Musculoskeletal System: Mechanical Properties of Bone Tissue

Mechanical Properties of Bone Tissue

Functions of Bone Tissue

• Support: Bones physically hold the body up, providing a rigid framework for soft tissue and organ attachment.

• Protection of Organs: Vital organs are shielded by bones (e.g., skull protects the brain, ribs protect the heart and lungs).

• Leverage: Bones act as levers, altering the magnitude and direction of forces generated by skeletal muscles (as discussed in MNB 7).

• Storage and Release of Minerals and Lipids: Bones primarily store and release calcium and phosphorus, and also store lipids.

• Blood Cell Production: Red and white blood cells are produced in the red bone marrow.

Bone Mechanical Properties

• The mechanical properties of bone are crucial for the skeleton's ability to support movement and protect vital organs.

• Bone possesses a hierarchical structure, meaning its mechanical properties vary significantly across different length scales.

  • Nanoscale: Properties of collagen and mineral are distinct.

  • Macroscale: Properties of compact or trabecular bone (e.g., strength, stiffness) are measured at this level and differ from nanoscale properties.

Bone Matrix Composition

Inorganic Material ($\sim$60% of bone weight)

• Primarily calcium phosphate, existing as insoluble hydroxyapatite crystals.

• Also includes calcium carbonate.

• Contains trace elements like fluoride, sulphate, potassium, and magnesium.

Organic Material ($\sim$35% of bone weight)

• Mainly Type I collagen.

• Glycoprotein: Helps bind fibers, inorganic material, and cells together.

• Proteoglycans: Crucial for providing hydration and swelling pressure, enabling bone to withstand compression.

• Glycosaminoglycans (GAGs): Play a role in regulating collagen formation and mineralization.

Water ($\sim$5% of bone weight)

Cells

• Osteoblasts: Bone-forming cells.

• Osteoclasts: Bone-resorbing cells.

• Osteocytes: Mature bone cells, embedded in the matrix.

• Bone lining cells: Found on bone surfaces where remodeling is not occurring.

Type I Collagen

• The most abundant structural protein in the human body.

• Found in skin, ligaments, tendons, and is the primary organic substance in bone.

• Imparts flexibility to bone.

• Contributes to bone's ability to resist tensile forces (pulling and stretching).

• Bone tissue with mineral removed (only collagen present) highlights its flexible nature.

• The alternating orientation of collagen fibers within concentric lamellae creates a fiber-reinforced composite, significantly enhancing bone strength.

Hydroxyapatite

• Hydroxyapatite crystals provide bone with much of its stiffness.

• Responsible for bone's resistance to compressive forces (pressing and squeezing).

• Bone tissue with collagen removed (only mineral present) demonstrates its rigid, brittle nature.

• Although larger-scale bone structures vary, the organization of mineralized fibrils is highly conserved across different bone types, serving as the bone's fundamental building block.

• During bone formation, collagen molecules first assemble into fibrils, which are then mineralized by the formation of hydroxyapatite crystals.

Elastic Properties of Biological Materials

• All materials undergo deformation (change of shape) when a force is applied.

• Biological materials like bone and tendons can be modeled as springs, assuming they are not stretched too far.

• A material behaves elastically if it returns to its original shape after the force is removed.

• Materials behaving elastically follow Hooke's Law, where the force is directly proportional to the extension ($F hinspace ext{µ} hinspace ext{ extDelta}L$).

Stress ($\sigma$)

• Stress is the force per unit area experienced by a material.

• It accounts for both the magnitude of the force and the area over which it is applied.

• Formula: $\sigma = rac{F}{A}$

  • $F$: applied force (Newtons, N)

  • $A$: cross-sectional area (square meters, m$^2$)

• Units: Newtons per square meter (N/m$^2$) or Pascals (Pa).

• Materials can be stressed in various ways (e.g., tension, compression, shear).

Strain ($\epsilon$)

• Strain is the fractional change in length of a material when subjected to compressional or tensional stress.

• Formula: $\epsilon = rac{ ext{ extDelta}L}{L}$

  • $ext{ extDelta}L$: change in length (meters, m)

  • $L$: original length (meters, m)

• Strain has no units because it is a ratio of two lengths.

Young's Modulus ($E$)

• Definition: The ratio of stress to strain.

• Formula: $E = rac{ ext{ extsigma}}{ ext{ extepsilon}}$

• Units: Newton-meters (Nm$^{-2}$) or Pascals (Pa).

• Linear Relationship (Hooke's Law): Up to a certain point (point A on a stress-strain graph), stress and strain are linearly related, and the material obeys Hooke's Law.

• Graphical Representation: Young's Modulus is the slope of the stress-strain graph in this linear region.

• Measure of Stiffness: A high Young's Modulus indicates a stiff material (e.g., Material 1 on a graph, which shows less strain for the same stress compared to Material 2).

  • A material with a lower Young's Modulus is more elastic (e.g., Material 2, exhibiting greater strain for the same stress).

Stress/Strain Relationship (Material Behavior)

• Elastic Region (up to Point B, the Elastic Limit):

  • Deformation is entirely elastic.

  • The material will return to its original shape when the applied force is removed.

  • Between point A and point B, stress and strain are no longer linearly proportional, meaning Young's Modulus cannot be precisely measured here.

• Plastic Deformation (Range from Point B to Point D):

  • Beyond Point B (also known as the Yield Point), the material experiences permanent deformation.

  • It will not return to its original shape once the force is removed.

• Maximum Permissible Stress (Point C):

  • This point on the curve represents the maximum stress the material can withstand before its internal structure begins to fail significantly.

  • Any stress beyond Point C indicates that the material is destined to fracture.

• Fracture Point (Point D, Ultimate Yield):

  • The point at which the material breaks or ruptures.

Compact vs. Trabecular Bone

• Compact (Cortical) Bone:

  • Description: Solid bone tissue.

  • Location: Forms the solid outer shell and the shaft (diaphysis) of long bones.

  • Strength: Can withstand significantly greater loads than trabecular bone.

• Trabecular (Cancellous or Spongy) Bone:

  • Description: A network of interconnecting bony struts called trabeculae, forming a porous structure.

  • Location: Commonly found at the ends (epiphyses) of long bones.

  • Support: Provides lightweight strength and acts as a shock absorber.

Compression vs. Tension in Bone

• Bone is generally stronger under compression than under tension.

• Compressive Strength: Values for compact bone range from 70-280 MPa (Mega Pascals).

• Tensile Strength: When subjected to tension, bone undergoes less strain before reaching its yield point.

• There is considerable variation in both tensile and compressive strength values among different bones due to the varied forces they must support throughout the body.

Anisotropy of Bone

• Definition: Bone is an anisotropic material, meaning its mechanical properties (e.g., ultimate strength, stiffness) vary depending on the direction in which a load is applied or tested.

• Relevance: This is critical for understanding how bone responds to forces in different anatomical orientations (e.g., longitudinal, transverse, or tilted relative to the bone's long axis).

  • For example, the femur's compact bone orientation will exhibit different properties depending on whether the test load is applied longitudinally along the shaft or transversely across it.

Bending Bones

• Most long bones (e.g., the femur) have a natural curvature and are hollow.

• Comparison: A straight, solid column of bone is theoretically stronger under pure compression than a curved, hollow bone.

• Evolutionary Advantages of Curved, Hollow Bones:

  1. Predictable Deformation: When loaded longitudinally, the bone deforms in a more predictable direction.

  2. Increased Bending with Load: A curved bone will bend more significantly as the load increases, allowing for adaptable response.

  3. Resistance to Bending: A curved and hollow structure offers greater resistance to bending stresses than a solid structure of the same mass, especially in non-axial loading conditions.

  • This design optimizes strength-to-weight ratio, allowing for efficient movement and load bearing without excessive mass.

Resisting Bending Forces

• During bending, one surface of the bone experiences compression (shortening), while the opposite surface experiences tension (lengthening).

• Internal Resistance: The bonds (e.g., hydrogen bonds, covalent crosslinks) holding bone constituents together are strained, resisting the change in material length.

• Stress Distribution: When a beam (like bone) bends under load, the fibers on the top surface become shorter (compression), and those on the bottom surface become longer (tension). The most extreme top and bottom fibers experience the greatest amounts of compression and tension, respectively.

• Therefore, the greatest resisting forces occur at these outer surfaces of the material.

Neutral Surface (or Neutral Axis)

• Definition: As you move from the outer surfaces towards the center of a bending bone, the change in length and the resisting forces decrease. At the very center, there is a surface that undergoes no change in length and experiences no resisting forces.

• Significance: Optimal resistance to bending forces is achieved when the material is distributed as far as possible from this neutral surface (i.e., a hollow structure).

• Trade-off: While hollowness helps resist bending, if the cylinder becomes too thin, it increases the risk of buckling under compressive forces.

Biomechanical Design of the Femur

• The femur's design exemplifies optimized resistance to compression and bending, providing both strength and flexibility in a lightweight, hollow tube.

• Strength and Flexibility: This duality is conferred by the combined presence of hydroxyapatite (HA) for stiffness and collagen for flexibility.

• Fiber-Reinforced Composite: The arrangement of collagen fibers creates a composite structure that provides additional strength to compact bone.

• Trabecular Bone Design: The specific orientation of trabecular rods and plates in the epiphysis offers lightweight strength and acts as a crucial shock absorber.

• Hollow Cylinder of Compact Bone: The diaphysis (shaft) is designed as a hollow cylinder of compact bone, providing maximum structural support efficiently for its weight.

Problem Calculation: Tibia Shortening

• Problem Statement: Calculate the shortening of a human tibia (length $L = 50 ext{ cm}$) with an average cross-sectional area ($A = 3 ext{ cm}^2$) when supporting an entire body mass ($m = 80 ext{ kg}$) in compression.

• Given: Young's Modulus of bone ($E = 1.8 imes 10^{11} ext{ Pa}$).

• Formulas:

  • Stress: $\sigma = rac{F}{A}$

  • Strain: $\epsilon = rac{ ext{ extDelta}L}{L}$

  • Young's Modulus: $E = rac{ ext{ extsigma}}{ ext{ extepsilon}}$

• Steps:

  1. Calculate Force ($F$): $F = mg = 80 ext{ kg} imes 9.8 ext{ m/s}^2 = 784 ext{ N}$

  2. Convert Area ($A$) to m$^2$: $A = 3 ext{ cm}^2 = 3 imes (10^{-2} ext{ m})^2 = 3 imes 10^{-4} ext{ m}^2$

  3. Convert Length ($L$) to m: $L = 50 ext{ cm} = 0.5 ext{ m}$

  4. Insert values into Young's Modulus formula rearranged for $ext{ extDelta}L$:

     • From $E = rac{ ext{ extsigma}}{ ext{ extepsilon}}$ we get $E = rac{F/A}{ ext{ extDelta}L/L} = rac{FL}{A ext{ extDelta}L}$

     • Rearranging for $ext{ extDelta}L$: $ext{ extDelta}L = rac{FL}{AE}$

     • Substitutiing values: $\text{ extDelta}L = rac{784 ext{ N} imes 0.5 ext{ m}}{(3 imes 10^{-4} ext{ m}^2) imes (1.8 imes 10^{11} ext{ Pa})}$

  5. Calculate $ext{ extDelta}L$:

     • $ext{ extDelta}L = rac{392}{5.4 imes 10^7} = 7.259 imes 10^{-6} ext{ m}$

     • Rounded Result: $\text{ extDelta}L hickapprox 7.26 imes 10^{-6} ext{ m}$

Learning Outcomes

Upon completion of this material, learners should be able to:

• Describe the comprehensive role of bone in supporting, protecting, and providing leverage within the body.

• Discuss how the mineral (hydroxyapatite) and collagen phases contribute to the mechanical properties of bone.

• Explain the concepts of stress and strain and identify the parameters used to characterize bone's biomechanical properties.

• Demonstrate how the elastic properties of bone differ under compression and tension.

• Accurately calculate and interpret the Young's Modulus of a material.

• Explain the anisotropic nature of compact bone and its implications.

• Describe the neutral surface and its significance in relation to bone's resistance to bending.

• Discuss the influence of both bone material composition and structural design on the optimal function of the femur in daily mechanical challenges.