The short-pulse subharmonic response of microbubbles based on a two-frequency approximation

The subharmonic response due to the nonlinear behavior of microbubble can be used to provide good discrimination between microbubble and surrounding tissue, especially in deep region. However, there is no proper analysis about the subharmonic response under short-pulse insonification. In this work, we extend the two-frequency approximated analytic solution of Newhouse et al. to derive the subharmonic response of microbubble under band-limited insonification. Based on Fourier theory, a band-limited signal can be synthesized by multiple sinusoids, with a two-frequency approximation being the simplest case. Our theoretical analysis illustrates that the amplitude of the subharmonics decrease with the transmitted fractional bandwidth. Moreover, under an applied pressure of 514 kPa, it approaches zero when the fractional bandwidth is increased to 8 %. In other words, this proves theoretically that only narrowband transmission can excite the microbubble to generate the subharmonics. The amplitude of low-frequency response can be derived to increase with the fractional bandwidth, which is different from that of subharmonics. The experimental data from free gas were used to verify the theoretical predictions. It can be shown that the amplitude of the subharmonics decrease with the transmitted fractional bandwidth being varied from 4 % to 18 % when the emitted frequency is 3.00 MHz and the acoustic pressure is 514 kPa. On the contrary, the low-frequency response increases with the transmitted bandwidth.