Coping with Time Scales in Disease Systems Analysis: Application to Bone Remodelling
Stephan Schmidt (1), Teun M. Post (2), Lambertus A. Peletier (3), Massoud A. Boroujerdi (1), Oscar E. Della Pasqua (1,4), Meindert Danhof (1)
(1)Division of Pharmacology, Leiden-Amsterdam Center for Drug Research, Leiden, The Netherlands; (2)Pharmacokinetics, Pharmacodynamics & Pharmacometrics (P3), Global Drug Metabolism & Pharmacokinetics, Merck Research Labs, Merck Sharp & Dohme, Oss, The Netherlands; (3)Mathematical Institute, Leiden University, Leiden, The Netherlands; (4)Clinical Pharmacology Modelling & Simulation, GlaxoSmithKline, Stockley Park, UK
Objectives: Models characterizing the dynamic behaviour of disease systems as well as the impact of therapeutic interventions can be established at different levels of complexity, ranging from data driven and descriptive to complex mechanistic approaches . While descriptive models may not be predictive beyond the data on which they were established, complex mechanistic approaches may face problems with parameter identifiability. To overcome this limitation, mechanism-based models capturing a system's behaviour rather than its complexity can be established using mathematical model reduction approaches. The aim of this study was to demonstrate the value of mathematical model reduction for characterizing complex dynamical systems using bone remodelling as an example.
Methods: The mechanistic bone cell interaction model proposed by Lemaire et al.  was mathematically reduced from a three-dimensional to a two-dimensional system. The dynamic properties of both the full Lemaire model and the reduced Lemaire model were then compared using simulations. In these simulations, the response of both models to changes in the underlying physiology and to therapeutic interventions was evaluated using four physiologically meaningful scenarios: 1) estrogen deficiency/estrogen replacement therapy, 2) Vitamin D deficiency, 3) ageing and 4) chronic glucocorticoid treatment/cessation of glucocorticoid treatment.
Results: On the time scale of disease progression and therapeutic intervention, the full and the mathematically reduced Lemaire model showed negligible differences in their dynamic properties. Both models were suitable for characterizing the impact of changes in the underlying physiology and/or therapeutic interventions on bone forming/resorbing cells for all four scenarios. Reduction to a two-dimensional system yielded new qualitative insight, such as the difference in times scales involved in the onset and washout of treatment effects, and brought down the number of parameters to be identified.
Conclusions: Mathematical model reduction is a valuable approach for analyzing disease systems and simplifying complex models while maintaining their dynamic properties. A significant decrease in the number of parameters to be identified and estimated in addition to an increased system transparency qualifies reduced models as tools to evaluate the impact of changes in physiological states and/or therapeutic interventions with respect to the different time scales involved.
 T.M. Post et al. Pharm Res 22(7) (2005) 1038-1049
 V. Lemaire et al. J Theor Biol 229 (2004) 293-309