Function approximations to accelerate 3-D beam predictions for thermal dose calculations

Robust and safe control of tissue heating requires predictions of beam location, temperature and thermal dose in order to optimize the spatial and temporal distribution of acoustic power. Such calculations are computationally intensive, and our objective is to identify methods to speed up such calculations. Monochromatic beam prediction relies heavily on complex exponentials. We have identified a fast 4^th order approximation for the evaluation of a complex exponential using a reduced-complexity polynomial that minimizes absolute error to a maximum of 0.001 over -¿ to ¿, and we quantified the effects of the approximation on beam intensity and total acoustic power (TAP) for a 128 physical element transducer array. We demonstrate that ultrasound beam predictions are accelerated by 13× using a fast approximation to evaluate complex exponentials compared to standard libraries. The approximation was optimized to minimize error in TAP, the relevant quantity for thermal diffusion-limited heating. The resulting error in TAP was found to be less than 8 × 10^-4%, and the error in intensity was found to be less than 3 × 10^-2% as compared to full precision calculations. The same methodology may be applied to other beam prediction algorithms to speed their execution at a relatively minor degradation in accuracy.