Parametric stability and dynamic buckling of an encapsulated microbubble subject to acoustic disturbances

Abstract

Stability analysis of the radial pulsations of a gas microbubble that is encapsulated by a thin viscoelastic shell and surrounded by an ideal incompressible liquid is carried out. Small axisymmetric disturbances in the microbubble shape are imposed and their long and short term stability is examined depending on the initial bubble radius, the shell properties, and the parameters, i.e., frequency and amplitude, of the external acoustic excitation. Owing to the anisotropy of the membrane that is forming the encapsulating shell, two different types of elastic energy are accounted for, namely, the membrane and bending energy per unit of initial area. They are used to describe the tensions that develop on the shell due to shell stretching and bending, respectively. In addition, two different constitutive laws are used in order to relate the tensions that develop on the membrane as a result of stretching, i.e., the Mooney-Rivlin law describing materials that soften as deformation increases and the Skalak law describing materials that harden as deformation increases. The limit for static buckling is obtained when the external overpressure exerted upon the membrane surpasses a critical value that depends on the membrane bending resistance. The stability equations describing the evolution of axisymmetric disturbances, in the presence of an external acoustic field, reveal that static buckling becomes relevant when the forcing frequency is much smaller than the resonance frequency of the microbubble, corresponding to the case of slow compression. The resonance frequencies for shape oscillations of the microbubble are also obtained as a function of the shell parameters. Floquet analysis shows that parametric instability, similar to the case of an oscillating free bubble, is possible for the case of a pulsating encapsulated microbubble leading to shape oscillations as a result of subharmonic or harmonic resonance. These effects take place for acoustic amplitude values that lie above a certain threshold but below those required for static buckling to occur. They are quite useful in providing estimates for the shell elasticity and bending resistance based on a frequency/amplitude sweep that monitors the onset of shape oscillations when the forcing frequency resonates with the radial pulsation, f = 0 , or with a certain shape mode, f =2 n . An acceleration based instability, identified herein as dynamic buckling, is observed during the compression phase of the pulsation, evolving over a small number of periods of the forcing, when the amplitude of the acoustic excitation is further increased. It corresponds to the RayleighTaylor instability observed for free bubbles, and has been observed with contrast agents as well, e.g., BR-14. Finally, phase diagrams for contrast agent BR-14 are constructed and juxtaposed with available experimental data, illustrating the relevance and range of the above instabilities.

Nonlinear response to ultrasound of encapsulated microbubbles

“…First, for safety considerations, the Mechanical Index must be restricted to small values. Second, similarly to free bubbles, above a certain threshold of the acoustic amplitude parametric instability appears, as it has been recently detected in experiments [31,32] and predicted theoretically [33]. Indeed, for a particular class of lipid-coated microbubbles of radii between 2 and 4 urn, at a frequency 1.7 MHz, an acoustic pressure of 100 kPa is sufficient to excite surface modes, and shell rupture occurs for small bubbles at pressure above 200 kPa [31].…”

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

“…(21). Finally, comparison with results of linear theory pertaining to the stability of radial pulsations with respect to axisymmetric disturbances, 14 constitutes an additional check for our calculations and is discussed in the next section, Sec. III A.…”

“…III the boundary element method is presented for studying axisymmetric oscillations and specific test runs are performed in order to validate the numerical methodology vs previous stability results. 14 Finally, in Sec. IV selected numerical results are presented and discussed in the context of outstanding issues in the literature of contrast agent dynamics, as outlined in the presentation provided in the previous paragraphs and, lastly, in Sec.…”

“…15 It is known that axisymmetric pulsations may arise as a result of parametric mode excitation for the appropriate amplitude and size range. 13,14 Static buckling and wrinkling of the interface can occur as a consequence of gas diffusion out of the shell and the subsequent development of large compressive stresses on the interface. 16 Alternatively, the presence of a nearby boundary, 17,18 or another contrast agent, may accelerate growth of interfacial instabilities and instigate phenomena like jet formation and, finally, splitting and break-up of the microbubble.…”

“…In particular, the onset of static buckling when the overpressure across the microbubble shell is increased needs to be studied in order to investigate the parameter range over which steady deformed shapes appear, the super or subcritical nature of the bifurcation, and how such shapes affect parametric mode stability when acoustic disturbances are imposed. Linear stability analysis 14 and optical observations of pulsating contrast agents 7,13 indicate the possibility for parametric excitation of axisymmetric modes. In the latter study 13 nonlinear saturation of the emerging modes was reported.…”