Hang Yao Advisor: Professor Wei Tong Department of Mechanical Engineering

1. Mechanics of Materials Laboratory MECHANICAL TESTING OF BONE AND BONE-LIKE MATERIALS USING IMAGE CORRELATION STRAIN MEASUREMENT TECHNIQUE Hang Yao Advisor: Professor Wei Tong Department of Mechanical Engineering Southern Methodist University Dallas, Texas August 1st, 2007

2. INTRODUCTION Bone Mechanics Digital Image Correlation Department of Mechanical Engineering

3. Cortical and Trabecular Bone • Cortical Bone • Compact Bone • Porosity 5% - 10% • Trabecular Bone • Cancellous Bone • Porosity 50% - 90% Department of Mechanical Engineering

4. Composition of Bone Treated with hydrochloric acid to dissolve mineral Leaves collagen component intact • On a weight basis • 60% inorganic, 30% organic, 10% water • On a volume basis • 40% inorganic, 35% organic, 25% water • The inorganic phase of bone is a ceramic crystalline type mineral, which is an impure form of hydroxyapatite: • The organic phase of bone consists primarily of type I collagen (90% by weight), some other minor collagen types, and a variety of noncollagenous proteins. Treated with bleach (hypochlorite) to digest collagen Leaves mineral component intact Department of Mechanical Engineering

5. Hierarchical Levels • 0.1 µm • Composite of mineralized collagen fibrils • 10 µm • Lamellar bone (stacked thin sheets) • Woven fibrils (randomly oriented) • 0.5 – 1.0 mm • Haversian bone (concentric lamellae around a central Haversian canal) • Lacunae (holes in 5 - 10 µm) • 1 – 2 mm • Cortical bone • Trabecular bone Department of Mechanical Engineering

6. Volume Fraction and Density • Material properties of bone, particularly stiffness and strength, are strongly dependent on the volume fraction and density. • Tissue density: about 2.0 g/cm3 • Apparent density: • Cortical bone: about 1.85 g/cm3 • Trabecular bone: from 0.10 g/cm3 to 0.50 g/cm3 • The wider variation in density for trabecular than cortical bone results in a much greater heterogeneity in its material properties compared with cortical bone. Department of Mechanical Engineering

7. Anisotrophy • Human cortical bone is generally assumed to be transversely isotropic. • Cortical bone also has asymmetric strengths. • Trabecular bone is also an anisotropic material. It can be considered as either transverse isotropic or orthotropic. Department of Mechanical Engineering

8. Heterogeneity • Heterogeneity can arise from variations in microstructural parameters such as porosity. • Cortical porosity can vary from less than 5% to almost 30%. • Both modulus and ultimate stress can be reduced by 50% when porosity is increased from 5% to 30%. • Due to the substantial heterogeneity in modulus and strength of trabecular bone, the modulus (or strength) of trabecular bone is not easily measured. Details such as anatomic site, species, age, loading direction, and disease state must be specified. Department of Mechanical Engineering

9. Aging and Disease • Aging and disease also affect the mechanical properties of cortical bone. • Tensile ultimate stress decreases at a rate of approximately 2% per decade. • Tensile ultimate strain decreases by about 10% per decade. Department of Mechanical Engineering

10. Digital Image Correlation in Bone Mechanics • How to get a precise and accurate knowledge of bone mechanical properties? • Need a measurement to carry out micromechanical testing of bone... • To refine and validate material models on bone damage and fracture due to injury, aging, and disease (such as osteoporosis); • To construct finite element models based on the bone micro-architecture measurements from 3D micro-CT scanning data; • To analyze micromechanical testing results of bone to define more meaningful property indexes. • Digital Image Correlation is a… • Non invasive … • Non destructive … • Measurement of local finite deformation. Department of Mechanical Engineering

11. Digital Image Correlation • A non-contacting optical technique. • Whole-field deformation measurement. • A simple system and direct sensing. • DIC compares an undeformed and deformed image pair of a solid specimen surface and extracts local displacements at a point by correlating the intensity distributionof a subset image region containing the point between the image pair. Department of Mechanical Engineering

12. FORMULATION OF DIGITAL IMAGE CORRELATION General Formulation Correlation Coefficient Mapping Functions Sub-Pixel Interpolation Post-Processing Department of Mechanical Engineering

13. General Formulation • The deformation equations in the Lagrangian formulation • The deformation gradient F of an arbitrary infinitesimal material vector • The displacement mapping u is often parameterized by a vector P • The match between the two images over the subset region for an assumed parameter set P can be measured by a correlation coefficient • The correlation coefficient is often defined to be an absolute minimum or maximum Department of Mechanical Engineering

14. Correlation Coefficient • The deformation mapping by image correlation can be stated as the solution to the following equations • In general, the above equations are highly nonlinear and solution of such equations may be obtained by iterative Newton-Raphson method • Where is Jacobian matrix, and is Hessian matrix. • The iteration process stops when • A generic sum-of-squared-difference correlation coefficient can be defined as Department of Mechanical Engineering

15. Mapping Functions • A general linearly parameterized local mapping P can be written as • Where a is a certain base vector of the same dimension of P. • 2D constant displacement gradients mapping function • 3D constant displacement gradients mapping function Department of Mechanical Engineering

16. Sub-Pixel Interpolation • Bilinear interpolation • Trilinear interpolation Department of Mechanical Engineering

17. Post-Processing • Deformation mapping u between the undeformed X to the deformed x can be written as • The deformation gradient tensor • The local right body rotation tensor • The right stretch tensor • The Lagrangian strain tensor • The Euler angles are given by Department of Mechanical Engineering

18. MATLAB IMPLEMENTATION AND EXECUTION Graphical User Interfaces Import Images Select an Image Region Set DIC Processing Parameters Inspect the DIC Results Department of Mechanical Engineering

19. Flowchart of DIC Program Department of Mechanical Engineering

20. Graphical User Interfaces • Importing images • Inputting DIC parameters • Displaying images • Inspecting results Department of Mechanical Engineering

21. Import Images • Loading a sequence of images from micro-CT scanning Department of Mechanical Engineering

22. Select an Image Region • Select region of interest • Define grid points • Use a coarse guess Department of Mechanical Engineering

23. Set DIC Processing Parameters • Subset size • Mapping function • Sub-pixel interpolation scheme • Convergence criterion Department of Mechanical Engineering

24. Inspect the DIC Results • 2D DIC For a single pointFor grid points Department of Mechanical Engineering

25. Inspect the DIC Results • 2D DIC • For various sizes of subset displacements versus x-coordinates vertical offsets from the best-fit line versus x-coordinates mean values of vertical offsets versus subset sizes Department of Mechanical Engineering

26. Inspect the DIC Results • 3D DIC For a single pointFor grid points Department of Mechanical Engineering

27. DEMONSTRATIVE EXPERIMENTS AND PRELIMINARY RESULTS Numerical Tests Demonstrative Experiments Preliminary Results Department of Mechanical Engineering

28. 2D Numerical Test • A 2.0º rotation angle is applied to the image Department of Mechanical Engineering

29. 3D Numerical Test • A 2.0º rotation angle and a 0.2 pixel displacement are applied to z-axis. • These errors may be caused by the method of numerical image generation, because in order to improve efficiency, the numerical image generation algorithm is relatively coarse. Department of Mechanical Engineering

30. 2D Demonstrative Experiment • A compression test for trabecular bone is processed by DIC program. • The sample is about 15 mm cube. • Image resolution is 1280 x 960 pixels • The average Lagrangian strain tensors are • The apparent Young’s modulus is yielded as 196.50 MPa Department of Mechanical Engineering

31. 3D Demonstrative Experiment • A high-resolution micro-CT scan is processed by 3D DIC. • Sample is a piece of trabecular bone. It is rescanned with a rigid body motion. • Images are cropped into 800 x 600 x 200 voxels for processing. • The Lagrangian strains and Euler angles are Department of Mechanical Engineering

32. Discussion of Results • The apparent Young’s modulus of trabecular bone • Computational speed • 2D • 3D Department of Mechanical Engineering

33. DISCUSSION AND CONCLUSION Discussion of DIC Conclusion Future Works Department of Mechanical Engineering

34. Discussion of DIC • As it is a non-contact measuring technique, it can minimize the affect to testing materials. • This technique can be performed under any scale, which is only dependent on image acquisition system. • DIC is a type of whole-field measurement method, and it is particularly good performed on highly heterogeneous materials such as bone. • In 3D DIC for micro-CT of bone, its microstructure can be utilized as natural random intensity distribution of an image, and this makes 3D deformation measurement much easier. Department of Mechanical Engineering

35. Conclusion • These 2D and 3D DIC programs perform robustly and precisely in all numerical tests and demonstrative experiments. The fact proves that those developed DIC programs can be fairly used to measure the mechanical properties of bone in the ongoing bone mechanics research project. • All the tests show that the mechanical properties of bone are very complicated to compare as too many factors affect them, especially those of trabecular bone. The Young’s modulus of trabecular bone is highly discrete, and it is hardly to find out a uniform value of it. Department of Mechanical Engineering

36. Future Works • The DIC programs still have great potential to be improved. • The DIC programs are expected to process a sequence of images. • For 3D DIC tricubic interpolation scheme can be added as a new choice. • More choices of output are needed for future version of DIC programs. • DIC programs can be utilized in large rotation applications. • More tests on bone mechanics are needed. They will focus on finding out relationship between microstructure, density, and mechanical properties of both trabecular bone and cortical bone. Besides compression test, tensile test and bending test will also be conducted in near future by using DIC measurement. Department of Mechanical Engineering

37. QUESTION? Thanks to Professor Tong Professor Hurmuzlu and Professor Kovacevic!