Median lethal dose (LD₅₀) is the amount of test agent that can cause death in 50% of test animals. The LD₅₀ concept was first introduced in 1927 by Trevan [1]. There are different methods used to estimate median lethal dose [2-10]. Up – and – down procedure was adopted several times [11-17]. The method involves the use of less number of animals (5 – 15), is faster and its findings are relatively comparable to the findings from other methods [18]. For all the methods, the study outcome is likely to be influenced by the choice of starting dose level(s), relative to the true LD₅₀ value, especially in the case of shallow slopes [19]. Therefore, information on toxic effects seen only at dose levels close to a lethal dose will not be obtained [20]. Upon all the benefits being derived from the most recent and the past revised up – and – down procedure, there is hurdle of mathematical complexity involved using up – and – down procedure. Such mathematical complexity involves rough estimation of LD₅₀ using geometric mean, maximum likelihood estimator, most confidence interval and a point estimate of the LD_50, which is a high descriptor of toxicity of a chemical to a population. In order to remove all the hurdles mentioned above and decrease the number of test animals, I hereby present a simplest arithmetic method for rough estimation of LD₅₀ using up – and – down procedure.

Revision of up – and – down procedure

Stitzel gave a hypothetical example [21-23] of how to estimate LD₅₀ in “Test Guide Lines 425”, using up – and – Down Procedure as follows:

Table 1: Stitzel hypothetical example.

Since there are three reversals, Stitzel calculated the geometric mean of all the doses and arrived at 209 mg/kg body weight [21]. My views about Stitzel estimation are: Arithmetic or harmonic mean can be used in place of geometric mean and that would give us 212.3 mg/kg body weight; Standard deviation (SD) of the either mean should serve as default dose progression which is logarithm of 41 mg/kg (1.6); and Standard error of the either mean (SEM) should also serve as lower and upper boundary of the LD₅₀. i.e. +16.7mg/kg realizing the fact that LD₅₀ is not fixed. So our LD₅₀ from Stitzel example should be 212+16.7 mg/kg body weight. Hence the LD₅₀ estimated by Stitzel [21] is within the range of my LD₅₀. Saganuwan et al. estimated LD₅₀ of aqueous extract of Abrus precatorius leaf using geometric mean10 as presented below:

The Geometric mean of the three reversals seen in Table 2 is 2558.8 mg/kg body weight. My views about Saganuwan et al. estimation [10] are: Arithmetic or harmonic mean can be used instead of geometric mean and that would give us 2823.7 mg/kg body weight; Standard deviation of the either mean should serve as default dose progression which is logarithm of 1193.1 mg/kg (3.1); Standard error of mean (SEM) should serve as lower and upper boundary of the LD50. i.e. + 487.2 mg/kg body weight, realizing the fact that LD₅₀ is not fixed. Therefore, the LD₅₀ (2823.7 + 487.2 mg/kg) reported by Saganuwan et al [10] should be between 2336.5 and 3310.9 mg/kg body weight. Hence, the LD₅₀ of Saganuwan et al. falls within the range of my LD_50.

Table 2: Toxicity pattern of Abrus precarious leaf extract

The LD₅₀ (209 mg/kg) estimated by Stitzel [23] is within the range of my estimated LD₅₀ (205 to 229.0 mg/kg) signifying that arithmetic mean of three reversals can be used as rough estimate of LD₅₀ and standard deviation can be used as default dose progression, whereas standard error of mean can serve as lower and upper boundary of the mean which may in turn represent confidence interval. But the LD₅₀ (2558.8 mg/kg) estimated from Saganuwan et al. [10] is within the estimated range of my LD₅₀ (2336.5 to 3310.9 mg/kg body weight). Saganuwan et al. [10, 22] and Saganuwan [9, 18] had earlier reported that median lethal dose of aqueous extract of Abrus precatorius leaf is between 2559.5 and 3011.4 mg/kg body weight. Therefore, there is precision, validity and reliability of using arithmetic mean as rough estimate of LD₅₀, standard deviation as default dose progression and standard error of mean as lower and upper boundary. The arithmetic, geometric and harmonic means are Pythagorean means. In their calculating, the arithmetic mean is greater than or equal to the geometric mean, which is greater than or equal to the harmonic mean. The geometric mean is particularly appropriate for exponential type of data. The harmonic mean is good for things like rates and ratios where an arithmetic mean would actually be incorrect. But when the sample size is unequal, the far and away most common procedure uses the harmonic mean of sample sizes. As a result, an unbalanced design will have less statistical power because the average samplesize will tend toward the least sample. The mean uses every value in the data and hence is a good representative of the data. Repeated samples drawn from the sample population tend to have similar means. The mean is therefore the measure of central tendency that best resists the fluctuation between different samples [25]. It is closely related to standard deviations, the most common measure of dispersion [24]. The important disadvantage of mean is that it is sensitive to extreme values/outliers, especially when the sample size is small [26]. Therefore, it is not an appropriate measure of central tendency for skewed distribution [27]. Harmonic mean is appropriate in situations where the reciprocals of values are more useful. It is used for determination of the average sample size of a number of groups, each of which has a different sample size. Geometric mean is an appropriate measure when values change exponentially and in case of skewed distribution that can be made symmetrical by a log transformation. It cannot be used if any of the values are zero or negative [24]. However, because small numbers of animals were used, the actual level of confidence was generally not exact [22]. The random stopping rule in UDP improves the ability of the test overall to respond to varying underlying conditions, but also causes the reported level of confidence and the actual level of confidence to differ somewhat [23]. Of all the principles of the three alternatives (fixed dose; acute toxic class and up – and - down), up – and – down, is the most suitable, because it gives endpoint – evident toxicity and obeys stopping criteria to limit number of animals used, using only 2 – 6 animals with possible 1 – 3 deaths [24]. If there is no death above 15,000 mg/kg body weight, the test agent is practically safe and that serves as the limit boundary for safety [28-30]. The revised method saves time, reserves large numbers of animals that may be waisted and ignores complex mathematical manipulation involved and so encouraging the principles of refinement, reduction and replacement (3R).

In conclusion, either arithmetic or harmonic mean can be used in place of geometric mean as rough estimate of median lethal dose (LD50) using up-and-down procedure since arithmetic, geometric and harmonic means go together harmoniously.