^{1}and Robert J. McGough

^{1,a)}

### Abstract

The angular spectrum approach is evaluated for the simulation of focused ultrasound fields produced by large thermal therapy arrays. For an input pressure or normal particle velocity distribution in a plane, the angular spectrum approach rapidly computes the output pressure field in a three dimensional volume. To determine the optimal combination of simulation parameters for angular spectrum calculations, the effect of the size, location, and the numerical accuracy of the input plane on the computed output pressure is evaluated. Simulation results demonstrate that angular spectrum calculations performed with an input pressure plane are more accurate than calculations with an input velocity plane. Results also indicate that when the input pressure plane is slightly larger than the array aperture and is located approximately one wavelength from the array, angular spectrum simulations have very small numerical errors for two dimensional planar arrays. Furthermore, the root mean squared error from angular spectrum simulations asymptotically approaches a nonzero lower limit as the error in the input plane decreases. Overall, the angular spectrum approach is an accurate and robust method for thermal therapy simulations of large ultrasoundphased arrays when the input pressure plane is computed with the fast nearfield method and an optimal combination of input parameters.

This work was supported in part by NIH Grant No. R21CA121235 and NSF Theoretical Foundations Grant No. 0634786.

I. INTRODUCTION

II. THEORY

A. Integral approaches

B. Phased array beamforming

C. Angular spectrum approach

D. Error evaluations

E. Temperature simulations

III. SIMULATION RESULTS

A. Reference pressure field generated by a element phased array

B. Evaluation of pressure and normal particle velocity inputs

C. Optimal parameters for the input plane

D. Evaluation of input and output errors

E. Temperature simulations

IV. DISCUSSION

A. Simulations with other arrays

B. The size and location of the input pressure plane

C. The spatial sampling rate

D. The spectral sampling rate

E. Computation time

V. CONCLUSION

### Key Topics

- Particle velocity
- 35.0
- Antenna arrays
- 20.0
- Spatial analysis
- 20.0
- Sound pressure
- 15.0
- Ultrasonics
- 14.0

## Figures

The discretized input plane, where the input pressure or normal particle velocity plane is initially computed in an grid and then zero-padded to an grid for angular spectrum calculations. The spectral propagator, which is not zero-padded, is then evaluated in an grid for angular spectrum calculations.

The discretized input plane, where the input pressure or normal particle velocity plane is initially computed in an grid and then zero-padded to an grid for angular spectrum calculations. The spectral propagator, which is not zero-padded, is then evaluated in an grid for angular spectrum calculations.

A planar ultrasound phased array comprised of square elements. The size of the array is ( for a driving frequency). The array consists of square transducers with a kerf between adjacent elements.

A planar ultrasound phased array comprised of square elements. The size of the array is ( for a driving frequency). The array consists of square transducers with a kerf between adjacent elements.

Reference pressure field generated by the element planar array in Fig. 2, where the array is focused at . The pressure, which is normalized by the maximum amplitude, is shown in the plane at . The excitation frequency for the array is , and the attenuation coefficient is .

Reference pressure field generated by the element planar array in Fig. 2, where the array is focused at . The pressure, which is normalized by the maximum amplitude, is shown in the plane at . The excitation frequency for the array is , and the attenuation coefficient is .

Simulated axial pressures generated by the element planar array in Fig. 2. The array is located at and electronically focused at . The reference pressure calculated by the spatial impulse response method is indicated by the solid line, the pressure computed with the angular spectrum approach using an input normal particle velocity plane is represented by the dash-dot line, and the pressure computed with the angular spectrum approach using an input pressure plane is represented by the dashed line.

Simulated axial pressures generated by the element planar array in Fig. 2. The array is located at and electronically focused at . The reference pressure calculated by the spatial impulse response method is indicated by the solid line, the pressure computed with the angular spectrum approach using an input normal particle velocity plane is represented by the dash-dot line, and the pressure computed with the angular spectrum approach using an input pressure plane is represented by the dashed line.

RMS output errors for the element array focused at evaluated in transverse planes for ranging from to . The pressure is calculated with the angular spectrum approach using an input normal particle velocity plane (dash-dot line) and an input pressure plane (dashed line). For this result, the input particle velocity and pressure planes are both located at and truncated with a square window.

RMS output errors for the element array focused at evaluated in transverse planes for ranging from to . The pressure is calculated with the angular spectrum approach using an input normal particle velocity plane (dash-dot line) and an input pressure plane (dashed line). For this result, the input particle velocity and pressure planes are both located at and truncated with a square window.

RMS output errors obtained with the element array evaluated in 3D volumes as a function of the input pressure plane location . The size of the input pressure plane is (a) and (b) . The location of the input pressure plane ranges between and with an increment of .

RMS output errors obtained with the element array evaluated in 3D volumes as a function of the input pressure plane location . The size of the input pressure plane is (a) and (b) . The location of the input pressure plane ranges between and with an increment of .

Axial pressures simulated with the angular spectrum approach using input pressure planes that are truncated by square windows of different sizes. The reference pressure is indicated by the solid line, the output pressure computed with the angular spectrum approach using a input pressure plane is represented by the dash-dot line, and the axial pressure computed with the angular spectrum approach using a input pressure plane is represented by the dashed line. The solid line and the dashed line are nearly coincident, which indicates that is sufficiently large for the phased array in Fig. 2.

Axial pressures simulated with the angular spectrum approach using input pressure planes that are truncated by square windows of different sizes. The reference pressure is indicated by the solid line, the output pressure computed with the angular spectrum approach using a input pressure plane is represented by the dash-dot line, and the axial pressure computed with the angular spectrum approach using a input pressure plane is represented by the dashed line. The solid line and the dashed line are nearly coincident, which indicates that is sufficiently large for the phased array in Fig. 2.

RMS output errors plotted as a function of the extent of the input pressure plane. The input pressure plane for angular spectrum simulations is located at and truncated by square windows, where ranges from to with an increment of . The two markers indicate the values of shown in Fig. 7.

RMS output errors plotted as a function of the extent of the input pressure plane. The input pressure plane for angular spectrum simulations is located at and truncated by square windows, where ranges from to with an increment of . The two markers indicate the values of shown in Fig. 7.

RMSE values in the input pressure plane plotted as a function of the number of abscissas. The input pressure plane is located at and truncated by a window. The input pressure is computed with the Rayleigh–Sommerfeld integral using to abscissas and with the fast nearfield method using two to ten abscissas.

RMSE values in the input pressure plane plotted as a function of the number of abscissas. The input pressure plane is located at and truncated by a window. The input pressure is computed with the Rayleigh–Sommerfeld integral using to abscissas and with the fast nearfield method using two to ten abscissas.

RMSE values for 3D pressure field outputs plotted as a function of the number of abscissas used for input pressure calculations. The input pressure is calculated with the Rayleigh–Sommerfeld integral using to abscissas and with the fast nearfield method using two to ten abscissas. The errors obtained from both methods approach the same limiting value, but the fast nearfield method achieves convergence with far fewer abscissas.

RMSE values for 3D pressure field outputs plotted as a function of the number of abscissas used for input pressure calculations. The input pressure is calculated with the Rayleigh–Sommerfeld integral using to abscissas and with the fast nearfield method using two to ten abscissas. The errors obtained from both methods approach the same limiting value, but the fast nearfield method achieves convergence with far fewer abscissas.

Axial temperatures computed with the BHTE for power depositions calculated with the angular spectrum approach using different input planes. The pressures are generated by the element planar phased array in Fig. 2, which is electronically focused at . The temperature obtained from the reference power deposition is indicated by the solid line, the temperature obtained from the power deposition calculated with the angular spectrum approach using the input normal particle velocity plane is indicated by the dash-dot line, the temperature obtained from the power deposition calculated with the angular spectrum approach when the input pressure is computed with the Rayleigh–Sommerfeld integral using abscissas is represented by the dashed line, and the temperature obtained from the power deposition calculated with the angular spectrum approach when the input pressure is computed with the fast nearfield method using two abscissas is represented by the dotted line with markers. The result obtained with the angular spectrum approach where the input pressure is computed with the fast nearfield method is nearly coincident with the reference temperature field, whereas the other simulated temperature fields contain noticeable errors.

Axial temperatures computed with the BHTE for power depositions calculated with the angular spectrum approach using different input planes. The pressures are generated by the element planar phased array in Fig. 2, which is electronically focused at . The temperature obtained from the reference power deposition is indicated by the solid line, the temperature obtained from the power deposition calculated with the angular spectrum approach using the input normal particle velocity plane is indicated by the dash-dot line, the temperature obtained from the power deposition calculated with the angular spectrum approach when the input pressure is computed with the Rayleigh–Sommerfeld integral using abscissas is represented by the dashed line, and the temperature obtained from the power deposition calculated with the angular spectrum approach when the input pressure is computed with the fast nearfield method using two abscissas is represented by the dotted line with markers. The result obtained with the angular spectrum approach where the input pressure is computed with the fast nearfield method is nearly coincident with the reference temperature field, whereas the other simulated temperature fields contain noticeable errors.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content