Adaptive Approach for Modelling Variability in Pharmacokinetics
We present an improved adaptive approach for studying systems of ODEs affected by parameter variability and state space uncertainty. Our approach is based on a reformulation of the ODE problem as a transport problem of a probability density describing the evolution of the ensemble of systems in time. The resulting multidimensional problem is solved by representing the probability density w.r.t. an adaptively chosen Galerkin ansatz space of Gaussian densities. Due to our improvements in adaptivity control, we substantially improved the overall performance of the original algorithm and moreover inherited to the numerical scheme the theoretical property that the number of Gaussian distributions remains constant for linear ODEs. We illustrate the approach in application to dynamical systems describing the pharmacokinetics of drugs and xenobiotics, where variability in physiological parameters is important to be considered.
KeywordsMonte Carlo PBPK Model Sparse Grid Adaptive Approach Modelling Variability
Unable to display preview. Download preview PDF.
- 2.Abraham, K., Mielke, H., Huisinga, W., Gundert-Remy, U.: Internal exposure of children by simulated acute inhalation of volatile organic compounds: the influence of chemical properties on the child/adult concentration ratio. Basic Clin. Pharmacol. Toxicol. 96(3), 242–243 (2005)CrossRefGoogle Scholar
- 3.Beresford, A.P., Selick, H.E., Tarbit, M.H.: The emerging importance of predictive adme simulation in drug discovery. DDT 7, 109–116 (2002)Google Scholar
- 4.Griebel, M., Zumbusch, G.W.: Adaptive sparse grids for hyperbolic conservation laws. In: Fey, M., Jeltsch, R. (eds.) Hyperbolic Problems: Theory, Numerics, Applications. 7th International Conference in Zürich, February 1998. International Series of Numerical Mathematics 129, pp. 411–422. Birkhäuser, Basel (1999), http://wissrech.ins.uni-bonn.de/research/pub/zumbusch/hyp7.pdf Google Scholar
- 7.Keese, A.: A review of recent developments in the numerical solution of stochastic partial differential equations (stochastic finite elements). Informationbericht 2003-6, Department of Computer Science, Technical University Braunschweig, Brunswick, Germany, Institute of Scientific Computing, TU Braunschweig (2002)Google Scholar
- 9.Kwon, Y.: Handbook of Essential Pharmacokinetics, Pharmacodynamics, and Drug metabolism for Industrial Scientists. Kluwer Academic/Plemun Publishers (2001)Google Scholar
- 10.Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd edn. Springer, Berlin (1995)Google Scholar
- 11.Schweitzer, M.A., Griebel, M.: Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2003)Google Scholar
- 12.Matthies, H.G., Meyer, M.: Nonlinear galerkin methods for the model reduction of nonlinear dynamical systems. Informationsberich 2002-3, Department of Computer Science, TU Braunschweig, Germany, pp. 2002–2003 (March 2002)Google Scholar
- 13.Neunzert, H., Klar, A., Struckmeier, J.: Particle methods: Theory and applications. In: ICIAM 1995: Proceedings of the Third International Congress on Industrial and Applied Mathematics held in Hamburg, Germany (1995)Google Scholar
- 15.Talay, D.: Probabilistic numerical methods for partial differential equations: elements of analysis. In: Talay, D., Tubaro, L. (eds.) Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol. 1627, pp. 48–196. Springer, Heidelberg (1996)CrossRefGoogle Scholar
- 17.Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations. Notes on Numerical Fluid Mechanics, vol. 31, pp. 241–251. Vieweg (1991)Google Scholar