Discussion of the application of finite Volterra series for the modeling of the oscillation behavior of ultrasound contrast agents

Ultrasound contrast agents consist of microbubbles with diameters in the micrometer range. Excited by ultrasound, these bubbles exhibit highly nonlinear oscillation. While well developed physical models for microbubble oscillation exist, the efficiency of pulse sequences for sensitive microbubble detection is discussed based on simple mathematical models of general nonlinearity. Typically, Taylor series are used to model microbubble nonlinearity for the development of detection schemes. Recently, pulse sequences were proposed which exploit nonlinear memory of microbubbles, a property that cannot be modeled by a Taylor series but can be explained using a Volterra series. Therefore, this paper discusses and evaluates the usage of Volterra series for the modeling of the scattering behavior of contrast agent microbubbles. A numerically stable linear estimation algorithm is implemented to determine a third order Volterra model for a free gas bubble with a resting radius r0=1μm. For insonification pressures up to 100 kPa, the identified model allowed for a mean-square error of less than −16 dB with respect to the reference signal. Analysis of the response to narrowband signals showed that the achievable mean-square error is further reduced for the bandwidth available to typical ultrasound transducers used for clinical diagnostics.