Mechanics of Bone BME 615

Why study bone mechanics? Why Test Bone? – Interesting Mechanical Properties Why study bone mechanics? Interesting Mechanical Properties In vivo load leads to complex stress distributions

Why study bone mechanics? Why Test Bone? – Interesting Mechanical Properties Why study bone mechanics? Important structural function Lots of pathologies cause mechanical compromise and clincial issues. Obvious mechano-transduction Important for fixation systems for trauma or joint prosthetics Good demo of “apparent” properties, orthotropic behaviors Why not?

Bone will remodel to adapt to loads Why Test Bone? – Interesting Mechanical Properties Wolff’s Law Bone will remodel to adapt to loads

Bone will remodel to adapt to loads Why Test Bone? – Interesting Mechanical Properties Wolff’s Law Bone will remodel to adapt to loads http://guide.stanford.edu/Publications/21-1.jpg J. Wolff

Ankylosed Knee – J. Wolff

Two types of bone to accommodate complex loading Why Test Bone? – Interesting Mechanical Properties Bone Types Two types of bone to accommodate complex loading 1. Compact (Cortical) Bone Relatively high Young’s modulus Higher resistance to torsion and bending Outer shell of bones Slow turnover

Two types of bone to accommodate complex loading Why Test Bone? – Interesting Mechanical Properties Bone Types Two types of bone to accommodate complex loading 1. Cancellous (trabecular) Bone Lower apparent modulus Higher resistance compression More elastic than cortical Inner portion of bone Higher turnover

Apparent Modulus “Apparent Modulus” Why Test Bone? – Interesting Mechanical Properties Apparent Modulus “Apparent Modulus” “Trabecular bone” does not fill entire space Spaces filled with bone marrow Mechanical properties reflect what’s going on for the entire sample “apparent” modulus  what the overall mechanical properties are doing (in this case, bone + what’s filling the gaps) How will the bone marrow affect the mechanical test results?

Bone Elastic Modulus, E Cortical Bone E ≈ 17.9 GPa 4/28/2017 4/28/2017 Bone Elastic Modulus, E Cortical Bone E ≈ 17.9 GPa σult ≈ 170 MPa (compression) σult ≈ 120 MPa (tension) Trabecular Bone E ≈ 0.076 GPa σult ≈ 2.2 MPa (compression) Note: Bone is non-homogeneous and assumed orthotropic (spatial & direction dependence)

Mechanics 9 independent parameters require 9 independent tests

How to test bone? Sample Selection Cancellous bone with different orientation Different locations may lead to different mechanical properties Try not to test off of principal axes Cortical bone In vivo load leads to complex stress distributions

Structural Testing Three-point Testing Advantages: How to test bone? – Three-point testing Structural Testing Three-point Testing Advantages: easy specimen preparation (relatively) easy testing Disadvantage: Sensitive to geometry (specimen-specific) Calculated parameters: flexural stress and strain, flexural modulus, fracture toughness

Structural Testing Three-point Testing For hollow cylinder: How to test bone? – Three-point testing Structural Testing Three-point Testing For hollow cylinder: parameters Stress intensity factor at crack tip:

Tensile Testing Apparent properties Apparent properties Bone cement to embed ends in metal cap Apparent properties Apparent properties

Compression Testing Advantages: Disadvantages: Calculated parameters: How to test bone? – Compression testing Compression Testing Advantages: Easy calculations Uniaxial Disadvantages: Careful specimen preparation Principal material axes? End conditions? Spatially changing properties Specimen size requirements Calculated parameters: Apparent compressive strength, Apparent modulus of elasticity

Behaviors from Testing How to test bone? – Compression testing Behaviors from Testing Cortical Bone Trabecular Bone Stronger in compression than tension Different properties with different bone types

Compression Testing Analysis How to test bone? – Compression testing Compression Testing Analysis Modulus Estimation Elastic modulus can be estimated from apparent density using a power law relationship α=2.5 for ρ<1.2 g/cm3 α=3.2 for ρ>1.2 g/cm3 ρ = Tissue Mass/Bulk Volume Reference: DR Carter, GS Beaupre, Skeletal Function and Form: Mechanobiology of Skeletal Development, Aging and Regeneration. Cambridge University Press 2007

Simulated Compression Test Virtual Lab Demo Simulated Compression Test 2cm x 2cm x 2cm cube of bone taken from inside of pig femoral head Cancellous/trabecular bone Bone marrow left in place inside the cube X Y Z How will this likely affect testing?

Simulated Compression Test Virtual Lab Demo Simulated Compression Test Steady Compression Applied Actuator Bone Sample Testing Stage Strain Time

Simulated Compression Test Virtual Lab Demo Simulated Compression Test Steady Compression Applied Actuator Bone Sample Testing Stage Specimen rotated until all three sides tested

Simulated Compression Test Virtual Lab Demo Simulated Compression Test Gather “apparent” mechanical properties Approximating properties as if homogeneous, continuous substance centimeters millimeters At what scale does the continuum assumption break down?

Indentation Nano-indentation Density measurements Ultrasound Other testing methods Equations given previously in BVP section Indentation Nano-indentation Density measurements Ultrasound Longitudinal wave propagation velocity in an elastic, homogeneous material

Structural Components Why Test Bone? – Interesting Mechanical Properties Structural Components Two major components to accommodate complex loading Collagen: 90% of the organic matrix (~36% of total dry weight) Provides tensile strength to bone Primarily type I collagen Calcium hydroxyapatite: inorganic matrix (~60% of total dry weight) Provides compressive strength to bone Responsible for bone mineralization http://www.nsbri.org/HumanPhysSpace/focus6/student2.html

Mixture Theory Voigt Upper Bound: Why Test Bone? – Interesting Mechanical Properties Mixture Theory Mi = Modulus of ith constituent fi = volume fraction of ith constituent Voigt Upper Bound: Model as parallel components in mixture Component 1 Component 2 HA C f1M1 f2M2 Not for apparent behaviors

Mixture Theory Reuss Lower Bound: Why Test Bone? – Interesting Mechanical Properties Mixture Theory Mi = Modulus of ith constituent fi = volume fraction of ith constituent Reuss Lower Bound: Model as serial components in mixture Component 1 Component 2 HA C f1M1 f2M2

Mixture Theory Apparent Modulus Volume Fraction of HA Why Test Bone? – Interesting Mechanical Properties Mixture Theory Mi = Modulus of ith constituent fi = volume fraction of ith constituent MHA Apparent Modulus Voigt Reuss Mc 1 Volume Fraction of HA

Expectations after this section Why study bone mechanics? Methods of testing to obtain orthotropic mechanical properties “Apparent” properties Mixture theory for homogeneous elastic composites What can you do with properties?

Further References Class Website: “Bone Introduction” BME 315 Notes: Bone, Beams, and Torsion Ref F 12.1-4 (Bone) Ref F 12.5-6 (more Bone Mechanics) “Bounds on bone stiffness” “Bone Issues and Pathology” “von Mises Stress” “Failure and Fatigue in Bone” R. Lakes’ Website: silver.neep.wisc.edu/~lakes/

Parameters σf = Stress in outer fibers at midpoint, (MPa) εf = Strain in the outer surface, (mm/mm) P = load at a given point on the load deflection curve, (N) L = Support span, (mm) y = distance from neutral axis ρ = radius of curvature back P is the applied load, B is the thickness of the specimen, a is the crack length, W is the width of the specimen