Median lethal dose
Median lethal dose, commonly referred to as LD50, is a critical measurement in toxicology that represents the dose of a substance required to kill 50% of a population of exposed organisms. The concept is rooted in the study of dose-response relationships, which illustrate how varying doses of a compound affect mortality rates. Typically determined through controlled experiments on animals, LD50 provides a straightforward, quantifiable metric for comparing the toxicity of different substances.
The analysis involves plotting mortality rates against doses to create a dose-response curve, from which the LD50 value is derived by identifying the dose at which 50% mortality occurs. This measurement assumes a monotonic relationship, meaning as the dose increases, mortality also increases. While LD50 is primarily associated with lethal outcomes, the underlying mathematical principles can be applied to other health effects, such as determining doses that induce non-lethal symptoms. Understanding LD50 is essential for risk assessment in fields like food safety and environmental health, offering a standardized way to evaluate the hazards posed by various compounds.
Median lethal dose
Summary: The median lethal dose of a compound is determined through experiment and statistical estimation.
Toxicity often needs to be compared across various chemical compounds and other noxes. The detailed and complex dose-response curve describes the relationship between the dose of a compound and its harmful effect. Frequently, a simple summary in the form of a single number is needed for practical purposes. Median lethal dose, or “LD50” is most popular in this context. It is defined as the dose at which 50% of exposed individuals die.
In order to be meaningful, such a definition implicitly assumes certain features of the dose-response relationship, namely its monotonicity, the fact that mortality increases with dosage. Although the concept is defined for a theoretical dose-response curve, its practical application is strongly related to statistical estimation of the dose-response curve model based on data obtained from an experiment with many animal or other nonhuman organisms randomly assigned to various doses.
Toxicological Testing
In toxicology and related disciplines, such as food safety and environmental risk assessment, one often needs to quantify how toxic or dangerous a substance is. A quantification of the harmful effect is needed for many practical comparisons; for instance, to compare the toxicity of different substances or to compare them with a standard. Although there are many possible aspects of how dangerous a compound is, survival of exposed individuals is frequently of interest. The survival is assessed experimentally in the “quantal response trial.”
It is based on a set of animal or other nonhuman organisms, whose randomly selected groups are exposed to different doses of the tested compound. The outcomes are summarized as the percentages or proportions of those that survived in each dose group. Mortality would be just the complement of the proportion of survivors, expressed as: Mortality = (number of individuals dead after exposure) ÷ (total number of exposed individuals).
Based on common sense, one would expect that the mortality would increase with the dose of a toxic compound. Most typically this is indeed the case and the mortality obtained from an experiment with a large total number of exposed is monotonic, meaning it increases with the dose.
Dose-Response Curve
When mortality is taken as a function of dose, one can plot the so-called dose-response curve. Dose-response curve has lower asymptote at 0 since no exposure-related death can occur when no exposure is applied. Similarly, it has upper asymptote at 1, since exposure-related death will always occur with a large enough dose. The asymptotes are shown as horizontal dashed lines.
Note that often one needs to go over several orders of magnitude of doses in order to observe transition from zero effect to the full effect, so the dose-response is then plotted as the mortality versus logarithm of the dose. Since the logarithm is a one-to-one function, nothing is lost by the transformation, and the plot is more readable.
LD50
Because the dose-response curve is a rather complex quantity, many possible features might be compared across different compounds. It might be cumbersome in practice to compare curves, however. A simple summary is often all what is needed. Median lethal dose, or LD50, is the most popular characteristic. It is defined as the dose at which 50% of exposed individuals die. When a dose-response curve is available, an LD50 is constructed by drawing a horizontal line at 0.5, finding its intersection with the dose-response curve, drawing vertical line at the intersection, and reading off the value where it crosses the horizontal axis.
Statistical Estimation
In practice, one does not have the dose-response curve at hand. It needs to be estimated from experimental data by statistical means. In fact, the mortalities obtained from two experiments with the same doses would be very likely somewhat different, as a result of random errors. For example, different randomly selected experimental animals would react differently to a given dose.
Nevertheless, when the size of the experiment increases, increasing both the number of animals in every dose group and increasing the number of different dose groups, random errors would tend to decrease in line with the law of large numbers. In fact, for a very large experiment, the mortality estimates get close to the probabilities of survival. Since not all of the infinite possible doses can be explored in a real experiment, a model relating the survival probability to the dose is assumed in order to be able to interpolate between the doses actually used in the experiment. An interpolation is typically needed when calculating LD50. Parameters of the model are then estimated by various statistical means. Very often, logistic regression is used to this end.
Other Uses of LD50
While the definition of LD50 is directly related to lethality, the mathematical concepts used in LD50 testing and modeling can be applied to many other less-dramatic outcomes. In general, these models are useful when the relationship being explored involves a binary response variable, like yes/no or pass/fail, predicted by a quantitative explanatory variable, as long as the relationship is bounded and monotonically increasing in the same manner as before. For example, rather than finding the dose that induces mortality, researchers may wish to model what dose of a medicine will cause 50% of exposed individuals to show a certain, nonlethal symptom.
Bibliography
Agresti, A. Categorical Data Analysis. Hoboken, NJ: Wiley, 2002.
Casarett, L. J., J. Doull, and C. D. Klaassen, eds. Casarett and Doull’s Toxicology: The Basic Science of Poisons. 6th ed. New York: McGraw-Hill, 2001.
Dixon, W. J. Design and Analysis of Quantal Dose-Response Experiments. Los Angeles: Dixon Statistical Associates, 1991.