Quantitative approach to the stochastics of bone remodeling

During life bones constantly adapt their structure to their mechanical environment via a mechanically controlled process called bone remodeling. For trabecular bone, this process modifies the thickness of each trabecula leading occasionally to full resorption. We describe the irreversible dynamics of the trabecular thickness distribution (TTD) by means of a Markov chain discrete in space and time. By using thickness data from adult patients, we derive the transition probabilities in the chain. This allows a quantification, in terms of geometrical quantities, of the control of bone remodeling and thus to determine the evolution of the TTD with age. Copyright c © EPLA, 2012 Introduction. – Architectural changes in trabecular bone, the porous bone found within vertebrae and at the ends of long bones [1], can lead to weakening and fracture of the entire bone organ [2]. In a human vertebra, the trabecular bone consists of a complex network of roughly horizontal and vertical rod and plate-like struts (trabeculae). This trabecular network changes continuously during life due to the regenerative process of bone remodeling. The process of bone remodeling consists in resorption or formation of discrete bone packets by specialized cells [3,4] thus altering the thickness of the trabeculae. Strong local bone resorption may even result in a complete loss of some of the trabeculae, with a preferential loss transverse to the main loading direction in vertebrae leading to an increase in structural anisotropy [5]. This deteriorates the local mechanical stability of the bone thus increasing the risk of spontaneous bone fractures, as seen especially in elderly people [2] and astronauts [6,7]. However, despite (a)Current address: Bernstein Center for Computational Neuroscience, Charité Universitätsmedizin Berlin Philippstr. 13, 10115 Berlin, Germany, EU; E-mail: [email protected] the frequent remodeling events the trabecular network is well preserved over the lifetime of a human. This observation suggests that a mechanism controlling remodeling induces an effective protection of thin trabeculae against complete resorption, although not completely avoiding it. In other words, based on this observation we expect that for thin trabeculae the probability of bone formation is larger than the probability of bone resorption. Such a control of bone remodeling can be realized by the action of mechanical forces. Indeed, in the standard mechanostat theory [8] of bone remodeling described by the Wolff-Roux law [9], new bone is locally formed where the local loading is high and removed where the local loading is low. This effectively leads to a protection of thin trabeculae because under the same external force thin trabeculae are more highly strained than thick trabeculae. Although such a mechano-biological law has been successfully implemented in simulations [10–13], the rule remains somewhat qualitative. Even the basic question about the mechanical stimulus that controls bone remodeling is still an open problem. In fact, the quantitative understanding of bone remodeling is hampered by experimental difficulties in