Biological organisms respond differently to the same dose of a toxicant. These differences are a result of age, sex, weight, diet, general health, and other factors. For example, consider the ef- fects of an irritant vapor on human eyes. Given the same dose of vapors, some individuals will barely notice any irritation (weak or low response), whereas other individuals will be severely irritated (high response).
Consider a toxicological test run on a large number of individuals. Each individual is ex- posed to the same dose and the response is recorded. A plot of the type shown in Figure 2-2 is prepared with the data. The fraction or percentage of individuals experiencing a specific re- sponse is plotted. Curves of the form shown in Figure 2-2 are frequently represented by a nor- mal or Gaussian distribution, given by the equation
where
f(x) is the probability (or fraction) of individuals experiencing a specific response, x is the response,
a is the standard deviation, and p is the mean.
L I
Low Average High
response response response
Figure 2-2 A Gaussian or normal distribution representing the biological response to exposure to a toxicant.
2-5 Dose versus Response 43
The standard deviation and mean characterize the shape and the location of the nor- mal distribution curve, respectively. They are computed from the original data fixi) using the equations
where n is the number of data points. The quantity u 2 is called the variance.
The mean determines the location of the curve with respect to the x axis, and the stan- dard deviation determines the shape. Figure 2-3 shows the effect of the standard deviation on the shape. As the standard deviation decreases, the distribution curve becomes more pro- nounced around the mean value.
The area under the curve of Figure 2-2 represents the percentage of individuals affected for a specified response interval. In particular, the response interval within 1 standard devia- tion of the mean represents 68% of the individuals, as shown in Figure 2-4a. A response inter- val of 2 standard deviations represents 95.5% of the total individuals (Figure 2-4b). The area under the entire curve represents 100% of the individuals.
Figure 2-3 Effect of the standard deviation on a normal distribution with a mean of 0. The dis- tribution becomes more pronounced around the mean as the standard deviation decreases.
44 Chapter 2 Toxicology
Figure 2-4 Percentage of individuals affected based on a response
p-20 ~1 p+2u between one and two
standard deviations of
(b) the mean.
Example 2-1
Seventy-five people are tested for skin irritation because of a specific dose of a substance. The responses are recorded on a scale from 0 to 10, with 0 indicating no response and 10 indicating a high response. The number of individuals exhibiting a specific response is given in the following table:
2-5 Dose versus Response 45
Number of individuals Response affected
a. Plot a histogram of the number of individuals affected versus the response.
b. Determine the mean and the standard deviation.
c. Plot the normal distribution on the histogram of the original data.
Solution
a. The histogram is shown in Figure 2-5. The number of individuals affected is plotted versus the response. An alternative method is to plot the percentage of individuals versus the response.
Response
Figure 2-5 Percentage of individuals affected based on response.
46 Chapter 2 Toxicology
b. The mean is computed using Equation 2-2:
The standard deviation is computed using Equation 2-3:
u 2 = [(I - 4.51)'(5)
+
(2 - 4.51)~(10)+
(3 - 4.51)~(13)c. The normal distribution is computed using Equation 2-1. Substituting the mean and stan- dard deviations, we find
The distribution is converted to a function representing the number of individuals affected by multiplying by the total number of individuals, in this case 75. The corresponding values are shown in Table 2-3 and Figure 2-5.
Table 2-3 Theoretical Frequency and Number of People Affected for Each Response for Example 2-1
2-5 Dose versus Response 47
Dose
Figure 2-6 Dose-response curve.
The bars around the data points represent the standard deviation in response to a specific dose.
The toxicological experiment is repeated for a number of different doses, and normal curves similar to Figure 2-3 are drawn. The standard deviation and mean response are deter- mined from the data for each dose.
A complete dose-response curve is produced by plotting the cumulative mean response at each dose. Error bars are drawn at +a around the mean. A typical result is shown in Fig- ure 2-6.
For convenience, the response is plotted versus the logarithm of the dose, as shown in Figure 2-7. This form provides a much straighter line in the middle of the response curve than the simple response versus dose form of Figure 2-6.
If the response of interest is death or lethality, the response versus log dose curve of Fig- ure 2-7 is called a lethal dose curve. For comparison purposes the dose that results in 50%
Logarithm of the dose
Figure 2-7 Response versus log dose curve.
This form presents a much straighter function than the one shown in Figure 2-6.
48 Chapter 2 Toxicology
a,
1
500 Q
(I)
F
.- 0 X
Figure 2-8 The various
10 types of response vs. log
dose curves. ED, effective dose; TD, toxic dose; LD, lethal dose. For gases, LC
EDlo ED50 TD50 LD50 (lethal concentration)
Logarithm of the dose is used.
lethality of the subjects is frequently reported. This is called the LD,, dose (lethal dose for 50%
of the subjects). Other values such as LD,, or LD, are sometimes also reported. For gases, LC (lethal concentration) data are used.
If the response to the chemical or agent is minor and reversible (such as minor eye irri- tation), the response-log dose curve is called the effective dose (ED) curve. Values for ED5,, ED,,, and so forth are also used.
Finally, if the response to the agent is toxic (an undesirable response that is not lethal but is irreversible, such as liver or lung damage), the response-log dose curve is called the toxic dose, or T D curve.
The relationship between the various types of response-log dose curves is shown in Figure 2-8.
Most often, response-dose curves are developed using acute toxicity data. Chronic toxi- city data are usually considerably different. Furthermore, the data are complicated by differ- ences in group age, sex, and method of delivery. If several chemicals are involved, the toxicants might interact additively (the combined effect is the sum of the individual effects), synergisti- cally (the combined effect is more than the individual effects), potentiately (presence of one in- creases the effect of the other), or antagonistically (both counteract each other).