Exposure reconstruction using a physiologically based toxicokinetic model with cumulative amount of metabolite in urine: a case study of trichloroethylene inhalation

Abstract

The use of a physiologically based toxicokinetic (PBTK) model to reconstruct chemical exposure using human biomonitoring data, urinary metabolites in particular, has not been fully explored. In this paper, the trichloroethylene (TCE) exposure dataset by Fisher et al. (Toxicol Appl Pharm 152:339–359, 1998) was reanalyzed to investigate this new approach. By treating exterior chemical exposure as an unknown model parameter, a PBTK model was used to estimate exposure and model parameters by measuring the cumulative amount of trichloroethanol glucuronide (TCOG), a metabolite of TCE, in voided urine and a single blood sample of the study subjects by Markov chain Monte Carlo (MCMC) simulations. An estimated exterior exposure of 0.532 mg/l successfully reconstructed the true inhalation concentration of 0.538 mg/l with a 95% CI (0.441–0.645) mg/l. Based on the simulation results, a feasible urine sample collection period would be 12–16 h after TCE exposure, with blood sampling at the end of the exposure period. Given the known metabolic pathway and exposure duration, the proposed computational procedure provides a simple and reliable method for environmental (occupational) exposure and PBTK model parameter estimation, which is more feasible than repeated blood sampling.

Appendix

1.1 Compartment TCE concentrations

Index the fat, slowly perfused (e.g., muscle), richly perfused, and liver compartments of a 4-compartment PBTK model by k = f, s, r, l, respectively. According to the law of mass-balance, the rate of change of the amount of TCE within a compartment equals the amount of unit time influx of TCE subtracted by efflux. The PBTK model is thus constructed based on the rate changes of TCE for the non-liver compartments

$$ \frac{{{\text{d}}A_{k}^{\text{TCE}} (t)}}{{{\text{d}}t}} = Q_{k} \times \left( {C_{\text{art}}^{\text{TCE}} (t) - \frac{{A_{k}^{\text{TCE}} (t)}}{{V_{k} \times p_{k} }}} \right),\;\;\;\;k = f,s,r $$

(4)

and the rate change for the liver compartment because a certain amount of TCE is metabolized

$$ \frac{{{\text{d}}A_{\text{l}}^{\text{TCE}} (t)}}{{{\text{d}}t}} = Q_{4} \times \left( {C_{\text{art}}^{\text{TCE}} (t) - \frac{{A_{\text{l}}^{\text{TCE}} (t)}}{{V_{\text{l}} \times p_{\text{l}} }}} \right) - \frac{{V_{\max }^{\text{TCE}} \frac{{A_{\text{l}}^{\text{TCE}} (t)}}{{V_{\text{l}} \times p_{\text{l}} }}}}{{K_{\text{m}}^{\text{TCE}} + \frac{{A_{\text{l}}^{\text{TCE}} (t)}}{{V_{\text{l}} \times p_{\text{l}} }}}} - k_{\text{f}}^{\text{TCE}} \times \frac{{A_{\text{l}}^{\text{TCE}} (t)}}{{V_{\text{l}} \times p_{\text{l}} }} \times V_{\text{l}} , $$

(5)

where

$$ C_{\text{art}}^{\text{TCE}} (t) = \left( {Q_{\text{c}} C_{\text{ven}}^{\text{TCE}} (t) + Q_{\text{alv}} C_{\text{inh}} \left( t \right)} \right)/\left( {Q_{\text{c}} + Q_{\text{alv}} /p_{\text{b}} } \right) $$

(6)

$$ C_{\text{ven}}^{\text{TCE}} (t) = \frac{{Q_{\text{f}} \frac{{A_{\text{f}}^{\text{TCE}} (t)}}{{V_{\text{f}} \times p_{\text{f}} }} + Q_{\text{s}} \frac{{A_{\text{s}}^{\text{TCE}} (t)}}{{V_{\text{s}} \times p_{\text{s}} }} + Q_{\text{r}} \frac{{A_{\text{r}}^{\text{TCE}} (t)}}{{V_{\text{r}} \times p_{\text{r}} }} + Q_{\text{l}} \frac{{A_{\text{l}}^{\text{TCE}} (t)}}{{V_{\text{l}} \times p_{\text{l}} }}}}{{Q_{\text{c}} }} $$

(7)

and C ^TCE_art (t) (C ^TCE_ven (t)) (mg/l) is the arterial (venous) blood TCE concentration, C _inh(t) (mg/l) is the inhalation TCE concentration, C ^TCE _k (t) (mg/l) is the TCE concentration in the kth compartment at time t, Q _alv (l/h) is the alveolar ventilation, and Q _k (l/h), V _k (l) and p _k are the blood flow rate, volume, and tissue–blood partition coefficient of the kth compartment, respectively. In addition, p _b is the blood–air partition coefficient, Q _c (l/h) is the cardiac flow, V ^TCE_max (mg/h) is the maximum rate of metabolism, and K ^TCE_m (mg/l) is the Michaelis–Menten kinetic constant of TCE. The conversion from C ^TCE_ven (t) to C ^TCE_art (t) that involves the blood–air partition coefficient P _b does not need an individual’s lung lumen volume. Therefore, a lung compartment is not necessary in this case.

1.2 Mass-balance equations of the metabolites TCOH and TCOG

Let the cumulative amount of the total metabolites in the liver at time t be A ^TCE(t) and the cumulative amounts of the metabolite TCOH (TCOG) in the liver and in the voided urine at time t be A ^TCOH(t)(A ^TCOG(t)) and A ^TCOG_u (t), respectively. The following mass balance equations mainly follow those of Clewell et al. (2000):

$$ \frac{{{\text{d}}A^{\text{TCE}} \left( t \right)}}{{{\text{d}}t}} = \frac{{V_{\max }^{\text{TCE}} \frac{{A^{\text{TCE}} \left( t \right)}}{{V_{\text{l}}\,\times\,p_{\text{l}} }}}}{{K_{\text{m}}^{\text{TCE}} + \frac{{A^{\text{TCE}} \left( t \right)}}{{V_{\text{l}}\,\times\,p_{\text{l}} }}}}, $$

(8)

$$ \frac{{{\text{d}}A^{\text{TCOH}} (t)}}{{{\text{d}}t}} = p_{\text{TCOH}}^{\text{TCE}} \times \frac{{{\text{d}}A^{\text{TCE}} (t)}}{{{\text{d}}t}} \times \frac{{{\text{MW}}_{\text{TCOH}} }}{{{\text{MW}}_{\text{TCE}} }} - \frac{{V_{{\max \_{\text{G}}}}^{\text{TCOH}} \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}}{{K_{{{\text{m}}\_{\text{G}}}}^{\text{TCOH}} + \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}} - \frac{{V_{{\max \_{\text{A}}}}^{\text{TCOH}} \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}}{{K_{{{\text{m}}\_{\text{A}}}}^{\text{TCOH}} + \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}} $$

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$$ \frac{{{\text{d}}A^{\text{TCOG}} (t)}}{{{\text{d}}t}} = \frac{{V_{\max \_G}^{\text{TCOH}} \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}}{{K_{{{\text{m}}\_{\text{G}}}}^{\text{TCOH}} + \frac{{A^{\text{TCOH}} (t)}}{{{\text{VD}}^{\text{TCOH}} }}}} \times \frac{{{\text{MW}}_{\text{TCOG}} }}{{{\text{MW}}_{\text{TCOH}} }} - k_{\text{u}}^{\text{TCOG}} \times A^{\text{TCOG}} (t) - k_{\text{e}}^{\text{TCOG}} \times A^{\text{TCOG}} (t) \, $$

(10)

and

$$ \frac{{{\text{d}}A_{\text{u}}^{\text{TCOG}} (t)}}{{{\text{d}}t}} = k_{\text{u}}^{\text{TCOG}} \times A^{\text{TCOG}} (t) $$

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where p ^TCE_TCOH is the proportion of TCE in liver that is converted to TCOH, k ^TCOG_u (k ^TCOG_e ) is the urinary (biliary) excretion rate for TCOG in the liver, V ^TCOH_{max_G} (V ^TCOH_{max_A} ) is the capacity for glucuronidation (oxidation) of TCOH to TCOG (TCA) (mg/h), K ^TCOH_{m_G} (K ^TCOH_{m_A} ) is the affinity for glucuronidation (oxidation) of TCOH to TCOG (TCA), VD^TCOH (kg) is the TCOH distribution volume, and MW _j is the molecular weight of chemical j.

1.3 Rescale the model parameters

The vector of the model and kinetic parameters is

$$ {\varvec{\Uptheta}} = \left( \begin{gathered} Q_{\text{c}} ,Q_{\text{rc}} ,Q_{\text{lc}} ,Q_{\text{sc}} ,Q_{\text{fc}} ,V_{\text{rc}} ,V_{\text{lc}} ,V_{\text{sc}} ,V_{\text{fc}} ,p_{\text{r}} ,p_{\text{l}} ,p_{\text{s}} ,p_{\text{f}} ,p_{\text{b}} ,p, \hfill \\ V_{{\max \_{\text{c}}}}^{\text{TCE}} ,K_{\text{m}}^{\text{TCE}} ,V_{{\max \_{\text{A}}\_{\text{c}}}}^{\text{TCOH}} ,K_{{{\text{m}}\_{\text{A}}}}^{\text{TCOH}} ,V_{{\max \_{\text{G}}\_{\text{c}}}}^{\text{TCOH}} ,K_{{{\text{m}}\_{\text{G}}}}^{\text{TCOH}} ,k_{\text{ec}}^{\text{TCOG}} ,k_{\text{uc}}^{\text{TCOG}} ,k_{\text{fc}} ,{\text{VD}}^{\text{TCOH}} \hfill \\ \end{gathered} \right)^{\prime } $$

The scaling of the parameters is as follows:

${\text{SV}} = {{(V_{\text{fc}} + V_{\text{rc}} + V_{\text{sc}} + V_{\text{lc}} )} \mathord{\left/ {\vphantom {{(V_{\text{fc}} + V_{\text{rc}} + V_{\text{sc}} + V_{\text{lc}} )} {0.894}}} \right. \kern-\nulldelimiterspace} {0.894}},\;\;\;\;V_{\text{f}} = \frac{{V_{\text{fc}} }}{\text{SV}} \times {\text{BW}}/0.92,\;\;\;\;V_{\text{l}} = \frac{{V_{\text{lc}} }}{\text{SV}} \times {\text{BW}},\;\;\;\;V_{\text{r}} = \frac{{V_{\text{rc}} }}{\text{SV}} \times {\text{BW}},\;\;\;\;V_{\text{s}} = \frac{{V_{\text{sc}} }}{\text{SV}} \times {\text{BW}},\;\;\;\;Q_{\text{c}} = Q_{\text{cc}} \times {\text{BW}}^{0.75} ,\;\;\;\;{\text{SQ}} = Q_{\text{fc}} + Q_{\text{rc}} + Q_{\text{sc}} + Q_{\text{lc}} ,\;\;\;\;Q_{\text{f}} = \frac{{Q_{\text{fc}} }}{\text{SQ}}Q_{\text{c}} ,\;\;\;\;Q_{\text{s}} = \frac{{Q_{\text{sc}} }}{\text{SQ}}Q_{\text{c}} ,\;\;\;\;{\text{and}}\;\;\;\;Q_{\text{r}} = \frac{{Q_{\text{rc}} }}{\text{SQ}}Q_{\text{c}} ,\;\;\;\;Q_{\text{l}} = \frac{{Q_{\text{lc}} }}{\text{SQ}}Q_{\text{c}} ,\;\;\;\;p_{\text{TCOH}}^{\text{TCE}} = \frac{1}{1 + p},$ where $Q_{\text{cc}}$ is the cardiac output, V _fc, V _sc, V _rc, V _lc and Q _fc, Q _sc, Q _rc, Q _lc are the scaling coefficients of the volumes and flow rates to the fat, slowly perfused, rapidly perfused, and liver compartments, respectively.