Introduction

The “material properties” of the musculoskeletal tissues

• The “material properties” of the musculoskeletal tissues depend on the underlying micro-structures
• They are typically inhomogeneous
• The depend on age, disease, and other factors
• Thus, we must consider
  • Tissue level (inhomogeneous, anisotropic)
  • Organ level (homogenized, isotropic or anisotropic)
  • Knowledge of the scale of interest is critical to our material modeling approach

Challenges in biological material testing

• Challenges in material testing include:
  • What is our scale of interest?
  • Isolation of homogeneous tissue
  • Measurement of deformation (porous tissue)
  • Control of environment (in-vivo vs ex-vivo)
  • Inter-specimen variability
• We attempt generalization based on:
  • Density
  • Organic composition
  • Mineralization
  • …

• Biological tissues adapt to their environment at the cellular, structural, and molecular level
  • ie bone density and geometry adapt to load
• Feedback loop (in time)
  • Mechanics respond to biology
  • Biology responds to mechanics
• An important aspect of modern implant design is prediction of in-vivo mechanobiologic response
  • How will the bone respond to the implant?
  • ie: will stress shielding be a problem?
• Currently, we have little control and seek to understand
• In future, we may also try to influence this response

Composition of bone

• Bone composed of organics and inorganics
  • Loosely defined, organic compounds contain carbon covalently bonded to hydrogen, oxygen or nitrogen, etc
• By mass, bone is approximately:
  • 60% inorganic material (calcium phosphate, etc)
  • 30% organic
  • 10% water

Collagen

Bone is a hierarchical composite material

• Bone is a hierarchical composite material

The four levels of bone microstructure, from the level of mineralized collagen fibrils to cortical and trabecular bone. It is generally assumed that at the former level, all bone is equal, although there can be subtle differences in the nature of the lamellar architecture and degree of mineralization between cortical and Trabecular bone.³

Lowest hierarchical level

• Collagen fibril (\(\approx 0.1\) micron)
• Sheet lamellae of uni-directional fibrils (most common) or blocks of pseudo random “woven” fibrils (\(\approx 10\) micron)

Highest hierarchical level

• Cortical bone
  • Tightly packed lamellar, Haversian, laminar, or woven bone
  • Lots of osteons
• Trabecular bone
  • Highly porous rods and plates interspersed with marrow spaces
  • Less well organized packets of lamellae
  • Very few osteons if any

Highest hierarchical level, diaphysis

Diagram of a sector of the shaft of a long bone, showing the different types of cortical bone, trabecular bone, and the various channels. The osteons are located between the outer and inner circumferential lamellae.⁴

Cortical bone

• Laminar bone (sandwich sheets of lamellar bone layers)
• Haversian bone (10-15 lamellae in a cylinder surrounding a Haversian canal)
  • Contains blood vessels, nerves, and lymphatics
• Osteon (substructure which includes the Haversian canal) 1-3 mm long by 200 microns in diameter
  • This is a “unit cell” (non-medical term) discrete structure for mechanical study
  • Osteons are continually being torn down and replaced
    • Process takes months for individual osteon, thus time is required (stress fracture if overused without adaptation time)
  • Osteons are bound to each other by a cement line
    • It is “weak”
    • It is analogous to matrix in a composite material
    • Passes shear loads, dissipates energy

Damage detection

• Pores exist in the bone cells
• Fluid resides around bone cells in holes (called lacunae)
• These lacunae are interconnected by tiny channels (canaliculi) and meet at gap junctions
• Small molecules including ions pass between the cells and are thought to help sense damage

Trabecular architecture

Differences between cortical and trabecular bone

• The biggest difference is the cellular-like (spongy structure of trabecular bone
  • Holes filled with marrow
• The rods and plates tend to remodel more often, thus, trabecular bone is less mineralized than the “older” cortical bone
• The material properties are thus slightly worse on the tissue level (even worse on the structural level due to cellular structures)

Elastic anisotropy

Thought experiment: isotropic material

Uniform rectangular block pulled on both ends

What strains do you expect for \({\sigma_{xx}}\ne 0\) (all others stresses = 0)?

• We are all familiar with Hooke’s Law: \[\begin{equation*} {\sigma_{xx}}= E {\varepsilon_{xx}} \end{equation*}\]
• Rearranging: \[\begin{equation*} {\varepsilon_{xx}}= \frac{{\sigma_{xx}}}{E} \end{equation*}\]
• But what are the other strains?

• But what are the other strains? \[\begin{align} {\varepsilon_{yy}}=& \, -\nu {\varepsilon_{xx}}\cr {\varepsilon_{zz}}=& \, -\nu {\varepsilon_{xx}}\cr =& \, -\nu \frac{{\sigma_{xx}}}{E} \cr {\varepsilon_{ij}}=& \, 0 \hspace{5mm} \mbox{ for } i\neq j \cr \end{align}\]

• Similarly, we can obtain similar equations in the other directions:

• We’ve found a pattern for the normal stress-normal strain response

• What about for shear?

For \({\tau_{yz}}\ne 0\), all others 0? \[\begin{align} {\gamma_{yz}}=& \, \frac{{\tau_{yz}}}{G} \cr {\gamma_{xy}}=& \, 0 \cr {\gamma_{xz}}=& \, 0 \cr {\varepsilon_{ii}}=& \, 0 \hspace{5mm} \mathrm{No \; sum} \cr \end{align}\]

For multiple simultaneous stresses: use superposition

Isotropic constitutive behavior

\[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2 G} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\] Note also: the shear modulus is: \[\begin{split} %% G =& \mu \\ G =& \frac{E}{2 (1+\nu)}\\ \end{split}\] Thus there are only two constants that describe the behavior of an isotropic material

Is this how all materials behave?

Origin of anisotropic behavior in bone

• In cortical bone, osteons align parallel to loads
• In trabecular bone, rods and plates also align with loads
• Bone properties have directionality (called anisotropy)

Principal material coordinate system

Principal material coordinate system for an orthotropic material. This coordinate system (right) is aligned with the mutually orthogonal “grain” axes of the material’s microstructure (left. As a class of anisotropic materials, orthotropic materials have their grain along three mutually perpendicular axes.

Spatial variations in the orientation of the principal material coordinate system can occur. In many cases, a local coordinate system in cylindrical coordinates can be used to describe such spatial variations.

Anisotropic behavior

Anisotropic behavior

• Fibrous composites exhibit a more complex constitutive response
• Consider the following material description: \[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36} \\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46} \\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56} \\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]
• This is generalized Hooke’s law (applicable to any linear elastic material–called anisotropic).

• For simplicity, we can write generalized Hooke’s law as: \[\{{\varepsilon}\} = [S] \{\sigma\}\]
• the values of \(S\) are called “elastic compliances”

Elastic constants

• It is important to be able to consider the inversion of this system: \[\begin{split} \{{\varepsilon}\} =& [S] \{\sigma\} \\ \{\sigma\} =& [S]^-1 \{{\varepsilon}\} \\ \{\sigma\} =& [C] \{{\varepsilon}\} \\ \end{split}\]
• The values of \(C\) are called Elastic constants
• \([C]\) and \([S]\) are fully populated for an anisotropic material

For isotropic material:

\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)} \left[ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & 1-2\nu & 0 & 0 \\ 0 & 0 & 0 & 0 & 1-2\nu & 0 \\ 0 & 0 & 0 & 0 & 0 & 1-2\nu \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]

Lamé constants

\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} 2 \mu + \lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2 \mu + \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda &2 \mu + \lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \mu \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]

Where: \[\begin{split} \mu =& \frac{E}{2 (1-\nu)} \\ \lambda =& \frac{\nu E}{(1+\nu)(1-2 \nu)} \\ \end{split}\]

• Finally, we can also use indicial notation to quickly write our constitutive relationship:

\[{\varepsilon_{ij}}= \frac{1}{E} \left[(1+\nu) {\sigma_{ij}}- \nu \delta_{ij} {\sigma_{kk}}\right]\]

\[{\sigma_{ij}}= 2 \mu {\varepsilon_{ij}}+ \lambda \delta_{ij} {\varepsilon_{kk}}\] \(\mu\) and \(\lambda\) are called the Lamé constants. They can be found in standard texts but will not be discussed further.

• This method of expressing the equations is powerful and worthy of study, however, we will not discuss it further

Other material descriptions

• There are materials that fit between total anisotropy (21 constants) and isotropic (2 constants).
• In the aerospace world, a critical one is “orthotropic” (9 constants) \[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E_{xx}} & -\frac{\nu_{yx}}{E_{yy}} & -\frac{\nu_{zx}}{E_{zz}} & 0 & 0 & 0 \\ -\frac{\nu_{xy}}{E_{xx}} & \frac{1}{E_{yy}} & -\frac{\nu_{zy}}{E_{zz}} & 0 & 0 & 0 \\ -\frac{\nu_{xz}}{E_{xx}} & -\frac{\nu_{yz}}{E_{yy}} & \frac{1}{E_{zz}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G_{yz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G_{zx}} & 0 \\ 0 & 0 & 0 & & 0 & \frac{1}{2 G_{xy}} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]

• Due to symmetry: \[\begin{split} \frac{\nu_{yx}}{E_{yy}} =& \frac{\nu_{xy}}{E_{xx}} \\ \frac{\nu_{zx}}{E_{zz}} =& \frac{\nu_{xz}}{E_{xx}} \\ \frac{\nu_{zy}}{E_{zz}} =& \frac{\nu_{yz}}{E_{yy}} \\ \end{split}\]

\[\left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ {} & C_{22} & C_{23} & 0 & 0 & 0 \\ {} & {}& C_{33} & 0 & 0 & 0 \\ {} & {}& {} & C_{44} & 0 & 0 \\ {} & {}& {} & {} & C_{55} & 0 \\ {} & {}& {} & {} & {} & C_{66} \\ \end{array} \right] \left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\}\]

• This is often the best description of a composite ply.
• It works for bone in some cases too

• Also:
  • Monoclinic (13 constants)
  • Transversely isotropic (5 constants-ex: unidirectional composites and bone)
  • Cubic (3 constants-ex: silicon)
• Finally, this entire description is referred to as “generalized Hooke’s law” (Robert Hooke, Late 17th century)

\[{\sigma_{ij}}= E_{ijkl} \, \varepsilon_{kl}\]

Cortical bone is well described as transversely isotropic

Transversely isotropic

5 constants

\[\left\{ \begin{array}{c} {\varepsilon_{xx}}\\ {\varepsilon_{yy}}\\ {\varepsilon_{zz}}\\ {\varepsilon_{yz}}\\ {\varepsilon_{xz}}\\ {\varepsilon_{xy}}\\ \end{array} \right\} = \left[ \begin{array}{cccccc} \frac{1}{E_T} & -\frac{\nu_T}{E_T} & -\frac{\nu_L}{E_L} & 0 & 0 & 0 \\ -\frac{\nu_T}{E_T} & \frac{1}{E_T} & -\frac{\nu_L}{E_L} & 0 & 0 & 0 \\ -\frac{\nu_L}{E_L} & -\frac{\nu_L}{E_L} & \frac{1}{E_L} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 G} & 0 \\ 0 & 0 & 0 & & 0 & \frac{1+\nu_T}{E_T} \\ \end{array} \right] \left\{ \begin{array}{c} {\sigma_{xx}}\\ {\sigma_{yy}}\\ {\sigma_{zz}}\\ {\sigma_{yz}}\\ {\sigma_{xz}}\\ {\sigma_{xy}}\\ \end{array} \right\}\]

• \(E_T\) and \(E_L\) are transverse (in plane) and longitudinal (out of plane) modulus
• \(\nu_T\) and \(\nu_L\) are transverse (in plane) and longitudinal (out of plane) Poisson’s ratios
• \(G\) is the out of plane shear modulus

• Note: stresses and strains must be transformed into the local coordinate system to apply the constitutive law
• Alternatively, the stiffness or compliance matrix must be transformed into the global coordinate system
• See Bartel’s book or me for additional info

Have we covered it all?

• Bone is anisotropic and inhomogeneous
• Bone is rate dependent
• Bone constitutive response depends on fatigue, age, damage, and plasticity
• In models of bone, it is often convenient to assume isotropic properties
  • Difficult to obtain a “better” description
• Transversely isotropic (and orthotropic) may be more appropriate

Mechanical properties of cortical bone

Asymmetric stiffness and strength

Asymmetric stiffness and strength

Stiffness and strength with age

Graph Showing Relationship Between Age and Bone Mass

Bone density peaks at about 30 years of age. Women lose bone mass more rapidly than men.

(slide credit: @OpenStaxAnatomy2020 Ch. 6)

Volume fraction

• Porosity is present in both cortical and trabecular bone
• Define volume fraction (\(V_f\)) as the volume of actual bone tissue to the bulk volume
• Cortical 70-95%
• Trabecular 5-60%
• Extremes are the young adult and elderly

Bone density

• Density is strongly dependent on porosity and volume fraction
• It is also a primary indicator of bone strength and stiffness
• Apparent density is mass per bulk volume
• Common measure of apparent density include:
  • Hydrated
  • De-hydrated
  • De-organified
• Tissue density is mass per volume of actual bone tissue (2.0g/cc)
  • Importantly, this volume excludes vascular pore spaces

Relationship between bone density and volume fraction

• The volume fraction, tissue density, and apparent density are related by \[\rho_{\mathrm{app}} = \rho_{\mathrm{tiss}} V_f\]
• Apparent densities
  • Cortical – \(\approx\) 1.85 g/cc
  • Trabecular – 0.10-0.50 g/cc
• Trabecular density decreases about 2% per decade after skeletal maturity
• Note also the cortical bone wall thickness decreases as you age

Heterogeneity and variability

Strength

Mineral content

• Mineral content is also important for mechanical properties
• It is measured after heating bone to 700C for 24 hours (de-organification and drying)
• Content increases during skeletal growth and remains fairly constant thereafter

Heterogeneity and variability

Stiffness

Density and strength

Average values

Regressions with age

• \(\sigma = a \rho + b\)
• \(\sigma = a \rho^b\)

Fatigue

Minor’s rule for fatigue

• Hypothesis is that the fractional fatigue lives sum together and predict failure
• \(\displaystyle\sum_{i=1}^n \frac{N_i}{N_{Fi}} = 1\)
  • Where there are \(n\) different load levels
• Works well in most brittle metals (implants!)
• Not validated for bone!!!
• In bone, common assumption to replace stress with strain
  • Account for various levels of porosity, etc (also reduces variability in the test data)
• What about bone remodeling?

Creep

• De-vitalized bone exhibits creep
  • Resistance better in compression then tension
  • Difficult to test in-vivo response
• Metals creep – may/may not be significant for ortho

Creep

Plasticity and micro-structural damage

• Note the change in slope after yield – microdamage!

Strain rate sensitivity

• Only a factor of two – mildly rate dependent
• Probably not critical for most physiologic loads

Mechanical properties of trabecular bone

Trabecular bone behavior and large variability

Stress-strain behavior for compressive and tensile loading of bovine (left) and human vertebral (right) trabecular bone, showing the wide range in strength that is typical for trabecular bone.

Trabecular bone apparent density

Dependence of compressive on-axis strength on apparent density for trabecular bone for two different sites bovine tibial (BPT) and human vertebral (HVB) trabecular bone. The difference in slopes in the linear relationship for each is due to the different architectures, being mainly plate-like in the bovine bone and rodlike in the human vertebral bone. When all the data are pooled, there is a strong squared power law relationship, with r 0.94. (Bone Mechanics Handbook. Editor SC Cowin. CRC Press Boca Raton, 2001)

Trabecular bone crush strength and age

Data from Mosekilde et al. (1987) Bone 8:79-85 and McCalden et al. (1997) J Bone Jt Surg 79A:421-427.

Trabecular bone yield asymmetry

From Morgan and Keaveny (2001) J Biomechanics 34: 569-57.

Trabecular bone yield anisotropy

Yield stress anisotropic – yield strain isotropic

Dependence of yield strain (left) and Young’s modulus (right) on specimen orientation in tension and compression, for dense bovine trabecular bone. For the off-axis orientation, the specimen axes were offset 30-40 degrees from the principal trabecular (on-axis) direction. Error bars show ±1 SD. In contrast to the yield strains, which were isotropic, but asymmetric, elastic modulus and yield stress (not shown) were clearly anisotropic. From Chang et al. (1999) J Orthop Res 124582-585

Fatigue of trabecular bone

From Haddock et al. (2004) J Biomechanics 37:181-187.

Post-yield damage of trabecular bone

2. Dependence of percent modulus reduction (between E and ERESIDUAL) and strength reduction on the plastic strain that develops with overloading.⁶

Failure prediction

• Von Mises stress criterion is not good for bone (particularly when shear stresses are high)

• Tsai-Wu is a much better metric of strength, however, requires significantly more experiments to establish the criterion \[\begin{split} F_1 \sigma_1 + F_2 \sigma_2 + F_3 \sigma_3 &\\ + F_{11} \sigma_1^2 + F_{22} \sigma_2^2 + F_{33} \sigma_3^2 &\\ + 2 F_{12} \sigma_1 \sigma_2 + 2 F_{13} \sigma_1 \sigma_3 + 2 F_{23} \sigma_2 \sigma_3 &\\ + F_{44} \sigma_4^2 + F_{55} \sigma_5^2 + F_{66} \sigma_6^2 &=1 \end{split}\] (Note vector representation of stress 1-6)

• Requires tension, compression, and torsion tests in longitudinal and transverse specimens

• This is a big challenge in practical prediction of bone failure

  • Most studies only compare stresses without predicting failure