Towards implementation of a Digital Volume Correlation method for measurement of displacements

and strain in trabecular bone

Bryant Roberts, Egon Perilli, Karen Reynolds

Project Context• A focus of MDRI research towards

development of μFEM from micro-CT

• Projects include– orthopaedic screw insertion into the trabecular

bone of the human femoral head; and– human vertebral body under compressive

load

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Problem

How accurate are these models?

How can we validate these models?

A technique for direct measurement of displacements and strain?

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ProblemTraditional methods…

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Extensometer observes strain across 20mm sample of trabecular bone (Adapted from [1])

[1] Perilli, E et al. 2008 Dependence of mechanical compressive strength on local variations in microarchitecture in cancellous bone of proximal human femur, J Biomech, 41, 438-446

L = 20 mm, Ø = 10 mm

Digital reconstruction of cancellous bone sample pre- and post- loading. Large strain across sample is observed (from [1])

Problem…impractical for single trabecula

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Single trabecula of ~1mm length within an aluminium foam sample (Adapted from [2])

[1] Verhulp, E et al. 2004 A three-dimensional digital image correlation technique for strain measurements in microstructures, J Biomech, 37, 1313-1320

0.91

mm

1.01 mm

Proposed SolutionDigital Volume Correlation (DVC)1

– Takes image volumes from micro-CT and tracks displacement of microstructural features within sample

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[1] Bay, B et al. 1999 Digital volume correlation: three-dimensional strain mapping using x-ray tomography, Exp Mech, 39(3), 217-226

5002 pixel μ-CT images of (left) unloaded bone sample and (right) deformed bone sample with feature tracked throughout

Aim

Identify, and implement a suitable DVC method for measurement of internal displacements and strains

within trabecular bone

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MethodCoarse-Fine search implementation1

1) Global whole pixel search using NCC2

2) Refined sub-pixel computations using Lucas-Kanade algorithm3

Capable of producing displacement measurements in 2D

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[1] Jandejsek et al. 2011 Precise strain measurement in complex materials using DVC and time lapse micro-CT, Procedia Eng, 10, 1730-1735[2] Lewis, J.P. n.d., Fast Normalized Cross-Correlation, Industrial Light & Magic[3] Baker, S. & Matthews, I. 2004, Lucas-Kanade 20 years on: a unifying framework, Int J Comput Vision, 56(3), 221-255

1 Global Search

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• Unloaded subset translated over all possible whole pixel positions of deformed image

mn

n

(m + n) - 1

(m +

n)

- 1

Unloaded image subsetDeformed image

Correlation matrix, stores values [-1, 1]

1 Global Search

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2 Sub-pixel refinement

• Lucas-Kanade algorithm

Gauss-Newton gradient descent algorithm minimising the sum-of-squared error between the subset and deformed image

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2 Sub-pixel refinement• Lucas-Kanade algorithm

– Warps pixel co-ordinates of the subset to corresponding positions in deformed image

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Displacement Accuracy

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12.5 pix

12.5

pix

Deformed image from digital translationw/ grid of measurement pointsUnloaded image

Results

Measurement

Points (nr)Accuracy ± Precision*

(pixels)

Computation Time (min:sec)

529x: 12.5074 ± 0.1195

y: 12.4964 ± 0.1091 9:02

1024x: 12.5035 ± 0.1151

y: 12.5007 ± 0.116315:54

2025x: 12.5036 ± 0.1115

y: 12.4984 ± 0.115032:24

*Accuracy reported as the average of displacement measurements and precision reported as the RMSE

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Range of all displacement measurementsx: [11.3440, 13.5290]y: [11.5681, 13.6017]

For displacements of 12.5 pixels along x- and y- axes

Conclusions• Measurements precision 0.11 pixels (1.914 μm)

– 1.23 μm error is reliable for mapping of elastic strain across whole sample1

– 2.0 μm error useful for strain in single trabecula beyond yield strain2

• Time linearly increasing with number of points– Hours/days required to compute dense fields

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[1] Bay, B et al. 1999 Digital volume correlation: three-dimensional strain mapping using x-ray tomography, Exp Mech, 39(3), 217-226[2] Verhulp, E et al. 2004 A three-dimensional digital image correlation technique for strain measurements in microstructures, J Biomech, 37, 1313-1320

Future Focus

• Extending function of current program– Computation of strain– Handling undesirable displacements

• For consideration– Handling of 3D images– More efficient Inverse Compositional LK

algorithm for improved performance

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Future Focus

Jandejsek et al. report maximal displacement errors within 0.001 pixel

Acceptable tool for validation of full range of strains in μFEM

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Additional Outcomes

ABEC 2012 Abstract Presentation in Brisbane

Future review article for submission- Journal of Biomechanics

- Computer Methods in Biomechanics and Biomedical Eng.

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Thank You

Questions?