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, UK

2.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: jie.tong@port.ac.uk

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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,[footnoteRef:1] [1: Corresponding author.]

1.Mechanical Behaviour of Materials Laboratory, School of Engineering, University of Portsmouth, Portsmouth PO1 3DJ, UK

2.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: jie.tong@port.ac.uk

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 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 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 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 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 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.

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 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. 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.

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 Szz 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) (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 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 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 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 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

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

BV/TV=0.226

Tb.Th=0.233mm

Tb.Sp=1.127mm

SMI=2.057

BV/TV=0.342

Tb.Th=0.286mm

Tb.Sp=0.636mm

SMI=1.176

(c) Bone-cement interface

(d) Cement

BV/TV=0.594

Tb.Th=0.427mm

Tb.Sp=0.57mm

SMI=0.891

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.

(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).

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 line.

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.

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

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 (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.

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).

(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.

(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 (b).

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

Materials

BV/TV

Tb.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

166

Bovine 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 work

2 Liu and Morgan, 2007

050100150200250300

-4

-2

0

2

4

Mean strain of S

zz

(

)

Sub-volume size in the last step (voxel)

10

-5

(a)

0.01.53.04.56.07.5

Ratio of sub-volume size to Tb.Sp

1.7

050100150200250300

-5

0

5

10

15

20

25

30

Ratio of sub-volume size to Tb.Sp

Standard deviation S

zz

(

)

Sub-volume size in the last step (voxel)

10

-4

(b)

0.01.53.04.56.07.5

1.7

0 100 200 300

0

5

10

15

20

25

30

35

10

-4

unloaded

Standard deviation of S

zz

(

)

Sub-volume size in the last step (voxel)

(a)

3.32

0 3 6 9

Ratio of sub-volume size to Tb.Sp

preloaded

Bone

0 100 200 300

0

5

10

15

20

25

30

10

-4

unloaded

Standard deviation of S

zz

(

)

Sub-volume size in the last step (voxel)

(b)

0 3 6 9

Ratio of sub-volume size to Tb.Sp

preloaded

bone-cement interface

3.71

0 100 200 300

0

5

10

15

20

unloaded

10

-4

Standard deviation of S

zz

(

)

Sub-volume size in the last step (voxel)

(c)

0 2 4 6

Sub-volume size in the last step (mm)

preloaded

Cement

0369121518

-24

-18

-12

-6

0

6

12

18

24

10

-4

Unconfined, unloaded

Confined, unloaded

Unconfined, preloaded

Mean strain of S

zz

(

)

Number of slices

(a)

Cement

Interface

Bone

0369121518

0

5

10

15

10

-4

Unconfined, unloaded

Confined, unloaded

Unconfined, preloaded

Standard deviation of S

zz

(

)

Number of slices

Cement

Interface

Bone

(b)

0246810

0

3

6

9

12

15

Norminal stress (MPa)

Norminal axial strain ( )

Foam

Bone-Cement

(a)

A

B

C

D

0 100 200 300 400

0.000

0.005

0.010

0.015

0.020

Mean compressive strain (

)

Number of slices

646464 voxels

969696 voxels

128128128 voxels

(b)

Foam

0 100 200 300 400

0.000

0.003

0.006

0.009

Mean compressive strain (

)

Number of slices

646464 voxels

969696 voxels

128128128 voxels

Bone

Cement

Interface

(c)

0 10 20 30 40

0

2

4

6

8

Precision error (%)

Mean compressive strain of S

zz

(m)

Foam

Bone-Cement

(d)

0 1 2 3 4

Mean compressive strain of S

zz

()