LD50 Dose-Response Curve Calculator

The log-logistic (Hill) dose-response model is the standard curve fit for most pharmacological and toxicological data. The response at a given dose D is:

E = D^n / (ED50^n + D^n)

where ED50 (or LD50 for lethal effects) is the dose producing 50% response, and n is the Hill coefficient controlling curve steepness.

Hill coefficient n. n = 1 gives a shallow, symmetric sigmoid. n = 2 means the curve climbs steeply. n close to 0.5 is a wide, gradual curve. Most drugs and toxins have n between 0.5 and 3. When n is exactly 1, this reduces to the standard hyperbolic Michaelis-Menten equation.

LD50 vs ED50. LD50 is the median lethal dose, meaning the dose that kills half a test population. ED50 is the median effective dose, giving a therapeutic response in half of subjects. The therapeutic index = LD50 / ED50. A ratio above 10 is generally considered safe; below 3 is worrying.

Dose units. LD50 is typically expressed in mg/kg body weight. Common reference values: aspirin LD50 is about 200 mg/kg in rats; caffeine is about 192 mg/kg; table salt (NaCl) is around 3000 mg/kg; botulinum toxin is about 0.001 mg/kg. That last figure is not a typo, and it is why botulinum is both the most toxic substance known and a routine cosmetic injection: the therapeutic dose is a tiny fraction of a milligram.

Reading the curve. The chart shows predicted response from 1% of the LD50 to 100 times it, on a log scale. The 50% intercept always falls at exactly the LD50.

How steepness translates into a number. “The steeper the curve, the narrower the dangerous range” is easy to say and easy to leave vague, so here it is as arithmetic. Rearranging the Hill equation, the dose producing response p is LD50 × (p/(1−p))^(1/n). Divide the 90% dose by the 10% dose and everything cancels except the Hill coefficient: the ratio is 81^(1/n).

That gives a clean ladder. At n = 1 the 90% dose is 81 times the 10% dose, a very wide band. At n = 2 it is 9 times. At n = 3 it is 4.3 times. At n = 0.5 it is 6,561 times, which is a curve so shallow it barely deserves the name. A steep curve means a small change in dose flips the outcome, which is exactly what makes a narrow therapeutic window dangerous in practice rather than merely on paper.