1. Bone Remodelling Lecture 7 16/04/2017 Andrian Sue AMME4981/9981 Week 9
Semester 1, 2015

2. Mechanical Responses of Bone
Body kinematics
Biomaterial mechanics
Biomaterial responses
Bone microstructure
Descriptions
Imaging
Bone constitutive models and relationships
Stiffness and apparent density
Bone remodelling mechanisms
Physiological responses to mechanical stimuli
Models of bone responses
Simple example calculations

3. Bone microstructure
Description and clinical imaging

4. Description of bone microstructure and composition
Five major functions: load transfer; muscle attachment; joint formation; protection; hematopoiesis
Composition and structure of bone heavily influences its mechanical properties
Collagen fibres
Bone mineral (hydroxyapatite)
Bone cells (osteoblasts, osteoclasts, osteocytes)
Two classes of bone structure
Cancellous (or spongy) bone
Cortical (or compact) bone

5. Description of cortical bone microstructure

6. Bone imaging modalities
DXA – Dual energy X-ray Absorptiometry
Provides bone mineral density (BMD)
Commonly used to diagnose osteoporosis (based on BMD)
Two beams are used and soft tissue absorption is subtracted
MRI – Magnetic Resonance Imaging
Ideal for soft tissues
Commonly used to examine joints for soft tissue injuries
fMRI for brain studies
CT – X-Ray Computed Tomography.
Ideal for hard tissues.
MRI scan of the knee joint

7. DXA – Dual Energy X-Ray Absorptiometry
GE Lunar DEXA Systems

8. Imaging of bone microstructure
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Imaging of bone microstructure
To examine the bone microstructure, high resolution scanning is required
Micro-CT and Nano-CT options have high resolution that allows visualisation of bone microstructure
Imaging Modality
Max. Resolution (in-vivo) in microns
Medical CT
200 x 200 x 200
3D-pQCT
165 x 165 x 165
MRI
150 x 150 x 150
Micro-MRI
40 x 40 x 40
Micro-CT
5 x 5 x 5
Nano-CT
0.05 x 0.05 x 0.05
pQCT is peripheral quantitative computed tomography.

9. Micro-CT of bone microstructure
MicroCT Scanner (SkyScan 1172).
Micro-CT scan  sectional image  segmentation  3D image (STL)
Limited field of view

10. Microstructural changes in cancellous bone
Right: decreasing bone density with age as indicated.
Bottom: comparison of bone density between a healthy 37-year-old male (L) and a 73-year-old osteoporotic female (R).
32-year-old male
59-year-old male
89-year-old male

11. Bone constitutive models
Stiffness and apparent density relationships

12. Power law relationship for cancellous bone stiffness
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Power law relationship for cancellous bone stiffness
Well-established relationship between apparent density (ρ) and cancellous bone Young’s modulus (E)
a, p are empirically determined constants
𝐸=𝑎 𝜌 𝑝
References p R2
Carter and Hayes (1977)
J Bone Joint Surg Am, 59:954
3.00
0.74
Rice et al. (1988),
J Biomech 21:155.
2.00
0.78
Hodgkinson and Currey (1992), J Mater Sci Mater Med 3:377
1.96
0.94
Kabel et al. (1999), Bone 24:115
1.93
Homogenisation used by Kabel and more recent studies to determine effective stiffness

13. Power law models for cancellous bone
Hodgkinson – Currey model (1992)
Kabel isotropic model (1999)
Yang (1999) – orthotropic cancellous model
φ – volume fraction and ρ – 1800φ kg/m3
𝐸 𝑚𝑒𝑎𝑛 = 𝜌 1.96
𝐸= 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 813 𝜑 1.93 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜌 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜌 1.93
𝐸 11 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜑 1.8 , 𝐸 22 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 885 𝜑 , 𝐸 33 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 529 𝜑 1.92
𝐺 23 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜑 , 𝐺 13 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜑 , 𝐺 12 = 𝐸 𝑡𝑖𝑠𝑠𝑢𝑒 𝜑 1.98
𝜈 23 =0.256 𝜑 −0.086 , 𝜈 13 =0.316 𝜑 −0.191 , 𝜈 12 =0.176 𝜑 −0.248

14. Orthotropy of bone 1 2 3
Material possesses symmetry about three orthogonal planes
9 independent components
3 Young’s moduli:
𝐸 1 , 𝐸 2 , 𝐸 3
3 Poisson’s ratios:
𝜈 12 = 𝜈 21 , 𝜈 23 = 𝜈 32 , 𝜈 31 = 𝜈 13
3 shear moduli:
𝐺 12 , 𝐺 23 , 𝐺 31
𝜖 11 𝜖 22 𝜖 𝜖 𝜖 𝜖 12 = 1 𝐸 1 − 𝜈 12 𝐸 1 − 𝜈 13 𝐸 − 𝜈 12 𝐸 𝐸 2 − 𝜈 23 𝐸 − 𝜈 13 𝐸 3 − 𝜈 23 𝐸 𝐸 𝐺 𝐺 𝐺 𝜎 11 𝜎 22 𝜎 33 𝜎 23 𝜎 13 𝜎 12

15. Orthotropic constitutive models of cortical bone
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Orthotropic constitutive models of cortical bone
Tibia
Young’s modulus
Poisson’s ratio
Shear modulus
E1=6.9GPa
12=0.49, 21=0.62
G12=2.41GPa
E2=8.5GPa
13=0.12, 31=0.32
G13=3.56GPa
E3=18.4GPa
23=0.14, 32=0.31
G23=4.91GPa
E3 (axial)
E2 (circumferential)
E1 (radial)
Femur
Young’s modulus
Poisson’s ratio
Shear modulus
E1=12.0GPa
12=0.376, 21=0.422
G12=4.53GPa
E2=13.4GPa
13=0.222, 31=0.371
G13=5.61GPa
E3=22.0GPa
23=0.235, 32=0.350
G23=6.23GPa
(Humpherey and Delange, An Introduction to Biomechanics, 2004)

16. Homogenisation techniques for bone microstructure
Voxel-based FE model
Solid-based FE model
(element size: 25 microns)
(element size: 150 microns)
Directly transfer voxel to element (Grid mesh)
Segmentation process
Smoothing, creation of geometry
Large number of elements
Control of meshing algorithms
Stress concentration
Perform static FEA under simple loading to determine the effective stiffness of these structures

17. Bone remodelling mechanisms
Physiological responses to mechanical stimuli

18. Concept of bone remodelling
Living tissues are subject to remodelling/adaptation, where the properties change over time in response to various factors
“every change in the function of bone is followed by certain definite changes in internal architecture and external conformation in accordance with mathematic laws.” – Julius Wolff (1892)
Cancellous bone aligns itself with internal stress lines

19. Structural remodelling in bones
Remodelling or adaptation can occur when there are changes in tissue environment, such as:
Stress/strain/strain energy density
Loading frequency
Temperature
Formation of micro-cracks
A good example is to look at the growth of trees around external structures
Thickening of branches near fences/gates
Natural optimisation
Mattheck (1998)

20. Variations in bone remodelling responses
Bone remodelling is thought to be mediated by osteocytes
Site dependency
Different bones provide different mechanical function
Different sites have different biological environment
Results in different remodelling equilibrium
Time dependency
High frequency (25 Hz) vs low frequency
Large load and low frequency can prevent mass loss
At higher frequencies, substantial bone growth is possible
Fading memory effects, i.e. adaptation of tissue to mechanical stimulation

21. Mechanical signal transduction in bones
Bone transduces mechanical strain to generate an adaptive response
Piezoelectric properties of collagen
Fatigue micro-fracture damage-repair
Hydrostatic pressure of extracellular fluid influences on bone cell
Hydrostatic pressure influences on solubility of mineral and collagenous component, e.g. change in solubility of hydroxyapatite
Creation of streaming potential
Direct response of cells to mechanical loading
These are all mechanisms which may allow mechanical strain to be detected by bone

22. Bone remodelling by micro-fracture repair
Increased stress causes cracks in bone microstructure
Triggers osteoclast formation to remove broken tissue
Osteoblasts are then recruited to help form new bone
Osteoblasts become osteocytes and maintain bone structure

23. Bone remodelling by dynamic loading
Dynamic loading is much more effective at stimulating bone growth.
Burr et al. (2002) found that dynamic loading forces fluids through canalicular channels, stimulating osteocytes by increased shear stresses.

24. Models of bone responses
Bone remodelling calculations

25. Mechanical stimuli - ψ Strain energy density
𝜓=𝑈= 𝜎 𝑇 𝜀 = 1 2 [ 𝜎 11 𝜀 11 + 𝜎 22 𝜀 22 + 𝜎 33 𝜀 33 + 𝜎 12 𝜀 12 + 𝜎 13 𝜀 13 + 𝜎 23 𝜀 23 ]
Strain energy density
Energy stress – linearise the quadratic nature of U
Mechanical intensity scalar (volumetric shrinking/swelling)
von Mises stress (σ1,σ2,σ3 – principal stresses)
Daily stresses (ni = number of cycles of load case i, N = number of different load cases, m = exponent weighting impact of load cycles)
𝜓= 𝜎 𝑒𝑛𝑒𝑟𝑔𝑦 = 2 𝐸 𝑎𝑣𝑔 𝑈
𝜓=𝜗=𝑠𝑖𝑔𝑛( 𝜀 11 + 𝜀 22 + 𝜀 33 ) 𝑈
𝜓= 𝜎 𝑣𝑚 = 𝜎 1 − 𝜎 𝜎 2 − 𝜎 𝜎 3 − 𝜎 1 2
𝜓= 𝜎 𝑑 = 𝑖=1 𝑁 𝑛 𝑖 𝜎 𝑖 𝑚 𝑚

26. Continuum model of bone remodelling
Surface remodelling – changes in external geometry of bone (cortical surface) over time, e.g. radius of bone
Ψ(t) – Mechanical stimulus. It may also change with time.
Ψ*(t) – Reference stimulus
Internal remodelling – changes in a material property over time, e.g. apparent density
𝑆 =[𝑆 𝑡 ]
𝐶 =[𝐶 𝑡 ]

27. Surface remodelling 𝑒=𝜓 𝑡 − 𝜓 ∗ (𝑡)
Stimulus Error: deviation of stimulus from some baseline/ref. value.
Surface Remodelling Rate: cs is a given constant.
Surface Remodelling Rule with Lazy Zone: ws defines the lazy zone.
𝑒=𝜓 𝑡 − 𝜓 ∗ (𝑡)
𝑑𝑠 𝑑𝑡 = 𝑠 = 𝑐 𝑠 𝜓 𝑡 − 𝜓 ∗ 𝑡 = 𝑐 𝑠 𝑒
𝑠 = 𝑐 𝑠 𝜓 𝑡 − 𝜓 ∗ 𝑡 − 𝑤 𝑠 0 𝑐 𝑠 [𝜓 𝑡 − 𝜓 ∗ 𝑡 + 𝑤 𝑠 ] for 𝜓 𝑡 − 𝜓 ∗ 𝑡 > 𝑤 𝑠 for − 𝑤 𝑠 <𝜓 𝑡 − 𝜓 ∗ 𝑡 < 𝑤 𝑠 for 𝜓 𝑡 − 𝜓 ∗ 𝑡 <− 𝑤 𝑠

28. The lazy zone model Dead zone Lazy zone
Dead zone
Lazy zone
Alternative remodelling rule without lazy zone:

29. Internal remodelling Dead zone Lazy zone
Dead zone
Lazy zone
Alternative remodelling rule without lazy zone:

30. Finite element model workflow
Update
Geometry or
Material
Property
Update
FEA
Model
Modify
surface or
Bone
density
Compute
stress/strain
tensor 
Apply
remodeling rule No
Equilibrium
Yes

31. Calculations!

32. Example: Bone loss in space
Human spaceflight to Mars could become a reality within the next 15 years, but not until some physiological problems are resolved, including an alarming loss of bone mass, fitness and muscle strength. Gravity at Mars' surface is about 38% of that on Earth. With lower gravitational forces, bones decrease in mass and density. The rate at which we lose bone in space is times greater than that of a postmenopausal woman and there is no evidence that bone loss ever slows in space. NASA has collected data that humans in space lose bone mass at a rate of c =1.5%/month. Further, it is not clear that space travelers will regain that bone on returning to gravity. During a trip to Mars, lasting between 13 and 30 months, unchecked bone loss could make an astronaut's skeleton the equivalent of a 100-year-old person.

33. Bone loss in space: derivation of bone density changes
Bone remodelling sans lazy zone
𝜓 is the Mars stress level, 𝜓* is the Earth stress level
𝜌 = 𝑑𝜌 𝑑𝑡 ≈ Δ𝜌 Δ𝑡 = 𝜌 𝑛+1 − 𝜌 𝑛 Δ𝑡 = 𝑐 𝜌 𝜓 𝑡 − 𝜓 ∗ 𝑡 𝜓 ∗ 𝑡
𝜌 𝑛+1 = 𝜌 𝑛 +𝑐 𝜌 𝜓 𝑡 − 𝜓 ∗ 𝑡 𝜓 ∗ 𝑡 Δ𝑡
𝜓 𝜓 ∗ =0.38, 𝑐 𝜌 =0.015 𝜌 𝑛
𝜌 𝑛+1 = 𝜌 𝑛 − 𝜌 𝑛 Δt= 𝜌 𝑛 (1−0.0093Δ𝑡)

34. Critical bone loss in space
How long could an astronaut survive in space if we assume the critical bone density to be 1.0 g/cm3? Initial bone density on Earth is ~1.79 g/cm3.

35. Example: simply loaded femurz
Section S-S
Cortical
Cancellous R r
Take strain energy density as mechanical stimulus, ψ
Surface remodelling rate equation with lazy zone
𝜓=𝑈= 𝜎 𝑇 𝜀 = 1 2 [ 𝜎 𝑥𝑥 𝜀 𝑥𝑥 + 𝜎 𝑦𝑦 𝜀 𝑦𝑦 + 𝜎 𝑧𝑧 𝜀 𝑧𝑧 + 𝜎 𝑥𝑦 𝜀 𝑥𝑦 + 𝜎 𝑦𝑧 𝜀 𝑦𝑧 + 𝜎 𝑧𝑥 𝜀 𝑧𝑥 ]
𝑠 = 𝑐 𝑠 𝜓 𝑡 − 𝜓 ∗ 𝑡 − 𝑤 𝑠 0 𝑐 𝑠 [𝜓 𝑡 − 𝜓 ∗ 𝑡 + 𝑤 𝑠 ] for 𝜓 𝑡 − 𝜓 ∗ 𝑡 > 𝑤 𝑠 for − 𝑤 𝑠 <𝜓 𝑡 − 𝜓 ∗ 𝑡 < 𝑤 𝑠 for 𝜓 𝑡 − 𝜓 ∗ 𝑡 <− 𝑤 𝑠

36. Spreadsheet calculation for simply loaded femur
Assume lazy zone half-width is 10% of ψ*, so: ψ- ψ* > 0.1 x ψ* Modelled for 18 months, bone growth slows and equilibrium is reached

37. Spreadsheet calculation for dynamically loaded femur
Force is increased by 10 N every month: F(m+1) = F(m) + 10; Reference stimulus increased by 5% per month: Ψ*(m+1) = 1.05Ψ*(m)

38. Static vs dynamic loading cases in the femur

39. Summary Bone microstructure and composition Bone constitutive models
Bone remodelling mechanisms
Bone remodelling calculations
Assignment 3 handed out today, due week 13
Group project reports and presentations are due in week 13
Quiz in week 12