Bone Tissue Mechanics - ULisboaBone Tissue Mechanics João Folgado Paulo R. Fernandes Instituto...

Biomecânica dos Tecidos, MEBiom, IST

Bone Tissue Mechanics

João FolgadoPaulo R. Fernandes

Instituto Superior Técnico, 2015

PART 4 and 5


Bone Biology


Bone cellsBone cells:

• osteoclasts (bone resorption)

• osteoblasts (bone formation)

• osteocytes

•lining cells

trabecula

osteoblast

osteoclast

lining cell

Osteocyte (in lacunae)

canaliculi


Bone Modelling vs. Remodelling

osteogenesis – bone production from soft tissue (fibrosis tissue or cartilage). Bone formation in a early stage of growth. It also happens in bone healing.

bone modelling – Modelling results in change of bone size and shape. The rate of modelling is greatly reduced after skeletal maturity. Involves independent actions of osteoclasts and osteoblasts.

• bone remodelling – Process of replacement of “old bone” by “new bone”. It repairs damage and prevent fatigue damage. Usually does not affect size and shape. Occurs throughout life, but is also substantially reduced after growth stops. A combined action of osteoclasts and osteoblasts (BMU - Basic multicellular unit).


Some features of bone tissue

Bone Surfaces

• periosteum (external surface of bone)• endosteum (internal surface of bone)• Haversian canals• Trabeculae

Bone surfaces are important because is there that bone activity take place in order to renew bone tissue.


Types of bone tissue and the shapes of bones

- different shapes for bones of the same individual-Similarities between bones of different animals (we can recognize a femur, regardless of what animal it came from)- Some specific characteristics for a specie. (it is possible to identify an isolated bone as a human bone).


Compact Bone (cortical)

Spongy Bone (trabecular)

Marrow

Types of bone tissue: Trabecular vs. Compact bone

- Compact bone – porosity 5%-10%- Trabecular bone – porosity 75%-95%


Composition of Bone – Quantitative representation

VT=Vm+VvVT –total volumeVm – Bone matrixVv – volume of voids

Bv – volume fraction (Bv = 0 ∼ 1) pv – porosity (pv = 0 ∼ 1)

ρ – apparent densityρm – density of bone tissue ≅ 2.0 g/mlρv – density of soft tissue in the void spaces

≅ 1.0 g/ml

, m vv v

T T

V VB pV V

= = 1.0 2.0 /

m m v v

T

V VV

g ml

ρ ρρ

ρ

+=

=

Porosity:• cortical bone → pv = 5% ~ 10%• trabecular bone → pv = 75% ~ 90%

- Water- Organic Matrix (mainly collagen)- Mineral (mainly hydroxyapatite)

Ash fraction: ratio between the ash mass and dry mass. It is a measure of the degree of mineralization of bone tissue


Bone is a hierarchical material


Types of bone tissue: Lamelar vs. Woven

Types of bone tissue: Primary vs. Secondary

Primary – appears during growth (modelling)Secondary – appears due to remodelling

At a refined scale we can distinguish between lamelar and woven bone.Lamelar – highly organized boneWoven – poorly organized bone


Woven bone

• Quickly formed, poorly organized tissue

Plexiform bone

•Quickly formed •“brick wall” appearence• layers of lamelar and woven bone• appears in large, fast-growing animals (cows, sheeps)


Compact bone vs trabecularpropertie Cortical trabecular

Volume fraction 0.95 0.20

surface / bone volume (mm2/mm3)

2.5 20

Total bone volume(mm3)

1.4×106

(80%)0.35×106

(20%)

Total internal surface(mm2)

3.5×106

(33%)7.0×106

(67%)

• the surface of trabecular bone is one reason to have more bone remodelling.• This can be a consequence of a higher necessity of remodelling.• the higher remodeling rate can explain the fact of trabecular bone being less stiff.


Cortical bone – hierarquical levels

level structure dimensions0 Solid material > 3000 μm1 Secondary osteons (A)

Primary osteons (B)plexiform (C)woven bone (D)

Ø=200 ~ 300μm

l ≈ 150 μm100 – 300 μm

2 lamellae (A,B*,C*)lacunae (A,B,C,D)cement lines (A)canaliculi

t ≈ 3 ~ 7 μm

Ømax =10~20μm

t ≈ 1 ~ 5 μm3 – 20 μm

3 Collagen-mineral composite (A,B,C,D)

0.06 – 0.6 μm


Primary bone – Primary osteons

• primary bone is also called as immature bone• primary osteons are formed by the mineralization of cartilage• primary osteons have less lamellae than the secondary osteons.


Secondary Cortical Bone


• The secondary osteons result from remodeling•Osteons affect the mechanical properties of cortical bone:

•By the replacement of highly mineralized bone matrix with less calcified material•Increasing the porosity•Altering the collagen fibers orientation•Introduction of a cement line interface.

Ø osteons = 200 ~ 300 μmØ canal = 50 ~ 90 μmØ vessel = 15 μmØmax lacuna = 10 ~ 20 μm

Secondary Cortical Bone


BMU – Basic Multicellular Unit

In compact bone the BMU works in order to create a secundary osteon


BMU – tunneling and forming an osteon


Cortical bone – secondary osteons


osteons –lamellar structure

•osteons are made of severallamellae


Orientation of colagen fibers

Ascenzi and Bonucci :

• fibers are paralel within lamellae• different orientation between lamellae

a = type T (transverse fiber orientation) b = type A (alternating)c = type L (longitudinal)


Cortical bone structure

1

2

3?


Trabecular bone

• here we have trabeculae instead od osteons.• trabeculae thickness less than 200μm and length about1000μm=1mm • trabecular bone is less stiff than cortical bone. • Bone turnover is faster in trabecular bone


Trabecular bone


Bone remodelling on trabecular surface

• remodelling on trabecular surface


Bone remodelling on trabecular bone


BMU activation and Turnover rate


Material properties of cortical bone

1

2

3?


Mechanical Properties - Anisotropy of Cortical Bone

• The stiffness and strength for the human and bovine secondary bone are similar in the transverse direction but in the longitudinal direction the values are higher for bovine bone.•The anisotropy ratios are less in human Haversian bone than in bovine Havarsian bone. •The ultimate stress is very different for compression and tension

• comparing the ultimate stress in the longitudinal, circumferential and radial directions for bovine bone the following ration were obtained:

• primary bone → 3 : 1 : 0.4• secondary bone (tension) → 3 : 1 : 0.7• secondary bone (compression) → 3 : 1 : 1

• osteons make whole bone transversely isotropic with respect to strength

1 - radial

3 - longitudinal

2 - circunferencial


Mechanical properties – elastic properties of bovine bone

• Secondary bone present a transversely isotropic behavior• Due to bone remodeling (haversian bone) the behavior goes from orthotropic to transversely isotropic

• ultrasonic measure

1 - radial

3 - longitudinal

2 - circunferencial

transversalmente isotrópico

( )

44

44

13

1

11

11

1

2

3

1

3

1

12

3

0 0 00 0 00 0 0

0 00

1sim.2

C

CC

C

C C

C

C

C

C

= −

C


Mechanical properties for compact bone

12

3

transversely isotropic

( )

44

44

13

1

11

11

1

2

3

1

3

1

12

3

0 0 00 0 00 0 0

0 00

1sim.2

C

CC

C

C C

C

C

C

C

= −

C

?


example – bone with implant

• usual loads do not lead to high transverse normal stresses• however, there are situations where the transverse normal stress can be high, for instance the introduction of an implant can induce this kind of stress and lead to bone failure.


Failure criteria for orthotropic materials

• for orthotropic materials (or transversely isotropic), failure criteria such as the Von Mises criterion are not adequate. • a failure criterion often used for this type of materials is the Tsai-Wu criterion. (it is applied for composite materials – laminates reinforced with fibers – it is adequate for cortical bone)

• theTsai-Wu criterion says the the failure occurs when the stress level, described by the following expression, is greater or equal to 1

F1σ1+ F2σ2+ F3σ3+ F11(σ1)2+ F22(σ2)2+ F33(σ3)2++2F12σ1σ2+ 2F13σ1σ3+ 2F23σ2σ3+ F44(σ4)2+ F55(σ5)2+ F66(σ6)2 =1

where σ1,σ2 ,σ3,σ4,σ5,σ6 are the stresses σ11,σ22,σ33,σ12,σ23,σ13 in the material reference system, and the 12 coefficients Fi,Fij are determined experimentally


How to obtain the coefficients for the Tsai-Wu criterion?Example• for instance, doing uniaxial tension and compression tests in direction 1 (direction of orthotropy) we have:

tension: F1σ1tf + F11(σ1

tf )2=1compression: F1σ1

cf + F11(σ1cf )2=1

where σ1tf e σ1

cf are the ultimate values for tension and compression, respectivelySolving the system for the two unknowns:

F1=1/σ1tf +1/σ1

cf F11= – 1/(σ1tf σ1

cf )

• similar test can be done for the remaining directions as well as shear tests to obtain the coefficients Fi e Fii

• the coefficients Fij, i≠j, must be obtained by biaxial tests.

Failure criteria for orthotropic materials


Cortical Bone – influence of porosity

Elastic Modulus:

From diverse studies (Schaffler and Burr, 1988, Carter and Hayes, 1977, Rice et al., 1988) we can consider that the elastic modulus is proportional to a power of the volume fraction:

Cortical bone→ E ∼ (1−p)7.4

Trabecular bone→ E ∼ (1−p)2

Cortical and trabecular bone→ E ∼ (1−p)3


• type L osteon have greater tensile strength and lower ultimate deformation than the type A osteons.• the degree of mineralization has little effect on type A osteons•the degree of mineralization has great effect on the type L osteons, both on stiffness and deformation until rupture. • with less mineralization the two types of osteons has similar behavior.

a = type T , b = type A , c = type L

ε = 0.01 = 1% = 10.000 με

Mechanical properties of osteons – tension test


Cortical bone – influence of the collagen fibers orientation

• on the left panels, the darker pixels represent more fibers oriented in a longitudinal direction, and the lighter pixels represent more fibers oriented in the transversal direction.

•The darker region is where bone is under tensile stresses where the lighter correspond to bone in compression

•It suggest that collagen fibers tends to align depending on mechanical stimulus.

• The left panels show that the compression side is more mineralized.

Fiber orientation mineralization


Mechanical properties of cortical bone - summary

In summary, the variables that have influence on properties of cortical bone are:•Porosity•Degree of mineralization•Orientation of collagen fibers

(the porosity and the degree of mineralization are the factors that define the bone apparent density).

These variables are affected by the histological bone structure (primary or secondary bone, lamellar vs woven bone, osteons, …).

Fatigue damage and rate of deformation are also import factors.


Mechanical Properties of trabecular bone – continous assumption

• trabecular thickness is about 100μm−300μm and the space between adjacent trabeculae is the order of 300μm−1500μm (these values don’t not depend on animal size).• to analyze bone in the context of continuum mechanics the samples have to have the minimum dimensions of 5mm−10mm (equivalent to 5 spaces between trabeculae).

• the study of trabecular bone for animals with small dimensions should consider the trabeculae as structures instead of a porous material in continuum mechanics


• for a scale of cm, trabecular bone is less stiff (more compliant) and less strength (weaker) than cortical bone.• there is a great dispersion of values – the values are strongly influenced by porosity.•The ultimate stress is equivalent under compression and under tension

Mechanical Properties of trabecular bone


• for a scale of cm , the compression test for trabecular bone is very different of the compact bone.• a stress peak is followed by a reduction and than a plateau occurs. Finally the stress eventually increases. This behavior is consequence of the collapse and densification of trabeculae.



• elastic properties for a tibia (Ashman et al., 1989) (the standard deviation is between parentheses)

Modulus Mean Value (MPa)E1 346.8 (218)

E2 457.2 (282)

E3 1107.1 (634)

G12 98.3 (66.4)

G13 132.6 (78.1)

G23 165.3 (94.4)

• note the high values for the standard deviation .• note the degree of anisotropy, far from transversely isotropy.

•1,2,3 are directions of orthotropy1 = anterior-posterior2 = medial-lateral3 = inferior-superior



The variables that influence the mechanical properties of trabecular bone are:• porosity (or apparent density)• trabeculae orientation• the properties of an individual trabecula

remark: • the third variable is less influent then the others two.• the apparent density depend on porosity and degree of mineralization

Values for apparent density:trabecular bone → ρ = 1.0 ~ 1.4 g/cm3

cortical bone → ρ = 1.8 ~ 2.0 g/cm3

remark:The apparent density, ρ, is measured considering the voids filled with soft tissue (ρ = 1.0 ~ 2.0 g/cm3). Some authors consider apparent density, d, considering empty voids (d = 0.0 ~ 2.0 g/cm3).



• the influence of the apparent density on elastic modulus is clear in the graphs

• in literature the elastic modulus is referred as being proportional to a power (2 or 3) of the apparent density.trabecular bone (only) → E ∼ d 2 , σU ∼ d 2

cortical and trabecular bone → E ∼ d 3

• apparent density (or porosity) is the parameter responsible for about 75% of the variability of mechanical properties of trabecular bone.

Mechanical Properties of trabecular bone – influence of apparent density


• are the properties of trabecular tissue equivalent to the tissue of cortical bone?

• the bone of trabeculae is less stiff than the cortical tissue.

• it can be because of a lower degree of mineralization, but also because there is no intact fibers (osteons) inside a trabecula.

• other tests with samples of small dimension, show a decrease on bone properties.

Mechanical Properties of trabecular bone – influence of the properties of trabecular tissue


• Are the properties of trabecular tissue equivalent to the tissue of cortical bone?

•The lamellar structure is identical.

•But due to the rate of remodeling the degree of mineralization is lower.

• There isn’t entities like osteons totally integrated in a matrix. There is only part of these structures resulting of bone remodeling , where the boundary is not all inside the trabecula.

Mechanical Properties of trabecular bone – influence of the properties of trabecular tissue


Modeling Cancellous Bone as a Cellular System of plates or Struts

• Gibson (1985) derived relationships between equivalent material properties and porosity, using open and close cell porous structures. • Results are consistent with previous ones, with stiffness varied as d3 for closed cell (dense bone) and as d2 for open cell (cancellous bone)


Bibliography

Skeletal Tissue Mechanics , R. Bruce Martin, David B. Burr, Neil A.Sharkey, Springer Verlag,1998.

Orthopaedic Biomechanics, Mechanics and Design inMusculeskeletal Systems, D. Bartel, D. Davy, T. Keaveny, PearsonPrentice Hall, 2006.

Bone Mechanics Handbook, 2nd Edition, S.C. Cowin, CRC Press,2001

Mechanics of Materials, 5th Edition, F. Beer, Jr., E. R. Johnston , J.DeWolf, D. Mazurek, McGraw Hill, 2009

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