Carter and Hayes tested 100 human tibial plateau and 24 bovine femoral condyle trabecular bone cylinders in compression at strain rates spanning 0.001 to 10.0/s, with and without marrow. The study asks whether unified power-law equations can describe compressive strength and stiffness across all bone — from porous cancellous to dense cortical — and what role marrow plays at different loading rates.
When you interpret a bone mineral density scan showing a 30% reduction in density, this paper tells you the mechanical consequence: compressive strength is not 70% of normal — it is approximately 49% of normal (0.7² = 0.49). That nonlinear relationship is why osteoporosis fracture risk rises so steeply with what looks like modest density loss on imaging.
The density exponents are the practical rule: strength scales with ρ², stiffness with ρ³. A patient who loses one-third of bone mass loses half their compressive strength and more than two-thirds of their stiffness. No amount of strain-rate modification (activity restriction) compensates for that, because strain rate contributes only a twofold change across the entire physiological range.
For implant surgery, the cubic modulus relationship explains why cancellous bone stock quality is so critical for cementless fixation. Regions with low apparent density are dramatically less stiff than they appear on X-ray, and micro-motion thresholds for osseointegration are easily exceeded.
This paper is the quantitative foundation behind modern fracture risk algorithms and the reason DXA-derived T-scores translate into clinically meaningful fracture probability. Carter and Hayes gave us the equations that connect a number on a scan to a force the bone can no longer tolerate.
Carter and Hayes tested 100 human tibial plateau and 24 bovine femoral condyle trabecular bone cylinders in compression at strain rates spanning 0.001 to 10.0/s, with and without marrow. The study asks whether unified power-law equations can describe compressive strength and stiffness across all bone — from porous cancellous to dense cortical — and what role marrow plays at different loading rates.
When you interpret a bone mineral density scan showing a 30% reduction in density, this paper tells you the mechanical consequence: compressive strength is not 70% of normal — it is approximately 49% of normal (0.7² = 0.49). That nonlinear relationship is why osteoporosis fracture risk rises so steeply with what looks like modest density loss on imaging.
The density exponents are the practical rule: strength scales with ρ², stiffness with ρ³. A patient who loses one-third of bone mass loses half their compressive strength and more than two-thirds of their stiffness. No amount of strain-rate modification (activity restriction) compensates for that, because strain rate contributes only a twofold change across the entire physiological range.
For implant surgery, the cubic modulus relationship explains why cancellous bone stock quality is so critical for cementless fixation. Regions with low apparent density are dramatically less stiff than they appear on X-ray, and micro-motion thresholds for osseointegration are easily exceeded.
This paper is the quantitative foundation behind modern fracture risk algorithms and the reason DXA-derived T-scores translate into clinically meaningful fracture probability. Carter and Hayes gave us the equations that connect a number on a scan to a force the bone can no longer tolerate.