Simulations of insonated contrast agents: Saturation and transient break-up

Abstract

Under insonation contrast agents are known to perform nonlinear pulsations and deform statically, in the form of buckling, or dynamically via parametric mode excitation, and often exhibit jetting and break-up like bubbles without coating. Boundary element simulations are performed in the context of axisymmetry in order to establish the nonlinear evolution of these patterns. The viscoelastic stresses that develop on the coating form the dominant force balance tangentially to the shell-liquid interface, whereas the dynamic overpressure across the shell balances viscoelastic stresses in the normal direction. Strain softening and strain hardening behavior is studied in the presence of shape instabilities for various initial conditions. Simulations recover the pattern of static buckling, subharmonic/harmonic excitation, and dynamic buckling predicted by linear stability. Preferential mode excitation during compression is obtained supercritically for strain softening phospholipid shells while the shell regains its sphericity at expansion. It is a result of energy transfer between the emerging unstable modes and the radial mode, eventually leading to saturated oscillations of shape modes accompanied by asymmetric radial pulsations in favor of compression. Strain softening shells are more prone to sustain saturated pulsations due to the mechanical behavior of the shell. As the sound amplitude increases and before the onset of dynamic buckling, both types of shells exhibit transient break-up via unbalanced growth of a number of unstable shape modes. The effect of pre-stress in lowering the amplitude threshold for shape mode excitation is captured numerically and compared against the predictions of linear stability analysis. The amplitude interval for which sustained shape oscillations are obtained is extended, in the presence of pre-stress, by switching from a strain softening constitutive law to a strain hardening one once the shell curvature increases beyond a certain level. This type of mechanical behavior models the formation of lipid bilayer structures on the shell beyond a certain level of bending, as a result of a lipid monolayer folding transition. In this context a compression only type behavior is obtained in the simulations, which is accompanied by preferential shape deformation during compression at relatively small sound amplitudes in a manner that bears significance on the interpretation of available experimental observations exhibiting similar dynamic behavior.

Dynamic simulation of a coated microbubble in an unbounded flow: response to a step change in pressure

“…They observed mainly subharmonic shape mode excitation when the forcing frequency was twice the natural frequency of the shape mode, in agreement with results based on linear stability analysis (Tsiglifis & Pelekasis 2011). Tsiglifis & Pelekasis (2013) performed boundary integral simulations of pulsating contrast agents, while ignoring viscous effects on the liquid side, in view of the relatively large shell viscosity and the typical small size of encapsulated bubbles. They captured harmonic and subharmonic shape mode excitation during the compression phase of the pulsation, for the parameter range predicted by linear stability (Tsiglifis & Pelekasis 2011).…”

“…Numerical simulations of pulsating coated microbubbles with the boundary element methodology (Tsiglifis & Pelekasis 2013), when potential flow is considered in the surrounding liquid, recover a tendency for preferential excursion from equilibrium during compression due to nonlinear interactions between emerging shape modes during compression, as a manifestation of harmonic or subharmonic excitation (see also Tsiglifis & Pelekasis (2011) for an extensive analysis of this mechanism of shape mode excitation of coated microbubbles) and the breathing mode of pulsation; the latter term refers to volume pulsations of the bubble. Nevertheless, the extent of the compression amplitude obtained by the simulations performed in the latter study was smaller than the one reported from optical measurements.…”

“…We are interested in examining the dynamic response of coated microbubbles to such disturbances and investigate the possibility for a static solution. In this context, the relative importance of shell and liquid viscosity in controlling the dynamics towards static equilibrium must be ascertained, since previous numerical studies assuming potential flow conditions (Liu et al 2011;Tsiglifis & Pelekasis 2013), or taking into account the viscosity of the surrounding fluid (Liu et al 2011), focus on the effect of shell and liquid viscosity, respectively. Furthermore, it is important to study the response of contrast agents to greater disturbances in the far-field pressure and determine whether jet formation occurs, as is the case with gas-filled bubbles (Blake et al 1997) in the vicinity of a solid boundary.…”

“…They were thus able to calculate the stresses that develop on the microvessel wall as result of the pulsating motion, which was also seen to increase the permeability of the vessel. More recently, numerical studies were conducted to address the above issues via the finite volume (Liu et al 2011) and boundary integral method (Tsiglifis & Pelekasis 2013) as a means to obtain a more detailed description of the velocity field in the surrounding fluid but also the elastic stresses on the coating. In particular, Liu et al (2011) explored numerically the shape oscillations in an unbounded flow of an encapsulated microbubble that obeys the Mooney-Rivlin constitutive law by solving the continuity and the Navier-Stokes equations with the finite volume method using a boundary fitted coordinate system.…”

Cell mechanics in biomedical cavitation

“…It is assumed to be infinitely thin and isotropic in their planes. The elasticity of the membrane can be described by the Mooney-Rivlin law [46] …”