IC50 and EC50 calculator with a four-parameter logistic fit

The calculator fits the four-parameter logistic model, also called the Hill equation (Gadagkar and Call 2015), with the logarithm of the IC50 as a parameter:

$$Y=\mathrm{Bottom}+\frac{\mathrm{Top}-\mathrm{Bottom}}{1+{10}^{(\log {\mathrm{IC}}_{50}-\log X)\cdot \mathrm{HillSlope}}}$$

$X$ is the concentration, Top and Bottom are the plateaus, and the Hill slope sets the steepness (negative when the response falls with concentration). Fitting log IC50 rather than IC50 is preferred for small datasets because the estimate is closer to normally distributed on the log scale (Ritz et al. 2015). Parameters are found by least squares with the Levenberg–Marquardt method; replicates are fitted as separate points.

Confidence interval. Standard errors come from the inverse of the information matrix scaled by the residual variance. The 95% interval of log IC50 is the estimate $\pm t(0.975,n-p)\times \mathrm{SE}$, where $n$ is the number of points and $p$ the number of free parameters, and it is back-transformed to concentration (Ritz et al. 2015), so it is not symmetric around the IC50.

Relative and absolute IC50. The relative IC50 is the fitted parameter: the concentration giving a response midway between the fitted top and bottom. The absolute IC50 is the concentration where the curve crosses the 50% control, the mean of the 0% and 100% controls (Sebaugh 2011). The calculator scales responses as $$100\times \frac{Y-\text{mean 0\% control}}{\begin{array}{c}\text{mean 100\% control}\\ -\text{mean 0\% control}\end{array}}$$ and solves the fitted curve for 50%.

Sources: Ritz C, Baty F, Streibig JC, Gerhard D. Dose-response analysis using R. PLoS One 2015;10:e0146021, doi:10.1371/journal.pone.0146021. Sebaugh JL. Guidelines for accurate EC50/IC50 estimation. Pharm Stat 2011;10:128–134, doi:10.1002/pst.426. Gadagkar SR, Call GB. Computational tools for fitting the Hill equation to dose-response curves. J Pharmacol Toxicol Methods 2015;71:68–76, doi:10.1016/j.vascn.2014.08.006.

The built-in example is a synthetic inhibition curve generated from Top 100, Bottom 4, IC50 40 nM and Hill slope −1.1, with nine concentrations (3000 nM down to 0.457 nM in 3-fold steps) in triplicate and random noise added.

With top and bottom free, the fit gives a relative IC50 of 35.29 nM (95% CI 29.59 to 42.08 nM), logIC50 1.548 ± 0.037, Hill slope −1.19 (−1.42 to −0.956), top 99.20 and bottom 6.30, with R² = 0.9901 from 27 points (23 degrees of freedom). The interval contains the true 40 nM.

Normalised to a 0% control of 2, 3 and 4 and a 100% control of 99, 100 and 101, the relative IC50 is unchanged at 35.29 nM, and the curve crosses 50% at an absolute IC50 of 36.92 nM. Fixing the top at 100 and the bottom at 0 on the raw data instead gives 41.30 nM (35.72 to 47.76 nM), Hill slope −1.00.