Simulation of nonlinear Westervelt equation for the investigation of acoustic streaming and nonlinear propagation effects.

This study investigates the influence of blood flow on temperature distribution during high-intensity focused ultrasound (HIFU) ablation of liver tumors. A three-dimensional acoustic-thermal-hydrodynamic coupling model is developed to compute the temperature field in the hepatic cancerous region. The model is based on the nonlinear Westervelt equation, bioheat equations for the perfused tissue and blood flow domains. The nonlinear Navier-Stokes equations are employed to describe the flow in large blood vessels. The effect of acoustic streaming is also taken into account in the present HIFU simulation study. A simulation of the Westervelt equation requires a prohibitively large amount of computer resources. Therefore a sixth-order accurate acoustic scheme in three-point stencil was developed for effectively solving the nonlinear wave equation. Results show that focused ultrasound beam with the peak intensity 2470 W/cm(2) can induce acoustic streaming velocities up to 75 cm/s in the vessel with a diameter of 3 mm. The predicted temperature difference for the cases considered with and without acoustic streaming effect is 13.5 °C or 81% on the blood vessel wall for the vein. Tumor necrosis was studied in a region close to major vessels. The theoretical feasibility to safely necrotize the tumors close to major hepatic arteries and veins was shown.

GPU-accelerated study of the inertial cavitation threshold in viscoelastic soft tissue using a dual-frequency driving signal.

Inertial cavitation thresholds under two forms of ultrasonic excitation (the single- and dual-frequency ultrasound modes) are studied numerically. The Gilmore-Akulichev model coupled with the Zener viscoelastic model is used to model the bubble dynamics. The threshold pressures are determined with two criteria, one based on the bubble radius and the other on the bubble collapse speed. The threshold behavior is investigated for different initial bubble sizes, acoustic signal modes, frequencies, tissue viscosities, tissue elasticities, and all their combinations. Due to the large number of parameters and their many combinations (around 1.5 billion for each threshold criterion), all simulations were executed on graphics processing units to speed up the calculations. We used our own code written in the C++ and CUDA C languages. The results obtained demonstrate that using the dual-frequency signal mode can help to reduce the inertial cavitation threshold (in comparison to the single-frequency mode). The criterion based on the bubble size gives a lower threshold than the criterion using the bubble collapse speed. With an increase of the elasticity, the threshold pressure also increases, whereas changing the viscosity has a very small impact on the optimal threshold, unlike the elasticity. A detailed analysis of the optimal ultrasound frequencies for a dual-frequency driving signal found that for viscosities less than 0.02 Pa·s, the first optimal frequency, in general, is much smaller than the second optimal frequency, which can reach 1 MHz. However, for high viscosities, both optimal frequencies are similar and varied in the range 0.01-0.05 MHz. Overall, this study presents a detailed analysis of inertial cavitation in soft tissue under dual-frequency signal excitation. It may be helpful for the further development of different applications of biomedical ultrasound.

A detailed study of ion transport through the SARS-CoV-2 E protein ion channel.

The envelope (E) protein encoded in the genome of an RNA virus is crucial for the replication, budding and pathophysiology of the virus. In the light of the ongoing pandemic, we explored similarities/differences between SARS-CoV-1 and SARS-CoV-2 E protein ion channels in terms of their selectivity. Further, we also examined the impact of variation of the bath concentration and introduction of potential and concentration gradients across the channel on the binding ratios of sodium and chloride ions for the SARS-CoV-2 E protein. Ion transport is described through the fourth-order Poisson-Nernst-Planck-Bikerman (4PNPBik) model which generalizes the traditional model by including ionic interactions between ions and their surrounding medium and non-ionic interactions between particles due to their finite size. Governing equations are solved numerically using the immersed boundary-lattice Boltzmann method (IB-LBM). The mathematical model has been validated by comparing analytical and experimental ion activity. The SARS-CoV-1 E protein ion channel is found to be more permeable to cationic ions, while the SARS-CoV-2 E protein has similar selectivity for both cationic and anionic species. For SARS-CoV-2, an increase in the bath concentration results in an increase in the binding ratio for sodium ions. Furthermore, the chloride binding ratio increases as the concentration gradient increases. A potential gradient has a minimal effect on the binding ratio. The SARS-CoV-2 E protein was found to support higher ionic currents than the SARS-CoV-1 E protein. Furthermore, the ionic current increased with increasing bath concentrations.

Investigation of the Efficiency of Mask Wearing, Contact Tracing, and Case Isolation during the COVID-19 Outbreak.

Case isolation and contact tracing are two essential parts of control measures to prevent the spread of COVID-19, however, additional interventions, such as mask wearing, are required. Taiwan successfully contained local COVID-19 transmission after the initial imported cases in the country in early 2020 after applying the above-mentioned interventions. In order to explain the containment of the disease spread in Taiwan and understand the efficiency of different non-pharmaceutical interventions, a mathematical model has been developed. A stochastic model was implemented in order to estimate the effectiveness of mask wearing together with case isolation and contact tracing. We investigated different approaches towards mask usage, estimated the effect of the interventions on the basic reproduction number (R0), and simulated the possibility of controlling the outbreak. With the assumption that non-medical and medical masks have 20% and 50% efficiency, respectively, case isolation works on 100%, 70% of all people wear medical masks, and R0 = 2.5, there is almost 80% probability of outbreak control with 60% contact tracing, whereas for non-medical masks the highest probability is only about 20%. With a large proportion of infectiousness before the onset of symptoms (40%) and the presence of asymptomatic cases, the investigated interventions (isolation of cases, contact tracing, and mask wearing by all people), implemented on a high level, can help to control the disease spread. Superspreading events have also been included in our model in order to estimate their impact on the outbreak and to understand how restrictions on gathering and social distancing can help to control the outbreak. The obtained quantitative results are in agreement with the empirical COVID-19 data in Taiwan.

Solution of Ion Channel Flow Using Immersed Boundary-Lattice Boltzmann Methods.

Poisson-Nernst-Planck (PNP) model has been extensively used for the study of channel flow under the influence of electrochemical gradients. PNP theory is a continuum description of ion flow where ionic distributions are described in terms of concentrations. Nonionic interparticle interactions are not considered in this theory as in continuum framework, their impact on the solution is minimal. This theory holds true for dilute flows or flows where channel radius is significantly larger than ion radius. However, for ion channel flows, where channel dimensions and ionic radius are of similar magnitude, nonionic interactions, particularly related to the size of the ions (steric effect), play an important role in defining the selectivity of the channel, concentration distribution of ionic species, and current across the channel, etc. To account for the effect of size of ions, several modifications to PNP equations have been proposed. One such approach is the introduction of Lennard-Jones potential to the energy variational formulation of PNP system. This study focuses on understanding the role of steric effect on flow properties. To discretize the system, Lattice Boltzmann method has been used. The system is defined by modified PNP equations where the steric effect is described by Lennard-Jones potential. In addition, boundary conditions for the complex channel geometry have been treated using immersed boundary method.

Investigating ion transport inside the pentameric ion channel encoded in COVID-19 E protein.

Ion flow inside an ion channel can be described through continuum based Born-Poisson-Nernst-Planck (BPNP) equations in conjunction with the Lennard-Jones potential. Keeping in mind the ongoing pandemic, in this study, an attempt has been made to understand the selectivity and the current voltage relation of the COVID-19 E protein pentameric ion channel. Two ionic species, namely Na^{+} and Cl^{-}, have been considered here. E protein is one of the smallest structural protein which is embedded in the outer membrane of the virus. Once the virus is inside the host cell, this protein is expressed abundantly and is responsible for activities such as replication and budding of the virus. In the literature, we can find a few experimental studies focusing on understanding the activity of the channel formed by E proteins of different viruses. Here, we attempt the same study for the COVID-19 E protein ion channel through mathematical modeling. The channel geometry is calculated from the protein data bank file which was provided by NARLabs, Taiwan, using the hole program. Further, it was used to obtain the charge distribution using the pdbtopqr online program. The immersed boundary-lattice Boltzmann method (IB-LBM) has been implemented to numerically solve the system of equations in the channel generated by the protein data bank file. Further, an in-house code which operates on multiple GPUs and uses the cuda platform has been developed to achieve the goal of performing the current investigation.

Experimental and Numerical Study on the Temperature Elevation in Tissue during Moxibustion Therapy.

Moxibustion is a thermal therapy in traditional Chinese medicine that relies on the heat from burning moxa to be transferred beneath the skin surface. Although moxibustion has long been in widespread practice, the mechanism of heat transfer modality and temperature distribution during this treatment is not yet well understood. The current paper presents the first examination by magnetic resonance imaging (MRI) of the three-dimensional temperature elevation during moxibustion treatment. A mathematical model for the prediction of temperature elevation during moxibustion therapy has been constructed and compared with the experimental data. Good agreement between the measured temperature and the results of numerical calculations has been found. Tissue up to 3 cm deep can be heated during the treatment. It was revealed that both heat conduction and radiation heat transfer play important roles during the treatment. The results presented in the current paper can be used for understanding the mechanisms of Chinese medicine and developing useful guidelines for Chinese medicine doctors.

Dynamics of bubble-bubble interactions experiencing viscoelastic drag.

The subject of the present theoretical study is the dynamics of bubble-bubble interactions in a viscoelastic medium. First, new equations for calculating the viscoelastic drag exerted on bubbles during their translational motion in a viscoelastic medium are derived. The drag equations are incorporated in the bubble-bubble interaction model in which, thereby, both the translational and radial motions of the bubbles are affected by the viscoelastic features of the medium. Second, the derived equations are applied to investigate how the viscoelastic properties of the medium can affect the dynamics of multiple bubbles, as well as how the bubbles can affect each other. It was discovered that the bubble-bubble interaction can significantly influence the dynamics of a single bubble. As the distance between the bubbles increases, their effect on each other decreases, and at a distance of several millimeters, this effect can be neglected. Moreover, it was concluded that with increasing elasticity and viscosity of the medium, as well with decreasing relaxation time, the effects of other bubbles on the current bubble's radial motion can become negligible. The translational motion of the bubbles was investigated for different viscoelastic models. The elasticity resists the motion of bubbles in space, resulting in a dynamical steady state of the distance between the bubbles at high elasticity values. The relaxation time of the medium was also found to be important in terms of the bubbles' translational movement.

Temperature elevation by HIFU in ex vivo porcine muscle: MRI measurement and simulation study.

PURPOSE: High-intensity focused ultrasound is a rapidly developing medical technology with a large number of potential clinical applications. Computational model can play a pivotal role in the planning and optimization of the treatment based on the patient's image. Nonlinear propagation effects can significantly affect the temperature elevation and should be taken into account. In order to investigate the importance of nonlinear propagation effects, nonlinear Westervelt equation was solved. Weak nonlinear propagation effects were studied. The purpose of this study was to investigate the correlation between the predicted and measured temperature elevations and lesion in a porcine muscle. METHODS: The investigated single-element transducer has a focal length of 12 cm, an aperture of 8 cm, and frequency of 1.08 MHz. Porcine muscle was heated for 30 s by focused ultrasound transducer with an acoustic power in the range of 24-56 W. The theoretical model consists of nonlinear Westervelt equation with relaxation effects being taken into account and Pennes bioheat equation. RESULTS: Excellent agreement between the measured and simulated temperature rises was found. For peak temperatures above 85-90 °C "preboiling" or cavitation activity appears and lesion distortion starts, causing small discrepancy between the measured and simulated temperature rises. From the measurements and simulations, it was shown that distortion of the lesion was caused by the "preboiling" activity. CONCLUSIONS: The present study demonstrated that for peak temperatures below 85-90 °C numerical simulation results are in excellent agreement with the experimental data in three dimensions. Both temperature rise and lesion size can be well predicted. Due to nonlinear effect the temperature in the focal region can be increased compared with the linear case. The current magnetic resonance imaging (MRI) resolution is not sufficient. Due to the inevitable averaging the measured temperature can be 10-30 °C lower than the peak temperature. Computational fluid dynamics can provide additional important information that is lost using a state of the art MRI device.

Prediction of strong-shock structure using the bimodal distribution function.

A modified Mott-Smith method for predicting the one-dimensional shock wave solution at very high Mach numbers is constructed by developing a system of fluid dynamics equations. The predicted shock solutions in a gas of Maxwell molecules, a hard-sphere gas, and in argon using the newly proposed formalism are compared with the experimental data, direct-simulation Monte Carlo (DSMC) solution, and other solutions computed from some existing theories for Mach numbers M<50. In the limit of an infinitely large Mach number, the predicted shock profiles are also compared with the DSMC solution. The density, temperature and heat flux profiles calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutions.

Prediction of shock structure using the bimodal distribution function.

A modification of Mott-Smith method for predicting the one-dimensional shock wave solution is presented. Mott-Smith distribution function is used to construct the system of moment equations to study the steady-state structure of shock wave in a gas of Maxwell molecules and in argon. The predicted shock solutions using the newly proposed formalism are compared to the experimental data, direct-simulation Monte Carlo (DSMC) solution, and the solutions predicted by other existing theories for Mach numbers M<11 . The density, temperature, heat flux profiles, and shock thickness calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutions. In addition, the predicted shock thickness is in good agreement with the DSMC simulation result at low Mach numbers.