Computers in Biology and Medicine 37 (2007) 1203 – 1209 www.intl.elsevierhealth.com/journals/cobm Parametric sensitivity analysis applied to a specific one-dimensional internal bone remodelling problem S. Ramtani ∗ Laboratoire PMTM, CNRS UPR 9001, Institut Galilée, Université Paris Nord, 99 Avenue J-B Clément, 93430 Villetaneuse, France Received 4 July 2005; received in revised form 27 March 2006; accepted 30 October 2006 Abstract The relative importance of the various parameters in inducing bone mass loss and osteoclastic perforations is still controversial. Therefore, there is a significant motivation to better understand the parameters behind such dynamic response, and great interest to carry out a parametric sensitivity study as it can provide useful information. As an application, the widely-accepted bone remodelling equation [M.G. Mullender, R. Huiskes, H. Weinans, A physiological approach to the simulation of bone remodeling as self organizational control process, J. Biomech. 27 (1994) 1389.] is investigated using the “n units” model [M. Zidi, S. Ramtani, Bone remodeling theory applied to the study of n unit-elements model, J Biomech. 32 (1999) 743.]. This analysis pointed out that the power in the modulus density relationship p and the power to which density is raised in normalizing the energy stimulus q, known as strongly implicated in the stability condition of the remodelling process, were also stated as insensitive parameters in the bone loss area.  2006 Elsevier Ltd. All rights reserved. Keywords: Bone remodeling; Sensitivity analysis; Computer simulation; Numerical analysis 1. Introduction Many engineering and scientific problems are described by systems of differential-algebraic equations. Parametric sensi- tivity study is used to determine those parameters having the most significant effect by their perturbation on the process out- puts [1–4]. Such investigation aids in selecting the parameters to be estimated for further analysis using simulation and/or ex- perimental data. It uses also in designing future experiments. Parameter sensitivity analysis is a well-developed area and has been applied in many engineering applications using different mathematical models [5–10]. During normal aging and menopause, cancellous bone is lost at all skeletal sites due to remodelling-related factors: negative formation balance; temporarily increased remodelling space; and osteoclastic perforations. The relative importance of the various parameters in inducing bone mass loss and perforations is still controversial. Therefore, there is significant motivation ∗ Tel.: +33 01 49 40 39 53; fax: +33 01 49 40 39 38. E-mail address: salah.ramtani@lpmtm.univ-paris13.fr. 0010-4825/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiomed.2006.10.013 to better understand the parameters behind such dynamic re- sponse, and great interest to perform a parametric sensitivity analysis as it can provide a useful information. Indeed, it can point out the weaknesses of the used model and allow us to identify its most important parameters, which the modeller must accurately know to provide reliable results. Remodeling involves changes in material properties where the bone may change its internal structure through reorientation of trabeculae. The fundamental hypothesis emitted by Cowin et al. [11] is that the osteocytes act as sensors of a mechanical signal or “mechanoreceptors” and regulators of bone mass. In particular, we manipulated a model based on this concept [1,2,12] and analyzed the sensitivity of its parameters which are not precisely known and not easily measurable. This issue is of a great importance and has never been investigated before using rigorous mathematical and numerical approaches. Con- sequently, adaptive bone remodeling, which uses parametric sensitivity analysis, can be viewed as a first step toward select- ing the most important parameters, designing critical response, and adopting the hypothesis that best explain such dynamical process. More generally, parametric sensitivity investigation of the differential–algebraic equation model may give a useful