Available online at www.sciencedirect.com
Medical Engineering & Physics 31 (2009) 108–115
Vibrational testing of trabecular bone architectures
using rapid prototype models
P. Mc Donnell a,b,∗ , M.A.K. Liebschner c , Wafa Tawackoli d , P.E. Mc Hugh a,b
a National Centre for Biomedical Engineering Science, National University of Ireland, Galway, Ireland
b Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland
c Department of Bioengineering, Rice University, Houston, TX, USA
d Cedars-Sinai Medical Center, Los Angeles, CA, USA
Received 8 November 2007; received in revised form 29 April 2008; accepted 30 April 2008
The purpose of this study was to investigate if standard analysis of the vibrational characteristics of trabecular architectures can be used to
detect changes in the mechanical properties due to progressive bone loss. A cored trabecular specimen from a human lumbar vertebra was
CT scanned and a three-dimensional, virtual model in stereolithography (STL) format was generated. Uniform bone loss was simulated
using a surface erosion algorithm. Rapid prototype (RP) replicas were manufactured from these virtualised models with 0%, 16% and 42%
bone loss. Vibrational behaviour of the RP replicas was evaluated by performing a dynamic compression test through a frequency range
using an electro-dynamic shaker. The acceleration and dynamic force responses were recorded and fast Fourier transform (FFT) analyses
were performed to determine the response spectrum. Standard resonant frequency analysis and damping factor calculations were performed.
The RP replicas were subsequently tested in compression beyond failure to determine their strength and modulus. It was found that the
reductions in resonant frequency with increasing bone loss corresponded well with reductions in apparent stiffness and strength. This suggests
that structural dynamics has the potential to be an alternative diagnostic technique for osteoporosis, although significant challenges must be
overcome to determine the effect of the skin/soft tissue interface, the cortex and variabilities associated with in vivo testing.
© 2008 IPEM. Published by Elsevier Ltd. All rights reserved.
Keywords: Structural dynamics; Resonant frequency; Cancellous bone; Osteoporosis; Bone micro-architecture
Current diagnostic methods for osteoporosis are imagebased
and focus on measurement of bone mineral density
(BMD) instead of bone integrity. Although, the mechanical
strength and stiffness of bone is highly correlated to BMD
[1,2], it has been reported that BMD alone can account for
less than 40% of the variation of the strength of trabecular
bone [1,3,4]. This is due to biological variability and the fact
that BMD is only one aspect of bone quality [5,6], which
does not fully account for the effect of changes in trabecular
architecture and variations in tissue level properties. A direct
∗ Corresponding author at: National Centre for Biomedical Engineering
Science, National University of Ireland, Galway, Ireland.
Tel.: +353 86 4067530.
E-mail address: firstname.lastname@example.org (P. Mc Donnell).
mechanical measurement of trabecular bone properties could
improve the diagnosis of osteoporosis and detect bone loss
at an early stage.
Vibrational testing provides a potential means of achieving
this goal in vivo. It is a non-destructive test, which is based
on the principle that the resonant response of a structure is
dependent on its overall size, stiffness, material property distribution
and density. For example, the resonant frequencies
of a solid beam which is vibrating in longitudinal mode along
its axis, are given by :
f j = 2j − 1 E
where f j is the jth natural frequency, L is the length of beam,
E is the Young’s modulus of material and ρ is the density
of material (for trabecular bone this is equivalent to
1350-4533/$ – see front matter © 2008 IPEM. Published by Elsevier Ltd. All rights reserved.
P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115 109
BV/TV multiplied by the tissue density). Studies have shown
that the modulus–density relationship for trabecular bone is
non-linear  which would be a requirement for the successful
application of vibrational techniques for evaluating bone
It was not until the 1970s that vibrational techniques were
investigated for directly determining the mechanical properties
of long bones [9–11]. However, clinical validation was
not performed. More recently, Weinhold et al.  performed
dynamic compression tests on excised cores of ovine vertebral
bone and found good correlation between the vibrational
modulus and the elastic modulus determined from a standard
compression test. In addition, theoretical models of trabecular
bone networks have been developed where the reduction
in effectiveness of the load transfer mechanism in the network
could be predicted from the ratio of the high frequency
vibrational response to the static response [13–15].
The objective of this study was to investigate the vibrational
characteristic of trabecular architectures to detect
changes in the strength and stiffness of trabecular bone, using
rapid prototype (RP) replicas. The use of RP replicas was
supported by the results of a prior study by the authors’
research group in which compression tests were performed
on 20 cored trabecular specimens from ovine vertebrae and
20 corresponding RP replicas . It was found that the modulus
and ultimate stress results for the RP replicas correlated
well with the real bone specimens and there were also striking
similarities in the failure mechanisms and failure locations. In
addition, other researchers have shown that RP replicas provide
a realistic representation of the mechanical behaviour
of trabecular bone . This approach allowed replicas with
different levels of artificially induced bone loss to be manufactured
from an original healthy architecture. It also allowed
models to be manufactured with the same material volume
fraction, but with their axes aligned in the cranio-caudal and
2. Materials and methods
An excised trabecular specimen from the T-10 vertebra
of an 82-year-old woman was used to generate multiple RP
models. A square cross-section specimen, with dimensions
5.7 mm × 5.7 mm × 5.5 mm, was extracted in the craniocaudal
direction and was scanned at 30 m resolution using
a micro-computed tomography (CT) scanner (CT 80,
SCANCO Medical AG, Switzerland). The bone volume fraction
(BV/TV) was determined to be 0.14. The scan images
were converted into a 3D STL reconstruction using Mimics
software (Mimics 9.0, Materialise RapidParts, UK). A
custom written surface erosion algorithm was used to uniformly
reduce the thickness of all trabeculae, resulting in
virtual models in stereolithography (STL) format with 0%,
16% and 42% bone loss. These are referred to as the Level
0, Level 2 and Level 4 models, respectively. The surface erosion
algorithm models secondary osteoporosis (e.g. during
bed-rest or extended space travel) by synthetically degrading
a bone sample by uniform resorption of material from
its surface [18,19]. This process imposes trabecular thinning
as well as perforation of trabecular bone. Specifically,
all voxels that have nine or fewer occupied neighbors are
removed at each level of surface erosion. It should be noted
that age-related osteoporosis is more appropriately modeled
by including a consideration of adaptive remodeling based
on strain levels in the trabecular network under normal loading
conditions [20–23]. However, the exact mechanism of
how bone formation and bone resorption counterplay during
senile osteoporosis is still under investigation, and therefore
was outside the scope of this study.
A software design program (DeskArtes Design Expert,
version 7.1, DeskArtes Oy, Finland) was used to convert the
models to a cylindrical cross-section of 5.5 mm diameter with
the axis in the cranio-caudal direction. Integrated end plates
were applied to reduce end-artifacts in the compression testing
and to allow uniform vibration conduction from the tip of
the shaker through the sample. An additional model was generated
with no bone loss and with the axis in the transverse
direction. This is referred to as the Transverse model.
Each model was imported into the proprietary software
for the RP laser sinterstation (SLS 2500 Plus, DTM Corporation,
USA). The maximum resolution of the sinterstation
is 0.1 mm. Therefore, the models were scaled up by 10 times
so that trabecular features with a size scale of the order of
0.01 mm in the real bone specimens could be captured in the
corresponding RP replica. Based on Eq. (1), the effects of
resonance frequency changes due to density loss and reduction
in stiffness can be scaled by the size of the sample. PA-12
powder was used as the raw material (3D Systems, UK). The
energy density of the sintering laser was set to 0.016 J/mm 2 .
Prior process parameter studies have shown that this level
of energy density results in modulus and strength properties,
which are very similar in the vertical and transverse directions
of the RP build . Four models were manufactured:
Level 0, Level 2, Level 4 and Transverse (Fig. 1).
The models were mounted on a custom-testing jig (Fig. 2).
A dynamic compression test was performed through a frequency
sweep on each model. The testing jig consisted of
an electro-dynamic shaker system (Model F4/F7, Wilcoxon
Research Inc., MD, USA) which was mounted on a rigid
frame. The shaker system incorporated an impedance head,
which contained a force gauge and an accelerometer. The
force gauge measured the dynamic force applied to the structure
and the accelerometer measured the resulting motion.
The signals were generated and acquired by LabView
(Labview 8.2, National Instruments, TX, USA) through a
PXI-DAQ system connected to a PC. A load cell measured
the static compression load of approximately 20 N that was
applied to the RP model before the shaker was activated to
prevent loss of contact during the test. A conical load applicator
applied the vibration load to the model under test.
The range of the frequency sweep was from 50 Hz to
500 Hz in steps of 3 Hz. This range was chosen based on the
110 P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115
Fig. 1. Rapid prototyped models with 0%, 16% and 42% bone loss. The initial bone volume fraction at Level 0 was 0.14 BV/TV.
results of initial trials, which showed that the first resonant
frequency always occurred in this range for the models tested
in this study. A frequency range from 500 Hz to 5000 Hz,
was also investigated for the low stiffness Transverse model
but the second resonant frequency could not be identified.
It was therefore decided to limit the current study to analysis
of the 1st resonant condition. At each frequency step,
the stimulus signal, acceleration response and force response
signals were sampled. Fast Fourier transform (FFT) analysis
was performed on this data set and the signal was normalised
to shaker input power to determine the power response spectrum.
The phase change and gain in amplitude of the response
signals relative to the stimulus signal were calculated and
plotted at each frequency step. The first resonant frequency
was determined from these frequency plots. Other measures,
such as half-power bandwidth and damping factor, were also
calculated to determine if they could be used as additional
parameters to resonant frequency for detecting changes in
Additional frequency sweep tests were performed on the
RP models to determine the dependence of the resonant frequency
results on testing conditions. The effect of varying
the position of the load applicator, the initial compression
load and the level of vibration stimulus were investigated.
Finally, the repeatability of the method was evaluated by
performing 5 test repetitions on the Level 2 model for
a given level of initial static load and vibration stimulus
The RP models were subsequently tested in compression
beyond failure using a 25 kN servo-hydraulic testing
machine (Model 8874, Instron UK Ltd.). Strain was determined
from the actuator displacement and was corrected
for the compliance of the testing machine. The ultimate
stress and apparent modulus values were calculated and
were compared with the resonant frequencies and half-power
Fig. 2. Vibration testing rig. The electro-dynamic shaker was rigidly
mounted to a testing frame. The sample was placed underneath the shaker and
dynamically compressed against the backside of a load cell, which monitored
the pre-load level.
The standard compression tests to failure revealed an
expected decrease in apparent modulus and ultimate stress
with increasing material loss starting from Level 0 to Level
2 and Level 4 models (Table 1 and Fig. 3). The apparent
modulus in the transverse direction (19.1 MPa) was found
to be 75% lower than the modulus for the Level 0 model in
P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115 111
DXA density, ultimate stress, apparent modulus and resonant frequency results for RP models of trabecular bone
Parameter Level 0 Level 2 Level 4 Transverse
Level of bone loss (%) 0 16 42 0
DXA density (g/cm 2 ) 0.72 (1.00) 0.61 (0.84) 0.42 (0.58) 0.78 (1.08)
Ultimate stress (MPa) 2.26 (1.00) 1.07 (0.47) 0.29 (0.13) 0.92 (0.41)
Apparent modulus (MPa) 76.4 (1.00) 45.3 (0.59) 17.9 (0.23) 19.1 (0.25)
Resonant frequency (Hz) 146 (1.00) 136 (0.93) 108 (0.74) 110 (0.75)
Normalised resonant frequency (Hz) 91 (1.00) 80.7 (0.89) 53 (0.58) 55 (0.60)
Half-power bandwidth (Hz) 20 (1.00) 16.1 (0.81) 15.5 (0.78) 16 (0.8)
Damping factor 0.14 (1.00) 0.12 (0.86) 0.14 (1.00) 0.15 (1.07)
Notes: (1) Values normalised to Level 0 are included in brackets. (2) DXA density values were estimated by multiplying the volumetric density by the square
root of the cross-sectional area of the model . (3) Normalised resonant frequency was calculated by subtracting the resonant frequency of the system (55 Hz)
measured with a zero stiffness specimen (i.e. air gap between the load applicator and the base of the test system). (4) The DXA density for the Transverse
model is slightly higher than the Level 0 value. This is due to the fact that the cylindrical models incorporated small differences in the region of interest (ROI).
the cranio-caudal direction (76.4 MPa). The ultimate stress
in the transverse direction (0.92 MPa) was found to be 59%
lower than the ultimate stress in the cranio-caudal direction
(2.26 MPa). These reductions in stiffness and strength
between the cranio-caudal and transverse directions are of
a similar magnitude to those reported in the literature for
human vertebral trabecular bone [25,26].
As expected, the resonant frequencies for the Level 0,
Level 2 and Level 4 models, determined from the frequency
sweep plots of response gain vs. frequency (Fig. 4), were
found to decrease with increasing material loss (146 Hz,
136 Hz and 108 Hz, respectively). This corresponded well
with a decrease in apparent modulus and ultimate stress. The
resonant frequency of the Transverse model (110 Hz) was
significantly less than that of the Level 0 model. DXA density
values were estimated for each model by multiplying the
volumetric density by the square root of the cross-sectional
area of the model  so that the relative change of the vibrational
measures could be compared with that of the current
standard method for diagnosis of osteoporosis. All measured
parameters were normalised with respect to the Level 0 val-
ues to facilitate direct comparison (Table 1 and Fig. 3). For
the Level 2 model, the relative change of the normalised resonant
frequency was 5% lower than that of DXA for detecting
bone loss (0.89 for resonant frequency compared to 0.84 for
DXA). For the Level 4 model, the relative change of the normalised
resonant frequency was the same as that of DXA
(0.58 for both measures).
Good repeatability was obtained in the resonant frequency
results when 5 test repetitions were performed on the Level
2 model (135.7 ± 1.1 Hz). As expected, there was a variation
between 125 Hz and 135 Hz in the resonant frequency for the
Level 2 model when the position of the load applicator was
offset from the centre of the model by 10 mm at five different
locations (Fig. 5) due to inhomogeneity of the bone microarchitecture.
The resonant frequencies were also dependent
on the initial compression static load that was applied prior
to commencement of the frequency sweep, increasing from
136 Hz to 159 Hz for the Level 0 model as the initial static
load was increased from 10 N to 60 N (Fig. 6). The level
of applied vibration stimulus also affected the resonant frequency
results, with a decrease from 144 Hz to 104 Hz for the
Level 0 model as the vibration stimulus signal was increased
from 10 mV to 300 mV (Fig. 6).
Fig. 3. Normalised resonant frequency, DXA density, ultimate stress and
apparent modulus results for RP models (note: values have been normalised
to Level 0 values to facilitate comparison of results).
The objective of this study was to evaluate the potential for
vibrational testing to detect changes in mechanical properties
of trabecular bone that occur due to micro-architectural deterioration
and progressive bone loss. The use of RP models
allowed the effect of increasing levels of artificially induced
material loss to be examined while starting from the same
initial micro-architecture. This approach would not be possible
with real bone specimens as biological variability would
prevent the separation of tissue level properties from microarchitecture.
The decrease in resonant frequency with decreasing apparent
modulus and strength (Table 1 and Fig. 3) indicate that this
system could detect the deterioration in mechanical proper-
112 P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115
Fig. 4. Frequency sweep plots of response gain for Level 0, Level 2, Level 4 and Transverse models.
P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115 113
Fig. 5. Variation in resonant frequency response for Level 2 model with change in position of load applicator.
ties that occurs in trabecular bone with increasing bone loss.
The relative change of the resonant frequency measurement
for detecting bone loss from a baseline normalised value of
1 for the Level 0 model was similar to that of the estimated
DXA value. In addition, the results from the tests on the
Transverse model and the Level 0 model demonstrated that
the testing system could detect a change in the apparent modulus,
which is solely due to architecture. This change could
not be detected by the DXA measurement (Fig. 3).
Although the vibrational test results were promising for
detecting changes in trabecular bone apparent strength and
stiffness, the magnitudes of resonant frequencies that were
obtained were not as expected. The Transverse and Level 4
models had similar resonant frequency and apparent modulus
values (Table 1), even though the density of the Transverse
model was much higher than that of the Level 4 model. This
indicates that the resonant response was dependent on the
apparent modulus but not on the density, in contradiction of
the theoretical solution given by Eq. (1). This result can be
explained by the fact that the resonant condition incorporates
both the model under test and the testing system. The 1st
resonant frequency of the test system is much lower than that
of the 1st axial resonant frequency of the model. When the
model is coupled to the load applicator of the test system, the
effect is to increase the resonant frequency of the system but
the magnitude of the resonant frequency is much less than
that for the model in isolation. The 1st resonant frequency
is then independent of the density of the model, because the
mass of the model is negligible compared to that of the overall
This has positive implications for the performance of
the testing system, with greater sensitivity for determining
changes in the apparent modulus due to the uncoupling of
the E/ρ term in Eq. (1). Nevertheless, the relative change
of the normalised resonant frequency measurements for the
Level 2, Level 4 and Transverse models, from a baseline
value of 1 for the Level 0 measurement, does not improve
upon the relative change of the calculated DXA values
(Table 1). For in vivo applications, where the influence of the
cortex and skin–tissue interface may further decrease sen-
Fig. 6. Variation in resonant frequency response for Level 0 model with change in level of initial static compression load and change in vibration stimulus level.
114 P. Mc Donnell et al. / Medical Engineering & Physics 31 (2009) 108–115
sitivity, it is likely that the design of the test system would
have to be improved in order to obtain a greater range of
change in resonant frequency with changing bone mechanical
properties. In addition, other dynamic measures, such
as the slope of the amplitude gain response at the halfpower
bandwidth and the ratio of static to dynamic elastic
response , should be investigated to determine if these
measures exhibit improved sensitivity compared to resonant
Parameter studies were performed to evaluate the effect
of testing variables on the dynamic response of the samples
and to determine the repeatability of the method. The
repetition tests showed a low standard deviation in the measured
resonant frequency (135.7 ± 1.1 Hz) for 5 consecutive
tests on the Level 2 model with around 0.8% standard error.
This compares with the percent errors expected for DXA in
vitro measurements, using bone phantoms, which have been
reported to be between 0.5% and 1.0% [28–30]. The resonant
frequency decreased as the magnitude of vibration stimulus
increased, indicating that a diagnostic test system would have
to be calibrated for a pre-determined stimulus level. The resonant
frequency magnitudes increased with the initial static
compression load (Fig. 6), suggesting that the system would
have to be calibrated for different levels of static load. As
expected, variation of the position of application of the vibration
stimulus resulted in changes in the resonant frequency
(Fig. 5). Since the utilized RP models represent only a small
section out of a whole bone, it is expected that such positional
changes are not reflected in whole bone analysis as the geometric
size of the bone is much larger than the contact area
with the shaker.
Limitations of this study include the fact that only a small
number of models were tested. However, because all of the
models were based on one baseline architecture, and it was
possible to closely control the intrinsic material properties
and loss of material volume, it is felt that the number of models
tested was sufficient to provide an initial indication of
the potential for vibrational testing to be used as a diagnostic
tool. More experiments will be necessary to separate out the
effects of variability in tissue level properties [31,32], microarchitecture
and anisotropy [33–35] on the structural dynamic
response. Another limitation of the study was that a cored
out trabecular architecture was tested rather than a whole
bone structure including the cortex. In addition, the effect of
the skin and soft tissue interface between the load applicator
and the bone, which would exist in vivo, was not taken into
account. However, soft tissue effects generally cause damping
but do not alter the characteristic dynamic profile of the
Despite these limitations, this study has shown that vibrational
testing of trabecular bone models can detect a reduction
in apparent modulus and strength with levels of material loss
and micro-architectural deterioration, which typically occur
due to osteoporosis [36,37]. In addition, testing the same samples
in two different planes of orthotropic symmetry revealed
the potential of this technique to detect reductions in stiffness
that are due to architecture changes and are independent of
bone density. However, significant challenges have yet to be
overcome to determine the effect of the skin/soft tissue interface,
the cortex and the inevitable variabilities associated with
in vivo testing.
Conflicts of interest
The authors confirm that there is no conflict of interest in
relation to any financial and personal relationships with other
people or organisations that could inappropriately influence
This work is part of the Bone for Life project, funded
by the Programme for Research in Third Level Institutions
(PRTLI), administered by the Higher Education Authority
in Ireland, and the authors acknowledge the collaboration
of the Trinity Centre for Bioengineering and the Royal College
of Surgeons in Ireland. Additional support was provided
through the innovations fund of the Rice Space Institute.
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