Sensitivity Analysis of a Physiological Model for 2,3,7,8-Tetrachlorodibenzo-p-dioxin (TCDD): Assessing the Impact of Specific Model Parameters on Sequestration in Liver and Fat in the Rat

Marina Villafañe Evans*,1 and Melvin E. Andersen{dagger}

* U.S. Environmental Protection Agency, National Health and Environmental Effects Research Laboratory, Research Triangle Park, North Carolina 27711; and {dagger} Department of Environmental Health, Colorado State University, Ft. Collins, Colorado 80523

Received August 18, 1999; accepted November 8, 1999


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
2,3,7,8-Tetrachlorodibenzo-p-dioxin (TCDD) has dose-dependent tissue distribution because of induction of CYP1A2, a TCDD-binding protein, in the liver. Induction requires transcriptional activation of the CYP1A2 gene product by TCDD and the Ah receptor. An empirical model for dose-dependent distribution (Carrier et al., 1995, Toxicol. Appl. Pharmacol. 131, 253–266) included two simple descriptors: one for the maximum liver sequestration (Fmax) and the other for body burden leading to half maximum sequestration (Kd). Physiologically based pharmacokinetic (PBPK) models include specific parameters for protein receptors, protein binding, tissue solubility, and protein induction. We have applied a PBPK model to define two macroscopic constants related to these dose-response curves, i.e., the inflection point, and the maximum values of these curves. The dose-response curves generated from the PBPK model were for the proportion sequestered in liver and the liver to fat concentration ratio. Our analysis assessed the specific biological factors in the PBPK model that governed the values of these two macroscopic constants. For the fraction in liver, the Hill coefficient (a shape exponent describing the relationship between the Ah receptor–TCDD complex with the DNA receptor) resulted in the largest shift in inflection when using PBPK model parameters specific for TCDD. For the liver to fat ratio, the inflection point was most affected by the number of available Ah receptors. Conventional normalized sensitivity coefficients for the liver-to-fat ratio at the maximum were highest for the fat-to-blood partition coefficient, CYP1A2 binding affinity, and maximum extent of induction of CYP1A2. A similar pattern was observed for the liver fraction, except that the sensitivity coefficients were much smaller. The behavior of different TCDD congeners was evaluated by altering the value of key parameters. Our results demonstrate that the inflection point is more related to characteristics of DNA binding/induction steps of the Ah receptor-DNA complex than by the CYP1A2 affinity of TCDD or concentrations of CYP1A2. Surprisingly, the maximum is more sensitive to changes in CYP1A2 concentrations and affinity for TCDD. In addition, the analysis showed that the liver-to-fat ratio is a more useful experimental measure than is proportion in liver because the ratio responds with similar sensitivity over a much wider range of input parameters.

Key Words: TCDD; pharmacokinetic modeling; sensitivity analysis.


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
TCDD (2,3,7,8-tetrachlorodibenzo-p-dioxin), a persistent environmental contaminant affecting multiple organs, exerts its toxic effects through interaction with the Ah (aryl hydrocarbon) receptor (Birnbaum, 1994Go). With increasing doses of TCDD, there is a dose-dependent increase in proportion of the total body burden of TCDD dose in liver and a concomitant decrease in proportion found in fat (Abraham, et al., 1988Go; Birnbaum 1986Go; Kociba et al., 1978Go; Van den Berg et al., 1994Go). This distribution is surprising for a lipophilic compound. The explanation for the dose-dependent distribution is Ah receptor-mediated induction of a TCDD-binding protein in the liver. Interestingly, the existence of a TCDD-binding protein was initially suggested on the basis of pharmacokinetic analysis with a PBPK model for TCDD in mice (Leung et al., 1988Go). The hepatic binding protein was then tentatively identified as CYP1A2 in studies by Voorman and Aust (1989) and by Poland et al. (1989). The identification of the binding protein as CYP1A2 was verified by the absence of dose-dependent distribution of TCDD in CYP1A2 knock-out mice (Diliberto et al., 1997Go).

Physiologically based pharmacokinetic (PBPK) models have been developed to provide quantitative information about tissue distribution for a given chemical (Bischoff et al, 1971Go). For TCDD, PBPK models first simply had CYP1A2 induction in response to Ah receptor occupancy (Leung et al., 1990Go). More recently, the induction was modeled with a Hill coefficient relating transcription to the occupancy of DNA sites by an Ah receptor-TCDD complex (Andersen et al., 1991, 1993Go; Kedderis et al., 1993Go; Kohn et al., 1994Go). These models have been applied to study both the time course and the dose-dependent disposition for TCDD, particularly using short-term exposure data collected by Abraham et al. (1988). Using a very different modeling approach, an empirical model has been developed by Carrier et al. (1995) that assumed a quasi-steady state during chronic TCDD intake conditions. Carrier et al. (1995) used this model to describe TCDD's dose-dependent behavior in the liver and fat for both rats and humans. This empirical model has two defining characteristics: a maximum fraction retained in the liver (Fmax) and the body burden at which half the maximum value is present in liver (Kd).

The first goal of this paper was to restructure the available rodent PBPK models in order to calculate TCDD dose-dependencies in terms of the total dose of TCDD in the animal. This objective was met by calculating fraction in liver or liver to fat concentration ratios for multiple doses to produce a typical dose-response representation of these variables. The resulting curves, although calculated from the PBPK model, can also be characterized in terms of a maximum point (equivalent to Fmax) and a midpoint (equivalent to Kd). Once the TCDD-specific parameters for the PBPK model were defined, the second goal of this paper was to develop a sensitivity analysis technique to describe the shift in maxima and the shift in inflection point in response to variations in specific biological parameters. This sensitivity analysis approach permitted evaluation of the biological factors that give rise to the two macroscopic factors from the Carrier et al. (1995) model. The results were counter-intuitive. The inflection is primarily determined by characteristics of the interaction of TCDD, the Ah receptor, and DNA binding sites for the Ah-TCDD complex. Concentrations of CYP1A2 and its affinity for TCDD are critical parameters for the maximum of the curves.


    MATERIALS AND METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Computer Models and Numerical Analysis
The PBPK model used to generate the dose-response representations for the fraction of TCDD in the liver and the liver-to-fat ratio was based on the TCDD model for the rat (Andersen et al., 1993Go) adapted for steady state conditions. The following assumptions were incorporated in this application: each tissue had a specified blood volume and partition coefficient (solubility), and reversible binding equations were used to describe binding of TCDD to the Ah receptor and to a liver-specific protein (CYP1A2). Based on previous models, the induction of CYP1A2 was described through the formation of the Ah-TCDD complex and its binding to DNA using the Hill equation (Andersen et al., 1993Go; Buckley, 1995Go; Kedderis, et al., 1993Go; Kohn, et al., 1994Go). Transport of TCDD into the cell by diffusion (permeability coefficient) was assumed very rapid and not included for steady-state conditions. The PBPK model assumed that all body stores of TCDD were in fat and liver, similar to the assumptions made by Carrier et al., 1995. In addition, the assumption at steady state was that the blood concentration leaving each compartment and reentering as arterial blood had achieved the same levels. A schematic representation of the model is presented in Figure 1Go. Copies of the software program can be obtained by writing to the first author.



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FIG. 1. A Schematic representation of the steady state PBPK model.

 
The set of equations describing the steady state conditions for TCDD were entered into a computer simulation package, Simusolv (Dow Chemical Company, Midland, MI, Version 3.0, 1993). The simulations were run with body burden as the independent variable and curves were generated for the fraction in liver and the liver-to-fat ratio (y-axes) as functions of the body burden (x-axis). These simulations uniquely identified both an inflection point and a maximum. The equations used to calculate the liver fraction (FH) and the liver-to-fat ratio (LTOF) are presented in the Appendix. The set of parameters for the initial analysis corresponded to parameters that had been used previously to model TCDD in short-term exposure studies (Andersen et al., 1993Go). These are called the "default" set of parameters when compared to other parameter sets in which a constant was set to an altered value to assess sensitivities for hypothetical compounds with slightly different binding properties.

The fraction TCDD in liver, FH (dose), is complexly related to the biochemical and physiological parameters listed in Figure 1Go and Table 1Go. Similarly, the liver to fat concentration ratio, LTOF(dose), is related to a different group of parameters. These calculated dose-response curves from the PBPK model can also be described by an inflection (related to the ED50) and a maximum. The inflection was calculated at the dose where the derivative of the dose-response function with respect to body burden reached a maximum value. The maximum point was estimated at the dose where the derivative was zero. Simusolv's® derivative function was used to calculate the numerical derivatives of FH and LTOF as a function of dose, respectively. Due to their relationship to the derivative, the concentration at which these 2 characteristic points occur were termed the inflection and maximum, respectively (Fig. 2Go).


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TABLE 1 Parameters Used in the Steady-State Model for TCDD
 


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FIG. 2. Simulation results for the liver fraction (FH) and the derivative of the liver fraction (FH`) using the default TCDD model. The simulations were repeated after changing Kd from 0.05 to 0.5 nM.

 
Sensitivity Analysis
One measure of the ability to estimate a given parameter p from a measurement m gathered over time is defined as its sensitivity coefficient, i.e., the partial derivative of the measurement with respect to the parameter, {partial}m/{partial}p (Beck et al., 1977). An approximation that we will use is the slope around the parameter. That is, the sensitivity coefficient = {partial}m/{partial}p {approx} {Delta}m/{Delta}p. The larger the sensitivity coefficient, the larger the rate of change in the measurement for a given {Delta}p. Given this assumption, the ability to estimate p increases with a larger sensitivity coefficient. In this context, the sensitivity coefficient is analogous to the idea of calculating the slope around a given point. Using that analogy, a sensitivity coefficient equal to zero implies that there is no change in measurement regardless of what value the parameter p assumes. Similarly, a positive sensitivity coefficient implies an increase in the measurement with a given increase in the parameter p. A negative sensitivity coefficient implies a decrease in measurement given an increase in the same parameter p. These sensitivity coefficients can then be used to assess the relative effect of different model parameters on the predicted outcome.

Dose Shift in Inflection
The resulting increase (or decrease) in the inflection point for a given change in a model parameter was represented in Figure 3Go. As seen in the figure, a shift to the right, representing an increase in the dose, was defined as positive ({Delta}m). Similarly, a shift to the left, representing a decrease in dose, was defined as negative (–{Delta}m). A 5% change in each model parameter was used to represent the change in parameters ({Delta}p). A 5% change was chosen after experimenting with a 1%, 2%, 5%, and 10% changes in the parameter. The 5% increments were the smallest changes selected that gave consistent results. The resulting {Delta}m/{Delta}p's were normalized using the same concept Simusolv® uses in order to facilitate comparison with the shift in the maximum points. The final coefficients were calculated using Excel (Excel 97, Microsoft Corporation, WA) with the following formula, Equation 1.



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FIG. 3. Dose-response plots for the liver fraction using the default model and after doubling the Hill coefficient.

 

Since the results of the sensitivity analysis are parameter-specific, the analysis was repeated with a different set of input parameters for the purpose of comparison. The alternate parameter sets were selected to reflect several-fold magnitude changes on the default model. The analysis was repeated 3 additional times and compared to the default parameters presented in Table 1Go. First, KB2 (binding affinity for CYP1A2) was decreased from 6.5 to 1 nM and then increased to 40 nM. In addition, KB1 (Ah receptor binding constant) was increased from 0.04 to 0.4 nM. Finally, Kd (Ah–DNA receptor complex binding affinity) was increased from 0.05 to 0.5 nM. The resulting sensitivity coefficients were plotted using bar graphs (Sigmaplot®, SPSS Inc., Chicago, IL, Version 4.0).

Response Shift at the Maximum
As represented in Figure 4Go, a decrease in the maximum for a given increase in a model parameter is again defined by a negative sensitivity coefficient. Conversely, an increase in the maximum for a given increase in the model parameter is defined as a positive sensitivity coefficient. A more detailed description of the application of sensitivity coefficients to pharmacokinetic models is included in Evans and Andersen (1995). Briefly, the normalized sensitivity coefficients for the liver and fat-to-liver functions with respect to each parameter of interest were calculated at the maximum points using Simusolv's default settings. The finite-difference method was used in all the sensitivity coefficient calculations. The same approach as that described for the shift in inflection points was used for the analysis of the maxima. The impact of different changes in model parameters was again analyzed by means of bar graphs.



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FIG. 4. Dose response plots for the liver-to-fat ratio using the default model and after increasing fat partition to 400.

 

    RESULTS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The remaining graphical representations for the results summarize our attempt to understand the behavior of the inflection and maxima points for different TCDD congeners. Figure 5Go shows that tighter binding of TCDD to CYP1A2 leads to an increase in the liver fraction maximum when compared to the TCDD parameter set. Decreased binding affinity to CYP1A2 (larger Kb2) causes an opposite response, with little change in the inflection point. In contrast, increasing either of the binding parameters KB1 (Ah receptor TCDD binding) or Kd (Ah-TCDD complex DNA complex binding) causes the liver fraction inflection to shift to a higher dose.



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FIG. 5. Dose response plots the liver fraction for the default model and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5. These curves are the base cases for the sensitivity analyses in Figure 7Go.

 
Figure 6Go summarizes a similar analysis for the liver-to-fat ratio (LTOF). In this case, higher TCDD affinity for binding to CYP1A2 (KB2 = 1) leads to a higher LTOF when compared to default, with a maximum around 30. As before, decreased binding affinity to CYP1A2 (KB2 = 40) has an opposite effect, also showing little change in inflection. Decreasing affinity for Ah receptor binding (Kb1) and for Ah-TCDD-DNA complex binding (Kd) shifts the inflection to the right, i.e., to a higher dose, when compared to the default.



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FIG. 6. Dose response plots for the liver-to-fat ratio using the default model and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5. These curves are the base case for the sensitivity analyses in Figure 8Go. Note that this plot is on semi-logarithmic scale in contrast to Figure 5Go, which is linear.

 
Dose Shift in Inflection
The sensitivity analysis results outlined in this section were summarized in Figures 7 and 8GoGo.



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FIG. 7. Normalized changes at the inflection for the liver fraction using the default model and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5.

 


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FIG. 8. Normalized changes at the inflection for the liver-to-fat ratio using the default model and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5.

 
Default model.
The sensitivity coefficients (SCs) were predominantly positive in magnitude for the liver fraction results, with the exception of BM1 (amount of Ah receptor) and BM2i (CYP1A2 induction). These results imply a predominant increase in the dose for the inflection points, due to an increase in the values for: KB1 (Ah receptor binding), Pl (liver partition coefficient), PF (fat partition coefficient), BM20 (basal CYP1A2 levels), n (Hill coefficient), Kd (Ah-TCDD and DNA complex binding), and KB2 (binding to CYP1A2). The SC with the largest positive magnitude was that for the Hill coefficient, n. The negative value for BM1 relates the shift of induction to lower body burdens as more receptor is available for binding. As affinity for binding decreases (larger Kd values) the curve shifts to the right.

In contrast, the results for the liver to fat ratio were predominantly negative in magnitude (indicating a smaller dose required for the inflection point), with the exception of Kd (Ah complex-DNA binding). The largest absolute magnitude in this case was for BM1 (amount of Ah receptor), indicating that an increase in this parameter would lead to the largest decrease in inflection dose, just as with the results for liver fraction.

When comparing both default sensitivity results for the liver fraction and the liver-to-fat ratio, we found that a number of model parameters had zero sensitivity coefficients, represented by spaces between bars. Since non-zero sensitivity coefficients can be related to variability, the liver-fraction inflection point appears to be more affected by a wider range of pharmacokinetic parameters when compared to the liver-to-fat ratio.

Tighter TCDD binding to CYP1A2 (Kb2 = 1.0 vs. 6.5 nM).
For the liver fraction, the SCs were predominantly positive in magnitude, indicating a higher dose required for the inflection point. Several of the positive SCs were larger in magnitude for several model parameters, when compared to the default: n, Kd, KB1, and BM20. The only exception was the SC for PF, which was equal to the default. The largest positive SC result was for n, since the Hill coefficient controls the slope of the dose-response curve. The only negative SC, indicating a decrease in the dose inflection dose, with a value equal to the default, was for BM1.

As before, the liver-to-fat ratio had SCs that were predominantly negative. In this case, two SCs were larger in absolute magnitude when compared to default: BM2i, and n. The exceptions were for BM1, which was equal to the default, and Kb2, which was smaller than the default. When the SC graphs for the liver fraction and liver-to-fat ratio were compared, the SCs for the liver fraction were larger in magnitude than those for the liver-to-fat ratio.

Relaxed TCDD binding to CYP1A2 (Kb2 = 40 vs. 6.5 nM).
For the liver fraction, the SCs were predominantly positive in magnitude. Only Kd was higher in magnitude than the default results. The following positive SCs were equal to the default: KB1, PF, BM2i. The Hill coefficient results were smaller than the default. For the liver-to-fat ratio, the results were identical to those summarized in the previous section for tighter binding to CYP1A2. When comparing the liver fraction and liver-to-fat ratio sensitivity results, the liver fraction SCs appeared smaller than their liver-to-fat ratio counterparts.

Model with decreased Ah receptor binding (Kb1 = 0.4 vs. 0.04 nM).
For the liver fraction, the predominant SC results were positive in magnitude. The SC results larger than default were for KB1, PF, n, and Kd. The following positive SCs were equal in magnitude to the default: BM20, and KB2. Two SCs were negative and equal in value to default results: BM1, and BM2i.

For the liver-to-fat ratio, the magnitude of the SCs was predominantly negative. Only BM1 had an SC equal to that of the default case. The SC for KB2 was smaller than default. The rest of the negative SCs were larger than default: Pl, PF, BM20, and n. The positive SC results were for KB1, which was larger than default, and Kd, which was smaller than default. When comparing the liver fraction and liver-to-fat ratio sensitivity coefficient plots, the SCs were again higher for the fraction than for the ratio.

Model with decreased Ah complex-DNA binding (Kd = 0.5 vs. 0.05 nM).
For the liver fraction, again the predominant magnitude of the resulting SCs is positive. However, in this case, the following positive SC results are equal in magnitude to the default: KB1, PF, BM20, n, and Kd. Only Pl has a smaller SC when compared to default, and is equal to zero. The remaining SC results for BM1 and BM2i were negative and equal in magnitude to default.

For the liver-to-fat ratio, the SC results were predominantly negative, with a similar pattern to that of the previous case. The only difference was in the SC for KB2, which was equal to the default. When comparing the liver fraction and liver-to-fat ratio results together, Kd, Kb1 and BM1 had consistent relationships. The first two had positive values; the last of the three had negative values. These parameters regulate the amount of Ah-TCDD complex and of occupancy of the Ah-TCDD complex–DNA binding site.

Response Shift at Maximum
In this section, we summarize the results for change in response for the liver fraction or the liver-to-fat ratio for each maxima identified. Figures 5 and 6GoGo suggested the general direction of changes in maxima for the different TCDD congeners simulated. The sensitivity analysis results are summarized in Figures 9 and 10GoGo.



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FIG. 9. Sensitivity coefficients at the maximum for the liver fraction using the default model and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5.

 


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FIG. 10. Sensitivity coefficients at the maximum for the liver-to-fat ratio using the default model, and after setting KB2 = 1, KB2 = 40, KB1 = 0.4 and Kd = 0.5.

 
Default model.
For the liver fraction, most SCs were close to zero, except for: Pf, BM2i, and KB2. Only the results for Bm2i were positive. In this case, a positive magnitude refers to an increase in response, or higher liver fraction required for the maximum to occur. Conversely, a negative SC in this case refers to a decrease in liver fraction expected for the maximum to be achieved.

For the liver-to-fat ratio, the SCs followed the same pattern, except the magnitudes were larger. The liver-to-fat ratio does not have a fixed upper bound, compared to similar changes predicted for the liver fraction, which can never exceed 1.0. The absolute values of the SCs for the liver-to-fat ratio reflect this difference in bounding of the function.

Tighter TCDD binding to CYP1A2.
For the liver fraction, all the SCs were very close to zero, indicating that the maximum has a very small potential for change under tight binding conditions for CYP1A2. These results are summarized in Figure 5Go, where the maximum in this case is very close to a fraction of 100% TCDD in liver. For the liver-to-fat ratio, the SCs retain practically the same values as for the default case. Also referring to Figure 6Go, the liver-to-fat ratio maximum increases drastically for the case of tighter binding, since this ratio can assume higher values than can the fraction results.

Relaxed TCDD binding to CYP1A2.
For the liver fraction, the resulting SCs were similar in value for the default case. For the liver-to-fat ratio, there was no change in Pf with respect to default, but there was a decrease in magnitude for the SCs for BM2i and KB2. There was an increase in the magnitude of the SCs for KB1 and BM1. The two latter changes compensated each other, since they occurred in opposite directions.

Model with decreased Ah receptor binding.
For the liver fraction, the SC results had the same magnitude as for the default case. These results also followed the same direction as before. When comparing liver fraction and liver-to-fat ratio results, SCs for liver-to-fat ratio were uniformly larger.

Model with decreased Ah complex-DNA binding.
For the liver fraction, the SCs had the same magnitude and followed the same changes in direction as for the default. Again, the same pattern was observed for the liver-to-fat ratio results, with the latter being larger in magnitude when compared to the liver fraction. For the maximum point on either curve the main contributors were parameters associated with sequestration in the two tissues. They were the fat partition coefficient with negative values and the CYP1A2 capacity (BM2I) and affinity (Kb2). The SCs were positive for capacity and negative for the binding constant.


    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The present study provides a steady state PBPK model that successfully described TCDD's characteristic dose-dependent distribution in fat and liver. Carrier et al (1995) emphasized that the advantage of an empirical model was its relative simplicity and the ease of estimating characteristic parameters such as midpoint and maximum points. A goal of this work was to provide a bridge between the empirical model and the PBPK models. If the quasi-steady-state assumption is correct, then PBPK models developed to describe the acute distribution of TCDD should predict similar results when extended to chronic conditions to those obtained using a steady-state model.

When comparing Carrier's empirical model with our present steady-state pharmacokinetic model, the liver fraction at the highest concentration (100 ng/kg) was slightly higher for the present model: Carrier's results were between 0.5–0.61, depending on the data set used; ours were 0.76. Our present model assumes a rat weight of 250 g and does not take into account increases in body weight occurring in the experimental animals during the course of a chronic exposure. Also, it has been previously shown that partition and permeability coefficients are most accurately determined from the early part of TCDD's tissue time-course curves (Wang et al., 1997Go). The main discrepancy arises from the choices of parameters and underlying assumptions regarding the organs available for dioxin sequestration. The two models can be forced to match by adjusting individual parameters, for example, by adjusting fat partition in the PBPK model to 1000. Although there are small differences in the predictions from the two models, their behaviors are consistent. We chose to use the parameters from Andersen et al. (1993) that have successfully described a wide range of disposition results rather than adjusting the PBPK model parameters to agree with Carrier et al. (1995).

Sensitivity analysis has been previously used to describe the relative importance of model parameters with respect to one another and to describe the outcome in predictions due to differences in the parameters (Clewell et al, 1994Go; Csanady et al, 1994Go; Evans and Andersen, 1995Go; Kohn et al, 1993). To our knowledge, this is the first time that sensitivity coefficients have been estimated for a pharmacokinetic model that predicts outcome versus dose rather than versus time. As a result of this novel approach, the corresponding numerical technique to determine sensitivity coefficients describing changes in concentration at the inflection point for a given change in parameters had to be developed for this application. The corresponding sensitivity coefficients describing changes in liver or liver to fat functions for the maximum were obtained using available computer software. In order to understand the impact of these sensitivity coefficients, we used computer simulations to demonstrate changes occurring for a given change in model parameters. Figure 3Go describes the increase in inflection for the liver fraction when the Hill coefficient is doubled, which is represented by a positive sensitivity coefficient. Figure 4Go describes the decrease in the maximum for the liver-to-fat ratio when the fat partition coefficient increases from 350 to 400, which is represented by a negative sensitivity coefficient.

The interpretation of the resulting distribution curves with their inflection and maximum points might be based on simplified characterizations of affinities and binding constants. However, these two macroscopic parameters, i.e., inflection point and maximum, are complexly related to multiple PBPK model parameters, as suggested by the sensitivity analysis results presented in Figures 5 and 6GoGo. While multiple parameters contribute to the change in response for the macroscopic parameters, the highest sensitivity coefficients found were for Ah-receptor binding for TCDD and Ah-TCDD-complex binding to DNA. Thus, even though the distribution of dioxin between the liver and fat depends on partition coefficients and CYP1A2 binding characteristics, the position of the inflection point is predominantly determined by the Ah-receptor and DNA binding characteristics associated with induction of CYP1A2. This dependence is most clearly observed with the inflection for the liver-to-fat ratio (Fig. 6Go). However, it is important to consider that the values of the SCs are highly dependent on the parameters selected for the default and modified cases presented here, and different sensitivity results would be expected for different parameters.

In contrast, the SCs for the maximum show dependence on mainly three parameters: the fat partition coefficient, maximum fold induction for CYP1A2, and TCDD's binding affinity for CYP1A2. The changes in maximum reflect individual parameters that affect tissue sequestration, while the inflection point appears more related to inducing characteristics of the Ah receptor and the affinity for binding with DNA. The SCs for the liver fraction are lower in magnitude than the liver-to-fat ratio, but the results show a similar pattern. The behavior of the liver fraction is consistent with the fact that this fraction cannot be larger than 1 (100% of the dose is present in the liver), and the overall potential to change is limited by this characteristic. The liver-to-fat ratio is not expressed as a fraction; therefore, the resulting ratio can change over a wide range, as reflected by the larger sensitivity coefficients (Figs. 9 and 10GoGo). For this reason, the liver fraction is not sensitive to increasing KB2 after a certain value, while LTOF continues to change in response to increases in KB2.

Our analysis illuminates the determinants that give rise to the macroscopic behaviors—the maximal sequestration and the inflection point: the change from basal liver sequestration to maximal sequestration following CYP1A2 induction. These results also point the way to interpreting the behavior of other compounds that induce and bind to CYP1A2. Andersen et al. (1991) speculated that a particular TCDD congener, 4-PeCDF (2,3,4,7,8-pentachlorodibenzofuran) would have a higher CYP1A2 binding affinity when compared to TCDD. This prediction was based on a liver-to-fat ratio of 20 to 30. Experimental results have shown 4-PeCDF has a high affinity for liver, as determined by liver-to-fat ratios in mice (DeVito et al., 1998Go). The analysis presented here suggests that a higher binding affinity for CYP1A2 (setting KB2 = 1) results in a liver-to-fat ratio of approximately 30 (Fig. 6Go). In this regard, the liver-to-fat ratio for these halogenated compounds under maximally induced conditions appears to be a fairly good measure of their binding affinity to CYP1A2.

In conclusion, a PBPK model was developed to describe TCDD distribution in rats under steady state conditions. Dose-dependent induction of CYP1A2 leads to curves where liver fractions increase with increasing body burden of TCDD. These curves have maximal and inflection points. Sensitivity analysis techniques showed the influence of individual model parameters on the maximum and inflection points. The inflection point is primarily affected by parameters associated with TCDD binding to the Ah receptor and with the Ah-TCDD complex binding to DNA sites. Maxima are affected by fat solubility, affinity of TCDD for CYP1A2, and the amount of CYP1A2 induced. The determination of the liver-to-fat ratio, a non-bounded description, is more informative of dose-dependent behaviors. The SCs for the liver-to-fat ratio were very consistent over a broad range of input parameters.


    APPENDIX
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The mathematical formulation to account for total TCDD in the liver takes into account partitioning as well as CYP1A2 induction by TCDD. The total induction of CYP1A2 in the liver, BM2T, was calculated in Equation 1 as:


where n represents the Hill coefficient, Ah-TCDD represents the Ah-TCDD-DNA complex, and BM2i is the maximum induction of CYP1A2.The expression for TCDD in the blood took into account TCDD amounts in fat and liver both due to partitioning and binding to receptors and proteins. TCDD blood concentration was defined as:


where dose = amount of TCDD administered based on BW, µg/kg; Cb = concentration of TCDD in blood, nM; VB = blood volume, as percent of BW; PL = liver partition coefficient, dimensionless; VL = liver volume, as percent of BW; PF = fat partition coefficient, dimensionless; VF = fat volume, as percent of BW; BM1 = Ah receptor maximum availability, nmol/liver; KB1 = Ah receptor binding constant, nM; BM2T = total amount of CYP1A2, nmol/liver; and KB2 = binding affinity for CYP1A2, nM.When an equation such as the blood equation has the variable of interest in both sides, it is called an implicit equation. We used Simusolv's IMPL algorithm to solve this implicit equation with the number of points set to 200, supplied an initial value for blood (zero), a convergence criterion (0.000001), and the number of allowed iterations (10). Because multiple solutions may be derived, we used the AMAX1 function to return the maximum solution (or root) only. These results were then used to calculate the fraction of TCDD in the liver, FH as:


followed by LTOF, the ratio of liver to fat concentrations, as:


Carrier et al. (1995) made the simplifying assumption that all volume mass was either in the liver or fat, since blood volume represents a small percentage of the total mass. The present authors made use of the same assumptions in the present computer simulations. Thus, the present model reflects a physiological counterpart of Carrier's empirical representation. The actual equations used in the code did not include terms having a blood volume component.


    ACKNOWLEDGMENTS
 
The authors wish to thank Ms. Pamela Majette for technical editing and Mr. Christopher R. Eklund for technical support. We would also like to thank Drs. L. S. Birnbaum, H. A. Barton, M. R. Easterling, M. J. DeVito and particularly T. Leavens for thoughtful manuscript review and many helpful suggestions.


    NOTES
 
1 To whom correspondence should be addressed at U.S. EPA, NHEERL, MD 74, RTP, NC 27711. E-mail: EVANS.MARINA{at}EPA.GOV. Back


    REFERENCES
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
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