Full length articleA mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation
Introduction
It is well known that bone takes on a number of vital roles, including provision of the vertebrate skeleton's load-carrying capacity. For that purpose, it is essential that the microstructural integrity of the bone tissue is continuously maintained. The mechanism concerned with this important task is bone remodeling, involving numerous biochemically and mechanically stimulated processes, in concert leading to removal of bone tissue by cells called osteoclasts, and to concurrent addition of bone tissue by cells called osteoblasts, while a third cell type, osteocytes, has been identified as bone remodeling “conductor” [1], [2], [3], [4], [5], [6]. Under normal physiological conditions, the activities of osteoclasts and osteoblasts are finely tuned, and the volumes of removed and added bone tissue are the same. However, disturbance of this balance (caused, e.g., by bone disorders or a changed mechanical loading regime) can lead to changes in the bone composition [7], [8], [9]; in the worst case, the load-carrying capacity becomes significantly impaired [10], [11], [12].
The focus of this paper is the presentation of a mathematical model that is able to quantify (in predictive fashion) the effects of changes in the mechanical loading environment on the bone composition. A key novelty of this paper is that it takes into account the different characteristic lengths at which mechanical forces are transduced and the occurrence of cells and biochemical factors are quantified. In particular, the proposed modeling concept involves consideration of the exact spaces within a representative volume element (RVE) where bone remodeling takes place. Both bone-forming and -resorbing cells at various differentiation stages are located in the vascular pores, where they are activated or inhibited by biochemical factors to initiate the remodeling process; at this stage of cell maturation, they are attached to the pore walls and work in basic multicellular units (BMUs), resorbing old and forming new bone [13], [14]. Moreover, osteocytes reside in the lacunar pores and release biochemical factors such as sclerostin (sclr); the latter is transported to the vascular pore space, where it upregulates osteoblast precursor proliferation via wnt [15], [16]. As concerns the mechanical stimuli of cell activities, it is well known that oscillating hydrostatic pressure in the order of tens of kPa activates a variety of different biological cells, including bone cells [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. Scheiner et al. [34] have recently shown that pressures of this order of magnitude may indeed occur in the lacunar and vascular pore spaces of bone under physiologically relevant loading conditions. The main aim of the research presented in this paper was to integrate these different aspects into a comprehensive mathematical multiscale model of bone remodeling, considering concentrations of bone cells and biochemical factors at the respective pore fluid scales, accounting for changes in concentration due to pore volume changes, as well as incorporating mechanical stimuli at the relevant length scales, in order to reasonably provide mechanobiological feedback for bone remodeling.
Section snippets
Model representation of bone tissue
As basis for developing a suitable model representation of bone tissue, we first discuss the pore spaces found in bone tissue which are most relevant for the bone remodeling process:
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The largest pore space in bone is formed by the blood vessel-hosting vascular pores, with characteristic diameters of approximately 50 to 80 × 10 − 6 m [35], [36], [37], [38]. In cortical bone, the vascular pores occur in form of a branching structure [36], with the main branches (often referred to as Haversian canals)
Model calibration: simulation of bone formation and bone resorption occurring upon loading of mouse tibiae
First, we consider the experimental study of Sugiyama et al. [105] on the behavior of trabecular bone of mouse tibiae under mechanical loading. In particular, the tested mouse tibiae were subjected to axial compression, at a frequency of 0.1 Hz, for 40 cycles per day, over a period of 16 days. Thereby, the magnitude of the applied compressive force was varied from 0 to 14 N, resulting in peak compressive strains ranging from 0 to 2600 × 10 −6, see Fig. 3 (a). A linear fit of the experimental data
Discussion
This section is devoted to highlighting and discussing the major potentials and drawbacks of the multiscale mathematical model introduced in Section 2 of this paper, thereby also accounting for the numerical studies of Section 3.
Conclusions
In this paper, a mathematical model of bone remodeling was presented, based on coupling a multiscale systems biology model with a multiscale bone poromechanics model. The conceptual cornerstones and novelties of this approach involve (i) thorough consideration of the hierarchical organization of bone tissue, explicitly including information on the size of the pore spaces (in terms of the corresponding volume fractions), and on the shapes of the pore spaces (based on choosing representative pore
Acknowledgments
Financial support by the European Research Council (ERC), in the framework of the project Multiscale poromicromechanics of bone materials, with links to biology and medicine (project number FP7–257023), is gratefully acknowledged. The first author acknowledges a travel stipend of the Vienna University of Technology (TU Wien), for a research stay at The University of Melbourne, Australia. The continuous cooperation between the Austrian and Australian scientists was also facilitated within COST
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