Four-parameters logistic function for ELISA-type method

Enzyme-linked immunosorbent assays (ELISA) are specific and highly sensitive procedures for identifying and quantifying analytes such as proteins in samples. ELISA is routinely used in many areas of biological research, it is based on binding a target molecule (analyte, antigen) to antibodies that recognize the compound. The presence of an antigen-antibody complex is detected using a secondary enzyme-conjugated antibody. Detection is obtained by the addition of a substrate that yields a measurable product.
The determination of the analyte concentration relies upon the construction of a calibration curve. The standard curve is prepared by performing a dilution series of known concentrations of the analyte across a range of concentrations near the expected unknown concentration. The calibration curves are then used to calculate the concentration of an unknown sample.
For most analyses, a plot of response vs. concentration will create a linear relationship, at least within a certain range of concentrations, and can be analyzed with simple linear regression. However, for sigmoidal calibration plots, performing a linear fit can lead to errors in estimating sample concentration. These errors are most significant at the extremes of the standard curve.
A four-parameters logistic function (4PL) is a nonlinear regression model that is often used to analyze bioassays such as ELISA. It has the following form:

y=D+((C-D)/(1+(x/E)$^{B}$)
where:
- $C$ is the minimum asymptote, which corresponds to the response value at zero concentration of the analyte.
- $D$ is the maximum asymptote, which corresponds to the response value at infinite concentration of the analyte.
- $E$ is the inflection point, which is the concentration of the analyte where the response is halfway between $A$ and $D$.
- $B$ is the Hill's slope, which measures the steepness of the curve. It can be positive or negative depending on whether the curve is increasing or decreasing.

The 4PL function has a sigmoidal or "S" shape that fits well the bottom and top plateaus of the curve, as well as the transition region where the response changes rapidly with respect to the concentration. This makes it suitable for characterizing bioassays that are only linear across a specific range of concentration magnitudes.
The 4PL model assumes that the amount of variability in the response data is homogeneous across the concentration levels and that the variability follows a normal distribution. In real bioassay outcomes, however, variability is often greater at higher response values. Log-transforming the response variable can help normalize the variability and is a highly recommended procedure [1].

To fit a 4PL function to data using R, we can use the `drc` and the `nplr` packages, which provide functions for dose-response analysis. Specifically, we can use the `LL.4()` function to define the 4PL model, and the `drm()` function to fit it to data. The R program here provided is a function called ELISA_4PL which takes as input three variables: x, y and yhat. The function fits a curve to the input data and return the parameters of the fitted curve. In addition, the function calculates the value of x corresponding to a specified value of y (yhat) using the inversion of the fitted curve.



#EXAMPLE
For example, suppose we have a data set that contains two columns: `conc`, which is the concentration of the analyte in ng/mL, and `resp`, which is the response value in optical density units. In addition, imagine we want to evaluate the concentration level associated with a response equal to 1.5.

conc<- conc<- c(1,0.5,0.25,0.125,0.0625,0.03125,0.015625,0.0078125,0.00390625,0.001953125)

resp<- c(2.912,2.579,2.13,1.651,1.073,0.585,0.463,0.266,0.228,0.176)

Let's run the function ELISA_4PL with the arguments conc, resp, and yhat=1.5

ELISA_4PL(conc,resp,yhat=1.5)

We get the following results:

Parameters	Estimate	Std. Error	t-value	p-value
Slope:(Intercept)	-1.0856	0.0856	-12.6814	0.0000
Lower:(Intercept)	0.1509	0.0385	3.9158	0.0078
Upper:(Intercept)	3.2089	0.1151	27.8853	0.0000
EC50:(Intercept)	0.1377	0.0120	11.5136	0.0000

The plot below shows the S-shaped curve of the response vs. concentration. The red dot corresponds to the couple (xhat, yhat), where xhat is the concentration calculated from a given response yhat by using the estimated 4PL model.

References
1. Karpinski K.F., Hayward S., TryphonaS H., (1987). Statistical considerations in the quantitation of serum immunoglobulin levels using the Enzyme-Linked Immunoabsorbent Assay (ELISA). J Immunol Methods, 1987, 103(2): 189-194.