Unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation

Ultrasound-mediated nano-sized drug delivery systems for cancer treatment: Multi-scale and multi-physics computational modeling

Computational modeling enables researchers to study and understand various complex biological phenomena in anticancer drug delivery systems (DDSs), especially nano-sized DDSs (NSDDSs). The combination of NSDDSs and therapeutic ultrasound (TUS), that is, focused ultrasound and low-intensity pulsed ultrasound, has made significant progress in recent years, opening many opportunities for cancer treatment. Multiple parameters require tuning and optimization to develop effective DDSs, such as NSDDSs, in which mathematical modeling can prove advantageous. In silico computational modeling of ultrasound-responsive DDS typically involves a complex framework of acoustic interactions, heat transfer, drug release from nanoparticles, fluid flow, mass transport, and pharmacodynamic governing equations. Owing to the rapid development of computational tools, modeling the different phenomena in multi-scale complex problems involved in drug delivery to tumors has become possible. In the present study, we present an in-depth review of recent advances in the mathematical modeling of TUS-mediated DDSs for cancer treatment. A detailed discussion is also provided on applying these computational models to improve the clinical translation for applications in cancer treatment. This article is categorized under: Nanotechnology Approaches to Biology > Nanoscale Systems in Biology Therapeutic Approaches and Drug Discovery > Nanomedicine for Oncologic Disease.

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.

Development of an ultrasonic nonlinear frequency compounding method with applications in tissue thermometry

Frequency compounding is an ultrasound imaging technique used to improve signal-to-noise ratio (SNR). In this work, a nonlinear frequency compounding (NLFC) method was introduced, and its application in noninvasive tissue thermometry investigated. The NLFC method was used to produce two-dimensional maps of the temperature sensitive change in backscattered energy of acoustic harmonics (hCBE), during heating of ex vivo porcine tissue with a low intensity focused ultrasound transducer. A hCBE-to-temperature calibration was performed, and temperature maps produced and compared with a theoretical COMSOL based model. Last, a comparative study of the NLFC and previously used nonlinear single frequency (NLSF) methods was completed to quantify the improvement in SNR of the produced hCBE maps. When using the NLFC method, a SNR of 6.06 ± 1.28 was found. SNR values of 3.70 ± 0.53 and 4.38 ± 0.84 were found while using central frequencies of 4.31 and 5.43 MHz, respectively, with the NLSF method. This translates to an improvement of (64.13 ± 4.16)% over the 4.31 MHz NLSF, and (38.72 ± 2.97)% over the 5.43 MHz NLSF methods overall. It was concluded that the NLFC method can produce hCBE and temperature maps with superior image SNR over the NLSF method.

Temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various frequencies

Ultrasound-based thermometry requires a temperature-sensitive acoustic parameter that can be used to estimate the temperature by tracking changes in that parameter during heating. The objective of this study is to investigate the temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various pulse transmit frequencies from 1 to 20 MHz. Simulations were conducted using an expanded form of the Khokhlov-Zabolotskaya-Kuznetsov nonlinear acoustic wave propagation model in which temperature dependence of the medium parameters was included. Measurements were performed using single-element transducers at two different transmit frequencies of 3.3 and 13 MHz which are within the range of frequencies simulated. The acoustic pressure signals were measured by a calibrated needle hydrophone along the axes of the transducers. The water temperature was uniformly increased from 26 °C to 46 °C in increments of 5 °C. The results show that the temperature dependence of the harmonic generation is different at various frequencies which is due to the interplay between the mechanisms of absorption, nonlinearity, and focusing gain. At the transmit frequencies of 1 and 3.3 MHz, the harmonic amplitudes decrease with increasing the temperature, while the opposite temperature dependence is observed at 13 and 20 MHz.

Modeling of wave propagation for medical ultrasound: a review

Numerical modeling of medical ultrasound has advanced tremendously in the past two decades. This opens up a great number of opportunities for medical ultrasound and associated technologies. Numerous new governing equations and algorithms have emerged and been applied to studying various medical ultrasound applications, including ultrasound imaging, photo-acoustic imaging, and therapeutic ultrasound. In addition, thanks to the rapid development of computers, modeling acoustic wave propagation in three-dimensional, large-scale domains has become a reality. This article will provide an indepth literature and technical review of recent progress on numerical modeling of medical ultrasound. Future challenges will also be discussed.