central SQL database. The program features the compa- ny’s TrueFit™ weighted data reduction and SmartQC™ quality control (QC) and method analysis. Assays are com- pared to their historical performance for a comprehensive statistical QC analysis of assay performance, comprehen- sive customizable assay and unknown acceptance criteria, and test method performance qualification (PQ) analysis. The program features a large number of graphs in 2D and 3D, many report templates, and a report designer for cus- tomizing reports, PQ reports, and other reports in Excel®, PDF, HTML, TIFF, and CSV formats. 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(Charles Griffin, London, 1978). Gerlach, R.W., White, R.J., Deming, S.N., Palasota, J.A. and Van Emon, J.M. An evaluation of five commercial immunoassay data analysis software systems. Anal. Biochem. 212, 185–193 (1993). Gottschalk, P.G. and Dunn, J.R. Determining the error of dose estimates and minimum and maximum acceptable concentrations from assays with nonlinear dose-response curves. Comput. Methods Programs Biomed. 80, 204–215 (2005a). Gottschalk, P.G. and Dunn, J.R. The five parameter logistic: a characterization and comparison with the four parameter logistic. Anal. Biochem. 343, 54–65 (2005b). Haven, M.C., Orsulak, P.J., Arnold, L.L. and Crowley, G. Data-reduction methods for immunoradiometric assays of thyrotropin compared. Clin. Chem. 33, 1207–1210 (1987). Healy, M.J.R. Statistical analysis of radioimmunoassay data. Biochem. J. 130, 207–210 (1972). Lynch, M.J. Extended standard curve stability on the CCD Magic Lite immunoas- say system using a two-point adjustment. J. Biolumin. Chemilumin. 4, 615–619 (1989). Maciel, R.J. Standard curve-fitting in immunodiagnostics: a primer. J. Clin. Immunoassay 8, 98–106 (1985). Malan, P.G., Cox, M.G., Long, E.M.R. and Ekins, R.P. A multi-binding site model-based curve-fitting program for the computation of RIA data. In: Radioimmunoassay and Related Procedures in Medicine, vol. I, 425–455 (IAEA, Vienna, 1973). Nisbet, J.A., Owen, J.A. and Ward, G.E. A comparison of five curve-fitting proce- dures in radioimmunoassay. Ann. Clin. Biochem. 23, 694–698 (1986). Nix, B. and Wild, D.G. Data processing. In: Immunoassays, (ed Gosling, J.P.), (Oxford University Press, Oxford, 2000). Peterman, J.H. Immunochemical considerations in the analysis of data from non- competitive solid-phase immunoassays. In: Immunochemistry of Solid-Phase Immunoassay, (ed Butler, J.E.), (CRC Press, Boca Raton, 1991). Plikaytis, B.D., Turner, S.H., Gheesling, L.L. and Carlone, G.M. Comparisons of standard curve-fitting methods to quantitate Neisseria meningitidis Group A polysaccharide antibody levels by enzyme-linked immunosorbent assay. J. Clin. Microbiol. 29, 1439–1446 (1991). Raab, G.M. Estimation of a variance function, with application to immunoassay. Appl. Stat. 3, 32–40 (1981). Raggatt, P.R. Data manipulation. In: Principles and practice of immunoassay, 2nd edn (eds Price, C.P. and Newman, D.J.), 269–297 (Macmillan, London, 1997). Rodbard, D. Statistical quality control and routine data processing for radioim- munoassay and immunometric assays. Clin. Chem. 20, 1255–1270 (1974). Rodbard, D. and Feldman, Y. Kinetics of two-site immunoradiometric (‘sandwich’) assays-I. Mathematical models for simulation, optimization and curve-fitting. Immunochemistry 15, 71–76 (1978). Rodbard, D. and Hutt, D.M. Statistical analysis of radioimmunoassays and immu- noradiometric (labeled antibody) assays: a generalized, weighted, iterative, least-squares method for logistic curve-fitting. In: Radioimmunoassay and Related Procedures in Medicine, vol. I , 165–192 (IAEA, Vienna, 1974). Rodbard, D., Munson, P.J. and De Lean, A. Improved curve-fitting, parallelism testing, characterization of sensitivity, validation and optimization for radioli- gand assays. In: Radioimmunoassay and Related Procedures in Medicine, Proceedings of the Symposium, West Berlin, 1977, (IAEA, Vienna, 1978). Rogers, R.P.C. Data analysis and quality control of assays: a practical primer. In: Practical Immunoassay, the State of the Art (ed Butt, W.R.) 253–308 (Marcel Dekker, New York, 1984). Wilkins, T.A., Chadney, D.C., Bryant, J., et al. Non-linear least-squares curve fit- ting of a simple theoretical model using a mini-computer. Ann. Clin. Biochem. 15, 123–135 (1978a). Wilkins, T.A., Chadney, D.C., Bryant, J., et al. 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