Bone Tissue Mechanics
João Folgado Paulo R. Fernandes
Instituto Superior Técnico, 2015
PART 4 and 5
Biomecânica dos Tecidos, MEBiom, IST Bone Biology
Biomecânica dos Tecidos, MEBiom, IST Bone cells Bone cells: • osteoclasts (bone resorption) • osteoblasts (bone formation) • osteocytes •lining cells osteoblast
canaliculi
Osteocyte trabecula (in lacunae) lining cell osteoclast Biomecânica dos Tecidos, MEBiom, IST 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).
Biomecânica dos Tecidos, MEBiom, IST 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.
Biomecânica dos Tecidos, MEBiom, IST 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).
Biomecânica dos Tecidos, MEBiom, IST Types of bone tissue: Trabecular vs. Compact bone
Spongy Bone (trabecular) Compact Bone (cortical) Marrow
- Compact bone – porosity 5%-10% - Trabecular bone – porosity 75%-95%
Biomecânica dos Tecidos, MEBiom, IST Composition of Bone – Quantitative representation
- Water - Organic Matrix (mainly collagen) - Mineral (mainly hydroxyapatite)
VT=Vm+Vv ρ – apparent density V –total volume ρ ≅ T m – density of bone tissue 2.0 g/ml V – Bone matrix ρ m v – density of soft tissue in the void spaces Vv – volume of voids ≅ 1.0 g/ml ∼ Bv – volume fraction (Bv = 0 1) ρρVV+ ∼ ρ = mm vv pv – porosity (pv = 0 1) VT VV ==mv ρ=1.0 2.0gml / Bpvv , VVTT
Porosity:
• cortical bone → pv = 5% ~ 10% • trabecular bone → pv = 75% ~ 90% Ash fraction: ratio between the ash mass and dry mass. It is a measure of the degree of mineralization of bone tissue Biomecânica dos Tecidos, MEBiom, IST Bone is a hierarchical material
Biomecânica dos Tecidos, MEBiom, IST Types of bone tissue: Lamelar vs. Woven
At a refined scale we can distinguish between lamelar and woven bone. Lamelar – highly organized bone Woven – poorly organized bone
Types of bone tissue: Primary vs. Secondary
Primary – appears during growth (modelling) Secondary – appears due to remodelling
Biomecânica dos Tecidos, MEBiom, IST 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)
Biomecânica dos Tecidos, MEBiom, IST Compact bone vs trabecular propertie Cortical trabecular Volume fraction 0.95 0.20
surface / bone volume 2.5 20 (mm2/mm3)
Total bone volume 1.4×106 0.35×106 (mm3) (80%) (20%) Total internal surface 3.5×106 7.0×106 (mm2) (33%) (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.
Biomecânica dos Tecidos, MEBiom, IST Cortical bone – hierarquical levels
level structure dimensions 0 Solid material > 3000 μm
1 Secondary osteons (A) Ø=200 ~ 300μm Primary osteons (B) 100 – 300 μm plexiform (C) l ≈ 150 μm woven bone (D)
2 lamellae (A,B*,C*) t ≈ 3 ~ 7 μm lacunae (A,B,C,D) Ø =10~20μm max 3 – 20 μm cement lines (A) t ≈ 1 ~ 5 μm canaliculi 3 Collagen-mineral 0.06 – 0.6 μm composite (A,B,C,D)
Biomecânica dos Tecidos, MEBiom, IST 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.
Biomecânica dos Tecidos, MEBiom, IST Secondary Cortical Bone
Biomecânica dos Tecidos, MEBiom, IST 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 μm Ø canal = 50 ~ 90 μm Ø vessel = 15 μm μ Ømax lacuna = 10 ~ 20 m
Biomecânica dos Tecidos, MEBiom, IST BMU – Basic Multicellular Unit
In compact bone the BMU works in order to create a secundary osteon
Biomecânica dos Tecidos, MEBiom, IST BMU – tunneling and forming an osteon
Biomecânica dos Tecidos, MEBiom, IST Cortical bone – secondary osteons
Biomecânica dos Tecidos, MEBiom, IST osteons –lamellar structure
•osteons are made of several lamellae
Biomecânica dos Tecidos, MEBiom, IST 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)
Biomecânica dos Tecidos, MEBiom, IST Cortical bone structure
? 3 2
1
Biomecânica dos Tecidos, MEBiom, IST 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
Biomecânica dos Tecidos, MEBiom, IST Trabecular bone
Biomecânica dos Tecidos, MEBiom, IST Bone remodelling on trabecular surface
• remodelling on trabecular surface
Biomecânica dos Tecidos, MEBiom, IST Bone remodelling on trabecular bone
Biomecânica dos Tecidos, MEBiom, IST BMU activation and Turnover rate
Biomecânica dos Tecidos, MEBiom, IST Material properties of cortical bone
? 3 2
1
Biomecânica dos Tecidos, MEBiom, IST 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
3 - longitudinal
2 - circunferencial • comparing the ultimate stress in the longitudinal, circumferential and radial 1 -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
Biomecânica dos Tecidos, MEBiom, IST Mechanical properties – elastic properties of bovine bone
3 - longitudinal • ultrasonic measure
2 - circunferencial
1 -radial
transversalmente isotrópico
C11 C12 C13 00 0 C C 00 0 11 13
C33 00 0 C = C44 00 C 0 44 1 sim. ()C − C 2 11 12
• Secondary bone present a transversely isotropic behavior • Due to bone remodeling (haversian bone) the behavior goes from orthotropic to transversely isotropic
Biomecânica dos Tecidos, MEBiom, IST Mechanical properties for compact bone
?
transversely isotropic
C11 C12 C13 00 0 3 C11 C13 00 0 2 C33 00 0 C = C44 00 1 C 0 44 1 sim. ()C − C 2 11 12 Biomecânica dos Tecidos, MEBiom, IST 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.
Biomecânica dos Tecidos, MEBiom, IST 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
σ σ σ σ 2 σ 2 σ 2 F1 1+ F2 2+ F3 3+ F11( 1) + F22( 2) + F33( 3) + σ σ σ σ σ σ σ 2 σ 2 σ 2 +2F12 1 2+ 2F13 1 3+ 2F23 2 3+ F44( 4) + F55( 5) + F66( 6) =1
σ σ σ σ σ σ σ σ σ σ σ σ where 1, 2 , 3, 4, 5, 6 are the stresses 11, 22, 33, 12, 23, 13 in the material reference system, and the 12 coefficients Fi,Fij are determined experimentally
Biomecânica dos Tecidos, MEBiom, IST Failure criteria for orthotropic materials
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:
σ tf σ tf 2 tension: F1 1 + F11( 1 ) =1 σ cf σ cf 2 compression: F1 1 + F11( 1 ) =1
σ tf σ cf where 1 e 1 are the ultimate values for tension and compression, respectively Solving the system for the two unknowns: σ tf σ cf σ tf σ cf F1=1/ 1 +1/ 1 F11= – 1/( 1 1 )
• 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.
Biomecânica dos Tecidos, MEBiom, IST 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
Biomecânica dos Tecidos, MEBiom, IST Mechanical properties of osteons – tension test a = type T , b = type A , c = type L
ε = 0.01 = 1% = 10.000 με
• 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. Biomecânica dos Tecidos, MEBiom, IST Cortical bone – influence of the collagen fibers orientation
Fiber orientation mineralization
• 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.
Biomecânica dos Tecidos, MEBiom, IST 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.
Biomecânica dos Tecidos, MEBiom, IST 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
Biomecânica dos Tecidos, MEBiom, IST Mechanical Properties of trabecular bone
• 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
Biomecânica dos Tecidos, MEBiom, IST 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. Biomecânica dos Tecidos, MEBiom, IST Mechanical Properties of trabecular bone
• elastic properties for a tibia (Ashman et al., 1989) (the standard deviation is between parentheses)
Modulus Mean Value (MPa) •1,2,3 are directions of orthotropy 346.8 (218) 1 = anterior-posterior E1 2 = medial-lateral 3 = inferior-superior 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.
Biomecânica dos Tecidos, MEBiom, IST Mechanical Properties of trabecular bone
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).
Biomecânica dos Tecidos, MEBiom, IST Mechanical Properties of trabecular bone – influence of apparent density
• 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. ∼ 2 σ ∼ 2 trabecular bone (only) → E d , U d 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.
Biomecânica dos Tecidos, MEBiom, IST Mechanical Properties of trabecular bone – influence of the properties of trabecular tissue
• other tests with samples of small dimension, show a decrease on bone properties.
• 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.
Biomecânica dos Tecidos, MEBiom, IST 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.
Biomecânica dos Tecidos, MEBiom, IST 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)
Biomecânica dos Tecidos, MEBiom, IST Bibliography
Skeletal Tissue Mechanics ,R.BruceMartin,DavidB.Burr,NeilA. Sharkey, Springer Verlag,1998.
Orthopaedic Biomechanics, Mechanics and Design in Musculeskeletal Systems, D. Bartel, D. Davy, T. Keaveny, Pearson Prentice Hall, 2006.
Bone Mechanics Handbook, 2nd Edition,S.C.Cowin,CRCPress, 2001
Mechanics of Materials, 5th Edition, F. Beer, Jr., E. R. Johnston , J. DeWolf, D. Mazurek, McGraw Hill, 2009
Biomecânica dos Tecidos, MEBiom, IST