COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE . APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING Lecture 1. Structure, properties and mechanical behaviour of bone tissue Manuel Doblaré Aragón Institute of Engineering Research (I3A) University of Zaragoza (Spain) [email protected]Lecture 1. Structure, properties and mechanical behaviour of bone tissue 2 COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE, APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING-BIOMAT-Granada June-2009 Lecture 1. Structure, properties and mechanical behaviour of bone tissue 1.1. Overview of the structure of connective tissues 1.2. Mechanical properties of bone tissue 1.3. Influence of age 1.4. Mechanisms and criteria of bone fracture 1.5. Computational prediction of bone fracture
Transcript
COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE . APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING
Lecture 1. Structure, properties and mechanical behaviour of bone tissue
Manuel DoblaréAragón Institute of Engineering Research (I3A) University of Zaragoza (Spain)[email protected]
Lecture 1. Structure, properties and mechanical behaviour of bone tissue 2
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Lecture 1. Structure, properties and mechanical behaviour of bone tissue
1.1. Overview of the structure of connective tissues
1.2. Mechanical properties of bone tissue
1.3. Influence of age
1.4. Mechanisms and criteria of bone fracture
1.5. Computational prediction of bone fracture
Lecture 1. Structure, properties and mechanical behaviour of bone tissue 3
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1.1. Overview of the structureof connective tissues
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STRUCTURE OF CONNECTIVE TISSUES
http://bioengineering.ucsf.edu/news.html
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PRINCIPAL COMPONENTS
Major structural matrix molecules (ECM):
• collagens
• proteoglycans
• fibronectin, laminin, elastin
Water + small molecules (ions, metabolites)
Cells: tissue-specific
Soluble factors in fluid or attached tomatrix:
• growth factors
• cytokines
• hormones
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Cortical bone
Elastic cartilage Hyaline cartilage
Trabecular bone
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Tendon (longitudinal section)
Skin
Tendon (transversal section)
Fibrocartilage
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Bone
• Cells: osteoblasts, osteoclasts and osteocytes
• Matrix: mainly collagen I and minerals (hydroxiapathite)
Cartilage
• Cells: chondrocytes
• Matrix: collagen II and large proteoglycans (agrecans)
Tendon/ligament
• Cells: fibroblasts
• Matrix: collagen I, elastin and small proteoglycans
Skin
• Cells: fibroblasts
• Matrix: dense, disorganized collagen (types I, III, IV)
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1.2. Mechanical properties ofbone tissue
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Bone has the following functions:
• Structural support
• Protection (skull)
• Assist movement
• Mineral “bank”
• Blood cell production: hemopoyesis (red marrow)
• Energy storage: adipose tissue (yellow marrow)
INTRODUCTION
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9Bone classification by shape: • Long bones
• Short bones
• Flat bones
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Parts of a long bone
• Epiphyses: expanded ends of a long bone that includes the articular surface
• Diaphysis: shaft of long bone
• Medullar canal
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Two different bone tissues: cortical and trabecular
• Both have the samecomposition andstructure
• Cortical bone has lowerporosity (10 %) thantrabeculae bone (50-90 %)
• Different distributionimplies differentmechanical properties
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Bone is a multiphasic, porous, heterogeneous and anisotropic material with different properties in tension and compression.
Cortical boneSpongy
bone
Bone microstructure has a strong influence of the macroscopic behaviour and therefore on the mechanical properties, i.e. cortical and spongy bone.
The microstructure is wonderfully adapted to the specific function of each bone, leading to very different mechanical properties.Bone is able to adapt its properties and structure in the long term to the specific mechanical environment it supports, achieving maximum stiffness and strength with minimum weight and reducing microdamage (Bone Remodelling).
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Flat bone (iliac crest) Short bone (body of the 4thhuman vertebrae)
Long bone (human phalanx)
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Simple load states (standard mechanical tests) like uniaxial tension and compression, three or four point bending, biaxial stress, etc., are used to obtain bone properties.
The obtained values are then extrapolated and averaged to obtain a continuous mathematically consistent constitutive model.
The specimen size is very important due to the average character of the continuum mechanical properties and the complex structure of bone tissue.
MAIN BONE MECHANICAL PROPERTIES
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Anisotropy is other important characteristic of bone tissue. Mechanical properties strongly depend on the direction.
Tension and compression tests on longitudinally and transversely oriented specimens from the medial human femur
Femoral dyaphisis (Frankel&Nordin)
have provided the following average values for cortical bone (Reilly y Burstein, 1975):
tension strength:
135 MPa (longitudinally)
53 MPa (transversely)
compression strength:
193 MPa (longitudinally)
133 MPa (transversely)
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Similar measures on the elastic modulus for the same cortical bone have given the following numbers in GPa:
– Reilly & Burstein, 1974, 1975; Ashman et al, 1984. Direct measures. Tension tests
Etransversal 11.5 18.8 12. ó 13.4 Elongitudinal 17. 27.4 20.
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With respect to the stress-strain relation, a typical curve for cortical bone is the following obtained for a strain rate of 0,001 s-1. For physiological strains the response is almost linear.
(Proubasta et al., 1997; Cowin et al., 1987)
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The strain rate affects the stress-strain curve (visco-elastic effect) but this effect is not much relevant for physiological rates.
Bone porous are filled with marrow. There is also fluid (mainly water) flowing through
Proubasta et al, 1997.
the different channels (Haversian cannals, canaliculi,..) that formed a complex flow network. This flow has an important mechanobiological function but its contribution to the mechanical behaviour of the tissue is negligible.
Bone is therefore usually modelled as a heterogeneous, anisotropic linear elastic material.
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Mechanical properties in trabecular bone strongly depend on the porosity and directional distribution of the trabeculae, changing within the same bone and from one bone to another.
This makes it difficult to determine the specific values of the elastic properties for trabecularbone.
QUANTIFICATION OF THE POROUS STRUCTURE OF CANCELLOUS BONE
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Structural average constitutive properties of trabecular bone have been also evaluated from the bulk properties of the composing material and micro-CT images by means of voxel-FEM approaches at the microstructural level performing virtually standard tests (Huiskes et al., Mueller et al.).
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9In spongy bone, the size of the trabeculae impedes modelling the whole microstructure (except for a very few cases), forcing to work with average or apparent properties thatdepend on the specific point and on the actual properties of the tissue.
Bone is formed by solid and porous phases. Porous are filled with marrow, that although very viscous, flows between them. There is also fluid (mainly water) flowing through the different channels (Haversian cannals, canaliculi,..) that formed a complex flow network inside bone.
This flow has important mechanobiological functions but its contribution to the mechanical behaviour of the tissue is negligible.
Bone is therefore usually modelled as a heterogeneous, anisotropic linear elastic material.
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• Following the principles of Continuum Mechanics, we work, therefore with properties averaged in a volume of enough size to collect the main statistical properties of the tissue (RVE).
• The microstructural distribution of mass is then averaged by the so-called apparent density (ρ), related to the actual tissue density ( ) and porosity (n).
VT
VhuecosVM
( ) ρ
ρ -
/ρM
)ρ/(M-
V
V -
V
VV
V
V n
T
T
T
M
T
MT
T
voids
ˆ1
ˆ11 ===
−==
ρ̂
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9In 1985,86 Cowin defined a second-order, positive definite, structural tensor, the “fabric tensor”, as a stereological measure of the directional distribution of bone mass, related therefore with the anisotropic properties of bone tissue.
Since then, different ways of measuring the fabric tensor have been defined: The mean interception length (MIL), the volume orientation (VO) and the star volume distribution (SVD) among others.
Mean interception length (MIL) Method.
Determination of the fabric tensor
MIL (ω) = L / I (ω)L is the total length, I thenumber of interceptions
and ω the orientation
1/2-jiij2
)(L
1MH =⇒= nnM
n
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The volume orientation (VO) Method
The star volume distribution (SVD) Method
At each point the VO measure is defined as the length associated to the orientation with the maximum interception length for a number of random points distributed in the representative volume element (RVE) associated to that point.
∑=
ω=ωn
1ii )( L
n
1 )( s
At each point the SVD measure is defined as:
being n a number of random points distributed in the representative volume associated to that point
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Different experimental studies (Odgaard et al, 1997) have shown that all these methods give rise to very similar results and that those results characterize sufficiently well the principal directions and principal values of the fabric tensor.
These measures have also shown that these directions and values correlate equally well with the local anisotropic properties of bone tissue, that is with the local elastic constitutive tensor (van Rietbergen et al, 1996b; Odgaard et al, 1997; Zysset et al, 1998).
What it seems clear is that cancellous bone can be considered with enough accuracy as a locally orthotropic material, such as its principal axes of orthotropy are aligned with those of the fabric tensor.
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9As age progresses, the walls of the trabecuale of cancellousbone become thinner, sometimes arriving to complete resorption. An important reduction of the mass of the spongy bone is progressively produced.
The cortical region also diminishes its thickness and the whole size of the bone may also reduce. Porosity of cortical bone alsoincreases for advanced ages.
Healthy bone Osteoporotic bone
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This decrease in the internal mass and size of bone organs, produces a reduction of the mechanical properties of bone tissue (Frankel y Nordin, 1980; Kelly et al, 1988; Buckwalter et al, 1995b; Evans, 1973).
Stress-strain curves for the tibia of two adults with different ages (Frankel y Nordin, 1980; Burstein et al, 1976).
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Experimental studies show that the process of bone remodelling is altered by age. Both the response to cyclic loads and the hormonal balance are modified also by age (Buckwalter et al., 1995a,b).
Regular exercise seems to help to keep bone mass or at least to reduce the rate of loss of bone mass in elder people and postmestrual women (Forwoodand Burr, 1993; Jacobson et al., 1984).
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1.4. Mechanisms and criteriaof bone fracture
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OVERLOAD
CREEP
FATIGUE
MECANISMS OF BONE FRACTURE
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Stress tensor
MACROSCOPIC CRITERIA OF BONE FRACTURE UNDER OVERLOADING
Porosity
F(σeq(σij), σu (Hij, n, α)) = 0
Fabric tensor Ash fraction
Equivalent stress Ultimate strength
strength ultimatestress equivalent
=≡ RFCβ
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The ultimate strength of bone depends on:
• porosity/apparent density;
• degree of mineralisation;
• microstructure: fabric tensor.
σu (Hij, n, α)
porosityanisotropy
Ultimate Strength
mineralisation
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BONE FRACTURE CRITERIA USED IN NUMERICAL SIMULATIONSE
Anisotropic criteria
Symmetric
Non-symmetric
• von Mises-Hencky(Lotz et al., 1991; Ford et al. 1996, Keyak et al. 1998, 2000, 2001)
• Maximum stress (criterio de Rankine)(Fenech y Keaveny, 1999; Keyak y Rossi, 2000)
• Mohr-Coulomb(Keyak y Rossi, 2000)
• Tsai-Wu (1971)
• Cowin (1986)
• Pietrusczak (1999)
• Zysset and Ricón (2004)
Isotropic criteria
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( ) ( ) ( )
( ) ( ) ( )c
ct
c
Assumes
σσσσσσσ
β
σσσσσσσσσ
221
213
232
221
213
232
−+−+−=
≈
=−+−+−
-200
-150
-100
-50
0
50
100
150
200
-200 -100 0 100 200
sig1(Mpa)si
g2(M
pa)
Ductile materials (metals)
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Hoffman
( ) ( ) ( )
( ) ( ) ( ) 3625142
2132
1322
321
654321
3625142
2132
1322
321
11
2
1
1
σσσσσσσσσβ
σσσσ
σσσσσσσσσ
CCCCCC
CCCCCC
CCCCCC
ctct
+++−+−+−=
−======
=+++−+−+−
-120
-80
-40
0
40
-200 -150 -100 -50 0
σ2σ
1
(Brittle materials)
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Maximum stress
(Rankine)
321 σσσ >>⎩⎨⎧
<>
c
t
σσσσ
3
1
Mohr-Coulomb
Modified Mohr-Coulomb
⎥⎦
⎤⎢⎣
⎡=
ct
maxσσ
σσβ 31 ,⇒
13
1
=−c
t
σσ
σσ
ct σσ
σσβ 31 −=⇒
⎪⎪⎩
⎪⎪⎨
⎧
=−−
−≤=
..1
11
31
3
1
1
ccctc
tc
t
σσσ
σσσσ
σσ
σσ
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Criterion of Pietruszczak (Concrete, frictional anisotropic materials)
( ) ( )
( ) ( ) ( )( )( ) ( ) ( ) ( )θ⋅−−+−−+
−−+=θ
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅θ
σ+⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅θ
σ=
3sina1K1a1a1K
Ka1a1g
0f
Ia
fga
fgaF
c3
2
c2
c1
Directional dependency: ( ) ( ) γγ
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ρρ
=0
i0c
00cic n1
ln1fflf
Criterion of Tsai-Wu (anisotropic materials)
( ) 1=+= jiijiik FFf σσσσ
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9Criterion of Zysset and Rincón
• Double quadratic criterion (traction and compression) based on the Hill criterion.
• It fulfills all the tensorial and thermodynamic requirements.
• It has some problems in identifying the criterion material parameters of the bone organ point wise.
• It has been only applied to the interpretation of experimental tests on small size specimens very well characterized.
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Criterion of Cowin (1986) Locally orthotropic materials
6,...,2,1,, para 1),,( =−+= kjiFGnf kmijijkmijij σσσσ Η
12 −++ jjiiiijjiiiiiiiiii FFG σσσσ
In the space of principal streses, it may be written as
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• Biaxial strength:
• Directional tension strength:
Constant 0<α< 1
• Directional compression strength: (Pietruszczak et al. , 1999)
Directional porosity
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1.5. Computational predictionof bone fracture
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Non-invasive techniques as dual energy absorciometry no (DEXA) o computerized axial tomography (TAC) allow obtaining the distribution of porosity point-wise but not the distribution of anisotropy.
ALTERNATIVE: Three-dimensional Micro-CT for small size specimens.
GENERAL ALTERNATIVE : BONE REMODELLING MODELS
PARAMETERS ESTIMATION
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THE FRACTURE CRITERION
New loads andboundaryconditions
Mesh of theproximal
femur
BONE REMODELING
Cycle of 100 days Fabric TensorDensity
Prediction ofthe type of
bone fracture
Risk to Fracture Coefficient of
(RFC)
stress ultimatestressequivalent
=RFC
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Acetabularcup
Femur Load cell
Belt
Support
Acetabularcup
Load
45º
Load cell
Shaft holder
Belt
Support of the femoral
head
Femur
Shaft holder
Experimental results (Yang et al., 1996)
Load at the lesser trochanter 7 neck fractures
Load at the greater trochanter 4 subtrochanteric and 3 intertrochantericfractures
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Iliopsoascontraction
Gluteusmediuscontraction
45º
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Von Misses Hoffman Maximal stress
Load at the lesser trochanter (Iliopsoas)
Neck fracture
P=3000N α = 0.2
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Modified Cowin
Neck fracture
P=3000N α = 0.2
Load at the lesser trochanter (Iliopsoas)
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Von Misses Hoffman Maximal Stress
P=6000N α = 0.4
Cowin
Neck fracture
Load at the lesser trochanter (Iliopsoas)
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Von Misses Hoffman Maximal stress
P=3000N α = 0.2
Load at the greater trochanter (Gluteus medius)
Sub and Inter trochanteric fractures
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Lecture 1. Structure, properties and mechanical behaviour of bone tissue 54
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The use of isotropic elastic behavior and isotropic criteria allows predicting neck fractures but not intertrochantericfractures.
Anisotropy plays a fundamental role in the probability and region of fracture appearance in bone organs.
To obtain accurate quantitative results, subject-specific simulations are needed, using the actual material properties of the specific bone organ. On the contrary, only qualitative conclusions can be obtained due to the strong inter-individual variability of the properties of bone tissue.
MAIN CONCLUSIONS
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• Different works seem to indicate that local bone failure may be driven by deformation (Nalla et al., 2003, Taylor, 2003).
• There is a growing consensus on the substantial isotropy of the yield strain and its invariance to density (Bayraktar, 2004, Currey, 2004, Cowin and He, 2005).
• Strain based criteria seem to reproduce accurately some uniand multiaxial experiments (Bayraktar, 2004)
• Some recent results (Viceconti, 2007) seem to support these ideas, proving that a maximum strain criterion produces a more localized risk fracture region and in some specific experiments an accurate reproduction of the fracture load.
STRAIN-DRIVEN CRITERIA
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STRESS FRACTURES
Bone is continuously cumulating microcracks (damage) that induce a progressive reduction of the strength and increase the risk of fracture under physiological loads.
Stress fractures are produced when damage accumulation and damage repair due to bone remodeling become unbalanced.
Pathologic bones (i.e osteoporotic) and bones under cyclic loads with high amplitude have a higher probability of developing stress fractures.
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Fatigue strength of materials is usually determined by an important amount of tests under different stress levels, determining the number of load cycles needed to get the fracture.
This results are represented in curves (S-N curves).
Fatigue behaviour under stress control for a bone “in vitro” under cyclic loads (Proubasta, 1997; Carter y Hayes, 1977).
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Some results show that fatigue life is much more dependent on the amplitude of the strain cycle than on its mean or peak value.
Fatigue behaviour under strain control for a bone “in vitro” (Proubasta et al., 1997).
These results suggest that fatigue life of bone is much lower than that previously thought. This means that bone is permanently accumulating fatigue damage during its normal activity, being only the bone remodelling process the responsible of maintaining the long-term structural integrity of the skeletal system.
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PREDICTION OF STRESS FRACTURES
First alternative: Simple models that predict “fatigue” life by means of experimental S-N curves.
However, more accurate models should take into account the simultaneous accumulation and repair of microdamage. In this case, damage repair has to be taken into account.
In fact, Frost (1960) and many other authors have proposed that ones of the main goals of bone remodeling is microcrackrepairing and therefore microdamage would be the actual bone remodeling mechanical stimulus.
Second alternative: To use coupled damage-remodeling approaches. We shall discuss this aspect in the next lectures.
COMPUTATIONAL MECHANOBIOLOGY OF HARD TISSUE . APPLICATIONS TO IMPLANT DESIGN AND TISSUE ENGINEERING
Lecture 1. Structure, properties and mechanical behaviour of bone tissue
Manuel DoblaréAragón Institute of Engineering Research (I3A) University of Zaragoza (Spain)[email protected]
