ULTPASONIC IMAGING 9, 203-219 (1987) A TAYLOR SERIES EXPANSION FOR TIME SAVINGS IN ACCTJRATE COMPUTATION OF FOCUSED ULTRASOUND PRESSURE FIELDS Timothy J. Hall, Ernest L. Madsen and James A. Zagzebski Department of Medical Physics University of Wisconsin Madison, WI 53706 A model for the continuous wave (cw) pressure beam dist,ribution of a focused axially symmetric ultrasonic radiator with constant radius of curvature involves integration of eikr/r over the surface of the radiator. (Ic is the complex wave number and T is the distance from a radiating area element to the field point.) A single integral form exists, and it is this form that, is expanded in a Taylor series in frequency. Thus, when representing a pulse as a superposition of cw beams, the need to do numerical integrations for each one of a large number of frequencies is eliminated. Accuracy of the truncated Taylor series depends on the coordinates of the field point as well as on the difference between the frequency of interest and the reference frequency. Accuracy criteria for a particular application are also presented. The computer time savings for our applications correspond to a factor of about 60 with accuracy being maintained. D 1987 RCademLCmess, Inc. Key words: Computations; focused; pressure field; transducer; ultrasound. 1. INTRODUCTION Much of the research in our laboratory includes calculation of the acoustic pres- sure field resulting from pulsed focused or nonfocused transducers. For example, in a quantitative B-mode image texture simulation [1,2], determination of the pressure for a matrix of spatial positions and over a broad range of frequencies is required. These calculations involve numerical integration and can be very time consuming. A 2 cm x 2 cm image texture simulation for a 3.5 MHz single element focused transducer excited by a broadband pulse involves calculating the continuous wave pressure distributions for a matrix of about 5,000 spatial positions and 110 frequencies in the range from 1.5 MHz to 5.5 MHz. Using an experimentally-verified single integral expression for the acoustic pressure at a field point [3], calculation of the 550,000 complex pressure values requires about four hours on a VAX 785. A similar pressure calculation is em- ployed in the method of data reduction for estimating acoustic backscatter coefficients used in our lab [4]. We previously developed and reported [5] a Taylor series expansion in frequency of an integral expression for the acoustic pressure at a point in the field of a nonfocused, disc-shaped transducer. In the present work, we present a more general Taylor series expansion of this expression for application to single element focused or nonfocused radiators with axial symmetry. A formula which allows easy determination of an un- limited number of terms is also presented. 0161-7346/87 $3.00 Copyright 0 1987 by Academic Press, Inc. 203 All rights of reproduction in any form reserved. HALL ET AL Time savings realized by applying the Taylor series depends on the specific appli- cation. For the texture simulation example above, pressure calculation time is reduced to about four minutes; a time savings factor of about sixty. Similar time savings result in determining backscatter coefficients. It should be noted that this time savings factor applies also to the work reported in reference [5] on nonfocused transducers. In the latter paper, a time savings factor of seven was estimated. Inefficiency in a comput,er program section, discovered after publication, led to this discrepancy. A set of criteria are reported for a specific a.pplication that allow systematic de- termination of the off-axis distance to which acoustic pressure must be calculated and the number of Taylor series terms required t.o adequately represent the correct pressure calculation. These criteria can be used as a model for other applications. It is important in computing pressure distributions in attenuating media to employ the attenuation coefficient at the frequency of interest. When accurate pulse shapes are required in attenuating media, accurat,ely accounting for the frequency dispersion of phase speeds is also required [6]; t,h is is true even though the percent shift, in phase speed per MHz may be very small. In most laboratories, it is much easier to determine the frequency dependence of the at,tenuation coefficient of a material than to determine the frequency dependence of its phase speed. In the Taylor series expansion developed in this work, the Kramers-Kronig relations, as developed by Ginzberg [i’], are employed to express the frequency-dependent phase speed in terms of the frequency-dependent attenuation coefficient. 2. THEORY 2.1 Representation of a Pulsed Pressure Beam The derivation of the expression for acoustic pressure which is expanded in a Taylor series is presented in reference [3]. For the convenience of the reader, a summary of the relevant aspects is presented below. Consider a pulsed single element piezoelectric transducer. When the (linear) Hehnholtz equation applies, the acoust,ic pressure p(F, 2) for a field point at position r’ and time t, can be represented as the superposition of a set of continuous wave, sinusoidally-varying beams spanning some range of frequencies as follows: p(F+, t) = Re ( lrn B~(w)Ao(F,w)e@tdw) . (1) Re means “Real part of,” B,(w) is a complex superposition coefficient at frequency w and Ao(+‘, w)eeiwt dw is a differential cont,inuous wave solution of the Helmholtz equation for the particular transducer geometry involved. Modeling the transducer as a set of equivalent. monopole sources uniformly dis- tributed across the face of the transducer and acting in unison results in A,,(F,w) having the form (2) 204 TAYLOR SERIES PRESSURE FIELD COMPUTATIONS Fig. 1 Diagram showing the division of the fo- cused transducer field into two regions, I and II. The axis of symmetry of the ra- diating element lies along the horizontal line. The origin of the (+,h) coordinat,e system lies on the axis of symmetry at the center of curvature. Also shown are various parameters involved in the single int,egral form of Ao( F, w) described in sub- section 2.1. where the integration is over the surface S of the transducer’s active element and ? terminates on area element dS”. (Note: Both r; and r’ are position vectors extending from the origin of the coordinate system to the points specified.) k = ICR + Ike is the complex wavenumber in the propagating medium with real and imaginary parts k~ = W/C(W) and ICI = a(w), respectively. The parameter C(W) is the ultrasonic speed and Q(W) is the amplitude attenuation coefficient at frequency w. If the radiating element has the form of a spherical cap with axial symmetry, the double integral in Eq. (2) can be reduced to the sum of a single integral and a nonin- tegral term having the form A is the radius of curvature of the radiating element and s is the distance from the radius of curvature to the field point Q at K (See figure 1 for a pictorial aid.) The quantities C’, D’, y(r’), and F depend only on geometrical parameters and are independent. of frequency and physical properties of the propagating medium. They are defined as follows: C’ E [A2 + sz + 2As cos(& + +)I’, D’ = [A’ + s2 + 2As cos($, - r$)] ’ , h + (4’ - a”): - (r” + s2 - A”)=&+ y(d) z cos-* , and %$k [4&f2 _ (+’ + sZ _ AZ),]; 1 0 for region I of figure 1 F= $ ( eik6 _ ,jkd) for region II of figure I where 6 z and d E A2 + s2 + 2As cos(& - 20.5 HALL ET AL Fig. 2 Diagram showing the simplified coordinate system. The axis of symmetry of the ra- . diating element lies along the horizontal Z line. Because of axial symmetry, a field point can be specified by its coordinates z and z where the origin is at the center of the radiating element. The parameters $+,,d,s, and h are shown in figure 1. Note that the h axis lies on the axis of symmetry of the radiating element and has its origin at the center of curvature of the radiating element. For simplicity, it. is convenient to define a new coordinate system, shown in figure 2, in which a new coordinate z E h - A replaces h. Note that z is not a Cartesian coordinate but is the distance from the z axis to Q and P =I r’I= (z” + .z’)f. In the limit as A -t 00, A/s -+ 1, h/ ( h 1 -+ -1 and the radiating element becomes a nonfocusing disc. In this case, C’ z [(a-z)2+z2]i, D’ c [(a+z~)~+z~]‘, y(4) Et cos-*-a2 2+ ( rfZ+ z2- z2 I c?-‘l -z”) 5 1 7 5 = [(a- z)’ + z”] r and d E z. 2.2 Relationship Between Ultrasonic Phase Speeds and Attenuation Coefficients Even though the fractional shifts are extremely small, the frequency dispersion of the ultrasonic speed in (attenuating) soft tissue is significant when accurate represen- tations of ultrasonic pulse shapes are needed [6]. Ginaberg [7] has developed integral expressions, resulting from causality, relating the frequency-dependent ultrasonic speed to the frequency-dependent attenuation coefficient. Using our notation, Ginzberg’s re- sult is 1 1 -----~“P~m~(~‘)[~-,,r~W02]~~ (4) C(W) c(w) 7r where P means “Principle value of”, C(W) is the speed of sound at frequency w and C(WO) is the speed of sound at reference frequency wO. This is the form of the Kramers- Kronig relations employed in reference [6] yielding excellent agreement between theory and experiment. 206 TAYLOR SERIES PRESSURE FIELD COMPUTATIONS A good first.