This paper is based on a presentation at the Conference on Current Issues on Chemical Mixtures held 11-13 August 1997 in Fort Collins, Colorado. Manuscript received at EHP 17 February 1998; accepted 1 June 1998.
Portions of this work were supported by Cooperative Research Agreement CR-816069 from the U.S. Environmental Protection Agency (U.S. EPA) to G.D. Stoner at the Medical College of Ohio, Toledo, Ohio, and the Ohio State University, Columbus, Ohio.
Address correspondence to S. Nesnow, Biochemistry and Pathobiology Branch (MD-68), National Health and Environmental Effects Research Laboratory, U.S. Environmental Protection Agency, Research Triangle Park, NC 27711. Telephone: (919) 541-3847. Fax: (919) 541-0694. E-mail: nesnow.stephen@epamail.epa.gov
Abbreviations used: B[a]P, benzo[a]pyrene; B[b]F, benzo[b]fluoranthene; CPP, cyclopenta[cd]pyrene; DBA, dibenz[a,h]anthracene; 5MC, 5-methylchrysene; PAH, polycyclic aromatic hydrocarbon.
The interactions of mixtures of chemicals in biologic systems have been studied extensively in pharmacology (1,2) and toxicology (3). In general, these interactions can be classified as enhancing (greater than additive), inhibitory (less than additive), or no interaction (additive). When administered to experimental animals, chemical carcinogens exhibit all three of these effects depending on the chemicals, route of administration, sex, species, and target organ. The majority of these interaction studies have been performed using only two administered agents and a database of binary carcinogen interactions has been reported (4,5).
Polycyclic aromatic hydrocarbons (PAHs) are a pervasive class of environmental pollutant formed by the incomplete combustion of organic materials. Humans are exposed to PAHs from cigarette smoke, combustion products from gasoline, diesel fuel, coal, and oil, as well as from broiled and smoked foods (6). Many PAHs are carcinogenic in experimental animals and several PAH-containing mixtures (i.e., coke oven emissions, cigarette smoke, and coal tars) are human carcinogens (7-9). Although there have been a number of studies of interactions of PAHs within binary mixtures of PAHs, little work has focused on larger component mixtures and using lung tissues as tumor targets.
We selected five environmental PAHs to construct quintary (five-component) mixtures to examine tumorigenic interactions among PAHs. The PAHs benzo[a]pyrene (B[a]P), benzo[b]fluoranthene (B[b]F), dibenz[a,h]anthracene (DBA), 5-methylchrysene (5MC), and cyclopenta-[cd]pyrene (CPP) were selected for the extent and pervasiveness of their environmental occurrence, structural diversity, metabolic diversity, and range of tumorigenic potency. Using the strain A/J mouse lung as a target organ, we sought to answer the following questions: Are the mouse lung tumorigenic activities of these five PAHs additive? What is the extent of the deviation from additivity? Can specific interaction parameters be calculated? What is the effect of a nontumorigenic PAH on the tumorigenic activity of a quintary mixture of PAHs?
Methods for analysis of interactions have been reviewed including isobolographic analyses (10), interaction indexes (11), and response surface approaches (12). Response surface approaches have found use in identifying and quantitating chemical and drug interactions (13-16). Response surface methods use regression techniques that are both descriptive and predictive and are not limited to number of independent variables or same mechanisms of action or dose-response slopes. We utilized a response surface methodology, using response addition, to both design and analyze a study that sought to identify and quantitate the interactions among five environmental PAH lung tumorigens. Using a 25 factorial experimental design (five PAHs at two doses each) strain A/J mice were treated with a series of quintary PAH mixtures and lung adenomas were enumerated after 8 months.
We report that PAH-induced lung adenoma formation in strain A/J mice can exhibit both greater than additive and less than additive interactions that are dose related. Interaction analyses by maximum likelihood methods using a response surface model identified statistically significant binary, ternary, and quaternary interactions.
Chemicals
B[b]F (99%), DBA (97%), urethane (99%), and pyrene were purchased from Aldrich Chemical Co. (Milwaukee, Wisconsin) and B[a]P(
98%) from Sigma Chemical Co. (St. Louis, Missouri). CPP (99%) was obtained from A. Gold (University of North Carolina, Chapel Hill, North Carolina) and 5MC (99%) from S. Amin (American Health Foundation, Valhalla, New York). Tricaprylin was purchased from Eastman Kodak (Rochester, New York). The crude pyrene was purified by column chromatography in hexane using silica gel, recrystallized from hexane, and sublimed in vacuo at 170°C to give a melting point of 149 to 151°C. The reported melting point was 149.6 to 150.3°C (17). Liquid chromatography-mass spectrometry indicated this product was 99.7% pure.
Tumor Studies
Male strain A/J mice 6 to 8 weeks of age were obtained from Jackson Laboratories (Bar Harbor, Maine). Mice were housed in laminar flow rooms in groups of four in polycarbonate cages. Mice were maintained under standard conditions (20±2°C; 50±10% relative humidity; 12-hr light/dark cycle) and received food and water ad libitum. In the first study, individual PAHs were administered to male strain A/J mice at several doses. On the day of treatment, PAHs were sonicated in tricaprylin until complete solution was achieved, then mice were injected ip (0.2 ml/mouse). Urethane- and tricaprylin-treated mice served as positive and negative controls, respectively. After 8 months, all mice were sacrificed by cervical dislocation, the lungs removed, fixed in 10% neutral buffered formalin, and the surface tumors counted. No detailed histopathology was performed, as previous studies have identified these lesions as adenomas (18). In the second study (mixture study), a 25 factorial experimental design (five agents each at two doses) was used. Thirty-two groups of mice were randomized (20 per group) and dosing was performed in the order of the randomized group number. PAH mixtures were prepared by transferring individually weighed PAHs into each of 32 vials to construct the 32 dosing vials of quintary mixtures of PAHs. Quality assurance analysis by HPLC verified that the target doses were achieved (data not shown). Animals were treated and tumors were scored as described previously. In the second study smaller numbers of animals (10 mice per dose) were treated with the high dose of each PAH as indicator controls. Animal care and treatment were conducted in accordance with the guidelines established in the Guide for the Care and Use of Laboratory Animals (19). All animals were treated humanely with due consideration to the alleviation of distress and discomfort.
Statistics
Normality was examined by the Kolmogorov-Smirnov test, multiple comparisons by the Bonferroni multiple comparison test or Students-Newman-Kuels multiple comparison test (SigmaStat, Jandel Scientific, San Rafael, California), log likelihood analysis by SAS (SAS Institute, Cary, North Carolina), and multiple linear regression analysis using Mathematica (Wolfram Research, Champaign, Illinois).
Dose Responses of Individual Polycyclic Aromatic Hydrocarbons
Dose-response studies were performed with each of the five PAHs at dose ranges that resulted in a survival of 75 to 100% (Table 1). Statistical analysis indicated that 15 of the 22 groups were significantly different (p<0.01) from the tricaprylin control group using a Bonferroni multiple comparison test on the square root transformed tumor response data. Both positive (urethane) and vehicle (tricaprylin) controls for lung adenoma response were in agreement with historical data (18).
Selecting Dose Levels for Quintary Mixtures of Polycyclic Aromatic Hydrocarbons
Dose levels were selected that would satisfy the following: <25% mortality; <10% reduction in weight gain at the end of the study; a predicted range of tumor response between 2 and 100 lung adenomas per mouse; and the ability to observe an overall 2-fold greater than additive and a 4-fold less than additive tumor response. In addition, the PAH dose levels were prepared in ratios similar to those found in environmental air and combustion samples. In cigarette smoke (9), coal gasification emissions (21), ambient air (22), coke oven emissions (23), gasoline exhaust, and diesel exhaust (24), B[b]F and B[a]P are found in approximately equal amounts. In ambient air (22), diesel exhaust (24), and coke oven emissions (25), DBA is found in amounts that are approximately 5 to 40% that of B[b]F or B[a]P. CPP environmental levels range from 250 to 2000% of that of B[b]F or B[a]P in ambient air (22), gasoline exhaust, and diesel exhaust (24). 5MC has been detected in gasoline exhaust at approximately 6 to 10% that of B[b]F or B[a]P (24). Therefore, based on the dose-response data in Table 1 and the environmental levels discussed previously, the following dose levels in milligrams per kilogram were selected (high dose/low dose): B[a]P, 75/30; B[b]F, 75/30; DBA, 10/2.5; 5MC, 30/10; and CPP 100/30. The dosing groups were selected according to a 25 factorial dosing scheme (12,13). This resulted in 32 PAH mixture groups representing combinations of five PAHs at either high or low dose (Table 2). This dose scheme would allow the calculation of five PAH dose parameters, ten binary interaction parameters, ten ternary interaction parameters, five quaternary interaction parameters, and one quintary parameter.
Analyses of Polycyclic Aromatic Hydrocarbon Mixture Tumor Data
Survival of mice in the 32 PAH mixture groups ranged from 70 to 100%, with a median of 85% and a mean (±SD) of 84.8±9.1 (Table 2). No dose dependency could be established between survival and doses of PAHs with the dose range tested. The mean body weights for the highest dosed group compared to tricaprylin control indicated a significant loss of weight at day 7 with a recovery to control values on day 14 and beyond (data not shown). The mean body weights for each group at sacrifice were indistinguishable from that of the tricaprylin control group (data not shown).
The 32 PAH mixture groups gave tumor responses that ranged from 16.8 to 63.8 lung adenomas per mouse (Table 2). Statistical analysis using the Kolmogorov-Smirnov test on the numbers of tumors per mouse for each group indicated that many groups were not normally distributed, a conclusion supported by the observation that the variances exceeded the means for many groups. Statistical comparison indicated that each group was significantly different (p<0.01) from the tricaprylin control group using a Bonferroni multiple comparison test on the square root transformed tumor response data.
Comparison of Observed Responses
to those Predicted Based on Additive Responses from Individual Polycylic Aromatic Hydrocarbon Dose-Response Data
The expected additive responses for each PAH mixture were calculated by summing the responses from individual PAHs that comprised each mixture, using data from Table 1. Tumor responses for intermediate doses not specifically tested were obtained by estimates derived using curve-fitting procedures from a power equation as described in Nesnow et al. (20). The relationship between the observed lung adenomas per mouse and the expected lung adenomas per mouse based on additive responses for each of the 32 PAH mixture groups indicated that many of the observed data points deviated from those expected based on additivity (Figure 1). Two groups exhibited a statistically significant increase in expected tumor responses and 13 groups exhibited a significant decrease (p<0.05) by the Students-Newman-Keuls multiple comparison test on the square root transformed data. Regressing the data to a linear function (R2=0.679) gave major deviations from the expected slope of unity (calculated slope=1.70) and the expected y-intercept of zero (calculated y-intercept=-13.53). Calculation of the percent increase or decrease of observed tumor responses over the expected tumor responses based on additivity {100 
[(observed additivity)/additivity]} gave a range of 97.4% superadditivity to 55% supraadditivity.
Figure 1. Correlation between observed lung adenomas per mouse and expected lung adenomas per mouse after treatment of male strain A/J mice with quintary mixtures of polycyclic aromatic hydrocarbons. The data points represent the mean responses for each of the 32 quintary mixtures. The calculation of the expected additive responses for each PAH mixture was performed by summing the individual responses for each PAH within each mixture group based on the individual PAH dose-response studies in Table 1 and analyzed as described in Nesnow et al. (20). The responses for individual PAHs (PAH, dose in milligrams per kilogram, lung adenomas per mouse): B[a]P, 30, 2.11; B[a]P, 75, 8.14; B[b]F, 30, 1.17; B[b]F, 75, 3.44; DBA, 2.5, 2.9; DBA, 10, 32.2; 5MC, 10, 1.75; 5MC, 30, 13.2; CPP, 30, 1.53; and CPP, 100, 32.2. The line represents the expected relationship if the data fit additivity. 
statistically significant difference (p<0.05) between the observed and expected additive responses based on the Students-Newman-Keuls multiple comparison test. 
no statistically significant difference between the observed and expected additive responses.
Estimation of the Response
Surface Model
A generalized linear model (26) was fit to the data parameterized according to Equation 1 (Appendix). The variance of the response was assumed to be of the form 
µ (i.e., a constant times the mean). The method of maximum likelihood was used to estimate the unknown model parameters using a ridge-stabilized Newton-Raphson algorithm (Proc Genmod, SAS version 6.09). A power link function was used for g(µ) in Equation 1 with the best power estimate determined from a plot of the log likelihood versus the power parameter (Figure 2). The peak of this plot was approximately 0.5. Therefore, the square root transformation of the response data (lung adenomas per mouse) was used in the subsequent analysis of the data. To determine the shapes of the dose-response curves using the square root transformation for each individual PAH, the tumor data in Table 1 was square root transformed and plotted against dose (data not shown). Each PAH yielded a linear relationship with correlation coefficients >0.90, adding support for the use of a square root-linear equation to describe the mixture data in the response surface analyses.
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Figure 2. Plot of the log likelihood value versus the power parameter given in Equation 1. The full model in Equation 1 was fit for each value of the power parameter. The peak value of the plot indicates the maximum likelihood estimate for the power parameter.
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Response Surface Analyses
A square root-linear equation (Equation 1) was used to describe the 32-group quintary PAH mixture data. The equation contained 5 dose parameters and 26 interaction parameters: 10 binary, 10 ternary, 5 quaternary, and 1 quintary. A regression matrix of 1001 equations was constructed representing the dose and response data from each of the 1001 mice tested. This regression matrix included data from the 32 quintary mixture groups, the individual PAH dose-response data, and the indicator controls. Multiple linear regression techniques solved this matrix of equations and gave estimated parameters, 95% confidence intervals, and their associated p values (Table 3). The estimated scale parameter, , was 2.13. The goodness of fit of the model was assessed via the scaled deviance compared to a chi-square distribution and was determined to be adequate (p<0.05). The test for the significance of each model parameter was based on a comparison of the likelihood values with the full model and the model excluding the parameter under study. The p value is derived from comparing the likelihood ratio test to a chi-square distribution with one degree of freedom.
This model produced statistically significant values for 16 parameters (Table 3). All of the linear terms were highly significant (p<0.001). Furthermore, 10 of the interaction terms were significantly different from zero at the 5% significance level. The five significant binary interaction terms were all negative, giving less than additive interactions: ß13 (B[a]P-DBA), ß15 (B[a]P-CPP), ß23 (B[b]F-DBA), ß34 (DBA-5MC), and ß35 (DBA-CPP) (Table 3). Four significant ternary terms were positive, giving greater than additive interactions: ß134 (B[a]P-DBA-5MC), ß135 (B[a]P-DBA-CPP), ß234 (B[b]F-DBA-5MC), and ß345 (DBA-5MC-CPP). One quaternary term was statistically significant and negative: ß1345 (B[a]P-DBA-5MC-CPP).
Additivity Calculated from
the Response Surface Model
Following the definition of additivity as given by Berenbaum (11) and the logic of Carter et al. (12), the model in Equation 1 was reparameterized under the hypothesis of additivity (i.e., all of the interaction terms were removed) by including only the intercept and linear terms (i.e., ß0, ß1, ß2, ß3, ß4, ß5). A likelihood ratio test was used to test the hypothesis of additivity by comparing the likelihood of the constrained model (no interaction parameters) to the full model, a global test for additivity. Departure from additivity was found (p<0.001). A comparison of the additive responses derived from the constrained model (no interaction parameters) indicated close agreement with the additive responses obtained from the sum of the individual PAH dose responses from Table 1 (data not shown).
Prediction of Individual Dose-Response Curves
To test the model and model parameters, predictions for the dose-response relationships for each of the five PAHs were calculated. Individual dose-response curves were generated with the model parameters (ß0, ß1, ß2, ß3, ß4, ß5) found in Table 3 and compared to the observed data found in Table 1 (Figure 3). There was excellent agreement between the two data sets. The dose-response data for each of the individual PAHs (Table 1), although used in combination with the mixture tumor data to derive the model parameters, represent less than 15% of the total data set and therefore were not expected to dominate these predictions.
Figure 3. Observed male strain A/J mouse lung adenoma data for (A) B[a]P and B[b]F, (B) DBA, (C) CPP, and (D) 5MC compared to dose-response curves for each polycyclic aromatic hydrocarbon generated by the response surface model. The curves are from Equation 1 using the dose parameters from Table 3 (i.e., ß0, ß1, ß2, ß3, ß4, ß5). The data points represent means±SD of observed lung adenomas/mouse from Table 1.
Prediction by the Response Surface Model of the Tumorigenicity
of the Quintary Mixtures
Using Equation 1 and the parameters found in Table 3, the predicted response for each quintary mixture was estimated and compared to the observed responses (Table 2) for each of the 32 quintary mixture groups (Figure 4). Even though only 10 of the possible 26 interaction parameters were significant (p<0.05), the model and model parameters predicted the observed responses to a high degree. The correlation coefficient R2 was 0.979, the slope of the line was 0.996, and the y-intercept was 0.167. The 95% prediction intervals encompassed almost all of the data points.
Effect of Pyrene on One
Quintary Mixture
An additional group of mice was treated with a mixture containing the following PAHs (milligrams/kilogram): B[a]P, 30; B[b]F, 30; DBA, 2.5; 5MC, 30; CPP, 100; and pyrene, 100, using the same experimental protocol as described for the 32 quintary PAH groups. Pyrene was not tumorigenic at doses between 10 and 200 mg/kg (Table 4). A similar model to that described in Equation 1 was used to fit a subset of the mixture data and the pyrene data. The subset consisted of the tricaprylin control group, the group exposed to the five PAHs, the group solely exposed to pyrene, and the group simultaneously exposed to the five PAHs and pyrene. Thus the data consisted of a 2*2 design and was analyzed accordingly. The effect of pyrene was not different from background (p=0.614), the effect of the five-PAH mixture was to increase the numbers of lung adenomas from background (p<0.001), and the interaction of the five-PAH mixture and pyrene was negative and significant (p=0.007). Therefore, pyrene exerted a 35% reduction in the lung tumorigenicity of the quintary mixture.
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Figure 4. Correlation between the observed tumorigenic responses of male strain A/J mice treated with quintary mixtures of polycyclic aromatic hydrocarbons and the predicted responses from the response surface model. The data points represent the mean responses for each of the 32 quintary mixtures. R2=0.979; slope=0.996; y-intercept=0.167. The dashed lines represent 95% prediction intervals. The predicted responses were derived using Equation 1 and the parameters found in Table 3. Sixteen of the 32 parameters used in these calculations were significant at p<0.05.
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Polycyclic aromatic hydrocarbons are ubiquitous environmental contaminants found in the air, soil, and water, and in hazardous waste sites. Since their discovery as carcinogens in 1915 (27), their toxicologic effects have been studied intensively. Many PAHs are carcinogenic in multiple species (28) and are suspected carcinogens in humans (29), which also makes them an important chemical class from a public health standpoint. The epidemiologic data on PAH-containing mixtures strongly suggests that they are human respiratory carcinogens. The experimental animal data also point to lung, subcutaneous tissue, mammary tissue, and liver (in newborn and juvenile rodents) as targets of PAHs by various routes of administration (ip, intramammary, intrapleural, oral, inhalation, dermal, and iv). Because humans are exposed to mixtures of PAHs, it is important to understand the interactions among PAHs in these mixtures to assess their risk to humans. Statistical methodology is available that allows the determination of specific interactions between groups of toxicologic and pharmacologic agents using response surface methods (12,30,31).
To determine some of these potential PAH interactions, a study was constructed using five environmentally relevant PAHs administered as quintary mixtures to strain A/J mice with lung adenoma formation as the toxicologic outcome. The PAHs selected were B[a]P, B[b]F, DBA, 5MC, and CPP, based on their environmental occurrence, range of tumorigenic activities (32-37), structural features (methylated vs nonmethylated, condensed vs linear, alternant vs nonalternant), and a diversity of routes of metabolic activation (38-46).
The strain A/J mouse system is a medium-term tumorigenesis bioassay where tumors can be detected and quantitated 8 months after treatment. This mouse carries a lung cancer susceptibility gene or genes that have not yet been identified (18). Studies have shown that lung adenomas in this mouse will progress to adenocarcinomas after 18 to 24 months, with some metastasis. In addition, alveologenic carcinomas in humans are similar in morphology to adenocarcinomas in mouse lung (18). Finally, studies have shown that lung tumors in strain A/J mice produced by PAHs exhibit high proportions of Ki-ras mutations with mutation spectra different from the spontaneous controls (47-50). From these facts, we conclude that tumor formation in the strain A/J mouse has some relevance in the study of human lung cancer.
The results of these investigations indicate significant deviations from additivity that were both greater than additive and less than additive. The extent of these deviations ranged from +97.4 to -55% of that expected from additivity. Significant deviations were observed that were dose related i.e., lower doses, greater than additive; higher doses, less than additive. However, less than additive interactions dominated under most mixture conditions. Response surface modeling using multilinear regression techniques identified 6 statistically significant dose parameters and 10 significant interaction parameters of 32 possible parameters. The model and the estimated model parameters predicted the observed responses as well as the individual PAH dose-response curves. In additional studies we examined the need of all of the interaction parameters and found that significant fits to the data could not be obtained with fewer than the full complement of 26 interaction parameters (data not shown).
An analysis of the binary carcinogen interaction literature that encompasses multiple species, organs, and routes of administration has identified both greater than additive and less than additive effects for PAH-PAH interactions depending on target tissue species and route of administration (4). In single subcutaneous injection studies in female NMRI mice, Pfeiffer (37) reported dose-response sarcoma data with B[a]P, DBA, and mixtures of B[a]P and DBA at 114 weeks. The authors found the dose-response curves of the DBA group and the binary mixtures of B[a]P and DBA were not statistically different. This suggested a less than additive interaction between B[a]P and DBA. Schmähl et al. (51) summarized lifetime dermal application studies (two applications/week) of B[a]P and mixtures of PAHs, including B[a]P, B[b]F, and DBA in NMRI mice. Significant less than additive responses from the B[a]P dose response were noted in the four carcinogenic PAH mixture group receiving B[a]P, B[b]F, DBA, and benz[a]anthracene. Therefore, the studies by Pfeiffer (37) and Schmähl et al. (51) on the inhibitory tumorigenic activity of DBA in mouse skin and subcutaneous tissues are consistent with the inhibitory lung tumorigenic activity of DBA with other PAHs, possibly suggesting competitive inhibition of the enzymes involved in the metabolic activation of these PAHs.
There are a number of seemingly conflicting reports on the interactive effects of PAH, either in binary mixtures or in combination with complex mixtures containing PAHs. For example, B[a]P and CPP exert a greater than additive effect toward the induction of mouse skin papillomas (4), whereas this mouse lung study identified a less than additive interaction. Similarly, B[a]P and pyrene induce a greater than additive effect in papilloma formation in mouse skin tumor initiation studies (36), whereas pyrene, which induces neither mouse skin tumors (36) nor mouse lung tumors in strain A/J mice, significantly inhibits the lung adenoma formation of a quintary mixture of B[a]P, B[b]F, DBA, 5MC, and CPP. Finally, in mouse skin tumor coinitiation studies, cotreatment (initiation) of mice with B[a]P and cigarette smoke condensate followed by 12-O-tetradecanoylphorbol-13-acetate promotion produced a greater than additive effect, whereas similar studies with B[a]P and diesel exhaust particulate extracts produced a less than additive effect (36). Certainly, mixture composition, target tissues, species, strain, sex, and route of administration must play a role in the tumor outcome. Moreover, carcinogenesis is a multistage process that can involve absorption, distribution, metabolism, detoxification, elimination, macromolecular damage, mutation and/or chromosomal damage, DNA repair, altered gene expression, cytotoxicity, tissue injury, cell proliferation, and apoptosis. Many of these processes can be altered by enzyme induction and inhibition (3). Gibb and Chen (52) suggested that in the multistage model, a multiplicative effect of two or more carcinogens is consistent where each carcinogen acts on a different stage, whereas additivity occurs when each carcinogen acts on the same stage. Synergism has also been defined as occuring when the rate-limiting step in the generation of a single type of tumor differs for each of the two interacting carcinogens (53).
One approach to teasing out the dominant factors is to examine each separately. Future studies include examining quantitative DNA adduct formation, persistence, and repair over time and comparing the extent of DNA adducts formed by the quintary mixtures with the levels of adducts expected from additivity for each of the PAHs. Mixtures of PAHs enhance and inhibit covalent DNA binding (54). These studies are currently in progress.
In conclusion, a response surface model has identified a number of PAH-PAH interactions in a quintary mixture that accurately accounted for all of the observed responses. The observation of greater than additive responses from lower exposures is significant. However, because the magnitudes of all of the interactions were relatively small, these data suggest that although interactions of PAHs do occur, they are limited in extent.
Disclaimer: This manuscript has been reviewed by the National Health and Environmental Effects Research Laboratory, U.S. EPA, and approved for publication. Mention of trade names or commercial products should not be construed as endorsement or recommendation for use.
Appendix
g(
µ)=
x´ß [1]
where: µ=the number of lung adenomas for the nth mouse
g(.)=a specified monotone function of the mean
x´ß=ß0+ß1x1+ß2x2+ß3x3+ß4x4+ß5x5 +ß12x1x2+ß13x1x3+ß14x1x4+ß15x1x5+ß23x2x3+ß24x2x4+ß25x2x5+ß34x3x4+ß35x3x5+ß45x4x5+ß123x1x2x3
+ß124x1x2x4+ß125x1x2x5+ß134x1x3x4+ß135x1x3x5
+ß145x1x4x5+ß234x2x3x4+ß235x2x3x5+ß245x2x4x5
+ß345x3x4x5+ß1234x1x2x3x4+ß1235x1x2x3x5
+ß1245x1x2x4x5+ß1345x1x3x4x5+ß2345x2x3x4x5
+ß12345x1x2x3x4x5
x1=dose of B[a]P (mg/kg)
x2=dose of B[b]F (mg/kg)
x3=dose of DBA (mg/kg)
x4=dose of 5MC (mg/kg)
x5=dose of CPP (mg/kg)
ß0=unknown parameter associated with the background number of tumors
ß1=unknown parameter associated with the effect of B[a]P on the number of tumors
ß2=unknown parameter associated with the effect of B[b]F on the number of tumors
ß3=unknown parameter associated with the effect of DBA on the number of tumors
ß4=unknown parameter associated with the effect of 5MC on the number of tumors
ß5=unknown parameter associated with the effect of CPP on the number of tumors
ß12=unknown parameter associated with the interaction of B[a]P and B[b]F on the number of tumors
ß13=unknown parameter associated with the interaction of B[a]P and DBA on the number of tumors
ß14=unknown parameter associated with the interaction of B[a]P and 5MC on the number of tumors
ß15=unknown parameter associated with the interaction of B[a]P and CPP on the number of tumors
ß23=unknown parameter associated with the interaction of B[b]F and DBA on the number of tumors
ß24=unknown parameter associated with the interaction of B[b]F and 5MC on the number of tumors
ß25=unknown parameter associated with the interaction of B[b]F and CPP on the number of tumors
ß34=unknown parameter associated with the interaction of DBA and 5MC on the number of tumors
ß35=unknown parameter associated with the interaction of DBA and CPP on the number of tumors
ß45=unknown parameter associated with the interaction of 5MC and CPP on the number of tumors
ß123=unknown parameter associated with the interaction of B[a]P, B[b]F, and DBA on the number of tumors
ß124=unknown parameter associated with the interaction of B[a]P, B[b]F, and 5MC on the number of tumors
ß125=unknown parameter associated with the interaction of B[a]P, B[b]F, and CPP on the number of tumors
ß134=unknown parameter associated with the interaction of B[a]P, DBA, and 5MC on the number of tumors
ß135=unknown parameter associated with the interaction of B[a]P, DBA, and CPP on the number of tumors
ß145=unknown parameter associated with the interaction of B[a]P, 5MC, and CPP on the number of tumors
ß234=unknown parameter associated with the interaction of B[b]F, DBA, and 5MC on the number of tumors
ß235=unknown parameter associated with the interaction of B[b]F, DBA, and CPP on the number of tumors
ß245=unknown parameter associated with the interaction of B[b]F, 5MC, and CPP on the number of tumors
ß345=unknown parameter associated with the interaction of DBA, 5MC, and CPP on the number of tumors
ß1234=unknown parameter associated with the interaction of B[a]P, B[b]F, DBA, and 5MC on the number of tumors
ß1235=unknown parameter associated with the interaction of B[a]P, B[b]F, DBA, and CPP on the number of tumors
ß1245=unknown parameter associated with the interaction of B[a]P, B[b]F, 5MC, and CPP on the number of tumors
ß1345=unknown parameter associated with the interaction of B[a]P, DBA, 5MC, and CPP on the number of tumors
ß2345=unknown parameter associated with the interaction of B[b]F, DBA, 5MC, and CPP on the number of tumors
ß12345=unknown parameter associated with the interaction of B[a]P, B[b]F, DBA, 5MC, and CPP on the number of tumors
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Last Update: December 4, 1998
