Pharmacokinetic Studies Described by Stochastic Differential Equations: Optimal Design for Systems with Positive Trajectories

  • Valerii V. Fedorov
  • Sergei L. Leonov
  • Vyacheslav A. Vasiliev
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


In compartmental pharmacokinetic (PK) modelling, ordinary differential equations (ODE) are traditionally used with two sources of randomness: measurement error and population variability. In this paper we focus on intrinsic (within-subject) variability modelled with stochastic differential equations (SDE), and consider stochastic systems with positive trajectories which are important from a physiological perspective. We derive mean and covariance functions of solutions of SDE models, and construct optimal designs, i.e. find sampling schemes that provide the most precise estimation of model parameters under cost constraints.


Optimal Design Covariance Function Stochastic Differential Equation Stochastic System Ordinary Differential Equation 
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  1. Anisimov, V., V. Fedorov, and S. Leonov (2007). Optimal design of pharmacokinetic studies described by stochastic differential equations. In J. López-Fidalgo, J. Rodriguez-Diaz, and B. Torsney (Eds.), mODa 8 - Advances in Model-Oriented Design and Analysis, pp. 9–16. Physica-Verlag, Heidelberg.CrossRefGoogle Scholar
  2. Fedorov, V., R. Gagnon, S. Leonov, and Y. Wu (2007). Optimal design of experiments in pharmaceutical applications. In A. Dmitrienko, C. Chuang-Stein, and R. D’Agostino (Eds.), Pharmaceutical Statistics Using SAS. A Practical Guide, pp. 151–195. SAS Press, Cary, NC.Google Scholar
  3. Fedorov, V. and P. Hackl (1997). Model-Oriented Design of Experiments. Springer-Verlag , New York.MATHGoogle Scholar
  4. Fedorov, V. and S. Leonov (2007). Population PK measures, their estimation and selection of sampling times. J. Biopharmaceutical Statistics 17(5), 919–941.CrossRefMathSciNetGoogle Scholar
  5. Gagnon, R. and S. Leonov (2005). Optimal population designs for PK models with serial sampling. J. Biopharmaceutical Statistics 15(1), 143–163.CrossRefMathSciNetGoogle Scholar
  6. Gardiner, C. (2003). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, Berlin.Google Scholar
  7. López-Fidalgo, J. and W. Wong (2002). Design issues for Michaelis-Menten model. J. Theoretical Biology 215, 1–11.Google Scholar
  8. Mentré, F., S. Duffull, I. Gueorguieva, A. Hooker, S. Leonov, K. Ogungbenro, and S. Retout (2007). Software for optimal design in population PK/PD: a comparison. In Abstracts of the Annual Meeting of the Population Approach Group in Europe (PAGE). ISSN 1871-6032,
  9. Overgaard, R., N. Jonsson, C. Tornoe, and H. Madsen (2005). Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm. J. Pharmacokinetics and Pharmacodynamics 32(1), 85–107.CrossRefGoogle Scholar
  10. Picchini, U., S. Ditlevsen, and A. De Gaetano (2006). Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations. J. Theoretical Biology 53, 771–796.MATHGoogle Scholar
  11. Tornoe, C., J. Jacobsen, O. Pedersen, T. Hansen, and H. Madsen (2004). Grey-box modelling of pharmacokinetic/pharmacodynamic systems. J. Pharmacokinetics and Pharmacodynamics 31(5), 401–417.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Valerii V. Fedorov
    • 1
  • Sergei L. Leonov
    • 1
  • Vyacheslav A. Vasiliev
    • 2
  1. 1.GlaxoSmithKlineCollegevilleU.S.A
  2. 2.Department of Applied Mathematics and CyberneticsTomsk State UniversityTomskRussia

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