Simulation of ultrasound two-dimensional array transducers using a frequency domain model Min Rao,a Tomy Varghese, and James A. Zagzebski Department of Medical Physics, The University of Wisconsin���Madison, 1300 University Avenue, 1530 MSC, Madison, Wisconsin 53706 Received 6 December 2007 revised 18 April 2008 accepted for publication 13 May 2008 published 18 June 2008 Ultrasound imaging with two-dimensional 2D arrays has garnered broad interest from scanner manufacturers and researchers for real time three-dimensional 3D applications. Previously the authors described a frequency domain B-mode imaging model applicable for linear and phased array transducers. In this paper, the authors extend this model to incorporate 2D array transducers. Further approximations can be made based on the fact that the dimensions of the 2D array element are small. The model is compared with the widely used ultrasound simulation program FIELD II, which utilizes an approximate form of the time domain impulse response function. In a typical application, errors in simulated RF waveforms are less than 4% regardless of the steering angle for distances greater than 2 cm, yet computation times are on the order of 1/35 of those incurred using FIELD II. The 2D model takes into account the effects of frequency-dependent attenuation, back- scattering, and dispersion. Modern beam-forming techniques such as apodization, dynamic aper- ture, dynamic receive focusing, and 3D beam steering can also be simulated. �� 2008 American Association of Physicists in Medicine. DOI: 10.1118/1.2940158 Key words: ultrasound, simulation, 2D array transducer, beam pattern, b-mode image I. INTRODUCTION Recently, two-dimensional 2D array transducers have been studied extensively for applications in real time three- dimensional 3D ultrasound imaging. A problem facing the development of 2D arrays is the complexity arising from the large number of array elements required in such transducers and the channel count needed for the ultrasound system. Thus, there have been a number of reports describing design, fabrication methods, and characterization of 2D transducer arrays.1���4 To optimize image quality, many different array layouts and imaging techniques have been studied and applied.5���8 Due to the complexity of implementing such sys- tems, however, accurate and efficient simulation models are needed to evaluate beam properties and to determine opti- mum parameters for these devices. Models for computing transducer field patterns are thus essential in 2D array design. Transducer field simulation methods can be divided into two categories: time domain and frequency domain models. Time domain simulations are based on utilizing the impulse response of a radiator at the chosen field points.9,10 Closed form expressions for the impulse response function have been derived for many different geometries.11,12 Since these functions are exact solutions, they should yield accurate re- sults for the acoustic field. However, the impulse response generally exhibits discontinuities, which leads to the need for high temporal sampling rates to obtain accurate results for the field. Jensen and Svendsen13 have simplified this method to reduce the computational complexity and have imple- mented their approach in the FIELD II program, a popular, linear propagation beam calculation program. Turnbull and Foster14 have also performed an extensive study of beam patterns from 2D arrays using the time domain method. Frequency domain simulation techniques are based on diffraction theory15 for continuous waves. Various approxi- mations can be made to reduce the computational complex- ity, and these lead to different degrees of accuracy and effi- ciency. Crombie et al.16 studied the accuracy and efficiency of several beam simulation schemes. They concluded that in the frequency domain, the Fresnel approximation15 yields the most accurate results for an unsteered array however, the accuracy degrades with increasing steering angle. Li and Zagzebski17 have developed a frequency domain B-mode imaging model for linear array transducers that is based on a less restrictive approximation than the Fresnel approxima- tion. They also showed that this model yields accurate simu- lation results throughout the near field and far field, even when the beam is steered at large angles. In this paper, we describe extensions of the 1D model developed by Li and Zagzebski17 to study beam patterns and properties of B-mode images for 2D transducer arrays. We briefly describe the frequency domain model used for the simulation of beam patterns from 2D arrays. The accuracy of the model is demonstrated by comparison with results from FIELD II simulations. This model takes into account the ef- fects of frequency-dependent attenuation, backscattering, and dispersion. Imaging techniques such as apodization, dynamic aperture, dynamic receive focusing, and 3D beam steering can also be simulated. II. THEORY II.A. Acoustic field of a single square element For 2D ultrasound arrays, we assume that both the trans- ducer element height and width are limited by the half- 3162 3162 Med. Phys. 35 ���7���, July 2008 0094-2405/2008/35���7���/3162/8/$23.00 �� 2008 Am. Assoc. Phys. Med.

wavelength constraint in order to avoid grating lobes in the 3D radiation field. In a homogeneous medium, if the trans- ducer surface is embedded in an infinite rigid baffle, and is vibrating at an angular frequency i.e., the velocity of the surface is v t =u e-i t , the pressure field from a square element can be written as18 pi r, = - i kcu 4 A0 r, , 1 where A0 r, = -a/2 +a/2 -a/2 +a/2 eik r-r r - r dx dy . 2 Here is the density of the medium, c is the speed of sound, k= / c is the wave number, r is the field point, x , y , or r denotes the coordinates of the source point on the surface of the transducer, and a is the dimension of the square element. The coordinate system for 2D array calculations is shown in Fig. 1. The task in any model for the beam from a transducer is to compute the 2D Rayleigh integral, such as described in Eq. 2 , as efficiently as possible. Define r to be the distance from the field point to the center of the element. Under the assumption that r a, r -r can be expanded by r - r = x - x 2 + y - y 2 + z2 r + 1 2r x 2 + y 2 - 2yy - 2xx , 3 where x, y, and z are the coordinates of the field point. We can take advantage of the fact that if a / 2, terms on the order of x 2 / 2r are negligible for the integration in Eq. 2 . This will introduce a phase error of k x 2 +y 2 / 2r ka2 / 4r / 8r at most, which is smaller than 0.01 rad in the case when r is greater than 1 cm and the wavelength is around 250 m 1500 ms-1 / 6 MHz . Therefore, r - r r - 1 r yy + xx . 4 Thus, Eq. 2 can be rewritten as A0 r, a r eikrsinc kxa 2 r sinc kya 2 r . 5 The 2D Rayleigh integral now can be calculated using sinc functions. The computation time is greatly reduced using this approximation. Note that the y integration must be done numerically when simulating 1D transducer arrays because the height of rectangular elements used in these devices usu- ally is on the order of 1 cm much larger than and the term y 2 / 2r cannot be ignored. II.B. Acoustic field from linear 2D transducer array elements The field from 2D square array elements, which consist of N N active elements, can be calculated using the following equations: A r, = m=1 N n=1 N a m,n A0m,n r, e-i t m,n,Fx,Fy,Fz , 6 where a m,n is an apodization factor for the nth element in the mth row. The apodization factor can be as simple as Gaussian shaped, with maximum weighting of signals picked up from the center of the active aperture and the weighted falling off for elements at increasing distances from the cen- ter of the aperture. The parameter t m,n,Fx ,Fy ,Fz is the time delay needed to steer and focus the beam at a spot Fx ,Fy ,Fz . This can be written as t m,n,Fx,Fy,Fz = 1 c n - N + 1 2 d - Fx 2 + m - N + 1 2 d - Fy 2 + Fz 2 - Fx 2 + F2 y + Fz 2 , 7 where d is the center-to-center distance between elements. II.C. Generation of the RF signal For pulse-echo imaging, under the assumption of local plane waves and the Born approximation, the total backscat- tered force from multiple scatterers detected by the trans- ducer can be calculated using the following equation:17 F = - i 4 kcu j=1 M AT rj, AR rj, , 8 where M is the total number of scatterers, quantifies the scattered amplitude, AT r, models the transmitted a a r r��� r r��� ��� z y x FIG. 1. Coordinate system used for computing the acoustic field from a square element. r is the field point, r denotes the coordinates of the source point on the surface of the transducer, and a is the dimension of the square element. 3163 Rao, Varghese, and Zagzebski: Simulation of 2D array transducers 3163 Medical Physics, Vol. 35, No. 7, July 2008