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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

Abstract

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

1. Introduction

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: p.mcdonnell1@nuigalway.ie (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 [7]:


f j = 2j − 1 E

4L ρ

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

(1)

1350-4533/$ – see front matter © 2008 IPEM. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.medengphy.2008.04.012


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 [8] which would be a requirement for the successful

application of vibrational techniques for evaluating bone

quality.

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. [12] 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 [16]. 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 [17]. 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

transverse directions.

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 [24]. 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

mechanical properties.

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

level.

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

bandwidths.

3. Results

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

Table 1

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 [27]. (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 [27] 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).

4. Discussion

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

testing system.

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 [15], should be investigated to determine if these

measures exhibit improved sensitivity compared to resonant

frequency.

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

bone structures.

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.

Acknowledgements

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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