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Population pharmacokinetic reanalysis of a Diazepam PBPK model: a comparison of Stan and GNU MCSim

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The aim of this study is to benchmark two Bayesian software tools, namely Stan and GNU MCSim, that use different Markov chain Monte Carlo (MCMC) methods for the estimation of physiologically based pharmacokinetic (PBPK) model parameters. The software tools were applied and compared on the problem of updating the parameters of a Diazepam PBPK model, using time-concentration human data. Both tools produced very good fits at the individual and population levels, despite the fact that GNU MCSim is not able to consider multivariate distributions. Stan outperformed GNU MCSim in sampling efficiency, due to its almost uncorrelated sampling. However, GNU MCSim exhibited much faster convergence and performed better in terms of effective samples produced per unit of time.

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  1. Macheras P, Iliadis A (2016) Modeling in pharmacokinetics and pharmacodynamics: homogeneous and heterogeneous approaches. Springer, New York

    Book  Google Scholar 

  2. Tsamandouras N, Rostami-Hodjegan A, Aarons L (2015) Combining the bottom up and top down approaches in pharmacokinetic modelling: fitting PBPK models to observed clinical data. Br J Clin Pharmacol 79(1):48–55.

    Article  CAS  PubMed  Google Scholar 

  3. Clewell HJ, Clewell RA, Andersen ME (2011) Physiologically-based pharmacokinetic (PBPK) modeling and risk assessment. In: Nriagu J (ed) Encyclopedia of environmental health. Elsevier, New York, pp 536–570.

    Chapter  Google Scholar 

  4. Nestorov I (2003) Whole body pharmacokinetic models. Clin Pharmacokinet 42(10):883–908.

    Article  CAS  PubMed  Google Scholar 

  5. Edginton AN, Theil FP, Schmitt W, Willmann S (2008) Whole body physiologically-based pharmacokinetic models: their use in clinical drug development. Expert Opin Drug Metab Toxicol 4(9):1143–1152.

    Article  CAS  PubMed  Google Scholar 

  6. Wendling T, Dumitras S, Ogungbenro K, Aarons L (2015) Application of a Bayesian approach to physiological modelling of mavoglurant population pharmacokinetics. J Pharmacokinet Pharmacodyn 42(6):639657.

    Article  CAS  Google Scholar 

  7. Zhuang X, Lu C (2016) PBPK modeling and simulation in drug research and development. Acta Pharmaceutica Sinica B 6(5):430440.

    Article  Google Scholar 

  8. Gelman A, Bois FY, Jiang JM (1996) Physiological pharmacokinetic analysis using population modeling and informative prior distributions. JASA 91(436):1400–12.

    Article  Google Scholar 

  9. Wakefield J (1996) The Bayesian analysis of population pharmacokinetic models. J Am Stat Assoc 91(433):6275.

    Article  Google Scholar 

  10. Bois FY, Jamei M, Clewell HJ (2010) PBPK modelling of inter-individual variability in the pharmacokinetics of environmental chemicals. Toxicology 278(3):256267.

    Article  CAS  Google Scholar 

  11. Krauss M, Schuppert A (2016) Assessing interindividual variability by Bayesian-PBPK modeling. Drug Dis Today Dis Model 22:15–19.

    Article  Google Scholar 

  12. Krauss M, Tappe K, Schuppert A, Kuepfer L, Goerlitz L (2015) Bayesian population physiologically-based pharmacokinetic (PBPK) approach for a physiologically realistic characterization of interindividual variability in clinically relevant populations. PLoS ONE 10(10):e0139423.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Mezzetti M, Ibrahim JG, Bois FY, Ryan LM, Ngo L, Smith TJ (2003) A Bayesian compartmental model for the evaluation of 1,3-butadiene metabolism. J R Stat Soc Ser C 52:291–305.

    Article  Google Scholar 

  14. Zurlinden TJ, Reisfeld B (2016) Physiologically based modeling of the pharmacokinetics of acetaminophen and its major metabolites in humans using a Bayesian population approach. Eur J Drug Metab Pharmacokinet 41(3):267–80.

    Article  CAS  PubMed  Google Scholar 

  15. Geyer CJ (2011) Introduction to Markov Chain Monte Carlo. In: Brooks S, Gelman A, Jones GL, Meng XL (eds) Handbook of Markov Chain Monte Carlo, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  16. Bois FY (2009) GNU MCSim: Bayesian statistical inference for SBML-coded systems biology models. Bioinformatics 25(11):1453–1454.

    Article  CAS  PubMed  Google Scholar 

  17. Carpenter B, Gelman A, Hoffman MD, Lee D, Goodrich B, Betancourt M, Brubaker M, Guo J, Li P, Riddell A (2017) Stan: a probabilistic programming language. J Stat Softw

    Article  Google Scholar 

  18. Hoffman MD, Gelman A (2014) The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J Mach Learn Res 15:1593–1623

    Google Scholar 

  19. Weber S, Gelman A, Lee D, Betancourt M, Vehtari A, Racine-Poon A (2018) Bayesian aggregation of average data. Ann Appl Stat

    Article  Google Scholar 

  20. Langdon G, Gueorguieva I, Aarons L, Karlsson M (2007) Linking preclinical and clinical whole-body physiologically based pharmacokinetic models with prior distributions in NONMEM. Eur J Clin Pharmacol 63(5):485–498.

    Article  CAS  PubMed  Google Scholar 

  21. Gueorguieva I, Aarons L, Rowland M (2006) Diazepam pharmacokinetics from preclinical to Phase I using a Bayesian population physiological model with informative prior distributions in WINBUGS. J Pharmacokinet Pharmacodyn 33(5):571594.

    Article  CAS  Google Scholar 

  22. Greenblatt DJ, Allen MD, Harmatz JS, Shader RI (1980) Diazepam disposition determinants. Clin Pharmacol Ther 27(3):301312

    Article  Google Scholar 

  23. Gueorguieva II, Nestorov IA, Rowland M (2004) Fuzzy simulation of pharmacokinetic models: case study of whole body physiologically based model of Diazepam. J Pharmacokinet Pharmacodyn 31(3):185213.

    Article  Google Scholar 

  24. Leggett RW, Williams LR, Eckerman KF (1995) A blood circulation model for reference man. Health Phys 69(2):187–201

    Article  CAS  PubMed  Google Scholar 

  25. Nestorov IA (2001) Modelling and simulation of variability and uncertainty in toxicokinetics and pharmacokinetics. Toxicol Lett 120(1–3):411–420

    Article  CAS  PubMed  Google Scholar 

  26. Luecke RH, Wosilait WD, Slikker W Jr, Young JF, Pearce BA (2007) Postnatal growth considerations for PBPK modeling. J Toxicol Environ Health A 70(12):1027–1037.

    Article  CAS  PubMed  Google Scholar 

  27. Mould DR, Upton RN (2013) Basic concepts in population modeling, simulation, and model-based drug development-part 2: introduction to pharmacokinetic modeling methods. CPT Pharmacometrics Syst Pharmacol 2(4):e38.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  28. Stan Development Team (2018) Stan modeling language users guide and reference manual. Version 2(18).

  29. Lewandowski D, Kurowicka D, Joe H (2009) Generating random correlation matrices based on vines and extended onion method. J Multivar Anal 100(9):1989–2001.

    Article  Google Scholar 

  30. Papaspiliopoulos O, Roberts GO, Skld M (2007) A general framework for the parametrization of hierarchical models. Sta Sci 22(1):59–73

    Article  Google Scholar 

  31. Neal RM (2011) MCMC using Hamiltonian dynamics. In: Brooks S, Gelman A, Jones GL, Meng XL (eds) Handbook of Markov Chain Monte Carlo, 2nd edn. CRC Press, Boca Raton, pp 113–162

    Google Scholar 

  32. Monnahan CC, Thorson JT, Branch TA (2018) Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo. Methods Ecol Evol 8:339–348.

    Article  Google Scholar 

  33. Betancourt M (2017) A conceptual introduction to Hamiltonian Monte Carlo. arXiv:1701.02434v2

  34. Betancourt M (2016) Diagnosing suboptimal cotangent disintegrations in Hamiltonian Monte Carlo. arXiv:1604.00695

  35. Margossian C, Gillespie B (2017) Differential equation based models in Stan. Stan Conference.

    Article  Google Scholar 

  36. Bois FY, Maszle DR (1997) MCSim: a simulation program. J Stat Softw

    Article  Google Scholar 

  37. Hindmarsh AC, Brown PN, Grant KE, Lee SL, Serban R, Shumaker DE, Woodward CS (2005) SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans Math Soft 31(3):363–96

    Article  Google Scholar 

  38. Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2001) Bayesian data analysis, 3rd edn. CRC Press, Boca Raton, pp 284–287

    Google Scholar 

  39. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472

    Article  Google Scholar 

  40. Betancourt MJ, Girolami M (2013) Hamiltonian Monte Carlo for hierarchical models. arXiv:1312.0906

  41. Garcia RI, Ibrahim JG, Wambaugh JF, Kenyon EM, Setzer RW (2015) Identifiability of PBPK models with applications to dimethylarsinic acid exposure. J Pharmacokinet Pharmacodyn 42(6):591–609.

    Article  CAS  PubMed  Google Scholar 

  42. Yates JWT (2006) Structural identifiability of physiologically based pharmacokinetic models. J Pharmacokinet Pharmacodyn 33(4):421–39.

    Article  CAS  PubMed  Google Scholar 

  43. Hsieh NH, Reisfeld B, Bois F, Chiu WA (2018) Applying a global sensitivity analysis workflow to improve the computational efficiencies in physiologically-based pharmacokinetic modeling. Front Pharmacol 9:588.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  44. Carpenter B, Hoffman MD, Brubaker M, Lee D, Li P, Betancourt M (2015) AThe Stan Math Library: reverse-mode automatic differentiation in C++. arXiv:1509.07164

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We would like to thank Gueorgieva I. for granting us access to the Diazepam data. H. Sarimveis and F. Bois acknowledge financial support by OpenRiskNet (Grant Agreement 731075), a project funded by the European Commission under the Horizon 2020 Programme. Periklis Tsiros acknowledges financial support by the NTUA internal reward Programme Numbered 95/0085.

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Correspondence to Haralambos Sarimveis.

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Appendix 1: Coefficients of organ weight polynomials for humans

See Tables 4 and 5.

Table 4 Coefficients of organ weight polynomials for male humans [26]
Table 5 Coefficients of organ weight polynomials for female humans [26]

Appendix 2: Posterior estimates

See Tables 6 and 7.

Table 6 Posterior estimates obtained with Stan
Table 7 Posterior estimates obtained with GNU MCSim

Appendix 3: Impact of LKJ prior on computational efficiency

See Table 8.

Table 8 Computational efficiency for different values of the shape parameter a of the LKJ prior

Appendix 4: Comparison of the univariate models using noninformative priors

See Figs. 13 and 14.

Fig. 13
figure 13

Comparison of individual parameter estimates between GNU MCSim and Stan univariate models using noninformative priors

Fig. 14
figure 14

Comparison of population estimates between GNU MCSim and Stan univariate models using noninformative priors. The red crosses represent the mean population estimates while the blue crosses represent the population variance estimates

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Tsiros, P., Bois, F.Y., Dokoumetzidis, A. et al. Population pharmacokinetic reanalysis of a Diazepam PBPK model: a comparison of Stan and GNU MCSim. J Pharmacokinet Pharmacodyn 46, 173–192 (2019).

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