* Department of Biostatistics, Virginia Commonwealth University, Richmond, Virginia; Solveritas, LLC, Richmond, Virginia; and
The Dow Chemical Company, Midland, Michigan
Received January 6, 2004; accepted March 29, 2004
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ABSTRACT |
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Key Words: ray design; synergy; antagonism; interaction index.
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INTRODUCTION |
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Because of the unlimited number of potential mixture combinations, testing mixtures separately is only practical for a limited number of high priority mixtures. An alternative approach is to focus on representative compounds that are known to exert toxicity via a common mechanism of action (e.g., a common receptor) or a common target organ, and conduct studies to better understand the nature of chemical interactions and their underlying mechanisms. These studies would then form the basis for making inferences about related mixtures. The most common regulatory approach used to date for compounds thought to act through a common mechanism are based on the dose-addition approach as the default assumption (EPA, 2001; Wilkinson et al., 2000
). Dioxins and furans exemplify one group of compounds regulated under the principle of dose-addition, which is manifest as the Toxic Equivalency Quotient (TEQ) approach. Each of the many dioxin and furan congeners is assigned a "Toxic Equivalency Factor" (TEF) based on its potency relative to the prototype compound, 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD). The TEFs and dose of each congener are then multiplied, and the resulting values for each congener summed to derive the total mixture TEQ (Safe, 1998
).
While dose-additivity approaches such as TEQs have practical advantages, they also have a number of fundamental limitations, particularly with their ability to predict mixture toxicity at low doses. First, individual TEF values are typically based on responses at relatively high doses (e.g., EC50 values), with the assumption that relative potency and mechanism of action are uniform across the dose-response range. For mixtures of chemicals in the environment, exposures typically are at relatively low levels, usually below individual chemical thresholds. This situation is distinctly different from drug interactions, which deal with pharmacologically active doses. Mechanisms of individual chemical action, as well as chemical interaction, are rarely uniform across the entire dose-response range and instead tend to be highly dose-dependent (Gennings et al., 2002). At very low exposures, dose-addition assumptions ignore the effectiveness of detoxification systems which often are overwhelmed at high doses, leading to overestimation of toxicity at low doses (Feron et al., 1995
; Groten, 2000
). Another problem with simple dose-addition approaches is that many receptor-based systems exhibit a high degree of tissue/organ specificity. For example, tamoxifen is an estrogen receptor (ER) agonist in the uterus, but an ER antagonist in the breast. For this and other reasons, Safe (1998)
has cautioned against the use of TEQ approaches for hormonally active chemicals. Finally, even chemicals acting through a common receptor exhibit different thresholds, slopes, and maximal effect levels, such that simplistic predictions of combined effects based on individual TEFs can be grossly in error (Scaramellini et al., 1997
). Using estrogenic chemicals as an example, many of the chemicals of concern are several orders of magnitude less potent than "natural estrogen" (i.e., estradiol 17ß or E2) and have a much lower maximal effect level as compared to E2. Chemicals exhibiting the latter property are referred to as partial agonists.
Recently an expert working group was convened by the Society of Toxicology to suggest future directions in chemical mixture research. The group highlighted the need for a specific focus on the key attributes of "real world" exposures, which include the examination of low dose exposures and a better understanding and incorporation of dose-response into mixture toxicity predictions (Teuschler et al., 2002). With this aim in mind, this article describes the development of a flexible statistical approach that seeks to address mixtures of compounds with differing response thresholds, slopes and maximal effect levels (i.e., full and partial agonists). Recently, statistical methods have been developed that permit the analysis of mixtures of many chemicals while maintaining feasible experimental designs (e.g., Casey et al., in press
; Gennings et al., 2002
; Meadows et al., 2002
) without the required constraint of parallel dose-response curves and while adequately accounting for biological variability in testing hypotheses of additivity. In short, single chemical concentration-effect data and mixture data at fixed-ratios are experimentally generated. An additivity model is estimated from the single chemical data, which is used for comparison to a model fit to the mixture data in terms of total concentration. The statistical comparison of the two models based on prediction along the specified fixed-ratio ray(s) is a test of additivity for the fixed ratio of the mixture of chemicals. This general strategy has been termed the "single chemicals required" (SCR) approach (Casey et al., in press
).
The definition of additivity (i.e., zero interaction) we use is given by Berenbaum (1985) and is based on the classical isobologram for the combination of two chemicals (e.g., Loewe, 1953
; Loewe and Muischnek, 1926). That is, in a combination of c chemicals, let Ei represent the concentration/dose of the ith component alone that yields a fixed response, y0, and let xi represent the concentration/dose of the ith component in combination with the c agents that yields the same response. According to this definition of additivity if the substances combine with zero interaction, then
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An important feature of the SCR approach is that the additivity model estimated from single chemical data can be used to predict the "zero interaction" or additivity response for any specified mixture. It is necessary for the single chemical data to be adequately represented by the additivity model. Such a model often makes assumptions about the overall shape and characteristics of the single chemical concentration effectiveness. For example, a common maximum effect level is often assumed for all of the chemicals in the mixture. As mentioned earlier, this is not a valid assumption for mixtures containing partial agonists. When fitting threshold additivity models, the common parameter that defines each chemical's threshold concentration may not be sufficiently flexible. Extending the logic of Dawson et al. (2000), who used Equation 1 with estimated single chemical dose-response curves to determine zero interaction, we have developed a new method of analysis useful for ray designs that permits more flexibility in modeling each single chemical concentration-effect curve. Prediction of the response along the fixed-ratio rays under additivity is determined indirectly using the interaction index with only single chemical parameters. A likelihood-ratio test of the hypothesis of additivity is then conducted by comparing a fully parameterized model of all of the data to the reduced model using only single chemical parameters.
Here we describe the general SCR approach using a sigmoid-shaped nonlinear threshold model, followed by a description of the more flexible approach, based on the interaction index for defining additivity. Both methods are demonstrated in an analysis of a mixture of six chemicals, which exhibit in vitro estrogenic activity, while possessing differing response thresholds and maximal effect levels.
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MATERIALS AND METHODS |
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Cell culture. For use in the ER- gene transcription assays, MCF-7 human breast cancer cells (obtained from Dr. Leigh Murphy, University of Manitoba, Canada) were maintained in phenol red-free DMEM supplemented with 10% FBS, 2 mM L-glutamine, 15 mM HEPES, penicillin/streptomycin (100 IU/ml/ 100 µg/ml), and amphotericin B (2.5 µg/ml). Cells were maintained at 37°C and 34% CO2.
Transfection/treatment. Cells were plated in triplicate in 96-well plates at a density of 68 x 103 cells/well in 5% FBS-DCC (Hyclone, Logan, UT). After attachment and growth for 6 h, the cells were transfected using LipofectinTM (Life Technologies). To each well was added 50 ng of the ß-galactosidase (ß-gal) expression vector pCH110 (Pharmacia, Piscataway, NJ), 150 ng of 17m5-G-Luc, the Gal4-regulated luciferase reporter vector, and 5 ng of Gal4-HEG0, an ER expression vector (both provided by P. Chambon, INSERM, France). The plasmids were transfected in serum-free, antibiotic-free DMEM supplemented with 2 mM L-glutamine. Cells were allowed to incubate overnight at 37°C in a humidified atmosphere of 3% CO2/air. Sixteen to eighteen h following transfection, cells were treated in triplicate with individual test compounds, mixtures, or the solvent (0.2% dimethylsulfoxide, DMSO, Sigma) in 5% FBS-DCC. The ER agonist, E2, was used at 108 M as a positive control. Twenty-four h following treatment, cells were lysed and aliquots from each well divided into two 96-well plates for luciferase and ß-gal activity determination using a Packard Topcount NXTTM luminescence counter (Packard Instrument Company, Meriden, CT). The reference plasmid pCH110 was cotransfected as an internal control in order to correct for variations in transfection efficiency. The data was assessed as units of luciferase activity normalized to the ß-gal activity from individual wells. Experiments were evaluated with fold induction being the endpoint of interest. Treatment regimens which resulted in a reduced ß-gal activity relative to that of transfected cultures exposed to E2 under unaltered media conditions were considered cytotoxic and were not used for further analysis.
Data Analysis
Two differing statistical methods for detecting and characterizing interaction in a mixture of six chemicals are illustrated in the Example/Results section.
SCR approach. This method has been described previously by Gennings et al. (2002) and Casey et al. (in press)
. An important feature of the SCR method is that single chemical dose-response data are used to fit to an additivity model which is algebraically equivalent to the definition of additivity as given in Equation 1. This model is used to predict under additivity along one or more fixed-ratio mixing rays of the chemicals in the mixture in terms of total dose or concentration. A mixture model is fit to observed mixture response data along the fixed-ratio mixture ray(s) and compared to that predicted by the additivity model. Lack of coincidence of the results of both the mixture and additivity models using an appropriate statistical test is considered to be a demonstration of departure from additivity. Although the additivity and mixture models have parameters incorporated to allow for a flexible model fit, these models are constrained to have, for example, a common maximum response. In some cases, this assumption (common maximal response) and the subsequent resulting models do not adequately represent the observed data.
Flexible alternative approach. In contrast to SCR, a new and more flexible approach is developed and illustrated here. In general, single chemical dose-response data are used to estimate dose-response curves for each single chemical. The increased flexibility is due to the inclusion of additional parameters to describe the dose-response relationship for the individual chemicals. These curves are assumed to have a common background parameter and are then allowed sufficient flexibility (i.e., enough parameters) to adequately fit the single chemical data. This combination of single chemical dose response curves is made into an additivity model by imposing the definition of additivity in Equation 1, thus creating an indirect additivity model. A test of additivity for a fixed-ratio mixture in terms of total dose/concentration is based on statistical comparisons of a "full" and "reduced" model. The full model allows a simultaneous fit of the single chemical and mixture data using sufficiently flexible model assumptions. These can allow for different maximum effect plateaus, flexibility in estimating dose or concentration thresholds, and different slope parameters. The reduced model uses only the models from the single chemical data and the constraint in Equation 1 to predict responses along the fixed-ratio mixture ray(s). Details of the statistical methods are provided in the Appendix. (The equation numbers referenced in the following sections refer to equations in the Appendix.) Direct inquiries regarding available software to the first author.
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RESULTS |
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Flexible Alternative Analysis
For comparison, the second and more flexible approach was used to analyze the same data. The second method allows for more flexibility in fitting individual chemical concentration-effect curves. For example, the additivity model in Equation 2b assumes a common maximum effect parameter () across the six chemicals. The response data for DDT (Fig. 2C) and OCT (Figure 2E) suggest plateau (i.e., partial agonist) effects that are below that observed as defined by the response of the positive control E2 alone (14-fold induction). Thus, the model in Equation 5 was fit to each of the single chemical concentration-effect data. This model allows for different maximum-effect parameters (
i), slope parameters (ßi), and threshold parameters (
i) for each chemical (i = 1, ..., 6). The model, however, assumes a common background response (
= 1.0) in the DMSO control group. After model reduction (i.e., dropping the concentration-threshold parameter for DDT, which was originally estimated to be less than zero, allowing plateau parameters for DDT and OCT, and fixing the maximum effect parameter for the other chemicals to a fold induction of 14 as observed for Equation 2), the resulting model parameter estimates, standard errors and p values are provided in Table 3 with the fitted responses plotted in Figure 3. From Figure 3, this more flexible model adequately fits the single chemical and mixture data and leads to visually improved fits compared to those presented in Figure 2 (from the additivity model for SCR). A goodness-of-fit test comparing the deviance for the model in Table 3 to a more general model with seven concentration threshold parameters did not indicate a lack of fit (p > 0.99). The maximum effect parameter (
) for MXC, DPN, ß-HCH, and BPA was set at 14.0 based on the E2 positive control data. From this model (Table 3), five of the six individual concentration-effect curves had positive and significant slope parameters (ß1, ß2, ß3, ß4, ß6) while the only exception OCT had a positive and borderline significant slope (p = 0.061). The positive and significant slope parameter (
1) for the mixture indicated an increase in fold response as the total concentration of the chemical mixture increased along the fixed-ratio ray (p < 0.001). Further, there were positive point estimates for the threshold parameters for MXC, DPN, ß-HCH, OCT, and BPA (p = 0.374, p = 0.071, p = 0.835, p = 0.137, and p = 0.034, respectively). The estimate for the threshold for the mixture in terms of total concentration was estimated as 1.60 µM (Table 3,
) which was significantly different from zero (p < 0.001).
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Upon closer inspection, since MXC and DDT cover 94% of the mixing ratio, their low concentration effects are influential in the shape of the predicted response under additivity. Since the fold induction increases so rapidly in the low concentration range of DDT, the curve under additivity also rises rapidly. In fact, the test concentrations of DDT were not low enough to identify the threshold (i.e., without a threshold parameter, the concentration of DDT associated with the background fold induction of 1.0 was 0.0). Hence, the threshold concentration under additivity was estimated to be essentially zero (Fig. 3H). Similar to the previous analysis, since the experimentally derived mixture data along the specified mixing ratio of the six chemicals fall below that predicted under additivity (Fig. 3H), there is evidence of departure from additivity associated with an overall less than additive (antagonistic) interaction.
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DISCUSSION |
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A novel approach for testing the hypothesis of additivity along a fixed-ratio ray of chemicals was described. It has the advantage of more flexible model fits for the single chemical data than imposed by the additivity model for the SCR method. It is an extension to the work of Dawson et al. (2000) using the interaction index to predict under additivity. The proposed method uses a direct search algorithm to estimate the parameters required under additivity with a bisection algorithm to predict the mean response for the mixture data. The approach concludes with a likelihood ratio test of additivity.
The flexibility of the second approach is important in allowing different maximum effect levels for individual chemicals, such as commonly found for receptor-mediated toxicants. This flexibility is of biological importance as it allows for mixtures of full and partial agonists to be simultaneously considered in combination (mixtures) toxicity assessments. Here, the concentration-effect curves for MXC, DPN, ßHCH, and BPA were apparently still increasing based on the observed data. These chemicals were assumed to plateau above the observed experimental region. By contrast, DDT and OCT apparently plateau at a fold-induction below that assumed for the other four chemicals, consistent with the notion that they are partial agonists.
Further, the flexibility of the second approach allows for more precise modeling of threshold regions for individual chemicals. This is particularly evident by comparison of the predicted models in Figures 2 versus that in Figure 3 for MXC, DPN, and OCT. The threshold region under additivity includes all concentrations below the plane that connects the estimated concentration thresholds for each single chemical and is associated with background response levels. Figure 4 is a schematic for this threshold additivity surface for three chemicals. When concentration thresholds are not detected for a subset of the chemicals in the mixture, then it implies that any increase in exposure to these chemicals results in an increase in response levels. These are important features of environmental mixtures which are accommodated by more flexibility in assessing additivity and departure from additivity as demonstrated by the second statistical approach to the analysis of multi-component mixture responses.
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APPENDIX |
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Testing for Interaction Using a Fixed-Ratio Ray Design: SCR Approach
It is of interest to determine whether the components in a specified mixing ratio of c chemicals interact. Interaction is detected in the SCR approach by comparison of mixture models to the corresponding additivity model. Carter et al. (1988) and Gennings (2002)
demonstrated that generalized linear models and nonlinear models, respectively, which can be linearized (i.e., h(µ) =
ißixi) are algebraically equivalent to the definition of additivity in Equation 1. Further, these models can be adjusted to be in the form of a threshold additivity model (Schwartz et al., 1995
). Consideration of threshold additivity models permits inference regarding low concentration regions that are associated with background response. Although the SCR approach is applicable to general forms of nonlinear and threshold models (e.g., Gennings et al., 2002
, 2003), we describe the method via the models used in the example in section 3.
Using single chemical data, a threshold additivity model (e.g., Gennings et al., 1997) for increasing concentration-effect curves is given by
![]() | (2a) |
If all estimates for the concentration thresholds are outside of the experimental region (where negative estimates indicate lack of evidence for a concentration threshold and estimates above the experimental region demonstrate no concentration-effect relationship), then the corresponding nonlinear model is considered, i.e.,
![]() | (2b) |
We are interested in detecting and characterizing an interaction among the c chemicals in the mixture for the fixed mixing ratio of the chemicals. Following Gennings et al. (2002) define t as the total concentration of the xeno mixture (i.e.,
). Along a given ray with total dose t, the amount of the ith chemical is xi = ait, where ai is the ratio of the ith chemical in the mixture for i = 1,...,c, and
. The slope associated with this mixture under the assumption of additivity is given by
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The mixture data along the fixed mixture ray is fit to a similarly parameterized mixture model of the form
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The hypothesis of additivity along the specified ratio of the chemicals is an hypothesis of coincidence between the additivity model in either Equations 3a or 3b and the mixture model given in either Equations 4a or 4b, i.e., for the threshold models, H0: {mix =
add and
mix =
add} versus H1: {
mix
add or
mix
add}. An F test is used to test this hypothesis of additivity (e.g., Casey et al., in press
).
Alternative Approach: Defining Additivity Using the Interaction Index in Equation 1 Set to One
Of concern in conducting the SCR method is that the additivity model is underparameterized so that it lacks the flexibility to accommodate various features of the individual curves (e.g., the existence of a threshold dose, or differing plateau levels). For this reason, we were motivated to develop a different strategy for testing for interaction on a ray. The general strategy is to use more flexible models for single chemical concentration curves combined with a common intercept parameter. In short, an additivity model is only implicitly defined by combining these model fits with the definition of additivity as given in Equation 1. In addition, the concentration-effect curve in terms of total concentration is fit using a mixture model along the mixture ray(s). A likelihood-ratio test is then constructed to test for interaction. The approach is subsequently defined in more detail.
A more general form of the additivity model in Equation 2a allows for different threshold parameters and range parameters for each chemical. In addition, we allow for different threshold parameters and range parameters for each fixed-ratio mixture. For convenience in notation, we define a model for each single chemical and for each fixed-ratio mixture ray with subscript i, i = 1,..., R(= c + r), i.e.,
![]() | (5) |
Thus, the model does not restrict the range of the concentration-effect curves for the chemicals in the mixture (i.e., range between and
+
i); the concentration-effect curves are allowed to have different slope parameters (ßi); and, each chemical is allowed to have a concentration threshold if the data support the parameter and otherwise the
i parameter is removed from the model.
The model in Equation 5 is fit simultaneously to all of the single chemical data and the mixture data along the fixed-ratio ray(s). In the example we assume the mean model in Equation 5 and that the variance is proportional to the mean, i.e., Var(Y) = µ, across concentration and chemicals. Under these assumptions, a quasi-likelihood criterion is used to estimate the model parameters using an iterative algorithm (e.g., the Gauss-Newton algorithm). Assuming the variance is proportional to the mean, the quasi-likelihood function of the "full model" given in Equation 5 to the single chemical and mixture data is denoted as Q(full;
), i.e.,
with the scale parameter estimated using the moment estimate,
, where N is the total sample size and p is the number of model parameters.
In order to test for interaction along the fixed-ratio ray of interest, a reduced additivity model is estimated for comparison. This is accomplished by using only the parameters necessary to estimate the single chemical data from Equation 5 with a constraint of additivity as given in Equation 1 to determine the predicted values along the mixture ray. That is, define the ECi(µ) as the concentration for the ith chemical alone that produces a response of µ. From the model in Equation 5,
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Note that Equation 6 can be used to estimate the threshold along a fixed-ratio mixture ray under additivity by setting µ0(add) = , i.e.,
. Notice that if the ith chemical in the mixture is not associated with a threshold concentration (i.e.,
i(add) = 0), its contribution to the summation dominates the sum as the denominator is zero making
be zero.
Under the assumption that the variance is proportional to the mean, a quasi-likelihood estimation criterion is used for estimation along the R rays (=c single chemical rays + r mixture rays) using only single chemical model parameters and the constraint in Equation 6, i.e.,
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A likelihood ratio test of additivity along the fixed-ratio mixture ray(s) of interest is constructed as
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ACKNOWLEDGMENTS |
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NOTES |
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1 To whom correspondence should be addressed. Chris Gennings, Dept of Biostatistics, Virginia Commonwealth University, 1101 E. Marshall Street, #B1-039-A, Richmond, VA 23298-0032. Fax: (804) 828-8900. E-mail: gennings{at}hsc.vcu.edu.
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