Haber's rule: a special case in a family of curves relating concentration and duration of exposure to a fixed level of response for a given endpoint.

The concept that the product of the concentration (C) of a substance and the length of time (t) it is administered produces a fixed level of effect for a given endpoint has been ascribed to Fritz Haber, who was a German scientist in the early 1900s. He contended that the acute lethality of war gases could be assessed by the amount of the gas in a cubic meter of air (i.e. the concentration) multiplied by the time in min that the animal had to breathe the air before death ensued (i.e. C x t=k). While Haber recognized that C x t=k was applicable only under certain conditions, many toxicologists have used his rule to analyze experimental data whether or not their chemicals, biological endpoints, and exposure scenarios were suitable candidates for the rule. The fact that the relationship between C and t is linear on a log-log scale and could easily be solved by hand, led to early acceptance among toxicologists, particularly in the field of entomology. In 1940, a statistician named Bliss provided an elegant treatment on the relationships among exposure time, concentration, and the toxicity of insecticides. He proposed solutions for when the log-log plot of C and t was composed of two or more rectilinear segments, for when the log-log plot was curvilinear, and for when the slope of the dosage-mortality curve was a function of C. Despite the fact that Haber's rule can underestimate or overestimate effects (and consequently risks), it has been used in various settings by regulatory bodies. Examples are presented from the literature of data sets that follow Haber's rule as well as those that do not. Haber's rule is put into perspective by showing that it is simply a special case in a family of power law curves relating concentration and duration of exposure to a fixed level of response for a given endpoint. Also shown is how this power law family can be used to examine the three-dimensional surface relating C, t, and varying levels of response. The time has come to move beyond the limited view of C and t relationships inferred by Haber's rule to the use of the broader family of curves of which this rule is a special case.