Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit

Abstract

During bone remodeling, bone-resorbing osteoclasts and bone-forming osteoblasts are organized in bone multicellular units (BMUs), which travel at a rate of 20-40 mm/d for 6-12 mo, maintaining a cylindrical structure. However, the interplay of local BMU geometry with biochemical regulation is poorly understood. We developed a mathematical model of BMU describing changes in time and space of the concentrations of proresorptive cytokine RANKL and its inhibitor osteoprotegerin (OPG), in osteoclast and osteoblast numbers, and in bone mass. We assumed that osteocytes surrounding a microfracture produce RANKL, which attracted osteoclasts. OPG and RANKL were produced by osteoblasts and diffused through bone, RANKL was eliminated by binding to OPG and RANK. Osteoblasts were coupled to osteoclasts through paracrine factors. The evolution of the BMU arising from this model was studied using numerical simulations. Our model recapitulated the spatio-temporal dynamics observed in vivo in a cross-section of bone. In response to a RANKL field, osteoclasts moved as a well-confined cutting cone. The coupling of osteoclasts to osteoblasts allowed for sufficient recruitment of osteoblasts to the resorbed surfaces. The RANKL field was the highest at the microfracture in front of the BMU, whereas the OPG field peaked at the back of the BMU, resulting in the formation of a RANKL/OPG gradient, which strongly affected the rate of BMU progression and its size. Thus, the spatial organization of a BMU provides important constraints on the roles of RANKL and OPG as well as possibly other regulators in determining the outcome of remodeling in the BMU.

A hybrid bioregulatory model of angiogenesis during bone fracture healing

“…It should be remarked that the validation is still limited due to the fact that the model is two-dimensional. However, starting with the investigation of lower-dimensional models is a well-established approach in applied mathematics since geometric complexity and computational costs make the study of 3D versions much more involved (Ryser et al 2009). …”

Physiologically based mathematical model of transduction of mechanobiological signals by osteocytes

“…Several published studies were developed by different authors to describe mathematically some cellular aspects of bone remodeling process based on description of BMUs activities without considering the mechanotransduction phase (Rattanakul et al 2003;Lemaire et al 2004;Martin and Buckland-Wright 2004;Komarova 2005;Moroz et al 2006;Maldonado et al 2006;Wimpenny and Moroz 2007;Pivonka et al 2008;Ryser et al 2009). Moreover, bone adaptation models based on considerations of the physiological and biological processes of bone were proposed by (Hart et al 1984;Heaney 1994;Martin 1995).…”

Mathematical Modeling in Wound Healing, Bone Regeneration and Tissue Engineering

“…The process of bone remodeling does not only take place after fractures but is a constantly ongoing, life-long process, actively restoring micro damage and renewing the bone matrix throughout the entire body. The mathematical models of bone remodeling described in Kroll et al (2000), Ryser et al (2009) and Pivonka et al (2008) are more focused on these latter applications of the process and are not discussed here.…”