5.1 Case Study I: A Zero Dimensional Physiological System

Uncertainty Analysis of a Physiologically Based PharmacoKinetic Model for Perchloroethylene

5.1.1 Description of the Case Study

There is often significant uncertainty associated with human Physiologically Based PharmacoKinetic (PBPK) model parameters since human in vivo experimental data are not usually available for toxic chemicals. Thus many of the parameters in human PBPK models are generally estimated by in vitro experimentation and by inter-species scale-up of animal PBPK model parameters. The uncertainty in human PBPK parameters includes a significant amount of natural variability, reflecting the interindividual variability inherent in human populations. It is desirable to estimate the effects that uncertainties (including variability) associated with model inputs and parameters have on output metrics such as internal and biologically effective doses.

Here, a PBPK model for perchloroethylene (PERC) [61] is considered for uncertainty analysis. Human exposure to PERC, and the subsequent metabolism of PERC in the body, is of concern since PERC is a potential carcinogen [61]. Further, PERC is used as a cleaning solvent in the dry cleaning industry, and hence human exposures to it are very common. Three dose surrogates for PERC inhalation are considered in this study: (a) area under the arterial blood concentration (AUCA), (b) area under venous blood from liver (AUCL), and (c) the cumulative amount metabolized in the liver (CML).

CML is considered to be an alternative indicator of risk than simply the total uptake of PERC, since the health risks are thought to be associated with the metabolism of PERC in the body. AUCA and AUCL are indicators of the amount of time PERC resides in the arterial blood and liver, respectively, and thus are also considered as alternative indicators of risk.

Appendix B presents the formulation of a PBPK model, and additional details can be found in the literature [176,78]. Figure 5.1 presents the schematic of the PERC PBPK model used in this case study; the human body is considered to consist of a set of compartments corresponding to different organs and tissues. These are modeled as a series of interconnected continuous stirred tank reactors (CSTRs).

The effect of uncertainty in the parameters of the PERC PBPK on the three dose surrogates is analyzed using the following:

• Standard Monte Carlo approach,
• Latin Hypercube Sampling (LHS),
• Deterministic Equivalent Modeling Method/Probabilistic Collocation Method (DEMM/PCM),
• Stochastic Response Surface Method with Efficient Collocation (ECM),
• Regression based SRSM, also referred to simply as SRSM, and
• Combined Application of the Stochastic Response Surface Method and Automatic Differentiation Method (SRSM-ADIFOR).

  

Figure 5.1: Schematic representation of a PBPK model for PERC

The uncertainty analysis consisted of two stages: In the first stage, the effect of uncertainty in the metabolic constants ( $\ensuremath{K_m}$ and $\ensuremath{V_{\mbox{\scriptsize max}}}$) and the partition coefficients are considered (see Table 5.1). The uncertainty in these six parameters are described by independent log-normally distributed random variables. In the second stage, the uncertainty in compartmental volume proportions are included in the analysis. These five additional parameters are described by a set of mutually dependent Dirichlet distributed random variables. In the first stage (6 parameters), the DEMM approach and the ECM are evaluated through comparison with the results from a standard Monte Carlo simulation. For the second stage (11 parameters), the ECM, Regression Based SRSM and SRSM-ADIFOR are evaluated by comparing the results from these methods with the results from the standard Monte Carlo and LHS methods.

 

= 0.5

 

Table 5.1: Deterministic and uncertain parameters used in the uncertainty analysis of the PERC PBPK model

		Preferred
Symbol	Description	Value	UF
BW	body weight [Kg]	70	--
Partition Coefficients
$P_{\mbox{\scriptsize blood/air}}$	blood:air partition coefficient	12	1.7
$P_{\rm fat/blood}$	fat:blood partition coefficient	102	2.15
$P_{\rm sp/blood}$	slowly perfused tissue:blood partition coefficient	2.66	11.0
$P_{\rm rp/blood}$	rapidly perfused tissue:blood partition coefficient	5.05	5.69
$P_{\rm liv/blood}$	liver:blood partition coefficient	5.05	9.37
Blood Flows
$Q_c$	cardiac output [liters/hr]	348	1.12
$Q_p$	alveolar ventilation [liters/hr]	288	1.50
$Q_{\rm fat}$	blood flow to fat [liters/hr]	17.4	1.09
$Q_{\rm sp}$	blood flow to slowly perfused tissue [liters/hr]	87.0	1.04
$Q_{\rm rp}$	blood flow to rapidly perfused tissue [liters/hr]	153	1.25
$Q_{\rm liv}$	blood flow to liver [liters/hr]	90.6	1.35
Compartment Mass Fractions
$V_{\rm fat}$	mass of fat compartment [Kg]	23.0%	1.09
$V_{\rm sp}$	mass of slowly perfused tissue compartment [Kg]	62.0%	1.04
$V_{\rm rp}$	mass of rapidly perfused tissue compartment [Kg]	5.0%	1.25
$V_{\rm liv}$	mass of liver compartment [Kg]	2.6%	1.35
Metabolic Constants
$\ensuremath{K_m}$	Michaelis-Menten constant for metabolism [mg/liter]	1.47	12.3
$V\!_{\mbox{\scriptsize max,}c}$	maximum rate of metabolism [mg/hr]	0.33	2.84

In an earlier study, Farrar et al. [67] examined the effect of PBPK model parameter uncertainty on the three dose surrogates, CML, AUCA, and AUCL. The PBPK model parameter uncertainties used in the present work are adopted from their work, and are specified by the preferred values (PVs) and uncertainty factors (UFs) given in Table 5.1. The preferred values for all parameters other than metabolic constants, correspond to the nominal values in the PERC PBPK model [61], updated as suggested by Reitz and Nolan [170]. The preferred values of metabolic constants are given by the geometric mean of values suggested by Hattis et al [93], and by Reitz and Nolan [170]. The uncertainty factor for a parameter is defined such that the interval from $X_P/{\mbox{UF}}$ to $X_P{\,\mbox{UF}}$ is the 95% confidence interval, where $X_P$ is the preferred value of parameter $X$.

5.1.2 Specification of Parameter Uncertainty

Partition coefficients and metabolic constants are assumed to be lognormally distributed, according to Farrar et al [67]. The pdf for a given lognormally distributed parameter is specified by assuming that the preferred value of the parameter corresponds to the median of the distribution. This is convenient, since the mean and standard deviation that specify the corresponding normal distribution are then given by $\ln X_P$ and $\ln(\mbox{UF})/1.96$ respectively.

Compartment volume fractions are assumed to be specified by a Dirichlet distribution [67]. This implies that each compartment volume is specified by a $\chi^{2}$distribution [201,114], the pdf of which is given by:

$$f(x;a) = \left\{ \begin{array}{l} \displaystyle{\frac{1}{\Gamma(a/2)2^{a/2... ...-1} e^{-x/2},\ x > 0}\\ \displaystyle{0,\ \mbox{elsewhere}} \end{array}\right.$$ (5.1)

The mean of the $\chi^{2}$ distributions is $a$, and is related to the preferred value and the Dirichlet distribution parameter $\Theta$, as follows:

$$\mbox{E}[X_{i}]= x_{ip}\Theta$$ (5.2)

where $X_i$ is the $i$th compartment volume, E$[X_{i}]$ is its mean, and $x_{ip}$ is the preferred value of the fraction $X_i$, as given in Table 5.1. From the values of $\Theta$ and the preferred values for each compartment volume fraction, the parameters of the pdfs of the compartment volumes can be obtained.

Compartment blood flows are assumed to be specified as deterministic functions of compartment volumes and preferred values of compartmental blood flows. In sleeping humans, the blood flow to compartment $j$ is given by:

$$Q_j = {Q}_{j,P} \frac{V_j}{V_{j,P}}\ ,$$

where the notation $X_P$ for the preferred value of the parameter $X$is employed. The total cardiac output is given by

$$Q_c = Q_{\rm liv} + Q_{\rm fat} + Q_{\rm sp} + Q_{\rm rp}\ ,$$

and the total cardiac output in waking humans is given by:

$$Q_{cw} = Q_c + (Q_{cw,P} - Q_{c,P})q_{c\epsilon}\ ,$$

where $q_{c\epsilon}$ is a lognormally distributed random variable, the preferred value and uncertainty factor for which are given in Table 5.1. The form of the expression for $Q_{cw}$ensures that $Q_{cw} > Q_c$ always. The pdf for $q_{c\epsilon}$ is constructed in a manner analogous to that of the other lognormally distributed parameters as described above.

The increase in cardiac output in waking humans, is accounted for by an increase in blood flow to the slowly perfused compartment, as follows:

$$Q_{spw}=Q_{sp,P} + (Q_{cw,P} - Q_{c,P})q_{c\epsilon}.$$

Alveolar ventilation rate is a deterministic function of the cardiac output, and the preferred values for ventilation are given in Table 5.1. Alveolar ventilation rates in sleeping and waking humans are given by:

$$Q_p = Q_{p,P}\ \frac{Q_c}{Q_{c,P}}\ ,\ \mbox{and\ \ \ } Q_{pw} = Q_{pw,P}\ \frac{Q_cw}{Q_{cw,P}}$$

  
5.1.3 Implementation of the PERC PBPK Model

The PERC PBPK model has been implemented in SimuSolv [58], a software environment for numerical modeling and simulation. The PBPK model consisted of a set of coupled ordinary differential equations, which describe the dynamic variation in the amount of PERC in each of the compartments shown in Figure 5.1, resulting from uptake, distribution, metabolism and excretion processes. The random parameters in the model have been sampled for the simulations using the pseudo-gaussian random number generator function in SimuSolv, RGAUSS. The representation of pdfs of uncertain parameters in terms of standard gaussian random variables, each with zero mean and unit variance, is described here. Daily values of dose surrogates, CML, AUCA, and AUCL, are computed using the PBPK model to simulate a 24 hour inhalation exposure to 10 ppb of PERC, which is a typical high concentration in urban/industrial areas [102].

A lognormally distributed random variable $X$, with median $\mu_{X}$, and an uncertainty factor UF$_{X}$, can be represented in terms of a standard normal random variable $\xi = \mbox{N}(0,1)$, having zero mean and unit variance, as follows [201]:

$$X=\exp\left(\ln(\mu_{X}) + \frac{\ln(\mbox{UF}_{X})}{1.96}\xi\right)\$$ (5.3)

A $\chi^{2}$ distributed random variable with parameter $a$, can be approximated in terms of a normal random variable $\xi$ as follows[217]:

$$Y = a\left(\xi\sqrt{\frac{2}{9a}} + 1 - \frac{2}{9a}\right)^{3}$$ (5.4)

5.1.4 Results for the PBPK case study

  

Figure 5.2: Evaluation of DEMM/PCM: uncertainty in the cumulative amount of PERC metabolized, over a period of one day, resulting from 6 uncertain parameters

  
5.1.4.1 Stage I (Six Uncertain Parameters): Evaluation of DEMM/PCM and ECM

The effect of six uncertain PBPK model parameters on the dose surrogate pdfs is studied in the first stage of uncertainty analysis. The number of model simulations used for DEMM/PCM and the ECM are equal, as both these methods differ only with respect to the selection of collocation points. Second and third order approximations are used for both DEMM/PCM and the ECM. The number of simulations required for both these methods for the case of six uncertain inputs are 28 and 84, for 2nd and 3rd order approximations, respectively. Additionally, 10,000 Monte Carlo simulations are used in the first stage for evaluation of the DEMM/PCM and ECM.

  

Figure 5.3: Evaluation of DEMM/PCM: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL), over a period of one day, resulting from 6 uncertain parameters

Figures 5.2 and 5.3 present the pdfs estimated from four different sets of collocation points selected from the same set of original 4096 points (DEMM Trial 1, 2, 3 and 4), for the dose surrogates AUCA, AUCL and CML, respectively. They also present the pdfs estimated by standard Monte Carlo approach. From these figures, it can be seen that the DEMM/PCM approach cannot consistently guarantee convergence of the approximation to the true probability density. Since the DEMM/PCM method failed to produce consistent results for a simple model with independent input probability distributions, this method was not used in other case studies, which involved more complex models or input distributions.

  

Figure 5.4: Evaluation of ECM: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL), over a period of one day, resulting from 6 uncertain parameters

  

Figure 5.5: Evaluation of ECM: uncertainty in the cumulative amount of PERC metabolized, over a period of one day, resulting from 6 uncertain parameters

A comparison of the pdfs of the AUCL, AUCL, and CML dose surrogates, obtained using 2nd and 3rd order ECM and Monte Carlo method, for the six uncertain parameter case is shown in Figures 5.4 and 5.5. The pdfs estimated by both 2nd and 3rd order DEM/ECM appear to agree well with each other and with the pdfs estimated using Monte Carlo simulation. Since the differences between the 2nd order and 3rd order approximations do not appear to be significant, the approximation is judged to have converged, and uncertainty analysis has not been performed with higher order approximations.

Although the pdfs for all six input parameters considered in the first stage of the uncertainty analysis are log-normally distributed, the shapes of the dose surrogate pdfs vary considerably, from the near symmetrical pdf for AUCA to the highly skewed pdfs for AUCL and CML. In all the cases, however, the ECM method produced results very close to those of the Monte Carlo method, but required substantially fewer model runs.

5.1.4.2 Stage II (Eleven Uncertain Parameters): Evaluation of ECM, Regression Based SRSM and LHS

  

Figure 5.6: Evaluation of ECM: uncertainty in the cumulative amount of PERC metabolized, over a period of one day, resulting from 11 uncertain parameters

  

Figure 5.7: Evaluation of ECM: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL), over a period of one day, resulting from 11 uncertain parameters

In the second stage, the uncertainty in compartmental volume proportions are included in the analysis, resulting in eleven uncertain parameters. These five additional parameters are described by a set of mutually dependent Dirichlet distributed random variables. The number of simulations required for the 2nd and 3rd order DEM/ECM are 78, and 364, compared to 100,000 Monte Carlo simulations used to generate the dose surrogate pdfs. Figures 5.6 and 5.7 show the comparison of the pdfs of the CML, AUCA, and the AUCL dose surrogates. Again, The pdfs estimated by both 2nd and 3rd order ECM appear to agree well with each other and with the pdfs estimated using Monte Carlo simulation.

  

Figure 5.8: Evaluation of SRSM (regression based) and LHS: uncertainty in the the CML resulting from 11 uncertain parameters

  

Figure 5.9: Evaluation of SRSM (regression based) and LHS: uncertainty in (a) AUCL and in (b) AUCA (Figure b) resulting from 11 uncertain parameters

The number of simulations used for the 2nd and 3rd order SRSM (regression based SRSM) are 130 and 600, respectively. On the other hand, 1,000 Latin Hypercube samples, and 100,000 Monte Carlo simulations used to generate the dose surrogate pdfs. For this case study, both the ECM and the regression based methods resulted in close approximations, indicating that for simple systems, the ECM method is preferable to the regression based SRSM. The pdfs estimated by both 2nd and 3rd order SRSM appear to agree well with each other and with the pdfs estimated using Monte Carlo simulation. Further, the SRSM approximations have a closer agreement with the Monte Carlo simulation results than the results from the Latin Hypercube sampling method. The results indicate that the SRSM is computationally significantly more efficient than the standard Monte Carlo and more accurate than the Latin Hypercube sampling method.

5.1.4.3 Evaluation of SRSM-ADIFOR

In order to apply the SRSM-ADIFOR, the PERC PBPK model has been re-implemented in standard FORTRAN and that model was used in conjunction with the ADIFOR system to obtain the derivative code. The derivative model was then run at a set of selected sample points for the inputs. The model outputs and derivatives obtained at those points were then used to approximate the pdfs of the three dose surrogates. The number of simulations used for the SRSM-ADIFOR approach for the third order was 70, compared to 600 simulations for the regression based SRSM.

  

Figure 5.10: Evaluation of SRSM-ADIFOR: uncertainty in the cumulative amount of PERC metabolized, over a period of one day, resulting from 11 uncertain parameters

  

Figure 5.11: Evaluation of SRSM-ADIFOR: uncertainty in the area under the PERC concentration-time curve in (a) arterial blood (AUCA) and in (b) liver (AUCL), over a period of one day, resulting from 11 uncertain parameters