Speed-accuracy trade-offs in computing spatial impulse responses for simulating medical ultrasound imaging

Medical ultrasound imaging can be simulated realistically using linear acoustics. One of the most powerful approaches is to employ spatial impulse responses. Hereby both emitted fields and pulse-echo responses from point scatterers can be determined. Also any kind of dynamic focusing and apodization can be incorporated, as has been done in the Field II simulation program.

Here the transducer is modeled through a set of either rectangles, triangles, or bounding lines, so that any geometry can be simulated. The response from the transducer is found by summing the spatial impulse responses from the individual elements. One of the problems in using spatial impulse responses is the abrupt changes in the responses due to the sharp transducer boundaries.

Sampling the responses directly therefore have to be done at very high sampling frequencies to keep the shape and energy of the response. The high sampling frequency is unnecessary in the final signals, since the transducers used in medical ultrasound are band limited. Approaches to reduce the sampling frequency are, thus, needed to make efficient simulation programs.

Field II uses time integration of the spatial impulse responses using a continuous rather than discrete time-axis. This preserves the energy in the responses and makes it possible to make sub-sample interval delays for focusing. The paper discusses the consequence of the integration for the rectangular elements that uses an approximative calculation of the spatial impulse responses.

Data for the accuracy as a function of sampling frequency is given, and it is shown how a sampling frequency of 100 MHz gives similar results to using 2 GHz sampling of the analytic solution for rectangular elements. The spatial impulse responses for the triangular and bounding line elements are found analytically, and an iterative integration routine has to be used.

The Romberg integration routine is used, and the accuracy versus sampling frequency for bounding line is shown. An increased accuracy is attained for the lines compared to the rectangles, but the simulation times are significantly higher. Line elements should therefore, in this implementation, only be used very close to the transducer, and if a very high precision is needed in the calculation.