Modeling Acute Toxicity Dose-Response

By definition, things happen quickly during acute toxic responses to chemical exposure.  Indeed, adverse health effects from inhaling an acute toxicant can happen in the time-frame of seconds to tens of minutes.   Haber’s Rule advises that the toxic effect should be a linearly combined function of both concentration and time.  The classic expression of Haber’s Rule is: 

  Toxic Response = f ((Concentration)(time)).   

This mathematical expression says that when the breathing zone concentration is twice as high (e.g., 200 ppm versus 100 ppm) the time of exposure needs only be half (50%) as long to get the same toxic response.

The majority of acute toxicity dose-response modeling that has been done to date deals with lethality from inhalation.  Here acute lethality inhalation testing is done with rats in a time frame of tens of minutes extending to hours in duration.  These data are then modeled with the intention that these models will be useful for predicting human risk.

Relative to acute inhalation toxicity, Haber’s rule needs some adjustment.  The modified relationship that fit reality better is:

  Lethality = f ((Concentration)ⁿ(time))    n > 1   (but typically < 4)

In this relationship, the inhaled concentration has a non-linear effect on lethality.  For example, at n = 2, when the concentration is twice as high the time of exposure only needs to be 25% as long to render the same response.

Some of the earliest work on this was done in the Netherlands and, I believe, that much of the modeling tools were developed by Dr. Wil tenBerge.    The current standard equation used to fit animal data is:

  Probit = a + b ln (Cⁿt)

a, b and n are coefficients.   C is expressed either as mg/m³ or ppmv.

You may or may not remember what a “probit” is but it is a very useful mathematical construct.    It is directly related to the standard Gaussian bell-shaped curve with the area-under-the-curve (AUC) describing the portion of a population included on any part of that curve.  I know what some of you are thinking but please stay with me on this!  Like a professor of mine once said, “If it’s foggy you’re definitely learning something!”    

The peak of the bell-shaped Gaussian curve is right at 50%.   That is, half the folks are in the area-under- the-curve (AUC) below (to the left of) the peak and half are in the AUC above (to the right) of the peak.   This is the average or mean value.   It is also Probit = 5.   Now I am going to ask you to remember a statistical construct you learned called the standard deviation.   The AUC from the value of one  standard deviation above the mean on the Gaussian curve is approximately 84%; that is, 84% of the population is in the AUC to the left of one standard deviation above the mean.   One standard deviation above the mean is also Probit = 6.    Because it symmetrical, Probit = 4 is one standard deviation below the mean and only 16% of the population are in the AUC to the left of this value.  

Still foggy?  Let’s put this in terms we all understand:  SAT scores.    The average or mean SAT test score is a Probit multiplied by 100; that is, Probit 5 x 100 = 500.  Half the folks taking the test got a higher SAT score than 500 and half lower.   If you got an SAT of 600 you did better than 84% of the folks taking the test.    If you got a 733 on the SAT you were better than 99% of the folks who took that test.   The computer stops at 800 because you are getting so close to 100% that it does not matter any more. How did I figure that a 733 SAT score beat 99% of the folks tested?  Ans: It is a simple function in Excel, just put in (=NORMSINV(0.99)  + 5) into a cell and you will get 7.33, multiply by 100 to get the SAT score.

We all probably remember living and dying with “the curve” in college.  The raw tests scores were converted to probits and, depending on the teacher, the grades assigned such that there was something like 10% “A”s, 20% “B”s, 50% “C”s and perhaps 10% “D”s or lower.   This is how you could get 40 out of 100 correct on a physics test and still get a “B”!  This is also why we all hated the person who “killed the curve” by scoring very high and dragging the mean upward and everyone with lower scores downward.

So let’s shift our thinking back to dose-response.   Probit = 7.33 (5 + 2.33) means that 99% of the population will respond with the toxic effect being measured, in this case death.   Since it is symmetrical, Probit = 2.67 (or 5 – 2.33) means that 1% of the exposed population will be predicted to die.  You can never get there but you can get as close to 0% as you like with smaller and smaller probit values.  So let reproduce the above equation here:

Probit = a + b ln (Cⁿt)

Given any value of breathing zone concentration (C) over any time interval (t) and the fitted values for a, b and n (from animal studies) we get a predicted percentage response expressed as a probit.   At Probit = 9 everyone is predicted to respond.  At Probit = 1 essentially no one is predicted to respond or be adversely effected by this concentration over this time interval.  Using another function in Excel (NORMSDIST()) we can easily convert the probits to percentages predicted to respond.

A previous blog here discussed bolus exposures to acute toxicants.  Exposures that occur in a time frame of seconds to minutes.   In most cases we are not dealing with lethality but we could be encountering serious respiratory irritation from this short term exposures.   If we had good toxicological data on these responses at various C,t points we could model the percentage of the exposed population predicted to have a respiratory irritation response.   

As I mentioned in a previous blog, Dr. Wil tenBerge is an incredibly generous colleague who shares all of his models and software on his web site.  Just put in “home page Wil tenBerge” into Google.   http://home.wxs.nl/~wtberge/   He also has a considerable amount of very good educational material explaining this further as well as quite a few data sets of C,t rat lethality for chemicals like ammonia.

Questions for Discussion:

In a previous blog I discussed that potent chronic carcinogens like nitrosamines can have serious acute inhalation irritation potential. Do you have “chronic” toxicants in your work place that you are controling to 8 hour OELs than might also be acute irritants?   How would you evaluate and control this acute risk?

Is anyone out there aware of some available in-vivo or in-vitro toxicological testing protocols that could evaluate acute C,t irritation potential?   If so please share.