Kenny S. Crump

Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation

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 Abates 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.

A Meta-Analysis of Asbestos-Related Cancer Risk That Addresses Fiber Size and Mineral Type

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 (KLs) or mesothelioma (KMs) based on PCM that were calculated for multiple epidemiological studies in our previous paper (). 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 KLs from 15 epidemiological studies and KMs 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 μ m, widths < 0.4 μ m, and widths < 0.2 μ m, 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 μ m and 10 μ m or longer than 10 μ m. Using these metrics three parameters were estimated for lung cancer and, separately, for mesothelioma: KLA, the potency of longer (length > 10 μ m) amphibole fibers; rpc, the potency of pure chrysotile (uncontaminated by amphibole) relative to amphibole asbestos; and rps, the potency of shorter fibers (5 μ m < length < 10 μ m) 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 cancer, the hypothesis that chrysotile and amphibole asbestos are equally potent (rpc = 1) was rejected (p ≤ .05) by the two metrics based on thin fibers (length < 0.4 μ m and < 0.2 μ m) but not by the metrics based on thicker fibers. The “all widths” and widths < 0.4 μ m metrics provide the best fits to both the lung cancer and mesothelioma data over the other metrics evaluated, although the improvements are only marginal for lung cancer. That these two metrics provide equivalent (for mesothelioma) and nearly equivalent (for lung cancer) fits to the data suggests that the available data sets may not be sufficiently rich (in variation of exposure characteristics) to fully evaluate the effects of fiber width on potency. Compared to the metric with widths > 0.2 μ m with both rps and rpc fixed at 1 (which is nominally equivalent to the traditional PCM metric), the “all widths” and widths < 0.4 μ m metrics provide substantially better fits for both lung cancer and, especially, mesothelioma. Although the best estimates of the potency of shorter fibers (5 < length < 10 μ m) is zero for the “all widths” and widths < 0.4 μ m metrics (or a small fraction of that of longer fibers for the widths > 0.2 μ m metric for mesothelioma), the hypothesis that these shorter fibers were nonpotent could not be rejected for any of these metrics. Expansion of these metrics to include a category for fibers with lengths < 5 μ m did not find any consistent evidence for any potency of these shortest fibers for either lung cancer or mesothelioma. Despite the substantial improvements in fit over that provided by the traditional use of PCM, neither the “all widths” nor the widths < 0.4 μ m metrics (or any of the other metrics evaluated) completely resolve the differences in potency factors estimated in different occupational studies. Unresolved in particular is the discrepancy in potency factors for lung cancer from Quebec chrysotile miners and workers at the Charleston, SC, textile mill, which mainly processed chrysotile from Quebec. A leading hypothesis for this discrepancy is limitations in the fiber size distributions available for this analysis. recently analyzed by TEM archived air samples from the South Carolina plant to determine a detailed distribution of fiber lengths up to lengths of 40 μ m and greater. If similar data become available for Quebec, perhaps these two size distributions can be used to eliminate the discrepancy between these two studies.

Update of potency factors for asbestos-related lung cancer and mesothelioma.

The most recent update of the U.S. Environmental Protection Agency (EPA) health assessment document for asbestos (, referred to as “the EPA 1986 update”) is now 20 years old. That document contains estimates of “potency factors” for asbestos in causing lung cancer (KLs) and mesothelioma (KMs) derived by fitting mathematical models to data from studies of occupational cohorts. The present paper provides a parallel analysis that incorporates data from studies published since the EPA 1986 update. The EPA lung cancer model assumes that the relative risk varies linearly with cumulative exposure lagged 10 years. This implies that the relative risk remains constant after 10 years from last exposure. The EPA mesothelioma model predicts that the mortality rate from mesothelioma increases linearly with the intensity of exposure and, for a given intensity, increases indefinitely after exposure ceases, approximately as the square of time since first exposure lagged 10 years. These assumptions were evaluated using raw data from cohorts where exposures were principally to chrysotile (South Carolina textile workers, ; mesothelioma only data from Quebec miners and millers, ) and crocidolite (Wittenoom Gorge, Australia miners and millers, ) and using published data from a cohort exposed to amosite (Paterson, NJ, insulation manufacturers, ). Although the linear EPA model generally provided a good description of exposure response for lung cancer, in some cases it did so only by estimating a large background risk relative to the comparison population. Some of these relative risks seem too large to be due to differences in smoking rates and are probably due at least in part to errors in exposure estimates. There was some equivocal evidence that the relative risk decreased with increasing time since last exposure in the Wittenoom cohort, but none either in the South Carolina cohort up to 50 years from last exposure or in the New Jersey cohort up to 35 years from last exposure. The mesothelioma model provided good descriptions of the observed patterns of mortality after exposure ends, with no evidence that risk increases with long times since last exposure at rates that vary from that predicted by the model (i.e., with the square of time). In particular, the model adequately described the mortality rate in Quebec chrysotile miners and millers up through >50 years from last exposure. There was statistically significant evidence in both the Wittenoom and Quebec cohorts that the exposure intensity-response is supralinear1 rather than linear. The best-fitting models predicted that the mortality rate varies as [intensity]0.47 for Wittenoom and as [intensity]0.19 for Quebec and, in both cases, the exponent was significantly less than 1 (p< .0001). Using the EPA models, KLs and KMs were estimated from the three sets of raw data and also from published data covering a broader range of environments than those originally addressed in the EPA 1986 update. Uncertainty in these estimates was quantified using “uncertainty bounds” that reflect both statistical and nonstatistical uncertainties. Lung cancer potency factors (KLs) were developed from 20 studies from 18 locations, compared to 13 locations covered in the EPA 1986 update. Mesothelioma potency factors (KMs) were developed for 12 locations compared to four locations in the EPA 1986 update. Although the 4 locations used to calculate KM in the EPA 1986 update include one location with exposures to amosite and three with exposures to mixed fiber types, the 14 KMs derived in the present analysis also include 6 locations in which exposures were predominantly to chrysotile and 1 where exposures were only to crocidolite. The KMs showed evidence of a trend, with lowest KMs obtained from cohorts exposed predominantly to chrysotile and highest KMs from cohorts exposed only to amphibole asbestos, with KMs from cohorts exposed to mixed fiber types being intermediate between the KMs obtained from chrysotile and amphibole environments. Despite the considerable uncertainty in the KM estimates, the KM from the Quebec mines and mills was clearly smaller than those from several cohorts exposed to amphibole asbestos or a mixture of amphibole asbestos and chrysotile. For lung cancer, although there is some evidence of larger KLs from amphibole asbestos exposure, there is a good deal of dispersion in the data, and one of the largest KLs is from the South Carolina textile mill where exposures were almost exclusively to chrysotile. This KL is clearly inconsistent with the KL obtained from the cohort of Quebec chrysotile miners and millers. The KLs and KMs derived herein are defined in terms of concentrations of airborne fibers measured by phase-contrast microscopy (PCM), which only counts all structures longer than 5 μm, thicker than about 0.25 μ m, and with an aspect ratio ≥3:1. Moreover, PCM does not distinguish between asbestos and nonasbestos particles. One possible reason for the discrepancies between the KLs and KMs from different studies is that the category of structures included in PCM counts does not correspond closely to biological activity. In the accompanying article () the KLs and KMs and related uncertainty bounds obtained in this article are paired with fiber size distributions from the literature obtained using transmission electron microscopy (TEM). The resulting database is used to define KLs and KMs that depend on both the size (e.g., length and width) and mineralogical type (e.g., chrysotile or crocidolite) of an asbestos structure. An analysis is conducted to determine how well different KL and KM definitions are able to reconcile the discrepancies observed herein among values obtained from different environments.

Dose response problems in carcinogenesis.

The estimation of risks from exposure to carcinogens is an important problem from the viewpoint of protection of human health. It also poses some very difficult dose-response problems. Two dose-response models may fit experimental data about equally well and yet predict responses that differ by many orders of magnitude at low doses. Mechanisms of carcinogenesis are not sufficiently understood so that the shape of the dose-response curve at low doses can be satisfactorily predicted. Mathematical theories of carcinogenesis and statistical procedures can be of use with dose-reponse problems such as this and, in addition, can lead to a better understanding of the mechanisms of carcinogenesis. In this paper, mathematical dose-response models of carcinogenesis are considered as well as various proposed dose-response procedures for estimating carcinogenic risks at low doses. Areas are suggested in which further work may be useful. These areas include experimental design problems, statistical procedures for use with time-to-occurrence data, and mathematical models that incorporate such biological features as pharmacokinetics of carcinogens, synergistic effects, DNA repair, susceptible subpopulations, and immune reactions.

What Role for Biologically Based Dose–Response Models in Estimating Low-Dose Risk?

Background Biologically based dose–response (BBDR) models can incorporate data on biological processes at the cellular and molecular level to link external exposure to an adverse effect. Objectives Our goal was to examine the utility of BBDR models in estimating low-dose risk. Methods We reviewed the utility of BBDR models in risk assessment. Results BBDR models have been used profitably to evaluate proposed mechanisms of toxicity and identify data gaps. However, these models have not improved the reliability of quantitative predictions of low-dose human risk. In this commentary we identify serious impediments to developing BBDR models for this purpose. BBDR models do not eliminate the need for empirical modeling of the relationship between dose and effect, but only move it from the whole organism to a lower level of biological organization. However, in doing this, BBDR models introduce significant new sources of uncertainty. Quantitative inferences are limited by inter- and intraindividual heterogeneity that cannot be eliminated with available or reasonably anticipated experimental techniques. BBDR modeling does not avoid uncertainties in the mechanisms of toxicity relevant to low-level human exposures. Although implementation of BBDR models for low-dose risk estimation have thus far been limited mainly to cancer modeled using a two-stage clonal expansion framework, these problems are expected to be present in all attempts at BBDR modeling. Conclusions The problems discussed here appear so intractable that we conclude that BBDR models are unlikely to be fruitful in reducing uncertainty in quantitative estimates of human risk from low-level exposures in the foreseeable future. Use of in vitro data from recent advances in molecular toxicology in BBDR models is not likely to remove these problems and will introduce new issues regarding extrapolation of data from in vitro systems.

The Future Use of in Vitro Data in Risk Assessment to Set Human Exposure Standards: Challenging Problems and Familiar Solutions

Background The vision of a National Research Council (NRC) committee (the Committee on Toxicity Testing and Assessment of Environmental Agents) for future toxicity testing involves the testing of human cells in in vitro assays for “toxicity pathways”—normal signaling pathways that when perturbed can lead to adverse effects. Risk assessments would eventually be conducted using mathematical models of toxicity pathways (TP models) to estimate exposures that will not cause biologically significant perturbations in these pathways. Objectives In this commentary we present our vision of how risk assessment to support exposure standards will be developed once a suitable suite of in vitro assays becomes available. Discussion Issues to be faced basing risk assessments on in vitro data are more complex than, but conceptually similar to, those faced currently when applying in vivo data. Absent some unforeseen technical breakthrough, in vitro data will be used in ways similar to current practices that involve applying uncertainty or safety factors to no observed adverse effect levels or benchmark doses. TP models are unlikely to contribute quantitatively to risk assessments for several reasons, including that the statistical variability inherent in such complex models severely limits their usefulness in estimating small changes in response, and that such models will likely continue to involve empirical modeling of dose responses. Conclusion The vision of the committee predicts that chemicals will be tested more quickly and cheaply and that animal testing will be reduced or eliminated. Progress toward achieving these goals will be expedited if the issues raised herein are given careful consideration.

Evaluation of an exposure assessment used in epidemiological studies of diesel exhaust and lung cancer in underground mines

NIOSH/NCI (National Institute of Occupational Safety and Health and National Cancer Institute) developed exposure estimates for respirable elemental carbon (REC) as a surrogate for exposure to diesel exhaust (DE) for different jobs in eight underground mines by year beginning in the 1940s—1960s when diesel equipment was first introduced into these mines. These estimates played a key role in subsequent epidemiological analyses of the potential relationship between exposure to DE and lung cancer conducted in these mines. We report here on a reanalysis of some of the data from this exposure assessment. Because samples of REC were limited primarily to 1998–2001, NIOSH/NCI used carbon monoxide (CO) as a surrogate for REC. In addition, because CO samples were limited, particularly in the earlier years, they used the ratio of diesel horsepower (HP) to the mine air exhaust rate as a surrogate for CO. There are considerable uncertainties connected with each of these surrogate-based steps. The estimates of HP appear to involve considerable uncertainty, although we had no data upon which to evaluate the magnitude of this uncertainty. A sizable percentage (45%) of the CO samples used in the HP to CO model was below the detection limit which required NIOSH/NCI to assign CO values to these samples. In their preferred REC estimates, NIOSH/NCI assumed a linear relation between C0 and REC, although they provided no credible support for that assumption. Their assumption of a stable relationship between HP and CO also is questionable, and our reanalysis found a statistically significant relationship in only one-half of the mines. We re-estimated yearly REC exposures mainly using NIOSH/NCI methods but with some important differences: (i) rather than simply assuming a linear relationship, we used data from the mines to estimate the CO—REC relationship; (ii) we used a different method for assigning values to nondetect CO measurements; and (iii) we took account of statistical uncertainty to estimate bounds for REC exposures. This exercise yielded significantly different exposure estimates than estimated by NIOSH/NCI. However, this analysis did not incorporate the full range of uncertainty in REC exposures because of additional uncertainties in the assumptions underlying the modeling and in the underlying data (e.g. HP and mine exhaust rates). Estimating historical exposures in a cohort is generally a very difficult undertaking. However, this should not prevent one from recognizing the uncertainty in the resulting estimates in any use made of them.

Use of threshold and mode of action in risk assessment

Under current guidelines, exposure guidelines for toxicants are determined by following one of two different tracks depending on whether the toxicant’s mode of action (MOA) is believed to involve an exposure threshold. Although not denying the existence of thresholds, this paper points out problems with how the threshold concept and MOA is used in risk assessment. Thresholds are frequently described using imprecise terms that imply some unspecified increase in risk, which robs them of any meaning (any reasonable dose response will satisfy such a definition) and tacitly implies a value judgment about how large a risk is acceptable. MOA is generally used only to inform a threshold’s existence and not its value. Often MOA is used only to conclude that the adverse effect requires an upstream cellular or biochemical response for which a threshold is simply assumed. Data to inform MOA often come from animals, which complicates evaluation of the role of human variation in genetic and environmental conditions, and the possible interaction of the toxicant with processes already producing background toxicity in humans. In response to these and other problems with the current two-track approach, this paper proposes a modified point of departure/safety factor approach to setting exposure guidelines for all toxicants. MOA and the severity of the toxic effect would be addressed using safety factors calculated from guidelines established by consensus and based on scientific judgment. The method normally would not involve quantifying low-dose risk, and would not require a threshold determination, although MOA information regarding the likelihood of a threshold could be used in setting safety factors.

Asymptotic theory for analyzing dose-response survival data with application to the low-dose extrapolation problem

Abstract A model of the form P(t, d)=1−;exp{-ʌ(t)σ K i =0q i d i }, q i ⩾ 0 , is proposed for analyzing dose-response survival data with right censoring. The qis in the dose polynomial are estimated by maximizing the Cox partial likelihood, and given these estimates. Λ(t) is estimated nonparametrically by an estimator proposed by Breslow. Large-sample properties of these estimators are established. Estimates and related large-sample properties are provided for the “virtually safe dose” and other parameters for assessing low-dose carcinogenic risk as a function of age, using data from animal carcinogenesis experiments.

The dispersion of a neutral allele considered as a branching process

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data from Drosophila populations are used to show the improbability of the same neutral allele occurring at widely separated localities. NEUTRAL ALLELE; BRANCHING PROCESS; MIGRATION