Bone Biomechanics

Structure and Function of Bone (Bone Biomechanics)

1. Structure

1.1 Microstructure

Bone is an extremely well-organized tissue characterized by a remarkable synergy between its molecular, cellular, and tissue arrangements. This structure provides a tensile strength that is nearly equivalent to that of cast iron, while achieving such strength with a surprisingly low weight considering its role as a supporting structure.

Microscopically, bone can be classified into two forms: woven bone and lamellar bone.

  • Woven Bone:

    • Features:

    • Course-fibered

    • Lacks a uniform orientation of collagen

    • Exhibits isotropic properties, meaning it has the same properties in all directions.

  • Lamellar Bone:

    • Development:

    • Replaces woven bone starting at one year of age

    • Characteristics:

    • Highly organized collagen structure

    • Anisotropic properties, meaning it has different properties in different directions.

Both woven and lamellar bones are structurally organized into two main types:

  • Trabecular Bone (also known as cancellous or spongy bone):

    • Location: Found principally in the metaphysis and epiphysis of long bones, and also in cuboid bones such as the vertebrae.

    • Structure: Exhibits a 3-D branching lattice.

  • Cortical Bone (also known as dense or compact bone):

    • Location: Envelopes cuboid bone and forms the diaphysis of long bones.

    • Types:

    • Plexiform Bone: Composed of layers of lamellar and woven bone.

    • Haversian Bone: Contains vascular channels surrounded by lamellar bone, forming cylindrical units known as osteons.

The various microscopic and structural forms of bone are illustrated in the following categorical scheme:

  • MICROSCOPIC

    • LAMELLAR

    • CANCELLOUS

    • COMPACT

  • STRUCTURAL

    • WOVEN

    • PLEXIFORM

    • HAVERSIAN

1.2 Bone Cells

Bone cells are the principal regulators of bone metabolism and are divided into three types:

  • Osteoblasts:

    • Function: Bone-forming cells

    • Location: Line the surface of bone, playing a crucial role in bone growth and mineralization.

  • Osteocytes:

    • Function: Mature bone-forming cells encased within the mineralized matrix of bone.

    • Role: Maintain bone tissue and participate in the repair process and mineral homeostasis.

  • Osteoclasts:

    • Function: Large, multinucleated cells that are responsible for bone resorption.

    • Location: Primarily found in regions where bone resorption is occurring.

1.3 Bone Matrix Composition

Bone is a composite material made up of various components:

  • Inorganic Phase:

    • Composition:

    • 60-70% of bone is made up of minerals, primarily calcium phosphate minerals, known as hydroxyapatite (these are plate-like crystals that confer compressive strength).

  • Organic Phase:

    • Composition:

    • 5-8% of bone is water.

    • 90% of the organic matrix is collagen (Type I), along with non-collagenous matrix proteins, which are instrumental for tensile properties.

Bone's matrix and mineral components play crucial roles in controlling cell-mediated processes. Both inorganic and organic components contribute to both structural and regulatory properties of bone. It is noteworthy that the composition of bone can vary based on factors such as the site within the body, the age of the individual, dietary history, and disease states.

1.4 Summary of Bone Composition

  • Inorganic Phase:

    • Calcium phosphate minerals (Hydroxyapatite) - Plate-like crystals providing compressive strength.

  • Organic Phase:

    • Type I Collagen - 90% of the organic phase, serves tensile strength.

The processes of bone mineralization and remodeling are continuous and remain active throughout life, although such processes are beyond the scope of this overview.

2. Biomechanics of Cortical Bone

2.1 Mechanical (Material) Testing

To evaluate the mechanical properties of bone, small uniform specimens are subjected to loads using a materials testing machine. The mechanical properties evaluated include:

  • Compressive strength

  • Tensile strength

  • Elastic modulus, denoted as $E = \frac{\sigma}{\varepsilon}$

    • Where:

    • $
      u$ is stress ($P/A$) and $ au$ is strain (deformation per unit length).

The relationships for ultimate strength ($SU$) and yield strength ($SY$) can also be derived from the testing findings.

2.2 Factors Affecting the Mechanical Properties of Cortical Bone
2.2.1 Strain (Loading) Rate

Bone has viscoelastic properties, meaning its mechanical behavior can depend on the rate of loading. The elastic modulus and strength increase when bone is loaded very quickly. Typical strain rates include:

  • $0.001$ - slow walking

  • $0.1$ - strenuous exercise

The characteristics observed on the stress-strain curves showcase a ductile to brittle transition at higher strain rates.

2.2.2 Anisotropy

The stress-strain response of cortical bone is significantly influenced by the orientation of its microstructure. Cortical bone typically displays greater strength and stiffness along the longitudinal direction compared to the transverse direction.

Loading transversely tends to lead to more brittle failure of specimens.

  • Modulus in longitudinal direction $EZ = 1.5E\theta$

  • Compressive Strength: $SZ = 1.45S\theta$

  • Tensile Strength: $SZ = 2.6S\theta$

2.2.3 Age

Aging leads to a decline in both modulus and strength properties of cortical bone at an average rate of approximately 2% per decade from ages 20 to 90 for both genders. This decline results in reduced capacity for energy absorption, primarily due to a decreased area under the load deformation curve, which is particularly significant from a fracture risk perspective.

3. Biomechanics of Trabecular Bone

3.1 Mechanical Testing

The mechanical testing of trabecular bone can follow methods similar to those used in cortical bone testing, specifically using cylindrical specimens.

An alternative method involves testing in the intact state using indentation tests, which negate the need for isolating a cylinder, making it effective especially for weaker specimens.

3.2 Factors Affecting the Mechanical Properties of Trabecular Bone
3.2.1 Apparent Density

Trabecular bone is distinguished from cortical bone by its porosity.

The apparent density is defined as:
ρ=Mass of Bone TissueBulk Volume Including Marrow Spaces\rho = \frac{\text{Mass of Bone Tissue}}{\text{Bulk Volume Including Marrow Spaces}}

Higher porosity leads to significantly different compressive stress-strain behavior in trabecular versus cortical bone. Typical densities are:

  • Cortical: $
    ho = 1.85$ g/cc

  • Trabecular: $
    ho = 0.9$ g/cc and $
    ho = 0.3$ g/cc

The relationship between apparent density and mechanical properties can be depicted as a power function. For compressive strength and modulus, relationships are as follows:

  • SC=AρBS_C = A\rho^B

  • EC=2915ρ3E_C = 2915\rho^3

These relationships underscore that even minor changes in apparent density can have significant impacts on mechanical properties, exemplified by vertebral bone density loss leading to substantial decreases in both compressive strength and modulus.

3.2.2 Strain Rate

Similar to cortical bone, trabecular bone exhibits strain rate dependency, showcasing viscoelasticity. An increase of 100 times in strain rate, from $0.001$ (slow walking) to $0.1$ (strenuous exercise), results in approximately a 30% increase in both strength and modulus.

The trabecular structure, which consists of trabecular (solid) material interspersed within a marrow (fluid) matrix, further complicates its viscoelastic effects.

3.2.3 Anisotropy and Inhomogeneity

Trabecular bone is anisotropic, meaning it behaves differently under load depending on the direction. For instance, in the proximal tibia, the strength and stiffness are generally higher in the direction aligned with anatomical loading.

Mechanical properties also display inhomogeneity across different positions; for example, the proximal tibia exhibits varying properties from medial to lateral. The implications for implant fixation are significant, as the load-transfer mechanism deviates from the bone's natural loading state.

4. Structural Properties of Bone

The bones of the appendicular skeleton are long, slender, and slightly curved, primarily subjected to compressive contact forces at joint surfaces and tensile muscle forces around articulating surfaces. The diaphysis is under axial, bending, and torsional loads.

4.1 Axial Loading

When subjected to a pure axial load aligned with the centroidal axis, the resulting compressive stress can be represented as:
σ=PA\sigma = \frac{P}{A}
Where:

  • $ ext{Stress Element}$ is $ au$

4.2 Bending

In bending scenarios, the normal (bending) stress arises as:
σ<em>b=M⋅CI\sigma<em>b = \frac{M \cdot C}{I} Where $I = \frac{\pi}{64}(D0^4 - D_I^4)$, and $M$ represents the moment and $C$ refers to the distance from the neutral axis to the outer surface.

4.3 Combined Bending and Axial Loading

In vivo, bones are subject to combined loading conditions, leading to superimposed axial and bending stresses:
σ=σ<em>a+σ</em>b\sigma = \sigma<em>a + \sigma</em>b

4.4 Torsional Loading

When exposed to torsion, shear stresses develop, which increase linearly from zero at the central axis to a maximum at the outer surface and are given by:
τ=T⋅rJ\tau = \frac{T \cdot r}{J}
Where $J = 2I$.

4.5 Combined Axial, Bending, and Torsional Loads

The collective interaction of axial, bending, and torsional loads can be calculated to yield resultant stress conditions as follows:
σ=σ<em>a+σ</em>b+τ\sigma = \sigma<em>a + \sigma</em>b + \tau

Principal stresses can be calculated when torsion is absent, or for scenarios requiring comprehensive analysis, Mohr's Circle or the following formula can be used:
σ<em>1,σ</em>2=σ+τ24+τ2\sigma<em>1, \sigma</em>2 = \sigma + \frac{\tau^2}{4} + \tau^2

Understanding the stress distribution in bone allows for a deeper insight into fracture patterns under various loading scenarios; common fracture types have been observed based on specific loading behaviors.