Fluid flow within cortical bone tissue is modeled through an upscaling approach of a local description of the fluid movement. At the pore scale, the coupled phenomena (Poiseuille effect, osmosis, and electro-osmosis) governing the interstitial fluid movement are considered. Thus, actions of electro-osmotic and osmotic motions, in addition to the classical Poiseuille flow, are studied at the canaliculus scale by deriving a coupled Darcy law. The addition of a Brinkman-like term in this macroscopic result helps us to take into account the influence of the pericellular matrix on the coupled transport phenomena. At the canaliculus scale, the general trends that can be drawn from this study are as follows: (i) The presence of the fibrous matrix tends to reduce the fluid flow considerably; (ii) the role of osmotic and electro-osmotic effects is no longer negligible for dense fibrous media.

1.
Cowin
,
S. C.
, 2001,
Bone Mechanics Handbook
,
CRC
, 2nd ed.,
Boca Raton, FL
.
2.
Sauren
,
Y. M.
,
Mieremet
,
R. H.
,
Groot
,
C. G.
, and
Scherft
,
J. P.
, 1992, “
An Electron Microscopic Study on the Presence of Proteoglycans in the Mineralized Matrix of Rat and Human Compact Lamellar Bone
,”
Anat. Rec.
0003-276X,
232
, pp.
36
44
.
3.
Shapiro
,
F.
,
Cahill
,
C.
,
Malatantis
,
G.
, and
Nayak
,
R. C.
, 1995, “
Transmission Electron Microscopic Demonstration of Viventim in Rat Osteoblast and Osteocyte Cell Bodies and Processes Using the Immunogold Technique
,”
Anat. Rec.
0003-276X,
241
, pp.
39
48
.
4.
You
,
L. D.
,
Weinbaum
,
S.
,
Cowin
,
S. C.
, and
Schaffler
,
M. B.
, 2004, “
Ultrastructure of the Osteocyte Process and its Pericellular Matrix
,”
Anat. Rec.
0003-276X,
278A
(
2
), pp.
505
513
.
5.
Swan
,
C. C.
,
Lakes
,
R. S.
,
Brand
,
R. A.
, and
Stewart
,
K. J.
, 2003, “
Micromechanically Based Poroelastic Modelling of Fluid Flow in Haversian Bone
,”
ASME J. Biomech. Eng.
0148-0731,
125
, pp.
25
37
.
6.
Pollack
,
S. R.
,
Petrov
,
N.
,
Salzstein
,
R.
,
Brankov
,
G.
, and
Blagoeva
,
R.
, 1984, “
An Anatomical Model for Streaming Potentials in Osteons
,”
J. Biomech.
0021-9290,
17
, pp.
627
636
.
7.
Cowin
,
S. C.
,
Weinbaum
,
S.
, and
Zeng
,
Y.
, 1995, “
A Case For Bone Canaliculi as the Anatomical Site of Strain Generated Potentials
,”
J. Biomech.
0021-9290,
28
(
11
), pp.
1281
1297
.
8.
You
,
L. D.
,
Cowin
,
S. C.
,
Schaffler
,
M. B.
, and
Weinbaum
,
S.
, 2001, “
A Model for Strain Amplification in the Actin Cytoskeleton of Osteocytes due to Fluid Drag on the Pericellular Matrix
,”
J. Biomech.
0021-9290,
34
(
1
), pp.
1375
1386
.
9.
Han
,
Y.
,
Cowin
,
S. C.
,
Schaffler
,
M. B.
, and
Weinbaum
,
S.
, 2004, “
Mechanotransduction and Strain Amplification in Osteocyte Cell Processes
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
101
(
47
), pp.
16689
16694
.
10.
Wang
,
L.
,
Ciani
,
C.
,
Doty
,
S. B.
, and
Fritton
,
S. P.
, 2004, “
Delineating Bone’s Interstitial Fluid Pathway In Vivo
,”
Bone
,
34
, pp.
499
509
.
11.
Weinbaum
,
S.
, 1998, “
Models to Solve Mysteries in Biomechanics at the Cellular Level: A New View of Fiber Matrix Layers
,”
Ann. Biomed. Eng.
0090-6964,
26
(
3
), pp.
627
643
.
12.
Wang
,
L.
,
Wang
,
Y.
,
Han
,
Y.
,
Henderson
,
S. C.
,
Majeska
,
R. J.
, and
Weinbaum
,
S.
, 2005, “
In Situ Measurement of Solute Transport in the Bone Lacunar-Canalicular System
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
102
, pp.
11911
11916
.
13.
Holmes
,
J. M.
,
Davies
,
D. H.
,
Meath
,
W. J.
, and
Beebe
,
R. A.
, 2006, “
Gas Adsorption and Surface Structure of Bone Mineral
,”
Biochemistry
0006-2960,
3
, pp.
2019
2024
.
14.
Lemaire
,
T.
,
Naili
,
S.
, and
Rémond
,
A.
, 2006, “
Multi-Scale Analysis of the Coupled Effects Governing the Movement of Interstitial Fluid in Cortical Bone
,”
Biomechanics and Modeling in Mechanobiology
,
5
, pp.
39
52
.
15.
Yasuda
,
I.
, 1964, “
Piezoelectricity of Living Bone
,”
J. Kyoto Pref. Med.
,
53
, pp.
2019
2024
.
16.
Basset
,
C. A. L.
, and
Becker
,
R. O.
, 1962, “
Generation of Electrical Potentials by Bone in Response to Mechanical Stress
,”
Science
0036-8075,
137
, pp.
1063
1064
.
17.
Pollack
,
S. R.
, 2001, “
Streaming Potentials in Bone
,”
Bone Mechanics Handbook
, 2nd ed.
S.
Cowin
, ed.
CRC
,
Boca Raton, FL
, Chap. 24, pp.
1
22
.
18.
Gu
,
W. Y.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1998, “
A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors
,”
ASME J. Biomech. Eng.
0148-0731,
120
, pp.
169
180
.
19.
Weinbaum
,
S.
,
Cowin
,
S. C.
, and
Zeng
,
Y.
, 1994, “
A Model for the Excitation of Osteocytes by Mechanical Loading-Induced Bone Fluid Shear Stresses
,”
J. Biomech.
0021-9290,
27
(
3
), pp.
339
360
.
20.
Jacobs
,
C. R.
,
Yellowley
,
C. E.
,
Davis
,
B. R.
,
Zhou
,
Z.
,
Cimbala
,
J. M.
, and
Donahue
,
H. J.
, 1998, “
Differential Effect of Steady Versus Oscillating Flow on Bone Cells
,”
J. Biomech.
0021-9290,
31
, pp.
969
976
.
21.
Burger
,
E. H.
, and
Klein-Nulend
,
J.
, 1999, “
Mechanotransduction in Bone: Role of the Lacuno-Canalicular Network
,”
FASEB J.
0892-6638,
13
(Suppl), pp.
S101
112
.
22.
Zhang
,
D.
,
Weinbaum
,
S.
, and
Cowin
,
S. C.
, 1998, “
On the Calculation of Bone Pore Water Pressure Due to Mechanical Loading
,”
Int. J. Solids Struct.
0020-7683,
35
(
34–35
), pp.
4981
4997
.
23.
Piekarski
,
K.
, and
Munro
,
M.
, 1977, “
Transport Mechanism Operating Between Blood Supply and Osteocytes in Long Bones
,”
Nature (London)
0028-0836,
269
(
5623
), pp.
80
82
.
24.
Cowin
,
S. C.
, 2002, “
Mechanosensation and Fluid Transport in Living Bone
,”
J. Musculoskel Neuron Interaction
,
2
(
3
), pp.
256
260
.
25.
Knothe Tate
,
M. L.
, 2003, “
Whither Flows The Fluid in Bone? An Osteocyte’s Perspective
,”
J. Biomech.
0021-9290,
36
, pp.
1409
1424
.
26.
Mak
,
A. F.
, and
Zhang
,
J. D.
, 2001, “
Numerical Simulation of Streaming Potentials due to Deformation-Induced Hierarchical Flows in Cortical Bone
,”
ASME J. Biomech. Eng.
0148-0731,
123
(
1
), pp.
66
70
.
27.
Rémond
,
A.
, and
Naili
,
S.
, 2004, “
Cyclic Loading of a Transverse Isotropic Poroelastic Cylinder: A Model for the Osteon
,”
C. R. Mec.
1631-0721,
332
(
9
), pp.
759
766
.
28.
Gururaja
,
S.
,
Kim
,
H. J.
,
Swan
,
C. C.
,
Brand
,
R. A.
, and
Lakes
,
R. S.
, 2005, “
Modeling Deformation-Induced Fluid Flow in Cortical Bone’s Canalicular-Lacunar System
,”
Ann. Biomed. Eng.
0090-6964,
33
, pp.
7
25
.
29.
Biot
,
M. A.
, 1941, “
General Theory of Three-Dimensional Consolidation
,”
J. Appl. Phys.
0021-8979,
12
(
2
), pp.
155
164
.
30.
Cowin
,
S. C.
, 2001, “
Bone Poroelasticity
,”
Bone Mechanics Handbook
,
S.
Cowin
, ed., 2nd ed.,
CRC
,
Boca Raton, FL
, Chap. 23, pp.
1
31
.
31.
Tsay
,
R. Y.
, and
Weinbaum
,
S.
, 1991, “
Viscous Flow in a Channel With Periodic Cross-Bridging Fibers: Exact Solutions and Brinkman Approximation
,”
J. Fluid Mech.
0022-1120,
226
, pp.
125
148
.
32.
Hunter
,
R. J.
, 2001,
Foundations of Colloid Science
.
Oxford University Press
,
New York
.
33.
Kim
,
Y. W.
,
Kim
,
J. J.
,
Kim
,
Y. H.
, and
Rho
,
J. Y.
, 2002, “
Effects of Organic Matrix Proteins on the Interfacial Structure at the Bone-Biocompatible Nacre Interface In Vitro
,”
Biomaterials
0142-9612,
23
, pp.
2089
2096
.
34.
Berreta
,
D. A.
, and
Pollack
,
S. R.
, 1986, “
Ion Concentration Effects on the Zeta Potential of Bone
,”
J. Orthop. Res.
0736-0266,
4
, pp.
337
341
.
35.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
, 1960,
Electrodynamics of Continuous Media
,
Pergamon
,
Oxford
.
36.
Dormienx
,
L.
,
Barboux
,
P.
,
Coussy
,
O.
, and
Dangla
,
P.
, 1995, “
A Macroscopic Model of the Swelling Phenomenon of a Saturated Clay
,”
Eur. J. Mech. A/Solids
0997-7538,
14
, pp.
981
1004
.
37.
Moyne
,
C.
, and
Murad
,
M. A.
, 2002, “
Electro-Chemo-Mechanical Couplings in Swelling Clays Derived From a Micro/Macro-Homogenization Procedure
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
6159
6190
.
38.
Lemaire
,
T.
, 2004, “
Couplages Électro-chimio-hydro-mécaniques dans les milieux argileux
,” Ph.D. thesis, Institut National Polytechnique de Lorraine, Nancy.
39.
Lemaire
,
T.
,
Moyne
,
C.
, and
Stemmelen
,
D.
, 2007, “
Modelling of Electro-Osmosis in Clayey Materials Including pH Effects
,”
Physics and Chemistry of the Earth
,
32
, pp.
441
452
.
40.
Sasidhar
,
V.
, and
Ruckenstein
,
E.
, 1981, “
Electrolyte Osmosis Through Capillaries
,”
J. Colloid Interface Sci.
0021-9797,
82
, pp.
439
457
.
41.
Israelachvili
,
J.
, 1991,
Intermolecular and Surface Forces
,
Academic
,
New York
.
42.
Sanchez-Palencia
,
E.
, 1980,
Non-Homogeneous Media and Vibration Theory
,
Lectures Notes in Physics
,
Springer Berlin
, Vol.
127
.
43.
Auriault
,
J.-L.
, 1991, “
Heterogeneous Medium. Is An Equivalent Macroscopic Description Possible?
,”
Int. J. Eng. Sci.
0020-7225,
29
, pp.
785
795
.
44.
Lemaire
,
T.
,
Moyne
,
C.
,
Stemmelen
,
D.
, and
Murad
,
M. A.
, 2002, “
Electro-Chemo-Mechanical Couplings in Swelling Clays Derived by Homogenization: Electroviscous Effects and Onsager’s Relations
,”
Poromechanics II, Proceedings of the Second Biot Conference on Poromechanics, Grenoble, France
,
J. L.
Auriault
,
C.
Geindreau
,
P.
Royer
,
J.-F.
Bloch
,
C.
Boutin
, and
J.
Lewandowska
, eds.,
Balkema
,
Lisse
, pp.
489
500
.
45.
Derjaguin
,
B. V.
,
Churaev
,
N.
, and
Muller
,
V.
, 1987,
Surface Forces
,
Plenum
,
New York
.
46.
Rémond
,
A.
, and
Naili
,
S.
, 2005, “
Transverse Isotropic Poroelastic Osteon Model Under Cyclic Loading
,”
Mech. Res. Commun.
0093-6413,
32
, pp.
645
651
.
47.
Wang
,
L.
,
Fritton
,
S. P.
,
Weinbaum
,
S.
, and
Cowin
,
S. C.
, 2003, “
On Bone Adaptation due to Venous Stasis
,”
J. Biomech.
0021-9290,
36
(
10
), pp.
1439
1451
.
48.
Starkenbaum
,
W.
,
Pollack
,
S. R.
, and
Korostoff
,
E.
, 1979, “
Microelectrode Studies of Stress Generated Potentials in Four Point Bending of Bone
,”
J. Biomed. Mater. Res.
0021-9304,
13
, pp.
729
751
.
49.
Burger
,
E. H.
,
Klein-Nulend
,
J.
, and
Smit
,
T. H.
, 2003, “
Strain-Derived Canalicular Fluid Flow Regulates Osteoclast Activity in a Remodelling Osteon—A Proposal
,”
J. Biomech.
0021-9290,
36
(
10
), pp.
1453
1459
.
50.
Hunter
,
R. J.
, 1981,
Zeta Potential in Colloid Science: Principles and Applications
,
Academic
,
New York
.
51.
Sherwood
,
J.
,
Rubio-Hernandez
,
F.
, and
Ruiz-Reina
,
E.
, 2000, “
The Primary Electroviscous Effect: Thin Double Layers and a Stern Layer
,”
J. Colloid Interface Sci.
0021-9797,
228
, pp.
7
13
.
You do not currently have access to this content.