Determination of 50% endpoint titer using a simple formula

Abstract

Two commonly used methods for calculating 50% endpoint using serial dilutions are Spearman-Karber method and Reed and Muench method. To understand/apply the above formulas, moderate statistical/mathematical skills are necessary. In this paper, a simple formula/method for calculating 50% endpoints has been proposed. The formula yields essentially similar results as those of the Spearman-Karber method. The formula has been rigorously evaluated with several samples.

Key Words: Endpoint dilution, TCID50, Spearman-Karber, Reed and Muench

Core tip: The formula described in this manuscript can be used to calculate 50% endpoint titre such as TCID50%, LD50, TD50, etc., in addition to the currently existing methods. The proposed formula can be applied without the help of calculator or computer.

TO THE EDITOR

Currently, there are two methods (formulas) viz., Reed and Muench[1] and Spearman-Karber[2,3] are commonly employed for the calculation of 50% endpoint by serial dilution. To understand/apply these methods, moderate mathematical skills along with calculator or computer are essential. Here, I have proposed a simple formula to calculate the 50% endpoint titre and this formula can be used in addition to Reed and Muench or Spearman-Karber, methods but not exclusively at this point. In the following section, the newly proposed method is compared with two commonly used methods viz., Reed and Muench and Spearman-Karber.

Reed and Muench method

log₁₀ 50% end point dilution = log₁₀ of dilution showing a mortality next above 50% - (difference of logarithms × logarithm of dilution factor).

Generally, the following formula is used to calculate “difference of logarithms” (difference of logarithms is also known as “proportionate distance” or “interpolated value”): Difference of logarithms = [(mortality at dilution next above 50%)-50%]/[(mortality next above 50%)-(mortality next below 50%)].

Spearman-Karber method

log₁₀ 50% end point dilution = - (x₀ - d/2 + d ∑ r_i/n_i)

x₀ = log₁₀ of the reciprocal of the highest dilution (lowest concentration) at which all animals are positive;

d = log₁₀ of the dilution factor;

n_i = number of animals used in each individual dilution (after discounting accidental deaths);

r_i = number of positive animals (out of n_i).

Summation is started at dilution x₀.

Newly proposed method

Formula 1:

log₁₀ 50% end point dilution = -[(total number of animals died/number of animals inoculated per dilution) + 0.5] × log dilution factor.

Formula 2 (if any accidental death occurred):

log₁₀ 50% end point dilution = -(total death score + 0.5) × log dilution factor.

Comparison of the newly proposed and existing methods with an example of virus titration in mice: For simplicity, it is assumed that 1 mL of each dilution was inoculated (Table 1, Table 2 and Table 3).

Table 1 Calculation of virus titre in mice using the Reed and Muench method.

Log10 virus dilution	Mice		Cumulative total			Percent mortality
	Died	Survived	Died	Survived	Total
-1	10	0	57	0	57	57/57 × 100 = 100
-2	10	0	47	0	47	47/47 × 100 = 100
-3	10	0	37	0	37	37/37 × 100 = 100
-4	10	0	27	0	27	27/27 × 100 = 100
-5	10	0	17	0	17	17/17 × 100 = 100
-6	6	4	7	4	11	7/11 × 100 = 63
-7	1	9	1	13	14	1/14 × 100 = 7

Difference of logarithms = (63-50)/(63-7) = 0.23; log₁₀ 50% end point dilution = -6 - (0.23 × 1) = -6.23; 50% end point dilution = 10^-6.23; the titre of the virus = 10^6.23 LD₅₀/mL.

Table 2 Calculation of virus titre in mice using the Spearman-Karber method.

Log10 virus dilution	Mice
	Died	Inoculated
-1	10	10
-2	10	10
-3	10	10
-4	10	10
-5	10	10
-6	6	10
-7	1	10

x₀ = 5; d = 1; log₁₀ of 50% endpoint dilution = - [5 - ½ + 1 (17/10)] = -6.2; 50% end point dilution = 10^-6.2; the titre of the virus = 10^6.2 LD₅₀/mL.

Table 3 Calculation of virus titre in mice using the new method.

Log10 virus dilution	Mice		Death score
	Died	Inoculated
-1	10	10	10/10 = 1
-2	10	10	10/10 = 1
-3	10	10	10/10 = 1
-4	10	10	10/10 = 1
-5	10	10	10/10 = 1
-6	6	10	6/10 = 0.6
-7	1	10	1/10 = 0.1
Total	57		5.7

By using formula 1: log₁₀ 50% end point dilution = - (57/10 + 0.5) × 1 = -6.2; 50% end point dilution = 10^-6.2; the titre of the virus = 10^6.2 LD₅₀/mL. By using formula 2: log₁₀ 50% end point dilution = - (5.7 + 0.5) × 1 = -6.2; 50% end point dilution = 10^-6.2.

The newly proposed formula has been intensively validated with several samples and essentially yields the same results as those by the Spearman-Karber method. Therefore, the newly proposed method can be used in addition to the existing methods but not exclusively at this point.