ELISA results come back as optical densities, but what you want is concentration. This tool fits a 4-parameter logistic curve to your standard series and inverts it to read off the concentration of each unknown, the same method used by plate-reader software.
How it works
The 4PL model relates concentration x to signal y:
y = d + (a - d) / (1 + (x / c)^b)where a is the response at zero analyte, d is the saturated response, c is the EC50, and b is the Hill slope. The tool fits these four parameters to your standards by least-squares (a damped Gauss-Newton / Levenberg-Marquardt iteration), then solves the equation for x to interpolate each unknown:
x = c * ((a - d) / (y - d) - 1)^(1/b)Finally it multiplies by the dilution factor to report the concentration in the original sample.
Worked example: a TNF-alpha sandwich ELISA
Imagine a standard sandwich ELISA run with the following points (concentrations in pg/mL, absorbance at 450 nm):
| Standard (pg/mL) | OD450 |
|---|---|
| 0 (blank) | 0.050 |
| 31.25 | 0.185 |
| 62.5 | 0.320 |
| 125 | 0.580 |
| 250 | 0.950 |
| 500 | 1.380 |
| 1,000 | 1.680 |
| 2,000 | 1.820 |
The curve flattens toward 1.85 at high concentrations (the upper asymptote d) and toward 0.05 at zero (the lower asymptote a). The inflection point c is near 300 pg/mL and the Hill slope b is near 1.5.
If an unknown sample reads OD450 = 0.720 and was diluted 1:4 before the assay, the 4PL inversion gives approximately 185 pg/mL in the diluted well. Multiplying by the dilution factor of 4 gives approximately 740 pg/mL in the original sample.
What a good R-squared looks like — and what to do when it is not
An R-squared above 0.98 indicates that the 4PL curve fits your standard points closely enough for reliable interpolation. When R-squared is lower, here is a systematic way to diagnose the problem:
- Drop-in or dropout standard. Plot the standards visually. A single outlier point — usually a pipetting error or bubble in a well — drags the fit. Remove it, re-fit, and if R-squared recovers, re-run that point or flag it in your notebook.
- Curve does not plateau at the top. If you have no standards above the EC50 inflection, the upper asymptote is unanchored and the fit is unstable. Add a higher-concentration standard or dilute the top standards less.
- Blank too high. A contaminated zero well elevates the lower asymptote. Re-run the blank on a clean well or subtract a correction if your plate reader software supports it.
Dilution correction and re-assay decisions
The tool multiplies the interpolated in-well concentration by the dilution factor you supply. Set it to 1 for a neat (undiluted) sample. If a sample OD falls above your top standard, it is out of the assayable range — dilute the sample further and re-run. If it falls below the blank, the analyte is either absent or below the detection limit; do not extrapolate from the lower asymptote.
Why not use a straight line?
A simple linear regression on concentration vs. OD works only in the middle, linear portion of the ELISA response curve. Near the detection limit and near saturation, the relationship curves away from linear. Points fitted by a straight line in those regions will give concentrations that appear plausible but are systematically biased. The 4PL model handles the full sigmoidal shape, which is why it is the standard for quantitative immunoassays.