Analytical basis for the determination of the lacunar–canalicular permeability of bone using cyclic loading

Abstract

An analytical model for the determination of the permeability in the lacunar-canalicular porosity of bone using cyclic loading is described in this contribution. The objective of the analysis presented is to relate the lacunar-canalicular permeability to a particular phase angle that is measurable when the bone is subjected to infinitesimal cyclic strain. The phase angle of interest is the lag angle between the applied strain and the resultant stress. Cyclic strain causes the interstitial fluid to move. This movement is essential for the viability of osteocytes and is believed to play a major role in the bone mechanotransduction mechanism. However, certain bone fluid flow properties, notably the permeability of the lacunar-canalicular porosity, are still not accurately determined. In this paper, formulas for the phase angle as a function of permeability for infinitesimal cyclic strain are presented and mathematical expressions for the storage modulus, loss modulus, and loss tangent are obtained. An accurate determination of the PLC permeability will improve our ability to understand mechanotransduction and mechanosensory mechanisms, which are fundamental to the understanding of how to treat osteoporosis, how to cope with microgravity in long-term manned space flights, and how to increase the longevity of prostheses that are implanted in bone tissue.

Dynamic permeability of the lacunar–canalicular system in human cortical bone

“…The analytical model developed in Benalla et al (2012) considers a transverse segment of an isolated osteon whose annular region (LCS) between the Haversian canal and cement line is subjected to an axial cyclic loading between two smooth, parallel, and impermeable plates. The contact between the specimen and the plates is assumed to be frictionless, and the applied load is considered to be uniformly distributed on the loaded surface.…”

“…Formulas for the loss tangent, tan[ δ ( ω̄ )], representing the frequency-dependent time-delay between the sinusoidal stress and strain in the LCS, were obtained as a function of the lacunar–canalicular permeability in the radial direction and loading frequency. In the present study, this frequency-dependent analytical model (Benalla et al 2012) and measurements of loss tangent on isolated human osteons under harmonic loading are used (1) to determine the intrinsic lacunar–canalicular permeability in the radial direction of fresh osteonal samples, (2) to assess the effect of time post-isolation of osteons on lacunar–canalicular permeability measurements, (3) to evaluate the effect of loading frequency on the dynamic permeability of the lacunar–canalicular system, and (4) to investigate the relationship between porosity and permeability in the LCS.…”

“…An analytical solution of a saturated compressible poroelastic annular cylinder under an unconfined stress relaxation test was developed, and predictions from this model were compared to experimental stress relaxation measurements made on isolated osteons in vitro, i.e., considering zero blood pressure (Gailani et al 2009; Gailani and Cowin 2008, 2011). Recently, a frequency-dependent analytical model of a fluid-saturated porous annular cylinder subjected to an axial cyclic strain loading was developed by Benalla et al (2012). Formulas for the loss tangent, tan[ δ ( ω̄ )], representing the frequency-dependent time-delay between the sinusoidal stress and strain in the LCS, were obtained as a function of the lacunar–canalicular permeability in the radial direction and loading frequency.…”

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

“…Curve fitting between data and experiments was used to obtain the LCP permeability without the influence of the vascular porosity in bovine bone, resulting in LCP permeability on the order of 10 −24 to 10 −25 m 2 (Gailani et al 2009). Recently, the theoretical/experimental approach was extended to the case of harmonic loading of osteons to investigate the frequency dependence of the LCP permeability (Benalla et al 2012). …”

Blood and interstitial flow in the hierarchical pore space architecture of bone tissue

“…A new method for the experimental determination of the permeability of a small sample of a fluid-saturated hierarchically structured porous material was described and applied to the determination of the lacunar–canalicular permeability in osteonal bone (Benalla et al, 2012, 2013). While the permeability of LCP has been estimated to be anywhere from 10 −17 m 2 to 10 −25 m 2 (Cardoso et al, 2013), it was shown by Benalla et al, (2013), in a combined experimental and analytical, that the LCP permeability is on the order of 10 −22 m 2 for human osteonal bone.…”