Humboldt State University ® Department of Chemistry

Richard A. Paselk

Chem 451

Biochemical Toxicology

Spring 2010

Lecture Notes:: 12 February

© R. Paselk 2008


Quantitation of Dose Response

Dose-Response Curves, cont.

2. For this discussion to be relevant to Dose, we must have a particular relationship between dose and concentration at the active site. Generally we assume tissue concentration is related to dose - but what if it isn't? What factors might affect it?

Notice that the dose itself may alter any one of the above processes. For example, the substance may be actively excreted, but it "kills" the kidney, leading to a complex dose-response behavior.

These various processes may prevent a compound from reaching the site of action at lo doses, particularly with irreversibly bound compounds, and thus contribute to a threshold phenomena. We will look in detail at some of these processes later with specific examples, demonstrating how they can drastically affect the concentration at the site of action.

3. Finally, the response may not be causally related to the compound. 

Quantitation of Dose Response

Gaussian (Normal) Distributions: For populations often assume a Gaussian distribution, that is we assume that we in fact have a completely random distribution of values around a central value, as in the Gaussian curve below.

gausian curve
Recall that for a Gaussian distribution the the mean (arithmetic average), median (central value), and mode (most frequent value) coincide. It is also useful to note that the area under the curve between ± 1s (1 sigma) includes 68% of all values (of the population), while ± 2s (2 sigma) includes 95% and ± 3s (3 sigma) includes 99% of all values.
Of course not all distributions are randomly distributed around a central point. For example the distribution of concentrations around an average value near zero will be skewed toward high values since we cannot have concentrations below zero. This is similar to the situation we see for the speeds of gas molecules etc.

Because of the wide occurrence and common knowledge, and mathematical exactness of the Gaussian function other distributions are often transformed to get a Gaussian shape. Thus a log transformation will often convert a skewed distribution into a Gaussian. One should exercise caution however. The Gaussian is too nice, it is seductive because of its convenience etc., but it is not always followed. If at all in doubt one should test a distribution to be certain it is in fact Gaussian. Linear transformations are very useful in this respect. Thus one may plot data on "probability paper" or in Probit units for frequency (below) vs. log value. In each case deviation from linearity indicates a non-Gaussian distribution. This deviation in turn can be analyzed statistically etc.

Sigmoidal plot of normal distributions

Sigmoidal plot of normal distributions: If the values in a normal distribution are plotted vs. cumulative frequency instead of frequency, a sigmoidal plot results (Ideally the curve should intersect 0 at 50%, artifacts in the data set used in preparing the figure resulted in the error seen in this representation.):

sigmoidal cumulative response curve

Linear Dose Response

Assume the dose-response passes through zero,zero, that is that there is no threshold above zero. This is a common assumption for radiation and for carcinogens based on the idea that it only takes a single mutation or base change in DNA to lead to a cancer, so any dose will have a potential impact.

Of course in reality there are repair mechanisms which may result in threshold type responses in reality. For example, most of us can take large doses of sushine, which we know produce many mutations and much DNA damage even with NO "sunburn" with no apparent harm because we can repair this damage.


Hormesis refers to the idea that low doses of toxins may actually be beneficial. This is very controversial, but there is evidence for hormesis for many toxins. Thus dioxin appears to be protective regarding carcinogenesis in some animals while being a potent carcinogen at high concentration. Of course many metals required at low concentrations, such as copper(II), are toxic at high concentrations.

Probit Analysis

Probit Analysis gives a linear response from sigmoidal (Gaussian) Dose-response curves. For probit analysis divide data into multiples of s ("Probit units") from the mean. But define mean = 5. Thus -1s = 4, 1s =6 etc. The standard deviation is sometimes called NED (normal equivalent deviation). So for 50% response (mean value in Gaussian) NED = 0, for 15.9% (50 - 68.2/2) response NED = -1, for 84.1% (68.2 + 15.9) NED =1. So Probit = NED + 5.

Plotting probit units against log dose will then give a linear function for populations following a normal distribution. Plots of data in terms of probit units are very convenient. Very easy to define LD50 etc. [T. Fig. 2.6] Can also do regression analysis, and the slope aids in judging hazards. A narrow range gives a steep slope, indicating little warning for serious effects.

Selection of Toxic Criteria

We have looked at one of the simplest measures of toxicity, the LD50. Lets define a few of these measures:

Sometimes the occurrence of parallelism between the probit curves for these three measures is interpreted as indicating an equivalence in their mechanisms - not necessarily true.

Therapeutic Index

Therapeutic index is a comparison of effective versus lethal doses - an important thing to know if you are giving drugs out!

A safer, more reliable and thus more valuable expression for TI in drug development is thus TI = TD1/ED99 or LD1/ED99. This should lead to a very safe situation.

Finally, one should keep in mind that the determination of LD50 etc. are not biologically constant, but in fact depend on many factors, such that we often see differences between labs and even within the same lab.

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Last modified 16 February 2010

© RA Paselk 2001