Incorporation of stochastic variability in mechanistic population pharmacokinetic models: handling the physiological constraints using normal transformations

  • Nikolaos TsamandourasEmail author
  • Thierry Wendling
  • Amin Rostami-Hodjegan
  • Aleksandra Galetin
  • Leon Aarons
Original Paper


The utilisation of physiologically-based pharmacokinetic models for the analysis of population data is an approach with progressively increasing impact. However, as we move from empirical to complex mechanistic model structures, incorporation of stochastic variability in model parameters can be challenging due to the physiological constraints that may arise. Here, we investigated the most common types of constraints faced in mechanistic pharmacokinetic modelling and explored techniques for handling them during a population data analysis. An efficient way to impose stochastic variability on the parameters of interest without neglecting the underlying physiological constraints is through the assumption that they follow a distribution with support and properties matching the underlying physiology. It was found that two distributions that arise through transformations of the normal, the logit-normal generalisation and the logistic-normal, are excellent for such an application as not only they can satisfy the physiological constraints but also offer high flexibility during characterisation of the parameters’ distribution. The statistical properties and practical advantages/disadvantages of these distributions for such an application were clearly displayed in the context of different modelling examples. Finally, a simulation study clearly illustrated the practical gains of the utilisation of the described techniques, as omission of population variability in physiological systems parameters leads to a biased/misplaced stochastic model with mechanistically incorrect variance structure. The current methodological work aims to facilitate the use of mechanistic/physiologically-based models for the analysis of population pharmacokinetic clinical data.


PBPK Variability Population pharmacokinetics Constraints Logistic-normal Logit-normal 



N.T. was the recipient of a PhD grant jointly awarded by the University of Manchester and Eli Lilly and Company. A.R-H. is an employee of the University of Manchester and parttime secondee to Simcyp Limited (a Certara Company). The authors would like to acknowledge the discussions and fruitful comments made by Dr Alison Margolskee and by the members of the Centre for Applied Pharmacokinetic Research at the University of Manchester.

Supplementary material

10928_2015_9418_MOESM1_ESM.pdf (321 kb)
Supplementary material 1 (PDF 321 kb)
10928_2015_9418_MOESM2_ESM.pdf (312 kb)
Supplementary material 2 (PDF 312 kb)
10928_2015_9418_MOESM3_ESM.pdf (309 kb)
Supplementary material 3 (PDF 309 kb)
10928_2015_9418_MOESM4_ESM.pdf (445 kb)
Supplementary material 4 (PDF 445 kb)
10928_2015_9418_MOESM5_ESM.pdf (477 kb)
Supplementary material 5 (PDF 476 kb)
10928_2015_9418_MOESM6_ESM.pdf (147 kb)
Supplementary material 6 (PDF 147 kb)
10928_2015_9418_MOESM7_ESM.pdf (377 kb)
Supplementary material 7 (PDF 377 kb)


  1. 1.
    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):48–55PubMedCrossRefGoogle Scholar
  2. 2.
    Tsamandouras N, Dickinson G, Guo Y, Hall S, Rostami-Hodjegan A, Galetin A, Aarons L (2015) Development and application of a mechanistic pharmacokinetic model for simvastatin and its active metabolite simvastatin acid using an integrated population PBPK approach. Pharm Res 32(6):1864–1883. doi: 10.1007/s11095-014-1581-2 PubMedCrossRefGoogle Scholar
  3. 3.
    Aitchison J, Shen SM (1980) Logistic-normal distributions: some properties and uses. Biometrika 67(2):261–272CrossRefGoogle Scholar
  4. 4.
    Lavielle M (2014) Mixed effects models for the population approach. Models, tasks, methods & tools. Chapman & Hall/CRC Biostatistics Series, Boca RatonGoogle Scholar
  5. 5.
    van der Walt JS, Hong Y, Zhang L, Pfister M, Boulton DW, Karlsson MO (2013) A nonlinear mixed effects pharmacokinetic model for dapagliflozin and dapagliflozin 3-o-glucuronide in renal or hepatic impairment. CPT: Pharmacomet Syst Pharmacol 2:e42Google Scholar
  6. 6.
    Gueorguieva I, Aarons L, Rowland M (2006) Diazepam pharamacokinetics from preclinical to Phase I using a Bayesian population physiologically based pharmacokinetic model with informative prior distributions in Winbugs. J Pharmacokinet Pharmacodyn 33(5):571–594PubMedCrossRefGoogle Scholar
  7. 7.
    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):485–498PubMedCrossRefGoogle Scholar
  8. 8.
    Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian data analysis, 1st edn. Chapman & Hall, LondonGoogle Scholar
  9. 9.
    Coleman T, Li Y (1996) An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optimiz 6(2):418–445CrossRefGoogle Scholar
  10. 10.
    Aitchison J (1982) The statistical analysis of compositional data. J Roy Stat Soc B Met 44(2):139–177Google Scholar
  11. 11.
    Blei DM, Lafferty JD (2007) A correlated topic model of science. Ann Appl Stat 1(1):17–35CrossRefGoogle Scholar
  12. 12.
    Minka T (2000) Estimating a Dirichlet distribution. Technical report. http://researchmicrosoftcom/en-us/um/people/minka/papers/dirichlet/minka-dirichletpdfGoogle Scholar
  13. 13.
    Read NW, Al-Janabi MN, Holgate AM, Barber DC, Edwards CA (1986) Simultaneous measurement of gastric emptying, small bowel residence and colonic filling of a solid meal by the use of the gamma camera. Gut 27(3):300–308PubMedCentralPubMedCrossRefGoogle Scholar
  14. 14.
    Read NW, Cammack J, Edwards C, Holgate AM, Cann PA, Brown C (1982) Is the transit time of a meal through the small intestine related to the rate at which it leaves the stomach? Gut 23(10):824–828PubMedCentralPubMedCrossRefGoogle Scholar
  15. 15.
    Zhang X, Lionberger RA, Davit BM, Yu LX (2011) Utility of physiologically based absorption modeling in implementing quality by design in drug development. AAPS J 13(1):59–71PubMedCentralPubMedCrossRefGoogle Scholar
  16. 16.
    Peters SA (2008) Evaluation of a generic physiologically based pharmacokinetic model for lineshape analysis. Clin Pharmacokinet 47(4):261–275PubMedCrossRefGoogle Scholar
  17. 17.
    Lartigue S, Bizais Y, Bruley des Varannes S, Murat A, Pouliquen B, Galmiche JP (1994) Inter- and intrasubject variability of solid and liquid gastric emptying parameters. Dig Dis Sci 39(1):109–115PubMedCrossRefGoogle Scholar
  18. 18.
    Jamei M, Turner D, Yang J, Neuhoff S, Polak S, Rostami-Hodjegan A, Tucker G (2009) Population-based mechanistic prediction of oral drug absorption. AAPS J 11(2):225–237PubMedCentralPubMedCrossRefGoogle Scholar
  19. 19.
    Yu LX, Crison JR, Amidon GL (1996) Compartmental transit and dispersion model analysis of small intestinal transit flow in humans. Int J Pharm 140(1):111–118CrossRefGoogle Scholar
  20. 20.
    Gertz M, Tsamandouras N, Säll C, Houston JB, Galetin A (2014) Reduced physiologically-based pharmacokinetic model of repaglinide: impact of OATP1B1 and CYP2C8 genotype and source of in vitro data on the prediction of drug-drug interaction risk. Pharm Res 31(9):2367–2382PubMedCrossRefGoogle Scholar
  21. 21.
    Kawai R, Mathew D, Tanaka C, Rowland M (1998) Physiologically based pharmacokinetics of Cyclosporine A: extension to tissue distribution kinetics in rats and scale-up to human. J Pharmacol Exp Ther 287(2):457–468PubMedGoogle Scholar
  22. 22.
    Price PS, Conolly RB, Chaisson CF, Gross EA, Young JS, Mathis ET, Tedder DR (2003) Modeling interindividual variation in physiological factors used in PBPK models of humans. Crit Rev Toxicol 33(5):469–503PubMedCrossRefGoogle Scholar
  23. 23.
    Jamei M, Marciniak S, Feng K, Barnett A, Tucker G, Rostami-Hodjegan A (2009) The Simcyp® population-based ADME simulator. Expert Opin Drug Metab Toxicol 5(2):211–223PubMedCrossRefGoogle Scholar
  24. 24.
    Wendling T, Ogungbenro K, Pigeolet E, Dumitras S, Woessner R, Aarons L (2014) Model-based evaluation of the impact of furmulation and food intake on the complex oral absorption of mavoglurant in healthy subjects. Pharm Res 32(5):1764–1778. doi: 10.1007/s11095-014-1574-1 PubMedCrossRefGoogle Scholar
  25. 25.
    Zhou H (2003) Pharmacokinetic strategies in deciphering atypical drug absorption profiles. J Clin Pharmacol 43(3):211–227PubMedCrossRefGoogle Scholar
  26. 26.
    Csajka C, Drover D, Verotta D (2005) The use of a sum of inverse Gaussian functions to describe the absorption profile of drugs exhibiting complex absorption. Pharm Res 22(8):1227–1235PubMedCrossRefGoogle Scholar
  27. 27.
    Rowland M, Peck C, Tucker G (2011) Physiologically-based pharmacokinetics in drug development and regulatory science. Annu Rev Pharmacol Toxicol 51(1):45–73PubMedCrossRefGoogle Scholar
  28. 28.
    Nestorov I (2003) Whole body pharmacokinetic models. Clin Pharmacokinet 42(10):883–908PubMedCrossRefGoogle Scholar
  29. 29.
    Brown RP, Delp MD, Lindstedt SL, Rhomberg LR, Beliles RP (1997) Physiological parameter values for physiologically based pharmacokinetic models. Toxicol Ind Health 13(4):407–484PubMedCrossRefGoogle Scholar
  30. 30.
    Hoff PD (2003) Nonparametric modeling of hierarchically exchangeable data. University of Washington Statistics Department, Technical Report 421Google Scholar
  31. 31.
    Huang J, Malisiewicz T (2009) Fitting a hierarchical logistic normal distribution. Technical Report, Carnegie Mellon UniversityGoogle Scholar
  32. 32.
    DiCiccio T, Efron B (1996) Bootstrap confidence intervals. Stat Sci 11(3):189–228CrossRefGoogle Scholar
  33. 33.
    Klotz U, Antonin KH, Bieck PR (1976) Pharmacokinetics and plasma binding of diazepam in man, dog, rabbit, guinea pig and rat. J Pharmacol Exp Ther 199(1):67–73PubMedGoogle Scholar
  34. 34.
    Greenblatt DJ, Allen MD, Harmatz JS, Shader RI (1980) Diazepam disposition determinants. Clin Pharmacol Ther 27(3):301–312PubMedCrossRefGoogle Scholar
  35. 35.
    Rowland M, Leitch D, Fleming G, Smith B (1984) Protein binding and hepatic clearance: discrimination between models of hepatic clearance with diazepam, a drug of high intrinsic clearance, in the isolated perfused rat liver preparation. J Pharmacokinet Biopharm 12(2):129–147PubMedCrossRefGoogle Scholar
  36. 36.
    Sorger PK, Allerheiligen SR, Abernethy DR, et al. (2011) Quantitative and systems pharmacology in the post-genomic era: New approaches to discovering drugs and understanding therapeutic mechanisms. An NIH white paper by the QSP workshop group: 1-48Google Scholar
  37. 37.
    Jusko WJ (2013) Moving from basic toward systems pharmacodynamic models. J Pharm Sci 102(9):2930–2940PubMedCentralPubMedCrossRefGoogle Scholar
  38. 38.
    Wulkersdorfer B, Wanek T, Bauer M, Zeitlinger M, Muller M, Langer O (2014) Using positron emission tomography to study transporter-mediated drug-drug interactions in tissues. Clin Pharmacol Ther 96(2):206–213PubMedCentralPubMedCrossRefGoogle Scholar
  39. 39.
    Plan EL, Elshoff JP, Stockis A, Sargentini-Maier ML, Karlsson MO (2012) Likert pain score modeling: a Markov integer model and an autoregressive continuous model. Clin Pharmacol Ther 91(5):820–828PubMedCrossRefGoogle Scholar
  40. 40.
    Petersson KF, Hanze E, Savic R, Karlsson M (2009) Semiparametric distributions with estimated shape parameters. Pharm Res 26(9):2174–2185PubMedCrossRefGoogle Scholar
  41. 41.
    Krewski D, Wang Y, Bartlett S, Krishnan K (1995) Uncertainty, variability, and sensitivity analysis in physiological pharmacokinetic models. J Biopharm Stat 5(3):245–271PubMedGoogle Scholar
  42. 42.
    Farrar D, Allen B, Crump K, Shipp A (1989) Evaluation of uncertainty in input parameters to pharmacokinetic models and the resulting uncertainty in output. Toxicol Lett 49(2–3):371–385PubMedCrossRefGoogle Scholar
  43. 43.
    Gelman A, Bois F, Jiang J (1996) Physiological pharmacokinetic analysis using population modeling and informative prior distributions. J Am Stat Assoc 91(436):1400–1412CrossRefGoogle Scholar
  44. 44.
    Bois FY (2000) Statistical analysis of Clewell et al. PBPK model of trichloroethylene kinetics. Environ Health Perspect 108(Suppl 2):307–316PubMedCentralPubMedCrossRefGoogle Scholar
  45. 45.
    Yang Y, Xu X, Georgopoulos PG (2010) A Bayesian population PBPK model for multiroute chloroform exposure. J Expo Sci Environ Epidemiol 20(4):326–341PubMedCentralPubMedCrossRefGoogle Scholar
  46. 46.
    Lindberg-Freijs A, Karlsson MO (1994) Dose dependent absorption and linear disposition of cyclosporin a in rat. Biopharm Drug Dispos 15(1):75–86PubMedCrossRefGoogle Scholar
  47. 47.
    Silber HE, Frey N, Karlsson MO (2010) An integrated glucose-insulin model to describe oral glucose tolerance test data in healthy volunteers. J Clin Pharmacol 50(3):246–256PubMedCrossRefGoogle Scholar
  48. 48.
    Bizzotto R, Zamuner S, Mezzalana E, De Nicolao G, Gomeni R, Hooker AC, Karlsson MO (2011) Multinomial logistic functions in Markov chain models of sleep architecture: internal and external validation and covariate analysis. AAPS J 13(3):445–463PubMedCentralPubMedCrossRefGoogle Scholar
  49. 49.
    Gelman A (1995) Method of moments using Monte Carlo simulation. J Comput Graph Stat 4(1):36–54Google Scholar
  50. 50.
    Willmann S, Hohn K, Edginton A, Sevestre M, Solodenko J, Weiss W, Lippert J, Schmitt W (2007) Development of a physiology-based whole-body population model for assessing the influence of individual variability on the pharmacokinetics of drugs. J Pharmacokinet Pharmacodyn 34(3):401–431PubMedCrossRefGoogle Scholar
  51. 51.
    McNally K, Cotton R, Hogg A, Loizou G (2014) Popgen: a virtual human population generator. Toxicology 315:70–85PubMedCrossRefGoogle Scholar
  52. 52.
    Krauß M, Tappe K, Burghaus R, Schuppert A, Kuepfer L, Goerlitz L (2014) Hierarchical Bayesian-PBPK modeling for physiological characterization and extrapolation of patient populations from clinical data. PAGE 23 (2014), Abstr 3151.

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Nikolaos Tsamandouras
    • 1
    Email author
  • Thierry Wendling
    • 1
  • Amin Rostami-Hodjegan
    • 1
    • 2
  • Aleksandra Galetin
    • 1
  • Leon Aarons
    • 1
  1. 1.Centre for Applied Pharmacokinetic Research, Manchester Pharmacy SchoolUniversity of ManchesterManchesterUK
  2. 2.Simcyp Limited, Blades Enterprise CentreSheffieldUK

Personalised recommendations