Bryant Roberts, Egon Perilli, Karen Reynolds 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

Problem How accurate are these models? How can we validate these models? A technique for direct measurement of displacements and strain?

Problem Traditional methods… L = 20 mm, Ø = 10 mm Digital reconstruction of cancellous bone sample pre- and post- loading. Large strain across sample is observed (from [1]) L = 20 mm, Ø = 10 mm 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

…impractical for single trabecula Problem …impractical for single trabecula 0.91 mm 1.01 mm 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

Proposed Solution Digital Volume Correlation (DVC)1 Takes image volumes from micro-CT and tracks displacement of microstructural features within sample 5002 pixel μ-CT images of (left) unloaded bone sample and (right) deformed bone sample with feature tracked throughout [1] Bay, B et al. 1999 Digital volume correlation: three-dimensional strain mapping using x-ray tomography, Exp Mech, 39(3), 217-226

Aim Identify, and implement a suitable DVC method for measurement of internal displacements and strains within trabecular bone

Method Coarse-Fine search implementation1 Global whole pixel search using NCC2 Refined sub-pixel computations using Lucas-Kanade algorithm3 Capable of producing displacement measurements in 2D [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 Unloaded subset translated over all possible whole pixel positions of deformed image m n (m + n) - 1 Unloaded image subset Deformed image Correlation matrix, stores values [-1, 1]

1 Global Search

2 Sub-pixel refinement Lucas-Kanade algorithm Gauss-Newton gradient descent algorithm minimising the sum-of-squared error between the subset and deformed image

2 Sub-pixel refinement Lucas-Kanade algorithm Warps pixel co-ordinates of the subset to corresponding positions in deformed image 12

Displacement Accuracy 12.5 pix Deformed image from digital translation w/ grid of measurement points Unloaded image

Computation Time (min:sec) Results For displacements of 12.5 pixels along x- and y- axes Measurement Points (nr) Accuracy ± Precision* (pixels) Computation Time (min:sec) 529 x: 12.5074 ± 0.1195 y: 12.4964 ± 0.1091 9:02 1024 x: 12.5035 ± 0.1151 y: 12.5007 ± 0.1163 15:54 2025 x: 12.5036 ± 0.1115 y: 12.4984 ± 0.1150 32:24 *Accuracy reported as the average of displacement measurements and precision reported as the RMSE Range of all displacement measurements x: [11.3440, 13.5290] y: [11.5681, 13.6017]

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 [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 For consideration Computation of strain Handling undesirable displacements For consideration Handling of 3D images More efficient Inverse Compositional LK algorithm for improved performance

Future Focus Jandejsek et al. report maximal displacement errors within 0.001 pixel Acceptable tool for validation of full range of strains in μFEM

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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