Simulation of ultrasound two-dimensional array transducers using a frequency domain model.

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Medical Physics ()
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Abstract

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, backscattering, and dispersion. Modern beam-forming techniques such as apodization, dynamic aperture, dynamic receive focusing, and 3D beam steering can also be simulated.

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Simulation of ultrasound two-dime...

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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

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