Theoretical estimation of attenuation coefficient of resonant ultrasound contrast agents

Acoustic characterization of ultrasound contrast agents (UCAs, coated microbubbles) relies on the attenuation theory that assumes the UCAs oscillate linearly at sufficiently low excitation pressures. Effective shell parameters of the UCAs can be estimated by fitting a theoretical attenuation curve to experimentally measured attenuation data. Depending on the excitation frequency and properties of the shell, however, an UCA may oscillate nonlinearly even at sufficiently low excitation pressures, violating the assumption in the linear attenuation theory. Notably, the concern over the estimation of the attenuation coefficient of a microbubble at resonance using linearized approximation has long been addressed. This article investigated the attenuation phenomenon through analyzing the energy dissipation of a single UCA and propagating waves in an UCA suspension, both of which employed a nonlinear Rayleigh-Plesset equation. Analytical formulas capable of estimating the attenuation coefficient due to the weakly nonlinear oscillations of the UCA were obtained with a relatively rigorous mathematical analysis. The computed results that were verified by numerical simulations showed the attenuation coefficient of the UCA at resonance was pressure-dependent and could be significantly smaller than that predicted by the linear attenuation theory. Polydispersity of the UCA population enlarged the difference in the estimation of attenuation between the linear and present second-order nonlinear theories.