Computationally efficient boundary element methods for high-frequency Helmholtz problems in unbounded domains

This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation

A fast boundary element method for the scattering analysis of high-intensity focused ultrasound

High-intensity focused ultrasound (HIFU) techniques are promising modalities for the non-invasive treatment of cancer. For HIFU therapies of, e.g., liver cancer, one of the main challenges is the accurate focusing of the acoustic field inside a ribcage. Computational methods can play an important role in the patient-specific planning of these transcostal HIFU treatments. This requires the accurate modeling of acoustic scattering at ribcages. The use of a boundary element method (BEM) is an effective approach for this purpose because only the boundaries of the ribs have to be discretized instead of the standard approach to model the entire volume around the ribcage. This paper combines fast algorithms that improve the efficiency of BEM specifically for the high-frequency range necessary for transcostal HIFU applications. That is, a Galerkin discretized Burton-Miller formulation is used in combination with preconditioning and matrix compression techniques. In particular, quick convergence is achieved with the operator preconditioner that has been designed with on-surface radiation conditions for the high-frequency approximation of the Neumann-to-Dirichlet map. Realistic computations of acoustic scattering at 1 MHz on a human ribcage model demonstrate the effectiveness of this dedicated BEM algorithm for HIFU scattering analysis

Boundary integral formulations for acoustic modelling of high-contrast media

The boundary element method is an efficient algorithm for simulating acoustic propagation through homogeneous objects embedded in free space. The conditioning of the system matrix strongly depends on physical parameters such as density, wavespeed and frequency. In particular, high contrast in density and wavespeed across a material interface leads to an ill-conditioned discretisation matrix. Therefore, the convergence of Krylov methods to solve the linear system is slow. Here, specialised boundary integral formulations are designed for the case of acoustic scattering at high-contrast media. The eigenvalues of the resulting system matrix accumulate at two points in the complex plane that depend on the density ratio and stay away from zero. The spectral analysis of the Calderón preconditioned PMCHWT formulation yields a single accumulation point. Benchmark simulations demonstrate the computational efficiency of the high-contrast Neumann formulation for scattering at high-contrast media

Benchmarking preconditioned boundary integral formulations for acoustics

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave scattering. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretisation of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave scattering at large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimised for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarking provide valuable information on the computational characteristics of several hundred different models for multiple scattering and transmission of acoustic wave fields

Benchmarking preconditioned boundary integral formulations for acoustics.

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves

Laser vibrometry characterisation of a microfluidic lab-on-a-chip device: a preliminary investigation

Since their original inception as ultrasound contrast agents, potential applications of microbubbles have evolved to encompass molecular imaging and targeted drug delivery. As these areas develop, so does the need to understand the mechanisms behind the interaction of microbubbles both with biological tissue and with other microbubbles. There is therefore a metrological requirement to develop a controlled environment in which to study these processes. Presented here is the design and characterisation of such a system, which consists of a microfluidic chip, specifically developed for manipulating microbubbles using both optical and acoustic trapping. A laser vibrometer is used to observe the coupling of acoustic energy into the chip from a piezoelectric transducer bonded to the surface. Measurement of the velocity of surface waves on the chip is investigated as a potential method for inferring the nature of the acoustic fields excited within the liquid medium of the device. Comparison of measured surface wavelengths with wave types suggests the observation of anti-symmetric Lamb or Love-Kirchhoff waves. Further visual confirmation of the acoustic fields through bubble aggregation highlights differences between the model and experimental results in predicting the position of acoustic pressure nodes in relation to excitation frequency

New Developments and Challenges in Liver Transplantation

Liver disease is increasing in incidence and is the third most common cause of premature death in the United Kingdom and fourth in the United States. Liver disease accounts for 2 million deaths globally each year. Three-quarters of patients with liver disease are diagnosed at a late stage, with liver transplantation as the only definitive treatment. Thomas E. Starzl performed the first human liver transplant 60 years ago. It has since become an established treatment for end-stage liver disease, both acute and chronic, including metabolic diseases and primary and, at present piloting, secondary liver cancer. Advances in surgical and anaesthetic techniques, refined indications and contra-indications to transplantation, improved donor selection, immunosuppression and prognostic scoring have allowed the outcomes of liver transplantation to improve year on year. However, there are many limitations to liver transplantation. This review describes the milestones that have occurred in the development of liver transplantation, the current limitations and the ongoing research aimed at overcoming these challenges

The interaction of shockwaves with a vapour bubble in boiling histotripsy: The shock scattering effect

Boiling histotripsy is a High Intensity Focused Ultrasound (HIFU) technique which uses a number of short pulses with high acoustic pressures at the HIFU focus to induce mechanical tissue fractionation. In boiling histotripsy, two different types of acoustic cavitation contribute towards mechanical tissue destruction: a boiling vapour bubble and cavitation clouds. An understanding of the mechanisms underpinning these phenomena and their dynamics is therefore paramount to predicting and controlling the overall size of a lesion produced for a given boiling histotripsy exposure condition. A number of studies have shown the effects of shockwave heating in generating a boiling bubble at the HIFU focus and have studied its dynamics under boiling histotripsy insonation. However, not much is known about the subsequent production of cavitation clouds that form between the HIFU transducer and the boiling bubble. The main objective of the present study is to examine what causes this bubble cluster formation after the generation of a boiling vapour bubble. A numerical simulation of 2D nonlinear wave propagation with the presence of a bubble at the focus of a HIFU field was performed using the k-Wave MATLAB toolbox for time domain ultrasound simulations, which numerically solves the generalised Westervelt equation. The numerical results clearly demonstrate the appearance of the constructive interference of a backscattered shockwave by a bubble with incoming incident shockwaves. This interaction (i.e., the reflected and inverted peak positive phase from the bubble with the incoming incident rarefactional phase) can eventually induce a greater peak negative pressure field compared to that without the bubble at the HIFU focus. In addition, the backscattered peak negative pressure magnitude gradually increased from 17.4 MPa to 31.6 MPa when increasing the bubble size from 0.2 mm to 1.5 mm. The latter value is above the intrinsic cavitation threshold of –28 MPa in soft tissue. Our results suggest that the formation of a cavitation cloud in boiling histotripsy is a threshold effect which primarily depends (a) the size and location of a boiling bubble, and (b) the sum of the incident field and that scattered by a bubble