The Cellular Dynamics of Bone Remodeling: A Mathematical Model

Abstract

Abstract. The mechanical properties of vertebrate bone are largely determined by a process which involves the complex interplay of three different cell types. This process is called bone remodeling, and occurs asynchronously at multiple sites in the mature skeleton. The cells involved are bone resorbing osteoclasts, bone matrix producing osteoblasts and mechanosensing osteocytes. These cells communicate with each other by means of autocrine and paracrine signaling factors and operate in complex entities, the so-called bone multicellular units (BMU). To investigate the BMU dynamics in silico, we develop a novel mathematical model resulting in a system of nonlinear partial differential equations with time delays. The model describes the osteoblast and osteoclast populations together with the dynamics of the key messenger molecule RANKL and its decoy receptor OPG. Scaling theory is used to address parameter sensitivity and predict the emergence of pathological remodeling regimes. The model is studied numerically in one and two space dimensions using finite difference schemes in space and explicit delay equation solvers in time. The computational results are in agreement with in vivo observations and provide new insights into the role of the RANKL/OPG pathway in the spatial regulation of bone remodeling.

A Systems Approach to Understanding Bone Cell Interactions in Health and Disease

“…At the cell and single BMU scale one is concerned with modelling different cell behaviour such as resorption and formation properties of bone cells and respective features of the BMU such as resorption speed, number, type and distribution of cells in the cutting and closing cone, blood vessel, etc.. A number of mathematical models have been developed to investigate single BMU behavior using either discrete (agent-based) approaches (Cacciagrano, Corradini et al 2010;Buenzli, Jeon et al 2012) or continuous approaches based on partial differential equations (Ryser, Komarova et al 2010;Buenzli, Pivonka et al 2011). …”

Primal-mixed formulations for reaction–diffusion systems on deforming domains

“…We focus our attention in the numerical approximation of the chemical interaction between species concentrations and the response of the deformable medium where they react. Such a general framework is relevant to a wide range of applications going from molecular to macroscopic biological systems, and including, for instance, chemotaxis [52], organogenesis [41], bone remodelling [50,61], swelling of porous materials [38], cardiac electromechanics [31], tumor growth [10], force generation in skeletal muscle [15], wound healing [8], collagen network generation [34], tissue engineering [40], and many others.…”

Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling

“…Experimental work on bone remodeling shows a compli- 14 cated interaction between several substances and receptors. Diverse 15 hypotheses have been proposed for the different signaling pathways 16 and the communication between osteoclast and osteoblast cells [3][4][5], 17 but the whole remodeling process is still not completely understood. 18 Although it is accepted that the bone remodeling cycle depends on 19 the interaction between osteoclasts and osteoblasts and that the pro-20 cess keeps repeating (see for example [6][7][8] due to different types of bones and animal species, but [8,9] men-24 tion a median of 120 days.…”

“…More recent related works 38 are given in [12][13][14]. With some modifications to the first models, 39 there also have been modeling applications such as bone diseases, for 40 example such as myeloma [15] [17,18]. Other works include extra pop-44 ulations and effects to obtain new nonlinear square root differential 45 models, see [19].…”