Shear strain from irrotational tissue displacements near bubbles

Abstract

Particle displacements can be much greater near bubbles than they would be in a homogeneous liquid or tissue when exposed to an acoustic wave. In a plane wave, shear and bulk strains are of the same order of magnitude. In contrast, for a bubble oscillating close to its resonance frequency, the shear strain in the medium near the bubble is roughly four orders of magnitude greater than the bulk strain. This can lead to shear strains of a few percent even with acoustic excitation pressures far below the pressure thresholds required to cause inertial cavitation. High shear strains near oscillating bubbles could potentially be the cause of bioeffects. After acoustic exposures at audio frequencies, hemorrhages in tissues as diverse as lung, liver, and kidney have been observed at shear strains on the order of 1%.

Biological Effects of Low-Frequency Shear Strain: Physical Descriptors

“…Spherical waves, however, are a practical reality. In fact, the longitudinal shear strains near bubbles in tissue are perhaps the primary concern for safety in the use of diagnostic ultrasound (Carstensen et al 2011). Furthermore, bubble fields, which involve only fast (~1500 m s −1 ) longitudinal waves, make a useful conceptual transition from idealized plane waves to more realistic low frequency waves, which, in general, involve both fast and slow longitudinal and transverse waves.…”

“…At low frequencies, resonant gas bodies in tissues are rarely perfectly spherical. However, treating them as such in studying their biological effects can provide useful, semi-quantitative approximations (Carstensen et al ., 2011). …”

“…The displacement field of a radially oscillating sphere is given by $$\overrightarrow{\xi }true(rtrue)={\xi }_0{e}^{jtrue(\omega t-ktrue(r-atrue)true)}\frac{a^2}{r^2}\frac{true(1+jkrtrue)}{true(1+jkatrue)}\overrightarrow{e_r},$$ where ξ 0 is the displacement amplitude of the source sphere, a is the radius of the sphere, r is the point of observation relative to the center of the sphere, $k=\frac{\omega }{c_f}$ as defined above and $\overrightarrow{e_r}$ is the unit vector in the radial direction (Graff 1975; Carstensen et al 2011). Waves produced in the medium are irrotational and longitudinal throughout, i.e ., displacements are in the direction of propagation.…”

“…With bubbles present, however, significant microvascular damage may occur at those exposure levels. The explanation, of course, is that tissue shear strains near oscillating bubbles are many orders of magnitude greater than strains in tissue exposed to the same acoustic stresses but in the absence of those bubbles (Carstensen et al 2011). …”