Uptake Kinetics, Species Differences, and the Determination of Equivalent Combinations of Air Concentration and Exposure Duration for Assessment of Acute Inhalation Toxicity

ABSTRACT

For acute inhalation toxicity assessment, I develop a conceptual framework for expressing combinations of intensity (air concentration) and duration that produce equivalent toxicity by examining how the shape of the body-burden uptake curve during a bout of inhalation interacts with various pharmacodynamic measures of the critical body burden needed to produce toxicity. If toxicity depends on attaining a critical tissue concentration, three existing empirical approaches—Haber's Law, the ten Berge equation, and pure air-concentration-dependence—are but local approximations to different parts of an overarching mathematical relationship. The compound-specific half-life of elimination determines the range of durations for which each applies: durations of one half-life or shorter follow Haber's Law, exposures of 4 or more half-lives follow pure air-concentration-dependence, and intermediate durations can be approximated by the ten Berge equation. Better animal-to-human extrapolation is achieved if exposure durations are expressed as number of species-specific half-lives. I consider several alternative pharmacodynamic criteria, such as the dependence of toxicity on time spent above a critical tissue concentration, or on the area under the tissue concentration curve, on the tissue concentration of a toxic metabolite, or on the imbalance of damage and repair processes.

Key Words:

• acute toxicity
• inhalation toxicity
• Haber's Law
• ten Berge equation
• acute emergency guideline levels
• cross-species extrapolation

ACKNOWLEDGMENTS

This work was funded by the USEPA under contracts 4C-R302-NALX and EP-06-C-000304. The ongoing support and advice of Dr. Gary L. Foureman of the USEPA is gratefully acknowledged.

The author declares no conflict of interest.

Notes

*Technically, steady-state is approached asymptotically.

¹ Notably, however, nowhere is the ten Berge equation exactly correct, since the value of n applies only locally and in fact drifts in value as one moves along the curve. The particular straight-line approximation and its slope depend on what part of the range one is approximating. Indeed, in principle one could describe the whole equitoxicity curve in terms of the pattern of changing values of n as it gradually changes from unity at the lower right to infinity at the asymptote. An important consequence of this is that n is not a property inherently characteristic of a chemical, as it is often treated as being, but rather a property of the particular set of exposure durations that have been tested for that chemical (and used to estimate a value of n that gives an approximately correct curve within that range only).

² The blood levels of a toxic metabolite depend on its rate of formation as well as its rate of clearance—indeed, the blood level will be dictated by the balance of the two. A quickly formed metabolite will be a significant source of removal of the parent compound, that is, it is a cause of the quickly cleared metabolite pattern in Figure 15. For a slowly formed metabolite, since we are discussing exposures that produce toxicity, the levels of metabolite formation are by definition sufficient to produce toxic blood levels; if metabolite clearance is quick, and the consequent low blood level is toxic, then the metabolite is evidently quite potent. But in any case, the shape of the metabolite blood curve over time, and hence the time-intensity tradeoffs for equitoxic exposures, is driven by the half-life of clearance of the metabolite.

*One could replace V _max/K _m with a single parameter for first-order metabolism, but the ratio is used to preserve generality if saturable metabolism were to be considered, as it is not in the present article. Because we are using a one-compartment model, V _max and K _m represent whole-body parameters rather than the liver-only intrinsic clearance parameters that would be necessary in a multicompartment model.

³ The value of k is best determined by fitting Eq. (A5) to data on the declining blood level after exposure has ceased. C _air* cam be found by measuring (or estimating by interpolation) the blood concentration C _crit that just produces toxicity in an experiment of defined C _air and T. These can be plugged in to Eq. (B3) and the result solved for M. Then C _crit/M gives C _air*.

^†Using human value of k.