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c 2001 by Annual Reviews. All rights reserved
Copyright °
BIOMECHANICS OF TRABECULAR BONE
Tony M. Keaveny1,2, Elise F. Morgan1, Glen L. Niebur3,
and Oscar C. Yeh1
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1
Orthopaedic Biomechanics Laboratory, Department of Mechanical Engineering,
University of California, Berkeley, California 94720-1740, 2Department of Orthopaedic
Surgery, University of California, San Francisco, California 94143 and 3Department of
Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana
46556; e-mail: [email protected]; emorgan@biomech3. me.berkeley.edu;
[email protected]; [email protected]
■ Abstract Trabecular bone is a complex material with substantial heterogeneity.
Its elastic and strength properties vary widely across anatomic sites, and with aging
and disease. Although these properties depend very much on density, the role of architecture and tissue material properties remain uncertain. It is interesting that the strains
at which the bone fails are almost independent of density. Current work addresses the
underlying structure-function relations for such behavior, as well as more complex
mechanical behavior, such as multiaxial loading, time-dependent failure, and damage
accumulation. A unique tool for studying such behavior is the microstructural class
of finite element models, particularly the “high-resolution” models. It is expected that
with continued progress in this field, substantial insight will be gained into such important problems as osteoporosis, bone fracture, bone remodeling, and design/analysis
of bone-implant systems. This article reviews the state of the art in trabecular bone
biomechanics, focusing on the mechanical aspects, and attempts to identify important
areas of current and future research.
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TRABECULAR BONE COMPOSITION AND MICROSTRUCTURE . . . . . . . . . . .
HETEROGENEITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ANISOTROPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RELATIVE ROLES OF DENSITY (VOLUME FRACTION),
ARCHITECTURE, AND TISSUE MATERIAL PROPERTIES
ON TRABECULAR ELASTIC BEHAVIOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STRENGTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
POSTYIELD AND DAMAGE BEHAVIOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1523-9829/01/0825-0307$14.00
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308
309
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314
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319
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CREEP AND FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
MICROMECHANICAL FINITE ELEMENT MODELING . . . . . . . . . . . . . . . . . . . . 322
CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
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INTRODUCTION
Research on the biomechanics of trabecular bone has been ongoing for over 30
years and is still intensely active. Motivated mostly by the need to understand the
role of trabecular bone in age-related bone fracture and the design of bone-implant
systems, this work has addressed characterization of mechanical properties as a
function of such variables as anatomic site, density, and age. A number of reviews
have summarized that work (1–3). As sophisticated engineering analysis tools
requiring more precise input data for optimal performance, such as bone-specific
finite element modeling (4–14), have been developed, emphasis is now turning to
a more complete characterization of the mechanical properties. In addition, there
is growing interest in the role of trabecular bone damage in both weakening whole
bones and stimulating biological remodeling. The focus of this review is to present
an overview of this latest body of work with an emphasis on the mechanical aspects
and to indicate important areas of future research.
TRABECULAR BONE COMPOSITION
AND MICROSTRUCTURE
Trabecular bone is the spongy, porous type of bone found at the ends of all long
bones and found within flat and irregular bones, such as the sternum, pelvis, and
spine (Figure 1). The microstructural struts or trabeculae that make up a specimen of trabecular bone are composed of trabecular tissue material. The trabeculae
enclose a three-dimensional, interconnected, open porous space, resulting in a
cellular solid (15) type of material. The pores are filled with bone marrow and
cells in vivo. The scale of these pores is on the order of 1 mm, and the scale
of the trabecular thickness is an order of magnitude lower. We are concerned
mostly with the behavior of small specimens of trabecular bone, on the order
of 5–10 mm in dimension, a scale at which the bone behaves as a continuum
(16, 17). The trabecular microstructure is typically oriented, such that there is
a “grain” direction along which mechanical stiffness and strength are greatest.
This microstructural directionality gives trabecular bone anisotropy of mechanical properties. The trabecular tissue material itself is morphologically similar to
cortical bone (an anisotropic composite of hydroxyapatite, collagen, water, and
trace amounts of other proteins) but is arranged in “packets” of lamellar bone (18).
Thus, trabecular bone is classified from an engineering materials perspective as a
composite, anisotropic, open porous cellular solid. Like many biological materials, it displays time-dependent behavior, as well as damage susceptibility during
cyclic loading.
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Figure 1 Volume rendering (20-µm resolution) of (a) bovine proximal tibial, (b) human proximal tibial, (c) human femoral neck, and (d ) human vertebral trabecular bone.
All specimens have the same bulk dimensions (3 × 3 × 1 mm3).
HETEROGENEITY
A critical issue that distinguishes trabecular bone from many other biological tissues is its substantial heterogeneity, which leads to wide variations in mechanical
properties1. This heterogeneity results from underlying variations in volume fraction, “architecture” (i.e. the three-dimensional arrangement of the individual trabeculae), and tissue properties, in that order of importance. For example, compressive modulus can vary 100-fold from one location to another within a single
proximal tibia (19) (Figure 2), and strength can vary fivefold within the proximal
femur (20). Across sites and species, mean values of modulus and strength can
differ by more than an order of magnitude (Table 1). Substantial loss of mechanical
1
Because trabecular bone spans multiple length scales, it is important to distinguish between
mechanical behavior at the level of the whole specimen—the apparent properties—as opposed to that at the level of individual trabeculae—the tissue properties. Thus, for example,
we talk of apparent versus tissue modulus for trabecular bone for the whole specimen and
trabecular tissue, respectively. Unless noted otherwise, we refer to material properties at
the apparent level.
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Figure 2 Elastic moduli from a proximal transverse section of a human tibia. Circled
values are measured moduli (in megapascals) of specimens taken from that region.
(From Reference 19, with permission.)
properties also occurs with aging in humans. For example, ultimate stress is reduced
by almost 7% and 11% per decade for the human proximal femur and spine, respectively, from ages 20–100 (21–23) (Figure 3). Strength does not decrease in
any significant manner until after about age 30, perhaps even later depending on
site (22–24).
Because of the substantial heterogeneity of trabecular bone, factors such as
age and site need to be designated when discussing the specifics of the mechanical
properties, i.e. trabecular bone from the aged spine is much different than that from
the young hip. This is a key concept in trabecular bone biomechanics and has direct
relevance to such fields as tissue engineering, where the goal is to replace damaged
trabecular bone with a substitute having appropriate mechanical properties for that
site.
ANISOTROPY
Trabecular bone is anisotropic in both modulus (22, 25–27) and strength (22, 28).
The extent of anisotropy is mild compared with such materials as fiber-reinforced
composites, but its biomechanical significance in terms of whole bone strength
TABLE 1 Mean values of modulus and ultimate strength for various anatomic sites
Modulus
(MPa)
Anatomic site (reference)
Age Range
Vertebra (22)
15–87
67 ± 45
Ultimate Strength
(MPa)
2.4 ± 1.6
Proximal tibia (63)
59–82
445 ± 257
5.3 ± 2.9
Proximal femur (132)
58–85
441 ± 271
6.8 ± 4.8
Calcaneus (48)
Not reported
68 ± 84
1.4 ± 1.3
Bovine proximal tibia (71)
Not applicable
2380 ± 777
24 ± 8.3
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BIOMECHANICS OF TRABECULAR BONE
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Figure 3 Dependence of compressive strength on age for human vertebral and proximal femoral trabecular bone cores. (Adapted from References 21 and 23, with permission.)
or bone-implant performance remains to be quantified. We know, for example,
that as trabecular bone becomes more porous, its compressive strength becomes
more anisotropic (Figure 4), but the significance of this in hip fracture etiology
is not known. Consistent with the concepts set forth in Wolff’s law (29–31), it
appears that the anisotropy develops as a form of adaptive response to functional
loading—bone is placed where it most needs to be. Trabecular bone possesses
at least orthotropic symmetry (17, 32), in some instances displaying transverse
isotropy (33). As an example of the degree of trabecular anisotropy, mean values of
strength and modulus of human vertebral bone in the superior-inferior direction are
higher than those in the transverse direction by factors of 2.8 and 3.4, respectively
(22). For a single specimen of bovine femoral bone, ratios of maximum modulus
to that in an orthogonal direction can be as high as 7.4 (34).
Mechanical testing of a specimen of trabecular bone in its principal material
coordinate system—which is required for measurement of the intrinsic anisotropic
material behavior—requires knowledge of the orientation of this coordinate system a priori. Strong evidence now exists that the principal material directions of
trabecular bone are aligned with the principal structural directions of the trabecular architecture (17, 33). Thus, mechanical testing should be performed along
the principal directions of the trabecular architecture. If this is not done, the resulting measurements are difficult to interpret because they will be influenced by such
multiple material constants as elastic modulus and Poisson’s ratio in the main and
transverse directions, as well as by shear moduli (35).
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Figure 4 Yield stress for longitudinal versus transverse loading, and the strength anisotropy ratio (SAR), plotted as a function of apparent
density for three different loading modes (compression, tension, and shear). (From Reference 80, with permission.)
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BIOMECHANICS OF TRABECULAR BONE
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Figure 5 A representation of the anisotropic nature of trabecular bone architecture
(left) and the measured fabric ellipse (right). The principal radii of the fabric ellipse
are denoted by a and b. θ is the angle between the measurement coordinate system and
the principal fabric directions. (From Reference 44, with permission.)
Efforts to quantify the principal structural directions and the structural anisotropy
of trabecular bone go back at least 25 years. In the early 1970s, Whitehouse (36, 37)
analyzed histological sections of trabecular bone using measurements of mean
intercept length (MIL), the mean length of a line segment passing through the
specimen that lies entirely within bone tissue. He found that a polar plot of MIL
versus the angle of the line segment can be fit to an ellipse (Figure 5). The major
and minor axes of the ellipse quantify the structural anisotropy of the specimen.
In the early 1980s, Harrigan & Mann (38) observed that the MIL measurements in
three dimensions are equivalent to a positive definite second-order tensor defining
an ellipsoid.
Cowin (39) classified the inverse of the MIL tensor as a type of fabric tensor, a
measure of local structural anisotropy. He developed a general theory relating the
fabric tensor to the anisotropic elastic behavior of a material. Based on this theory,
the ratios of the eigenvalues of the fabric tensor quantify the degree of structural
anisotropy. The specimen is orthotropic if there are three distinct eigenvalues,
transversely isotropic if there are two repeated eigenvalues, and isotropic if they
are all equal. This theory was then applied to trabecular bone, using the inverse
square root of the MIL ellipsoid as the fabric tensor (31). The 1990s saw specific
applications of this theory to trabecular bone (34, 40, 41), as well as other empirical
studies relating fabric-derived measures of structural anisotropy to mechanical
properties (17, 25, 42–44).
By the mid-1990s, the fabric tensor had become, and still is, the standard quantitative descriptor of trabecular orientation and structural anisotropy. Indeed, it is
now routine to construct this ellipsoid from a three-dimensional image of a specimen derived from sophisticated imaging techniques, such as micro-computed
tomography (45–47), micro-magnetic resonance imaging (48, 49), or automated
serial sectioning (50, 51). Other tensorial fabric measures have been developed
and applied to trabecular bone, most notably volume orientation, star volume
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distribution, and star length distribution methods (52–54). This collective research
has been driven by the need to better understand the role of trabecular architecture
in loss of mechanical properties with aging and osteoporosis, an important issue
discussed in more detail below.
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RELATIVE ROLES OF DENSITY (VOLUME FRACTION),
ARCHITECTURE, AND TISSUE MATERIAL PROPERTIES
ON TRABECULAR ELASTIC BEHAVIOR
The elastic modulus (and failure stress) of trabecular bone depends primarily on
apparent density (the product of volume fraction and trabecular tissue density,
the latter being essentially constant at about 2 g/cm3). However, the precise form
of this relationship remains controversial. In retrospect, it appears that most of
the controversy stems from the dependence of this relationship on anatomic site
(55) and loading direction (39, 56), and from the imprecision introduced by ignoring anisotropy and end-artifact effects (57–64) in the mechanical tests. End
artifacts arise from damage incurred at the ends of machined specimens when
they are tested in compression between platens with no other means of attachment to the load frame. If strains are computed from the relative displacement of
the platens, substantial systematic underestimation and random errors can occur
(59). The known dependence of the modulus-density relationship on loading direction (39, 56) implies that experimental protocols must either control or correct
for specimen anisotropy. Although cubic (65) and squared (66) modulus-density
relationships have been reported for multiple sites pooled together, the relationship appears to be linear for on-axis loading (loading along the main trabecular
orientation) within a single site (28, 55, 67) (Figure 6). The limited range in density found within a single anatomic site renders the differences in the predicted
modulus values between linear and power law relationships negligible.
A full understanding of the modulus-density relation requires an equal understanding of the role of architecture, and a number of studies have been performed
that quantified the relative effects of density and architecture. In one such study,
104 specimens of human trabecular bone from a variety of anatomic sites were
machined and tested along anatomic axes (42), resulting in a large variation of
trabecular orientation across specimens. Correlations between elastic modulus in
the different anatomic directions and (a) bone volume fraction, (b) trabecular orientation (defined by the MIL values in the directions of the specimen geometric
axes), and (c) anisotropy ratio (defined from the ratio of maximum to minimum
eigenvalues of the MIL ellipsoid) revealed that these variables combined explained
about 90% of the observed variation in mechanical properties. In a similar series
of studies on both bovine and human bone (25, 43), it was found that up to 94% of
the observed variation in measured elastic modulus could be explained by a composite measure of bone volume fraction, trabecular orientation, and anisotropy
ratio (Figure 7a). Similarly, 94% of the variation in the 21 components of the
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BIOMECHANICS OF TRABECULAR BONE
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Figure 6 On-axis elastic modulus plotted as a function of apparent density for trabecular bone specimens taken from the human vertebral body (HVB) and bovine proximal
tibia (BPT). (Adapted from Reference 55, with permission.)
elasticity tensor calculated using microstructural finite element models was captured by volume fraction combined with fabric (41) (Figure 7b). In all these studies,
bone volume fraction alone explained significantly less of the variance in elastic
properties. However, despite the high r2 values obtained in these studies, there
remains substantial scatter. For example, even with an r2 value of 0.94, values of
modulus can vary by over 60% for a given value of the composite explanatory
variable (25) (Figure 7a).
These results represent notable improvements in the ability of researchers to
predict trabecular bone anisotropic elastic behavior, and the early consensus was
that interspecimen variations in tissue properties were not important. However,
as discussed above, the summary statistics suggest a level of precision that may
not be reflective of the true magnitude of the unexplained variance. The important issue now is to understand why some specimens can have such low values
of modulus (and strength) at a given volume fraction. One plausible explanation
is intra-specimen variations of tissue or structural properties within the specimen.
A number of studies using nanoindentation to measure the tissue elastic modulus
(68, 69) have indicated that substantial variations in tissue modulus can exist both
within and across specimens. The structure-function studies performed on computer models (17, 41) do not account for any effects of intraspecimen variations in
tissue properties and the biomechanical consequences of such variations remain
unknown. Idealized computational models have shown that intraspecimen variations in trabecular thickness can alter apparent modulus to the equivalent extent
of 10 years of bone loss (70). It is expected that with the availability of powerful analytical tools, we will soon understand the potentially important effects of
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Figure 7 (a) Regression of elastic modulus and a composite explanatory variable of density
and fabric provided a squared correlation coefficient of 0.94. However, for a given value of
the composite explanatory variable, modulus varied by approximately 60%. (b) The complete
fabric tensor combined with volume fraction results in a similar level of correlation between
predicted and numerically calculated elastic properties. (Figure 7a adapted from Reference
25, Figure 7b from Reference 41, with permission.)
intraspecimen variations in structure and material properties on the apparent level
behavior. This in turn should provide substantial insight into why some specimens
can have low strength but normal levels of bone volume fraction and specimenaveraged measures of architecture.
STRENGTH
The same early studies that characterized the dependence of elastic modulus on
density and site generally addressed strength as well and found similar trends.
Namely, strength can vary by an order of magnitude across sites (Table 1), is
anisotropic, and depends on density with either a linear or a power law relation.
The density-architecture interaction is also an issue. Although Ciarelli et al (26)
reported a dependence of the regression coefficients in the power law relation on
anatomic site, no specific statistical comparisons were given, and no other studies
have compared strength across anatomic sites using the same test protocols and
controlling for anisotropy for human bone. Thus, although substantial data from
separate studies (Table 2) have provided a basis for biomechanical analyses of
whole bones, additional work is required to provide more controlled comparisons
across sites.
A further complexity with the strength behavior is the lower strength in tension
compared with compression (67, 71), and the still lower strength in shear (72).
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TABLE 2 Power law regressions between ultimate stress and apparent density for
compressive loading of human trabecular bone specimens from a range of anatomic
sitesa
Cadavers
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Study (reference)
Number
σ = Aρ
ρB
Age
Specimens
(No.)
A
B
r2
Proximal tibia (63)
9
59–82
121
34.2
1.56
0.79
Proximal femur (133)
4
25–82
49
25.0
1.80
0.93
Lumbar spine
Hansson et al (134)
Mosekilde et al (22)
Kopperdahl &
Keaveny (67)b
3
42
11
71–84
15–87
32–65
231
40
22
50.3
24.9
33.2
2.24
1.80
1.53
0.76
0.83
0.68
σ , in megapascals; ρ, in grams per cubic centimeters.
a
b
0.2% offset yield stress is reported instead of ultimate stress because the latter was not measured.
These trends depend on the loading direction, and the relative differences in these
strengths increase with increasing modulus (see Figure 4). Thus, from a strength
perspective, the properties of trabecular bone are heterogeneous (vary with age,
site, disease, etc), anisotropic (depend on loading direction), and asymmetric (differ
in tension versus compression versus shear). Because most available data on such
complex characteristics of trabecular bone are for bovine tissue, there remains a
need to measure many of these characteristics for human tissue. Such data should
improve the ability of finite element models of whole bones to more accurately
predict fracture loads for a variety of loading conditions, an important issue for
many clinical problems.
Despite all this complexity, if failure is characterized by measures of strain, it
can be seen that nature may have provided a material that has remarkably simple
failure behavior. There is a strong linear correlation between the stress at which
trabecular bone fails and the corresponding elastic modulus (e.g. 19, 20, 71, 73)
(Figure 8). Because the ratio of stress to modulus is strain, at least for a linearly
elastic material, this correlation suggests that failure strains for trabecular bone are
relatively constant. Experiments that made direct measurements of the strains at
failure found that they have only a slight, if any, dependence on density (e.g. 67, 71
74, 75). Also, the (0.2% offset) yield strains for bovine trabecular bone are higher
in compression than tension (67, 71), and even more interesting, the yield strains
are isotropic (75, 76). Furthermore, yield strains are relatively uniform within each
site but can vary across sites (67, 77). The simplicity of this strain-based description
of failure is extremely important in trabecular bone biomechanics. It implies, for
example, that if the elastic properties of trabecular bone are known, then regardless
of bone density, strength can be estimated with a high degree of accuracy for any
loading axes based only on the tensile and compressive yield strains.
Multiaxial behavior is important clinically because multiaxial stresses can occur
during falls, during trauma, and at the bone-implant interface. In contrast to the
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Figure 8 Yield stress versus modulus for bovine tibial trabecular bone, loaded in
an on-axis configuration in tension and compression. (From Reference 71, with permission.)
multitude of studies on the uniaxial strength behavior of trabecular bone, few studies have addressed multiaxial strength behavior (78–81). Two aspects of the multiaxial behavior have been addressed experimentally: the response to axial-shear
loads and the response to triaxial compressive loads. In addition, theoretical formulations have been proposed (80, 81) based on the Tsai-Wu quadratic theory, which
was originally developed for engineering composite materials (82, 83), and on
cellular solid theory (79). Collectively, these studies have shown that the Tsai-Wu
theory, although a good choice for axial-shear loading, does not work well for triaxial loading because indications are that the failure envelope for the latter does not fit
the ellipsoidal shape obtained from the quadratic formulation of the theory (80).
Based on the observation that many specimens failed during the triaxial tests
at stresses close to their uniaxial values (80), a simple cellular solid analytical
theory was developed for the special case of axial-shear loading (79). The result was a triangular failure envelope, built on the assumption that at the level
of individual trabeculae, bending stresses would primarily develop in response to
the apparent level shear loads, and axial stresses would primarily develop in response to the apparent level axial loads. The theory was calibrated using data from
previous uniaxial experiments, and the resulting predictions of the axial-shear
behavior matched the experimental data to within 7.7%. In addition, when the
applied stresses were normalized (divided by modulus), the effects of density disappeared and the behavior of all specimens could be described by a single criterion
(Figure 9). This is an excellent performance considering the complexity of the trabecular bone strength properties and the loading conditions. It is expected therefore
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BIOMECHANICS OF TRABECULAR BONE
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Figure 9 Axial-shear behavior of bovine proximal tibial trabecular bone. When the
stress axes are normalized by modulus, data from many specimens fit a single failure
envelope. Both criteria shown were calibrated to fit through the uniaxial tension, compression, and shear failure points. (Gray bars) ±1 standard deviation of those data.
(From Reference 79, with permission.)
that the appropriate multiaxial failure criterion for trabecular bone should be based
on strains, and that it should allow for microstructural failure mechanisms that are
different and perhaps independent for the different loading directions.
POSTYIELD AND DAMAGE BEHAVIOR
A relatively new area of trabecular bone biomechanics is the damage behavior at
both apparent and tissue levels. Damage and repair of individual trabeculae are
now recognized as normal physiologic processes (84, 85) that tend to increase
with age (86–88), and which may have clinical and biological relevance. It has
been proposed that damage to trabeculae could increase osteoporotic fracture risk
(73, 84, 89), act as a stimulus for remodeling (90), occur during implantation of
prostheses, particularly in the elderly spine where the bone is fragile (89), or be
involved in aseptic necrosis of the femoral head, degenerative joint disease (91),
and other bone disorders (92).
Experiments on machined specimens of bovine tibial (93, 94) and human vertebral (89) trabecular bone, as well as the whole vertebral body (95), have established that when trabecular bone is loaded past its yield point, it unloads to a
residual strain at zero stress, reloads with a modulus equal to its initial modulus, but
then develops a reduced modulus characteristic of a perfectly-damaging material
(Figure 10). For human vertebral bone, residual strains of up to 1.05% occur
with compressive loading of up to 3.0% strain and increase in a slightly nonlinear
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Figure 10 Load-unload-reload postyield behavior of human vertebral trabecular
bone. For multiple specimens, the secant modulus (dashed line) is statistically similar to the slope of the main linear region in the reloading curve, and the initial slope of
the reloading curve is statistically similar to the modulus of the initial loading cycle.
(From Reference 89, with permission.)
fashion with increasing total applied strain (89). Clinical spine fractures are defined
in terms of permanent deformations, but many fractures are not associated with
any specific traumatic event (96, 97). Thus, it is possible that isolated overloads,
or fatigue (see below), that do not cause overt fracture of the bone do cause subtle
but cumulative permanent deformations that may result in clinical fractures after
a number of years (95). This hypothesis remains to be tested.
The reductions in modulus and strength that occur for reloading after monotonic
overloading are substantial and depend strongly on the magnitude of the applied
strain but are mostly independent of volume fraction. In an experiment performed
on machined specimens of human vertebral trabecular bone (89), modulus reductions (between the intact Young’s and the residual moduli) were over 85% for
applied apparent strains of up to 3.0% (Figure 11). Using concepts of continuum
damage mechanics for brittle materials (98), these modulus reductions can be interpreted as quantitative measures of effective mechanical damage in the specimen.
Thus, a modulus reduction of 85% corresponds to 85% damage from a mechanical
perspective. Such behavior has also been found for bovine bone (93, 94) and for
the entire vertebral body (95) and appears therefore to be independent of density,
anatomic site, and even species. Indeed, the similarity of this behavior with that of
cortical bone (99) suggests that the damage occurs at the nanometer scale of the
collagen and hydroxyapatite.
At the trabecular tissue level, examination of the physical damage that occurs
with overloading (Figure 12) has confirmed that subtle damage within trabeculae (versus fracture of entire trabeculae) can cause large reductions in apparent
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Figure 11 Dependence of percent stiffness reduction on level of initial applied “plastic” strain for trabecular cores from the bovine proximal tibia and human vertebral body.
(Adapted from Reference 95, with permission.)
modulus (100). Consistent with this, Fyhrie & Schaffler (73) reported that for human vertebral specimens loaded in compression to 15% strain, the primary mechanism of failure was microscopic cracking rather than overt fracture of individual trabeculae. Complete fracture of trabeculae was confined to elements oriented
transversely to the loading direction. Laser scanning confocal microscopy (101) has
shown that cross-hatch shear band type staining and more diffuse (102) staining observed with basic fuchsin included “ultramicrocracks” about 10 µm in length. The
implication is that cracking can occur at very small scales. Zysset & Curnier (94)
came to a similar conclusion after observing similarities in postyield and damage
behaviors between bovine trabecular bone they tested and cortical bone from the
literature.
CREEP AND FATIGUE
Following up on preliminary reports on time-dependent failure modes (103, 104),
it has been demonstrated that bovine trabecular bone has fatigue and creep characteristics similar to those of cortical bone. The creep characteristics exhibit the
three classical phases of an initial rapid response, a steady state creep at a constant
creep rate, and a rapid increase in strain just before fracture (105). Cyclic loading
results in cumulative creep deformations in addition to loss of stiffness (106, 107).
Standard “S-N” (applied stress versus number of cycles to failure) (Figure 13) and
creep stress-time curves, using nondimensional measures of stress, have been reported that can serve as input into whole bone and bone-implant structural analyses.
These curves indicate that the compressive strength of devitalized trabecular bone
can be reduced by up to 70% after 106 cycles of loading. Of course, in vitro creep
or fatigue experiments preclude biological healing, and thus the resulting fatigue
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Figure 13 Traditional “S-N” fatigue curve for bovine trabecular bone for zerocompression cyclic loading. The applied stress 1σ has been normalized by the initial
modulus Eo. (From Reference 107, with permission.)
S-N or creep stress-time curves might best be considered as lower bounds on the
specimen life, i.e. we can expect a longer life if biological healing of the fatigueor creep-related damage occurs. However, it has been suggested that osteoclastic
resorption during the remodeling process can in some situations reduce strength if
the resulting Howslip’s lacunae serve as significant stress concentrations (108). If
that is the case, then the in vitro fatigue characteristics may represent upper bounds
on the in vivo fatigue life.
The creep and fatigue characteristics of human trabecular bone are not known.
Tensile creep or fatigue characteristics, how any of these behaviors may depend on
aging or disease, and how previous mechanical overloads or biological remodeling
may potentially accentuate fatigue damage are also not known. More important,
all studies reported to date on trabecular bone have used relatively high loading
levels that are probably outside the range of repetitive, habitual load levels. Thus,
despite novel work over the past 10 years on characterization of the time-dependent
behavior of trabecular bone, there remains much to be done. This is particularly
relevant to osteoporotic fractures in the spine that are typically not associated with a
specific trauma (96, 97). Accumulation of damage from overloads, fatigue, or creep
loading may also help explain the 10%–20% of spontaneous hip fractures that occur
without a fall, and why only about 2% of those who fall actually fracture their hips.
MICROMECHANICAL FINITE ELEMENT MODELING
Because of the complexity of the empirical data described above, a number
of computational studies have been performed to gain insight into the underlying micromechanics of the bone. In particular, two strategies have been
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BIOMECHANICS OF TRABECULAR BONE
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developed to perform micromechanical finite element analyses of trabecular bone
specimens.
The first class of models is based on the cellular solid paradigm. The strategy is
to account for some of the complexity of trabecular architecture while maintaining the computational efficiency that allows for the development of an intuitive
understanding of the micromechanics. One method of idealizing the trabecular
geometry has been to incorporate the most salient features of the trabecular architecture based on published histomorphometric studies. Models have incorporated
statistical distributions of spacing, angular orientation, and thickness (70, 109). Alternatively, Voronoi techniques can be used to generate an idealized nonperiodic
mesh representative of trabecular bone (110, 111).
Various lattice-type finite element models have been used to examine the architectural manifestations of aging. Using age-related regressions of mean thickness and spacing for vertebral trabecular bone, trends of elastic modulus reduction with aging correlated reasonably well with experimental observations (109).
Other analyses demonstrated the potentially important role of stark interruptions
in the trabecular network versus more uniform thinning (110, 112). Whereas these
studies considered only randomly placed defects in the trabecular network, one
study (113) reported on a more mechanistic basis for removal of trabeculae by
considering both fatigue and creep. The study demonstrated that after reductions
of apparent modulus of about 15%, further removal of trabeculae from fatigue
crack propagation produced apparent modulus reductions that were greater than
for random removal of trabeculae. This occurred presumably because of a stressconcentration effect of the accumlated fractures, or load redistribution around the
accumulated fractures, and overloading and accelerated failure of the adjacent
trabeculae.
The other class of microstructural finite element models, the high-resolution
model, represents a critical methodological advance in trabecular bone biomechanics. This technique uses a high-resolution, three-dimensional image of a specific
specimen at up to 10-µm spatial resolution. The strategy is to directly convert the
digital image into a finite element mesh, thereby avoiding the need to generate
more traditional isoparametric meshes of the topographically complex trabecular
architecture. In the more popular implementation, a finite element mesh is generated directly from a three-dimensional image by applying a one-to-one mapping of
image voxels to eight-node hexahedral finite elements (114, 115) (Figure 14). This
method is particularly attractive for trabecular bone because an explicit mathematical representation of the geometry is not required. The resulting models can contain
millions of elements (116, 117). The congruence of the elements in the model is
then exploited to produce specialized solution algorithms that substantially reduce
memory requirements (118, 119). A number of studies have been reported that
characterize the numerical convergence behavior of this class of models, where
it is recommended that at least four elements span the mean trabecular thickness
(120–122).
The mechanical properties of the trabecular tissue are usually assumed to
be homogeneous and isotropic within the high-resolution finite element model.
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Figure 14 A 20-µm, high-resolution image of a 5-mm cube specimen of bovine tibial
trabecular bone (left), and a portion of the resulting “high-resolution” finite element model
(right). The finite element mesh (top front corner) is approximately 1.25 mm on each side,
with an element size of approximately 60 µm on a side.
Although recent studies using nanoindentation indicate that the tissue elastic modulus may vary substantially throughout the specimen (68), it has been demonstrated
that the use of isotropic and homogeneous properties can result in good predictions
of the apparent elastic properties (123). Linear elasticity is also assumed, resulting
in the overall stiffness of the model being a linear function of the assumed trabecular tissue modulus. Using an inverse technique, an “effective” modulus for the
trabecular tissue can be calculated by determining the ratio of the tissue to apparent
elastic moduli predicted by the finite element results, and multiplying this ratio
by the experimentally measured apparent modulus for the same specimen (115).
Using this approach, a large range (mean values from 5.6–18.7 GPa) of effective
tissue moduli has been reported (123–126). Although this large range may reflect
differences in anatomic site, age, and the experimental methods used to determine
the apparent elastic properties, a similar range of values has been reported using
other measurement techniques (for a review, see 127). Even so, recent nanoindentation studies for human bone have reported values of the tissue modulus in the
range of only 11 (68) to 18 (128, 129) GPa, which suggests that the lower values
reported for the effective tissue modulus may be more reflective of experimental
or computational artifacts. Thus, it remains a challenge to use the high-resolution
finite element technique to estimate absolute values of tissue properties.
Because it is so difficult to measure the orthotropic elastic constants of trabecular bone in a purely experimental setting, the high-resolution finite element
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BIOMECHANICS OF TRABECULAR BONE
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method provides a powerful means to address this issue. Using the concept of a
representative volume element (130), the 21 unique values in the apparent stiffness
tensor can be found from six uniaxial strain simulations. The calculation involves
no a priori assumption of the elastic symmetry of the bone and, in this sense, is superior to actual mechanical tests. Results are typically normalized by the assumed
tissue modulus, thereby eliminating this variable from consideration. The resulting
elastic stiffness matrix can be rotated into a coordinate system that minimizes the
nonorthotropic terms in the elasticity tensor. Using this technique, three studies
performed on bone taken from multiple sites have found that the orthotropic coefficients of the stiffness matrix were more than two orders of magnitude larger than
any remaining term (17, 32, 33), i.e. trabecular bone can be considered to have
orthotropic symmetry. As discussed earlier, this technique has also been used to
provide substantial insight into relationships between elastic modulus and both
volume fraction and fabric, as well as determination of the principal material and
structural directions (17, 33, 131).
Most recently, the high-resolution finite element technique has been extended
to address failure properties (126). The main assumption used was that the tensile
and compressive yield strains of the trabecular tissue were similar to those of cortical bone and were the same for all trabecular bone specimens. After calibrating
these properties for one specimen, and by using a specimen-specific value of effective tissue modulus, the compressive, tensile, and shear yield apparent properties
were predicted for six other specimens and were found to be statistically indistinguishable from those measured in previous experiments (71, 72). This level
of performance is outstanding considering that only two material constants were
used to characterize the failure properties for all specimens and loading directions.
Thus, this technique has great potential for exploring strength behavior. For example, because it is difficult to perform real multiaxial experiments on sufficient
numbers of human trabecular bone specimens to account for heterogeneity, it is
likely that the multiaxial strength problem will only be tractable if addressed using
a combined experimental-computational approach. Because the high-resolution
finite element models are microstructurally based, they should be able to predict
multiaxial failure behavior without introducing any new assumptions. In addition, these models can provide predictions of regions of tensile and compressive failure of tissue within the specimen (Figure 15). Such data now provide
a basis for addressing possible strain-specific biological responses to damage in
bone, an important issue in understanding the mechano-biology of trabecular bone
(84).
CONCLUDING REMARKS
The field of trabecular bone biomechanics has grown steadily over the past 30
years and has reached a level of maturity resulting in substantial collaborations between engineers, biologists, radiologists, and clinicians. We have reviewed some
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of the highlights of this work, mostly from a mechanics perspective, and have
attempted to note the important achievements and exciting areas of future work.
Trabecular bone is highly heterogeneous in microstructure, which results in substantial variations of elastic modulus and strength both within and across sites.
Even so, the strains at which the bone fails are relatively constant, suggesting a
biological control mechanism targeted toward apparent strains. Trabecular bone is
also anisotropic, and sophisticated computer modeling and micron-level imaging
techniques have been developed that provide much insight into the relationships
between such aniostropy and the underlying architecture and how these relations
may change with aging, disease, and drug treatment. The same types of models
promise the ability to better understand the role of damage in degradation of mechanical properties and offer a way to tackle so far intractable problems such as
the multiaxial strength behavior.
It should be noted that we did not review many areas that are important to
trabecular bone biomechanics2, namely, the role of trabecular bone mechanical
behavior in whole bone mechanics nor did we review any biological effects explicitly. Examples of the latter include bone adaptation, tissue engineering, and the
in vivo response to drug treatments, disuse, space flight, and exercise regimens.
These are important areas intimately related to the subject matter reviewed here.
In terms of understanding the clincal effects of aging and disease, a number
of challenges remain within the domain of trabecular bone mechanics. Most importantly perhaps, we need to understand how some trabecular bone can be so
weak, or so susceptible to fatigue failure, despite having normal levels of bone
density, and perhaps even a normal measure of architecture (by current standards
of the latter). Indeed, this may constitute a more rigorous definition of osteoporotic
bone as current definitions cannot predict who will fracture with much precision.
Factors such as damage accumulation, possible changes in tissue ducitility from
changes in the collagen, alterations in the biological remodeling dynamics, and
various types of intraspecimen variations in material and structural properties are
currently under investigation. These studies will require new types of model input
such as statistical distributions of tissue mineralization, damage, and trabecular
thickness, as well as new assays of collagen and mineral status and perhaps their
spatial distribution. Knock-out animal models also promise a way to alter the system to produce a specific phenotype characteristic of aging or disease and this
strategy is becoming more widespread.
At the whole bone level, a number of issues remain. The relative role of the
cortical shell versus trabecular bone remains controversial for bones with thin
cortices such as the vertebral body. The effects of trabecular anisotropy on whole
bone behavior in bones such as the proximal femur remain unknown. Multiaxial
2
Although a biologist might argue that we have reviewed the mechanics as opposed to the
biomechanics of trabecular bone in this review because we have not addressed biology per
se, the engineering perspective accepts the term biomechanics as we are addressing the
mechanics of a biological system.
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BIOMECHANICS OF TRABECULAR BONE
327
failure criteria need to be developed to enable fracture analysis of whole bones
for such loading conditions as a fall, sports impact, or vehicular trauma and for
more detailed analysis of bone-implant systems. Understanding the development
of damage over time in whole bones from creep, repetitive loading, or intermittent
falls—and the biological responses to such damage—represents a complex structural analysis, requiring development of time-dependent damage accumulation and
biological repair constitutive models for the trabecular bone and correspondingly
large scale, materially nonlinear and time-dependent finite element models. The
in vivo animal experiments required to validate such models also present technical
challenges.
In the future, we expect that with the substantial body of current knowledge
about trabecular bone mechanics, and with the highly sophisticated experimentalcomputational techniques that have been developed, there will be significant advances made in many areas both at the tissue and whole bone levels. The field
of trabecular bone biomechanics is intensely rich. We believe that the best is yet
to come and that the key to continued successlies in multidisciplinary approaches
that can successfully address the complexity of this important biological tissue
ACKNOWLEDGMENTS
The authors acknowledge support from the National Institutes of Health (grants
AR41481, AR43784), the National Science Foundation (grant BES-9625030 and
a Graduate Student Fellowship), and the Miller Institute for Basic Research in
Science, Berkeley. Computer resources were provided in part by the National
Partnership for Advanced Computing Infrastructure and Lawrence Livermore
National Laboratory (ISCR B291837, 97-06 and 98-04).
Visit the Annual Reviews home page at www.AnnualReviews.org
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June 26, 2001
Figure 12 Various types of physical damage patterns observed in bovine trabecular bone
viewed with light microscopy at 200X magnification: (a) transverse cracks; (b) shear bands;
(c) parallel cracks; and (d ) complete fracture. (e) Diffuse damage in human vertebral trabecular bone viewed under polarized light and using basic fuchsin stain. (Pattern 1) Uniform
staining of a trabecular packet on the bone surface; (pattern 2) diffuse staining characterized
by linear and cross-hatched streaks. ( f ) Pattern 2 shown at higher magnification under
confocal microscopy. (Figure 12a–d from Reference 100; Figure 12e, f from Reference
102, with permission.)
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Figure 15 The central 3-mm cubic portion of a high-resolution finite element model of a bovine tibial trabecular bone specimen shaded to show
regions of yielded tissue at the apparent tensile (left) and compressive (right) 0.2% offset yield points. (Red) Regions exceeding the tensile tissue
yield strain; (blue) regions exceeding the compressive tissue yield strain. (From Reference 126, with permission.)
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Annual Review of Biomedical Engineering
Volume 3, 2001
CONTENTS
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THOMAS MCMAHON: A DEDICATION IN MEMORIAM, Robert D. Howe
and Richard E. Kronauer
BIOMECHANICS OF CARDIOVASCULAR DEVELOPMENT, Larry A. Taber
FUNDAMENTALS OF IMPACT BIOMECHANICS: PART 2—BIOMECHANICS
OF THE ABDOMEN, PELVIS, AND LOWER EXTREMITIES, Albert I. King
CARDIAC ENERGY METABOLISM: MODELS OF CELLULAR
RESPIRATION, M. Saleet Jafri, Stephen J. Dudycha, and Brian O’Rourke
THE PROCESS AND DEVELOPMENT OF IMAGE-GUIDED PROCEDURES,
Robert L. Galloway, Jr.
i
1
27
57
83
CAN WE MODEL NITRIC OXIDE BIOTRANSPORT? A SURVEY OF
MATHEMATICAL MODELS FOR A SIMPLE DIATOMIC MOLECULE
WITH SURPRISINGLY COMPLEX BIOLOGICAL ACTIVITIES, Donald G.
Buerk
VISUAL PROSTHESES, Edwin M. Maynard
MICRO- AND NANOMECHANICS OF THE COCHLEAR OUTER HAIR CELL,
W. E. Brownell, A. A. Spector, R. M. Raphael, and A. S. Popel
NEW DNA SEQUENCING METHODS, Andre Marziali and Mark Akeson
VASCULAR TISSUE ENGINEERING, Robert M. Nerem and Dror Seliktar
COMPUTER MODELING AND SIMULATION OF HUMAN MOVEMENT,
Marcus G. Pandy
STEM CELL BIOENGINEERING, Peter W. Zandstra and Andras Nagy
BIOMECHANICS OF TRABECULAR BONE, Tony M. Keaveny, Elise F.
Morgan, Glen L. Niebur, and Oscar C. Yeh
109
145
169
195
225
245
275
307
SOFT LITHOGRAPHY IN BIOLOGY AND BIOCHEMISTRY, George M.
Whitesides, Emanuele Ostuni, Shuichi Takayama, Xingyu Jiang, and
Donald E. Ingber
IMAGE-GUIDED ACOUSTIC THERAPY, Shahram Vaezy, Marilee Andrew,
Peter Kaczkowski, and Lawrence Crum
335
375
CONTROL MOTIFS FOR INTRACELLULAR REGULATORY NETWORKS,
Christopher V. Rao and Adam P. Arkin
391
vii
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viii
Annual Reviews
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CONTENTS
RESPIRATORY FLUID MECHANICS AND TRANSPORT PROCESSES, James
B. Grotberg
421
INDEXES
Subject Index
Cumulative Index of Contributing Authors, Volumes 1–3
Cumulative Index of Chapter Titles, Volumes 1–3
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ERRATA
An online log of corrections to Annual Review of Biomedical
Engineering chapters (1997 to the present) may be found at
http://bioeng.AnnualReviews.org/errata.shtml.
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483