Fast Pressure
Field Simulations of Large Ultrasound Phased Arrays


Results

Introduction
Spherical
ultrasound phased arrays comprised of circular transducers are being evaluated
in simulation studies for thermal therapy. These arrays, which have a
diameter of 16cm, are harmonically excited at 1MHz. The apertures of these
arrays are sampled every 1.5 wavelength to avoid grating lobes. The pressure
fields are sampled twice per wavelength to avoid spatial aliasing during
temperature calculations. These result in a compact array model with more
than ten thousand sources and over ten million observation points in a three
dimensional field. The Fast Nearfield Method (FNM) is combined with the
Angular Spectrum Approach (ASA) to quickly and accurately compute the
acoustic pressure fields generated by these arrays. The fields are propagated
through the computational grids using a 2D FFT, where the source fields are
computed from FNM. This approach achieves a significant reduction in the
computation time. The numerical errors
in simulations using ASA are determined by the sampling and extent of the
spatial grids and by the propagation distance. The errors are then
characterized in terms of the corresponding changes in the computed
temperature response encountered in bioheat transfer simulations. Array Model
A spherical phased
array of 16cm diameter and
15cm curvature is populated with 11,845 circular sources with diameter λ/2. The centercenter spacing between
adjacent elements is λ. The excitation
frequency is 1MHz.
Modeling methods
Fast near field method (FNM)
The formula to
computes the pressure field generated by a harmonically excited circular
transducer with radius a is
The 1D integral over ψ is evaluated by Angular Spectrum Approach (ASA)
This approach takes the known twodimensional field generated by the array at plane A, evaluates the Fourier transform, multiplies by a propagation term H, and obtains the spectrum in plane B parallel to plane A, assuming the wave propagation satisfies the Helmholtz equation. An inverse Fourier transform of the result reconstructs the field at plane B. H also contains the linear attenuation effect through the trigonometric function in the exponent.
Figure 3. Source plane in ASA calculation. The red circle illustrates the array. The dark blue grid indicates the square plane A where the pressure field is known. The light blue region is the zeropadding area, discretized at the same rate as in plane A. The pressure is
evaluated at MXM points at source plane A. The dimension L of the source
plane determines how much spatial information is included. Zero padding
prevents the wrapround error caused by circular convolution and increases
the spatial samples to N in each direction. Small sampling δ (L/M) is required to avoid spatial
aliasing. The discretized angular frequencies are
Results
The ASA error increases
as the field propagates away from the source plane. By including multiple
source planes the error is reduced. Two methods are used to evaluate the
error at each plane parallel to z axis. The results are plotted in Figs 7 and
8. One is the peak normalized error, which is the absolute value of
the difference between the ASA result and the reference normalized by the
peak value in 3D field. The other is the root mean square error, which
is the root mean square error in each plane normalized by the peak value in
that plane. Fig 4. Pressure field generated by the spherical Fig 5. Temperature field generated by the spherical phased array evaluated in focal plane (z=15). phased array
evaluated in focal plane (z=15). Fig 6. Pressure field generated by the spherical Fig 7. Temperature field generated by the spherical phased array evaluated in depth plane (y=0). phased array evaluated in depth plane (y=0).
Fig 8. The peak normalized error of the pressure Fig 9. The peak normalized error of temperature in the depth plane (y=0) using five source planes. in the depth plane (y=0) using five source planes. The error is restricted to 4%. The error is restricted to 2%. Fig 10. ASA error in pressure field computation. Left: rms error, right: peak normalized error. The error increases in ASA as the distance from the source plane increases. 5 source planes are placed at z = 7, 10.45, 13.1, 14.6 and 19.7cm, and the error is evaluated in each plane normal to the z direction. The rms error and the normalized peak error are both restricted to 4%. Fig 11. ASA error in temperature field computation. Left: rms error, right: peak normalized error. The 3D pressure field computed with the ASA using multiple source planes is used as the input to the BHTE. The error is evaluated in each plane normal to the z direction. The rms error and the normalized peak error are both under 2%. In a
16cmX16cmX17cm computational
volume with 217X217X227 field points, computing five source planes using the
fast nearfield method takes 1.6 hours. Reconstructing the 3D fields using
angular spectrum approach takes 2 seconds. The error is 4% compared with the
reference computed with the FNM. The errors of temperature response calculated by the bioheat transfer
equation with different thermal parameters are consistently smaller than 2%. Conclusion
Simulations of the
threedimensional pressure field and the temperature response generated by
large ultrasound phased arrays are very timeconsuming. By combining the fast
nearfield method and the angular spectrum approach, the simulation time is
dramatically reduced. In the modeling
of a spherical phased array of 11845 circular sources with 10 million
observation points, the combined approach is 56 times faster than the direct
integral approach, while the pressure result and the temperature response
only differ by 4% and 2% compared with the reference. Reference
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