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Poroelastic Evaluation of Fluid Movement Through the Lacunocanalicular System

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Abstract

A poroelastic lacunocanalicular model was developed for the quantification of physiologically relevant parameters related to bone fluid flow. The canalicular and lacunar microstructures were explicitly represented by a dual-continuum poroelastic model. Effective material properties were calculated using the theory of composite materials. Porosity and permeability values were determined using capillaric and spherical-shell models for the canalicular and lacunar microstructures, respectively. Pore fluid pressure and fluid shear stress were calculated in response to simulated mechanical loading applied over a range of frequencies. Species transport was simulated with convective and diffusive flow, and osteocyte consumption of nutrients was incorporated. With the calculated parameter values, realistic pore fluid pressure and fluid shear stress responses were predicted and shown to be consistent with previous experimental and theoretical studies. Stress-induced fluid flow was highlighted as a potent means of species transport, and the importance of high-magnitude low-frequency loading on osteocyte nutrition was demonstrated. This new model can serve as the foundation for future hierarchical modeling efforts that may provide insight into the underlying mechanisms of mechanotransduction and functional adaptation of bone.

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Abbreviations

B :

Skempton’s coefficient; Eq. (8); see Table 1

D :

Species diffusion coefficient; D g = 1.0 × 10−10 m2 s−1

E d :

Drained Young’s modulus; Eq. (6); see Table 1

E s :

Young’s modulus of solid phase; E s = 17.51 GPa

G d :

Drained shear modulus; Eq. (4)

G s :

Shear modulus of solid phase; G s = E s/2(1 + ν s) = 6.56 GPa

J con :

Convective species flux; Eq. (24)

J diff :

Diffusive species flux; Eq. (25)

K d :

Drained bulk modulus; Eq. (3)

K f :

Bulk modulus of the pore fluid; K f = 2.3 GPa

K s :

Bulk modulus of solid phase; K s = E s/3(1 − 2ν s) = 17.66 GPa

L :

Distance from the osteonal canal wall to the cement line; L = r o − r i = 100 μm

R i :

Radius of the osteocyte; R i = 4 μm

R o :

Radius of the lacuna; R o = 5 μm

a :

Radius of the osteocytic process; a = 0.1 μm

b :

Radius of the canaliculus; b = 0.23 μm

c :

Pressure diffusion coefficient (consolidation coefficient)

d :

Cross-sectional width and height of bone section; d = 21 μm

k c :

Bulk permeability of the canalicular material; Eq. (12); see Table 1

k l :

Bulk permeability of the lacunar material; Eq. (13); see Table 1

l :

Lacuna-to-lacuna distance; l = 30 μm

n c :

Canalicular density on the osteonal canal surface; n c = 5.5/100 μm2

p :

Pore fluid pressure

q :

Darcy fluid velocity; Eq. (19)

r i :

Inner radius of the osteon (radius of osteonal canal); r i = 25 μm

r o :

Outer radius of the osteon; r o = 125 μm

α :

Biot’s effective stress coefficient; Eq. (7); see Table 1

μ :

Dynamic viscosity of the pore fluid; μ = 0.001 Pa s

ν s :

Poisson’s ratio of solid phase; ν s = 0.335

ν d :

Drained Poisson’s ratio; Eq. (5); see Table 1

τ :

Shear stress; Eq. (20)

ϕc :

Porosity of the canalicular section with osteocyte process; Eq. (1); ϕc = 0.0074

ϕl :

Porosity of the lacunar section with osteocyte body; Eq. (2); ϕl = 0.119

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Acknowledgments

The authors extend thanks to the Natural Sciences and Engineering Research Council of Canada, Alberta Heritage Foundation for Medical Research, and Wood Professorship in Joint Injury Research for partial funding support.

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Authors and Affiliations

  1. Schulich School of Engineering, Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada

    Grant C. Goulet, Robert J. Martinuzzi & Ronald F. Zernicke

  2. Computer Modelling Group Ltd., Calgary, AB, Canada

    Dennis Coombe

  3. Departments of Orthopaedic Surgery, Kinesiology, and Biomedical Engineering, University of Michigan, 24 Frank Lloyd Wright Drive, Lobby A, Ann Arbor, MI, 48105, USA

    Ronald F. Zernicke

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  1. Grant C. Goulet
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Correspondence to Ronald F. Zernicke.

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Goulet, G.C., Coombe, D., Martinuzzi, R.J. et al. Poroelastic Evaluation of Fluid Movement Through the Lacunocanalicular System. Ann Biomed Eng 37, 1390–1402 (2009). https://doi.org/10.1007/s10439-009-9706-1

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