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

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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.

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Abbreviations

\({\hat{A}_r},\, {\hat{A}_\theta},\, {\hat{A}_z}\) :

Three components of the 6D vector representing the Biot effective stress coefficient for PLC

r :

Arbitrary radius of the osteon

r o :

Outer radius of the osteon

r i :

Inner radius of the osteon

a :

Non-dimensional inner radius of the Osteon (a = r i / r o )

\({\hat{{\bf C}}^{i}}\) :

(i = m, d, u) elasticity or stiffness matrix for the matrix material, drained elastic constants and undrained elastic constants (2nd order tensor in 6D)

c :

Pore fluid pressure diffusion constant in the lacunar–canalicular porosity

\({\hat{{\bf E}}}\) :

Strain, a vector in 6D, equivalent to the strain tensor in 3D

E :

Modulus of elasticity

f o :

Constant of integration determined in Eq.(16)

I o , I 1 :

Modified Bessel functions of the first kind

i :

Imaginary number \({\left(i=\sqrt{-1}\right)}\)

K f :

Compressibility of the fluid

\({K_{\rm Reff}^i}\) :

Reuss lower bound on the effective (isotropic) bulk modulus of the anisotropic elastic material (i = m, d)

K o , K 1 :

Modified Bessel functions of the second kind

K rr :

Radial permeability

\({\tilde{p}(r,t)}\) :

Pore fluid pressure in the PLC

\({\tilde{u}(r,t)}\) :

Displacement vector

\({\varepsilon_o}\) :

Strain amplitude of the cyclic applied strain

\({\phi}\) :

Porosity

λ:

Ratio of r/r o (non-dimensional)

ω :

Angular frequency associated with the cyclic loading

\({\bar{\omega}}\) :

Dimensionless angular frequency

μ :

Viscosity of the pore fluid

ν ij :

Poisson’s ratios (i, j = r, θ, z)

σ * :

Average resultant stress due the applied cyclic strain

σ o :

Magnitude of the average resultant stress

\({\Upsilon}\) :

Constant of dimension force

Λ:

Solid–fluid compliance contrast coefficient

\({\tilde{\zeta}(r,t)}\) :

Variation of fluid content and

C * :

Dynamic elastic modulus

δ :

Phase angle

d :

The drained condition of the porous solid

u :

The undrained condition of the porous solid

f :

The fluid component

m :

The matrix material of the porous solid

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Authors and Affiliations

  1. The Department of Biomedical Engineering, The School of Engineering of The City College and The Graduate School of The City University of New York, New York, NY, 10031, USA

    M. Benalla, L. Cardoso & S. C. Cowin

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  1. M. Benalla
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  2. L. Cardoso
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  3. S. C. Cowin
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Correspondence to S. C. Cowin.

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Benalla, M., Cardoso, L. & Cowin, S.C. Analytical basis for the determination of the lacunar–canalicular permeability of bone using cyclic loading. Biomech Model Mechanobiol 11, 767–780 (2012). https://doi.org/10.1007/s10237-011-0350-y

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