Elsevier

Bone

Volume 107, February 2018, Pages 208-221
Bone

Full length article
A mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation

https://doi.org/10.1016/j.bone.2017.11.009Get rights and content

Highlights

  • A multiscale bone cell population model of bone remodeling is coupled with a multiscale poromicromechanical model.

  • Biochemical and mechanical regulation is considered on different, relevant length scales.

  • Lacunar and vascular pore pressures are considered as mechanical stimuli of bone remodeling.

  • Physiologically relevant scenarios involving disuse and overuse can be satisfactorily reproduced.

  • Consideration of mechano-biological regulation in different pore spaces allows for simulating the behavior of anosteocytic bone.

Abstract

While bone tissue is a hierarchically organized material, mathematical formulations of bone remodeling are often defined on the level of a millimeter-sized representative volume element (RVE), “smeared” over all types of bone microstructures seen at lower observation scales. Thus, there is no explicit consideration of the fact that the biological cells and biochemical factors driving bone remodeling are actually located in differently sized pore spaces: active osteoblasts and osteoclasts can be found in the vascular pores, whereas the lacunar pores host osteocytes – bone cells originating from former osteoblasts which were then “buried” in newly deposited extracellular bone matrix. We here propose a mathematical description which considers size and shape of the pore spaces where the biological and biochemical events take place. In particular, a previously published systems biology formulation, accounting for biochemical regulatory mechanisms such as the rank-rankl-opg pathway, is cast into a multiscale framework coupled to a poromicromechanical model. The latter gives access to the vascular and lacunar pore pressures arising from macroscopic loading. Extensive experimental data on the biological consequences of this loading strongly suggest that the aforementioned pore pressures, together with the loading frequency, are essential drivers of bone remodeling. The novel approach presented here allows for satisfactory simulation of the evolution of bone tissue under various loading conditions, and for different species; including scenarios such as mechanical dis- and overuse of murine and human bone, or in osteocyte-free bone.

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:

  • 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

References (157)

  • D.M.L. Cooper et al.

    Age-dependent change in the 3D structure of cortical porosity at the human femoral midshaft

    Bone

    (2007)
  • A. Fritsch et al.

    ‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity

    J. Theor. Biol.

    (2007)
  • L. Cardoso et al.

    Advances in assessment of bone porosity, permeability and interstitial fluid flow

    J. Biomech.

    (2013)
  • R.K. Fuchs et al.

    Bone anatomy, physiology and adaptation to mechanical loading

  • G.D. Roodman

    Cell biology of the osteoclast

    Exp. Hematol.

    (1999)
  • B.F. Boyce et al.

    Functions of RANKL/RANK/OPG in bone modeling and remodeling

    Arch. Biochem. Biophys.

    (2008)
  • P. Pivonka et al.

    Model structure and control of bone remodeling: a theoretical study

    Bone

    (2008)
  • P. Pivonka et al.

    Theoretical investigation of the role of the RANK-RANKL-OPG system in bone remodeling

    J. Theor. Biol.

    (2010)
  • S. Scheiner et al.

    Coupling systems biology with multiscale mechanics, for computer simulations of bone remodeling

    Comput. Methods Appl. Mech. Eng.

    (2013)
  • F. Padilla et al.

    Relationships of trabecular bone structure with quantitative ultrasound parameters In vitro study on human proximal femur using transmission and backscatter measurements

    Bone

    (2008)
  • S. Tatsumi et al.

    Targeted ablation of osteocytes induces osteoporosis with defective mechanotransduction

    Cell Metab.

    (2007)
  • D.B. Jones et al.

    Biochemical signal transduction of mechanical strain in osteoblast-like cells

    Biomaterials

    (1991)
  • D. Kaspar et al.

    Proliferation of human-derived osteoblast-like cells depends on the cycle number and frequency of uniaxial strain

    J. Biomech.

    (2002)
  • R. Hill

    Elastic properties of reinforced solids: some theoretical principles

    J. Mech. Phys. Solids

    (1963)
  • R. Hill

    Continuum micro-mechanics of elastoplastic polycrystals

    J. Mech. Phys. Solids

    (1965)
  • W.R. Drugan et al.

    A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

    J. Mech. Phys. Solids

    (1996)
  • C. Kohlhauser et al.

    Ultrasonic contact pulse transmission for elastic wave velocity and stiffness determination: influence of specimen geometry and porosity

    Eng. Struct.

    (2013)
  • A. Fritsch et al.

    Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength

    J. Theor. Biol.

    (2009)
  • Q. Grimal et al.

    A two-parameter model of the effective elastic tensor for cortical bone

    J. Biomech.

    (2011)
  • V. Lemaire et al.

    Modeling of the interactions between osteoblast and osteoclast activities in bone remodeling

    J. Theor. Biol.

    (2004)
  • N. Nakagawa et al.

    RANK Is the essential signaling receptor for osteoclast differentiation factor in osteoclastogenesis

    Biochem. Biophys. Res. Commun.

    (1998)
  • V. Kartsogiannis et al.

    Localization of RANKL (receptor activator of NF kappa B ligand) mRNA and protein in skeletal and extraskeletal tissues

    Bone

    (1999)
  • M.H. Kroll

    Parathyroid hormone temporal effects on bone formation and resorption

    Bull. Math. Biol.

    (2000)
  • C.H. Turner

    Three rules for bone adaptation to mechanical stimuli

    Bone

    (1998)
  • C. Morin et al.

    A multiscale poromicromechanical approach to wave propagation and attenuation in bone

    Ultrasound

    (2014)
  • G. Bergmann et al.

    Hip joint loading during walking and running, measured in two patients

    J. Biomech.

    (1993)
  • B. Mikić et al.

    Bone strain gage data and theoretical models of functional adaptation

    J. Biomech.

    (1995)
  • D.B. Burr et al.

    In vivo measurement of human tibial strains during vigorous activity

    Bone

    (1996)
  • I. Kutzner et al.

    Loading of the knee joint during activities of daily living measured in vivo in five subjects

    J. Biomech.

    (2010)
  • J.A. Buckwalter et al.

    Bone biology. Part II. Formation, form, modeling, remodeling, and regulation of cell function

    J. Bone Joint Surg.- Series A

    (1995)
  • R.B. Martin et al.

    Skeletal Tissue Mechanics

    (1998)
  • E. Ozcivici et al.

    Mechanical signals as anabolic agents in bone

    Nat. Rev. Rheumatol.

    (2010)
  • L.F. Bonewald

    The amazing osteocyte

    J. Bone Miner. Res.

    (2011)
  • L. Vico et al.

    Microgravity and bone adaption at the tissue level

    J. Bone Miner. Res.

    (1992)
  • S.C. Manolagas

    Birth and death of bone cells: basic regulatory mechanisms and implications for the pathogenesis and treatment of osteoporosis

    Endocr. Rev.

    (2000)
  • E. Martin et al.

    Osteogenesis imperfecta: epidemiology and pathophysiology

    Curr. Osteoporos. Rep.

    (2007)
  • P. Chavassieux et al.

    Insights into material and structural basis of bone fragility from diseases associated with fractures: how determinants of the biomechanical properties of bone are compromised by disease

    Endocr. Rev.

    (2007)
  • O. Brennan et al.

    Temporal changes in bone composition, architecture, and strength following estrogen deficiency in osteoporosis

    Calcif. Tissue Int.

    (2012)
  • H.M. Frost

    Dynamics of bone remodeling

  • K. Imamura et al.

    Continuously applied compressive pressure induces bone resorption by a mechanism involving prostaglandin E2 synthesis

    J. Cell. Physiol.

    (1990)
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