Abstract
Mechanical loads which are macroscopically acting onto bony organs, are known to influence the activities of biological cells located in the pore spaces of bone, in particular so the signaling and production processes mediated by osteocytes. The exact mechanisms by which osteocytes are actually able to “feel” the mechanical loading and changes thereof, has been the subject of numerous studies, and, while several hypotheses have been brought forth over time, this topic has remained a matter of debate. Relaxation times reported in a recent experimental study of Gardinier et al. (Bone 46(4):1075–1081, 2010) strongly suggest that the lacunar pores are likely to experience, during typical physiological load cycles, not only fluid transport, but also undrained conditions. The latter entail the buildup of lacunar pore pressures, which we here quantify by means of a thorough multiscale modeling approach. In particular, the proposed model is based on classical poroelasticity theory, and able to account for multiple pore spaces. First, the model reveals distinct nonlinear dependencies of the resulting lacunar (and vascular) pore pressures on the underlying bone composition, highlighting the importance of a rigorous multiscale approach for appropriate computation of the aforementioned pore pressures. Then, the derived equations are evaluated for macroscopic (uniaxial as well as hydrostatic) mechanical loading of physiological magnitude. The resulting model-predicted pore pressures agree very well with the pressures that have been revealed, by means of in vitro studies, to be of adequate magnitude for modulating the responses of biological cells, including osteocytes. This underlines that osteocytes may respond to many types of loading stimuli at the same time, in particular so to fluid flow and hydrostatic pressure.
Highlights
Quite recently, Gardinier et al (2010) presented a brilliant modification of the seminal work of Qin et al (2002), allowing for the first time ever direct experimental access to the permeability of the lacunar-canalicular system of bone—they reported pressurization and relaxation times of around 8 s, relating to filling or drainage across the osteonal thickness, typically measuring about 65 microns (Gardinier et al 2010)
The mechanical impact of the aforementioned pore spaces is studied within the framework of continuum micromechanics (Hill 1963, 1965; Suquet 1997; Zaoui 1997, 2002; Dormieux et al 2006), where a material is understood as a macrohomogeneous, but microheterogeneous body filling a representative volume element (RVE) with characteristic length RVE, fulfilling the following separation-of-scales conditions: (i) RVE dRVE, dRVE representing the characteristic length of inhomogeneities within the RVE, and (ii) RVE {L, P}, L representing the characteristic length of the geometry and P representing the characteristic length of the loading of a structure built up by the material defined on the RVE
As has been derived in theoretical detail in Dormieux et al (2006), Hellmich et al (2009) and Pichler and Hellmich (2010). This multilinearity is expressed by the “drained” cortical stiffness tensor Cmacro, and by the pore space-specific Biot tensors bvmaascro and blmacacro, which quantify the macroscopic stresses Σ arising in an undeformed cortical RVE, from pressures acting in the two considered pore spaces
Summary
Quite recently, Gardinier et al (2010) presented a brilliant modification of the seminal work of Qin et al (2002), allowing for the first time ever direct experimental access to the permeability of the lacunar-canalicular system of bone—they reported pressurization and relaxation times of around 8 s, relating to filling or drainage across the osteonal thickness, typically measuring about 65 microns (Gardinier et al 2010). While the in vitro stimulation of bone cells by hydrostatic pressure seems to be a generally accepted fact, there seems to be some doubt on whether the hydrostatic pressures identified as mechanical stimuli in vitro are occurring in vivo This doubt is exemplified by a quotation from the famous paper of Duncan and Turner (1995), reading “hydrostatic pressure almost never occurs in mineralized bone”. The setting changes, at the tens-of-microns length scale of a single lacunar pore (and of the osteocyte it hosts), where the millimeter-sized gra-
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