A dose-response model for teratological experiments involving quantal responses

This paper introduces a dose-response model for teratological quantal response data where the probability of response for an offspring from a female at a given dose varies with the litter size. The maximum likelihood estimators for the parameters of the model are given as the solution of a nonlinear iterative algorithm. Two methods of low-dose extrapolation are presented, one based on the litter size distribution and the other a conservative method. The resulting procedures are then applied to a teratological data set from the literature.     

Combining multiple comparisons and modeling techniques in dose-response studies

The analysis of data from dose-response studies has long been divided according to two major strategies: multiple comparison procedures and model-based approaches. Model-based approaches assume a functional relationship between the response and the dose, taken as a quantitative factor, according to a prespecified parametric model. The fitted model is then used to estimate an adequate dose to achieve a desired response but the validity of its conclusions will highly depend on the correct choice of the a priori unknown dose-response model. Multiple comparison procedures regard the dose as a qualitative factor and make very few, if any, assumptions about the underlying dose-response model. The primary goal is often to identify the minimum effective dose that is statistically significant and produces a relevant biological effect. One approach is to evaluate the significance of contrasts between different dose levels, while preserving the family-wise error rate. Such procedures are relatively robust but inference is confined to the selection of the target dose among the dose levels under investigation. We describe a unified strategy to the analysis of data from dose-response studies which combines multiple comparison and modeling techniques. We assume the existence of several candidate parametric models and use multiple comparison techniques to choose the one most likely to represent the true underlying dose-response curve, while preserving the family-wise error rate. The selected model is then used to provide inference on adequate doses.     

Regression models for overdispersed jejunal surviving crypts data

The dose-response model concerns to establish a relationship between a dose and the magnitude of the response produced by the dose. A common complication in the dose-response model for jejunal crypts cell surviving data is overdispersion, where the observed variation exceeds that predicted from the binomial distribution. In this study, two different methods for analyzing jejunal crypts cell survival after regimens of several fractions are contrasted and compared. One method is the logistic regression approach, where the numbers of surviving crypts are predicted by the logistic function of a single dose of radiation. The other one is the transform-both-sides approach, where the arcsine transformation family is applied based on the first-order variance-stabilizing transformation. This family includes the square root, arcsine, and hyperbolic arcsine transformations, which have been used for Poisson, binomial, and negative binomial count data, as special cases. These approaches are applied to a data set from radiobiology. Simulation study indicates that the arcsine transformation family is more efficient than the logistic regression when there exists moderate overdispersion.     

A solution to the problems of cytolysis assays with additional applications to other immunological and biochemical assays

After cellular immunoassays are compared with classical bioassays, conventional methods and consequent problems of data analysis for cytolysis assays are reviewed and a new solution is proposed. This solution incorporates new methods, called dose-response surface assays and analysis (DRSA), which estimate cytolytic activity coefficients on a surface in a three-dimensional space with two dose variables (killers and targets) and one response variable (counts). These new methods based on dose-response surfaces are demonstrated to be more informative and reliable than classical methods based on dose-response curves. In a test of the methods' robustness (sensitivity of parameter estimates to changes in the dose levels of the assay design), cytolytic activity coefficients estimated by DRSA varied by less than or equal to 30% over a reduction of three to four orders of magnitude in the dose levels. This remarkable robustness should be compared with the corresponding figures of as much as 500% over less than 1 order of magnitude for previously published results of coefficients estimated by conventional methods. DRSA is distinguished from replot-of-plots methods such as those used for enzyme inhibition assays in biochemistry, and is recommended as a more efficient method that should replace replot-of-plot methods now antiquated by the advent of microcomputers. DRSA can be applied to any experimental system that requires an activity coefficient to be estimated on a dose-response surface in a space of greater than or equal to 3 dimensions (greater than or equal to 2 dose variables and one response variable), regardless of the mathematical model and statistical estimators used to analyze the dose-response interaction. Finally, DRSA is compared with the methods known as response surface methodology (RSM), and is described as a new class of methods to be added to those that constitute RSM.     

A new model of thermal inactivation and its application to clonogenic survival data for WiDr human colonic adenocarcinoma cells

Based on the analysis of clonogenic survival data for human colonic adenocarcinoma cells (WiDr) after a single heating, a new model is proposed to describe cell survival after hyperthermia quantitatively. The effects of heat are explained as heat-induced cell damage assuming a first-order (single-hit) and a second-order (cumulative damage) process. Thus cell survival at a specified temperature can be described by the linear-quadratic (LQ) model. The proposed model is based on an alternative definition of the (single) thermal dose, given as the (normalized) product of heating time and a specified nonlinear function of the increase in temperature (relative to a threshold temperature) to be interpreted as the thermal dose rate. In further analogy to the modeling of the effects of low-dose-rate radiation, an inherent capacity of the cells to repair sublethal damage is assumed, and these effects are quantified by the usual g factor measuring incomplete repair effects. The model defines thermal dose-response and isoeffect dose relationships, enabling a direct (i. e. single-step) analysis of the available thermal response data. Additionally, the analysis of our data based on heating times in the range from 0 to 360 min and temperatures from 41 to 46 degrees C and covering a broad spectrum of different densities of cells seeded for colony formation did not yield any evidence of the existence of a breaking point usually derived from Arrhenius plots based on the single-hit, multitarget model and the Arrhenius equation. The model includes no specific assumptions describing the development of thermotolerance, which can be assumed to be negligible under our experimental conditions. The proposed thermal dose-response model correlates satisfactorily with the in vitro survival data for WiDr adenocarcinoma cells.     

Tumor induction in BALB/c female mice after fission neutron or gamma irradiation

This study was designed to examine the dose-response relationships for tumor induction after neutron irradiation in female BALB/c mice, with emphasis on the response in the dose range 0 to 50 rad. Tumors induced after radiation exposure included ovarian tumors, lung adenocarcinomas, and mammary adenocarcinomas. For comparison the dose responses for induction of these tumors after 137Cs gamma irradiation were also examined. As previously described for the female RFM mouse, the data for ovarian tumor induction after neutron and gamma irradiation were consistent with a threshold model. For lung and mammary tumors the dose-response curve after neutron irradiation appeared to "bend over" in the dose range 10 to 20 rad. The factors responsible for this bend-over and their relative contributions to the overall form of the dose-response relationship are not presently known. However, these data strongly indicate that extrapolation from data above 50 rad could result in a significant underestimate of risks. Further, it is clear that current models of neutron carcinogenesis are inadequate, since such a bend-over is not predicted at these low dose levels.     

Cell survival probability under ionizing radiation.

The survival probability of a living cell exposed to ionizing radiation in an experimental setup is derived. The survival of a cell depends on the severity of the radiation damage and efficiency of the cellular repair. The formula of the survival probability is expressed as a function of dose, nonlinear rate of lesion induction, nonlinear rate of cellular repair, and a key experimental parameter--the holding time. The result is an extension of the Markovian dose-response model developed by Yang and Swenberg.     

Mixed-effects nonlinear regression for unbalanced repeated measures.

Repeated measures data, such as clinical pharmacokinetic data, growth data, and dose-response data, are often inherently nonlinear with respect to a given response function and are frequently incomplete and/or unbalanced. Nonlinear random-effects models together with a variety of estimation procedures have been proposed for the analysis of such data. This paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed-effects nonlinear regression model. The asymptotic properties of the proposed estimators are given and large-sample tests of hypothesis provided. The results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure.     

A systems-based mathematical modelling framework for investigating the effect of drugs on solid tumours

Background and Purpose

Most information on the dose-response of radiation-induced cancer is derived from data on the A-bomb survivors. Since, for radiation protection purposes, the dose span of main interest is between zero and one Gy, the analysis of the A-bomb survivors is usually focused on this range. However, estimates of cancer risk for doses larger than one Gy are becoming more important for radiotherapy patients. Therefore in this work, emphasis is placed on doses relevant for radiotherapy with respect to radiation induced solid cancer.

Materials and methods

For various organs and tissues the analysis of cancer induction was extended by an attempted combination of the linear-no-threshold model from the A-bomb survivors in the low dose range and the cancer risk data of patients receiving radiotherapy for Hodgkin's disease in the high dose range. The data were fitted using organ equivalent dose (OED) calculated for a group of different dose-response models including a linear model, a model including fractionation, a bell-shaped model and a plateau-dose-response relationship.

Results

The quality of the applied fits shows that the linear model fits best colon, cervix and skin. All other organs are best fitted by the model including fractionation indicating that the repopulation/repair ability of tissue is neither 0 nor 100% but somewhere in between. Bone and soft tissue sarcoma were fitted well by all the models. In the low dose range beyond 1 Gy sarcoma risk is negligible. For increasing dose, sarcoma risk increases rapidly and reaches a plateau at around 30 Gy.

Conclusions

In this work OED for various organs was calculated for a linear, a bell-shaped, a plateau and a mixture between a bell-shaped and plateau dose-response relationship for typical treatment plans of Hodgkin's disease patients. The model parameters (α and R) were obtained by a fit of the dose-response relationships to these OED data and to the A-bomb survivors. For any three-dimensional inhomogenous dose distribution, cancer risk can be compared by computing OED using the coefficients obtained in this work.     

A note on the use of a stochastic mammillary compartmental model as an environmental safety model

Environmental safety testing typically requires procedures for extrapolating from the relatively high experimental to the very low use doses of potentially harmful substances. In the present paper, a stochastic mammillary compartmental model for environmental safety testing is proposed and extrapolation procedures based on its dose-response relationship are developed. The proposed model is a direct generalization of one of the basic safety models, the one-hit model, in that a harmful reaction is assumed to occur if at any time any of the peripheral compartments attains a specified threshold of particles. Consideration of a closed model yields an upper bound on the probability of attaining a certain threshold level, thus providing a conservative procedure for extrapolating to a low dose, while a lower bound obtained from a related open model provides a useful monitoring device as to the sharpness of the upper, bound. The extrapolation procedure is illustrated with simulated data and approximations for initial values are developed.     

Statistical Dose-Response Models with Hormetic Effects

Monotonically increasing or decreasing functions are often used to model the relationship between the response of an experimental unit and the dose of a given substance. Of late, there has been an increased interest in dose-response relationships that exhibit hormetic effects. These effects may be characterized by an increase in response at low doses instead of the expected decrease in response that is observed at higher doses. Herein, we study the statistical implications of hormesis in several ways. First, we present a broad class of parametric mathematical-statistical models, constructed from standard dose-response models, that allow the incorporation of hormetic effects in such a way that the presence of hormesis can be tested statistically. Second, we consider the impact of model misspecification on effective dose estimation, such as the ED50 and the limiting dose for stimulation, when the hormetic effect is present but ignored in the dose-response model by the researcher (model underspecification) and when an hormetic effect is not present but incorporated into the dose-response model (model overspecification). Our simulation study reveals that it is more damaging to the estimation of effective dose to ignore the hormetic effect through model underspecification than to include the hormetic effect in the model through model overspecification. Third, we develop a nonpara-metric regression technique useful as an exploratory procedure to indicate hormetic effects when present. Finally, both parametric and nonparametric methods are illustrated with an example.     

Dose-Response Envelope for Escherichia coli O157:H7

Escherichia coli O157:H7 is an emerging food and waterborne pathogen in the U.S. and internationally. The objective of this work was to develop a dose-response model for illness by this organism that bounds the uncertainty in the dose-response relationship. No human clinical trial data are available for E. coli O157:H7, but such data are available for two surrogate pathogens: enteropathogenic E. coli (EPEC) and Shigella dysenteriae. E. coli O157:H7 outbreak data provide an initial estimate of the most likely value of the dose-response relationship within the bounds of an envelope defined by beta-Poisson dose-response models fit to the EPEC and S. dysenteriae data. The most likely value of the median effective dose for E. coli O157:H7 is estimated to be approximately 190[emsp4 ]000 colony forming units (cfu). At a dose level of 100[emsp4 ]cfu, the median response predicted by the model is six percent.     

Development and Validation of Dose-Response Relationship for Listeria monocytogenes

Listeria monocytogenes is a foodborne pathogen internationally and in the U.S. The objective of this work was to develop and validate a dose-response model for infection by this organism. Only animal data was available in the literature. The beta-Poisson dose response model provided good fit to the data, and one of the two data sets was found to be concordant with attack rates noted in human outbreaks. There are differences, however, between the dose-response relationship and endemic illness rates computed from market basket surveys of the prevalence of L. monocytogenes. Further work to elucidate the bases for this difference is necessary.     

Effects of the Number of Genome Segments on Primary and Systemic Infections with a Multipartite Plant RNA Virus

Multipartite plant viruses were discovered because of discrepancies between the observed dose response and predictions of the independent-action hypothesis (IAH) model. Theory suggests that the number of genome segments predicts the shape of the dose-response curve, but a rigorous test of this hypothesis has not been reported. Here, Alfalfa mosaic virus (AMV), a tripartite Alfamovirus, and transgenic Nicotianatabacum plants expressing no (wild type), one (P2), or two (P12) viral genome segments were used to test whether the number of genome segments necessary for infection predicts the dose response. The dose-response curve of wild-type plants was steep and congruent with the predicted kinetics of a multipartite virus, confirming previous results. Moreover, for P12 plants, the data support the IAH model, showing that the expression of virus genome segments by the host plant can modulate the infection kinetics of a tripartite virus to those of a monopartite virus. However, the different types of virus particles occurred at different frequencies, with a ratio of 116:45:1 (RNA1 to RNA2 to RNA3), which will affect infection kinetics and required analysis with a more comprehensive infection model. This analysis showed that each type of virus particle has a different probability of invading the host plant, at both the primary- and systemic-infection levels. While the number of genome segments affects the dose response, taking into consideration differences in the infection kinetics of the three types of AMV particles results in a better understanding of the infection process.     

DNA double-strand breaks in human lymphocytes after single irradiation by low doses of pulsed X-rays: non-linear dose-response relationship

Effects of ionizing radiation registered in cells after low dose irradiation are still poorly understood. A pulsed mode of irradiation is even more problematic in terms of predicting the radiation-induced response in cells. Thus, the aim of this paper was to study and analyze the effects of dose and frequency of pulsed X-rays on the frequency of radiation-induced DNA double-strand breaks and their repair kinetics in human peripheral blood lymphocytes in vitro. Analysis of radiation-induced gammaH2AX and 53BP1 repair foci was used to assess the DNA damage in these cells. The dose-response curve of radiation-induced foci of both proteins has shown deviations from linearity to a higher effect in the 12-32 mGy dose range and a lower effect at 72 mGy. The dose-response curve was linear at doses higher than 100 mGy. The number of radiation-induced gammaH2AX and 53BP1 foci depended on the frequency of X-ray pulses: the highest effect was registered at 13 pulses per second. Moreover, slower repair kinetics was observed for those foci induced by very low doses with a nonlinear dose-response relationship.     

More Accurate Dose-Response Estimation Using Monte-Carlo Uncertainty Analysis: The Data Cube Approach

The traditional q₁ ^* methodology for constructing upper confidence limits (UCLs) for the low-dose slopes of quantal dose-response functions has two limitations: (i) it is based on an asymptotic statistical result that has been shown via Monte Carlo simulation not to hold in practice for small, real bioassay experiments (Portier and Hoel, 1983); and (ii) it assumes that the multistage model (which represents cumulative hazard as a polynomial function of dose) is correct. This paper presents an uncertainty analysis approach for fitting dose-response functions to data that does not require specific parametric assumptions or depend on asymptotic results. It has the advantage that the resulting estimates of the dose-response function (and uncertainties about it) no longer depend on the validity of an assumed parametric family nor on the accuracy of the asymptotic approximation. The method derives posterior densities for the true response rates in the dose groups, rather than deriving posterior densities for model parameters, as in other Bayesian approaches (Sielken, 1991), or resampling the observed data points, as in the bootstrap and other resampling methods. It does so by conditioning constrained maximum-entropy priors on the observed data. Monte Carlo sampling of the posterior (constrained, conditioned) probability distributions generate values of response probabilities that might be observed if the experiment were repeated with very large sample sizes. A dose-response curve is fit to each such simulated dataset. If no parametric model has been specified, then a generalized representation (e.g., a power-series or orthonormal polynomial expansion) of the unknown dose-response function is fit to each simulated dataset using “model-free” methods. The simulation-based frequency distribution of all the dose-response curves fit to the simulated datasets yields a posterior distribution function for the low-dose slope of the dose-response curve. An upper confidence limit on the low-dose slope is obtained directly from this posterior distribution. This “Data Cube” procedure is illustrated with a real dataset for benzene, and is seen to produce more policy-relevant insights than does the traditional q₁ ^* methodology. For example, it shows how far apart are the 90%, 95%, and 99% limits and reveals how uncertainty about total and incremental risk vary with dose level (typically being dominated at low doses by uncertainty about the response of the control group, and being dominated at high doses by sampling variability). Strengths and limitations of the Data Cube approach are summarized, and potential decision-analytic applications to making better informed risk management decisions are briefly discussed.     

Uncertainties in the risk assessment of three mycotoxins: aflatoxin, ochratoxin, and zearalenone

The risk assessment of mycotoxins is made up of two major components: an exposure assessment and a hazard assessment. There are many uncertainties in both of these components. This paper will briefly discuss the various aspects of the risk assessment process as it applies to mycotoxins and will then focus mainly on some of the uncertainties in the hazard assessment component of several carcinogenic mycotoxins. To arrive at an estimated "safe dose" (end point of the hazard assessment), we have previously used two major approaches: the no observed effect level (NOEL) divided by a safety factor approach and a mathematical (robust linear) extrapolation to a "virtual safe dose." Both of these approaches use only points from the no observed effect region of the dose-response curve and ignore valuable data from the response region. It is proposed to use the dose at which 50% of the animals would have developed tumors (the TD50) divided by a large safety factor of 50,000 as an additional estimate of "safe dose". For many studies, the TD50 lies within the observed response region of the dose-response curve and may have more validity. It is also suggested in certain cases that some of the uncertainties regarding the NOEL can be reduced if one uses a statistically derived no effect level (NEL).     

Fitting nonlinear and constrained generalized estimating equations with optimization software

In this article, we present an estimation approach for solving nonlinear constrained generalized estimating equations that can be implemented using object-oriented software for nonlinear programming, such as nlminb in Splus or fmincon and lsqnonlin in Matlab. We show how standard estimating equation theory includes this method as a special case so that our estimates, when unconstrained, will remain consistent and asymptotically normal. To illustrate this method, we fit a nonlinear dose-response model with nonnegative mixed bound constraints to clustered binary data from a developmental toxicity study. Satisfactory confidence intervals are found using a nonparametric bootstrap method when a common correlation coefficient is assumed for all the dose groups and for some of the dose-specific groups.     

Maximum likelihood estimation for cytogenetic dose-response curves

In vitro dose-response curves are used to describe the relation between chromosome aberrations and radiation dose for human lymphocytes. The lymphocytes are exposed to low-LET radiation, and the resulting dicentric chromosome aberrations follow the Poisson distribution. The expected yield depends on both the magnitude and the temporal distribution of the dose. A general dose-response model that describes this relation has been presented by Kellerer and Rossi (1972, Current Topics on Radiation Research Quarterly 8, 85-158; 1978, Radiation Research 75, 471-488) using the theory of dual radiation action. Two special cases of practical interest are split-dose and continuous exposure experiments, and the resulting dose-time-response models are intrinsically nonlinear in the parameters. A general-purpose maximum likelihood estimation procedure is described, and estimation for the nonlinear models is illustrated with numerical examples from both experimental designs. Poisson regression analysis is used for estimation, hypothesis testing, and regression diagnostics. Results are discussed in the context of exposure assessment procedures for both acute and chronic human radiation exposure.