LD50 Dose-Response Simulator: Lethal Dose Estimation

simulator intermediate ~10 min
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P = 36% — mortality at 150 mg/kg with LD50 = 200 mg/kg

At 150 mg/kg (75% of the LD50) with Hill slope 2, the predicted mortality rate is approximately 36%. The sigmoidal curve shows the steepest change near the LD50 inflection point.

Formula

P(D) = D^b / (LD50^b + D^b) (Hill equation for mortality)
LD_x = LD50 × (x / (100-x))^(1/b) (dose for x% mortality)
Probit(P) = a + b × log(D) (probit regression model)

The Dose Makes the Poison

Paracelsus established the foundational principle of toxicology in the 16th century: every substance is toxic at some dose, and the dose alone determines whether it acts as a remedy or a poison. Modern toxicology quantifies this through the dose-response relationship — a sigmoidal curve relating the administered dose to the probability of an adverse effect, with the LD50 marking the dose lethal to half the exposed population.

The Sigmoidal Curve

The dose-response curve follows a characteristic S-shape (sigmoid) when plotted on linear axes, or a straight line when mortality is expressed as probits (probability units) against log dose. The Hill equation captures this shape with two key parameters: the LD50 (the curve's midpoint) and the Hill slope b (its steepness). A steep slope means a narrow range between harmless and lethal doses — critical information for setting safety margins.

From LD50 to Safety Margins

Regulatory toxicology uses the LD50 as a starting point for risk assessment, but the full dose-response curve matters more than the midpoint alone. The safety margin — the ratio of toxic dose to therapeutic or expected exposure dose — determines how much buffer exists between normal use and danger. Substances with steep slopes and narrow safety margins (like warfarin) require careful dose monitoring.

Beyond Lethality

Modern toxicology recognizes that death is a crude endpoint. Sub-lethal effects — organ damage, reproductive toxicity, carcinogenicity, endocrine disruption — may occur at doses far below the LD50. The No Observed Adverse Effect Level (NOAEL) and Benchmark Dose (BMD) approaches capture these subtler endpoints, forming the basis for establishing acceptable daily intakes and occupational exposure limits.

FAQ

What is LD50?

LD50 (Lethal Dose 50) is the dose of a substance that kills 50% of a test population. It is expressed in mg per kg of body weight and serves as a standard measure of acute toxicity. Lower LD50 values indicate more toxic substances — botulinum toxin has an LD50 of ~1 ng/kg, while table salt is ~3,000 mg/kg.

What is the Hill equation in toxicology?

The Hill equation models the sigmoidal dose-response relationship: P = D^b / (LD50^b + D^b), where b is the Hill coefficient (slope). It was originally developed for oxygen-hemoglobin binding but applies broadly to any cooperative or threshold response.

Why is LD50 controversial?

LD50 testing requires large numbers of animals and has been criticized on ethical grounds. Modern alternatives include the Fixed Dose Procedure, Acute Toxic Class method, and in vitro assays. The EU banned LD50 testing for cosmetics in 2009, and regulatory agencies increasingly accept alternative methods.

How does body weight affect toxicity?

Toxicity is expressed per kg body weight because larger organisms generally tolerate larger absolute doses. However, allometric scaling means dose-response doesn't scale linearly — metabolic rate, surface area, and organ function ratios differ across species, requiring interspecies scaling factors.

Sources

Embed

<iframe src="https://homo-deus.com/lab/toxicology/ld50-estimation/embed" width="100%" height="400" frameborder="0"></iframe>
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