Mast T D, Souriau L P, Liu D L, Tabei M, Nachman A I, Waag R C
Applied Research Laboratory, Pennsylvania State University, University Park, PA 16802, USA.
IEEE Trans Ultrason Ferroelectr Freq Control. 2001 Mar;48(2):341-54. doi: 10.1109/58.911717.
Large-scale simulation of ultrasonic pulse propagation in inhomogeneous tissue is important for the study of ultrasound-tissue interaction as well as for development of new imaging methods. Typical scales of interest span hundreds of wavelengths; most current two-dimensional methods, such as finite-difference and finite-element methods, are unable to compute propagation on this scale with the efficiency needed for imaging studies. Furthermore, for most available methods of simulating ultrasonic propagation, large-scale, three-dimensional computations of ultrasonic scattering are infeasible. Some of these difficulties have been overcome by previous pseudospectral and k-space methods, which allow substantial portions of the necessary computations to be executed using fast Fourier transforms. This paper presents a simplified derivation of the k-space method for a medium of variable sound speed and density; the derivation clearly shows the relationship of this k-space method to both past k-space methods and pseudospectral methods. In the present method, the spatial differential equations are solved by a simple Fourier transform method, and temporal iteration is performed using a k-t space propagator. The temporal iteration procedure is shown to be exact for homogeneous media, unconditionally stable for "slow" (c(x) < or = c0) media, and highly accurate for general weakly scattering media. The applicability of the k-space method to large-scale soft tissue modeling is shown by simulating two-dimensional propagation of an incident plane wave through several tissue-mimicking cylinders as well as a model chest wall cross section. A three-dimensional implementation of the k-space method is also employed for the example problem of propagation through a tissue-mimicking sphere. Numerical results indicate that the k-space method is accurate for large-scale soft tissue computations with much greater efficiency than that of an analogous leapfrog pseudospectral method or a 2-4 finite difference time-domain method. However, numerical results also indicate that the k-space method is less accurate than the finite-difference method for a high contrast scatterer with bone-like properties, although qualitative results can still be obtained by the k-space method with high efficiency. Possible extensions to the method, including representation of absorption effects, absorbing boundary conditions, elastic-wave propagation, and acoustic nonlinearity, are discussed.
非均匀组织中超声脉冲传播的大规模模拟对于超声与组织相互作用的研究以及新成像方法的开发都非常重要。感兴趣的典型尺度跨越数百个波长；当前大多数二维方法，如有限差分法和有限元法，无法以成像研究所需的效率在这个尺度上计算传播。此外，对于大多数现有的模拟超声传播的方法，大规模的三维超声散射计算是不可行的。以前的伪谱法和k空间法克服了其中一些困难，这些方法允许使用快速傅里叶变换执行大部分必要的计算。本文给出了一种针对声速和密度可变介质的k空间法的简化推导；该推导清楚地表明了这种k空间法与过去的k空间法和伪谱法之间的关系。在本方法中，空间微分方程通过简单的傅里叶变换方法求解，时间迭代使用k-t空间传播子进行。时间迭代过程对于均匀介质是精确的，对于“慢”（c(x) ≤ c0）介质是无条件稳定的，对于一般的弱散射介质是高度精确的。通过模拟入射平面波在几个仿组织圆柱体以及模型胸壁横截面中的二维传播，展示了k空间法在大规模软组织建模中的适用性。k空间法的三维实现也用于通过仿组织球体传播的示例问题。数值结果表明，k空间法对于大规模软组织计算是准确的，其效率比类似的蛙跳伪谱法或2-4时域有限差分法高得多。然而，数值结果也表明，对于具有类骨特性的高对比度散射体，k空间法不如有限差分法精确，尽管k空间法仍能高效地获得定性结果。讨论了该方法可能的扩展，包括吸收效应的表示、吸收边界条件、弹性波传播和声学非线性。
