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Abstract:
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 1⁄4 1 lm. 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.

Keywords:
Ultrasound contrast agent, Microbubble, Volterra series, Rayleigh–Plesset, System identification, Nonlinear oscillation

Abstract:
Ultrasound contrast agents consist of gas-filled microbubbles stabilized by a shell. Under ultrasound insonification, these bubbles oscillate nonlinearly with resonance frequencies being well within the diagnostic range. Currently, different detection methods are proposed, often with a heuristic reasoning based on the bubble nonlinearity being modeled by a time-invariant polynomial characteristic. However, it has been demonstrated [1] that microbubbles exhibit the behavior of a nonlinearity with memory. To optimize detection schemes, we propose to take this into account by ultrasound contrast agent modeling with a Wiener series. With these models, which can be identified from acoustic measurements, nonlinear system theory can be applied to improve detection methods. The feasibility of contrast agent modeling by Wiener series was evaluated on a contrast agent simulation, implemented by a modified Rayleigh-Plesset differential equation. For a sinusoidal input, the Wiener series approximated contrast agent behavior with a mean square error of 7.6% of the power of the contrast agent signal. The Wiener series approach was subsequently validated in an experimental setup where the nonlinear characteristics of a commercially available contrast agent were identified. The model obtained allowed for a mean square prediction error of 2.6% of the power of the measured signal for a pseudo-random multilevel sequence. With these experiments, it has been shown that the modeling of the oscillation behavior of ultrasound contrast agents with a Wiener series is feasible.

Abstract:
The driving of contrast microbubbles towards a boundary by means of primary radiation (Bjerknes) forces has been of interest for ultrasound-assisted drug delivery. Secondary radiation forces, resulting from oscillating microbubbles under ultrasound insonification, may cause the mutual attraction and subsequent coalescence of contrast microbubbles. This phenomenon has been less studied. Microbubbles with a negligible shell can be forced to translate towards each other at relatively low mechanical indices (MI). Thick-shelled microbubbles would require a higher MI to be moved. However, at high MI, microbubble disruption is expected. We investigated if thick-shelled contrast agent microbubbles can be forced to cluster at high-MI. Two thick-shelled contrast agents, inserted through a cellulose capillary, were subjected to 3 MHz, high- MI pulsed ultrasound from a commercial ultrasound machine, and synchronously captured through a high numerical aperture microscope. The agent QuantisonTM did not translate, but showed a small percentage of disrupted bubbles. The agent M1639 showed the ultrasound-induced formation of bubble clusters, and the translation thereof towards the capillary boundary. It is concluded, that forced translation and clustering of thick-shelled contrast microbubbles is feasible.