W. Hans Carter

Detecting Interaction(s) and Assessing the Impact of Component Subsets in a Chemical Mixture Using Fixed-Ratio Mixture Ray Designs

An important environmental and regulatory issue is the protection of human health from potential adverse effects of cumulative exposure to multiple chemicals. Earlier literature suggested restricting inference to specific fixed-ratio rays of interest. Based on appropriate definitions of additivity, single chemical data are used to predict the relationship among the chemicals under the zero-interaction case. Parametric comparisons between the additivity model and the model fit along the fixed-ratio ray(s) are used to detect departure from additivity. Collection of data along reduced fixed-ratio rays, where subsets of chemicals of interest are removed from the mixture and the remaining compounds are at the same relative ratios as considered in the full ray, allow researchers to make inference about the effect of the removed chemicals. Methods for fitting simultaneous confidence bands about the difference between the best fitting model and the model predicted under additivity are developed to identify regions along the rays where significant interactions occur. This general approach is termed the “single chemicals required” (SCR) method of analysis. A second approach, termed “single chemicals not required” (SCNR) method of analysis, is based on underlying assumptions about the parameterization of the response surface. Under general assumptions, polynomial terms for models fit along fixed-ratio rays are associated with interaction terms. Consideration is given to the case where only data along the mixture rays are available. Tests of hypotheses, which consider interactions due to subsets of chemicals, are also developed.

Erratum: Detection of Departures from Additivity in Mixtures of Many Chemicals with a Threshold Model

We have recently discovered an error in the dosing values of one of the chemicals used in the mixture studied in Gennings, Schwartz, Carter, and Simmons (1997). This note provides corrected tables and figures for that manuscript. Table 1 provides a listing of dose levels (mMoles/kg), mean SDH responses, standard deviations of the responses, and sample sizes. The correction is that the dose levels of CDBM in this Table 1 are twice those reported in the original paper. With the corrected dose levels, the additivity model (Gennings et al. 1997, Equation (1.1)) is estimated to have a lower background parameter (3o), which also affects the slope parameters for the other three chemicals. The quasi-likelihood ratio test of no dose response for any of the chemicals was rejected (p < 0.001). The threshold parameter, 6, is not significantly different from zero (Table 2, p = 0.953). The point estimates and 95% confidence intervals for the threshold for each chemical are provided in Table 3. All four intervals include zero. The fitted curves from the additivity model for each single chemical are provided in a new Figure 3. Using the additivity model given in Equation (1.1) and parameter estimates in Table 2, the predicted SDH response under additivity at x = (0.208,0.084,0.568, 0.012) for (BDCM, CDBM, CHCl3, CHBr3) (i.e., the Krasner mixture) is y = 40.5 with a standard deviation of 2.46. The large sample 95% interval constructed under the assumption of additivity associated with x is [31.9, 51.9]. Here, the observed sample mean response, y = 43.9, is included in the prediction interval. Therefore, these data provide no evidence of departure from additivity at the combination point of interest.

Phase I Study of Bryostatin-1 and Fludarabine in Patients with Chronic Lymphocytic Leukemia and Indolent Non-Hodgkin's Lymphoma

Purpose: Preclinical studies suggested that bryostatin 1 might potentiate the therapeutic effects of fludarabine in the treatment of hematologic malignancies. We undertook a phase I study to identify appropriate schedules and doses of bryostatin 1 and fludarabine to be used in phase II studies. Experimental Design: Patients with chronic lymphocytic leukemia (CLL) or indolent lymphoma received fludarabine daily for 5 days and a single dose of bryostatin 1 via a 24-hour continuous infusion either before or after the fludarabine course. Doses were escalated in successive patients until recommended phase II doses for each sequence were identified on the basis of dose-limiting toxic events. Results: Bryostatin 1 can be administered safely and tolerably with full dose fludarabine (25 mg/m2/d × 5). The recommended bryostatin 1 phase II dose is 50 μg/m2 for both sequences, bryostatin 1 → fludarabine and fludarabine → bryostatin 1. The combination is active against both CLL and indolent lymphomas with responses seen in patients who had been previously treated with fludarabine. Correlative studies do not support the hypothesis that bryostatin 1 potentiates fludarabine activity through down-regulation of protein kinase C in target cells. Conclusions: Bryostatin 1 can be administered with full dose fludarabine, and the combination is moderately active in patients with persistent disease following prior treatment. In view of the activity of monoclonal antibodies such as the anti-CD20 monoclonal antibody rituximab in the treatment of CLL and indolent lymphomas, the concept of combining bryostatin 1 and fludarabine with rituximab warrants future consideration.

Utilizing Concentration-Response Data from Individual Components to Detect Statistically Significant Departures from Additivity in Chemical Mixtures

The classical approach for detecting interactions in a combination of drugs or chemicals is that of the isobologram, quantified and generalized by Berenbaum (1981, Advances il Cancer Research 35, 269-335). In this formulation it is assumed that contours of constant response of the dose-response surface are planar if the compounds do not interact. Building upon this approach, this paper develops methodology for detecting and characterizing departures from additivity. Reflecting the local rather than global nature of departure from additivity, this methodology only requires doseresponse data for the individual components and the specific combinations(s) of interest. This is in contrast to the larger experiments required to estimate the multidimensional dose-response surface for the combination. Procedures for incorporating data from multiple control groups are developed for a fixed-effects model, a random-effects model, and through use of a generalized estimating equations approach. An example is given that illustrates the application of these techniques to the analysis of a mixture of polycyclic aromatic hydrocarbons found in kerosene soot.

Relating isobolograms to response surfaces

The interaction index, isobologram and the appropriate contour of constant response of a dose response surface each offer essentially equivalent information regarding departures from additivity in a chemical combination. The benefit of relating the interaction index and isobolograms to a contour of a fitted dose response surface is that departures from additivity can be related to parameters of a statistical model. This permits an assessment of the statistical significance of the parameters and conclusions regarding the nature of departures from additivity which account for the variability inherent in the experiments used to generate the data. Relationships between statistical models and experimental designs can be exploited to yield economical designs for studying chemical combinations.

A statistical test for detecting and characterizing departures from additivity in drug/chemical combinations

In this paper, we propose a test of additivity in a combination of drugs/chemicals based on the interaction index. The test is developed for data resulting from an experiment involving single-agent exposure groups and s combination groups of interest. The testing procedure begins with an overall size a test of additivity with s degrees of freedom. When this test is rejected, single degree-of-freedom tests combined with post hoc corrections for multiple testing are developed to determine which dose combinations are associated with departure from additivity. The method is illustrated using data associated with a patent application for a two-drug combination.

The impact of exposure to a mixture of eighteen polyhalogenated aromatic hydrocarbons on thyroid function : Estimation of an interaction threshold

When an interaction has been detected among the chemicals in a mixture, it may be of interest to predict the interaction threshold. A method is presented for estimation of an interaction threshold along a mixture ray which allows differences in the shapes of the dose-response curves of the individual components (e.g., mixtures of full and partial agonists with differing response maxima). A point estimate and confidence interval for the interaction threshold may be estimated. The methods are illustrated with data from a study of a mixture of 18 polyhalogenated aromatic hydrocarbons (PHAHs) in rats exposed by oral gavage for four consecutive days. Serum total thyroxine (T4) was the response variable. Previous analysis of these data demonstrated a dose-dependent interaction among the 18 chemicals in the mixture, with additivity suggested in the lower portion of the dose-response curve and synergy (greater than additive response) in the higher portion of the dose-response curve. The present work builds on this analysis by construction of an interaction threshold model along the mixture ray. This interaction threshold model has two components: an implicit additivity region and an explicit region that describes the departure from additivity; the interaction threshold is the boundary between the two regions. Estimation of the interaction threshold within the observed experimental region suggested evidence of additivity in the low dose region. Total doses of the mixture that exceed the upper limit of the confidence interval on the interaction threshold were associated with a greater-than-additive interaction.

Evaluation of the Effect of Ammonia on Nicotine Pharmacokinetics Using Rapid Arterial Sampling

INTRODUCTION The nicotine bolus theory states that the dependence-producing potential of cigarettes relates to a rapid increase in nicotine at brain receptor sites. It has been suggested that ammonia, a compound typically found in tobacco products, further increases the amount of nicotine absorbed and its absorption rate. The aim of this study was to determine whether different ammonia yields in cigarettes affected the rate or amount of nicotine absorption from the lungs to arterial circulation. METHODS 34 adult smokers received 3 separate puffs from each of 2 test cigarettes with different ammonia yields (ammonia in smoke: 10.1 μg per cigarette vs. 18.9 μg per cigarette), followed by rapid radial arterial blood sampling (maximum one sample per second) with 30 min between puffs. Arterial blood samples were assayed for nicotine by liquid chromatography tandem mass spectrometry. Pharmacokinetic modeling was performed and the two test cigarettes were assessed for bioequivalence. RESULTS No significant differences were found in area under the curve, C(max), or T((max)) and the 2 test cigarettes were found to be bioequivalent based on 2 one-sided tests at a significance level of 5%. In addition, the zero-order rate constant (k(0)) obtained from the initial slope of the curves and the model-dependent first-order rate constant (k(a)) were not significantly different. CONCLUSIONS This study provides strong evidence that the different ammonia yields of the test cigarettes had no impact on nicotine pharmacokinetics; thus, the ammonia did not increase the rate or amount of nicotine absorption from a puff of cigarette smoke.

Sample Size and Power Determination for Detecting Interactions in Mixtures of Chemicals

In the analysis of mixtures of drugs/chemicals it is often of interest to test for the presence of interaction. If the hypothesis of no interaction (additivity) is not rejected, then the analyst may reasonably claim additivity if and only if the study is powered to a desired (e.g., biologically meaningful) level. The objective of this article is to address the sample size and power issues related to testing the hypothesis of additivity at specified mixture points. The study of disinfectant by-products (DBPs) found in drinking water, described in earlier literature, is used to illustrate the procedures for estimating power and sample sizes for detecting interactions at specified mixtures. The four trihalomethanes used in the study are bromodichloromethane (BDCM), chlorodibromomethane (CDBM), chloroform (CHCl3), and bromoform (CHBr3)

A Quasi-Likelihood Approach for Overdispersed Binomial Data When N is Unobserved

Several methods for the analysis of binomial data when the denominator, N, is unknown have been developed. Each of these methods requires that the mean of the distribution of N is known. In this article, we develop a quasi-likelihood technique that allows for the estimation of the means of the distributions needed to define the expected value and variance of the observed response and suggest a different form of the variance function. We illustrate the results of the proposed analysis and the results obtained when the mean of the distribution of N is assumed known through the analysis of a surviving jejunal crypt data set. Although the proposed method shows inflated standard errors of the parameter estimates in the cited example, the proposed method performs as well as a previously published method in all simulated conditions. Moreover, in cases where E(N) is misspecified, the proposed method outperforms the previously published method.