Numerical Simulations of the Nonlinear Interaction of a Bubble Cloud and a High Intensity Focused Ultrasound Field

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller-Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity supresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.

Weakly nonlinear propagation of focused ultrasound in bubbly liquids with a thermal effect: Derivation of two cases of Khokolov-Zabolotskaya-Kuznetsoz equations

A physico-mathematical model composed of a single equation that consistently describes nonlinear focused ultrasound, bubble oscillations, and temperature fluctuations is theoretically proposed for microbubble-enhanced medical applications. The Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation that has been widely used as a simplified model for nonlinear propagation of focused ultrasound in pure liquid is extended to that in liquid containing many spherical microbubbles, by applying the method of multiple scales to the volumetric averaged basic equations for bubbly liquids. As a result, for two-dimensional and three-dimensional cases, KZK equations composed of the linear combination of nonlinear, dissipation, dispersion, and focusing terms are derived. Especially, the dissipation term depends on three factors, i.e., interfacial liquid viscosity, liquid compressibility, and thermal conductivity of gas inside bubbles; the thermal conduction is evaluated by using four types of temperature gradient models. Finally, we numerically solve the derived KZK equation and show a moderate temperature rise appropriate to medical applications.

A multi-pillar piezoelectric stack transducer for nanodroplet mediated intravascular sonothrombolysis

We aim to develop a nanodroplet (ND)-mediated intravascular ultrasound (US) transducer for deep vein thrombosis treatments. The US device, having an efficient forward directivity of the acoustic beam, is capable of expediting the clot dissolution rate by activating cavitation of NDs injected onto a thrombus. We designed and prototyped a multi-pillar piezoelectric stack (MPPS) transducer composed of four piezoelectric stacks. Each stack was made of five layers of PZT-4 plates, having a dimension of 0.85 × 0.85 × 0.2 mm3. The transducer was characterized by measuring the electrical impedance and acoustic pressure, compared to simulation results. Next, in-vitro tests were conducted in a blood flow mimicking system using the transducer equipped with an ND injecting tube. The miniaturized transducer, having an aperture size of 2.8 mm, provided a high mechanical index of 1.52 and a relatively wide focal zone of 3.4 mm at 80 Vpp, 0.96 MHz electric input. The mass-reduction rate of the proposed method (NDs + US) was assessed to be 4.1 and 4.6 mg/min with and without the flow model, respectively. The rate was higher than that (1.3-2.7 mg/min) of other intravascular ultrasound modalities using micron-sized bubble agents. The ND-mediated intravascular sonothrombolysis using MPPS transducers was demonstrated with an unprecedented lysis rate, which may offer a new clinical option for DVT treatments. The MPPS transducer generated a high acoustic pressure (~3.1 MPa) at a distance of approximately 2.2 wavelengths from the small aperture, providing synergistic efficacy with nanodroplets for thrombolysis without thrombolytic agents.