Mechanic stress generated by a time-varying electromagnetic field on bone surface

Abstract

Bone cells sense mechanical load, which is essential for bone growth and remodeling. In a fracture, this mechanism is compromised. Electromagnetic stimulation has been widely used to assist in bone healing, but the underlying mechanisms are largely unknown. A recent hypothesis suggests that electromagnetic stimulation could influence tissue biomechanics; however, a detailed quantitative understanding of EM-induced biomechanical changes in the bone is unavailable. This paper used a muscle/bone model to study the biomechanics of the bone under EM exposure. Due to the dielectric properties of the muscle/bone interface, a time-varying magnetic field can generate both compressing and shear stresses on the bone surface, where many mechanical sensing cells are available for cellular mechanotransduction. I calculated these stresses and found that the shear stress is significantly greater than the compressing stress. Detailed parametric analysis suggests that both the compressing and shear stresses are dependent on the geometrical and electrical properties of the muscle and the bone. These stresses are also functions of the orientation of the coil and the frequency of the magnetic field. It is speculated that the EM field could apply biomechanical influence to fractured bone, through the fine-tuning of the controllable field parameters.

Appendix - Determining unknown coefficients C n, D n in Eq. (3) using boundary conditions (A)-(D)

Since V is bounded at r = 0 and r → ∞, from Eq. (3),

$$ {D}_A=0\kern0.5em {A}_B=0 $$

Therefore, expressions for the potential distribution in the extracellular media, the membrane, and in the cytoplasm are:

$$ {V}_A=\frac{A_A}{r}\sin \theta $$

(A-1)

$$ {V}_M=\left(\frac{A_M}{r}+{D}_Mr\right)\sin \theta $$

(A-2)

$$ {V}_B={D}_Br\sin \theta $$

(A-3)

Substitution of A_0r (Eq. 9) and the $\overset{\rightharpoonup }{r}$ components of ∇V in the three regions into (1) yielded the expressions of the normal components of the electric fields in the three regions:

$$ {E}_{Ar}=-\frac{j\omega {B}_0C}{2}\sin \theta +\frac{A_A}{r^2}\sin \theta $$

(A-4)

$$ {E}_{Mr}=-\frac{j\omega {B}_0C}{2}\sin \theta +\left(\frac{A_M}{r^2}-{D}_M\right)\sin \theta $$

(A-5)

$$ {E}_{Br}=-\frac{j\omega {B}_0C}{2}\sin \theta -{D}_B\sin \theta $$

(A-6)

Following boundary condition (A), V is continuous at the air/limb interface (r = R_M) and muscle/bone interface (r = R_B),

$$ \frac{A_A}{R_M}=\frac{A_M}{R_M}+{D}_M{R}_M $$

(A-7)

$$ \frac{A_M}{R_B}+{D}_M{R}_B={D}_B{R}_B $$

(A-8)

From the boundary condition (B), that the normal components of the current densities are continuous between two different media (Eqs. 1 and 2), we can obtain the following equations:

$$ {S}_A\left(-\frac{j\omega {B}_0C}{2}+\frac{A_A}{R_M^2}\right)={S}_M\left(-\frac{j\omega {B}_0C}{2}+\frac{A_M}{R_M^2}-{D}_M\right) $$

(A-9)

$$ {S}_M\left(-\frac{j\omega {B}_0C}{2}+\frac{A_M}{R_B^2}-{D}_M\right)={S}_B\left(-\frac{j\omega {B}_0C}{2}-{D}_B\right) $$

(A-10)

Equations (A-7) through (A-10) yielded the last four unknown coefficients:

$$ {\displaystyle \begin{array}{l}{A}_A=\frac{j\omega {B}_0C}{2}\frac{R_M^2\left[\left({S}_A+{S}_M\right)\left({S}_M-{S}_D\right){R}_B^2+\left({S}_A-{S}_M\right)\left({S}_M+{S}_B\right){R}_M^2\right]}{\left({S}_A-{S}_M\right)\left({S}_M-{S}_B\right){R}_B^2+\left({S}_A+{S}_M\right)\left({S}_M+{S}_B\right){R}_M^2}\\ {}{A}_M=\frac{j\omega {B}_0C}{2}\frac{2{R}_B^2{R}_M^2{S}_A\left({S}_M-{S}_B\right)}{\left({S}_A-{S}_M\right)\left({S}_M-{S}_B\right){R}_B^2+\left({S}_A+{S}_M\right)\left({S}_M+{S}_B\right){R}_M^2}\\ {}{D}_M=-\frac{j\omega {B}_0C}{2}\frac{\left({S}_A-{S}_M\right)\left[\left({S}_M-{S}_B\right){R}_B^2-\left({S}_M+{S}_B\right){R}_M^2\right]}{\left({S}_A-{S}_M\right)\left({S}_M-{S}_B\right){R}_B^2+\left({S}_A+{S}_M\right)\left({S}_M+{S}_B\right){R}_M^2}\\ {}{D}_B=-\frac{j\omega {B}_0C}{2}\frac{\left({S}_A-{S}_M\right)\left({S}_M-{S}_B\right){R}_B^2+\left[{S}_A\left({S}_B-3{S}_M\right)+{S}_M\left({S}_M+{S}_B\right)\right]{R}_M^2}{\left({S}_A-{S}_M\right)\left({S}_M-{S}_B\right){R}_B^2+\left({S}_A+{S}_M\right)\left({S}_M+{S}_B\right){R}_M^2}\end{array}} $$