Biomechanics and Modeling in Mechanobiology

, Volume 13, Issue 1, pp 215–225 | Cite as

The influence of load repetition in bone mechanotransduction using poroelastic finite-element models: the impact of permeability

Original Paper

Abstract

Experimental evidence suggests that interstitial fluid flow is a stimulus for mechanoadaptation in bone. Bone adaptation is sensitive to the frequency of loading and rest insertion between load cycles. We investigated the effects of permeability, frequency and rest insertion on fluid flow in bone using finite-element models to understand how these parameters affect the mechanical stimulus. A simplified 3D poroelastic finite-element model of a beam in bending was developed, in order to simulate the behavior of interstitial fluid flow in the lacunar-canalicular system of mouse cortical bone. Two different load sets were considered: (1) a continuous haversine sinusoid, with frequency ranging from 1 to 30 Hz, and (2) a 10 Hz haversine with rest-insertion times ranging from 0 to 10 s. For both load sets, a range of intrinsic permeability from \(10^{-23}\) to \(10^{-18}\, \mathrm m^2 \) was tested, and fluid flow was determined. Models with permeabilities down to \(10^{-21}\, \mathrm m^2 \) follow a dose–response relationship between fluid flow and sinusoidal frequency. Smaller orders of magnitude of permeability proved to be relatively insensitive to frequency. Our results also suggest that there is a minimum time of rest between load cycles that is required to maximize fluid motion, which depends on the order of magnitude of the intrinsic permeability. We show that frequency and rest insertion may be optimized to deliver maximal mechanical stimulus as a function of permeability.

Keywords

Bone Mechanotransduction Poroelasticity Load frequency Load rest insertion 

List of symbols

\(B\)

Skempton pore pressure coefficient

\(c\)

Diffusivity coefficient

\(d\)

Characteristic length

\(E\)

Drained elastic modulus

\(f_{\mathrm{LW}}\)

Load wave frequency

\(f_{\mathrm{LR}}\)

Load repetition frequency

\(f^*\)

Dimensionless frequency

\(G\)

Drained shear modulus

\(i,j\)

Tensor components

\(k\)

Intrinsic permeability

\(K\)

Drained bulk moduli

\(K_{\mathrm{f}}\)

Bulk modulus of the fluid phase

\(K_{\mathrm{s}}\)

Bulk modulus of the solid matrix

\(p\)

Pore pressure

\(p_0\)

Pore pressure at the time of step load removal

\(t_{\mathrm{a}}\)

Total time of analysis

\(t_{\mathrm{RI}}\)

Rest-insertion time interval

\(\varvec{V}\)

Average effective fluid velocity

\(V_{\mathrm{averaged}}\)

Fluid velocity magnitude averaged over time

\(V_{\mathrm{peak}}\)

Peak fluid velocity magnitude

\(V_\mathrm{S}\)

Steady-state fluid velocity amplitude

\(V^*\)

Dimensionless fluid velocity amplitude

\(\alpha \)

Biot effective stress coefficient

\(\epsilon _{ij}\)

Strain tensor

\(\mu \)

Interstitial fluid viscosity

\(\nu \)

Drained Poisson’s ratio

\(\nu _{\mathrm{u}}\)

Undrained Poisson ratio

\(\xi \)

Fluid content

\(\sigma _{ij}\)

Stress tensor

\(\tau \)

Characteristic pore pressure relaxation time

\(\tau _{\mathrm{RI}}\)

Characteristic time for rest-insertion fluid motion

\(\phi \)

Pore volume fraction of the PLC

\(\Psi \)

Mechanical stimulus—amount of fluid motion

\(\hat{\Psi }\)

Normalized fluid motion

\(\hat{\Psi }_0\)

Normalized mechanical stimulus for no rest inserted between load cycles

\(\Psi _{\mathrm{max}}\)

Maximum fluid motion

\(\Psi _{\mathrm{10 cycles}}\)

First 10 cycles maximum fluid motion

\(\nabla p\)

Pore pressure gradient

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of BioengineeringImperial College LondonLondonUK

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