Modeling of nonlinear viscous stress in encapsulating shells of lipid-coated contrast agent microbubbles

Abstract

A general theoretical approach to the development of zero-thickness encapsulation models for contrast microbubbles is proposed. The approach describes a procedure that allows one to recast available rheological laws from the bulk form to a surface form which is used in a modified Rayleigh-Plesset equation governing the radial dynamics of a contrast microbubble. By the use of the proposed procedure, the testing of different rheological laws for encapsulation can be carried out. Challenges of existing shell models for lipid-encapsulated microbubbles, such as the dependence of shell parameters on the initial bubble radius and the "compression-only" behavior, are discussed. Analysis of the rheological behavior of lipid encapsulation is made by using experimental radius-time curves for lipid-coated microbubbles with radii in the range 1.2-2.5 microm. The curves were acquired for a research phospholipid-coated contrast agent insonified with a 20 cycle, 3.0 MHz, 100 kPa acoustic pulse. The fitting of the experimental data by a model which treats the shell as a viscoelastic solid gives the values of the shell surface viscosity increasing from 0.30 x 10(-8) kg/s to 2.63 x 10(-8) kg/s for the range of bubble radii, indicated above. The shell surface elastic modulus increases from 0.054 N/m to 0.37 N/m. It is proposed that this increase may be a result of the lipid coating possessing the properties of both a shear-thinning and a strain-softening material. We hypothesize that these complicated rheological properties do not allow the existing shell models to satisfactorily describe the dynamics of lipid encapsulation. In the existing shell models, the viscous and the elastic shell terms have the linear form which assumes that the viscous and the elastic stresses acting inside the lipid shell are proportional to the shell shear rate and the shell strain, respectively, with constant coefficients of proportionality. The analysis performed in the present paper suggests that a more general, nonlinear theory may be more appropriate. It is shown that the use of the nonlinear theory for shell viscosity allows one to model the "compression-only" behavior. As an example, the results of the simulation for a 2.03 microm radius bubble insonified with a 6 cycle, 1.8 MHz, 100 kPa acoustic pulse are given. These parameters correspond to the acoustic conditions under which the "compression-only" behavior was observed by de Jong et al. [Ultrasound Med. Biol. 33 (2007) 653-656]. It is also shown that the use of the Cross law for the modeling of the shear-thinning behavior of shell viscosity reduces the variance of experimentally estimated values of the shell viscosity and its dependence on the initial bubble radius.

Fig. 1

Best-fit values of the shell viscosity κ_s and the shell elasticity χ versus the initial bubble radius. Bubbles are insonified with a 20-cycle, 3.0 MHz, 100 kPa acoustic pulse. The values of κ_s and χ were evaluated by the de Jong shell model. Each circle represents the best fit for one experimental radius-time curve.

Fig. 2

(a) Shell viscosity versus maximum shear rate. (b) Shell elasticity versus deformation strength. The values of κ_s and χ were evaluated by the de Jong shell model. The maximum shear rate and the deformation strength were calculated directly from experimental radius-time curves.

Fig. 3

Simulated radius-time curves for a 2.03-μm-radius encapsulated bubble insonified with a 6-cycle, 1.8 MHz, 100 kPa acoustic pulse. (a) The Kelvin-Voigt shell model. (b) The model with the shell viscosity specified by equation (17).

Fig. 4

Fitting of an experimental radius-time curve showing “compression-only” behavior. (a) The experimental radius-time curve for a 1.4-μm-radius bubble with phospholipid coating. The excitation is a 20-cycle, 3.0 MHz, 100 kPa acoustic pulse. (b) The best fit given by the Kelvin-Voigt shell model. (c) The best fit given by the model with the shell viscosity specified by equation (17).

Fig. 5

Best-fit values of the shell viscous coefficients κ₀ and κ₁, and the shell elasticity χ versus the initial bubble radius. The values of the shell parameters were evaluated by using (16) and (17).

Fig. 6

Best-fit values of the shell viscous coefficients κ₀ and κ₁ versus the initial bubble radius. The values of κ₀ and κ₁ were evaluated by the model with the Cross law, equation. (18).