Journal of Biomechanics 31 (1998) 601 — 608
`
`Yield strain behavior of trabecular bone
`
`David L. Kopperdahl!,*, Tony M. Keaveny!,"
`
`!Orthopaedic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740, U.S.A.
`"Department of Orthopaedic Surgery, University of California, San Francisco, CA 94143, U.S.A.
`
`Received in (cid:222)nal form 27 March 1998
`
`Abstract
`
`If bone adapts to maintain constant strains and if on-axis yield strains in trabecular bone are independent of apparent density,
`adaptive remodeling in trabecular bone should maintain a constant safety factor (yield strain/functional strain) during habitual
`loading. To test the hypothesis that yield strains are indeed independent of density, compressive (n"22) and tensile (n"22) yield
`strains were measured without end-artifacts for low density (0.18$0.04 g cm~3) human vertebral trabecular bone specimens. Loads
`were applied in the superior—inferior direction along the principal trabecular orientation. These (cid:212)on-axis(cid:213) yield strains were compared
`to those measured previously for high-density (0.51$0.06 g cm~3) bovine tibial trabecular bone (n"44). Mean ($S.D.) yield strains
`for the human bone were 0.78$0.04% in tension and 0.84$0.06% in compression; corresponding values for the bovine bone were
`0.78$0.04 and 1.09$0.12%, respectively. Tensile yield strains were independent of the apparent density across the entire density
`range (human p"0.40, bovine p"0.64, pooled p"0.97). By contrast, compressive yield strains were linearly correlated with
`apparent density for the human bone (p(0.001) and the pooled data (p(0.001), and a suggestive trend existed for the bovine data
`(p"0.06). These results refute the hypothesis that on-axis yield strains for trabecular bone are independent of density for compressive
`loading, although values may appear constant over a narrow density range. On-axis tensile yield strains appear to be independent of
`both apparent density and anatomic site. ( 1998 Elsevier Science Ltd. All rights reserved.
`
`Keywords: Trabecular bone; Yield strain; Safety factor; Remodeling; Spine
`
`1. Introduction
`
`It has been hypothesized that bone adapts to produce
`uniform functional apparent strains in both cortical and
`trabecular bone in response to habitual loads (Turner
`et al., 1997). If yield strains were also uniform, this would
`imply that the set point for remodeling is some ratio of
`the yield strain to the functional strain or a (cid:212)safety factor(cid:213).
`Wol⁄ (cid:213)s law implies that trabecular orientation aligns
`itself to the direction of the functional principal stresses
`(Cowin, 1986). As a consequence, the relevant yield
`strains in consideration of bone adaptation are those for
`(cid:212)on-axis(cid:213) loading of the bone (i.e. along the principal
`trabecular orientation). It has also been hypothesized
`that aging and disease may decrease the yield strain of
`bone and thus its safety factor (Biewener et al., 1993). One
`prerequisite to establishing the above hypotheses is to
`
`*Corresponding author. Tel.: (510) 642-3787; fax: (510) 642-6163;
`e-mail: kopper@euler.me.berkeley.edu
`
`0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.
`P I I S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 0 5 7 - 8
`
`determine the dependency of the on-axis yield strains of
`trabecular bone on apparent density and to do this for
`a range of anatomic sites. A more complete understand-
`ing of the yield strains in trabecular bone is also funda-
`mental to continued progress in computer modeling of
`whole bones (Keyak et al., 1993; Lotz et al., 1991; Silva et
`al., 1996), which in turn may improve diagnosis and
`treatment of pathologies that weaken trabecular bone
`such as osteoporosis.
`While there is mounting evidence that apparent failure
`strains in trabecular bone are independent of apparent
`density, the data are not conclusive. A number of studies
`have shown no dependence of compressive failure strains
`on apparent density (Ford and Keaveny, 1996; Hansson
`et al., 1987; Keaveny et al., 1994; Lindahl, 1976; Rohl et
`al., 1991), but a number of others have shown failure
`strains to increase (Hvid et al., 1989; Keaveny et al., 1994;
`Turner, 1989) or decrease (Hvid et al., 1985; Mosekilde
`et al., 1987) with increasing apparent density. It is not
`clear whether the former studies lacked statistical power
`to show a real dependence or if di⁄erences in anatomic
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`
`site, trabecular orientation, de(cid:222)nition of the failure strain,
`or experimental testing techniques accounted for these
`di⁄erent (cid:222)ndings. For example, accurate data on appar-
`ent failure strains for trabecular bone are diƒcult if not
`impossible to measure if end-artifacts are present since
`strain measures are highly sensitive to this artifact
`(Keaveny et al., 1993; 1997; Odgaard and Linde, 1991). In
`the context of understanding trabecular bone failure as it
`pertains to bone adaptation, very few data exist for
`on-axis apparent failure strains since specimens are usu-
`ally machined along anatomic directions which rarely
`align with the principal trabecular orientation. Thus,
`there is a need for a study on trabecular failure strains
`that minimizes end-artifact errors, uses bone from di⁄er-
`ent anatomic sites, and tests the bone in the on-axis
`orientation.
`Based on predictions of an axial strut cellular solid
`model for trabecular bone (Fig. 1) (Christensen, 1986;
`Gent and Thomas, 1959; Gibson, 1985; Gibson and
`Ashby, 1988; Rajan, 1985), the hypothesis was developed
`in this study that, for on-axis loading, compressive appar-
`ent yield strains should be positively correlated with
`apparent density due to underlying buckling mecha-
`nisms. Since the slenderness (length/thickness) ratio of
`individual trabeculae decreases as apparent density in-
`creases (Snyder et al., 1993), the signi(cid:222)cance of this rela-
`tionship should diminish as density increases due to
`a lower propensity for trabeculae to buckle. Since buck-
`ling cannot occur in tension, it was also hypothesized
`that the tensile apparent yield strains should remain
`constant regardless of anatomic site or species due to
`axial yielding of trabeculae. The following questions were
`addressed speci(cid:222)cally: (1) What are the relationships be-
`
`Fig. 1. Axial strut cellular solid model with strut thickness („), vertical
`strut length (‚) and horizontal strut length (H). This model was used to
`motivate the hypothesis that compressive yield strains should be posit-
`ively correlated with apparent density due to buckling of trabeculae,
`and that tensile yield strains should be independent of density due to
`axial yielding of trabeculae.
`
`tween the compressive and tensile yield strains vs appar-
`ent density for human vertebral trabecular bone? (2) Are
`these yield strains and strain—density relationships di⁄er-
`ent from those measured previously for bovine tibial
`trabecular bone (Keaveny et al., 1994)? and (3) For
`a single anatomic site, is the variance in yield strains
`small enough to reasonably assume constant yield strains
`in bone adaptation simulations? The results were then
`discussed in the context of bone adaptation, aging, and
`disease.
`
`2. Methods
`
`Seventeen fresh frozen vertebrae, T10-L4, without
`radiographic evidence of bone pathologies, were ob-
`tained from 11 cadavers (Table 1). Forty-eight cylindrical
`specimens (8 mm diameter, 25 mm length nominally)
`were cored in water with marrow in situ along the su-
`perior—inferior direction. With the central portion
`(&15 mm) wrapped in damp gauze, marrow was re-
`moved from the ends (&5 mm), which were then glued
`into pre-aligned brass end-caps. The specimens were then
`randomly assigned to a tension or compression group.
`This protocol eliminates end-artifact errors during test-
`ing (Keaveny et al., 1997). The data from a previous
`experiment (Keaveny et al., 1994) were obtained from 44
`specimens from the bovine proximal tibia. These speci-
`mens were also cored parallel to the trabecular orienta-
`tion, but with a reduced cross-section.
`Uniaxial compressive and tensile mechanical tests
`were performed on the vertebral specimens at room
`temperature using a servohydraulic load frame (858
`mini-bionix, MTS, Eden Prairie, MN). The end-capped
`specimens were cycled (cid:222)ve times nondestructively
`between $0.1% strain (fully reversed compression-ten-
`sion) at 0.5% strain per second, allowing a paired com-
`parison between the compressive and tensile moduli.
`
`Table 1
`Seventeen fresh frozen vertebrae, T10—L4, and 44 cylindrical trabecular
`bone specimens, none of which showed radiographic evidence of bone
`pathologies, were obtained from 11 human cadavers, ages 32—65 yr
`(mean "54, SD"11 yrs)
`
`Cadaver
`ID
`
`Segment
`level
`
`No. of
`Cylindrical
`specimens
`
`Age
`
`Sex
`
`A
`B
`C
`D
`E
`F
`G
`H
`I
`J
`K
`
`L4
`T11, T12
`L2
`T12
`T12, L2
`T11, T12
`T12, L1
`L4
`L4
`T10—T12
`L4
`
`4
`6
`3
`2
`6
`4
`6
`2
`2
`6
`3
`
`50
`63
`58
`61
`64
`61
`58
`65
`37
`50
`32
`
`M
`F
`F
`F
`F
`M
`M
`F
`F
`M
`M
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`D.L. Kopperdahl, T.M. Keaveny / Journal of Biomechanics 31 (1998) 601—608
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`603
`
`Specimens were then loaded destructively to 3% strain in
`either compression or tension. Strains were measured by
`a 25 mm gage length extensometer (632.11F-20, MTS,
`Eden Prairie, MN) attached to the end-caps. (The bovine
`specimens had been tested previously with a miniature
`extensometer attached to the reduced-section, but our
`subsequent work on this bone has shown that this
`method produces strains in agreement with the current
`method.) The e⁄ective gage length was assumed to be the
`length of the bone exposed between the end-caps plus
`half the length of bone embedded in the end-caps. This
`general
`(cid:212)end-cap(cid:213)
`technique has been validated for
`measurement of elastic properties without end-artifacts
`(Keaveny et al., 1997). A miniature 5 mm extensometer
`was attached to the central region of 29 specimens. These
`
`Fig. 2. Typical compression and tension stress—strain curves for the
`(cid:222)nal destructive loading ramp for two human vertebral specimens of
`di⁄erent apparent density (o). The lower density of the tensile specimen
`accounts for its lower modulus and strength. The absence of any
`nonlinear toe region indicates that end-artifacts were successfully elimi-
`nated. Note that the stress—strain curves for both specimens have
`clearly entered the nonlinear region at strains of less than 1.0%, while
`ultimate strains were less than 2.0%. X indicates fracture of the tension
`specimen; the compression specimen did not fracture.
`
`data veri(cid:222)ed that the end-cap technique was valid for
`measuring failure properties also, since specimens did not
`fail preferentially at their ends due to potential stress
`concentrations at the end-caps. Overall, 44 of the 48
`specimens were successfully tested (n"22, compression;
`n"22, tension). After testing, specimens were sectioned
`from the end-caps and cleaned of marrow. Water was
`removed from the marrow space with an air jet. Hy-
`drated apparent densities were calculated as the wet mass
`divided by bulk volume.
`Analysis of variance, t-tests and regression analysis
`were performed (Systat, Version 5.2, Systat Inc., Evan-
`ston, IL) to analyze the data. Elastic modulus was de(cid:222)ned
`as the slope of the best (cid:222)t straight line to the stress—strain
`data over a range of 0.02—0.24% strain (Fig. 2). The yield
`point was de(cid:222)ned using a 0.2%-strain o⁄set method; and
`the ultimate point was de(cid:222)ned at the point of maximum
`stress. Similar protocols were used for the bovine tibial
`bone, except the modulus was de(cid:222)ned over the range of
`0.1—0.4% strain (Keaveny et al., 1994). These strain
`ranges were chosen in order to sample data over as much
`of the elastic range as possible, which was found to be
`larger for the bovine bone.
`
`3. Results
`
`Tensile yield strains for the human vertebral bone were
`independent of apparent density (p"0.31) while com-
`pressive yield strains showed a weak but highly signi(cid:222)-
`correlation with density (r2"0.52,
`cant positive
`p"0.0002) (Table 2). Ultimate strains were independent
`of apparent density in both compression (p"0.64) and
`tension (p"0.19), although these data showed consider-
`ably more scatter than the yield strains. As expected,
`elastic modulus (Fig. 3), yield stress (Fig. 4), and ultimate
`stress for the human vertebral bone demonstrated strong
`positive correlations with apparent density.
`
`Table 2
`Linear and power law regressions relating mechanical properties (‰) to apparent density (o, g cm~3) for the human vertebral trabecular bone. Yield
`strains were linearly related to apparent density in compression, but independent of apparent density in tension. Exponents for the strength vs
`apparent density power laws tended towards 1.6 in compression and 1.0 in tension. (n"22 unless otherwise indicated)
`
`‰"a#bo
`
`Compression
`
`‰"ao"
`
`Tension
`
`Compression
`
`a
`
`0.66
`
`Yield strain (%)
`Ultimate strain (%)
`!1.40
`Yield stress (MPa)
`Ultimate stress (MPa) !1.46
`Modulus (MPa)!
`—
`
`b
`
`1.09
`NS
`19.6
`21.9
`
`2100
`
`r2
`
`0.49
`
`0.73
`0.71
`
`0.61
`
`a
`
`—
`—
`
`b
`
`NS
`NS
`10.1
`13.2
`
`r2
`
`a
`
`0.51
`0.47
`
`1.24
`
`32.6
`33.2
`
`2350
`
`b
`
`0.21
`NS
`1.60
`1.53
`
`1.20
`
`r2
`
`0.48
`
`0.70
`0.68
`
`0.60
`
`Tension
`
`a
`
`10.0
`13.3
`
`b
`
`NS
`NS
`1.04
`1.07
`
`r2
`
`0.51
`0.47
`
`—indicates intercept in the linear regression was not signi(cid:222)cantly di⁄erent from zero (p’0.05); NS indicates regression was not signi(cid:222)cant
`(p’0.05).
`!Regressions for modulus are for pooled compression-tension data (n"44).
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`D.L. Kopperdahl, T.M. Keaveny / Journal of Biomechanics 31 (1998) 601—608
`
`Fig. 3. Pooled compressive and tensile elastic moduli for the human
`vertebral specimens (n"44) were linearly proportional to apparent
`density. The intercept was not signi(cid:222)cantly di⁄erent
`from zero
`(p"0.10).
`
`Fig. 4. Compressive and tensile yield strains were strongly correlated
`with apparent density. When buckling dominates compressive axial
`failure of trabeculae, the axial strut cellular solid model predicts
`a squared relationship between compressive yield strain and density.
`A least-squares power-law (cid:222)t to this data predicts an exponent of
`1.6 (Table 2). Since buckling cannot occur in tension, axial yielding
`dominates tensile failure and the axial strut model predicts a linear
`relationship.
`
`When the low-density human vertebral and high-
`density1 bovine tibial specimens were pooled, yield
`strain behavior remained similar across the entire range
`of densities for tension but di⁄ered in compression
`(Fig. 5). Tensile yield strains
`for
`the pooled data
`(mean$S.D."0.78$0.04%) were independent of ap-
`parent density (p"0.97), and values were not di⁄erent
`(p"0.99) for the two types of bone. By contrast, the
`
`1Apparent densities for the bovine trabecular bone were originally
`reported based on QCT mineral densities (Keaveny et al., 1994). We
`noticed in retrospect that values were too large with respect to similar
`specimens for which apparent densities were directly measured. There-
`fore, QCT-based apparent densities were scaled down 17% to produce
`a correct mean apparent density (mean $S.D."0.51$0.06 g cm~3,
`range "0.39—0.65 g cm~3).
`
`Fig. 5. Compressive and tensile yield strains vs wet apparent density
`for both human vertebral (current study) and bovine proximal tibial
`(Keaveny et al., 1994) trabecular bone specimens tested on-axis without
`end-artifacts. Tensile yield strains were constant at approximately
`0.78% strain across the entire range of densities (dotted line). At low
`densities, compressive yield strains were linearly related to density
`(p"0.0003, solid line), but approximately constant at 1.09% strain at
`high densities (horizontal dash—dot line) although a positive trend did
`exist (p"0.06, dashed line).
`
`Table 3
`Mean values ($S.D.) and ranges of compressive and tensile mechan-
`ical properties for the human vertebral specimens. Yield strains were
`signi(cid:222)cantly higher in compression. All other measures were statist-
`ically similar in compression and tension. Corresponding data for the
`bovine tibial specimens are reported elsewhere (Keaveny et al., 1994)
`
`Wet apparent density
`(g cm~3)
`Modulus (MPa)
`
`Yield strain (%)
`
`Ultimate strain (%)
`
`Yield stress (MPa)
`
`Ultimate stress (MPa)
`
`Compression Tension
`(n"22)
`(n"22)
`
`p!
`
`0.17$0.04
`0.11!0.26
`291$113
`90!536
`0.84$0.06
`0.75!0.95
`1.45$0.33
`0.96!2.30
`1.92$0.84
`0.56!3.71
`2.23$0.95
`0.70!4.33
`
`0.19$0.04
`0.12!0.27
`301$100
`139!472
`0.78$0.04
`0.71!0.88
`1.59$0.33
`1.09!2.51
`1.75$0.65
`0.77!2.75
`2.23$0.76
`1.33!3.53
`
`0.12
`
`0.76"
`
`0.0003
`
`0.18
`
`0.46
`
`0.99
`
`! p-values for comparison of compressive vs tensile group means,
`using unpaired Student(cid:213)s t-test.
`" A paired Student(cid:213)s t-test (n"44) for compressive vs tensile moduli
`for each specimen indicated that tensile modulus was signi(cid:222)cantly
`higher than compressive modulus (p(0.001), but the di⁄erence was
`negligible ((1%). Thus, the compressive and tensile moduli of each
`specimen were considered as equal. The mean ($S.D.) modulus for the
`pooled data was 309$109 MPa.
`
`mean ($S.D.) compressive yield strains for the bovine
`tibial bone (1.09$0.12%) were larger than values for
`the human vertebral bone (0.84$0.06%) by 30%
`(p(0.0001). The pooled compressive yield strain data
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`D.L. Kopperdahl, T.M. Keaveny / Journal of Biomechanics 31 (1998) 601—608
`had a strong positive linear correlation (p"0.0003) with
`apparent density, although the high density bovine bone,
`when considered alone, had only a marginal correlation
`(p"0.06).
`The variation in tensile yield strains was small for the
`entire density range considered, while the variation in
`compressive yield strain was small only for a single ana-
`tomic site (Table 3). For the entire density range, the
`percentage coeƒcient of variation for tensile yield strains
`was only 5.5%. Despite the dependence of compressive
`yield strains on apparent density for the low density
`human vertebral bone, the coeƒcient of variation for
`compressive yield strains was only 6.8% (Table 3). This
`was lower than the coeƒcient of variation for the higher
`density bovine bone (11.3%), and for the pooled data
`(16.1%). Thus, the error induced by assuming constant
`compressive yield strains increased as both the density
`and the density range increased.
`
`4. Discussion
`
`The goal of this work was to provide comprehensive
`data on the failure strains of trabecular bone, with
`speci(cid:222)c reference to their role in bone adaptation,
`disease, and aging. Our results provide strong evidence
`that
`(1)
`the on-axis
`compressive apparent yield
`strains correlate positively but weakly with apparent
`density, the correlation being stronger in less dense bone;
`and (2) the on-axis tensile apparent yield strains represent
`a uniform failure property independent of anatomic site
`and apparent density. With respect to analysis of bone
`adaptation and failure, these results demonstrate that
`over a narrow density range, and particularly at high
`densities, trabecular yield strains can reasonably be as-
`sumed to be constant. If the range of densities in a region
`of interest is not known a priori, tensile yield strains may
`still be considered constant, but compressive yield strains
`should be assigned as a function of density as reported
`here.
`Biewener (1993) has suggested that safety factors are
`lower in osteoporotic bone due to a decrease in the yield
`strain with decreased density. Results of this study sug-
`gest otherwise, namely that an increase in functional
`strains — and not a decrease in failure strains — is mostly
`responsible for age related decreases in the safety factor
`of whole bones. Changes in yield strain due to changes in
`density (and presumably age) at a speci(cid:222)c anatomic site
`are minor, and therefore may have only a minor e⁄ect on
`reductions in safety factors for trabecular bone. Instead,
`the strong correlation between modulus and density indi-
`cates that the decrease in density that accompanies aging
`(Mosekilde et al., 1987) results in bone that is more
`compliant, i.e. that strains more for a given load. Thus,
`functional strains will tend to approach the yield strain as
`density decreases with aging.
`
`605
`
`The validity of these conclusions is largely supported
`by the accuracy of the mechanical test data and the wide
`range of bone analyzed. The data sets for each anatomic
`site were generated using similar techniques that ensured
`elimination of end-artifacts and on-axis loading. For the
`human vertebral trabecular bone, the mean modulus
`(309$109 MPa), while 4.5—13.5 times higher than pre-
`vious studies that mechanically tested human vertebral
`bone between platens (Hansson et al., 1987; Lindahl
`1976; Mosekilde et al., 1987), is similar to values reported
`where end-artifacts errors were eliminated (Ashman et
`al., 1987; Neil et al., 1983). The mean compressive ulti-
`mate strain (1.45%) is (cid:222)ve times lower than those from
`studies using a platens test con(cid:222)guration (Hansson et al.,
`1987; Mosekilde et al., 1987). Although no previous stud-
`ies have reported ultimate strains free of end-artifact
`errors for human vertebral bone, mean values of
`1.11—1.86% strain have been reported for other anatomic
`sites (Keaveny et al., 1994; Rohl et al., 1991) and thus are
`consistent with these new data. The use of bone from two
`anatomic sites that exhibited large di⁄erences in apparent
`density and architecture (rod for human vertebral bone
`vs plate for bovine tibial bone) also provides substantial
`generality to the results.
`Despite these advantages, some caveats exist. In for-
`mulating our hypothesis, an assumption was made that
`the failure properties of the tissue comprising individual
`trabeculae were uniform. The small coeƒcient of vari-
`ation in tensile apparent yield strains (5.5%) and their
`similarity between human vertebral vs bovine tibial spec-
`imens indicates that age- or site-related di⁄erences in
`trabecular tissue morphology must have only a minor
`e⁄ect on tissue yield strains. Another caveat is that these
`results only apply to on-axis loads. Thus, there is direct
`application to biomechanical analysis of habitual activ-
`ities in which loads presumably act along the principal
`trabecular orientation by implication of Wol⁄(cid:213)s law
`(Cowin, 1986). Traumatic fractures resulting from o⁄-
`axis loads, such as a fall to the side of the hip, result in
`multiaxial stresses with respect to the on-axis coordinate
`system, and the trends reported here may not apply to
`that situation.
`With the hindsight provided by this study, the data
`from the literature are consistent with the principle that
`yield strains in trabecular bone demonstrate a weak
`dependence on apparent density in compression, but not
`in tension (Table 4). Our data indicate that the small
`slope of the density-failure strain regression may go un-
`detected statistically if considerable scatter exists in the
`failure strain data, particularly if ultimate strains are used
`instead of yield strains. This would explain why studies
`with relatively small sample sizes found no signi(cid:222)cant
`relationship between ultimate
`strains and density
`(Keaveny et al., 1994; Rohl et al., 1991), while others did
`(cid:222)nd a signi(cid:222)cant relationship by using a large sample size
`(Hvid et al., 1989) or by de(cid:222)ning failure at the yield point
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`606
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`D.L. Kopperdahl, T.M. Keaveny / Journal of Biomechanics 31 (1998) 601—608
`
`Table 4
`Summary of yield and ultimate strains reported in the literature including the current study. Mean ($S.D.) for failure strains are reported as well as
`correlation coeƒcients (r) and signi(cid:222)cance levels (p) for linear regressions between failure strain vs apparent density
`
`Reference
`
`Compression:
`Lindahl (1976)!
`
`Bone type
`
`Human vertebral body
`
`Human proximal tibia
`
`Hvid et al. (1985)
`
`Canine proximal tibia
`
`Hansson et al. (1987)
`
`Human vertebral body
`
`Mosekilde et al. (1987)"
`Hvid et al. (1989)
`Turner (1989)
`Rohl et al. (1991)
`Keaveny et al. (1994)
`
`Human vertebral body
`Human proximal tibia
`Bovine distal femur
`Human proximal tibia
`Bovine proximal tibia
`
`Current study
`
`Human vertebral body
`
`„ension:
`Rohl et al. (1991)
`Keaveny et al. (1994)
`
`Human proximal tibia
`Bovine proximal tibia
`
`Current study
`
`Human vertebral body
`
`n
`
`32
`32
`32
`32
`192
`
`231
`
`40
`94
`61
`15
`20
`
`22
`
`15
`20
`
`22
`
`Measure
`
`Strain (%)
`
`r
`
`[p]
`
`Yield
`Ultimate
`Yield
`Ultimate
`Yield
`Ultimate
`Yield
`Ultimate
`Ultimate
`Ultimate
`Yield
`Ultimate
`Yield
`Ultimate
`Yield
`Ultimate
`
`Ultimate
`Yield
`Ultimate
`Yield
`Ultimate
`
`6.1$2.3
`9.0$4.5
`6.9$4.5
`11.6$6.2
`NR
`NR
`6.0$2.2
`7.4$2.4
`7.4$1.3
`NR
`1.24$0.197
`1.11$0.63
`1.09$0.12
`1.86$0.49
`0.81$0.06
`1.45$0.3
`
`1.55$0.49
`0.78$0.04
`1.37$0.33
`0.78$0.04
`1.59$0.33
`
`—
`—
`—
`—
`!0.18
`!0.14
`—
`—
`!0.37
`0.26
`0.55
`
`0.42
`
`0.72
`
`—
`
`—
`
`—
`
`—
`—
`—
`—
`—
`
`NS
`NS
`NS
`NS
`0.0153
`0.0467
`NS
`NS
`(0.05
`NR
`(0.001
`NS
`0.062
`NS
`(0.001
`
`NS
`NS
`NS
`NS
`NS
`
`!Failure strains were not regressed against apparent density, but no correlation was found between failure strain vs age. Values shown here are for
`female subjects, but were similar for male subjects.
`"Values reported here are for the vertical direction.
`NS: Regression was not signi(cid:222)cant (p’0.05); NR: Value was not reported.
`
`(Keaveny et al., 1994; Turner, 1989). Studies reporting
`failure strains in excess of 6% presumably had too much
`experimental error to provide reliable failure strain data
`(Hansson et al., 1987; Lindahl, 1976; Mosekilde et al.,
`1987). Cellular solid analysis (Gibson and Ashby, 1988;
`Turner, 1989) also suggests that the relationship between
`yield strain and apparent density can vary from a positive
`linear correlation, when failure is dominated by buckling
`of trabeculae aligned with the applied load, to a negative
`correlation due to bending of trabeculae loaded o⁄-axis.
`While loading in the inferior—superior direction assures
`inferior—su-
`on-axis loading for vertebral specimens,
`perior-oriented specimens from other locations such as
`the tibia and femur may experience o⁄-axis loads. Thus,
`a negative correlation between failure strain vs apparent
`density for the canine proximal tibia (Hvid et al., 1985)
`may be due to bending of trabeculae due to o⁄-axis loads.
`Our results are also consistent with all available tensile
`strain data, which consistently indicate a lack of depend-
`ence on density (Keaveny et al., 1994; Rohl et al., 1991).
`Since the slenderness ratio of individual trabeculae
`decreases as apparent density increases (Snyder et al.,
`1993), there should exist some density below which buck-
`ling dominates on-axis compressive failure, and above
`which axial yielding dominates. The intersection of the
`
`two regression lines for the high and low-density bone
`(Fig. 5, solid and dash—dot lines) suggests that this
`transition occurs at a density of about 0.40 g cm~3. This
`agrees well with the value of 0.38 g cm~3 predicted using
`the Euler buckling theorem (Appendix A). This is also the
`density around which trabecular structure is observed to
`change from mainly a rod to a plate architecture (Gibson,
`1985). In an actual specimen, there is a distribution in the
`slenderness ratios of trabeculae. Thus, both buckling in
`slender rods and yielding in stout rods or plates may be
`present simultaneously, with one mechanism dominating
`failure depending on the apparent density. This di⁄erence
`in compressive failure mechanisms provides a plausible
`explanation for the increased variance in compressive
`yield strains at higher densities.
`One additional biological implication of this study
`relates to the possible role of trabecular tissue damage in
`bone remodeling (Carter, 1984; Mori and Burr, 1993;
`Prendergast and Taylor, 1994). Although the yield strains
`reported here for the human vertebral bone were in the
`range of 0.71—0.95%, subtle damage initiates at lower
`strains. In a pilot experiment, human vertebral trabecular
`bone specimens were cyclically loaded without end-
`artifacts to 0.30% strain in either on-axis compression
`or tension. Permanent deformation occurred in all
`
`ALPHATEC HOLDINGS, INC., ALPHATEC SPINE INC., -
`IPR2019-00362, Ex. 1037, p. 6 of 8
`
`

`

`D.L. Kopperdahl, T.M. Keaveny / Journal of Biomechanics 31 (1998) 601—608
`
`607
`
`Appendix B
`
`In vivo apparent strains for vertebral trabecular bone
`can be approximated using a simple elliptical model with
`major and minor diameters of 46 and 32 mm, respectively
`(Berry et al., 1987); a cortical shell with a thickness of
`0.4 mm (Silva et al., 1994) and modulus of 5 GPa (Choi et
`al., 1990); and a trabecular centrum with a modulus of
`300 MPa. Wilson and Meyers (1996) recently estimated
`the compressive load on the L2 vertebra of a woman of
`average build lifting an 8 kg mass to be 1320 N. Assum-
`ing uniform load sharing between the cortical shell and
`centrum (i.e. assuming the cortical and trabecular bone
`behave as springs in parallel), this model predicts strains
`in the trabecular centrum of 0.23%.
`
`References
`
`Ashman, R.B., Corin, J.D., Turner, C.H., 1987. Orthotropic elastic
`properties of the human spine. In Transactions of the Orthopaedic
`Research Society 12, 368.
`Berry, J.L., Moran, J.M., Berg, W.S., Ste⁄ee, A.D., 1987. A morphomet-
`ric study of human lumbar and selected thoracic vertebrae. Spine 12,
`362—367.
`Biewener, A.A., Fyhrie, Par(cid:222)tt, Davy, Scha§er, Heaney, 1993. Safety
`factors in bone strength. Calci(cid:222)ed Tissue Research International 53,
`S68—S74.
`Carter, D.R., 1984. Mechanical loading histories and cortical bone
`remodeling. Calci(cid:222)ed Tissue Research International 36 Suppl 1,
`S19—S24.
`Choi, K., Kuhn, J.L., Ciarelli, M.J., Goldstein, S.A., 1990. The elastic
`moduli of human subchondral, trabecular, and cortical bone tissue
`and the size-dependency of cortical bone modulus. Journal of Bio-
`mechanics 23, 1103—1113.
`Christensen, R.M., 1986. Mechanics of low density materials. Journal of
`the Mechanics of the Physics of Solids 34, 563—578.
`Cowin, S., 1986. Wol⁄(cid:213)s Law of
`trabecular architecture at
`remodeling equilibrium. Journal of Biomechanical Engineering 108,
`83—88.
`Ford, C.M., Keaveny, T.M., 1996. The dependence of shear failure
`properties of trabecular bone on apparent density and trabecular
`orientation. Journal of Biomechanics 29, 1309—1317.
`Gent, A.N., Thomas, A.G., 1959. The deformation of foamed elastic
`materials. Journal of Applied Polymer Science 1, 107—113.
`Gibson, L.J., 1985. The mechanical behavior of cancellous bone. Jour-
`nal of Biomechanics 18, 317—328.
`Gibson, L.J., Ashby, M.F., 1988. Cellular Solids: Structures & Proper-
`ties, Pergamon Press, Oxford, UK.
`Hansson, T.H., Keller, T.S., Panjabi, M.M., 1987. A study of the
`compressive properties of lumbar vertebral trabeculae: E⁄ects of
`tissue characteristics. Spine 11, 56—62.
`Hvid, I., Bentzen, S.M., Linde, F., Mosekilde, L., Pongsoipetch, B.,
`1989. X-ray quantitative computed tomography: The relations to
`physical properties of proximal tibial trabecular bone specimens.
`Journal of Biomechanics 22, 837—844.
`Hvid, I., Jensen, N.C., Bunger, C., Solund, K., Djurhuus, J.C., 1985.
`Bone mineral assay: Its relation to the mechanical strength of cancel-
`lous bone. Engineering Medicine 14, 79—83.
`Keaveny, T.M., Borchers, R.E., Gibson, L.J., Hayes, W. C., 1993. Theor-
`etical analysis of the experimental artifact in trabecular bone com-
`pressive modulus. Journal of Biomechanics 26, 599—607.
`
`specimens indicating that some type of tissue level dam-
`age occurred at this strain level. In vivo apparent strains
`have not been measured for vertebral trabecular bone,
`but can be approximated using a simple elliptical model
`(Appendix B) to be about 0.23%. This suggests that
`during strenuous activities, trabecular apparent strains in
`the spine may be high enough to cause damage. While
`this damage may have little direct mechanical conse-
`quence at the apparent level,
`it may interrupt the
`canalicular network or produce other ultrastuctural dis-
`continuities and therefore stimulate bone turnover.
`
`Acknowledgements
`
`Funding was provided by NIH AR41481 and
`AR43784. Bone tissue was provided by the Harvard
`Anatomical Gifts Program, the UCSF School of Medi-
`cine, and the Southwestern Medical Center at the Uni-
`versity of Texas, Dallas. Thanks to Peter Kim and Albert
`Lou for technical assistance.
`
`Appendix A
`
`The Euler Buckling equation can be used to relate the
`critical buckling stress (p
`#3) of an individual trabecula to
`its length : thickness slenderness ratio (‚/„):
`" Cn2E
`(2‚/„)2
`
`p
`#3
`
`,
`
`