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Spatial Resolution and Measurement Uncertainty of Strains in Bone and Bone-Cement

Interface using Digital Volume Correlation

Ming-Liang Zhu1, 2, Qing-Hang Zhang1, Colin Lupton1, Jie Tong1, *

1.Mechanical Behaviour of Materials Laboratory, School of Engineering, University of

Portsmouth, Portsmouth PO1 3DJ, UK2.Key Laboratory of Pressure Systems and Safety, Ministry of Education; School of Mechanical

and Power Engineering, East China University of Science and Technology, Shanghai 200237,

China

*Corresponding authorTel: +44(0) 23 9284 2326

Fax: +44(0) 23 9284 2351

Email: [email protected]

Spatial Resolution and Measurement Uncertainty of Strains in Bone and Bone-Cement

Interface using Digital Volume Correlation

Ming-Liang Zhu1, 2, Qing-Hang Zhang1, Colin Lupton1, Jie Tong1,*

1.Mechanical Behaviour of Materials Laboratory, School of Engineering, University of

Portsmouth, Portsmouth PO1 3DJ, UK2.Key Laboratory of Pressure Systems and Safety, Ministry of Education; School of Mechanical

and Power Engineering, East China University of Science and Technology, Shanghai 200237,

China

*Corresponding author: Tel: +44 (0) 23 9284 2326; Fax: +44 (0) 23 9284 2351; Email: [email protected]

Abstract

The measurement uncertainty of strains has been assessed in a bone analogue (sawbone),

bovine trabecular bone and bone-cement interface specimens under zero load using the

Digital Volume Correlation (DVC) method. The effects of sub-volume size, sample constraint

and preload on the measured strain uncertainty have been examined. There is generally a

trade-off between the measurement uncertainty and the spatial resolution. Suitable sub-

volume sizes have been be selected based on a compromise between the measurement

uncertainty and the spatial resolution of the cases considered. A ratio of sub-volume size to

a microstructure characteristic (Tb.Sp) was introduced to reflect a suitable spatial resolution,

and the measurement uncertainty associated was assessed. Specifically, ratios between 1.6

and 4 appear to give rise to standard deviations in the measured strains between 166 and

620 in all the cases considered, which would seem to suffice for strain analysis in pre as

well as post yield loading regimes.

A microscale finite element (FE) model was built from the CT images of the sawbone, and

the results from the FE model and a continuum FE model were compared with those from

the DVC. The strain results were found to differ significantly between the two methods at

* Corresponding author.

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mailto:[email protected]

tissue level, consistent in trend with the results found in human bones, indicating mainly a

limitation of the current DVC method in mapping strains at this level.

Keywords: DVC; sawbone; bone-cement interface; spatial resolution; FE

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1. Introduction

Digital Volume Correlation (DVC) has experienced rapid growth in recent years as a result of

more accessible volumetric imaging methods such as high-resolution X-ray computed

tomography (CT) and greatly increased computational power. DVC was first introduced as a

method of mapping three dimensional through-volume strains in trabecular bone (Bay et al.,

1999), as an extension from the 2D digital image correlation (DIC) (Sutton et al., 1983) for

surface strain mapping. The essence of the approach is tracking the displacements of small

regions of pixel subsets (DIC) or volume voxel subsets (DVC) in successive images taken in

both unloaded and loaded states, and mapping the strains from the displacements using

appropriate correlation algorithms. Unlike DIC, which relies on surface speckle patterns

applied externally to track in-plane displacements, DVC utilises naturally occurring

microstructure features for tracking 3D displacements from which strains are estimated. A

spatial-domain DVC process (Bay, 2008) involves: i) The specification of a region of interest

(ROI) populated by discrete points where displacement values are sought; ii) estimation of

the displacement vector at each measurement point by correlation of a reference and a

target image volume and iii) an objective function defined to quantify the degree of match

between the original and trial subvolumes using algorithms such as the sum-of-squares

coefficient (SSCC) or the normalized cross-correlation coefficient (NCCC). A global minimum

within a ROI is found through searches and strain tensor estimated at each measurement

point from a cloud of neighbouring points using a least squares fit to a series expansion of

the displacement vector field.

One of the most common DVC procedures is based on 3D reconstructed images from X-ray

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micro-Computed Tomography (CT) (Bay, 2008; Bay et al., 1999), where images with

resolutions down to micrometer level can be obtained (Durand and Rüegsegger, 1992). The

DVC technique has since been applied in a number of biomaterials for 3D measurement of

displacements and strains, including trabecular bones (Bay, 1995; Bay et al., 1999; Gillard et

al., 2014; Liu and Morgan, 2007; Madi et al., 2013; Smith et al., 2002), scaffolds (Madi et al.,

2013), cortical bones (Almer and Stock, 2005; Dall'Ara et al., 2014; Hoc et al., 2006), whole

bones (Hardisty and Whyne, 2009; Hussein et al., 2012), corneas (Fu et al., 2013) and a

biomaterial composite (Tozzi et al., 2014).

The knowledge of measurement uncertainty using the DVC approach is of vital importance

in the application of the DVC method to quantify strains in materials with rich

microstructural details. The accuracy and precision of DVC measurement of displacements

and strains are significantly influenced by the characteristics of the material texture under

study. The influence of a number of parameters on the measurement uncertainty has been

examined (Roberts et al., 2014), including objective function, shape function, image subset

size and voxel size. The voxel-size of a volume is known to be strongly dependent on the

scanning parameters of CT and is considered an important indicator for image quality and

noise level (Liu and Morgan, 2007). X-ray hardening and scattering in bones may lead to

dimensional errors thus strain errors, an influence which may be reduced by filtration

measures (Meganck et al., 2009). A pre-set of rigid body displacement was found to have a

strong influence on the measured strain precision in trabecular bone samples (Gillard et al.,

2014). As opposed to controlled application of speckles on surfaces in DIC (Lecompte et al.,

2006), DVC predominantly relies on the naturally occurring micro-texture to resolve full-field

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displacement and strain distributions (Bay, 2008), hence the measurement accuracy and

precision are largely dictated by the micro-texture of the material.

Although a fine micro-texture is essential for a DVC analysis, microstructural features also

give rise to noise during imaging and correlation processes; hence the choice of a suitable

sub-volume size is important for the DVC analysis. There is generally a trade-off between the

choice of sub-volume size and the spatial resolution. Too small a sub-volume size is more

susceptible to noise; whilst an excessively large sub-volume is unable to capture the

microstructural details leading to inadequate spatial resolution (Gillard et al., 2014; Sun and

Pang, 2007). The chosen sub-volume size should therefore be a compromise between the

measurement uncertainty and the spatial resolution, and it may be estimated based on the

relationship between the standard deviation of the strain measured and the sub-volume size

as well as the required spatial resolution. For biomaterial composites, such as a bone-

cement interface (Tozzi et al, 2014), strains will need to be resolved in a microstructure with

both constituents. This presents a challenge in choosing the appropriate micro-CT scanning

parameters and a sub-volume size suitable for multiple constituents simultaneously. In this

type of materials, the chosen sub-volume size should be able to capture the microstructure

details whilst the strain uncertainty remains low or acceptable. The porous phase should be

considered primarily but scanning parameters should also be suitable for both phases.

A balance between high spatial resolution and acceptable measurement uncertainty is

particularly important when tissue level information is needed. Micro-finite element models

built from CT images have been used to simulate the mechanical behaviour of bones at

tissue-level (Eswaran et al., 2007; Niebur et al., 2000) and continuum-level (van Rietbergen

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et al., 1995; Verhulp et al., 2006); whereas DVC has been used as a tool for validation

purposes (Zauel et al., 2006). However, information obtained at tissue level from FE models

is highly sensitive to the input material parameters, constitutive laws and meshing methods;

whilst results from DVC are volume averages and highly sensitive to processing parameters

such as sub-volume size. Further work is required to improve the fidelity of tissue level

analysis for a range of material systems so that quantitative analysis at this level could be

carried out with confidence.

The main objectives of the present work are: i) To provide a detailed assessment of the

measurement uncertainty in a bone analogue (Sawbone), bovine trabecular bone and bone-

cement interface in relation to spatial resolution; ii) to evaluate the effects of selected

operational parameters on the measurement uncertainty using DVC and iii) to compare the

results from a FE model and DVC in a model cellular material sawbone at both continuum

and tissue levels. Suitable ratios of sub-volume size to a microstructure dimension (Tb.Sp)

were chosen to be reflective of the spatial resolution for the three cases studied, as well as

for the cases from Liu and Morgan (2007), and the measurement uncertainties were

estimated. Additionally, it is the first time a biomaterial composite, in the form of bone-

cement interface, a generic case of interest in implant fixation of many types, was assessed

using the DVC method, and the results are compared with those of the constituent materials.

2. Materials and methods

2.1 Materials

A model cellular foam, Sawbone (Pacific Research Laboratories,

http://www.sawbones.com/), in a rectangular block (20 mm 15 mm 25 mm), was used

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http://www.sawbones.com/

in this study. Bovine trabecular bones were also used to interdigitate with acrylic bone

cement (Simplex P, Stryker, UK) to create bone-cement interface samples. The bovine bone

was taken from trabecular-rich locations of acetabular region, machined into small

rectangular blocks, cleaned and fatty tissues removed using compressed air. The bone blocks

were then put into the lower half of a mold, and cement injected onto the top to form a

bone-cement block with a constant pressure applied to facilitate cement penetration.

Cylindrical bone-cement specimens were then drilled from the bone-cement block with a

diameter of 15 mm and a height of 16.5 mm.

2.2 X-ray micro-tomography and in situ testing

Micro-computed tomography was carried out on the foam and the bone-cement interface

specimens. Scanning was carried out on a XT H 225 CT System (X-Tek Systems Ltd), and this

was followed by data acquisition and reconstruction. For the foam specimen, the scanner

was set at a voltage of 85 kV and current of 84 A. Approximately two hours were required

for imaging and reconstruction using a voxel size of 30 m. For the bone-cement sample,

scanning was conducted at a voltage of 101 kV and current of 100 A. A total of 70 mins was

required for each scan and an isotropic voxel size of 22 m was obtained. The 3D

reconstruction of the samples was carried out using VG Studio MAX 2.0 (Volume Grapics,

Heidelberg, Germany). Fig. 1 shows the morphologies from the micro-tomography of the

foam (Fig. 1a), the bovine trabecular bone (Fig. 1b), the bone-cement interface area (Fig. 1c)

and the cement (Fig. 1d). The main microstructure characteristics were analysed using

ImageJ. The values of volume fraction (BV/TV), mean trabecular thickness (Tb.Th), mean

trabecular spacing (Tb.Sp) and Structural model index (SMI) are presented in Fig. 1.

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A customised micro-mechanical loading device (Deben Ltd., UK) with a 3 kN miniature load

cell was used for in situ uniaxial compression. The details of the micro-mechanical testing

device and procedures were reported elsewhere (Madi et al., 2013). Before testing, the

samples were positioned in the lower compression platen in the loading stage without

bonding. Two successive scans were carried out under zero load for both foam and bone-

cement specimens. For the bone-cement sample, two further successive scans were carried

out under zero load with the sample inside of a confinement chamber and under a preload

of 5N, respectively, to assess the effects of specimen confinement and preload on the

measurement uncertainty. The specimens were subsequently tested in step-wise

compression at a constant displacement rate of 0.2 mm/min till failure. The specimens were

positioned on, but not glued to, the lower platen of the loading stage, and a small preload

was applied on the top platen connected to the actuator to ensure a good end contact. At

each step, the specimen was allowed to relax about 15 mins before imaging and data

acquisition. The load and displacement data were recorded and presented as nominal stress

versus nominal strain curves.

2.3 Digital volume correlation

Volumes reconstructed from CT images using VG Studio Max were imported into the

StrainMaster (DaVis 8.1.3, LaVision, Goettingen, Germany) for digital volume correlation

(DVC). The dimensions (length width height) of the volumes of interest (VOI) were

chosen for the foam as: 440 560 400 voxels and for the bone-cement specimen as: 703

703 400 voxels. The DVC was carried out based on a cross correlation approach of the

Fast Fourier Transform (FFT) utilized in LaVision. Baseline data of accuracy and precision for

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the strain measurements were obtained from the successive scans under zero load; whereas

the correlation between the volumes from two consecutive load levels provided the

incremental results of the displacements and the strains. A geometrical masking process

with a radius of 350 voxel was applied to the reference volume, and the final size of the VOI

was 700 700 400 voxels for the bone-cement specimen.

To achieve the best correlation results, a multi-pass approach was adopted, where the

displacement gradient information from the previous passes was used to inform the sub-

volumes on the subsequent passes (Gillard et al., 2014; Madi et al., 2013; Quinta et al.,

2005). In addition, intermediate sub-volume was also utilised to reduce the displacement

and strain uncertainty (Madi et al., 2013), as part of the multi-step calculation strategy.

Three steps were used in this study, and each step has three passes. The first and second

step have an overlap of 50% while the third step, which is critical to the final results, has an

overlap of 75%. A final sub-volume of 32 voxels 32 voxels 32 voxels overlapped by 75%

was reached after successive steps using sub-volumes of 96 voxels 96 voxels 96 voxels

overlapped by 50% with three passes, and 64 voxels 64 voxels 64 voxels overlapped by

50% with three passes. In order to investigate the effects of sub-volume size on the accuracy

and the precision of the measured strains, the sub-volume size in all the steps was increased

in a step of 32 voxels.

2.4 Finite Element Modelling

The micro-CT images of a central column of 9mm × 9mm × 15mm from the VOI of the foam

specimen were imported into AVIZO 6.3 (Visualization Sciences Group, Mérignac, France) for

three dimensional reconstruction and micro-FE mesh generation using tetrahedral elements.

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The element edges were set to be between 60 and 100µm to achieve a compromise

between numerical accuracy and computational cost. The resulting mesh of VOI consisted of

1,753,211 elements and 477,713 nodes, as shown in Fig. 2a. For the micro-FE analysis, the

mechanical behaviour of the foam cell was assumed to be homogeneous and isotropic. The

elastic modulus of the cell was estimated by matching the resultant force (Fz) at the bottom

surface of the VOI under the applied boundary conditions with that obtained from the

experiment. For the boundary conditions, the displacement fields on the top and the

bottom surfaces of the VOI were extracted from the DVC, with a sub-volume size of 64 voxels

64 voxels 64 voxels at the final step, and mapped onto the nodes of the corresponding

surfaces of the FE model by a linear interpolation. The Poisson’s ratio was assumed to be 0.3

(Zauel et al., 2006).

The FE analysis was conducted at both tissue and continuum levels. For the tissue level

analysis, the micro-FE model was analysed under the boundary conditions described above

and the displacements and strains were calculated and presented at the trabecular level. For

the continuum analysis, a rectangular block with the same dimensions of the VOI was built

and meshed using brick elements, as shown in Fig. 2b, where the element size was 0.5mm.

The results of displacements and strains from the micro-FE model were then mapped onto

the continuum model by averaging the corresponding values of all the micro-FE elements

within each brick element of the solid block FE model. The resulting displacements and

strains were then compared with those calculated by the DVC method using a sub-volume

size of 64 voxels on both 3D exterior and a 2D mid-plane.

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3. Results and discussion

3.1 Accuracy and precision

The accuracy (mean) and the precision (standard deviation) of the measured displacements

and strains were examined for all samples under zero load. For clarity only the results in the

strain component Szz are presented here. The variation of the mean strain and the standard

deviation of the strain component Szz with sub-volume size is shown in Fig. 3 for the foam.

For the range of sub-volume sizes considered, the mean strain of S zz varies between 210-5

and 210-5, and it seems to stabilise when the sub-volume size is above 96 voxels (Fig. 3a).

The standard deviation of Szz, however, decreases sharply as the sub-volume size increases

from 32 voxels to 64 voxels, stabilising thereafter (Fig. 3b). This trend is consistent with the

previous reports (Dall'Ara et al., 2014; Gillard et al., 2014). Generally, small sub-volume sizes

are more susceptible to noise, whilst large sub-volume size may lead to insufficient spatial

resolution (Sun and Pang, 2007). For the foam, a sub-volume size of 64 voxels 64 voxels

64 voxels would seem to give reasonable accuracy and precision, whilst retaining a good

spatial resolution with the ratio of the sub-volume size to the average cell spacing about 1.7,

as indicated in Fig. 3.

Fig. 4 shows the variation of the standard deviation of Szz with sub-volume size for the

bone-cement interface specimen under zero load and preloaded conditions. Similar in trend

to that in the foam (Fig. 3b), the standard deviation in the bone (Fig. 4a), the bone-cement

interface (Fig. 4b) and the cement (Fig. 4c) decreases with the increase of subvolume size.

Low standard deviations appear to be reached when a subvolume size is about 96 voxel or

greater. The corresponding ratios of the sub-volume size to the average cell spacing (Th.Sp)

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(Figs. 4a and 4b), or the sub-volume size in mm (Fig. 4c), are indicated by the dash lines. Of

the three parts within the bone-cement interface specimen, the cement has the best strain

precision, followed by the interface area, whilst the bone part has the lowest strain

precision. The influence of preload on the standard deviation seems more notable in the

bone than in the other parts of the specimen at a subvolume size of 96 voxel or greater.

3.2 Influence of confinement and preload

The influence of specimen confinement and preload on the measurement uncertainty of the

bone-cement interface specimen was assessed. Both methods aim to reduce any involuntary

movements of the sample during the imaging process. In confinement, the bone-cement

specimen was inserted into a well-fitted plastic cylinder; whilst a small compressive load was

applied to the specimen during the imaging in preload. The combined effects of the two on

the mean strain and the standard deviation are presented in Fig. 5. A sub-volume size of 96

voxels 96 voxels 96 voxels was used and the regions of cement, bone-cement interface

and bone are labelled. It seems that the effects of confinement and preload on the mean

strain and the standard deviation in the cement are not particularly significant, as opposed

to in the bone, where the largest bias and standard deviation are obtained in the case of

unconfined and unloaded. Confinement and preload appear to have a similar positive effect

on the reduction of the bias and the standard deviation, most evidently in bone but in bone-

cement interface region also.

3.3 Strain measurement and error analysis of specimens under compression

The foam and the bone-cement specimens were loaded in step-wise compression and the

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nominal stress and strain are plotted in Fig. 6a. It is apparent that predominantly elastic

behaviour is observed for the bone-cement specimen up to a strain value of about 4%, whilst

the foam becomes non-linear at a strain of about 5%. A convergence study was carried out

using three sub-volume sizes on volumes from two consecutive loading stages. The volumes

used for the convergence analysis are indicated in Fig. 6a (A and B for the foam; C and D for

the bone-cement specimen). Fig. 6b and Fig. 6c present the mean compressive strain

distribution within the foam and the bone-cement specimen, respectively. For the foam

sample, a sub-volume size 64 voxels 64 voxels 64 voxels was chosen for the DVC

analysis; whilst for the bone-cement specimen, a sub-volume size of 96 voxels 96 voxels

96 voxels was used. Both microstructure and end effect might have affected the results in the

initial slices, although further into the sample the difference in the strains with the change of

sub-volume size appears to be small (Fig. 6b). For the bone-cement specimen, the variation

of sub-volume size seems mainly affect the mean and the standard deviation of the strains in

the bone (Fig. 6c), and partly the interfacial region (Fig. 6c). Using a suitably chosen sub-

volume size in the final step, a precision error, defined as the ratio of the baseline precision

under zero load to that of under the mean compressive strain (loading steps shown in Fig.

6a), is plotted in Fig. 6d as a function of mean compressive strain for both the foam and the

bone-cement interface samples. The precision error decreases drastically with the increase

of mean compressive strain initially, then gradually. It seems that the precision error tends to

be stabilised in the foam, but not quite so in the stiffer bone-cement interface specimen

within the range of load applied. Nevertheless both precision errors of foam and bone-

cement specimens are well below 10% even in the worst cases, suggesting that the

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measurement errors are acceptable for both foam and bone-cement interface in the elastic

loading regime as well as the post yield regime of bone (Gillard et al., 2014).

3.4 Sensitivity of strain errors to microstructure detail

It is known that measurement uncertainty is dependent on the microstructure features,

hence the choice of a suitable sub-volume size will vary for materials with different

microstructures/textures. For single phase porous materials, a sub-volume sensitivity

analysis, such as that in Dall'Ara et al. (2014), would suffice where strain precision decreases

with the increase of sub-volume size to a stabilised value. However, for bi-phase materials,

such as bone-cement composite, the micro-texture characteristics are more complex and all

constituents need to be considered in the choice of a suitable sub-volume size.

The sensitivity of the measurement errors to microstructural characteristics in selected

cases of animal and human bones is summarised in Table. 1, where data from Liu and

Morgan (2007) as well as from the present study are included. Note that only the data from

the DVC performed using the normalised cross-correlation (NCCC) algorithm are included to

remove the influence of objective function (Roberts et al., 2014). A summary of the bias and

the standard deviation in the strain measurements is also given in Fig. 7. The values of the

mean strain and the standard deviation from the present study are generally lower than

those from (Liu and Morgan, 2007), possibly due to the improved imaging and processing

techniques.

From Table 1, it seems that the standard deviation in the measured strains of bovine

trabecular bone is about twice that of foam, possibly due to the more uniformly distributed

pores in the foam, as opposed to the irregular microstructure of the bone; also, more bone

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mass (BV/TV) in bone than that in foam might have also contributed to the higher standard

deviation in bone. This is so despite that the chosen ratio of sub-volume size to Tb.Sp for the

bone (3.3) is nearly double that of the foam (1.7). A suitably chosen sub-volume size should

generally be larger than the cell spacing for DVC analysis, and a ratio of sub-volume size and

trabecular spacing in the range of 1.5 to 4 appears to produce standard deviations in the

measured strains from 200 to 600 for the cellular materials examined.

3.5 Spatial resolution and measurement accuracy

It is generally accepted that a compromise is needed between the spatial resolution and the

accuracy of the strain measurement. Previous research showed that although favourable

comparisons were obtained between the finite element prediction and the DVC

measurement at continuum level (Madi et al., 2013; Verhulp et al., 2006; Zauel et al., 2006),

comparisons at tissue level present the greatest challenge. In this work, we have compared

the distributions of displacement (Vz) and strain field (Szz) in the loading direction in the VOI

(Fig. 2) between the results from the FE at tissue level and at continuum level and those from

the DVC, and presented them in Figs. 8a and 8b on the exterior surfaces of the VOI. It seems

that the displacement fields predicted by both the continuum and the micro FE models

compared reasonably well with that obtained from the DVC, for both the range and the

distribution. However, the strain fields obtained from the two FE models and that of DVC

differ greatly, where the strain value ranges between -0.018 and 0 from the DVC and

between -0.049 and 0.004 from the micro-FE, with significantly different distribution

patterns also (Fig. 8b). Further studies show that there are significant differences even

between the average strains from the FE and the DVC in the loading direction, despite that

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the average displacements compare well between the FE and the DVC. Fig. 9 illustrates the

distribution of the strain component in the loading direction on a mid-slice within the VOI,

where the discrete values (Fig. 9a) from the micro-FE model (-0.6 – 0%) differ significantly

from the continuum strain map (-0.87% -0.214%) obtained from the DVC (Fig. 9b). These

results are consistent in trend with those of Zauel et al. (2006) on human bones, although

the authors attributed the difference to the scanning technology. The current results show

that, although with improved scanning technology, the differences between the two

methods remain. A more fundamental reason may be due to the difference in the length

scale of the basic unit for the analyses. In the DVC, the sub-volume size must be sufficiently

large to encompass the microstructural details in these cellular materials to obtain an overall

good correlation. This means a sub-volume size about 1mm in both the current work and in

that of Zauel et al. (2006). The finite element sizes in the micro-FE models, on the other

hand, are between 60 and 100 microns in the current work, and about 44 microns in Zauel et

al. (2006). It is not difficult to see why the strain results from these two methods differ, as

the sub-volume size in DVC is dictated by the microstructure characteristics (i.e. cell spacing)

whilst the size of the finite elements is limited mainly by the computing power. Furthermore,

the results from the FE analysis are also influenced by the materials parameters and

constitutive laws, both are approximate. Further work is needed in the development of new

algorithms for voxel level DVC analysis, high resolution advanced scanning techniques as well

as more accurate modelling strategies.

4. Conclusions

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The accuracy and the precision of the measured strains in sawbone, bovine trabecular bone,

bone cement and cement-bone interface have been examined under zero load using the

DVC method, and the role of preload and sample confinement on the accuracy and the

precision of the measured strains have also been assessed with regard to sub-volume size.

Suitably chosen sub-volume sizes, sufficient to capture the microstructural characteristics,

were used to assess the cases, and the precisions in the measured strains are found to be

about 166, 302 and 383 for sawbone, bovine trabecular bone and bone-cement interface,

respectively. The precisions of the measured normal strains in bone, cement and bone-

cement interface are found to be within 2.6%, 0.77% and 3.3%, respectively, of the

maximum strain in the elastic loading regime, giving confidence to strain measurements by

the DVC in pre-yield loading regime.

Comparisons have also been made between the strain results in sawbone obtained from the

FE models and the DVC at continuum and tissue levels. Although the displacements

compare well between the results of the FE models and the DVC, the strain results differ

significantly between the two methods at tissue level, suggesting a challenge for further

development in methods for both the DVC and the micro-FE.

Acknowledgements

ML Zhu is grateful for a Visiting Scholarship from China Scholarship Council.

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(a) Foam (b) Bovine trabecular bone

21

BV/TV=0.226Tb.Th=0.233mmTb.Sp=1.127mmSMI=2.057


(c) Bone-cement interface (d) Cement


Fig. 1. Images from micro-computed tomography of (a) foam (sawbone); (b) bovine trabecular bone; (c) bone-cement interface and (d) bone cement. BV/TV: Volume fraction; Tb.Th: Mean trabecular thickness; Tb.Sp: Mean trabecular spacing; SMI: Structural model index.

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(a) (b)Fig. 2. The FE mesh of the volume of interest (VOI) extracted from the central column of the foam specimen (9mm×9mm×15mm). (a) The micro-FE model of the VOI; (b) The continuum FE model down-graded from the micro-FE model (a).

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0 50 100 150 200 250 300-4

-2

0

2

4

Mea

n st

rain

of S

zz (

)

Sub-volume size in the last step (voxel)

10-5

(a) 0.0 1.5 3.0 4.5 6.0 7.5Ratio of sub-volume size to Tb.Sp

1.7

0 50 100 150 200 250 300-5

0

5

10

15

20

25

30

Ratio of sub-volume size to Tb.Sp

Sta

ndar

d de

viat

ion

Szz

()


10-4

(b) 0.0 1.5 3.0 4.5 6.0 7.5

1.7

Fig. 3. The variation of the mean strain and the standard deviation of strain component Szz with sub-volume size in the foam specimen under zero load. A suitable ratio of the sub-volume size to the cell spacing (Tb.Sp), representing a compromise between the spatial resolution and the measurement errors, is indicated by the dash

24

line.

25

0 100 200 3000

5

10

15

20

25

30

351

0-4

unloaded

Sta

ndar

d de

viat

ion

of S

zz(

)


(a)3.32

0 3 6 9Ratio of sub-volume size to Tb.Sp

preloaded

Bone

0 100 200 3000

5

10

15

20

25

30

10-4

unloaded

Sta

ndar

d de

viat

ion

of S

zz(

)


(b) 0 3 6 9Ratio of sub-volume size to Tb.Sp

preloaded

bone-cement interface3.71

26

0 100 200 3000

5

10

15

20 unloaded1

0-4S

tand

ard

devi

atio

n of

Szz

()


(c) 0 2 4 6Sub-volume size in the last step (mm)

preloaded

Cement

Fig. 4. The variation of the standard deviation of Szz with sub-volume size for (a) bone; (b) bone-cement interface and (c) cement of the bone-cement specimen under zero load and preloaded conditions. Suitable ratios of the sub-volume size to the cell spacing (Tb.Sp) (bone, bone-cement), or the sub-volume size in mm (cement), are indicated by the dash lines.

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0 3 6 9 12 15 18-24

-18

-12

-6

0

6

12

18

2410-4

Unconfined, unloaded Confined, unloaded Unconfined, preloaded

Mea

n st

rain

of S

zz(

)

Number of slices

(a)

Cement Interface Bone

0 3 6 9 12 15 18

0

5

10

1510-4

Unconfined, unloaded Confined, unloaded Unconfined, preloaded

Sta

ndar

d de

viat

ion

of S

zz(

)

Number of slices

Cement Interface Bone

(b)

Fig. 5. The influence of specimen confinement and preload on the mean strain and the standard deviation of Szz for the bone-cement specimen.

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0 100 200 300 4000.000

0.005

0.010

0.015

0.020

Mea

n co

mpr

essi

ve s

train

()

Number of slices

646464 voxels 969696 voxels 128128128 voxels

(b)Foam

29

0 100 200 300 400

0.000

0.003

0.006

0.009

Mea

n co

mpr

essi

ve s

train

()

Number of slices

646464 voxels 969696 voxels 128128128 voxels

Bone CementInterface

(c)

Fig. 6. (a) The nominal stress versus strain curves from the stepped compression tests of a bone-cement and a foam specimen, where strain increments (from A to B) and (from C to D) were selected for convergence analyses; (b) the mean compressive strains increment in the foam calculated from the two consecutive loadings (A: 1.00 MPa and B: 2.82 MPa); (c) a convergence analysis of the mean compressive strain increment in the bone-cement specimen calculated from the two consecutive loadings

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(C: 7.14 MPa and D: 8.01 MPa); (d) precision errors as a function of the mean compressive strain of the bone-cement and the foam samples.

31

Fig. 7. A comparison of (a) the mean strain and (b) the standard deviation of Szz for

the selected porous materials (Note only results using NCCC objective function and

comparable spatial characteristics are included; details are given in Table 1).

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(i) (ii) (iii)

Fig. 8a. Comparison of the displacement (Vz) distributions in the loading direction of the VOI obtained from: (i) DVC calculation with a sub-volume of 64 voxels; (ii) average FE results from the continuum displacement mapping and (iii) micro-FE results.

(i) (ii) (iii)

Fig. 8b. Comparison of the strain (Ezz) distributions in the loading direction of the VOI obtained from: (i) DVC calculation with a subvolume of 64 voxels; (ii) continuum strain mapping by averaging FE results and (iii) micro-FE results.

33

(a) DVC

(b) FE

Fig. 9. The distributions of the strain component in the loading direction in a middle

slice of the VOI: Comparison of results from the DVC (a) and the micro-FE model

34

(b).

Table 1. The sensitivity of strain errors under zero load to spatial characteristics.

Materials BV/TVTb.Th (mm)

Tb.Sp (mm)

Ratio of sub-volume size to

Tb.Sp

Mean strain of Szz ()

Standard deviation of

Szz ()Foam1 0.226 0.233 1.13 1.7 3.1 166Bovine

trabecular bone1 0.342 0.286 0.636 3.32 87 302

Bone-cement interface1 0.594 0.427 0.57 3.71 301 383

Bovine distal femur2 0.203 0.154 0.61 2.36 520 184

Bovine proximal tibia2 0.413 0.271 0.533 2.70 971 515

Rabbit distal femur2 0.352 0.214 0.455 3.16 1285 528

Rabbit proximal tibia2 0.577 0.360 0.355 4.05 1181 620

Rabbit vertebral body2 0.332 0.310 0.770 1.87 910 436

Human vertebral body2 0.158 0.177 0.906 1.59 571 233

1 Current work2 Liu and Morgan, 2007

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