Humans metabolize benzene via two pathways model provides a much better fit to the data. (Note that the weight of evidence supporting the two-pathway model depends strictly upon the difference in AIC values between the two models rather than the relative magnitudes of the AIC values under the two models.) This conclusion is supported by the Akaike weights of w1 = 0.061 and w2 = 0.939 for the one- pathway and two-pathway models, respec- tively, which indicate that, given the available data, Equation 2 offers a statistically robust depiction of the true benzene metabolism. Table 2 also summarizes results from bootstrap resampling of the data under the one-pathway and two-pathway models. In both cases, the medians of the bootstrap real- izations are very similar to the parameters estimated by the original models. The uncer- tainties in parameter estimates are indicated by the range between the 2.5th and 97.5th percentile values from the 1,000 bootstrap samples in each case. To visualize predicted metabolite lev- els free of background effects, the curves in Figure 3 represent the background-adjusted benzene metabolite levels, given by ( Yˆx–Yˆ0) where Yˆx and Yˆ0 represent the predicted levels of benzene metabolites at benzene concentra- tions X and 0 ppm, respectively, using the estimated parameters from Equation 1 (blue curve) and Equation 2 (black curve) from the original regressions (Table 2). Metabolite levels predicted from the two models are similar for benzene exposures in the range of 3–100 ppm but differ substantially for lower exposure concentrations. Exposure-specific metabolism of ben- zene at parts per billion exposure levels. The surrogate for the maximum metabolic rate of benzene (vmax/Km) in our analyses is the exposure-specific level of benzene metabolites (micromolar per parts per million) produced at nonsaturating (parts per billion) air con- centrations. Under the one-pathway model (Equation 1), this exposure-specific metabo- lite level would be Ymax/X50 = 68.6 µM/ ppm, whereas under the two-pathway model (Equation 2) it would be the sum Ymax,1/X50,1 = 48.7 µM/ppm plus Ymax,2/X50,2 = 135 µM/ ppm, or 184 µM/ppm (Table 2). Based on AIC values given above, the fit of the two-pathway model was superior to that of the one-pathway model. Thus, we sought inde- pendent confirmation that the exposure-specific Table 2. Parameter estimates for models of benzene metabolites (corresponding to Equations 1 and 2) for 263 nonsmoking female subjects. Parameter Y0 (µM) Ymax (µM) X50 (ppm) Ymax:X50 (µM/ppm) Y0 (µM) Ymax,1 (µM) X50,1 (ppm) Ymax,2 (µM) X50,2 (ppm) Ymax,1:X50,1 (µM/ppm) Ymax,2:X50,2 (µM/ppm) Original model estimate (SE) 94.64 (4.774) 8,253 (2,877) 120.3 (47.49) 68.60 87.20 (5.244) 14,637 (10,321) 300.6 (260.6) 80.06 (67.26) 0.5938 (0.9616) 48.69 134.8 Bootstrap resampling (n = 1,000) 2.5th, 97.5th Median percentiles 94.68 8,252 119.7 69.08 86.30 14,742 301.2 77.75 0.5756 48.91 142.7 85.76, 104.1 4,602, 15,169 58.76, 251.6 56.01, 84.84 60.13, 95.94 7,554, 60,992 121.5, 1,743 32.17, 209.6 0.0030, 2.765 32.98, 68.23 45.84, 18,672 2002). Because the models represented by Equations 1 and 2 are nested, and the number of observations (n = 263) exceeds 40 times the number of model parameters (three or five), Akaike’s information criterion (AIC) provides an appropriate means for judging the better fit; that is, the model with the smaller AIC is better. The difference in AIC values (∆ i) between the ith model and the best model and the associated Akaike weights, ⎛⎜ exp(− 12 Δi ) ⎟⎞ ⎜ wi = r ⎟, ⎜ ∑ exp (− 1 Δ ) ⎟ ⎝ i=l 2 i ⎠ [3] (where r is the number of models for compar- ison; r = 2 here) provide information about the weights of evidence supporting the com- peting models. By definition, the best model (i.e., the one with the smallest AIC) has a value of ∆ min = 0. Therefore, wi varies between exp(–∆ i/2) and 1 and represents the odds that the ith model is, in fact, the best depiction of truth, given the available data. Thus, a model having wi > 0.9 offers a compelling depiction of truth with the data at hand, and a model with wi < 0.1 offers a poor depiction of truth (Burnham and Anderson 2002). All statistical analyses were performed using SAS software for Windows version 9.13 (SAS). Results Figure 2 shows a scatterplot of total metabo- lite levels for the 263 subjects versus the shift- long air concentration of benzene on the day of urine collection. At benzene concentrations < 0.1 ppm, the data primarily reflect back- ground sources of the metabolites. The effect of benzene exposure on metabolite production becomes increasingly apparent at air concentra- tions > 1 ppm, and at levels > 100 ppm there is evidence that metabolite levels are approaching a plateau. There is a roughly 5-fold range of metabolite levels among nonsmoking females at a given air concentration of benzene. Estimated parameters for the Michaelis- Menten-like models. Also shown in Figure 2 are the mean trends for metabolite levels predicted from the parameters of the one- pathway model (dashed curve) and the two- pathway model (solid curve). Although both models offer qualitatively similar fits to the data in log scale, there are important quan- titative differences, as shown by the esti- mated parameters in Table 2 and the ratios Ymax,1:X50,1 and Ymax,2:X50,2. Thus, the ques- tion of which model provides a better fit to the data is extremely important. Because the AIC values were –313.755 for the one-path- way model (Equation 1) and –319.320 for the two-pathway model (Equation 2), result- ing in a difference of –313.755 – (–319.320) = 5.473, we conclude that the two-pathway Environmental Health Perspectives • volume 117 | number 6 | June 2009 949 Model One pathway (Equation 1) Two pathways (Equation 2) 10,000 )M1,000 µ ( s e t i lob 100 a t e m tla 10 o T A Combined pathways Pathway 1 Pathway 2 B 100 s 90 itle 80 bo 70 a te 60 m lta 50 fto 40 o tn 30 e rc 20 e P10 0 0.001 Abbreviations: Y, level of total metabolites (µM), X, benzene air concentration (ppm); Y0, background level of Y; Ymax,i, maximum value of Y given the ith metabolic pathway; X50,i, X corresponding to 50% of Ymax,i. 1 0.001 0.01 0.1 1 Benzene (ppm) 10 100 0.01 0.1 1 Benzene (ppm) 10 100 Figure 3. Predicted contributions of two metabolic pathways toward production of total metabolites at dif- ferent levels of benzene exposure. (A) Predicted mean trend (metabolic pathway 1 + metabolic pathway 2) and partial trends (metabolic pathway 1 or metabolic pathway 2) representing background-adjusted levels of benzene metabolites for a typical nonsmoking female subject. Predictions were based on parameters estimated for the two-pathway model (Equation 2) from the original regression (see Table 2). (B) Predicted percentages of benzene metabolites from the trends shown in (A) for the individual metabolic pathways.