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Accounting for inter-correlation between enzyme abundance: a simulation study to assess implications on global sensitivity analysis within physiologically-based pharmacokinetics


Physiologically based pharmacokinetic (PBPK) models often include several sets of correlated parameters, such as organ volumes and blood flows. Because of recent advances in proteomics, it has been demonstrated that correlations are also present between abundances of drug-metabolising enzymes in the liver. As the focus of population PBPK has shifted the emphasis from the average individual to theoretically conceivable extremes, reliable estimation of the extreme cases has become paramount. We performed a simulation study to assess the impact of the correlation between the abundances of two enzymes on the pharmacokinetics of drugs that are substrate of both, under assumptions of presence or lack of such correlations. We considered three semi-physiological models representing the cases of: (1) intravenously administered drugs metabolised by two enzymes expressed in the liver; (2) orally administered drugs metabolised by CYP3A4 expressed in the liver and gut wall; (3) intravenously administered drugs that are substrates of CYP3A4 and OATP1B1 in the liver. Finally, the impact of considering or ignoring correlation between enzymatic abundances on global sensitivity analysis (GSA) was investigated using variance based GSA on a reduced PBPK model for repaglinide, substrate of CYP3A4 and CYP2C8. Implementing such correlations can increase the confidence interval for population pharmacokinetic parameters (e.g., AUC, bioavailability) and impact the GSA results. Ignoring these correlations could lead to the generation of implausible parameters combinations and to an incorrect estimation of pharmacokinetic related parameters. Thus, known correlations should always be considered in building population PBPK models.

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The authors would like to thank Eleanor Savill for her assistance in the submission of the manuscript.

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Correspondence to Nicola Melillo.

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Amin Rostami-Hodjegan is an employee of Certara UK Limited (Simcyp Division).

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The differential equations of the model represented in Fig. 1b are reported in (5). The model describes the bioavailability of a generic drug following oral administration. Our purpose was to derive analytically the expression of the \(F_{oral}\) following a bolus in the compartment representing the small intestine.

The solution of \(x_{lum}\) is the one of a single compartment with two linear clearances, so:

$$x_{lum} = D_{0} e^{{ - \left( {k_{t} + k_{a} } \right)t}}$$

with \(D_{0}\) the dose. It is possible to substitute the expression of \(x_{lum}\) in the equation representing the dynamics of \(x_{ent}\) in the equation system (5). By multiplying \({ \exp }\left( {\left( {Q_{ent} + CL_{ent} } \right)/V_{ent} \cdot t} \right)\) in both the sides of that equation, is possible to derive the expressions below and thus the analytical solution of \(x_{ent}.\)

$$\frac{d}{dt}\left( {x_{ent} e^{{\left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)t}} } \right) = D_{0} k_{a} e^{{\left( { - \left( {k_{t} + k_{a} } \right)t + \left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)t} \right)}}$$
$$x_{ent} e^{{\left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)t}} = D_{0} k_{a} \mathop \int \limits_{0}^{t} e^{{\left( { - \left( {k_{t} + k_{a} } \right)t + \left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)t} \right)}} dt$$
$$x_{ent} = \frac{{D_{0} k_{a} }}{{\left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right) - \left( {k_{t} + k_{a} } \right)}}\left[ {e^{{ - \left( {k_{t} + k_{a} } \right)t}} - e^{{ - \left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)t}} } \right]$$

For readability purpose, let us define the following variables.

$$\begin{aligned} k_{lum} & = \left( {k_{t} + k_{a} } \right) \\ k_{ent} &= \left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right) \\ k_{liv} &= \left( {\frac{{Q_{liv,ven} }}{{V_{liv} }} + \frac{{CL_{liv} }}{{V_{liv} }}} \right) \\ \psi_{1} &= \frac{{D_{0} k_{a} }}{{k_{ent} - k_{lum} }} \end{aligned}$$

Now, by substituting the \(x_{ent}\) expression in \(x_{liv}\) differential equation and multiplying both equation sides for \({ \exp }\left( {k_{liv} \cdot t} \right)\), is possible to derive the analytical solution of \(x_{liv}\).

$$\frac{d}{dt}\left( {x_{liv} e^{{\left( {k_{liv} t} \right)}} } \right) = \frac{{Q_{ent} }}{{V_{ent} }}\psi_{1} \left( {e^{{\left( { - k_{lum} t + k_{liv} t} \right)}} - e^{{\left( { - k_{ent} t + k_{liv} t} \right)}} } \right)$$
$$x_{liv} e^{{k_{liv} t}} = \frac{{Q_{ent} }}{{V_{ent} }}\psi_{1} \mathop \int \limits_{0}^{t} \left( {e^{{\left( { - k_{lum} t + k_{liv} t} \right)}} - e^{{\left( { - k_{ent} t + k_{liv} t} \right)}} } \right)dt$$
$$x_{liv} = \frac{{Q_{ent} }}{{V_{ent} }}\psi_{1} \left[ {\frac{1}{{k_{liv} - k_{lum} }} \left( {e^{{ - k_{lum} t}} - e^{{ - k_{liv} t}} } \right) - \frac{1}{{k_{liv} - k_{ent} }}\left( {e^{{ - k_{ent} t}} - e^{{ - k_{liv} t}} } \right)} \right]$$

Now, by substituting \(x_{liv}\) in the last differential equation of system (5), is possible to derive directly the analytical solution of \(x_{sys}.\)

$$x_{sys} = \frac{{Q_{liv,ven} }}{{V_{liv} }}\frac{{Q_{ent} }}{{V_{ent} }}\psi_{1} \mathop \int \limits_{0}^{t} \left[ {\frac{1}{{k_{liv} - k_{lum} }} \left( {e^{{ - k_{lum} t}} - e^{{ - k_{liv} t}} } \right) - \frac{1}{{k_{liv} - k_{ent} }}\left( {e^{{ - k_{ent} t}} - e^{{ - k_{liv} t}} } \right)} \right]dt$$
$$x_{sys} = \frac{{Q_{liv,ven} }}{{V_{liv} }}\frac{{Q_{ent} }}{{V_{ent} }}\psi_{1} \left[ {\frac{1}{{k_{liv} - k_{lum} }}\left( {\frac{1}{{k_{lum} }}\left( {1 - e^{{ - k_{lum} t}} } \right) - \frac{1}{{k_{liv} }}\left( {1 - e^{{ - k_{liv} t}} } \right)} \right) - \frac{1}{{k_{liv} - k_{ent} }}\left( {\frac{1}{{k_{ent} }}\left( {1 - e^{{ - k_{ent} t}} } \right) - \frac{1}{{k_{liv} }}\left( {1 - e^{{ - k_{liv} t}} } \right)} \right)} \right]$$

Finally, it is possible to derive \(F_{oral}.\)

$$\begin{aligned} F_{oral} & = \mathop {\lim }\limits_{t \to + \infty } \frac{{x_{sys} }}{{D_{0} }} = \frac{{k_{a} \left( {Q_{ent} /V_{ent} } \right)\left( {Q_{liv,ven} /V_{liv} } \right)}}{{k_{ent} - k_{lum} }}\left[ {\frac{1}{{k_{liv} - k_{lum} }}\left( {\frac{1}{{k_{lum} }} - \frac{1}{{k_{liv} }}} \right) - \frac{1}{{k_{liv} - k_{ent} }}\left( {\frac{1}{{k_{ent} }} - \frac{1}{{k_{liv} }}} \right) } \right] \\ F_{oral} & = \frac{{k_{a} \left( {Q_{ent} /V_{ent} } \right)\left( {Q_{liv,ven} /V_{liv} } \right)}}{{k_{ent} - k_{lum} }} \cdot \frac{{k_{ent} - k_{lum} }}{{k_{lum} k_{ent} k_{liv} }} \\ F_{oral} & = \frac{{k_{a} \cdot \left( {Q_{ent} /V_{ent} } \right) \cdot \left( {Q_{liv,ven} /V_{liv} } \right)}}{{(k_{a} + k_{t} )\left( {\frac{{Q_{ent} }}{{V_{ent} }} + \frac{{CL_{ent} }}{{V_{ent} }}} \right)\left( {\frac{{Q_{liv,ven} }}{{V_{liv} }} + \frac{{CL_{liv} }}{{V_{liv} }}} \right) }} \\ \end{aligned}$$

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Melillo, N., Darwich, A.S., Magni, P. et al. Accounting for inter-correlation between enzyme abundance: a simulation study to assess implications on global sensitivity analysis within physiologically-based pharmacokinetics. J Pharmacokinet Pharmacodyn 46, 137–154 (2019).

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  • Enzymes
  • Correlation
  • Simulation
  • Physiologically based pharmacokinetics
  • Global sensitivity analysis