Journal of Biomechanics 47 (2014) 923–934

Contents lists available at ScienceDirect

Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com

Review

Application of the digital volume correlation technique for the measurement of displacement and strain fields in bone: A literature review Bryant C. Roberts, Egon Perilli, Karen J. Reynolds n Medical Device Research Institute, School of Computer Science, Engineering and Mathematics, Flinders University, GPO Box 2100, Adelaide 5001, South Australia, Australia

art ic l e i nf o

a b s t r a c t

Article history: Accepted 3 January 2014

Digital volume correlation (DVC) provides experimental measurements of displacements and strains throughout the interior of porous materials such as trabecular bone. It can provide full-field continuumand tissue-level measurements, desirable for validation of finite element models, by comparing image volumes from subsequent mCT scans of a sample in unloaded and loaded states. Since the first application of DVC for measurement of strain in bone tissue, subsequent reports of its application to trabecular bone cores up to whole bones have appeared within the literature. An “optimal” set of procedures capable of precise and accurate measurements of strain, however, still remains unclear, and a systematic review focussing explicitly on the increasing number of DVC algorithms applied to bone or structurally similar materials is currently unavailable. This review investigates the effects of individual parameters reported within individual studies, allowing to make recommendations for suggesting algorithms capable of achieving high accuracy and precision in displacement and strain measurements. These recommendations suggest use of subsets that are sufficiently large to encompass unique datasets (e.g. subsets of 500 mm edge length when applied to human trabecular bone cores, such as cores 10 mm in height and 5 mm in diameter, scanned at 15 mm voxel size), a shape function that uses full affine transformations (translation, rotation, normal strain and shear strain), the robust normalized cross-correlation coefficient objective function, and high-order interpolation schemes. As these employ computationally burdensome algorithms, researchers need to determine whether they have the necessary computational resources or time to adopt such strategies. As each algorithm is suitable for parallel programming however, the adoption of high precision techniques may become more prevalent in the future. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Digital volume correlation Trabecular bone Strain measurement Displacement

Contents 1. 2. 3.

n

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 3.1. Overview of developments and advancements of DVC in bone:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 3.1.1. Generation of image volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 3.1.2. Calculation of discrete displacement measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930 3.1.3. Calculation of strain field measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 3.2. The effect of individual parameters variations, and of sample micro-structure, on displacement and strain measurement errors . . . . 931 3.2.1. Influence of subset size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 3.2.2. Influence of objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 3.2.3. Influence of shape function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 3.2.4. Influence of image voxel size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932 3.2.5. Influence of sample micro-structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932

Corresponding author. Tel.: +61 8 8201 5190; fax: þ 61 8 8201 2904. E-mail address: karen.reynolds@flinders.edu.au (K.J. Reynolds).

0021-9290/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2014.01.001

924

4. Discussion . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . Conflict of interest statement . References . . . . . . . . . . . . . . . .

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

1. Introduction X-ray micro-computed tomography (μCT) imaging enables three-dimensional (3D) investigation of materials with a spatial resolution in the micro-meter range allowing researchers to nondestructively visualize and characterize the internal microarchitecture of specimens, such as bone (Kuhn et al., 1990; Rüegsegger et al., 1996; Perilli et al., 2007). Growth in computer performance and memory has permitted researchers to utilize powerful numerical tools such as micro-finite element models (mFEM) developed from these μCT images (van Rietbergen et al., 1995). Simulations of experimental tests can be performed on sample-specific FE meshes, ranging from tissue-level (mFE) to continuum-level (FE) models, of excised cores up to whole bones, to predict stresses and strains within these structures whilst under mechanical loads (van Rietbergen et al., 1995; Niebur et al., 2000; Verhulp et al., 2006; Eswaran et al., 2007). Strain gages or extensometers are used to obtain accurate experimental measurements of strains, in order to validate predictions obtained from FE-analysis (Keaveny et al., 1994a, 1994b; Taddei et al., 2007; Perilli et al., 2008; Keyak et al., 1993). However, strain gages only measure surface strains at the discrete points to which they are attached, and extensometers measure the strain across the whole sample on which they are fixed (excised bone core or entire whole bone), rather than providing local measurements within the individual trabecular struts, so desired for validation of mFEM (Bay, 2008; Currey, 2009). In 1999, Bay et al. described a novel technique called “subsetbased digital volume correlation” (DVC), for the direct measurement of strain in bone tissue, where the “digital volume” is generated by a 3D imaging method such as mCT (Bay et al., 1999). DVC is a 3D-extension of the 2D digital image correlation (DIC) technique (Sutton et al., 1983, 1986), an extensively researched technique for the measurement of in-plane displacement and strain fields, in numerous materials subjected to mechanical or thermal loads (Pan et al., 2009). For a given material sample under load, DIC provides a surface- or in-plane measurement of deformation and strain, whereas DVC, in its initial implementation, was able to provide full continuum-level displacement and strain fields throughout the interior of the sample in 3D. DVC is able to use naturally occurring micro-structures to track changes in material features, as well as speckle patterns as often applied by users in DIC (Pan et al., 2009). Since its application to cylindrical samples (e.g., 15 mm diameter and 18 mm length) of trabecular bone under compressive load (Bay et al., 1999), DVC has since found application in the analysis of strain in a variety of materials including: collagen (Roeder et al., 2004), agarose gel (Franck et al., 2007), metal foams (Smith et al., 2002), woods (Forsberg et al., 2010), argillaceous rock (Lenoir et al., 2007), and whole bones (Hardisty and Whyne, 2009; Hussein et al., 2012). Briefly, DVC is based on tracking the deformation of microstructural features observed within image volumes, by optimizing an objective function used to compare small subsets of image data from two subsequent scans of a sample, in both an unloaded and a loaded state. A product of advanced computational capabilities, DVC whilst closely replicating the DIC method, was in its initial implementation limited in algorithm complexity by the additional computational burden – significant growth of image data due to

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

932 933 933 933

the additional third image dimension – inherent in working with digital volumes (Bay et al., 1999). During the past decade however, subsequent investigations utilizing DVC have resulted in improvements to the technique, quantification of measurement accuracy and diversification in application, facilitated by advancements in computational resources. It has become evident with each study that a number of parameters within the DVC algorithm, as well as the microstructure of the investigated specimen, influence the performance of this technique. Changes to the DVC parameters, such as DVC objective function, shape function, and image subset size, and changes in image contrast and voxel size, can affect the accuracy and precision of displacement and strain measurements, and the computation time required for each displacement calculation. Although scientific literature broadly discussing the DVC technique for application across various rigid and highly deformable materials has been published (Bay, 2008), a systematic review that focuses explicitly on the increasing number of DVC algorithms applied to bone, or structurally similar materials, is currently unavailable. Bone imaging is among the main applications in mCT analysis, and DVC has increasingly been used as a tool for validation of μFEM derived from mCT (Zauel et al., 2006; Basler et al., 2011). Subsequent implementations of DVC have resulted, not in a single method, but rather in a variety of algorithms, from which an optimal set of procedures for measurement of strain in bone, in particular at the tissue level, still remains unclear. A systematic review, which has not yet been undertaken, may assist in developing guidelines in the choice for a high performance DVC method. This paper aims to provide (1) an overview of the developments and advances of the DVC method reported in the literature specifically towards application in bone, from excised trabecular cores up to whole bones, and (2) an analysis of the measurement errors in displacement and strain reported in these papers, by considering the influence of the experimental design as well as the changes to parameters within the DVC algorithm. The analysis of the displacement and strain measurement errors is in terms of accuracy and precision, defined as the mean and standard deviation of the differences between the measurements obtained by DVC and the displacements and strains applied to the image volume, respectively (Liu and Morgan, 2007). This review may prove useful to investigators seeking to apply such a technique towards their own research. It may serve as a single resource that provides an overview of the DVC methods applied to bone, and a set of procedures for minimizing measurement errors with corresponding computation time, helpful in the choice of a high performance DVC algorithm.

2. Methods An extensive review of English or English translated literature was undertaken to locate studies that explicitly reported on the developments of the subset-based DVC method first described by Bay et al. (1999) for the estimation of displacement and strain fields in trabecular bone, aluminum foams, and whole bones. Aluminum foams have been included, as the anisotropic and inhomogeneous nature of such a porous alloy is very similar to human trabecular bone (Nazarian and Müller, 2004). A computer search of the online bibliographic databases PubMed and IEEE was undertaken in July 2012 using the keywords “digital volume correlation” OR “digital volumetric correlation” OR “three-dimensional digital image correlation” OR “3D digital image correlation”, and confined to results published since January 1999 (year of publication of Bay et al.). This search produced 47 articles, which were

Table 1 Digital volume correlation procedures and measurement error for algorithms applied to trabecular bone or similar micro-structures in the cited studies. Study

Site

Sample geometry (mm)a

Techniques computing continuum-level strain Bay et al. (1999)l HVB Cylind. 15  18 HPT HDF Smith et al. (2002)m,d,e

HVB

Cube, 5.6

Image volume geometry (voxels)

Voxel size (lm)

Image contrast Subset size (voxels)

Displacement measurement points

Shape function

Interpolation

520  520  580

35

N/A

61

5500

Transl.

Tricubic

160  160  160

35

N/A

51

125

Transl. Transl. Transl. Transl. Transl. Transl. Transl. Transl.

PCc PC PC PC l,f

Tricubic and rot. and rot. and rot. and rot.

Cylind. 15  18

520  520  580

35

N/A

30

N/A

Transl.

Tricubic

BDF BPT RDF RPT RVB HVB

Cube, 4.3

N/A

36

N/A

40

N/A

N/A

N/A

Cube, 10

N/A

20

N/A

31

7000

N/A

N/A

Techniques computing tissue-level strain Al foam Verhulp et al. (2004)l,n,i

Cube, 5

28  16  26

12, 20, 36

N/A

7, 13, 17, 21

917, 2130, 9459

Full affine

Tricubic

Basler et al. (2011)m,j

HFH

Height: 50

400  400  600 cylind., 200  200

82

N/A

38, 75, 150

N/A

N/A

N/A

Jandejsek et al. (2011)k

Trab. bone Metal foam

Cylind., 5  10 10  10  20

50  50  50 50  50  50

15

N/A

35 35

35,000 47,000

Full affine

N/A

Jiroušek et al. (2011)k

PPF

N/A

50  50  50

15

N/A

17

35,344

N/A

N/A

298  305  411

17.5

N/A

N/A

N/A

Full affine

Spline

N/A

37

N/A

N/A

N/A

N/A

Linear

Liu and Morgan (2007)l,m,g,h

BDF BPT RDF RPT RVB HVB BDF BPT RDF RPT RVB HVB

Brémand et al. (2008)

HFH

Techniques applied to whole bones (continuum-level strain) Hardisty and Whyne (2009)m RTV 5.2  5.3  7.2 Hussein et al. (2012)

l,m

HVB

L1 human vertebra

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

HVB HDF

Zauel et al. (2006)

925

Table 1 (continued )

926

Study

Objective function

Optimization

Displacement (voxels)

Displacement (lm)

Strain (l-strain)

Precisionb

Accuracy

Precisionb

Accuracy

Precisionb

Precision (range)

0.035 0.035

N/A

1.23 1.23

N/A

302 289

211–401 211–457

1.869–117.201 0.224–0.242 1.873–116.470 0.182–0.200 11.771–113.803 12.145–12.201 1.873–116.949 0.186–0.203

N/A

N/A

N/A

N/A

N/A

69

39–100

N/A

N/A

Techniques computing continuum-level strain Bay et al. (1999)l SSCC

LM

N/A

Smith et al. (2002)m,d,e

SSCC SSCC NCCC NCCC SSCC SSCC NCCC NCCC

BFGS

0.0534–3.4861 0.0064–0.0069 0.0535–3.3277 0.0052–0.0057 0.3363–3.2515 0.347–0.3486 0.0535–3.3414 0.0053–0.0058

Zauel et al. (2006)l,f

SSCC

Steepest desc.

N/A

0.005

N/A

0.175

Liu and Morgan (2007)l,m,g,h

CC CC CC CC CC CC

N/A

 0.154  0.408  0.2  0.279  0.323  0.266 Avg.  0.272 0.056  0.231 0.025 0.025 0.012  0.002 Avg.  0.019 0.002 0.000  0.005  0.005  0.019 0.004 Avg.  0.004

0.065 (0.052) 0.05 (0.091) 0.114 (0.084) 0.096 (0.097) 0.137 (0.073) 0.069 (0.075) 0.089 (0.079) 0.085 (0.066) 0.072 (0.161) 0.141 (0.107) 0.143 (0.138) 0.167 (0.102) 0.084 (0.093) 0.115 (0.111) 0.069 (0.052) 0.046 (0.071) 0.111 (0.077) 0.099 (0.094) 0.14 (0.073) 0.072 (0.076) 0.090 (0.074)

 5.53  14.70  7.2  10.05  11.63  9.58  9.78 2.02  8.31 0.88 0.9 0.44  0.08  0.69 0.08  0.01  0.18  0.19  0.68 0.16  0.14

2.34 (1.87) 1.81 (3.27) 4.11 (3.04) 3.47 (3.49) 4.92 (2.62) 2.50 (2.70) 3.19 (2.83) 3.05 (2.38) 2.59 (5.79) 5.07 (3.85) 5.15 (4.96) 6.01 (3.68) 3.04 (3.35) 4.15 (4.00) 2.47 (1.86) 1.67 (2.52) 4.00 (2.76) 3.57 (3.39) 5.05 (2.62) 2.61 (2.75) 3.23 (2.65)

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

BFGS

N/A

0.056

N/A

2.0

N/A

10,000

N/A

N/A

N/A

N/A

N/A 0.089

N/A

N/A 7.30

N/A

N/A 43,810

N/A

Jandejsek et al. (2011)k

NCCC/SSCC

N/A

N/A 0.0011

N/A

N/A 0.0165

N/A

N/A 39,230

N/A

N/A

Jiroušek et al. (2011)k

NCCC/SSCC

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Techniques applied to whole bones (continuum-level strain) Hardisty and Whyne (2009)m Mutual information Steepest desc.

N/A

N/A

N/A

N/A

300

400

N/A

Hussein et al. (2012)l,m

0.58

1.12

21.46

41.44

740

630

N/A

NCCC NCCC NCCC NCCC NCCC NCCC MLE MLE MLE MLE MLE MLE

Brémand et al. (2008)

NCCC

Techniques computing tissue-level strain Verhulp et al. (2004)l,n,i CC Basler et al. (2011)

m,j

MLE

N/A

345–794

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

Accuracy

927

examined for inclusion in this review according to the following inclusion criteria: (1) the DVC method was applied either to trabecular bone or to aluminum foam, or to whole bones; (2) the strain mapping techniques used a subset-based DVC method described by Bay et al. (1999). All reference lists within the included studies were examined and papers that fulfilled the selection criteria were included for review.

b

For cube samples single side length is reported, whilst cylindrical samples are reported by diameter  height. Displacement precision is reported as the average error across the x, y and z axes. c Images of poor contrast were generated by halving the 8-bit gray-levels of the original data set (grey level range: 0–127). d Displacement accuracy reported is the range of errors reported through 0.5–151 of artificial rotation. e Precision not specified, but noted to equal approximately half the accuracy for all measurements. f Spherical subsets, the value reported under subset size represents radius length. g The average value reported is the average across all bone types for each DVC method. h Displacement accuracy and precision is reported for artificially translated image volumes, with displacement precision for repeated scans in parentheses. i Image geometry reported for the single trabecula upon which computations were performed, equivalent physical dimensions of 1.008  0.575  0.91 mm3. j DVC was applied to a cylindrical image subvolume of 200 voxels length and diameter for validation. k NCCC utilized for predicting initial whole voxel deformations, followed by the Lucas–Kanade algorithm that is based on the minimisation of the sum-of-squares error. l Repeated scans. m Artificially translated or rotated images. n Physically rotated samples.

3. Results

a

Abbreviations: N/A – entries have not been reported in the paper; Material/anatomical site: HVB – human vertebral body, HPT – human proximal tibia, HDF – human distal femur, BDF – bovine distal femur, BPT – bovine proximal tibia, RDF – rabbit distal femur, RVB – rabbit vertebral body, HFH – human femoral head, and PPF – porcine proximal femur, RTV - rat tail vertebra; Objective functions: SSCC – sum-of-squares correlation coefficient, NCCC – normalised cross-correlation coefficient, CC – cross correlation, MLE – maximum likelihood estimation and PC – poor contrast images; Optimization algorithm: BFGS – Broydon–Fletcher–Goldfarb–Shanno, LM – Levenberg– Marquardt, and steepest desc. – steepest descent.

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

Nine studies were deemed to have fulfilled the review criterion for application to trabecular bone or aluminum foam microstructures (Table 1): Bay et al. (1999); Smith et al. (2002); Verhulp et al. (2004); Zauel et al. (2006); Liu and Morgan (2007); Brémand et al. (2008); Basler et al. (2011); Jandejsek et al. (2011), and Jiroušek et al. (2011). These DVC algorithms measured either continuum-level strain or tissue-level strain throughout the specimen considered. Two studies, i.e. Hardisty and Whyne (2009), and Hussein et al. (2012), have applied DVC to wholes bones, capturing full-field continuum-level strains, for the validation of predicted strains from FE analyses, and for detecting onset and progression of bone failure. Within this review, the distinction between DVC techniques reporting “continuum-” or “tissue-level” strain fields was made according to Verhulp et al. (2004) as follows: the DVC methods reporting “continuum-level” strain were those that measured the distribution of strain throughout bone structures including both the trabecular bone struts and the pores between them. Specifically, these techniques computed strains from displacements of discrete measurement points placed within both the trabecular bone struts and the marrow spaces. DVC techniques computing “tissue-level” strains instead quantified strain only in the trabecular bone strut itself. That is, the strain measurements were computed from displacement measurements for discrete measurement points distributed only throughout the trabecular strut (the actual bone tissue of the trabecular strut, no measurement points in marrow cavities), where the measurement grid is often generated with FE-meshing techniques fitted to the trabecular struts (Verhulp et al., 2004). For each implementation of DVC, some papers reported both the accuracy and precision of displacement and strain measurements for the applied algorithm. Methods for validation of the DVC technique found in the reviewed literature include the following: – reporting precision of the displacements by application of DVC to successive mCT scans of a single bone sample in the unloaded condition, without removing the sample from mCT between scans; – reporting accuracy and precision for both displacements and strains by simulating displacement either by digitally rotating or digitally translating the sample, along a single axis or multiple axes, by a known measure; – reporting precision of strains following physical rotation of a sample about the z-axis through a known range of motion. In those studies where the displacement was simulated by translation, the images were shifted by whole voxels and subvoxels, such that in the subvoxel case the effect of the interpolation could be observed (Basler et al., 2011; Liu and Morgan, 2007; Hussein et al., 2012). The resulting measurement accuracy and precision (where given) are presented in Table 1, and compared with computation time (when reported) in Figs. 1 and 2. The accuracy and precision (reported in both voxels and mm for corresponding voxel size) of continuum-level displacement measurements, including application to whole bones, are within the ranges 0.000–3.486 voxels (0.01 and 117.2 mm respectively) (Liu and Morgan, 2007; Smith

928

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

Fig. 1. Displacement accuracy error vs. computation time for a single displacement measurement point. Error bars in accuracy or computation time represent the range for the corresponding variable, where reported by the authors. The figure legend reports respectively: lead author (year) – objective function and shape function; clock frequency of the central processing unit (processor type); and image voxel size. Smith et al. (2002) applied the DVC algorithm to mCT images that were digitally rotated (01, 11, 51, 101 and 151), using subsets with 51 voxels edge length; Jandejsek et al. (2011) validated their DVC algorithm on images that digitally underwent full affine transformations, using subsets with 35 voxels edge length. SSCC: sum of squares correlation coefficient, and NCCC: normalized cross-correlation coefficient. The DOF listed here refer to the DOF-capability of the implemented DVC method; 3 DOF: translation transformations, 6 DOF: translation and rotation transformations, and 12 DOF: full affine transformations (translation, rotation, normal strain and shear strain). Data from global DVC techniques (Hussein et al., 2012) have not been included in graph, as such techniques compute displacements for nodes within a mesh all at once, and thus only the time taken to compute the entire displacement field was reported in that paper (see Table 2). To compare the performance of DVC algorithms reported in papers purely based on computation time is difficult also because of differences in computer resources among studies; thus, where applicable, computer details have been listed, with more details provided in Table 2.

Fig. 2. Displacement precision error vs. computation time for one single displacement measurement point. Error bars in computation time represent the range where reported by the authors (no range reported for precision). The figure legend reports respectively: lead author (year) – objective function and shape function; computer processing unit clock frequency (processor type); image voxel size. SSCC: sum of squares correlation coefficient, and CC: cross-correlation coefficient. The DOF listed here refer to the DOF-capability of the implemented DVC method; 3 DOF: translation transformations, and 12 DOF: full affine transformations (translation, rotation, normal strain and shear strain). Bay et al. (1999) and Zauel et al. (2006) applied their respective DVC algorithms to repeated mCT scans of the material of interest, captured without removing the material from the scanner between scans; Verhulp et al. (2004) used both repeated scans and scans of the material after it was physically rotated about the z-axis. Data from global DVC techniques (Hussein et al., 2012) have not been included in the graph, as such techniques compute displacements for nodes within a mesh all at once, and thus only the time taken to compute the entire displacement field was reported in that paper (see Table 2). To compare the performance of DVC algorithms reported in papers purely based on computation time is difficult also because of differences in computer resources among studies; thus, where applicable, computer details have been listed, with more details provided in Table 2.

et al., 2002) and 0.005–1.12 voxels (0.175 and 41.44 mm) respectively (Zauel et al., 2006; Hussein et al., 2012), whilst strain precision is within the range 39–630 μ-strain (Zauel et al., 2006; Bay et al., 1999; Hussein et al., 2012). Precision of tissue-level displacement and strain measurements is within the ranges 0.056–0.089 voxels (2.0 and 7.3 mm) and 10,000–43,810 m-strain (Verhulp et al., 2004; Basler et al., 2011). Application of DVC for repeated scans as opposed to artificially displaced image volumes

gave variations in precision errors, depending also on objective function and on anatomical sites (Table 1) (Liu and Morgan, 2007); DVC applied to bovine proximal tibia bone resulted in smaller precision errors (0.046 voxels, 1.67 mm) for artificial displacements compared with repeated mCT scans (0.071 voxels, 2.52 mm); conversely, for the rabbit vertebral body, errors were bigger for artificial displacement (0.140 voxels, 5.05 mm) compared with repeated scans (0.073 voxels, 2.62 mm). Bone specimen geometry

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

ranged from excised cylindrical, cubical (Fig. 3) and parallelepiped bone cores, to human femoral heads, and rat tail and human vertebrae. DVC was applied to image volumes ranging in size from 28  16  26 voxels to 520  520  580 voxels, with voxel sizes from 15 mm to 82 mm, equivalent to physical samples from 1.008  0.576  0.910 mm3 (single trabecular strut) to 32.8  32.8  49.2 mm3 in size (partial femoral head) (Verhulp et al., 2004; Basler et al., 2011). Whole bone structures, to which DVC has been applied for measurement of continuum-level strain, range in size from a rat tail vertebra (5.2  5.3  7.2 mm3) to a human lumbar (L1) vertebral body (Hardisty and Whyne, 2009; Hussein et al., 2012). Displacement for a single measurement point can be computed within 0.47–40 s (Fig. 4) (Zauel et al., 2006; Verhulp et al., 2004), with decreases in computation time attributed to better optimized

929

algorithms, improvements in computer processing power (133 MHz to 3.0 GHz) and increases in memory (128 MB to 16 GB) within a single workstation. Whether or not this computation time considered the time taken to read images was not stated within the reviewed studies. Hussein et al. (2012) computed displacement measurements using a “global DVC” approach that performed registration of an entire image dataset containing a whole human vertebra, as opposed to the registration of small subsets of image data as occurs in “local (subset-based) DVC” techniques. Image datasets of whole bones at high spatial resolutions can result in data file size (hard drive and memory space) in excess of gigabytes (e.g. 3.3 GB at 34 μm voxel size, or 26.4 GB at 17 µm voxel size) (Perilli et al., 2012). While the use of a personal workstation may be sufficient for registration of small subsets of image data (Bay et al., 1999; Smith et al., 2002; Verhulp et al., 2004), global DVC applied to large image volumes such as entire vertebra required the use of a supercomputing system, capable to compute all displacements in about 5 h for each image dataset pair (Hussein et al., 2012) (Table 2). 3.1. Overview of developments and advancements of DVC in bone: DVC, despite variations in its implementation found among the reviewed papers, can be generalized into three main stages as follows: (i) generation of 3D images of material samples in both unloaded and loaded conditions; (ii) measurement of a displacement field represented by discrete measurement points distributed throughout the sample; and (iii) calculations of strain tensors from the computed displacement field. In this section, an overview of the three main stages of the DVC algorithm applied to bone is given. The effect of changes in individual parameters and of micro-structure, on displacement and strain measurement errors, is addressed in the next section.

Fig. 3. Renderings of trabecular bone structure derived from micro-CT scans (human vertebra). Cylindrical (15 mm diameter, 18 mm long, left) and cubic (5  5  5 mm3, right) are common sample geometries to which DVC has been applied.

3.1.1. Generation of image volumes X-ray mCT remains the predominant imaging modality used in the development of DVC, generating a sequential stack of 2D images to produce an image volume. Recent work has also employed confocal microscopy (Franck et al., 2007), micro-magnetic resonance imaging

Fig. 4. Computation time for a single displacement measurement point (s/point), where reported, for each DVC algorithm presented in the reviewed studies. SSCC: sum of squares correlation coefficient, CC: cross-correlation coefficient, and NCCC: normalized cross-correlation coefficient. The DOF listed here refer to the DOF-capability of the implemented DVC method; 3 DOF: translation transformations, 6 DOF: translation and rotation transformations, and 12 DOF: full affine transformations (translation, rotation, scaling and shear). Bay et al. (1999), Smith et al. (2002), and Zauel et al. (2006) computed continuum-level strain measurements on trabecular bone or aluminum foam cores; Verhulp et al. (2004) and Jandejsek et al. (2011) computed tissue-level strain on a single trabecular strut and trabecular bone core, respectively; and Hardisty and Whyne (2009) computed continuum-level strain on a whole rat tail vertebra. To compare the performance of DVC algorithms reported in papers purely based on computation time is difficult also because of differences in computer resources among studies; thus, where applicable, computer details have been listed, with more details provided in Table 2.

930

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

Table 2 Summary of computer resources utilized and validation tests undertaken in application of digital volume correlation. Study

DVC method

Computational resources

Computation time (s/point)

Time per 5500 points (h)

Validation tests

6.5–7

10–10.5

SSCC, transl. SPARC10 ROSS 133 MHz process. NCCC, transl. SSCC, transl. and rot. NCCC, transl. and rot. SSCC, transl. 1.8 GHz Pentium 4 PC

5.3–10.5 5.9–10.0 11.5–16.7 13.0–18.6 0.47

8–16 9–15 17.5–25.5 20–28.5 0.33

Repeated scans Compressive strain Digital rotation, 0.51, 11, 51, 101 and 151

CC NCCC MLE CC, 9 DOF

35 42 192–482 N/A

N/A N/A N/A N/A

2.0 and 0.5 voxel tri-axis disp. and repeated scans

40

61

2 AMD Opteron process. and 16 GB RAM

N/A

N/A

Repeated scans Real rotation, 8.51 about z-axis Voxel sizes, 12, 20 and 36 mm Compressive strain 5 and 4.5 voxel z-axis transl.

2.8 GHz Intel Xeon process. (5500 series)

1.3

2

N/A

N/A

N/A

20

30.6

5h

N/A

Techniques computing continuum-level strain Bay et al. (1999) SSCC, transl. SPARC10 166 MHz process. 128 MB RAM Smith et al. (2002)a

Zauel et al. (2006) Liu and Morgan (2007)b Brémand et al. (2008)

3.0 GHz Pentium 4 w/3 GB RAM 12-process. SGI Altix with 36 GB RAM N/A

Techniques computing tissue-level strain Verhulp et al. CC, affine 800 MHz PC (2004)

Basler et al. (2011) Jandejsek et al. (2011) Jiroušek et al. (2011)

Demon0 s image registration NCCC and SSCC, affine NCCC and SSCC, affine

Techniques applied to whole bones (continuum-level strain) Hardisty and MI, affine N/A Whyne (2009)c Hussein et al. (2012)d

MLE

IBM LS21 blade servers, each w/2 quad-core 3.0 GHz process. and sharing 16 GB RAM

Repeated scans

N/A

Artificial compression and rigid body motion N/A

Repeated scans Application of known strain fields Repeated scans

a

Digital rotation uses trilinear interpolation to predict between voxel intensities. Paper does not specify whether computation time is for single or all displacement measurement points. c DVC technique applied to entire rat tail vertebra. d Computation time is for all points throughout the image volume of an entire human vertebral body (global DVC method). b

(Benoit et al., 2009) and high-resolution peripheral quantitative computed tomography (HR-pQCT) (Basler et al., 2011). Using the desired imaging modality, volumes of the sample are generated in both an undeformed (reference) state and in a deformed state, either whilst under load or after the load has been removed. DVC was first applied for measurement of strain in bone tissue to images from two successive mCT scans of cylindrical trabecular bone cores (15 mm diameter and 18 mm length) extracted from human vertebrae, tibiae and femurs. Each mCT scan produced a reconstructed image volume of 520  520  580 voxels at 35 μm voxel size (Bay et al., 1999). DVC has since been applied to image volumes from mCT or HRpQCT scans of bone equivalent to physical samples of size from 1.008  0.576  0.91 mm3 (single trabecular strut, imaged with 12–35 mm voxel size) to 32.8  32.8  49.2 mm3 (partial femoral head, 82 mm voxel size) (Verhulp et al., 2004; Basler et al., 2011), up to whole human organs (lumbar L1 vertebra (Hussein et al., 2012)).

3.1.2. Calculation of discrete displacement measurements 3.1.2.1. Definition of discrete measurement points. Within the reference image, a “dense” grid of points of interest has to be defined, at which the calculation of displacement values will then occur. These discrete measurement points may either be userdefined and evenly distributed throughout the sample (Bay et al., 1999; Smith et al., 2002), else generated through meshing techniques readily available in finite element post-processing

software (Verhulp et al., 2004; Jiroušek et al., 2011; Jandejsek et al., 2011), in which case the nodes of the meshes become the measurement points of interest. Bay et al. (1999), with the former method, defined a measurement field containing 5500 points throughout their cylindrical volume. Measurement fields containing 125–7000 discrete user-defined points have since been utilized (Smith et al., 2002; Brémand et al., 2008), whilst the use of FE model nodes has produced measurement fields containing between 917 and 47,000 discrete points (Verhulp et al., 2004; Jandejsek et al., 2011). Depending on the FE cube edge length (36–108 mm), Verhulp et al. (2004) produced between 917 and 9459 nodes within one single trabecular strut.

3.1.2.2. Mapping of the reference image and the deformed image (definition of shape function, optimization algorithm, objective function, interpolation). Shape function: At each displacement measurement point, a 3D image subset of user-defined size is generated and centered about this point. The DVC procedure then determines the deformation required to best correlate (map) the subset in the reference image with a subset in the deformed (loaded) image. This correlation requires the undeformed subvolume to undergo a deformation characterized by a function, called “shape function” that encompasses the affine transformations: translation, rotation, normal strain and shear strain. Each of these transformations adds three degrees-offreedom (DOF) and significant computational burden. Due to

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

computational limitations, in earlier studies only translational transformations were considered (three DOF) (Bay et al., 1999; Zauel et al., 2006). In subsequent developments of the DVC technique applied to bone, the rotational transformations have been included (three additional DOF), up to the full set of affine transformations (up to 12 DOF in total), with equivalent or improved computation time facilitated by an increase in computer processing speeds (Smith et al., 2002; Verhulp et al., 2006; Hardisty and Whyne, 2009). Optimization algorithm: The correlation algorithm for determining displacement fields generally operates first by undertaking a coarse (integer voxels) search for subsets within the deformed image using only translation DOF. This provides a whole-voxel estimate of displacement. This displacement measurement is then used as an initial guess within an optimization algorithm, utilizing additional DOF for measuring subvoxel displacements. Either the steepest descent (Zauel et al., 2006; Hardisty and Whyne, 2009), the Levenberg–Marquardt variation of the Gauss–Newton method (Bay et al., 1999) or the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method (Smith et al., 2002; Verhulp et al., 2006) have been utilized as an optimization algorithm for minimization of an objective function. Objective function: Mapping the reference subset to the corresponding position within the deformed image requires a correlation function, also called “objective function”, to be defined. This function quantifies the degree of match between the reference subset and corresponding deformed images. The sum-of-squares correlation coefficient (SSCC), cross-correlation (CC), normalized cross-correlation (NCCC) and mutual information are typical functions reported within the literature (Bay, 2008; Liu and Morgan, 2007; Hardisty and Whyne, 2009). Interpolation: The affine transformations applied to the reference subset volume may result in subvoxel deformations. Thus, some form of interpolation is required in order to estimate image data at between voxel locations. Tricubic interpolation is prevalent within the literature (Bay, 2008; Smith et al., 2002; Verhulp et al., 2004; Zauel et al., 2006), though advancements with DIC suggest that high-order interpolation schemes be implemented, if adequate computational resources are available, for more faithful representation of image voxel intensities (Schreier et al., 2000).

3.1.3. Calculation of strain field measurements The strain measurements can then be estimated from the dense displacement fields. The displacement data may first be smoothed by fitting a quadratic function to a displacement coordinate and its nearest neighbors (Bay et al., 1999). From either filtered or unfiltered data the gradient deformation tensor is estimated. Tensor measurements are calculated either following a leastsquares fit of a tensor to a cloud of displacement values (Peters, 1987), or by fitting of a second-order approximation of the strain tensor to this data (Geers et al., 1996).

3.2. The effect of individual parameters variations, and of sample micro-structure, on displacement and strain measurement errors With the subsequent developments to the DVC algorithm it became evident that, as it is with DIC, sources of error can be attributed to both the limitations of imaging tools, such as the introduction of noise during image acquisition and digitization, and to changes to individual parameters within the DVC algorithm, as well as the sample micro-structure (Pan et al., 2009). This section highlights how these affect the measurement errors reported in the literature.

931

3.2.1. Influence of subset size Subset size has been indicated as possibly the most influential parameter in terms of measurement precision (Table 1) (Jandejsek et al., 2011). It is a requirement for successful measurements of displacement that the selected subset, used to track changes between images of loaded and unloaded bone, be large enough so that during the correlation procedure, the intensity pattern is sufficiently unique in order to distinguish itself from all other subsets. Jandejsek et al. (2011) determined that a cubic subset with 35 voxels (0.525 mm) side length was adequate to produce optimal measures of displacement when applied to both trabecular bone and metal foam samples. Increases in subset size beyond 35 voxels did not produce significant improvements in these measures, whereas smaller subsets, which had side lengths between 13 and 20 voxels (0.2–0.3 mm), resulted in much larger maximal displacement measurement errors (between 0.01 and 0.035 voxels, i.e. 0.15 and 0.525 mm) compared with the 35 voxel side length subset (displacement errors 0.001 voxels, i.e. 0.016 mm). 3.2.2. Influence of objective function Smith et al. (2002) compared the measurement accuracy when using either a sum-of-square correlation coefficient (SSCC) or normalized cross-correlation coefficient (NCCC) as an objective function (Fig. 1). The NCCC provided consistently more accurate displacements (0.0054 vs. 0.0066 voxels, i.e. 0.19 vs. 2.31 mm) throughout the range of rotations they had applied to each sample. When the NCCC algorithm was applied to low contrast images, no change in measurement accuracy was observed, whereas the error increased fifty-fold (from 0.006 to 0.348 voxels, i.e. 2.10 to 12.18 mm) with the SSCC algorithm. The NCCC however was more computationally expensive, with displacements calculated in 13.0– 18.6 s per point across all rotations, compared with 11.5–16.7 s per point when using the SSCC. Liu and Morgan (2007) reported lower errors in displacement measurements using a SSCC algorithm (identified as maximum likelihood estimation (MLE) in that study) compared with NCCC and cross correlation (CC) techniques (respectively,  0.004 70.09,  0.272 70.089, and  0.0419 7 0.115 voxels, i.e.  0.144 73.240,  9.7927 3.204, and  0.684 7 4.14 mm). The MLE algorithm however (which computed displacement measurements within 192–482 s compared with 35 and 42 s for CC and NCCC, respectively), represents a global DVC algorithm. A global correlation approach has been shown to reduce displacement measurement error compared with local techniques, by imposing continuity requirements such that mapping of a single subset depends also on mapping of adjacent regions. This minimizes aberrant displacement measurements that are often found when mapping subset independently, as occurs with local approaches (Hild and Roux, 2012). Therefore, with subset-based DVC algorithms, although the SSCC offers a compromise between computational burden and correlation accuracy, the NCCC, while slower than the SSCC, is more accurate, in particular when the images exhibit low gray level contrast. 3.2.3. Influence of shape function Since the earliest developments of DVC (Smith et al., 2002) it was shown that, when non-translational deformations such as rigid body rotations are present within the image of loaded bone, the errors in accuracy and precision of displacement measurements are relatively high, when implementing only the three translational DOF in the DVC algorithm, compared with algorithms implementing the additional affine transformations (rotation, and normal strain and shear strain) (Fig. 1). For example, when applying 151 of uniaxial rotation about the z-axis to an image volume of trabecular bone, Smith et al. (2002) reported a displacement accuracy of 3.328 voxels (116.47 mm) using only

932

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

translational transformations, while the addition of rotational DOF resulted in displacements with an accuracy of 0.0057 voxels (0.20 mm). The inclusion of rotational DOF in that paper, however, resulted in need for additional computer memory and processing time, increasing the maximum time to compute a displacement measurement point almost two-fold (10.0–18.6 s/point). Similarly, Verhulp et al. (2004), who measured tissue-level strain on a single trabecula, were the first to apply the full set of affine transformations (12 DOF). The algorithm described in that study computed displacement measurements in 40 s per point, more than 2–8 times the computation time utilized by previous algorithms, despite greater processing power (800 MHz compared with 133 MHz), for displacement precision of 0.056 voxels (2.0 mm). However, as demonstrated recently by Jandejsek et al. (2011) utilizing a 2.8 GHz processor, displacement measurements using full affine transformations can be computed in much less time, 1.3 s per measurement point, and with high displacement accuracy (0.001 voxels, i.e. 0.016 mm) (Fig. 1 and Fig. 4). 3.2.4. Influence of image voxel size Increases in voxel size from 12 to 20 and 36 mm, when the physical dimensions of the subset are held constant (252  252  252 mm3), have been shown to increase the error in tissuelevel displacement precision by less than 1 mm, validated on a single trabecula (Verhulp et al., 2004). Liu and Morgan (2007) reported similar outcomes at the continuum-level, observing an increase in error from 1.22 mm to 2.47 mm when the voxel size was increased from 20 to 36 mm with a constant subset size of 1.44  1.44  1.44 mm3. Similarly, when error was expressed in terms of voxels in the latter study, minimal differences in error are observed (0.061 and 0.069 voxels, for voxel sizes of 20 and 36 mm, respectively) suggesting that such changes in voxel size, at high resolutions, have minimal effect on error values (Liu and Morgan, 2007). Imaging structures at relatively low spatial resolution (82 mm voxel size), however may introduce errors due to the inability to accurately resolve thin trabecular struts (for example, owing to partial volume effects) (Basler et al., 2011). 3.2.5. Influence of sample micro-structure By applying their DVC algorithm to specimens of trabecular bone (cubes, 4.3 mm side length) taken from human, bovine and rabbit bones, Liu and Morgan (2007) observed that changes in bone micro-structure (bone volume fraction (BV/TV), trabecular number (Tb.N), trabecular separation (Tb.Sp), and structure model index (SMI)), affect accuracy and precision of displacement and strain measurements (Table 1). Their most accurate DVC algorithm, with objective function based on MLE, indicated that strain accuracy error ranged between 345 and 794 μ-strain (human vertebral body and rabbit distal femur, respectively), being smaller for samples having lower BV/TV (from 0.158 to 0.203), lower Tb.N (from 1.057 to 1.563 1/mm) and higher Tb.Sp (from 0.610 to 0.906 mm) and SMI (from 0.954 to 1.102) compared with samples of higher BV/TV (from 0.332 to 0.577) and Tb.N (from 1.222 to 2.163 l/mm), and smaller Tb.Sp (from 0.355 to 0.770 mm) and SMI (from  2.479 to 0.514).

4. Discussion From the reviewed articles various developments upon the initial implementation of DVC have been observed. DVC has been applied to aluminum foam, an individual trabecular strut, excised cores of trabecular bone, and whole bones, including rat tail and human vertebra. Recent developments of DVC have produced continuum-level displacement precision errors as small as 0.005

voxels (0.175 mm) (Zauel et al., 2006), with displacement accuracy error reported as low as 0.00028 voxels (0.01 mm) (Liu and Morgan, 2007). This is an improvement compared to its first implementation, where DVC produced continuum-level displacement measurements with a precision of 0.035 voxels (1.23 mm) (Bay et al., 1999). At the tissue-level, displacement measurement accuracy and precision errors as low as 0.001 and 0.056 voxels (0.016 and 2.0 mm) respectively, have been reported (Jandejsek et al., 2011; Verhulp et al., 2004). Initially, computational resources (e.g. workstation with a 166 MHz clock-frequency processor and 128 MB RAM) limited the DVC algorithms complexity, where 5500-point displacement measurements required more than 10 h of computation time, using a shape function with only translational DOF. While differences in computer hardware (e.g. clock frequency, processors, memory size and type, etc.) between the years 1999 and 2012 make it difficult to compare algorithm performance among studies by looking purely at the computation time required, the improvements in computer processing power (e.g. workstation with a 2.8 GHz processor and gigabytes of RAM), in addition to better optimized algorithms, have since enabled researchers to compute displacements in a 5500 measurement point grid in only 2 h (Table 2), and by using full affine transformations (Jandejsek et al., 2011). The application of the DVC technique has also diversified, from validation of mFE predictions of strain (Zauel et al., 2006), to validation of continuum-level FE data (Hardisty and Whyne, 2009), and to detection of the onset and progression of vertebral fractures using continuum-level measurements of strain, obtained from application of DVC to mCT images of entire human vertebrae (Hussein et al., 2012). It is up to the individual researchers, depending on the specific application, to determine the magnitude of measurement error that they are willing to accept within their algorithm and whether they have suitable computer resources and time available to utilize computationally burdensome algorithms to achieve their desired goals. Precision errors in strain measurements equal to or smaller than 10% of the nominal strain have been considered adequate for quantifying strain (Bay et al., 1999; Liu and Morgan, 2007). With this in mind, if one was only interested in the measurements of strain in the post-yield regions during mechanical loading (strain greater than  9000 m-strain (Kopperdahl and Keaveny, 1998)), then a strain precision error between or greater than 450–900 m-strain would be acceptable. If however one needs to accurately measure the full-range (elastic, yield and post-yield range) of strains, then considerably smaller precision errors would be required. So far, algorithms capable of achieving this level of precision error (69–302 m-strains) have been reported only for continuum-level strain distributions from 3D images of trabecular bone cores (Bay et al., 1999; Zauel et al., 2006). This level of measurement error is yet to be reported for strain measurements observed at the tissue-level (10,000–43,810 m-strain (Verhulp et al., 2004; Basler et al., 2011)). These relatively larger errors in strain measurements from current tissue-level techniques, when compared to continuum-level measurements, may be attributed to increased displacement measurement point density (e.g. 2130 measurement points for a single trabeculae (28  16  26 voxel image volume) (Verhulp et al., 2004) compared to 5500 points throughout a cylindrical bone core (15 mm diameter and 18 mm length, 520  520  580 voxels image volume) containing many trabeculae (Bay et al., 1999)). As strain measurements are a function of displacement gradients, any error in displacement measurements is increasingly amplified with decreasing distance between measurement points, i.e., tissue-level techniques are in principle more likely to give a larger strain measurement error, compared to continuum-level methods, as the measurement points are closer together (Roeder et al., 2004). Accuracy and precision of displacement and strain measurements obtained using DVC are dependent on a number of factors. The presented list of parameters is by no means exhaustive where the

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

effect of changes in interpolation scheme has not been discussed within the relevant literature, though inferences could be made from literature data obtained using DIC (Pan et al., 2009). How much a change in an individual parameter would improve (minimize) the measurement error from one study to another is not easy to determine, because of numerous differences in parameters between each of the algorithms presented, and the methods by which measurements of the error were obtained. Also, whilst digitally translating or rotating an image removes errors attributed to imaging hardware (movement of the material sample, fluctuations in X-ray flux intensity, and detector noise), other errors are introduced. These include interpolation errors in the cross-section image that would not be present in repeated scans or in scans of physically displaced or deformed material samples. Variations in these errors can also be dependent on micro-architecture or anatomical site (Table 1) (Liu and Morgan, 2007). From the papers examined in this review, general recommendations can be made to minimize displacement and strain measurement errors in bone, where a good DVC method should attempt to use: – An image subset size large enough to contain a gray level intensity pattern sufficiently unique to distinguish itself from all other subsets. It is up to the individual researcher, depending on the specific application, to determine the suitable subset size. For example, cubic subsets of approximately 500 mm edge length have been found suitable for human trabecular bone and aluminum foam for cylindrical cores 10 mm in height and 5 mm in diameter, scanned with 15 mm voxel size, that underwent compressive loading (Jandejsek et al., 2011). – A shape function utilizing the full set of affine transformations (12 DOF; translation, rotation, normal strain and shear strain) (Smith et al., 2002). – A robust objective function such as the NCCC to minimize effects due to changes in image contrast (Smith et al., 2002). – A high-order interpolation scheme for more accurate representations of subvoxel intensities (from recommendations in DIC literature (Pan et al., 2009)) A good experimental design should also consider imaging with high spatial resolutions (e.g. voxel size smaller than 36 mm) to avoid errors resulting from the inability to accurately resolve thin trabeculae (Basler et al., 2011). Large localized deformations are often observed for rigid materials such as trabecular bone (strains greater than 14% at failure in individual trabeculae (Szabo et al., 2011)) while other portions of a sample remain relatively intact (Nazarian et al., 2005; Nazarian and Müller, 2004; Perilli et al., 2008). Thus, as recommended by Bay et al. (1999) a “dense” field of measurement points should be used (e.g., 5500 points throughout a 15 mm diameter and 18 mm long cylinder of trabecular bone for measurement of a continuum-level strain field). For such material types, Bay (2008) further suggests that scans of few loading increments should be captured rather than many loading increments and the computer resources should be best allocated towards the application of algorithms with high measurement precision. As the algorithms within each DVC method are suitable for parallel programming solutions, the adoption of high precision DVC techniques, which use computationally burdensome highorder interpolation schemes and shape functions, may become more prevalent in future, with the opportunity for parallelization with access to modern graphics processing units (GPUs) (Jandejsek et al., 2011). Improved computing capabilities may also permit increasing employment of DVC for validation of mFE data from models as large as entire human bones in future that thus far has only been applied to continuum FE models of entire human vertebrae and mFE models of the partial human femoral head.

933

5. Conclusions Since Bay et al. first applied digital volume correlation for measurement of strain in bone tissue in 1999, 10 subsequent reports of its application to bone structures have appeared within the literature. Comparing performance between algorithms is difficult. The parameters implemented in the technique, in particular the subset size, the images to which a DVC algorithm is applied, and the reported methods in which accuracy and precision in displacement and strain are estimated (e.g. application to images of repeated scans or artificially displaced images), all influence the magnitude of error. Investigating the effects of individual parameters reported within individual studies however has allowed us to make recommendations for suggesting algorithms capable of achieving high accuracy and precision in displacement and strain measurements. These recommendations suggest use of subsets that are sufficiently large to encompass unique datasets (e.g. 500 mm edge length, when applied to human trabecular bone cores, such as cores 10 mm in height and 5 mm in diameter, imaged with 15 mm voxel size), a shape function that uses full affine transformations (translation, rotation, normal strain and shear strain), the robust NCCC objective function, and high-order interpolation schemes. As these recommendations employ computationally complex and burdensome algorithms, it is up to the individual researchers to determine whether they have the necessary resources or time to adopt such strategies. As each algorithm is suitable for parallel programming however, the adoption of high precision techniques may become more prevalent in the future.

Conflict of interest statement The authors have no conflict of interest to declare. References Basler, S.E., Mueller, T.L., Christen, D., Wirth, A.J., Müller, R., van Lenthe, G.H., 2011. Towards validation of computational analyses of peri-implant displacements by means of experimentally obtained displacement maps. Comput. Methods Biomech. Biomed. Eng. 14, 165–174. Bay, B.K., 2008. Methods and applications of digital volume correlation. J. Strain Anal. Eng. Des. 43, 745–760. Bay, B.K., Smith, T.S., Fyhrie, D.P., Saad, M., 1999. Digital volume correlation: threedimensional strain mapping using X-ray tomography. Exp. Mech. 39, 217–226. Benoit, A., Guérard, S., Gillet, B., Guillot, G., Hild, F., Mitton, D., Périe, J.N., Roux, S., 2009. 3D analysis from micro-MRI during in situ compression on cancellous bone. J. Biomech. 42, 2381–2386. Brémand, F., Germaneau, A., Doumalin, P., Dupré, J.C., 2008. Study of mechanical behavior of cancellous bone by digital volume correlation and X-ray microcomputed tomography. In: Proceedings of the XIth International Congress and Exposition on Experimental and Applied Mechanics. Orlando, Florida. http://www. sem.org/Proceedings/ConferencePapers-Paper.cfm?ConfPapersPaperID=20425. Currey, J., 2009. Measurement of the mechanical properties of bone: a recent history. Clin. Orthop. Relat. Res. 467, 1948–1954. Eswaran, S.K., Gupta, A., Keaveny, T.M., 2007. Locations of bone tissue at high risk of initial failure during compressive loading of the human vertebral body. Bone 41, 733–739. Forsberg, F., Sjödahl, M., Mooser, R., Hack, E., Wyss, P., 2010. Full three-dimensional strain measurements on wood exposed to three-point bending: analysis by use of digital volume correlation applied to synchrotron radiation micro-computed tomography image data. Strain 46, 47–60. Franck, C., Hong, S., Maskarinec, S., Tirrell, D., Ravichandran, G., 2007. Threedimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation. Exp. Mech. 47, 427–438. Geers, M.G.D., De Borst, R., Brekelmans, W.A.M., 1996. Computing strain fields from discrete displacement fields in 2D-solids. Int. J. Solids Struct. 33, 4293–4307. Hardisty, M.R., Whyne, C.M., 2009. Whole bone strain quantification by image registration: a validation study. J. Biomech. Eng. 131, 064502-1–064502-6. Hild, F., Roux, S., 2012. Comparison of local and global approaches to digital image correlation. Exp. Mech. 52, 1503–1519. Hussein, A.I., Barbone, P.E., Morgan, E.F., 2012. Digital volume correlation for study of the mechanics of whole bones. Proc. IUTAM 4, 116–125. Jandejsek, I., Jiroušek, O., Vavřík, D., 2011. Precise strain measurement in complex materials using digital volumetric correlation and time lapse micro-CT data. Proc. Eng. 10, 1730–1735.

934

B.C. Roberts et al. / Journal of Biomechanics 47 (2014) 923–934

Jiroušek, O., Jandejsek, I., Vavřík, D., 2011. Evaluation of strain field in microstructures using micro-CT and digital volume correlation. J. Instrum. 6, C01039. Keaveny, T.M., Guo, X.E., Wachtel, E.F., McMahon, T.A., Hayes, W.C., 1994a. Trabecular bone exhibits fully linear elastic behavior and yields at low strains. J. Biomech. 27, 1127–1136. Keaveny, T.M., Wachtel, E.F., Ford, C.M., Hayes, W.C., 1994b. Differences between the tensile and compressive strengths of bovine tibial trabecular bone depend on modulus. J. Biomech. 27, 1137–1146. Keyak, J.H., Fourkas, M.G., Meagher, J.M., Skinner, H.B., 1993. Validation of an automated method of three-dimensional finite element modelling of bone. J. Biomed. Eng. 15, 505–509. Kopperdahl, D.L., Keaveny, T.M., 1998. Yield strain behavior of trabecular bone. J. Biomech. 31, 601–608. Kuhn, J.L., Goldstein, S.A., Feldkamp, L.A., Goulet, R.W., Jesion, G., 1990. Evaluation of a microcomputed tomography system to study trabecular bone structure. J. Orthop. Res. 8, 833–842. Lenoir, N., Bornert, M., Desrues, J., Bésuelle, P., Viggiani, G., 2007. Volumetric digital image correlation applied to X-ray microtomography images from triaxial compression tests on argillaceous rock. Strain 43, 193–205. Liu, L., Morgan, E.F., 2007. Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone. J. Biomech. 40, 3516–3520. Nazarian, A., Müller, R., 2004. Time-lapsed microstructural imaging of bone failure behavior. J. Biomech. 37, 55–65. Nazarian, A., Stauber, M., Müller, R., 2005. Design and implementation of a novel mechanical testing system for cellular solids. J. Biomed. Mater. Res. Part B: Appl. Biomater. 73B, 400–411. Niebur, G.L., Feldstein, M.J., Yuen, J.C., Chen, T.J., Keaveny, T.M., 2000. Highresolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone. J. Biomech. 33, 1575–1583. Pan, B., Qian, K., Xie, H., Asundi, A., 2009. Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review. Meas. Sci. Technol. 20, 062001. Perilli, E., Baleani, M., Öhman, C., Baruffaldi, F., Viceconti, M., 2007. Structural parameters and mechanical strength of cancellous bone in the femoral head in osteoarthritis do not depend on age. Bone 41, 760–768. Perilli, E., Baleani, M., Öhman, C., Fognani, R., Baruffaldi, F., Viceconti, M., 2008. Dependence of mechanical compressive strength on local variations in

microarchitecture in cancellous bone of proximal human femur. J. Biomech. 41, 438–446. Perilli, E., Parkinson, I.H., Reynolds, K.J., 2012. Micro-CT examination of human bone: from biopsies towards the entire organ. Ann. Ist. Super. Sanità 48, 75–82. Peters, G., 1987. Tools for the Measurement of Stress and Strainfields in Soft Tissue (Ph.D. thesis). Eindhoven University of Technology, Eindhoven. Roeder, B.A., Kokini, K., Robinson, J.P., Voytik-Harbin, S.L., 2004. Local, threedimensional strain measurements within largely deformed extracellular matrix constructs. J. Biomech. Eng. 126, 699–708. Rüegsegger, P., Koller, B., Müller, R., 1996. A microtomographic system for the nondestructive evaluation of bone architecture. Calcif. Tissue Int. 58, 24–29. Schreier, H.W., Braasch, J.R., Sutton, M.A., 2000. Systematic errors in digital image correlation caused by intensity interpolation. Opt. Eng. 39, 2915–2921. Smith, T.S., Bay, B.K., Rashid, M.M., 2002. Digital volume correlation including rotational degrees of freedom during minimization. Exp. Mech. 42, 272–278. Sutton, M.A., Mingqi, C., Peters, W.H., Chao, Y.J., McNeill, S.R., 1986. Application of an optimized digital correlation method to planar deformation analysis. Image Vis. Comput. 4, 143–150. Sutton, M.A., Wolters, W.J., Peters, W.H., Ranson, W.F., McNeill, S.R., 1983. Determination of displacements using an improved digital correlation method. Image Vis. Comput. 1, 133–139. Szabo, M.E., Taylor, M., Thurner, P.J., 2011. Mechanical properties of single bovine trabeculae are unaffected by strain rate. J. Biomech. 44, 962–967. Taddei, F., Schileo, E., Helgason, B., Cristofolini, L., Viceconti, M., 2007. The material mapping strategy influences the accuracy of CT-based finite element models of bones: an evaluation against experimental measurements. Med. Eng. Phys. 29, 973–979. van Rietbergen, B., Weinans, H., Huiskes, R., Odgaard, A., 1995. A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models. J. Biomech. 28, 69–81. Verhulp, E., van Rietbergen, B., Huiskes, R., 2004. A three-dimensional digital image correlation technique for strain measurements in microstructures. J. Biomech. 37, 1313–1320. Verhulp, E., van Rietbergen, B., Huiskes, R., 2006. Comparison of micro-level and continuum-level voxel models of the proximal femur. J. biomech. 39, 2951–2957. Zauel, R., Yeni, Y.N., Bay, B.K., Dong, X.N., Fyhrie, D.P., 2006. Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements. J. Biomech. Eng. 128, 1–6.