A k-space method for acoustic propagation using coupled first-order equations in three dimensions

Abstract

A previously described two-dimensional k-space method for large-scale calculation of acoustic wave propagation in tissues is extended to three dimensions. The three-dimensional method contains all of the two-dimensional method features that allow accurate and stable calculation of propagation. These features are spectral calculation of spatial derivatives, temporal correction that produces exact propagation in a homogeneous medium, staggered spatial and temporal grids, and a perfectly matched boundary layer. Spectral evaluation of spatial derivatives is accomplished using a fast Fourier transform in three dimensions. This computational bottleneck requires all-to-all communication; execution time in a parallel implementation is therefore sensitive to node interconnect latency and bandwidth. Accuracy of the three-dimensional method is evaluated through comparisons with exact solutions for media having spherical inhomogeneities. Large-scale calculations in three dimensions were performed by distributing the nearly 50 variables per voxel that are used to implement the method over a cluster of computers. Two computer clusters used to evaluate method accuracy are compared. Comparisons of k-space calculations with exact methods including absorption highlight the need to model accurately the medium dispersion relationships, especially in large-scale media. Accurately modeled media allow the k-space method to calculate acoustic propagation in tissues over hundreds of wavelengths.

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

“…At each time step, the operations performed consist of four forward and six inverse 3D FFTs, and around 100 element-wise matrix operations. Previously, this type of model has been implemented using C++ and parallelised using either OpenMP, with simulations reported up to 1024 × 1024 × 1024 grid points [9], or MPI, with simulations reported up to 4096 × 2048 × 2048 grid points [2,10,11]. In both cases, the 3D FFTs are calculated over the entire 3D domain, which requires an all-to-all global communication for each FFT.…”

“…This approach was first proposed in [6] and further developed in [7][8][9]. The parallel implementation of the k-space pseudospectral method has previously been described by several groups [2,[10][11][12][13]. Aside from a number of element-wise matrix operations, the most significant operations performed at each time step are multiple real-to-complex and complex-toreal 3D fast Fourier transforms (FFTs).…”

Comparison of temporal and spectral scattering methods using acoustically large breast models derived from magnetic resonance images

“…Ultrasound waves within biological tissue are governed in three dimensions by the first-order acoustic equations 21 qðrÞ@ t uðr; tÞ ¼ Àrpðr; tÞ; (1a) ð t 0 ds jðr; t À sÞ@ s pðr; sÞ ¼ Àr Á uðr; tÞ;…”

“…Similar studies of the k-space method have been previously described. 20,21 No known prior attempts have been made to validate either method using models of comparable acoustic size or complexity. Both of these aspects of the computed models present unique challenges that require novel formulations and implementations of acoustic scattering solvers.…”

“…Both of these aspects of the computed models present unique challenges that require novel formulations and implementations of acoustic scattering solvers. The FMM 22,23 and the k-space method 21,24,25 have each been substantially enhanced to improve efficiency and also to extend the scope of solvable problems characterized by segmented magnetic resonance images. Combinations of these prior improvements made possible the case studies presented here of previously infeasible FMM and k-space solutions to large-scale, realistic scattering problems.…”