A benchmark dose (BMD) is the dose of a substance that corresponds to a prescribed increase in the response (called the benchmark response or BMR) of a health effect. A statistical lower bound on the benchmark dose (BMDL) has been proposed as a replacement for the no-observed-adverseeffect-level (NOAEL) in setting acceptable human exposure levels. A method is developed in this paper for calculating BMDs and BMDLs from continuous data in a manner that is consistent with those calculated from quantal data. The method involves defining an abnormal response, either directly by specifying a cutoff x,, that separates continuous responses into normal and abnormal categories, or indirectly by specifying the proportion Po of abnormal responses expected among unexposed subjects. The method does not involve actually dichotomizing individual continuous responses into quantal responses, and in certain cases can be applied to continuous data in summarized form (e.g., means and standard deviations of continuous responses among subjects in discrete dose groups). In addition to specifylng the BMR and either xo or Po, the method requires specification of the distribution of continuous responses, including specification of the dose-response 8(d) for a measure of central tendency. A method is illustrated for selecting 6(d) to make the probability of an abnormal response any desired dose-response function. This enables the same dose-response model (Weibull, log-logistic, etc.) to be used for the probability of an abnormal response, regardless of whether the underlying data are continuous or quantal. Whenever the continuous responses are normally distributed with standard deviation u (independent of dose), the method is equivalent to defining the BMD as the dose corresponding to a prescribed change in the mean response relative to u.

A method is presented for numerically inverting a Laplace transform that requires, in addition to the transform function itself, only sine, cosine, and exponential functions. The method is conceptually much like the method of Dubner and Abate, which approximates the inverse function by means of a Fourier cosine series. The method presented here, however, differs from theirs in two important respects. First of all, the Fourier series contains additional terms involving the sine function selected such that the error in the approximation is less than that of Dubner and Abate and such that the Fourier series approximates the inverse function on an interval of twice the length of the corresponding interval in Dubner and Abate's method. Second, there is incorporated into the method in this paper a transformation of the approximating series into one that converges very rapidly. In test problems using the method it has routinely been possible to evaluate inverse transforms with considerable accuracy over a wide range of values of the independent variable using a relatively few determinations of the Laplace transform itself.

The presence of perchlorate (ClO(4) (-)) in some U.S. drinking water supplies has raised concern about potential adverse thyroidal health effects, because ClO(4) (-) is known to competitively inhibit iodide uptake at the sodium iodide symporter (NIS). Humans are nutritionally and environmentally exposed to other competitive inhibitors of iodide uptake, including thiocyanate (SCN(-)) and nitrate (NO(3) (-)). The joint inhibiting effects of these three anions was studied by exposing Chinese hamster ovary cells stably expressing human NIS to varying concentrations of each anion separately, and in combination, and conducting measurements of (125)I(-) uptake. The entire data set was fit to a single Hill equation using maximum likelihood. The relative potency of ClO(4) (-) to inhibit (125)I(-) uptake at the NIS was found to be 15, 30 and 240 times that of SCN(-), I(-), and NO(3) (-) respectively on a molar concentration basis, with no evidence of synergism. These results are consistent with a common mode of action by these anions of simple competitive interaction, in which a concentration of any one of ClO(4) (-) SCN(-), and NO(3) (-), occurring either individually or as part of a mixture of the three anions, is indistinguishable from a concentration or dilution of either one of the remaining two ions in inhibiting iodine uptake at the NIS.

Quantitative estimates of the risk of lung cancer or mesothelioma in humans from asbestos exposure made by the U.S. Environmental Protection Agency (EPA) make use of estimates of potency factors based on phase-contrast microscopy (PCM) and obtained from cohorts exposed to asbestos in different occupational environments. These potency factors exhibit substantial variability. The most likely reasons for this variability appear to be differences among environments in fiber size and mineralogy not accounted for by PCM.In this article, the U.S. Environmental Protection Agency (EPA) models for asbestos-related lung cancer and mesothelioma are expanded to allow the potency of fibers to depend upon their mineralogical types and sizes. This is accomplished by positing exposure metrics composed of nonoverlapping fiber categories and assigning each category its own unique potency. These category-specific potencies are estimated in a meta-analysis that fits the expanded models to potencies for lung cancer (K L 's) or mesothelioma (K M 's) based on PCM that were calculated for multiple epidemiological studies in our previous paper (Berman and Crump, 2008). Epidemiological study-specific estimates of exposures to fibers in the different fiber size categories of an exposure metric are estimated using distributions for fiber size based on transmission electron microscopy (TEM) obtained from the literature and matched to the individual epidemiological studies. The fraction of total asbestos exposure in a given environment respectively represented by chrysotile and amphibole asbestos is also estimated from information in the literature for that environment. Adequate information was found to allow K L 's from 15 epidemiological studies and K M 's from 11 studies to be included in the meta-analysis.Since the range of exposure metrics that could be considered was severely restricted by limitations in the published TEM fiber size distributions, it was decided to focus attention on four exposure metrics distinguished by fiber width: "all widths," widths >0.2 µm, widths <0.4 µm, and widths <0.2 µm, each of which has historical relevance. Each such metric defined by width was composed of four categories of fibers: chrysotile or amphibole asbestos with lengths between 5 µm and 10 µm or longer than 10 µm. Using these metrics three parameters were estimated for lung cancer and, separately, for mesothelioma: K L A , the potency of longer (length >10 µm) amphibole fibers; rpc, the potency of pure chrysotile (uncontaminated by amphibole) relative to amphibole asbestos; and rps, the potency of shorter fibers (5 µm < length < 10 µm) relative to longer fibers.For mesothelioma, the hypothesis that chrysotile and amphibole asbestos are equally potent (rpc = 1) was strongly rejected by every metric and the hypothesis that (pure) chrysotile is nonpotent for mesothelioma was not rejected by any metric. Best estimates for the relative potency of chrysotile ranged from zero to about 1/200th that of amphibole asbestos (depending on metric). For lung can...

Copper (Cu) and its alloys are used extensively in domestic and industrial applications. Cu is also an essential element in mammalian nutrition. Since both copper deficiency and copper excess produce adverse health effects, the dose-response curve is U-shaped, although the precise form has not yet been well characterized. Many animal and human studies were conducted on copper to provide a rich database from which data suitable for modeling the dose-response relationship for copper may be extracted. Possible dose-response modeling strategies are considered in this review, including those based on the benchmark dose and categorical regression. The usefulness of biologically based dose-response modeling techniques in understanding copper toxicity was difficult to assess at this time since the mechanisms underlying copper-induced toxicity have yet to be fully elucidated. A dose-response modeling strategy for copper toxicity was proposed associated with both deficiency and excess. This modeling strategy was applied to multiple studies of copper-induced toxicity, standardized with respect to severity of adverse health outcomes and selected on the basis of criteria reflecting the quality and relevance of individual studies. The use of a comprehensive database on copper-induced toxicity is essential for dose-response modeling since there is insufficient information in any single study to adequately characterize copper dose-response relationships. The dose-response modeling strategy envisioned here is designed to determine whether the existing toxicity data for copper excess or deficiency may be effectively utilized in defining the limits of the homeostatic range in humans and other species. By considering alternative techniques for determining a point of departure and low-dose extrapolation (including categorical regression, the benchmark dose, and identification of observed no-effect levels) this strategy will identify which techniques are most suitable for this purpose. This analysis also serves to identify areas in which additional data are needed to better define the characteristics of dose-response relationships for copper-induced toxicity in relation to excess or deficiency.