Simulating Focused Ultrasound Transducers Using Discrete Sources on Regular Cartesian Grids

Accurately representing the behavior of acoustic sources is an important part of ultrasound simulation. This is particularly challenging in ultrasound therapy where multielement arrays are often used. Typically, sources are defined as a boundary condition over a 2-D plane within the computational model. However, this approach can become difficult to apply to arrays with multiple elements distributed over a nonplanar surface. In this paper, a grid-based discrete source model for single- and multielement bowl-shaped transducers is developed to model the source geometry explicitly within a regular Cartesian grid. For each element, the source model is defined as a symmetric, simply connected surface with a single grid point thickness. Simulations using the source model with the open-source k-Wave toolbox are validated using the Rayleigh integral, O'Neil solution, and experimental measurements of a focused bowl transducer under both quasi-continuous wave and pulsed excitations. Close agreement is shown between the discrete bowl model and the axial pressure predicted by the O'Neil solution for a uniform curved radiator, even at very low grid resolutions. Excellent agreement is also shown between the discrete bowl model and experimental measurements. To accurately reproduce the near-field pressure measured experimentally, it is necessary to derive the drive signal at each grid point of the bowl model directly using holography. However, good agreement is also obtained in the focal region using uniformly radiating monopole sources distributed over the bowl surface. This allows the response of multielement transducers to be modeled, even where measurement of an input plane is not possible.