1. MEGN 536 – Computational Biomechanics Prof. Anthony J. Petrella Bone Material Properties

2. Bone Macrostructure  Long bone  Epiphysis  Diaphysis  Compact bone (cortical)  Spongy bone (cancellous) 1 www.agen.ufl.edu/~chyn/age2062/lect/lect_19/lect_19.htm 2 webschoolsolutions.com/patts/systems/skeleton.htm 12

3. 1 academic.wsc.edu/faculty/jatodd1/351/ch4outline.html 2 castlefordschools.com/Kent/Lessons/Advanced%20Biology%20Lessons/chapter%2037/… Advanced%20Biology%20Chapter%2037%20Introduction%20to%20Body%20Structure_files/image026.jpg 2 Bone Microstructure  Cortical bone  Note circumferential layers  Structure influences the material properties 1

4. Bone Microstructure  Cancellous Bone  Trabeculae – struts  Notice axial alignment  Some plate-like structures 40x 1 academic.wsc.edu/faculty/jatodd1/351/ch4outline.html 2 www.gla.ac.uk/ibls/fab/public/docs/xbone1x.html 1 2 2

5. Bone Constituents  Red marrow  Red blood cells, platelets, most white blood cells arise in red marrow  Found in flat bones (sternum, pelvis) and epiphyses  Yellow marrow  Some white blood cells arise here  Color comes from much higher fat content  Found in medullary canals of diaphyses in long bones  Both types of marrow contain numerous vessels  Lots of “squishy” stuff here

6. Bone Properties  Like many biological tissues with “squishy” stuff, bone can behave viscoelastically -- Guedes et al., J.Biomech, 2006  Some studies have shown tensile and compressive behavior similar and linear elastic -- Keaveny et al., J.Biomech, 1994  Many studies have shown that bone is inhomogeneous and anisotropic  Inhomogeneous – properties vary with location  Anisotropic – properties vary with direction of loading  Modulus for cortical bone usually in the 15-20 GPa range, cancellous bone in the 100-500 MPa range

7. Example: Inhomogeneous Strength

8. Inhomogeneity  The inhomogenous nature of bone suggests that it’s important to model the material properties with correct spatial variation  A recent study shows that patient-specific models are inaccurate without a correct inhomogeneous mapping of material properties -- Taddei et al., J.Biomech, 2006  One of the advantages of Simpleware… the software can automate this inhomogeneous mapping

9. Hooke’s Law  Recall Hooke’s law for a linear elastic, isotropic material:  = E   We also need to know Poisson’s ratio:  Isotropic elastic requires only two constants: E,  Many studies have shown that bone is transversely isotropic, which means the axial direction behaves differently than the radial direction  Transverse isotropic materials exhibit properties that are invariant under axial rotation  Recall axial alignment of bone structure…

10. Constitutive Models for Bone  A transverse isotropic model requires five elastic constants: E z, E xy, xz = yz, xy, G xz = G yz  These constants can be found experimentally, but most basic bone models in the literature still use an isotropic model for simplicity  A transverse isotropic model also cannot be easily parameterized using CT data z x y

11. Bone Density  Bone contains many internal structures/spaces and constituents besides calcified tissue  Some density metrics try to account for this  Apparent density (range: 0.05 – 2.0 g/cm 3 )  Your usual density measure  Mass of sample divided by total volume of sample  Ash density (range: 0.03 – 1.2 g/cm 3 )  Seeks to eliminate non-calcified tissue  Mass of bone ash divided by volume of bone only  Bone ash created by drying out bone and incinerating

12. Modulus Relationship to Density  Density can be expressed as linear function of Hounsfield units  = a + b * HU (g/cm 3 )  Modulus and strength have been shown to obey a power-law relationship to density E = c + d *  e (GPa) S = f + g *  h (MPa)  Coefficients vary among different studies, but exponents are usually in the 1-3 range -- Keller, J.Biomech, 1994

13. Modulus-Density Relation also Inhomogeneous

14. Mapping Properties with Simpleware  For simplicity, we stay with a linear elastic, isotropic constitutive model  Use Simpleware automatic mapping to account for inhomogeneity  Necessary number of materials depends on the specific model  How much density variation is there?  How large is the domain?  Typical numbers of distinct materials in validation studies are in the 100- 500 range --Taddei, J.Biomech, 2006  For simplicity, we will use 10 materials

15. A Note about Density and Units in FEA  Abaqus does not use explicit units  Unit system must be consistent…  We are using N, mm, sec  Stiffness is in terms of N/mm 2 → MPa  Similarly… mass * accel → must give Newtons  Density must be units of tonne/mm 3 where tonne = 1000 kg, this compensates for accel in units of mm/s 2 rather than m/s 2

16. A Note about Density and Units in FEA  Density of tissue similar to water… ~ 1 g/cm 3  Stable time increment of FE simulation approx… where h is an element edge length and c is the wave speed in the relevant material, but note that

17. A Note about Density and Units in FEA  Estimate the following…  It is common to artificially increase density in order to increase the stable time increment and decrease total solution time