Cortical Bone Vibrations Induced by Electromagnetic Field Pulse

Abstract

The purpose of our study was to test the hypothesis that the electromagnetic pulse (EMP) is capable of inducing mechanical vibrations in bone <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex vivo</i> . A thin segment of human femur diaphysis (from a tissue repository) suspended on a tensioned line (range <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T =$ </tex-math></inline-formula> 2.2–123 N) was exposed to EMP (mean <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B =0.64$ </tex-math></inline-formula> T, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dB</i> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dt</i> = 5877 T/s, and the mean <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> -field gradient of 127 T/m) from a solenoid with axis orthogonal to tensioning line, forming a harmonic oscillator whose mechanical vibrations were measured using laser Doppler vibrometry (LDV, noise floor 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> /s). Calculated mean Maxwell stress and Lorentz forces acting on a weakly conducting, diamagnetic bone slice point away from the solenoid for maximum sensitivity of LDV measurement. The electromechanical origin of the LDV signal was confirmed by the order-of-magnitude agreement between calculated (range from 12 to 50 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> /s) and measured initial bone velocity amplitudes (e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$35.5~\mu \text{m}$ </tex-math></inline-formula> /s ± <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$7.5~\mu \text{m}$ </tex-math></inline-formula> /s at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T =22.2$ </tex-math></inline-formula> N and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$17.7~\mu \text{m}$ </tex-math></inline-formula> /s ± <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.5~\mu \text{m}$ </tex-math></inline-formula> /s at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T =58.2$ </tex-math></inline-formula> N) and the increasing frequency (25–180 Hz) of decaying oscillations with the square root of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> over the range of line tensions ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r^{2} =0.978$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p &lt; 10^{-4}$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n =17$ </tex-math></inline-formula> ). Theory and experiment show that magnetic field impulses are capable of exerting measurable mechanical forces on bone <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex vivo</i> . The results raise an interesting question if the electromechanical effect could be sufficiently large to contribute to bone remodeling, reportedly sensitive to vibration amplitudes as small as 1 nm, and considering long duration of orthopedic therapy using repetitive EMP (months).