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

Abstract

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.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

“…This makes the implementation of this analytic preconditioner rather easy with a low additional computational cost. Furthermore, it can easily be combined with fast methods such as FMM [39] or H-matrices techniques [67] since it simply consists in the application of the preconditioner at each iteration of the iterative solver. This preconditioner has been shown to yield a very fast convergence of GMRES solver and in particular with a number of iterations independent of the frequency and the mesh density.…”

Computationally Efficient Boundary Element Methods for High-Frequency Helmholtz Problems in Unbounded Domains

“…In the case of multi-element transducers, the incident acoustic pressure field is commonly modeled as a superposition of plane circular piston sources [24]. The spatial component of the acoustic field of such a circular source may be represented by the Rayleigh integral, which can be solved using numerical quadrature techniques [48].…”

“…The boundary element method (BEM) is an efficient and competitive tool to solve large-scale high-frequency Helmholtz problems in bounded or unbounded domains. Recent developments in fast matrix compression and preconditioning for boundary integral operators have pushed the computational limit of high-frequency boundary element computations such that problems in three dimensions with over a hundred wavelengths across the domain can be solved on a single workstation [48]. Furthermore, the availability of high-level software libraries allows for a convenient implementation of different boundary integral formulations [42].…”

“…We will apply these state-of-the-art techniques to a challenging problem arising from medical high-intensity focused ultrasound simulations [25]. In [48] we have published an earlier version of some of the techniques presented in this chapter. There, more details about the engineering application can be found.…”

Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

“…Some of the other analytic preconditioning strategies that can be incorporated in the FMM use the pseudo-inverse of the hypersingular operator [27,28,29].…”