Lethal Dose (LD 50 Calculator)

🧪

LD50 Calculator

Fit a sigmoidal dose-mortality curve and calculate LD50 (or LC50) with 95% confidence intervals, LD10, LD90, and goodness-of-fit. Choose from nonlinear regression (Hill/4PL), probit analysis, or logit analysis. Supports Abbott's correction for background mortality.

Dose-Mortality Table (min. 4 dose groups)
Dose / Conc. N Tested N Dead % Mortality
💡 Quick Summary

Calculate LD50 (lethal dose for 50% of a population) or LC50 (lethal concentration) by fitting a dose-mortality sigmoidal curve to your data. Supports three fitting methods — nonlinear regression (Hill/4PL), probit analysis, and logit analysis — plus Abbott's correction for background mortality. Reports LD50 with 95% confidence intervals, LD10, LD90, and goodness-of-fit statistics.

📋 How to Use
  1. Select Mode (LD50 for dose-based toxicity, LC50 for concentration-based), dose unit, and fitting method from the options row.
  2. For Nonlinear Regression: also choose 3PL (baseline fixed at 0% mortality) or 4PL (baseline fitted freely, for partial lethality data).
  3. Enter your data in the table: Dose, N Tested (animals per group), and N Dead. The % Mortality column calculates automatically.
  4. If untreated control animals died naturally, enable Abbott's Correction and enter the control % mortality. The calculator adjusts all dose groups before fitting.
  5. Click Paste from Excel/Sheets to import tab-separated data (Dose | N Tested | N Dead), or click Load example data for a sample dataset.
  6. Click Fit Curve & Calculate LD50. Results include LD50 with 95% CI, LD10, LD90, slope, goodness-of-fit, the fitted equation, and a sigmoidal dose-response chart.
🧮 Formulas & Logic
Hill equation (3PL)
% Mortality = 100 / (1 + (LD50 / x)n)
Hill equation (4PL)
% Mortality = Bottom + (Top − Bottom) / (1 + (LD50 / x)n)
Probit transform
z = Φ−1(p)  —  z = a + b × log10(dose)
Logit transform
logit(p) = ln[p / (1−p)]  —  logit(p) = a + b × log10(dose)
LD50 from probit / logit
LD50 = 10(−a / b)
LD<sub>p</sub> from probit
LDp = 10((Φ−1(p) − a) / b)
Abbott's correction
pcorrected = (pobs − pcontrol) / (1 − pcontrol)
Chi-square goodness-of-fit
χ2 = ∑ Ni (pobs − pfit)2 / [pfit(1 − pfit)]
📊 Result Interpretation
LD50 value

The dose (or concentration) that kills 50% of the test population. Lower LD50 = more toxic. Compare across compounds only when the same species, route of exposure, and assay conditions are used.

95% Confidence Interval

The range within which the true LD50 is expected to fall with 95% probability. Narrow CI = high precision (large N per group, steep curve). Wide CI indicates uncertainty — increase group sizes or add dose levels near the LD50.

Hill slope (n) / Probit slope (b)

Steepness of the dose-response curve. A steep slope (high n or b) means a narrow range of doses transitions from low to high lethality — common for single-target mechanisms. A shallow slope suggests a heterogeneous population or multiple mechanisms.

LD10 and LD90

The doses killing 10% and 90% of the population. The ratio LD90/LD10 characterises the steepness of the response. A steep curve gives a ratio close to 1; a shallow curve gives a much larger ratio.

R² (NLR method)

Proportion of variance in % mortality explained by the fitted Hill curve. R² > 0.95 is good; R² < 0.90 suggests the model does not fit well — consider switching to probit/logit or checking for outliers.

Chi-square / p-value (probit / logit)

Tests whether the observed mortalities deviate significantly from the fitted curve. p > 0.05: no significant heterogeneity (good fit). p ≤ 0.05: significant heterogeneity — check for outlier dose groups, inadequate sample sizes, or a non-sigmoidal dose-response.

Abbott's correction

Adjusts for natural background mortality in the control group. Required when untreated animals die at a non-trivial rate (>5%). Without correction, LD50 is underestimated because background deaths are falsely attributed to the test agent.

🔬 Applications
  • Acute toxicology: determining oral, dermal, or inhalation LD50 for regulatory submissions (OECD 401, 423, 425)
  • Ecotoxicology: LC50 for aquatic organisms (fish, daphnia, algae) per OECD 203/202
  • Pesticide and herbicide registration: establishing safety margins and hazard classification
  • Pharmacology: estimating toxic dose range before therapeutic window studies
  • Radiation biology: calculating lethal radiation dose (LD50/30 for 30-day mortality)
  • Antibiotic / antifungal research: MBC (minimum bactericidal concentration) dose-response characterisation
  • Venom research: snake, spider, and scorpion venom lethality ranking
⚠️ Common Mistakes & Warnings
LD50 requires data spanning the full sigmoidal range

You need at least one dose group with 80% mortality (upper plateau). If no group reaches 100% lethality, the upper asymptote is extrapolated and the LD50 confidence interval becomes unreliable. Add higher dose groups.

Minimum group size affects statistical precision

Groups of ≥ 10 animals per dose level are generally required for reliable probit analysis (OECD 425 guideline). Very small groups (n=3–5) produce wide confidence intervals and unreliable chi-square tests.

LD50 is route- and species-specific

LD50 values cannot be directly compared across different routes of administration (oral vs. inhalation vs. dermal) or across species. Always report the species, strain, age, sex, and route alongside the LD50.

Abbott's correction assumes independence of effects

Abbott's formula assumes that background mortality and treatment mortality are independent (multiplicative model). If the test compound interacts with the cause of background mortality, the correction introduces bias. Use with caution when control mortality exceeds 20%.

❓ Frequently Asked Questions

What is the difference between LD50 and LC50?
LD50 (Lethal Dose 50) is the dose per unit body weight (e.g. mg/kg) that kills 50% of a test population — used for solid/liquid compounds administered by a specific route (oral, IV, dermal). LC50 (Lethal Concentration 50) is the concentration in air or water (e.g. mg/L, ppm) that kills 50% — used for inhalation or aquatic toxicity studies. The underlying mathematics and curve fitting are identical; the difference is only in the unit and experimental setup.
Which method should I use — NLR, probit, or logit?
For regulatory submissions (EPA, WHO, EU), probit analysis is the accepted standard and should be used first. Logit analysis gives nearly identical results and is common in ecotoxicology. Nonlinear regression (Hill/4PL) is familiar to researchers coming from IC50 analysis and is appropriate for in-house research or when the data deviates from a normal (probit) or logistic (logit) distribution. If the three methods give very different LD50 estimates, your data likely does not follow a simple sigmoidal model.
When should I use Abbott's correction?
Apply Abbott's correction when your negative control (zero-dose) group shows mortality > 0%, indicating natural background death. This is common in long-duration studies or in fragile animal strains. The correction adjusts all treated group mortalities downward before fitting. If control mortality is very high (>20%), consider redesigning the experiment rather than relying on the correction.
What does a steep vs. shallow dose-response curve mean biologically?
A steep curve (high Hill coefficient n > 2, or steep probit slope b > 5) means a very narrow dose range separates low from high lethality — typical of single-target mechanisms with cooperative binding. A shallow curve (n < 1, or b < 2) suggests heterogeneous sensitivity in the population, multiple targets, or mixed mechanisms. For risk assessment, a shallow slope means a wider margin of safety between the no-effect level and the lethal dose.
How do I calculate a safety factor or therapeutic index from LD50?
Therapeutic Index (TI) = LD50 / ED50, where ED50 is the effective dose for 50% of subjects. A TI > 10 is generally considered acceptable for drug candidates. The Certain Safety Factor (CSF) = LD01 / ED99 is a more conservative measure. This calculator provides LD10 and LD90 directly; ED values come from a separate efficacy experiment.
What do the 95% CI Lower and Upper values mean?
The 95% confidence interval (CI) is the range within which the true LD50 most likely falls. For example, if LD50 = 78.33 mg/kg with CI 70.83–86.56 mg/kg, it means that if you repeated the experiment many times, 95% of those intervals would contain the true population LD50. A narrow CI indicates a precise estimate — achieved with large group sizes, a steep dose-response curve, and dose levels that closely bracket the LD50. A wide CI signals uncertainty; the practical fix is to increase animals per group or add dose levels near the expected LD50. For Nonlinear Regression, the CI is computed by profile likelihood (finding where the sum-of-squares worsens significantly as LD50 is moved away from its best-fit value). For Probit and Logit, the delta method (Fieller approximation) is used.
What does R² mean, and what is a good value?
R² (coefficient of determination) measures how well the fitted curve explains the variance in your observed mortality data. A value of 1.0 means the curve passes through every data point perfectly; 0.0 means it explains nothing. For LD50 analysis, R² > 0.95 is generally considered a good fit, 0.85–0.95 is acceptable, and below 0.85 suggests the model may be a poor description of your data — consider switching fitting methods or checking for outlier dose groups. R² is reported for Nonlinear Regression; for Probit and Logit the equivalent goodness-of-fit check is the chi-square p-value shown in the equation panel — p > 0.05 means no significant deviation from the fitted line.

Hill 3PL: % Mortality = 100 / (1 + (LD50/x)n) · Probit: Φ−1(p) = a + b·log10(dose) · Logit: ln[p/(1−p)] = a + b·log10(dose) · Abbott: pcorr = (pobs − pctrl) / (1 − pctrl)