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Median lethal dose
Median lethal dose
Median lethal dose
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In toxicology, the median lethal dose, LD50 (abbreviation for "lethal dose, 50%"), LC50 (lethal concentration, 50%) or LCt50 is a toxic unit that measures the lethal dose of a given substance.[1] The value of LD50 for a substance is the dose required to kill half the members of a tested population after a specified test duration. LD50 figures are frequently used as a general indicator of a substance's acute toxicity. A lower LD50 is indicative of higher toxicity.

The term LD50 is generally attributed to John William Trevan.[2] The test was created by J. W. Trevan in 1927.[3] The term semilethal dose is occasionally used in the same sense, in particular with translations of foreign language text, but can also refer to a sublethal dose. LD50 is usually determined by tests on animals such as laboratory mice. In 2011, the U.S. Food and Drug Administration approved alternative methods to LD50 for testing the cosmetic drug botox without animal tests.[4][5]

Conventions

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The LD50 is usually expressed as the mass of substance administered per unit mass of test subject, typically as milligrams of substance per kilogram of body mass, sometimes also stated as nanograms (suitable for botulinum toxin), micrograms, or grams (suitable for paracetamol) per kilogram. Stating it this way allows the relative toxicity of different substances to be compared and normalizes for the variation in the size of the animals exposed (although toxicity does not always scale simply with body mass). For substances in the environment, such as poisonous vapors or substances in water that are toxic to fish, the concentration in the environment (per cubic metre or per litre) is used, giving a value of LC50. But in this case, the exposure time is important (see below).

The choice of 50% lethality as a benchmark avoids the potential for ambiguity of making measurements in the extremes and reduces the amount of testing required. However, this also means that LD50 is not the lethal dose for all subjects; some may be killed by much less, while others survive doses far higher than the LD50. Measures such as "LD1" and "LD99" (dosage required to kill 1% or 99%, respectively, of the test population) are occasionally used for specific purposes.[6]

Lethal dosage often varies depending on the method of administration; for instance, many substances are less toxic when administered orally than when intravenously administered. For this reason, LD50 figures are often qualified with the mode of administration, e.g., "LD50 i.v."

The related quantities LD50/30 or LD50/60 are used to refer to a dose that without treatment will be lethal to 50% of the population within (respectively) 30 or 60 days. These measures are used more commonly within radiation health physics, for ionizing radiation, as survival beyond 60 days usually results in recovery.

A comparable measurement is LCt50, which relates to lethal dosage from exposure, where C is concentration and t is time. It is often expressed in terms of mg-min/m3. ICt50 is the dose that will cause incapacitation rather than death. These measures are commonly used to indicate the comparative efficacy of chemical warfare agents, and dosages are typically qualified by rates of breathing (e.g., resting = 10 L/min) for inhalation, or degree of clothing for skin penetration. The concept of Ct was first proposed by Fritz Haber and is sometimes referred to as Haber's law, which assumes that exposure to 1 minute of 100 mg/m3 is equivalent to 10 minutes of 10 mg/m3 (1 × 100 = 100, as does 10 × 10 = 100).

Some chemicals, such as hydrogen cyanide, are rapidly detoxified by the human body, and do not follow Haber's law. In these cases, the lethal concentration may be given simply as LC50 and qualified by a duration of exposure (e.g., 10 minutes). The material safety data sheets for toxic substances frequently use this form of the term even if the substance does follow Haber's law.

For disease-causing organisms, there is also a measure known as the median infective dose and dosage. The median infective dose (ID50) is the number of organisms received by a person or test animal qualified by the route of administration (e.g., 1,200 org/man per oral). Because of the difficulties in counting actual organisms in a dose, infective doses may be expressed in terms of biological assay, such as the number of LD50s to some test animal. In biological warfare infective dosage is the number of infective doses per cubic metre of air times the number of minutes of exposure (e.g., ICt50 is 100 medium doses - min/m3).

Limitation

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As a measure of toxicity, LD50 is somewhat unreliable and results may vary greatly between testing facilities due to factors such as the genetic characteristics of the sample population, animal species tested, environmental factors and mode of administration.[7]

There can be wide variability between species as well; what is relatively safe for rats may very well be extremely toxic for humans (cf. paracetamol toxicity), and vice versa. For example, chocolate, comparatively harmless to humans, is known to be toxic to many animals. When used to test venom from venomous creatures, such as snakes, LD50 results may be misleading due to the physiological differences between mice, rats, and humans. Many venomous snakes are specialized predators on mice, and their venom may be adapted specifically to incapacitate mice; and mongooses may be exceptionally resistant. While most mammals have a very similar physiology, LD50 results may or may not have equal bearing upon every mammal species, such as humans, etc.

Examples

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Note: Comparing substances (especially drugs) to each other by LD50 can be misleading in many cases due (in part) to differences in effective dose (ED50). Therefore, it is more useful to compare such substances by therapeutic index, which is simply the ratio of LD50 to ED50.[8]

The following examples are listed in reference to LD50 values, in descending order, and accompanied by LC50 values, {bracketed}, when appropriate.

Substance Animal, route LD50
{LC50}
LD50 : g/kg
{LC50 : g/L}
standardised 		Reference
Water (H2O) rat, oral >90,000 mg/kg >90 [9]
Sucrose (table sugar) rat, oral 29,700 mg/kg 29.7 [10]
Corn syrup rat, oral 25,800 mg/kg 25.8 [11]
Glucose (blood sugar) rat, oral 25,800 mg/kg 25.8 [12]
Monosodium glutamate (MSG) rat, oral 16,600 mg/kg 16.6 [13]
Stevioside (from stevia) mice and rats, oral 15,000 mg/kg 15 [14]
Gasoline (petrol) rat 14,063 mg/kg 14.0 [15]
Vitamin C (ascorbic acid) rat, oral 11,900 mg/kg 11.9 [16]
Glyphosate (isopropylamine salt) rat, oral 10,537 mg/kg 10.537 [17]
Lactose (milk sugar) rat, oral 10,000 mg/kg 10 [18]
Aspartame mice, oral 10,000 mg/kg 10 [19]
Urea (OC(NH2)2) rat, oral 8,471 mg/kg 8.471 [20]
Cyanuric acid rat, oral 7,700 mg/kg 7.7 [21]
Cadmium sulfide (CdS) rat, oral 7,080 mg/kg 7.08 [22]
Ethanol (CH3CH2OH) rat, oral 7,060 mg/kg 7.06 [23]
Sodium isopropyl methylphosphonic acid (IMPA, metabolite of sarin) rat, oral 6,860 mg/kg 6.86 [24]
Melamine rat, oral 6,000 mg/kg 6 [21]
Taurine rat, oral 5,000 mg/kg 5 [25]
Melamine cyanurate rat, oral 4,100 mg/kg 4.1 [21]
Fructose (fruit sugar) rat, oral 4,000 mg/kg 4 [26]
Sodium molybdate (Na2MoO4) rat, oral 4,000 mg/kg 4 [27]
Sodium chloride (table salt) rat, oral 3,000 mg/kg 3 [28]
Paracetamol (acetaminophen) rat, oral 2000 mg/kg 2 [29]
Aspirin (acetylsalicylic acid) rat, oral 1,600 mg/kg 1.6 [30]
Delta-9-tetrahydrocannabinol (THC) rat, oral 1,270 mg/kg 1.27 [31]
Cannabidiol (CBD) rat, oral 980 mg/kg 0.98 [32]
Methanol (CH3OH) human, oral 810 mg/kg 0.81 [33]
Trinitrotoluene (TNT) rat, oral 790 mg/kg 0.790
Arsenic (As) rat, oral 763 mg/kg 0.763 [34]
Ibuprofen rat, oral 636 mg/kg 0.636 [35]
Formaldehyde (CH2O) rat, oral 600–800 mg/kg 0.6 [36]
Solanine (main alkaloid in the several plants in Solanaceae amongst them Solanum tuberosum) rat, oral (2.8 mg/kg human, oral) 590 mg/kg 0.590 [37]
Atropine (from Atropa bella-donna, Datura stramonium, Mandragora officinarum and Brugmansia) rat, oral 500 mg/kg 0.500 [38]
Alkyl dimethyl benzalkonium chloride (ADBAC) rat, oral
fish, immersion
aquatic invertebrates, immersion 		304.5 mg/kg
{0.28 mg/L}
{0.059 mg/L} 		0.3045
{0.00028}
{0.000059} 		[39]
Coumarin (benzopyrone, from Cinnamomum aromaticum and other plants) rat, oral 293 mg/kg 0.293 [40]
Psilocybin (from psilocybin mushrooms) mouse, oral 280 mg/kg 0.280 [41]
Hydrochloric acid (HCl) rat, oral 238–277 mg/kg 0.238 [42]
Ketamine rat, intraperitoneal 229 mg/kg 0.229 [43]
Caffeine rat, oral 192 mg/kg 0.192 [44]
Arsenic trisulfide (As2S3) rat, oral 185–6,400 mg/kg 0.185–6.4 [45]
Sodium nitrite (NaNO2) rat, oral 180 mg/kg 0.18 [46]
Methylenedioxymethamphetamine (MDMA) rat, oral 160 mg/kg 0.18 [47]
Uranyl acetate dihydrate (UO2(CH3COO)2) mouse, oral 136 mg/kg 0.136 [48]
Dichlorodiphenyltrichloroethane (DDT) mouse, oral 135 mg/kg 0.135 [49]
Uranium (U) mice, oral 114 mg/kg (estimated) 0.114 [48]
Bisoprolol mouse, oral 100 mg/kg 0.1 [50]
Cocaine mouse, oral 96 mg/kg 0.096 [51]
Cobalt(II) chloride (CoCl2) rat, oral 80 mg/kg 0.08 [52]
Cadmium oxide (CdO) rat, oral 72 mg/kg 0.072 [53]
Thiopental sodium (used in lethal injection) rat, oral 64 mg/kg 0.064 [54]
Demeton-S-methyl rat, oral 60 mg/kg 0.060 [55]
Methamphetamine rat, intraperitoneal 57 mg/kg 0.057 [56]
Sodium fluoride (NaF) rat, oral 52 mg/kg 0.052 [57]
Nicotine mouse and rat, oral

human, smoking

50 mg/kg 0.05 [58]
Pentaborane human, oral 50 mg/kg 0.05 [59]
Capsaicin mouse, oral 47.2 mg/kg 0.0472 [60]
Vitamin D3 (cholecalciferol) rat, oral 37 mg/kg 0.037 [61]
Piperidine (from black pepper) rat, oral 30 mg/kg 0.030 [62]
Heroin (diamorphine) mouse, intravenous 21.8 mg/kg 0.0218 [63]
Lysergic acid diethylamide (LSD) rat, intravenous 16.5 mg/kg 0.0165 [64]
Arsenic trioxide (As2O3) rat, oral 14 mg/kg 0.014 [65]
Metallic arsenic (As) rat, intraperitoneal 13 mg/kg 0.013 [66]
Coniine (from Conium maculatum) mouse, intravenous 8 mg/kg 0.008 [67]
Sodium cyanide (NaCN) rat, oral 6.4 mg/kg 0.0064 [68]
Chlorotoxin (CTX, from scorpions) mice 4.3 mg/kg 0.0043 [69]
Hydrogen cyanide (HCN) mouse, oral 3.7 mg/kg 0.0037 [70]
Carfentanil rat, intravenous 3.39 mg/kg 0.00339 [71]
Nicotine (from various Solanaceae genera) mice, oral 3.3 mg/kg 0.0033 [58]
White phosphorus (P) rat, oral 3.03 mg/kg 0.00303 [72]
Strychnine (from Strychnos nux-vomica) human, oral 1–2 mg/kg (estimated) 0.001–0.002 [73]
Aconitine (from Aconitum napellus and related species) human, oral 1–2 mg/kg 0.001–0.002 [74]
Mercury(II) chloride (HgCl2) rat, oral 1 mg/kg 0.001 [75]
Cantharidin (from blister beetles) human, oral 500 μg/kg 0.0005 [76]
Aflatoxin B1 (from Aspergillus flavus mold) rat, oral 480 μg/kg 0.00048 [77]
Plutonium (Pu) dog, intravenous 320 μg/kg 0.00032 [78]
Bufotoxin (from Bufo toads) cat, intravenous 300 μg/kg 0.0003 [79]
Brodifacoum rat, oral 270 μg/kg 0.00027 [80]
Caesium-137 (137
Cs) 		mouse, parenteral 21.5 μCi/g 0.000245 [81]
Sodium fluoroacetate (CH2FCOONa) rat, oral 220 μg/kg 0.00022 [82]
Chlorine trifluoride (ClF3) mouse, absorption through skin 178 μg/kg 0.000178 [83]
Sarin mouse, subcutaneous injection 172 μg/kg 0.000172 [84]
Robustoxin (from Sydney funnel-web spider) mice 150 μg/kg 0.000150 [85]
VX human, oral, inhalation, absorption through skin/eyes 140 μg/kg (estimated) 0.00014 [86]
Venom of the Brazilian wandering spider rat, subcutaneous 134 μg/kg 0.000134 [87]
Amatoxin (from Amanita phalloides mushrooms) human, oral 100 μg/kg 0.0001 [88][89]
Dimethylmercury (Hg(CH3)2) human, transdermal 50 μg/kg 0.000050 [90]
TBPO (t-Butyl-bicyclophosphate) mouse, intravenous 36 μg/kg 0.000036 [91]
Fentanyl monkey 30 μg/kg 0.00003 [92]
Venom of the inland taipan rat, subcutaneous 25 μg/kg 0.000025 [93]
Ricin (from castor oil plant) rat, intraperitoneal
rat, oral 		22 μg/kg
20–30 mg/kg 		0.000022
0.02 		[94]
2,3,7,8-Tetrachlorodibenzodioxin (TCDD, in Agent Orange) rat, oral 20 μg/kg 0.00002
Tetrodotoxin from the blue-ringed octopus intravenous 8.2 μg/kg 0.0000082 [95]
CrTX-A (from Carybdea rastonii box jellyfish venom) crayfish, intraperitoneal 5 μg/kg 0.000005 [96]
Latrotoxin (from widow spider venom) mice 4.3 μg/kg 0.0000043 [97][self-published source?]
Epibatidine (from Epipedobates anthonyi poison dart frog) mouse, intravenous 1.46-13.98 μg/kg 0.00000146 [98]
Batrachotoxin (from poison dart frog) human, sub-cutaneous injection 2–7 μg/kg (estimated) 0.000002 [99]
Abrin (from rosary pea) mice, intravenously
human, inhalation
human, oral 		0.7 μg/kg
3.3 μg/kg
10–1000 μg/kg 		0.0000007
0.0000033
0.00001–0.001 		[citation needed]
Saxitoxin (from certain marine dinoflagellates) human, intravenously
human, oral 		0.6 μg/kg
5.7 μg/kg 		0.0000006
0.0000057 		[99]
Pacific ciguatoxin-1 (from ciguateric fish) mice, intraperitoneal 250 ng/kg 0.00000025 [100]
Palytoxin (from Palythoa coral) mouse, intravenous
[What do the 2 different figures represent?] 		45 ng/kg
2.3–31.5 μg/kg 		0.000000045
0.0000023 		[101]
Maitotoxin (from ciguateric fish) mouse, intraperitoneal 50 ng/kg 0.00000005 [102]
Polonium-210 (210
Po) 		human, inhalation 10 ng/kg (estimated) 0.00000001 [103]
Diphtheria toxin (from Corynebacterium) mice 10 ng/kg 0.00000001 [104]
Shiga toxin (from Shigella bacteria) mice 2 ng/kg 0.000000002 [104]
Tetanospasmin (from Clostridium tetani) mice 2 ng/kg 0.000000002 [104]
Botulinum toxin (from Clostridium botulinum) human, oral, injection, inhalation 1 ng/kg (estimated) 0.000000001 [105]
Ionizing radiation human, irradiation 3–5 Gy (Gray) — [106][107][108]

Poison scale

[edit]
Negative values of the decimal logarithm of the median lethal dose LD50 (−log10(LD50)) on a linearized toxicity scale encompassing 11 orders of magnitude. Water occupies the lowest toxicity position (1) while the toxicity scale is dominated by the botulinum toxin (12).[109]

The LD50 values have a very wide range. The botulinum toxin as the most toxic substance known has an LD50 value of 1 ng/kg, while the most non-toxic substance water has an LD50 value of more than 90 g/kg; a difference of about 1 in 100 billion, or 11 orders of magnitude. As with all measured values that differ by many orders of magnitude, a logarithmic view is advisable. Well-known examples are the indication of the earthquake strength using magnitude scales, the pH value, as a measure for the acidic or basic character of an aqueous solution or of loudness in decibels. In this case, the negative decimal logarithm of the LD50 values, which is standardized in kg per kg body weight, is considered −log10(LD50).

The dimensionless value found can be entered in a toxin scale. Water as the baseline substance is nearly 1 in the negative logarithmic toxin scale.

Procedures

[edit]

A number of procedures have been defined to derive the LD50. The earliest was the 1927 "conventional" procedure by Trevan, which requires 40 or more animals. The fixed-dose procedure, proposed in 1984, estimates a level of toxicity by feeding at defined doses and looking for signs of toxicity (without requiring death).[110] The up-and-down procedure, proposed in 1985, yields an LD50 value while dosing only one animal at a time.[111][112]

See also

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Other measures of toxicity

[edit]
[edit]

References

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Further reading

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Median lethal dose

View on Grokipedia
from Grokipedia
The median lethal dose (LD50), also known as the 50, is a statistical estimate in representing the single dose of a substance that causes in 50% of a test population, usually laboratory animals such as rats or mice, when administered via a specified route (e.g., oral, dermal, or ) and observed over a defined period, typically 14 days. This measure quantifies by providing a standardized endpoint for comparing the potency of chemicals, drugs, or other agents, with values expressed in units like milligrams per kilogram of body weight (mg/kg). Introduced in by British pharmacologist J. W. Trevan to address inconsistencies in early assessments of pharmaceuticals, the LD50 became a foundational tool for evaluation in regulatory contexts, including and industrial chemical classification. Historically, the classical LD50 test involved administering graded doses to groups of 40–100 animals to generate a dose-response curve, from which the median lethal point is interpolated using methods like probit analysis; however, ethical concerns over and variability in results across species led to refinements. Today, the LD50 remains integral to systems like the Globally Harmonized System (GHS) for chemical labeling, categorizing substances into classes (e.g., LD50 < 5 mg/kg indicates extreme acute toxicity) to inform safety data sheets and risk assessments, though it primarily reflects short-term effects rather than chronic exposure or human relevance. Despite its utility, the LD50 has limitations, including interspecies extrapolation challenges, high animal usage in traditional protocols, and poor prediction of non-lethal effects like morbidity; as a result, regulatory bodies like the FDA and have promoted alternatives such as the fixed-dose procedure ( 420), acute toxic class method ( 423), and up-and-down procedure ( 425), which use fewer animals (6–15) and focus on observable toxicity signs without requiring 50% mortality. Emerging and approaches, including cell-based assays and computational modeling, are gaining traction to further reduce while maintaining reliable hazard identification.

Core Concepts

Definition

The median lethal dose (LD50), or 50, is defined as the dose of a substance that is expected to cause in 50% of a test , such as animals, within a specified observation period under controlled conditions. This metric is typically expressed in units of milligrams per of body weight (/kg), allowing for normalization across different sizes and species. The LD50 concept was introduced in by toxicologist J. W. Trevan to provide a reliable benchmark for comparing substance potencies. The primary purpose of the LD50 is to standardize the assessment of , enabling consistent comparisons of hazardous potential among diverse chemicals, regardless of variations in test species or administration methods like oral ingestion, dermal contact, or . By quantifying the dose at which half the population succumbs, it facilitates regulatory classification and risk evaluation in . Conceptually, the LD50 represents the on a sigmoidal dose-response , which describes the nonlinear relationship between increasing doses of a and the proportion of mortality in the population, rising gradually at low doses, steeply around the midpoint, and plateauing toward 100% lethality at high doses. This underscores the probabilistic nature of outcomes in biological systems. Unlike measures of , which evaluate long-term effects from repeated exposures, the LD50 focuses exclusively on acute responses to a single dose, typically observed over 14 days.

Conventions

The median lethal dose, denoted as LD50, is standardized in scientific literature using notations that specify the administration route, such as oral LD50, dermal LD50, or intravenous LD50, to distinguish variations in absorption and based on exposure method. For inhalation exposures, the analogous measure is the median lethal concentration, or LC50, which quantifies the airborne concentration causing 50% mortality. Observation periods are often indicated in the notation, such as LD50/14 to denote a 14-day post-exposure monitoring window, ensuring deaths are attributed to the acute effect of the single dose rather than delayed responses. LD50 values are typically expressed in milligrams per kilogram of body weight (mg/kg), a unit that normalizes for animal size and facilitates cross-study comparisons, while LC50 values use mass concentration units like milligrams per liter (mg/L) for vapors and aerosols or parts per million (ppm) for gases. Route-specific notations, such as intravenous LD50, retain the mg/kg unit but reflect faster systemic distribution compared to oral or dermal routes. To account for experimental variability, reports routinely include 95% confidence intervals around the LD50 or LC50 estimate, calculated via methods like maximum likelihood, enhancing the reliability and interpretability of the data. Reporting conventions mandate explicit specification of the test species, such as or , along with details like age, (often nulliparous females), and status to ensure reproducibility. Exposure conditions must also be detailed, including acute single-dose administration via gavage for oral tests or controlled chamber exposure for , with environmental parameters like temperature (22 ± 3°C) and humidity (30-70%) noted. International standards, particularly the Organisation for Economic Co-operation and Development () Test Guidelines (e.g., No. 425 for acute oral toxicity and No. 403 for inhalation), govern LD50 and LC50 reporting in regulatory contexts, requiring comprehensive data summaries including dose levels, mortality rates, clinical observations, and necropsy findings to support hazard classification under systems like the Globally Harmonized System (GHS). These guidelines emphasize consistency by prohibiting limit tests above 5,000 mg/kg unless justified for human health protection, promoting ethical reductions in animal use while maintaining scientific rigor.

Measurement and Analysis

Experimental Procedures

The determination of the median lethal dose (LD50) involves standardized laboratory protocols using controlled , primarily with , to assess . Test populations typically consist of rats or mice, aged 8 to 12 weeks, selected for their sensitivity and to human . Animals must be healthy, with body weights varying no more than ±20% from the group to ensure uniformity, and they are randomly allocated to treatment groups to minimize . A minimum of 10 to 20 per dose group is common in traditional designs, though modern sequential methods reduce this to 6 to 10 animals total across the experiment. Prior to dosing, animals undergo an acclimation period of at least 5 days in standard housing conditions, including controlled (19–25°C), (30–70%), and a 12-hour light-dark cycle, with unrestricted access to and except during for oral studies. Dose administration follows the route most relevant to anticipated exposure, such as oral, dermal, , or injection, to mimic potential contact. For , the test substance is delivered via gavage using a tube or , with a maximum volume of 1 mL per 100 g body weight (up to 2 mL for aqueous solutions). Injection routes include subcutaneous, intramuscular, intraperitoneal, or intravenous, using needles appropriate to animal size, while testing employs whole-body or nose-only chambers to expose animals to aerosolized or vaporized substances at controlled concentrations. Dose levels are spaced logarithmically for efficiency, often using factors of 2 to 3.2 between levels (e.g., 100, 316, 1000 mg/kg for oral), starting below the estimated LD50 to avoid excessive mortality. In the up-and-down approach, doses are administered sequentially to single animals at 48-hour intervals, increasing the dose if the previous animal survives or decreasing it if death occurs, allowing estimation with fewer animals. Observation protocols span a standard 14-day period post-dosing to capture delayed effects, with animals housed individually or in small groups to facilitate monitoring. Endpoints include mortality as the primary measure, alongside clinical signs of such as changes in (e.g., , tremors), respiratory distress, convulsions, or , recorded at least once within the first 30 minutes, frequently during the initial 4 hours, and daily thereafter. Body weights are measured pretest, on days 7 and 14 (or at death), and all animals undergo gross necropsy to identify target organs. Humane endpoints are enforced to minimize suffering: moribund animals or those showing severe distress (e.g., inability to reach /, prolonged seizures) are euthanized via methods like CO2 or cervical dislocation and counted as deaths for LD50 calculation. The foundational up-and-down method, developed by Dixon and Mood in 1948, revolutionized LD50 testing by using sequential dosing on small samples of to bracket the lethal threshold efficiently, replacing larger fixed-group designs. This approach involves predefined dose steps and alternating directions based on outcomes, typically observing for survival over a short interval before the next dose. Modern refinements, such as Test Guideline 425 adopted in 1998 and updated in 2022, build on this by specifying the half-log progression and integrating limit tests at 2000 or 5000 mg/kg for low-toxicity substances, further reducing animal use while maintaining procedural rigor.

Statistical Methods

The primary statistical method for estimating the median lethal dose (LD50) from quantal dose-response data—where outcomes are binary (death or survival) across dose levels—is , which linearizes the typically sigmoid-shaped mortality curve. In this approach, observed mortality percentages p are transformed into using the inverse of the : probit(p) = Φ-1(p), where Φ is the standard normal CDF; to facilitate computation, "working probits" are often employed by adding 5 to this value, yielding Y = 5 + Φ-1(p), such that Y = 5 corresponds to 50% mortality. These are then regressed against the logarithm of the dose using a with a link: probit(p)=a+b⋅log⁡10(dose)\text{probit}(p) = a + b \cdot \log_{10}(\text{dose}) where a is the intercept and b is the . The parameters a and b are estimated via maximum likelihood, assuming independent binomial responses at each dose level. The LD50 is solved by setting (p) = 0 (or Y = 5 in working probits), giving: LD50=10−a/b\text{LD}_{50} = 10^{-a/b} This method, formalized by Finney, accounts for the assumed underlying of tolerances in the population and is widely applied in for its robustness to moderate sample sizes. Alternative models to probit analysis include the and Weibull distributions, which offer different assumptions about the shape of the tolerance distribution. The model transforms mortality probabilities using the : (p) = ln(p / (1 - p)) = a + b · log10(dose), with LD50 = 10-a/b; it assumes a symmetric , producing curves very similar to but with slightly steeper central slopes, and is preferred when computational simplicity or compatibility with software is prioritized. The Weibull model, in contrast, uses a cumulative distribution of the form p = 1 - exp(-(dose / α)β), where α scales the doses and β shapes the curve (β > 1 yields sigmoid shapes); solving for p = 0.5 gives LD50 = α · (ln 2)1/β, and it is suitable for data exhibiting asymmetry or heavy tails, such as in chronic exposure scenarios where response variability increases at higher doses. remains the standard in studies due to its historical precedence and alignment with biological normality assumptions, while provides nearly identical estimates in practice, and Weibull is selected for datasets deviating from symmetry. Confidence intervals for the LD50 are typically constructed using maximum likelihood estimation of the model parameters, with asymptotic standard errors derived from the inverse Hessian matrix of the log-likelihood. For the probit model, the variance-covariance matrix provides Var(a), Var(b), and Cov(a, b); the standard error of log10(LD50) is then approximated via the delta method: SE(log⁡10LD50)≈Var(a)b2+[log⁡10LD50]2⋅Var(b)b2−2log⁡10LD50⋅Cov(a,b)b3\text{SE}(\log_{10} \text{LD}_{50}) \approx \sqrt{ \frac{\text{Var}(a)}{b^2} + \frac{[\log_{10} \text{LD}_{50}]^2 \cdot \text{Var}(b)}{b^2} - 2 \frac{\log_{10} \text{LD}_{50} \cdot \text{Cov}(a,b)}{b^3} }
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