LD50 calculator
Median lethal dose from dose, deaths and group size, with a confidence interval.
Median lethal dose from dose, deaths and group size, with a confidence interval.
The LD50, or median lethal dose, is the single dose that kills half of a treated population. It is the standard summary of acute toxicity, normally quoted in milligrams of substance per kilogram of body weight along with the species and the route of administration, because all three change the number. A smaller LD50 means a more acutely toxic substance.
LD50 is a point on a curve rather than a threshold. Doses well below it still kill some animals and doses above it spare some, so the value is always reported with a confidence interval and with the slope of the dose-response curve, which says how quickly mortality rises as the dose increases.
Mortality plotted against dose forms an S-shaped curve. Plotting it against thelogarithm of the dose and transforming the proportion killed straightens that curve, so the LD50 can be read off a straight line. The probit transform uses the inverse normal distribution:
Setting the probit to 5, which is 50 percent mortality, and solving for the dose gives the estimate:
This calculator does not fit that line by eye or by ordinary regression on the transformed points. It fits the binomial model directly by maximum likelihood, using iteratively reweighted least squares, which weights each dose group by its size and uses the groups with 0 and 100 percent mortality instead of discarding them. The result matches what a statistics package reports for a probit or logistic regression of the same data.
The three methods answer the same question with different assumptions, and on well-behaved data they agree closely.
| Method | Assumes | Best for |
|---|---|---|
| Probit | Tolerances are log-normally distributed | The classical toxicology default, comparable with published values |
| Logit | Tolerances follow a logistic distribution | Nearly identical to probit, with slightly heavier tails |
| Spearman–Kärber | Nothing about the curve shape | When the curve shape is doubtful, but it needs a 0 percent and a 100 percent dose |
Spearman–Kärber estimates the mean of the log tolerance distribution directly from the observed steps in mortality. It needs the dose range to run all the way from no deaths to total mortality, and it enforces a non-decreasing response by pooling any dose group that goes backwards. The calculator states when either adjustment applies.
The slope is the change in probit or logit units per tenfold increase in dose. Asteep slope, roughly 3 or more on the probit scale, means the gap between a dose that harms nobody and one that kills everybody is narrow, so small dosing errors matter. Ashallow slope means a wide, gradual transition and a much less precise LD50 unless many animals are used. The chi-square figure beside the slope tests whether the observed mortalities scatter more than binomial sampling alone explains; a value well above its degrees of freedom points to heterogeneity between groups rather than a bad LD50.
Reed–Muench is an older graphical method that accumulates deaths and survivors across doses and interpolates where the two cumulative curves cross. It is still common for viral TCID50 and ID50 work, but it uses each animal more than once, which makes its confidence interval hard to justify. The Kärber method, implemented here in its Spearman–Kärber form, answers the same need without that problem and comes with a usable standard error, so it is the better choice for a modern lethality dataset.
For a lethal concentration in air or water rather than a dose per body weight, use theLC50 calculator. For a half-maximal effect rather than lethality, use the EC50 calculator, and for inhibition of a target use the IC50 calculator.
Fit a dose-response curve to the fraction of animals killed at each dose and read off the dose where the fitted mortality is 50 percent. This calculator fits the curve by maximum likelihood on a log10 dose scale, using either a probit or a logit link, and also offers the non-parametric Spearman-Karber estimate.
Probit analysis transforms each mortality proportion with the inverse normal distribution, which turns a sigmoid dose-response curve into a straight line against log dose. LD50 is the dose where the probit equals 5, that is where mortality is 50 percent. The slope of that line describes how sharply the response rises with dose.
At least three doses, and ideally four or five, with two or more producing partial mortality between 0 and 100 percent. Doses that kill none or all of the animals carry little information about where the curve crosses 50 percent, so a design that brackets the midpoint gives a much tighter confidence interval.
The LD50 is then extrapolated beyond the tested range and should be reported as greater than the highest dose, or less than the lowest. The calculator warns when the data do not bracket 50 percent, because an extrapolated value depends entirely on the assumed curve shape.
LD50 is a median lethal dose, an amount given per unit of body weight, usually mg/kg. LC50 is a median lethal concentration in the surrounding air or water that the organism is exposed to, usually mg/L or ppm. The statistics are identical; only the exposure unit differs.