A TimeSpace Decomposition Method for Fast Calculations of Transient Pressure Fields Generated by Ultrasound Phased Arrays
James F. Kelly and Robert J. McGough^{1}
^{1}Dept. of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA kellyja8@msu.edu
INTRODUCTION
Computing the
transient pressure field generated by large phased arrays is helpful in
designing ultrasound imaging systems.
Methods for computing the transient nearfield pressure of a planar
aperture include the pointsource method [1] and the spatial impulse response
(SIR) method [2,3,4]. Recently, a rapid
single integral approach has been developed for computing the time domain
pressure generated by a baffled circular piston [5]. This solution is applied to a 129 element focused phased array
and compared to
a similar computation made using Field II [6].
THEORY
A
time domain solution to the lossless wave equation can be derived subject to an
input pulse v(t), which models the uniform normal velocity of the
piston. Consider a baffled rigid piston
with radius a radiating into a homogeneous fluid with density _{} and sound speed c. Solving
the wave equation in cylindrical coordinates (r,z) yields the
singleintegral solution:
_{}
Although Eq. (1) can be directly evaluated in
the timedomain using Gauss quadraure, the computational complextiy is significantly
reduced by decoupling the temporal and spatial dependence in the integrand of
Eq. (1). Consider a Hanningweighted
pulse v(t) with duration W, which is decomposed via
_{}
where the delay t depends on the spatial coordinates (r,z) and the variable of
variable of integration y.
Inserting Eq. (2) into Eq. (1) allows the temporal and spatial
dependence to be seperated. The
expansion functions in Eq. (2) are computed via trigonometric expansions. This decompositon reduces the number of
integrations per observation point from the number of time samples to 6 without
introducing any additional error.
A linear array of 129 circular elements is simulated with pistons of radius a = 0.30 mm (half –wavelength) and interelement spacing d = 0.90 mm. Fig. 1 shows the array geometry. A computational grid using quarterwavelength (0.15 mm) spatial sampling in both the lateral and axial dimensions was employed. The Hanning excitation pulse has a central frequency of 2.5 MHz and duration of 1.2 microseconds. Beam steering and focusing is achieved by application of temporal delays to each element [7]. The total pressure is then synthesized via superposition. Beam steering and focusing are achieved by applying temporal delays. To focus on axis at distance F = 50 mm, quadratic delays are employed.
Fig 1: Densely
sampled linear array. The array used in
these computations has 129 circular pistons with radius a = 0.30 mm
(half –wavelength) and interelement spacing d = 0.90 mm on a computational
grid extending 288 wavelengths in lateral direction by 167 wavelengths in axial
direction.
RESULTS
To
determine the correct number of quadrature points to apply to the decomposition
technique given by Eq. (1), an error analysis is shown in Table 1. As Table 1 shows, Eq, (1) requires 4 Gauss
abscissas to achieve a 1 % peak error.
Field II, which subdivides the aperture into rectangular subelements,
requires 484 subelements to achieve this error level.
Simulation
pressure fields are shown in Fig. 2 at three successive times, where the
pressure has been normalized with respect to peak pressure. The total computation time for this array
system was 11 minutes. In comparison,
similar computations using Field II software[a]
[6] took approximately 8 hours to achieve commensurate accuracy. The peak field error is computed relative
to a 1000 point sampling frequency of 32 Hz (compared to
Field’s 100 reference
field; a 4point quadrature yields a peak error below 1% at all points in the
computational grid. Since the present
decomposition technique utilizes a temporal MHz sampling), less memory is used.
Fig 2: Normalized pressure field at three successive times for the phased array defined in Fig. 1. Pressure is normalized with respect to the peak value Onaxis focusing at 50 mm is employed via quadratic time delays.
Table 1. Singleelement error analysis.

10 % Error 
1 % Error 
TimeSpace Decompositon 
3 abscissas 
4 abscissas 
Field II 
12 subelements 
484 subelements 
DISCUSSION
Fast and accurate incident pressure
field compuations are necessary in several applications. Of particular importance
is the interative design of imaging arrays (both 1 and 2D). Array geometry and parameters can be
optimized by computing transmitted and pulseecho pressure fields. Largescale modeling of wave propagaton and
scattering can also benefit from the fast method presented. Timedomain scattering methods, such as
generalizations of the fast multipole method (FMM) [8] require incident field
data on large, unstructed grids as an input.
CONCLUSION
A simulation scheme for pulsed computations with linear phased arrays has been proposed. Unlike previous methods [6], far field and aperture approximations are not used; instead, an exact timedomain solution forms the basis for array simulations, which is accelerated by decomposing the spatial and temporal dependence of the integrand.
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ACKNOWLEDGEMENTS
This research was funded in part by NIH grant 5R01CA093669. The authors thank Xiaozheng Zeng, Shanker Balasubramaniam, and Liyong Wu for helpful discussions and comments.