The convergence behavior of finite element models depends on the size of elements used, the element polynomial order, and on the complexity of the applied loads. For high-resolution models of trabecular bone, changes in architecture and density may also be important. The goal of this study was to investigate the influence of these factors on the convergence behavior of high-resolution models of trabecular bone. Two human vertebral and two bovine tibial trabecular bone specimens were modeled at four resolutions ranging from 20 to 80 μm and subjected to both compressive and shear loading. Results indicated that convergence behavior depended on both loading mode (axial versus shear) and volume fraction of the specimen. Compared to the 20 μm resolution, the differences in apparent Young’s modulus at 40 μm resolution were less than 5 percent for all specimens, and for apparent shear modulus were less than 7 percent. By contrast, differences at 80 μm resolution in apparent modulus were up to 41 percent, depending on the specimen tested and loading mode. Overall, differences in apparent properties were always less than 10 percent when the ratio of mean trabecular thickness to element size was greater than four. Use of higher order elements did not improve the results. Tissue level parameters such as maximum principal strain did not converge. Tissue level strains converged when considered relative to a threshold value, but only if the strains were evaluated at Gauss points rather than element centroids. These findings indicate that good convergence can be obtained with this modeling technique, although element size should be chosen based on factors such as loading mode, mean trabecular thickness, and the particular output parameter of interest.

1.
Ashman
R. B.
, and
Rho
J. Y.
,
1988
, “
Elastic modulus of trabecular bone material
,”
Journal of Biomechanics
, Vol.
21
, pp.
177
781
.
2.
Beck
J. D.
,
Canfield
B. L.
,
Haddock
S. M.
,
Chen
T. J. H.
,
Kothari
M.
, and
Keaveny
T. M.
,
1997
, “
Three-dimensional imaging of trabecular bone using the Computer Numerically Controlled Milling technique
,”
Bone
, Vol.
21
, pp.
281
287
.
3.
Fenech
C.
, and
Keaveny
T. M.
,
1999
, “
A cellular solid criterion for predicting the axial-shear failure properties of trabecular bone
,”
ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol.
121
, pp.
414
422
.
4.
Goulet
R. W.
,
Goldstein
S. A.
,
Ciarelli
M. J.
,
Kuhn
J. L.
,
Brown
M. B.
, and
Feldkamp
L. A.
,
1994
, “
The relationship between the structural and orthogonal compressive properties of trabecular bone
,”
Journal of Biomechanics
, Vol.
27
, pp.
375
389
.
5.
Guldberg
R. E.
,
Hollister
S. J.
, and
Charras
G. T.
,
1998
, “
The accuracy of digital image-based finite element models
,”
ASME JOURNAL OF BIOMECHANICAL ENGINEERING
, Vol.
120
, pp.
289
295
.
6.
Hollister
S. J.
,
Brennan
J. M.
, and
Kikuchi
N.
,
1994
, “
A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress
,”
Journal of Biomechanics
, Vol.
27
, pp.
433
444
.
7.
Hughes
T.
,
Ferencz
R.
, and
Hallquist
J.
,
1987
, “
Large-scale vectorized implicit calculation in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
61
, pp.
215
248
.
8.
Jacobs, C. R., Mandell, J. A., and Beaupre, G. S., 1993, “A comparative study of automatic finite element mesh generation techniques in Orthopaedic biomechanics,” Proc. ASME Bioengineering Division, ASME BED-Vol. 24, pp. 512–514.
9.
Jacobs, C. R., Davis, B. R., Rieger, C. J., Francis, J. J., Saad, M., and Fyhrie, D. P., 1999, “The impact of boundary conditions and mesh size on the accuracy of cancellous bone tissue modulus determination using large scale finite element modeling,” Journal of Biomechanics, Vol. 32.
10.
Keaveny
T. M.
,
1997
, “
Mechanistic approaches to analysis of trabecular bone
,”
Forma
, Vol.
12
, pp.
267
275
.
11.
Keyak
J. H.
,
Meagher
J. M.
,
Skinner
H. B.
, and
Mote
C. D.
,
1990
, “
Automated three-dimensional finite element modelling of bone: a new method
,”
Journal of Biomedical Engineering
, Vol.
12
, pp.
389
397
.
12.
Keyak
J. H.
, and
Skinner
H. B.
,
1992
, “
Three-Dimensional finite element modelling of bone — effects of element size
,”
Journal of Biomedical Engineering
, Vol.
14
, pp.
483
489
.
13.
Kinney
J. H.
, and
Ladd
A. J.
,
1998
, “
The relationship between three-dimensional connectivity and the elastic properties of trabecular bone
,”
Journal of Bone and Mineral Research
, Vol.
13
, pp.
839
845
.
14.
Kopperdahl
D. L.
, and
Keaveny
T. M.
,
1998
, “
Yield strain behavior of trabecular bone
,”
Journal of Biomechanics
, Vol.
31
, pp.
601
608
.
15.
Kothari
M.
,
Keaveny
T. M.
,
Lin
J. C.
,
Newitt
D. C.
,
Genant
H. K.
, and
Majumdar
S.
,
1998
, “
Impact of spatial resolution on the prediction of trabecular architecture parameters
,”
Bone
, Vol.
22
, pp.
437
443
.
16.
Ladd
A. J. C.
, and
Kinney
J. H.
,
1997
, “
Elastic constants of cellular structures
,”
Physica A
, Vol.
240
, pp.
349
360
.
17.
Ladd
A. J. C.
,
Kinney
J. H.
, and
Breunig
T. M.
,
1997
, “
Deformation and failure in cellular materials
,”
Physical Review E
, Vol.
55
, pp.
3271
3275
.
18.
Ladd
A. J.
, and
Kinney
J. H.
,
1998
, “
Numerical errors and uncertainties in finite-element modeling of trabecular bone
,”
Journal of Biomechanics
, Vol.
31
, pp.
941
945
.
19.
Ladd
A. J.
,
Kinney
J. H.
,
Haupt
D. L.
, and
Goldstein
S. A.
,
1998
, “
Finite-element modeling of trabecular bone: comparison with mechanical testing and determination of tissue modulus
,”
Journal of Orthopædic Research
, Vol.
16
, pp.
622
628
.
20.
Odgaard
A.
,
Kabel
J.
,
an Rietbergen
B.
,
Dalstra
M.
, and
Huiskes
R.
,
1997
, “
Fabric and elastic principal directions of cancellous bone are closely related
,”
Journal of Biomechanics
, Vol.
30
, pp.
487
495
.
21.
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
,”
Journal of Biomechanics
, Vol.
21
, pp.
155
168
.
22.
Simmons
C. A.
, and
Hipp
J. A.
,
1997
, “
Method-based differences in the automated analysis of the three-dimensional morphology of trabecular bone
,”
Journal of Bone and Mineral Research
, Vol.
12
, pp.
942
947
.
23.
Taylor
R. L.
, and
Wilson
E. L.
,
1976
, “
A non-conforming element for stress analysis
,”
International Journal for Numerical Methods in Engineering
, Vol.
10
, pp.
1211
1219
.
24.
Ulrich
D.
,
Hildebrand
T.
,
Van Rietbergen
B.
,
Mu¨ller
R.
, and
Ruegsegger
P.
,
1997
, “
The quality of trabecular bone evaluated with micro-computed tomography, FEA and mechanical testing
,”
Stud Health Technol Inform
, Vol.
40
, pp.
97
112
.
25.
Ulrich
D.
,
Van Rietbergen
B.
,
Weinans
H.
, and
Ruegsegger
P.
,
1998
, “
Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques
,”
Journal of Biomechanics
, Vol.
31
, pp.
1187
1192
.
26.
Van Rietbergen
B.
,
Weinans
H.
,
Huiskes
R.
, and
Odgaard
A.
,
1995
, “
A new method to determine trabecular bone elastic properties and loading using micromechanical finite element models
,”
Journal of Biomechanics
, Vol.
28
, pp.
69
81
.
27.
Van Rietbergen
B.
,
Odgaard
A.
,
Kabel
J.
, and
Huiskes
R.
,
1996
a, “
Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture
,”
Journal of Biomechanics
, Vol.
29
, pp.
1653
1657
.
28.
Van Rietbergen
B.
,
Weinans
H.
,
Huiskes
R.
, and
Polman
B. J. W.
,
1996
b, “
Computational strategies for iterative solutions of large FEM applications employing voxel data
,”
International Journal for Numerical Methods in Engineering
, Vol.
39
, pp.
2743
2767
.
29.
Van Rietbergen
B.
,
Odgaard
A.
,
Kabel
J.
, and
Huiskes
R.
,
1998
, “
Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions
,”
Journal of Orthopædic Research
, Vol.
16
, pp.
23
28
.
30.
Van Rietbergen
B.
,
Mu¨ller
R.
,
Ulrich
D.
,
Ru¨egsegger
P.
, and
Huiskes
R.
,
1999
, “
Tissue stresses and strain in trabeculae of a canine proximal femur can be quantified from computer reconstructions
,”
Journal of Biomechanics
, Vol.
32
, pp.
165
173
.
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