*2.1. Traditional (interpolation) approach to ELISA analysis*

The traditional approach proceeds as follows. First a dose-response model is fitted to the reference standard. Usually either a linear or a four-parameter logistic (4PL) model is used, though other models are also possible. In this paper we will consider only the 4PL, but everything we will say applies to other methods as well.

The 4PL model for the reference standard is:

$$

\text {response}=D+\frac{A-D}{1+\exp \left(B\left(\ln (\text { concentration })-C_{\mathrm{ref}}\right)\right)}

$$

Here A is the left asymptote, D is the right asymptote, B is the slope parameter, and C_{ref} is the midpoint (EC_{50}) of the curve. The best fitting parameters A, B, C_{ref} and D are found by fitting all the reference data simultaneously to this model, usually using statistical software.

The model is then used to estimate concentrations for each test sample. If a test sample has multiple dilutions and/or replicates, each of these is treated separately. The concentration of the test sample is estimated by interpolation. That is, the concentration estimate is the point on the reference standard curve which gives the observed response. This is illustrated in Fig. 1.

In the case of the 4PL, the interpolated concentration, x_{i}, corresponding to a test response y_{i} is given by

$$

\ln \left(x_{i}\right)=C_{r e f}+\frac{1}{B} \ln \left(\frac{A-y_{i}}{y_{i}-D}\right)

$$

To estimate the original test sample concentration, x_{i} is then multiplied by the test sample dilution d_{i}. For example, in the left panel of Fig. 1, x_{i} for the first point is 0.0232; this is multiplied by 64 to give a concentration estimate of 1.48. If there are multiple test sample dilutions and/or replicates, the geometric mean should then be taken to give an overall estimate (although in practice the arithmetic mean is often used, which is incorrect). Therefore the estimate for U is

$$

\begin{aligned}

\widehat{U} &=\exp \left\{\frac{1}{N_{\text {test }}} \sum_{i} \ln \left(d_{i} x_{i}\right)\right\} \\

&=\exp \left\{\frac{1}{N_{\text {test }}} \sum_{i}\left[\ln \left(d_{i}\right)+\left(C_{r e f}+\frac{1}{B} \ln \left(\frac{A-y_{i}}{y_{i}-D}\right)\right)\right]\right\}

\end{aligned}

$$

where N_{test} is the number of test sample responses.

The interpolated concentration, xi cannot be calculated if the test response is outside the range of the reference standard. Therefore any such values are excluded from the calculation. In practice, any values close to the asymptotes are often excluded as well. This means a cut-off distance must be chosen; responses that are closer to the asymptote than the cut-off are excluded, while those further than the cut-off are included. The cut-off is typically some fraction (say 5%) of the difference between the asymptotes |D − A|. The fraction is typically chosen indirectly by assessing its effect on the accuracy of results for samples of known concentration. However any such choice leads to an abrupt transition between including a response and giving it equal influence with all the other data on the one hand, and not including it at all on the other.

Sometimes after removing responses near the asymptotes there are not enough responses left to get a valid result so a retest is needed, possibly using different dilutions. This is particularly a problem when the unknown concentrations of the test samples vary over a wide range. A related issue is that responses on the flatter parts of the curve are more uncertain but influence the estimate equally with others (if not excluded), so can have an excessive influence on the result.

The same issues with the traditional approach were noted by Cheung et al., 2015. They proposed using a weighted average of the estimates at each dilution to address these. The idea of using a parallel relative potency model, which is closely related to our new method, is mentioned briefly in United States Pharmacopeia. Second Supplement to USP 35–NF 30, 2012; however, no details of how this should be done are given.

Confidence intervals are not usually reported for concentrations estimated using interpolation analysis. It is however possible to calculate them (Daly et al., 2005); we describe how this can be done in the Appendix.