CC BY
180Electronic Journal «Technical Acoustics» http://www .ejta.org
2007, 14
Hamid Behnam1, Elham Mokhtar Tajvidi2
1 Biomedical Engineering group, Electrical Engineering Dept., Iran University of Science and Technology, Tehran, Iran, email: [email protected]
2 Biomedical Engineering Dept., Islamic Azad University, Science and Research branch, Tehran, Iran
Simulation of acoustic fields from 2D-array transducers with continuous wave excitation in homogeneous medium
Received 29.05.2007, published 09.07.2007
2D-Array transducers are used in real-time 3D ultrasound imaging. Appropriate design of the received and emitted beam pattern from 2D-Array transducers is the aim of simulation of the beam pattern. In most methods of the simulation of acoustic fields, the far-field approximation was applied in order to increase the speed of the calculation. In this paper a new method that without applying any approximation converts the two dimensional Rayleigh integral over the transducer surface to one dimensional integral expression is used. It has shown proper speed and good accuracy both in near-field and in far-field. So it will be useful in intravascular sonography or low depth imaging, where the desired tissue is very near to the transducer. Also it can be used for optimizing array parameters in designing stage and also for education and research purposes. Simulation results are shown for different parameter settings to evaluate the effects of every parameter on the field of 2-D arrays.
INTRODUCTION
The role of the acoustic transducer in high quality image acquisition from medical ultrasound scanners is very important. In fact it determines the quality of an image. There are considerable efforts in designing transducers and determining the characteristics of the emitted field.
The goal of the 3D imaging is observing the real structure of the body organs. 2D-Array transducers are used to acquire real-time 3D images; in echocardiography, in monitoring fetus and in urology. Using 2D-Array eliminates the need of physical movement of 1D array for acquiring 3D images. It is able to focus acoustic beam in the elevation direction in comparison with 1D array that use lenses to do this. Calculating and simulating the acoustic fields are simpler and more economical than experimental tasks in order to determine the field pattern emitted from acoustic transducers.
There are different methods [1] for simulating the acoustic fields that most of them are based on the Rayleigh integral [2-12].
Approximate methods for quickly predicting the behavior of transducers especially when the observation point is beyond the near-field zone, have traditionally played an important role in design. Two approximations that have been widely used for sinusoidal excitation are based on the Fraunhoffer (far-field) and Fresnel (mid- and far-field) approximations.[13]
In this paper a new software will be introduced, which is more efficient and has no approximation for near-field, that calculates the ultrasound field from array transducers. Here we calculate the Rayleigh integral by considering the points on the transducer surface that have the same distance from the observation point, so it reduce the double integral to a onedimensional integral expression. This technique has been described by Schoch [14] and Ohtsuki [15] for impulse response and continuous wave excitation, respectively.
In sonography sometimes we interested in imaging the tissues very near to the transducer, especially in intravascular sonography, so it will be very helpful to develop a calculating method that has a good accuracies both in near-field and in far-field.
1. “RING FUNCTION” AND CALCULATING THE ACOUSTIC FIELD
To determine the pressure field produced by a vibrating source, the appropriate wave equation with proper constrains must be solved to meet the boundary conditions defined by the problem. The Rayleigh equation, a mathematical expression for the Huygens' principle, is one of the solutions for the wave equation in a homogeneous media. In this equation a flat transducer with piston like vibration is assumed. The Rayleigh integral (1) means that the source can be considered to consist of a set of point sources that produce spherical waves whose superposition construct the resulted signal at the observation point:
r ds
P = (t - ^ (»
where P is acoustic pressure, p is the density of the medium in which the ultrasound wave propagates, c is the sound speed in the medium, r is the distance of the observation point from the transducer, V' the is time derivative, ST is the area of the transducer.
It states that the acoustic pressure at the observation point can be found by integrating the time derivative of the normal component of the vibrating velocity over the entire surface of the transducer. Direct use of the Rayleigh integral requires the numerical evaluation of a double integral over the transducer surface. Above equation can also be used approximately in cases of transmitter with curved surfaces, if the dimension of the vibrating surface is smaller than its radius of curvature.
In case of a transmitter with a plane or spherical surface, Eq. (1) can be transformed into one-dimensional integral of r as
r2
r
c
P = p [ V'(t —)R(r)dr . (2)
j r
The Ring function is defined as a weight function proportional to the transducer arc that has the same distance from the observation point [14]:
(3)
In Eq. 2, r1 and r2 are the maximum and the minimum distance from an observing point O to the transmitter, respectively.
When the orthogonal from the observation point to the plane of the transmitter in fig.1 placed into the transmitter, the circle is contained in it, and so the R(r) is unity.
Adding some terms, this method can be applied to the case of absorptive media. The ring function can be used for transmitter with linear and circular boundary or combination of them.
2. SIMULATION
We should be able to change the transducer parameters and graphically show the beam pattern pressures that produce by these transducers, in order to optimize the design of the 1D and 2D array transducers for the particular application.
The calculation are performed by C program and because of the flexibility of the Matlab in presenting data in 2D and 3D figures, data have been shown in this environment.
According to the symmetry of the fields around the element (in order to decrease the amount of calculation), the results of calculations from single rectangular element were shifted in elevation and azimuth directions by considering the number of elements, size of elements and element spacing in both directions and the acoustic field from 1D or 2D arrays were calculated.
It will be possible for the user to set the desired parameters of the transducer and run the simulation process, then study the emitted beam pattern with high resolution in the main, side and grating lobes in both near-field and far-field.
transducer
Fig. 1. Plane of the single element disk transmitter and the observation point and definition of the Ring function
3. EFFECTIVE AND VARIABLE PARAMETERS IN THIS SOFTWARE
- Number of elements (1D & 2D array) : they are variable in both elevation and azimuth directions.
- Size of the elements: width and length of the individual piezoelectric elements.
- Element spacing: spaces between two adjacent elements in both elevation and azimuth directions.
- Frequency: the frequency of the emitted ultrasound wave.
- Sound velocity of the medium: the velocity of the sound propagation in the medium.
- Amplitude of excitation.
4. RESULTS
The array is located on the xy plane, and the results of the acoustic field are shown in the xz plane, which is orthogonal to the middle of xy plane, Fig. 2. The calculated pattern in the results are from 2D-Array transducer, which is defined in table 1 unless the other parameters are stated.
Table 1. Parameters of 2-D transducer array
Number of elements 16x16
Size of Element 0.4x0.4 mm
Element spacing 0 mm
Sound velocity in the medium 1540 m/s
Density of the medium 1000 kg/m3
Frequency 3.5 MHz
\
\/ |lhlHL
z
Fig. 2. 2D-Array transducer on the xyz coordinate and the plane of receiving acoustic field
4.1. The effect of increasing frequency and size of the transducer
In this section we just increase the frequency, without changing the other parameters of table 1 in order to study the effect of the frequency on the beam pattern. The amount of divergence of the far filed [16] is shown as
— c
sin# = 1.22— = 1.22, (4)
D fD yV
where — is the wavelength, c is the sound velocity in the medium, f is the frequency and D
shows the diameter of the transducer.
By increasing the frequency in the above equation, the main lobe wide will reduce.
The length of the near filed [16] can be calculated by
D2 D 2f
x «-----=--------, (5)
4— 4c
which shows that, by increasing the frequency, the maximum pressure point distance from the transducer will increase.
Fig. 3 shows the results for 1 MHz, 3.5 MHz, 10 MHz and 15 MHz frequencies, which are increased respectively. Fig. 4 shows the results of increasing the size of the element. The size of the element in fig 4 are 0.2x0.2, 0.4x0.4 and 0.6x0.6, respectively. As Eq. 5 shows by increasing the size of the element, increase in the maximum pressure distance were observed.
'20: 40 60 80 100 120 140 160 180
z(.4mm)
Fig. 3. Calculated pattern of 2D-Array transducer: effect of increasing frequency. The frequencies from top left to button right are 1 MHz, 3.5 MHz, 10 MHz and 15 MHz, respectively
20 40 60 80 100 12D 140 160 180
z(.4mm)
'10 20' -30 40 50 60 70 80 90 100 110
z(mm)
.20 40 60 80 100 120 140 160 180
z(.4min)
Fig. 4.
Effect of increasing the size of the elements, which from top to bottom are 0.2x0.2, 0.4x0.4 and 0.6x0.6 mm, respectively
4.2. The effect of increasing the number of elements and elements spacing
By increasing the number of elements, the effective transducer diameter will increase so the main and side lobes and the distance of the maximum pressure point were increased. The results are shown in fig. 5.
Space between two adjacent elements is not a variable parameter in Ultrasim software (as a reference for calculating the acoustic fields which is available in IEEE site). Existence of grating lobes which is shown in fig. 6 is the effect of increasing this parameter. Number of elements are 64 and the other parameters are chosen from table 1. The element spacings were set to 0.2, 0.3 and 0.4 mm, respectively.
50 100 150 200 250 300
41mm)
140
120
100
f 00 E
H 60 40 20
10 20 30 40 50 60
z(15mm)
20 40 60 80 100 120 140 160 100
z(.4mm)
Fig. 5.
The effect of increasing the number of elements which from top to bottom are 8x8, 16x16 and 32x32, respectively
4.3. The effect of increasing sound velocity in the medium
Considering the Eq. 5, by increasing the sound velocity in the medium, the maximum pressure point position will move toward the transducer. Fig. 7 illustrates this effect.
(a) (b)
Fig. 7. Calculated pattern of 2D-Array transducer considering different sound velocity in the
medium
(a): soft human tissue with c = 1540m / s , (b): hard tissue with c = 4080m / s
5. DISPLAYING OF ACOUSTIC FIELD
The results of calculation are saved totally for 3D space, so they can be shown parallel to the array in every distance from it and orthogonal to the array in desired plane as well as its 3D shape, Fig. 8.
z=0.3 mm
z=l mm
z=2 mm
z=5 mm
Fig. 8. Calculated pattern for defined array in Table 1, for 64 elements, with the size of every element 0.2 mm. In left side from top to bottom the parallel plane to the array are shown, the distances are z=0.3 mm, z=1 mm, z=2 mm and z=5 mm respectively.
In the right side, the 3D shapes of these planes are shown
CONCLUSIONS
The presented paper introduced simulation method for calculating the ultrasound field of arrays. The method uses the exact calculation of the Rayleigh integral for a transducer element in an array and then by proper shifting of the result obtains the field of other transducer elements and then adds the effects of all elements to construct the whole array field.
This software can simulate the emitted beam pattern pressure from 1D, 1.5D [17] (have the same size of elements in rows), and 2D rectangular arrays. It is able to show the effect of changing the parameters of the array including the number and size of elements, inter element space, frequency, excitation amplitude, sound velocity and density of the medium in which the sound propagates.
The effects of changing the array parameters are studied. By increasing the size of each array element, main and side lobes and the distance of the maximum pressure point were increased, and the side lobes angles with the main lobe are reduced. When frequency increased, reduction of the main lobe wide, increase of the distance of the maximum pressure point and reduction of the side lobes angles with the main lobe were observed. The distance of the maximum pressure point and the wide of the main lobe were increased and the wide of side lobes were decreased, by increasing the number of elements of the 2D-array. Increasing the inter elements space will cause production of grating lobes.
This software can be used for optimizing array parameters in designing stage and also for education and research purposes. Further works are under way for considering pulse wave excitation of the transducer, steering, focusing and apodization.
REFERENCES
[1] IEEE Ultrasonics, Ferroelectrics and Frequency Control web site for software package and utilities for ultrasound research:
http://www.ieee-uffc.org/index.asp?page=ultrasonics/software/index.html&Part=3
[2] Jorgen Arendt Jensen. Users’ guide for the Field II program. August 2001.
[3] Interactive ultrasound field simulation from the University of Oslo http://www.ifi.uio.no/~ultrasim
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[13] Foundations of Biomedical Ultrasound (Biomedical Engineering Series) Oxford University Press, USA; 1st edition, 2006.
[14] Schoch, A. Betrachtungen uber das schallfeld einer kolbenmembran. Akust. Zeitschr., 1941, 6, 318-326.
[15] Ohtsuki, S. Ring function method for calculating near field of sound source, Bull. of the Tokyo Inst. of Tech., N°123, 23-27, 1974.
[16] Thomas S. Curry, James E. Dowdey, Robert C. Murry, Edward E. Christensen. Christensen’s Physics of Diagnostic Radiology. Lea & Febiger Publishing, 4-th edition, 1990.
[17] G. Rizzatto. Evolution of Ultrasound Transducers: 1.5-D and 2-D Arrays. European Radiology, 9 (Suppl. 3), S304-S306. Springer-Verlag, 1999.
