Eccentric Loading & Asymmetric Callus Differentiation (2D Cross-Section)

Fixation Method

Mechanical Conditions

Axial load 400 N

Load eccentricity 0.0 mm

Fracture gap size 2.0 mm

Simulation speed 2.0 days/s

Live Readouts

Days post-fracture

0.0

Mean bridging

Peak strain (tension side)

100.0%

Min strain (compression side)

100.0%

Compression side

Granulation / hematoma

Tension side

Granulation / hematoma

How it works (2D cross-section model)

This is an end-on view of the fracture plane, discretized into 24 independent tissue sectors arranged in a ring around the bone's cross-section. Unlike a single lumped stiffness, each sector grows its own local callus and experiences its own local strain — because a real off-axis (eccentric) load applies a bending moment as well as an axial force, which a single scalar can't represent.

Assuming plane sections remain plane (standard beam theory), the gap-opening displacement at angle θ is:

δ(θ) = δ₀ + φ·R·cos(θ − θₑ)

Force balance:   Σ kᵢ·δ(θᵢ)            = F
Moment balance:  Σ kᵢ·δ(θᵢ)·R·cos(θᵢ)  = F·e

This 2×2 linear system for (δ₀, φ) is solved every frame — it is the same statically-indeterminate composite-section analysis used for reinforced concrete or bolted-joint design, applied here to a healing fracture. Each sector's own local strain εᵢ = δ(θᵢ)/gap then drives its own Perren tissue-differentiation state and its own callus growth rate, using the identical Gaussian mechanobiological window as the 3D simulator:

growthRate(ε) ∝ exp(−(ε − 15%)² / (2·13²))

Tissue-tolerance thresholds (Perren; Claes & Heitemeyer): ε ≤ 2% bone, 2–10% cartilage, 10–30% fibrocartilage, 30–100% fibrous tissue, >100% disrupted/nonunion risk.

Load eccentricity is the new control this 2D model adds: at e = 0 every sector sees identical strain and the ring behaves exactly like the 3D simulator's single scalar. As e grows, the near (compression) side sees reduced gap strain — often crossing into the bone-forming window early — while the far (tension) side sees amplified strain and can stall as fibrous tissue or nonunion, even while the near side has already bridged. This asymmetric healing pattern is a real, clinically documented consequence of bending loads (e.g. muscle pull, off-center weight-bearing) on a healing fracture.

end-on cross-section
at the fracture plane

Bone (ε ≤ 2%) Cartilage (2–10%) Fibrocartilage (10–30%) Fibrous tissue (30–100%) Disrupted (>100%)