Loading bone beyond its yield point creates microdamage, leading to reduction in stiffness. Previously, we related microdamage accumulation to changes in mechanical properties. Here, we develop a model that predicts stiffness loss based on the presence of microdamage. Modeling is done at three levels: (1) a single trabecula, (2) a cellular solid consisting of intact, damaged, and fractured trabeculae, and (3) a specimen with a localized damage band. Predictions of a reduced modulus agree well with experimental measured modulus reductions of post-yield compression of bovine trabecular bone. The predicted reduced modulus is relatively insensitive to changes in the input parameters.

1.
Keaveny
,
T. M.
,
Wachtel
,
E. F.
,
Guo
,
X. E.
, and
Hayes
,
W. C.
,
1994
, “
Mechanical Behavior of Damaged Trabecular Bone
,”
J. Biomech.
,
27
, pp.
1309
1318
.
2.
Keaveny
,
T. M.
,
Wachtel
,
E. F.
, and
Kopperdahl
,
D. L.
,
1999
, “
Mechanical Behavior of Human Trabecular Bone After Overloading
,”
J. Biomech.
,
17
, pp.
346
353
.
3.
Wachtel
,
E. F.
, and
Keaveny
,
T. M.
,
1997
, “
Dependence of Trabecular Damage on Mechanical Strain
,”
J. Orthop. Res.
,
15
, pp.
781
787
.
4.
Wachtel
,
E. F.
, and
Keaveny
,
T. M.
,
1995
, “
The Dependence of Trabecular Damage on Applied Strain Level for Bovine Trabecular Bone
,”
Trans. Annu. Meet. — Orthop. Res. Soc.
,
20
, p.
132
132
.
5.
Burr
,
D. B.
,
Turner
,
C. H.
,
Naick
,
P.
,
Forwood
,
M. R.
,
Ambrosius
,
W.
,
Hasan
,
M. S.
, and
Pidaparti
,
R.
,
1998
, “
Does Microdamage Accumulation Affect the Mechanical Properties of Bone?
,”
J. Biomech.
,
31
, pp.
337
345
.
6.
Yeh
,
O. C.
, and
Keaveny
,
T. M.
,
2000
, “
Roles of Microdamage and Microfracture in the Mechanical Behavior of Trabecular Bone
,”
Trans. Annu. Meet. — Orthop. Res. Soc.
,
25
, p.
34
34
.
7.
Guo
,
X. E.
,
McMahon
,
T. A.
,
Keaveny
,
T. M.
,
Hayes
,
W. C.
, and
Gibson
,
L. J.
,
1994
, “
Finite Element Modeling of Damage Accumulation in Trabecular Bone Under Cyclic Loading
,”
J. Biomech.
,
27
, pp.
145
155
.
8.
Schaffner
,
G.
,
Guo
,
X. E.
,
Silva
,
M. J.
, and
Gibson
,
L. J.
,
2000
, “
Modelling Fatigue Damage Accumulation in Two-Dimensional Voronoi Honeycombs
,”
Int. J. Mech. Sci.
,
42
, pp.
645
656
.
9.
Silva
,
M. J.
, and
Gibson
,
L. J.
,
1997
, “
Modeling the Mechanical Behavior of Vertebral Trabecular Bone: Effects of Age-Related Changes in Microstructure
,”
Bone
,
21
, pp.
191
199
.
10.
Silva
,
M. J.
, and
Gibson
,
L. J.
,
1997
, “
The Effects of Non-Periodic Microstructure and Defects on the Compressive Strength of Two-Dimensional Cellular Solids
,”
Int. J. Mech. Sci.
,
39
, pp.
549
563
.
11.
Vajjhala
,
S.
,
Kraynik
,
A. M.
, and
Gibson
,
L. J.
,
2000
, “
A Cellular Solid Model for Modulus Reduction Due to Resorption of Trabeculae in Bone
,”
ASME J. Biomech. Eng.
,
122
, pp.
511
515
.
12.
Vajjhalla, S., 1999, “Finite Element Analysis of Voronoi Cellular Solids,” SM Thesis, Massachusetts Institute of Technology, Cambridge, MA.
13.
Taylor
,
D.
,
1998
, “
Microcrack Growth Parameters for Compact Bone Deduced From Stiffness Variations
,”
J. Biomech.
,
31
, pp.
587
592
.
14.
Budiansky
,
B.
, and
O’Connell
,
R. J.
,
1976
, “
Elastic Moduli of a Cracked Solid
,”
Int. J. Solids Struct.
,
12
, pp.
81
97
.
15.
Arthur Moore, T. L., and Gibson, L. J., 2002, “Microdamage Accumulation in Bovine Trabecular Bone in Uniaxial Compression,” submitted to ASME J. Biomech. Eng..
16.
Fazzalari
,
N. L.
,
Forwood
,
M. R.
,
Manthey
,
B. A.
,
Smith
,
K.
, and
Kolesik
,
P.
,
1998
, “
Three-Dimensional Confocal Images of Microdamage in Cancellous Bone
,”
Bone
,
23
, pp.
373
378
.
17.
Stauffer, D., and Aharony, A., 1992, Introduction to Percolation Theory, Taylor and Francis, London.
18.
Gibson, L. J., and Ashby, M. F., 1997, Cellular Solids, Cambridge University Press, Cambridge.
19.
Grenestedt
,
J. L.
, and
Bassinet
,
F.
,
2000
, “
Influence of Cell Wall Thickness Variations on Elastic Stiffness of Closed Cell Cellular Solids
,”
Int. J. Mech. Sci.
,
42
, pp.
1327
1338
.
20.
Yeh
,
O. C.
, and
Keaveny
,
T. M.
,
1999
, “
Biomechanical Effects of Intraspecimen Variations in Trabecular Architecture: A Three-Dimensional Finite Element Study
,”
Bone
,
25
, pp.
223
228
.
21.
Ashman
,
R. B.
, and
Rho
,
J. Y.
,
1998
, “
Elastic Modulus of Trabecular Bone Material
,”
J. Biomech.
,
21
, pp.
177
181
.
22.
Carter
,
D. R.
, and
Hayes
,
W. C.
,
1977
, “
The Compressive Behavior of Bone as a Two-Phase Porous Structure
,”
J. Bone Jt. Surg. Am. Vol.
59
, pp.
954
962
.
23.
Ciarelli
,
M. J.
,
Goldstein
,
S. A.
,
Kuhn
,
J. L.
,
Cody
,
D. D.
, and
Brown
,
M. B.
,
1991
, “
Evaluation of Orthogonal Mechanical Properties and Density of Human Trabecular Bone From the Major Metaphyseal Regions With Materials Testing and Computed Tomography
,”
J. Orthop. Res.
,
9
, pp.
674
682
.
24.
Ding
,
M.
,
Dalstra
,
M.
,
Danielsen
,
C. C.
,
Kabel
,
J.
,
Hvid
,
I.
, and
Linde
,
F.
,
1997
, “
Age Variations in the Properties of Human Tibial Trabecular Bone
,”
J. Bone Jt. Surg. Br. Vol.
,
79
, pp.
995
1002
.
25.
Ducheyne
,
P.
,
Heymans
,
L.
,
Martens
,
M.
,
Aernoudt
,
E.
,
de Meester
,
P.
, and
Mulier
,
J. C.
,
1977
, “
The Mechanical Behavior of Intracondylar Cancellous Bone of the Femur at Different Loading Rates
,”
J. Biomech.
,
10
, pp.
747
762
.
26.
Goldstein
,
S. A.
,
Goulet
,
R.
, and
McCubbrey
,
D.
,
1993
, “
Measurement and Significance of Three-Dimensional Architecture to the Mechanical Integrity of Trabecular Bone
,”
Calcif. Tissue Int.
,
53
, pp.
S127–S133
S127–S133
.
27.
Hvid
,
I.
,
Jensen
,
N. C.
,
Bunger
,
C.
,
Solund
,
K.
, and
Djurhuus
,
J. C.
,
1985
, “
Bone Mineral Assay: Its Relation to the Mechanical Strength of Cancellous Bone
,”
Eng. Med.
,
14
, pp.
79
83
.
28.
Hvid
,
I.
,
Bentzen
,
S. M.
,
Linde
,
F.
,
Mosekilde
,
L.
, and
Pongsoipetch
,
B.
,
1989
, “
X-Ray Quantitative Computed Tomography: The Relations to Physical Properties of Proximal Tibial Trabecular Bone Specimens
,”
J. Biomech.
,
22
, pp.
837
844
.
29.
Rice
,
J. C.
,
Cowin
,
S. C.
, and
Bowman
,
J. A.
,
1988
, “
On the Dependence of the Elasticity and Strength of Cancellous Bone on Apparent Density
,”
J. Biomech.
,
21
, pp.
155
168
.
30.
Keaveny
,
T. M.
, and
Hayes
,
W. C.
,
1993
, “
A 20-Year Perspective on the Mechanical Properties of Trabecular Bone
,”
ASME J. Biomech. Eng.
,
115
, pp.
534
542
.
31.
Bastawros
,
A.-F.
,
Bart-Smith
,
H.
, and
Evans
,
A. G.
,
2000
, “
Experimental Analysis of Deformation Mechanisms in a Closed-Cell Al Alloy Foam
,”
J. Mech. Phys. Solids
,
48
, pp.
301
322
.
32.
McCullough
,
K. Y. G.
,
Fleck
,
N. A.
, and
Ashby
,
M. F.
,
1999
, “
Uniaxial Stress–Strain Behavior of Aluminum Alloy Foams
,”
Acta Mater.
,
47
, pp.
2323
2330
.
33.
Bart-Smith
,
H.
,
Bastawros
,
A.-F.
,
Mumm
,
D. R.
,
Evans
,
A. G.
,
Sypeck
,
D. J.
, and
Wadley
,
H. N. G.
,
1998
, “
Compressive Deformation and Vielding Mechanisms in Cellular Al Alloys Determined Using X-Ray Tomography and Surface Strain Mapping
,”
Acta Mater.
,
46
, pp.
3583
3592
.
34.
Tantillo, M., 1999, “Image-Guided Assessment of Local Bone Failure,” Bachelor of Science Thesis, Boston University, Boston, MA.
35.
Muller
,
R.
,
Gerber
,
S. C.
, and
Hayes
,
W. C.
,
1998
, “
Micro-compression: A Novel Technique for the Nondestructive Assessment of Local Bone Failure
,”
Technol. Health Care
,
6
, pp.
433
444
.
36.
Turner
,
C. H.
,
Cowin
,
S. C.
,
Rho
,
J. Y.
,
Ashman
,
R. B.
, and
Rice
,
J. C.
,
1990
, “
The Fabric Dependence of the Orthotropic Elastic Constants of Cancellous Bone
,”
J. Biomech.
,
23
, pp.
549
561
.
37.
Oden
,
Z. M.
,
Sevitelli
,
D. M.
,
Hayes
,
W. C.
, and
Myers
,
E. R.
,
1998
, “
The Effect of Trabecular Structure on DXA-Based Predictions of Bovine Bone Failure
,”
Calcif. Tissue Int.
,
63
, pp.
67
73
.
38.
Taylor
,
D.
, and
Lee
,
T. C.
,
1998
, “
Measuring the Shape and Size of Microcracks in Bone
,”
J. Biomech.
,
31
, pp.
1177
1180
.
39.
Fazzalari
,
N. L.
,
Forwood
,
M. R.
,
Smith
,
K.
,
Manthey
,
B. A.
, and
Herreen
,
P.
,
1998
, “
Assessment of Cancellous Bone Quality in Severe Osteoarthrosis: Bone Mineral Density, Mechanics, and Microdamage
,”
Bone
,
22, pp.
381
388
.
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