Shear Wave Elasticity

Shear Wave Elasticity describes the propagation of transverse mechanical waves in viscoelastic tissue. When a harmonic or pulsed excitation at frequency $\omega$ induces displacement $u(x,t)=A e^{i(\omega t - kx)}$, the complex wave number $k$ encodes both the shear modulus $\mu$ and the material’s attenuation. The fundamental relation between wave speed and stiffness is

$$c_s = \sqrt{\frac{\mu}{\rho}},$$

with typical brain densities $\rho\approx1000\,\mathrm{kg/m^3}$ and shear moduli $\mu\approx1\text{–}4\,\mathrm{kPa}$ yielding $c_s\approx1\text{–}2\,\mathrm{m/s}$.

In a time-harmonic inversion, one measures $k=\omega/c_s + i\alpha$, and computes

$$\mu = \rho\bigl(\tfrac{\omega}{k}\bigr)^2,$$

where $\alpha$ (attenuation) in brain tissue is ~0.5–1.5 dB/cm at 50 Hz. Wavelengths $\lambda=c_s/f$ span 10–60 mm, setting a spatial resolution limit of a few millimetres after regularized inversion.

Shear wave attenuation and dispersion are strongly frequency-dependent; higher frequencies (e.g., 160–200 Hz in ARF-SWE) produce shorter wavelengths (λ≈5–7 mm) but suffer increased loss in skull and parenchyma, limiting penetration depth to ~7–10 cm.