Bone Remodelling Lecture 8

Transcription

1 Bone Remodelling Lecture 8 Andrian Sue AMME4981/9981 Week 9 Semester 1, 2016 The University of Sydney Page 1

2 Mechanical Responses of Bone Biomaterial mechanics Body kinematics 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 The University of Sydney Page 2

3 Bone microstructure Description and clinical imaging The University of Sydney Page 3

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 The University of Sydney Page 4

5 Description of cortical bone microstructure Circumferential lamellae Osteon (haversian system) Periosteum Concentric lamellae Interstitial lamellae Blood vessel within the periosteum Blood vessels within a perforating (Volkmann s) canal Blood vessels within a central (haversian) canal Canaliculi Blood vessel within a perforating (Volkmann s) canal between osteons Osteocytes in lacunae (a) The University of Sydney Page 5

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 The University of Sydney Page 6

7 DXA Dual Energy X-Ray Absorptiometry GE Lunar DEXA Systems The University of Sydney Page 7

8 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 The University of Sydney Page 8

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

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 The University of Sydney Page 10

11 Bone constitutive models Stiffness and apparent density relationships The University of Sydney Page 11

12 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 = aρ % References p R 2 Carter and Hayes (1977) J Bone Joint Surg Am, 59:954 Rice et al. (1988), J Biomech 21:155. Hodgkinson and Currey (1992), J Mater Sci Mater Med 3: Kabel et al. (1999), Bone 24: The University of Sydney Page 12

13 Power law models for cancellous bone Hodgkinson Currey model (1992) Kabel isotropic model (1999) E = E 34556' 813φ 0.19 = E 34556' 813 Yang (1999) orthotropic cancellous model ρ E &'() = ρ 0.12 = E 34556' ρ 0.19 E 00 = E 34556' 1240φ 0.<, E >> = E 34556' 885φ 0.<1, E 99 = E 34556' 529φ 0.1> G >9 = E 34556' 533.3φ >.AB, G 09 = E 34556' 633.3φ 0.1D, G 0> = E 34556' 972.6φ 0.1< ν >9 = 0.256φ FA.A<2, ν 09 = 0.316φ FA.010, ν 0> = 0.176φ FA.>B< φ volume fraction and ρ 1800φ kg/m 3 The University of Sydney Page 13

14 Orthotropy of bone 3 Material possesses symmetry about three orthogonal planes 9 independent components 3 Young s moduli: E 0, E >, E 9 3 Poisson s ratios: ν 0> = ν >0,ν >9 = ν 9>, ν 90 = ν 09 3 shear moduli: G 0>,G >9,G 90 ε 00 ε >> ε 99 2ε >9 2ε 09 2ε 0> = 1 1 E 0 ν 0> E 0 ν 09 E ν 0> E > 1 E > ν >9 E > ν 09 E 9 ν >9 E 9 1 E G > G G 0> 2 σ 00 σ >> σ 99 σ >9 σ 09 σ 0> The University of Sydney Page 14

15 Orthotropic constitutive models of cortical bone Tibia Young s modulus Poisson s ratio Shear modulus E 1 (radial) E 1 =6.9GPa ν 12 =0.49, ν 21 =0.62 G 12 =2.41GPa E 3 (axial) E 2 (circumferential) Femur E 2 =8.5GPa ν 13 =0.12, ν 31 =0.32 G 13 =3.56GPa E 3 =18.4GPa ν 23 =0.14, ν 32 =0.31 G 23 =4.91GPa Young s modulus Poisson s ratio Shear modulus (Humpherey and Delange, An Introduction to Biomechanics, 2004) E 1 =12.0GPa ν 12 =0.376, ν 21 =0.422 G 12 =4.53GPa E 2 =13.4GPa ν 13 =0.222, ν 31 =0.371 G 13 =5.61GPa E 3 =22.0GPa ν 23 =0.235, ν 32 =0.350 G 23 =6.23GPa The University of Sydney Page 15

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

17 Bone remodelling mechanisms Physiological responses to mechanical stimuli The University of Sydney Page 17

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 The University of Sydney Page 18

19 Structural remodelling in bones Mattheck (1998) 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 The University of Sydney Page 19

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 The University of Sydney Page 20

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 The University of Sydney Page 21

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 The University of Sydney Page 22

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. The University of Sydney Page 23

24 Models of bone responses Bone remodelling calculations The University of Sydney Page 24

25 Mechanical stimuli - ψ Strain energy density ψ = U = 1 2 σ L ε = 1 2 [σ 00ε 00 + σ >> ε >> + σ 99 ε 99 + σ 0> ε 0> + σ 09 ε 09 + σ >9 ε >9 ] Energy stress linearise the quadratic nature of U ψ = σ ')'QRS = 2E (TR U Mechanical intensity scalar (volumetric shrinking/swelling) ψ = θ = sign(ε 00 + ε >> + ε 99 ) U von Mises stress (σ 1,σ 2,σ 3 principal stresses) ψ = σ T& = 1 2 σ 0 σ > > + σ > σ 9 > + σ 9 σ 0 > Daily stresses (n i = number of cycles of load case i, N = number of different load cases, m = exponent weighting impact of load cycles) _ ψ = σ \ = ] n 4 σ^ & 4 4`0 0 & The University of Sydney Page 25

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

27 Surface remodelling Stimulus Error: deviation of stimulus from some baseline/ref. value. e = ψ t ψ (t) Surface Remodelling Rate: c s is a given constant. ds dt = s = c 5 ψ t ψ t = c 5 e Surface Remodelling Rule with Lazy Zone: w s defines the lazy zone. s = i c 5 ψ t ψ t w 5 0 for ψ t ψ t > w 5 for w 5 < ψ t ψ t < w 5 c 5 [ψ t ψ t + w 5 ] for ψ t ψ t < w 5 The University of Sydney Page 27

28 The lazy zone model for surface remodelling Surface remodelling rate (Ṡ) ψ* w s ψ* Lazy zone ψ*+w s s = c 5 [ψ t ψ t ] c 5 ψ t ψ t w 5 s = i 0 c 5 [ψ t ψ t + w 5 ] Mechanical stimulus (ψ) Alternative remodelling rule without lazy zone: s = c 5 ψ t ψ t ψ t The University of Sydney Page 28

29 The lazy zone model for internal remodelling (ρ ) ρ = c q [ψ t ψ t ] Internal remodelling rate ψ* w ρ ψ* Lazy zone ψ*+w ρ c q ψ t ψ t w 5 s = i 0 c q [ψ t ψ t + w 5 ] Mechanical stimulus (ψ) Alternative remodelling rule without lazy zone: s = c q ψ t ψ t ψ t The University of Sydney Page 29

30 Calculations! The University of Sydney Page 30

31 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. The University of Sydney Page 31

32 Bone loss in space: derivation of bone density changes Bone remodelling sans lazy zone ρ = dρ dt Δρ Δt = ρ )t0 ρ ) Δt = c q ψ t ψ t ψ t ρ )t0 = ρ ) + c q ψ t ψ t ψ t Δt ψ is the Mars stress level, ψ* is the Earth stress level ψ ψ = 0.38, c q = 0.015ρ ) ρ )t0 = ρ ) 0.015ρ ) 0.62 Δt = ρ ) ( Δt) The University of Sydney Page 32

33 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/cm 3? Initial bone density on Earth is ~1.79 g/cm 3. The University of Sydney Page 33

34 Example: simply loaded femur F x y l S S l F Section S-S y Cancellous R z x r Cortical Take strain energy density as mechanical stimulus, ψ ψ = U = 1 2 σ L ε = 1 2 [σ vvε vv + σ SS ε SS + σ ww ε ww + σ vs ε vs + σ Sw ε Sw + σ wv ε wv ] Surface remodelling rate equation with lazy zone s = i c 5 ψ t ψ t w 5 0 for ψ t ψ t > w 5 for w 5 < ψ t ψ t < w 5 c 5 [ψ t ψ t + w 5 ] for ψ t ψ t < w 5 The University of Sydney Page 34

35 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 The University of Sydney Page 35

36 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) The University of Sydney Page 36

37 Static vs dynamic loading cases in the femur Radius (m) Months The University of Sydney Page 37

38 FEA! The University of Sydney Page 38

39 Finite element model workflow Update FEA Model Update Geometry or Material Property Compute stress/strain tensor s Modify surface or Bone density Equilibrium No e = ψ ( t) ψ *( t) > w Apply remodeling rule Yes The University of Sydney Page 39

40 Subroutines Many classic FE solvers have components written in Fortran Custom behaviour by the development of Fortran subroutines Subroutines are executed during the solution The University of Sydney Page 40

41 ANSYS Classic/APDL ANSYS Classic is the old GUI for ANSYS Workbench adds convenience, often at the cost of control Classic is closely linked with the ANSYS Parametric Design Language (APDL) APDL can be used as an input language to talk directly to the ANSYS solver To set-up and run subroutines, we will use APDL to talk to ANSYS The University of Sydney Page 41

42 Programming workflow Programming languages (C variants, Fortran, Java, etc.) require a compiler code compile execute Scripting languages (MATLAB, APDL, Python, etc.) are interpreted at runtime code interpret/execute The University of Sydney Page 42

43 USERFLD subroutine Pre-compiled using Intel Fortran compiler Called by ANSYS by specifying the location of compiled binaries on the command line User-defined field (UF01) across all nodes in the model set ANS_USER_PATH = C:\test echo %ANS_USER_PATH% ansys161.exe -b -i input -o output Make a.bat file for ease of use ANSYS command can be generated from the ANSYS Launcher (launcher161.exe) Solve and update UF01 using the USERFLD subroutine UF01 = Young s modulus (YM) The University of Sydney Page 43

44 APDL script Generate your own APDL script based on the template Get nodes, elements, and components from Workbench (FE Modeler) Use a good text editor (Notepad++ is available on PCLAB computers) Post-process or use ANSYS Classic GUI Import geometries into Workbench (Design Modeler) Mesh as a mechanical model Use FE Modeler to export mesh and components The University of Sydney Page 44

45 Summary Bone microstructure and composition Bone constitutive models Bone remodelling mechanisms Bone remodelling calculations External remodelling Internal remodelling Assignment 3 handed out today, due week 13 Group project reports and presentations are due in week 13 Quiz in week 12 The University of Sydney Page 45