Unit 2: Bone and cartilage

Bone is a living, mineralised composite that is simultaneously a structural material and a metabolically active organ. It remodels continuously in response to mechanical load, stores calcium and phosphate, and houses marrow. Cartilage is its avascular partner at the joints, providing a low-friction, load-distributing surface. Analysing either mechanically requires treating them as time-dependent, direction-dependent composites rather than simple elastic solids.

• Two length scales: macroscopic (cortical vs. trabecular architecture) and microscopic (osteon, lamella, mineralised collagen fibril).
• Composite nature: stiff mineral phase (hydroxyapatite) reinforcing a compliant organic matrix (type I collagen) — a two-phase system whose behaviour lies between the two constituents.
• Time dependence: stress and strain responses depend on loading rate; both bone and cartilage are viscoelastic.
• Direction dependence: properties vary with the axis of loading — bone is anisotropic.
• Living response: mechanical stimulus drives adaptation (Wolff's law), unlike inert engineering materials.

II. Bone Structure and Composition

The material basis of bone's mechanical function.

Bone's properties follow directly from what it is made of and how those constituents are arranged.

A. Bone structure & composition

Bone combines a mineral phase for stiffness with an organic phase for toughness, arranged hierarchically.

• Mineral phase (~60–70% by weight): hydroxyapatite, Ca₁₀(PO₄)₆(OH)₂, provides compressive stiffness and hardness; contributes most of the elastic modulus.
• Organic phase (~20–25%): ~90% type I collagen plus non-collagenous proteins (osteocalcin, osteonectin); provides tensile strength and ductility.
• Water (~10%): bound to collagen and mineral; strongly influences viscoelastic response.
• Cortical (compact) bone: dense outer shell, porosity ~5–10%; organised into osteons (Haversian systems) — concentric lamellae around a central canal.
• Trabecular (cancellous) bone: porous interior, porosity ~50–90%; a lattice of struts (trabeculae) aligned along principal stress lines.
• Cell types: osteoblasts (form matrix), osteocytes (embedded mechanosensors), osteoclasts (resorb bone).

B. Blood circulation in bone

Bone is well vascularised despite appearing inert, and its blood supply governs remodelling and healing.

• Nutrient artery: enters the diaphysis through the nutrient foramen; supplies the inner two-thirds of cortex and marrow.
• Periosteal vessels: supply the outer one-third of the cortex.
• Metaphyseal and epiphyseal arteries: feed the bone ends.
• Haversian and Volkmann's canals: the microvascular network — Haversian canals run longitudinally within osteons; Volkmann's canals connect them transversely and to the periosteum.
• Flow direction: predominantly centrifugal (medulla → cortex → periosteum), which matters in fracture healing and infection spread.

III. Mechanical Properties of Bone

How bone carries and resists load.

Bone's stiffness, strength and directional behaviour arise from its composite, oriented microstructure.

A. Mechanical properties of bone

Bone is stiff and strong in compression, weaker in tension and weakest in shear.

• Elastic modulus (E): cortical bone ≈ 15–20 GPa longitudinally; trabecular bone ≈ 0.1–2 GPa depending on density.
• Strength asymmetry: cortical bone ~ 190 MPa in compression, ~130 MPa in tension, ~70 MPa in shear.
• Stress–strain curve: near-linear elastic region → yield → short plastic region → fracture; cortical bone strains only ~1–3% before failure (brittle-tending).
• Density dependence: trabecular modulus scales roughly with the square of apparent density, E ∝ ρ².

σ = E · ε
σ = stress (Pa)   ε = strain (dimensionless)   E = elastic modulus (Pa)

B. Anisotropy and composite models for bone

Bone's stiffness depends on loading direction, and composite theory bounds its effective modulus.

• Anisotropy: stiffer along the long (osteon) axis than transversely; often modelled as transversely isotropic — one axis of symmetry, requiring 5 independent elastic constants.
• Composite analogy: mineral = stiff fibres/particles, collagen = compliant matrix; effective modulus lies between two bounds.
  1. Voigt (parallel) model: phases share equal strain; gives the upper bound.
  2. Reuss (series) model: phases share equal stress; gives the lower bound.

Voigt:  E_c = Vf·Ef + Vm·Em         (isostrain, upper bound)
Reuss:  1/E_c = Vf/Ef + Vm/Em       (isostress, lower bound)
Vf, Vm = volume fractions of fibre (mineral) and matrix (collagen)
Ef, Em = their moduli;  Vf + Vm = 1

• Interpretation: measured longitudinal modulus sits near the Voigt bound; transverse modulus near the Reuss bound — explaining the anisotropy directly.

C. Bone response to stress

Bone remodels to match its mechanical environment, a living-material property with no engineering analogue.

• Wolff's law: bone lays down material where stress is high and resorbs it where stress is low; trabeculae align with principal stress trajectories.
• Mechanotransduction: osteocytes sense strain (and fluid flow in canaliculi) and signal osteoblasts/osteoclasts.
• Adaptation examples: cortical thickening in a tennis player's dominant arm; disuse osteopenia in bed-rest or microgravity.
• Strain thresholds: habitual strains ~1000–1500 µε maintain balance; very low strains promote resorption, high strains promote formation.

IV. Viscoelastic Behaviour of Bone and Cartilage

Time- and rate-dependent response of both tissues.

Because both tissues contain fluid and long-chain molecules, their response combines elastic (spring) and viscous (dashpot) elements.

A. Viscoelastic properties of bone

Bone's stiffness and strength increase with loading rate, and it exhibits creep, relaxation and hysteresis.

• Rate dependence: higher strain rate → higher modulus and higher failure stress (bone is stronger in a fast impact than slow load).
• Creep: slow increase in strain under constant stress.
• Stress relaxation: decay of stress under constant strain.
• Hysteresis: loading and unloading curves differ; enclosed area = energy dissipated per cycle.

B. Maxwell & Voight models

Two spring–dashpot arrangements bracket idealised viscoelastic behaviour.

1. Maxwell model (spring + dashpot in series): captures stress relaxation but predicts unbounded creep, so it models a viscoelastic fluid.
2. Voigt (Kelvin–Voigt) model (spring + dashpot in parallel): captures creep toward a limit but cannot represent instantaneous elastic strain or stress relaxation, so it models a viscoelastic solid.

Maxwell (series):   dε/dt = (1/E)·dσ/dt + σ/η
   relaxation:      σ(t) = σ0·exp(−t/τ),   τ = η/E

Voigt (parallel):   σ = E·ε + η·(dε/dt)
   creep:           ε(t) = (σ0/E)·[1 − exp(−t/τ)]

E = spring modulus (Pa)   η = dashpot viscosity (Pa·s)
τ = relaxation/retardation time (s)   σ0 = applied constant

• Combined model: a standard linear solid (three elements) is needed to reproduce real bone — instantaneous response and bounded creep and relaxation.

C. Viscoelastic properties of articular cartilage

Cartilage is a biphasic, fluid-filled tissue whose time dependence comes chiefly from fluid flow, not just matrix.

• Composition: ~70–80% water, type II collagen, and negatively charged proteoglycans (aggrecan) that draw in water osmotically.
• Biphasic behaviour: load is initially borne by pressurised interstitial fluid; over time fluid exudes through the low-permeability matrix and load transfers to the solid collagen–proteoglycan network.
• Creep under constant load: rapid initial deformation, then slow consolidation as fluid flows out until equilibrium.
• Stress relaxation under constant deformation: high initial peak stress from fluid pressure, decaying as fluid redistributes.
• Function: this fluid pressurisation gives very low friction and spreads joint contact stress; permeability decreasing with compression is self-protective.

V. Osteoporosis

Loss of bone mass and microarchitecture leading to fragility.

Osteoporosis is a skeletal disorder of reduced bone strength — low bone mineral density plus degraded microarchitecture — raising fracture risk, especially at hip, spine and wrist.

A. Causes

An imbalance in remodelling, with resorption exceeding formation, drives bone loss.

• Post-menopausal (Type I): oestrogen deficiency increases osteoclast activity; rapid trabecular loss.
• Age-related (Type II): reduced osteoblast function and calcium/vitamin D deficiency; cortical and trabecular loss.
• Secondary causes: corticosteroids, hyperthyroidism, immobilisation, smoking, alcohol, low body weight.
• Mechanism: thinning and perforation of trabeculae reduce E ∝ ρ² sharply, so modest density loss causes disproportionate strength loss.

B. Diagnosis

Diagnosis rests on quantifying bone mineral density (BMD) against a young-adult reference.

• DEXA (dual-energy X-ray absorptiometry): gold standard; measures BMD at hip and spine.
• T-score: standard deviations from young-adult mean — ≤ −2.5 = osteoporosis, −1 to −2.5 = osteopenia, ≥ −1 normal.
• Z-score: compares with age-matched peers; used in younger patients.
• Adjuncts: quantitative CT, ultrasound of calcaneus, and fracture-risk tools (e.g., FRAX) combining BMD with clinical risk factors.

C. Treatment

Treatment aims to restore the resorption–formation balance and reduce fracture risk.

• Antiresorptives: bisphosphonates (alendronate) inhibit osteoclasts; denosumab (anti-RANKL); oestrogen/SERMs (raloxifene).
• Anabolics: teriparatide (PTH analogue) stimulates osteoblastic bone formation.
• Nutrition: calcium (~1000–1200 mg/day) and vitamin D supplementation.
• Lifestyle: weight-bearing and resistance exercise exploiting Wolff's law; smoking and alcohol cessation; fall prevention.