Compurerized Medical Imaging and Graphics, Vol. 17, Nos.415, pp. 339-343, 1993 Printed in the U.S.A. All xi&s reserved. 089.561 I l/93 $6.00 + .Xl copyright 0 1993 Per&mm Press Ltd. 3D MODELING OF PHASED ARRAY GENERATED ULTRASOUNDS IN LOSSY MEDIA Alessandro Sarti, Paolo Bassi, and Claudio Lamberti* Dipartimento di Elettronica, Informatica e Sistemistica, Universita’ di Bologna.Viale Risorgimento 2., 40 136 Bologna, Italy. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO (Received 25 March, 1993) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO Abstract-A general algorithm for three-dimensional (3D) modeling of acoustic fieldsgenerated by phased array transducers and propagating in uniformly lossy media is introduced and illustrated also with the help of specific examples. Applications of the method are foreseen in the analysis and the design of transducers for echographic applications in order to evaluate and/oroptimize their performances. Key Words: Ultrasounds, Numerical simulation, Echocardiography, Transducer modeling, Acoustic fields, 3D visualization, Lossy media INTRODUCTION Detailed knowledge of shape and amplitude of ultra- sonic fields propagating through biological tissues could substantially improve transducer design strategies for better quality and accuracy of the echo images. Global image quality mainly depends on two parameters, namelythe geometrical resolution, i.e. the minimal distance between two surfaces that can be detected by different echoes, and the densitometric resolution, i.e. the ability to recognize two different non-echogener- ating structures. Such parameters are highly influenced by the spatial shape of the generated ultrasonic field and by the material properties whichaffect the field shape during propagation. These considerations suggest the importance of ultrasound propagation modelling to simulate different design conditions. Modelling of ultrasound propagation has been deeply investigated in the past years. Computation of the pressure field radiated by a flat piston source, vi- brating in an infinite rigid baffle, has been investigated in the literature, usingdifferent degrees of approxi- mation.The simplest model assumes the transducer to be much bigger than the wave length so that, ne- glecting diffraction, the geometrical optics rules can be adopted. A less approximate model includes diffraction. Suchanalysis assumes steady state purelysinusoidal excitation. Solutions have been obtained through Ray- leigh double integrals in frequency domain (l), double * Correspondence should be addressed to Claudio Lamberti, Dipartimento di Elettronica, Informatica e Sistemistica, Universita’di Bologna. Viale Risorgimento 2. 40 136 Bologna, Italy. Ring integrals when circular sources have been adopt (2), Schock integrals (3) or convolution integrals (4). These methods have been exhaustively analyzed from a theoretical pointof view (5, 6). Finally, sinusoidal excitation, modulated by short pulses with arbitrary temporal dependence, has been considered. This is the case of recently developed B- modeechography devices. Solutions can be found in the frequency domain by using Fourier transforms and the previously introduced harmonic methods, or in the time domain solving a surface Rayleigh integral (4,7). Literature on the ultrasonic field propagating through non-attenuating media is exhaustive, while only few papers consider attenuation. In the latter case, results have been obtained considering transducers with simple anular or circular geometry (8-lo), while only recently transducers with arbitrary geometry have been considered (11). We have considered phased array transducers, because of their versatility and reconfigu- rability. Theirbeamshape and direction can, in fact, be controlled electronically and then be easily adjusted according to the user’s requirements. The papers mentioned have produced no more than the visualization of a section of the ultrasonic field, even though the field computation can easily be performed with reference to a 3D volume instead of a 2D section. Unfortunately, such visualization is satis- factory only in the case of circular and anular trans- ducers producing an axially symmetric field. For rec- tangular or phased-array transducers and mainly for multilayered media with curved surfaces, a full under- standing of field generation can only be obtained through a 3D calculation and visualization. 339