Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: The effect of membrane constitutive law

Abstract

The nonlinear radial oscillations of bubbles that are encapsulated in an elastic shell are investigated numerically subject to three different constitutive laws describing the viscoelastic properties of the shell: the Mooney-Rivlin (MR), the Skalak (SK), and the Kelvin-Voigt (KV) models are used in order to describe strain-softening, strain-hardening and small displacement (Hookean) behavior of the shell material, respectively. Due to the isotropic nature of the acoustic disturbances, the area dilatation modulus is the important parameter. When the membrane is strain softening (MR) the resonance frequency decreases with increasing sound amplitude, whereas the opposite happens when the membrane is strain hardening (SK). As the amplitude of the acoustic disturbance increases the total scattering cross section of a microbubble with a SK membrane tends to decrease, whereas that of a KV or a MR membrane tends to increase. The importance of strain-softening behavior in the abrupt onset of volume pulsations, that is often observed with small insonated microbubbles at moderately large sound amplitudes, is discussed.

Comparison between maximum radial expansion of ultrasound contrast agents and experimental postexcitation signal results

“…Albumin-coated UCAs were first modeled by adding damped linear oscillator terms to describe the effect of the shell, 22 while later models more rigorously derived shell terms assuming shell behavior as an elastic solid 23,24 or with an infinitesimally thin Newtonian rheology. 25 Increasingly complicated behaviors observed with UCAs containing more flexible lipid shells have prompted the introduction of a rich variety of UCA models, incorporating descriptions such as Maxwell rheology, 26 shear thinning, 27 and strain-softening, 28,29 among others. The earliest model to capture a wide range of lipid-based UCA responses was the Marmottant model, which postulated that the UCA shell behaves as a two-dimensional monolayer with varying surface tension.…”

Material characterization of the encapsulation of an ultrasound contrast microbubble and its subharmonic response: Strain-softening interfacial elasticity model

“…13,24 Following our investigation of a constant elasticity model of Sonazoid, 4 constitutive models appropriate for membranes such as Mooney-Rivlin ͑strain-softening͒ and Skalak ͑strain-hardening͒ models were applied to the encapsulation of contrast microbubbles, where the authors performed a parametric investigation of the effects of the various constitutive parameters on the response of a bubble surrounded by such membranes. 25 These models have been widely used for membranes of capsules and biological cells, 26,27 that contain incompressible liquid in contrast to the compressible gas in a microbubble, and therefore do not undergo volume change. Membrane models ͑e.g., MooneyRivlin model for rubbery material͒ find the membrane stresses using the generalized strain energy, which is a function of the invariants of the finite deformation membrane strains.…”

Excitation threshold for subharmonic generation from contrast microbubbles

“…There, such linear models might be inappropriate, warranting more sophisticated nonlinear constitutive models for the encapsulation. Recently, considerable effort has been devoted toward developing rigorous models for the encapsulation (Chatterjee and Sarkar, 2003;Chatterjee et al, 2005b;Doinikov and Dayton, 2007;Hoff et al, 2000;Marmottant et al, 2005;Sarkar et al, 2005;Tsiglifis and Pelekasis, 2008). We have developed Newtonian (Chatterjee and Sarkar, 2003) and viscoelastic (Sarkar et al, 2005) (zero-thickness) interfacial rheological models of the encapsulation; the molecular nature of the encapsulation and thereby the directional anisotropy justified such a two-dimensional continuum model in contrast to finite thickness models, e.g., Church (1995); Hoff et al (2000); Doinikov and Dayton (2007).…”