Abstract
The structural complexity of a PBPK model is usually accompanied with significant uncertainty in estimating its input parameters. In the last decade, the global sensitivity analysis, which accounts for the variability of all model input parameters simultaneously as well as their correlations, has gained a wide attention as a powerful probing technique to identify and control biological model uncertainties. However, the current sensitivity analysis techniques used in PBPK modeling often neglect the correlation between these input parameters. We introduce a new strategy in the PBPK modeling field to investigate how the uncertainty and variability of correlated input parameters influence the outcomes of the drug distribution process based on a model we recently developed to explain and predict drug distribution in tissues expressing P-glycoprotein (P-gp). As direct results, we will also identify the most important input parameters having the largest contribution to the variability and uncertainty of model outcomes. We combined multivariate random sampling with a ranking procedure. Monte–Carlo simulations were performed on the PBPK model with eighteen model input parameters. Log-normal distributions were assumed for these parameters according to literature and their reported correlations were also included. A multivariate sensitivity analysis was then performed to identify the input parameters with the greatest influence on model predictions. The partial rank correlation coefficients (PRCC) were calculated to establish the input–output relationships. A moderate variability of predicted Clast and Cmax was observed in liver, heart and brain tissues in the presence or absence of P-gp activity. The major statistical difference in model outcomes of the predicted median values has been obtained in brain tissue. PRCC calculation confirmed the importance for a better quantitative characterisation of input parameters related to the passive diffusion and active transport of the unbound drug through the blood-tissue membrane in heart and brain. This approach has also identified as important input parameters those related to the drug metabolism for the prediction of model outcomes in liver and plasma. The proposed Monte–Carlo/PRCC approach was aimed to address the effect of input parameters correlation in a PBPK model. It allowed the identification of important input parameters that require additional attention in research for strengthening the physiological knowledge of drug distribution in mammalian tissues expressing P-gp, thereby reducing the uncertainty of model predictions.
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Acknowledgments
This work has been supported by FRSQ through PhD scholarship held by Frédérique Fenneteau. FCAR and MITACS are also acknowledged for their support. Financial support of the NSERC is held by Dr. Fahima Nekka.
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Appendix
Appendix
The general framework of the PBPK model retained at the end of the modeling process [23] is presented in Fig. 10. This PBPK model comprises a mechanistic transport based (MTB) model for heart and brain tissues, and a well-stirred WS model for all other tissues. The MTB model is illustrated in Fig. 11. When applied to heart tissue, the MTB model involves apparent passive diffusion and P-gp-mediated transports. For brain, the MTB model involves apparent passive diffusion, P-gp mediated transports and a potential additional efflux transport. However, this assumption should be further studied through a sensitivity analysis and additional in vitro and in vivo experiments.
The parameters used in the figures and equations presented in this section refer to concentration (C), volume (V), blood flow to tissue (Q), tissue–plasma partition coefficient (Ptp), blood–plasma ratio (BP), unbound fraction of drug (fu), clearance (CL), and permeability-surface area product (PSA). The subscripts refer to cardiac output (co), tissue (t), kidneys (k), spleen (sp), gut (g), plasma (p), liver (li), lung (lg), heart (h), arterial blood (ab), venous blood (vb), blood in equilibrium with tissue (bl), venous blood living tissue (v, t), unbound fraction (u), bound fraction (b), intracellular water (iw), extracellular water (ew), neutral lipid (nl), neutral phospholipid (np), and microsomal binding (mic). Some subscripts refer to active transport processes, such as P-gp mediated transport (P-gp), as well as other transporters (OT) such as influx transporters (in,OT) and additional efflux transporters (out, OT).
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Extravascular space of brain or heart
$$ {\text{V}}_{\text{t}} \times {\frac{{{\text{dC}}_{\text{t}} }}{\text{dt}}} = {\text{PSA}}_{\text{t}} \times \left({{\text{fu}}_{\text{p}} \times {\text{C}}_{\text{p,t}} - {\text{fu}}_{\text{t}} \times {\text{C}}_{\text{t}} } \right) - {\text{fu}}_{\text{t}} \times {\text{C}}_{\text{t}} \times \left( {{\text{CL}}_{\text{Pgp,t}} + {\text{CL}}_{\text{out,OT}} } \right) $$(4) -
Vascular space of brain and heart
$$ \begin{gathered} {\text{V}}_{\text{bl,t}} \times {\frac{{{\text{dC}}_{\text{v, t}} }}{\text{dt}}} = {\text{Q}}_{\text{t}} \times \left( {{\text{ C}}_{\text{ab}} - {\text{C}}_{\text{v,t}} } \right) + {\text{PSA}}_{\text{t}} \times \left( {{\text{ fu}}_{\text{t}} \times {\text{C}}_{\text{t}} - {\text{fu}}_{\text{p}} \times {\text{C}}_{\text{p,t }} } \right) \\ + {\text{fu}}_{\text{t}} \times {\text{C}}_{\text{t}} \times \left( {{\text{CL}}_{\text{Pgp,t}} + {\text{CL}}_{\text{out,OT}} } \right) \\ \end{gathered} $$(5)where CLout,OT represents the efflux clearance of the drug due to additional transporter. This parameter was adjusted to experimental data for brain tissue, whereas it was set to 0 for heart tissue.
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Other non-eliminating tissues
$$ {\text{V}}_{\text{t}} \times {\frac{{{\text{dC}}_{\text{t}} }}{\text{dt}}} = {\text{Q}}_{\text{t}} \times \left( {{\text{C}}_{\text{ab}} - {\text{C}}_{\text{v,t}} } \right) $$(6) -
Eliminating tissues (liver)
$$ \begin{gathered} {\text{V}}_{\text{li}} \times {\frac{{{\text{dC}}_{\text{li}} }}{\text{dt}}} = \left( {{\text{Q}}_{\text{li}} - {\text{Q}}_{\text{sp}} - {\text{Q}}_{\text{g}} } \right) \times {\text{C}}_{\text{ab}} + {\text{Q}}_{\text{spl}} \times {\text{C}}_{\text{v,spl}} + {\text{Q}}_{\text{g}} \times {\text{C}}_{\text{v,g}} \\ - {\frac{{{\text{fu}}_{\text{p}} }}{{{\text{fu}}_{\text{mic}} }}}{\text{CL}}_{\text{int}} \cdot {\text{C}}_{\text{v,li}} - {\text{Q}}_{\text{li}} \times {\text{C}}_{\text{v,li}} \\ \end{gathered} $$(7)where CLint = Vmax/Km × NCYP450 and fumic is the fraction unbound to liver microsomes.
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Arterial blood
$$ {\text{V}}_{\text{ab}} \times {\frac{{{\text{dC}}_{\text{ab}} }}{\text{dt}}} = {\text{Q}}_{\text{co}} \times \left( {{\text{C}}_{\text{v,lg}} - {\text{C}}_{\text{ab}} } \right) $$(8) -
Venous blood
$$ {\text{V}}_{\text{vb}} \times {\frac{{{\text{dC}}_{\text{vb}} }}{\text{dt}}} = \sum\limits_{\text{t}} {\left( {{\text{Q}}_{\text{t}} \times {\text{C}}_{\text{v,t}} } \right)} - {\text{Q}}_{\text{co}} \times {\text{C}}_{\text{vb}} $$(9) -
Lung
$$ {\text{V}}_{ \lg } \times {\frac{{{\text{dC}}_{ \lg } }}{\text{dt}}} = {\text{Q}}_{\text{co}} \times \left( {{\text{C}}_{\text{vb}} - {\text{C}}_{\text{v,lg}} } \right) $$(10)with
$$ {\text{C}}_{\text{v,x}} = {\frac{{{\text{C}}_{\text{x}} \times {\text{BP}}}}{{{\text{P}}_{\text{tp,x}} }}}\quad {\text{where x stands for t}},{\text{sp}},{\text{ li and lg}} $$(11)
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Fenneteau, F., Li, J. & Nekka, F. Assessing drug distribution in tissues expressing P-glycoprotein using physiologically based pharmacokinetic modeling: identification of important model parameters through global sensitivity analysis. J Pharmacokinet Pharmacodyn 36, 495–522 (2009). https://doi.org/10.1007/s10928-009-9134-8
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DOI: https://doi.org/10.1007/s10928-009-9134-8