Bayesian Concentration Estimation from ELISA Measurements

Working Group 8.42

Content

• Introduction
• Aim
• Model
• Results
• Collaboration
• Publications

Immunoassays are biochemical tests applied to measure even very small amounts of substance using the highly specific bindings between an antibody and its antigen. Immunoassays thus have a wide range of applications, for example to detect the presence of an infection, of hormones or drugs.

This work focuses on an enzyme-linked immunosorbent assay, called sandwich ELISA. It allows detection of antigens by ‘sandwiching’ them between two antibodies – one of them is linked to an enzyme to generate a detectable signal (such as fluorescence, see Figure 1).

 

While a popular example is the home-pregnancy test, we aim at estimating the concentration of human interferon IFN alfa-2b (a protein which is involved in innate immune response against viral infection) using a fluorescent sandwich ELISA. We develop a method for estimating concentrations and their associated uncertainties, which is generally adoptable or adaptable to any ELISA.

In fluorescent sandwich ELISAs, the concentration of a solution is to be estimated from a set of fluorescence measurements, which are generated by repeatedly diluting and chemically preparing the original solution. In order to better determine the relation between concentration and fluorescence intensity, a calibration is performed on each ELISA plate, i.e. the same protocol steps are performed with a solution of known concentrations. This is illustrated in Figure 2.

 

ELISAs typically involve a high number of protocol steps, each susceptible to perturbations. A recent publication [Noble et al., 2008] has highlighted the variability in concentration estimates in the scope of an international comparability study. Some laboratories estimated an average concentration twice as high as other laboratories. However, little is published on the uncertainties of individual laboratory estimates. We reanalyse these data to quantify uncertainties of individual laboratory estimates. This enables us to assess the consistency of concentration estimates within and between laboratories, and will facilitate key comparisons in metrology [BIPM, 1999].

Let us assume a heteroscedastic nonlinear Gaussian model for all pairs ($\boldsymbol{Y}$, $\boldsymbol{X}$) of fluorescence intensity and concentration:

\begin{equation} Y_i=f(X_i,\boldsymbol{\beta})+\varepsilon_i \quad  \text{mit}  \quad f(X_i,\boldsymbol{\beta}) = \beta_1+\frac{\beta_2-\beta_1}{1+\left(\frac{X_i}{\beta_3}\right)^{\beta_4}} \end{equation}

 $f$ being the sigmoid function displayed in Figure 3 and  $\varepsilon_i \sim \text{N}(0, a x_i+ c)$ being the error model.

In order to estimate the unknown concentration  $\boldsymbol{\widetilde{X}}$, Bayes’ Theorem is applied twice.

For Calibration:

$$ \color{blue}{P(\boldsymbol{\beta},a,c|\boldsymbol{Y},\boldsymbol{X})} \color{black} {\propto P(\boldsymbol{Y}|\boldsymbol{X},\boldsymbol{\beta},a,c)P(\boldsymbol{\beta},a,c)} $$

For Estimation:

$$ P(\boldsymbol{\widetilde{X}},\boldsymbol{\beta},a,c|\boldsymbol{Y^{m}},\boldsymbol{Y},\boldsymbol{X}) \propto P(\boldsymbol{Y^{m}}|\boldsymbol{\beta},a,c,\boldsymbol{\widetilde{X}}) P(\widetilde{X}) \color{blue}{P(\boldsymbol{\beta},a,c|\boldsymbol{Y},\boldsymbol{X})} $$

To perform Bayesian inference, prior knowledge needs to be quantified for all unknown quantities. Let us assign uniform nonnegative prior distributions to all parameters a, c, ß, X, but the upper asymptote ß₁. From control samples carried out on each ELISA plate (i.e. fluorescein), it is known that the maximum intensity (maximum ß₁) is approximately 100 times the intensity of these control wells. We thus specify a uniform prior distribution for ß₁ with an upper bound driven by the intensity measurements of these control wells as displayed in Figure 4.

Calibration Step

Applying Model (1) and MCMC with the help of WinBUGs [Lunn et al., 2000] to calibration data from the PTB, results in the posterior distributions for parameters a, c of the error model and the parameters ß of the calibration function ƒ which are depicted in grey (assuming the priors depicted in black).

Concentration Estimate

The approach described above consisting of Bayesian calibration and Bayesian concentration estimation is currently applied to all data sets of the international comparability study. Preliminary results suggest clearly larger uncertainties than previously (i.e. in Noble et al., 2008) associated with ELISA concentration estimates. The concentratiom estimated for an ELISA performed at the PTB is displayed as an example in Figure 6.

 

Bayesian inference offers several advantages for ELISA concentration estimation. It:

In contrast to traditional frequentist approaches as well as metrological standards, Bayesian inference allows the inclusion of prior knowledge and gives proper uncertainties despite a nonlinear calibration curve and unknown error model parameters.

• Working group 8.31 (Tissue Optics and Molecular Imaging)