Effects of Nonlinearity on the Estimation of In Situ Values of Acoustic Output Parameters

Transcription

1 Effects of Nonlinearity on the Estimation of In Situ Values of Acoustic Output Parameters Thomas L. Szabo, PhD, Frances Clougherty, MS, Charles Grossman, BS Water can generate extreme waveform distortion compared to tissue, as indicated by the Gol dberg number for water, which is 20 times larger than that of tissue at typical diagnostic ultrasound levels. This result was demonstrated by using tofu as a tissue mimicking material. By adjusting transducer voltage drive levels in water to match the peak rarefactional pressures in water to those of waveforms in tofu, a close correspondence was obtained for the peak compressional pressure and time average intensity with depth. A poorer correspondence was found by comparing tofu waveforms with water waveforms that were compensated for broadband attenuation and driven at the same voltage level as tofu. A simplified broadband derating factor, allowing for bandwidth adjustment, was shown to be more accurate than the standard monochromatic derating. Several new indicators for quantifying the degree of observed nonlinearity are suggested: a field based nonlinearity parameter, a peak pressure ratio p c /p r, and a second harmonic to fundamental frequency spectral ratio. These indicators may have the potential for more consistent characterization of nonlinear relationships among output parameters and drive levels. KEY WORDS: Nonlinearity; Attenuation; Ultrasonic exposimetery; Harmonic; Hydrophone. ABBREVIATIONS ODS, Output Display Standard; PVDF, Polyvinylidene difluoride; FFT, Fast Fourier transform Received from the Imaging Systems Division, Hewlett Packard, Andover, Massachusetts. Address correspondence and reprint requests to Thomas L. Szabo, Hewlett Packard, MS-095, 3000 Minuteman Road, Andover, MA Acoustic output measurements from diagnostic imaging equipment are made in water as prescribed by existing regulatory standards, with the hope that the data will be useful in estimating output levels in tissue (in situ values) and relevant bioeffect parameters. Because the measurement medium, water, is significantly different from tissue, water waveforms do not resemble those obtained in vivo. Can water data be used to estimate values expected in tissue? What are the best methods for estimation? In this work, we drew on the science of nonlinear acoustics to guide us in developing and evaluating practical, more accurate methods for predicting in situ acoustic output levels from measurements of acoustic parmeters made in water. The ODS 1 links acoustic output data to algorithms for estimating and displaying bioeffect parameters (such as the mechanical index and thermal index) that give imaging system users feedback information for making risk-benefit decisions. We assumed the use of the present ODS algorithms and focused on the development of more accurate in situ prediction methods by the American Institute of Ultrasound in Medicine J Ultrasound Med 18:33 41, /99/$3.50

2 34 IN SITU VALUES OF ACOUSTIC OUTPUT PARAMETERS J Ultrasound Med 18:33 41, 1999 A number of approaches have been proposed for making water based acoustic output measurements closer to the values found in tissue. The simplest of these methods is a constant derating factor multiplier for a measured pressure waveform. The derating factor takes the form e (α of c z), where αo is an attenuation factor, f c is the transducer center frequency, and z is the distance from the transducer to the measurement point. In the United States, a conservative in situ value of α o = neper/(mhz cm) [equivalent to 0.3 db/(mhz cm)] 1 is used for derating output data. In contrast to this monochromatic derating factor, Schafer 2 has proposed a broadband derating factor, based on multiplication of the spectrum of a measured waveform by the attenuation α(f) and velocity dispersion ν(f) of typical tissue values at each frequency, e [α(f)z+i2πfz/ν(f)]. Other possible ways of introducing tissuelike attenuation in the acoustic path include the use of plastic disk attenuators 3,4 or the substitution of a tissue mimicking absorbing fluid for water. The term electric attenuation usually is applied to either the reduction of transducer voltage drive levels or the insertion of electric attenuation circuits between the imaging system and transducer 3,4 to achieve derating. An additional complication in obtaining tissuelike values of acoustic output from water values is the nonlinear relationship between the voltage drive level at the transducer and pressure because of the water medium; therefore, it is not enough to estimate comparable tissue values from water data at maximum drive levels but also at intermediate drive levels as well. The process of approximating this highly nonlinear relationship between acoustic output and applied drive levels for estimation of typical in situ values is called characterization. This approximation is used by imaging systems to manage acoustic output levels and to produce display indices via ODS algorithms. Christopher and Carstensen 5 suggest a linear or quasi-linear relationship for a conservative estimate or a small signal linear extrapolation corrected for nonlinearity. Because of the variability of nonlinearity encountered in practice, universal ways of identifying the level of nonlinear distortion are needed to obtain consistent results. In this study we began by comparing the nonlinearity of water to that of tissues. Later we suggested ways of quantifying the degree of nonlinearity encountered in focused fields. Alternative approaches for estimating tissue values were proposed and compared to data measured with a tissue mimicking material. Our emphasis throughout was the development of approaches that are more accurate, simple and practical to implement, robust, consistent, and transportable (independently verifiable). NONLINEARITY OF TISSUES Almost all materials can be characterized by nonlinear equations of state; these relations show that beyond a certain excitation level, each material no longer responds in a linear fashion. For example, the relationship between pressure and density for fluids can be expressed as a Taylor series expansion. Most often, for design convenience and simplicity, only the first or linear term with a coefficient A is used. To describe nonlinear behavior, the second coefficient for density squared, B/2, is needed as well. How nonlinear is a material? The coefficient of nonlinearity β is an indication: β = 1 + (B/2)/A (1) Typical values for β are 3.5 for water, 4 for blood, and 6.2 for fatty tissue. 6 An extensive body of literature, models, and theories have increased our understanding of acoustic propagation in nonlinear media. 7 9 For plane wave propagation, a normalized nonlinearity parameter σ is useful: where and σ = βεkz (2) ε = u o /c o = p o /(ρ o c o 2 ) = 2I o /(ρ o c o 2 ) (3) k = ω/c o (4) Here k = wave number, ω = angular excitation frequency, c o = speed of sound, ρ o = density, u o = particle velocity on transducer surface, p o = pressure on transducer surface, I o = time average intensity on transducer surface, and z = axial distance from transducer. In classic theories, various nonlinear output parameters are given as functions of σ as an independent variable. Nonlinear distortion increases with σ or, in other words, with amplitude, excitation frequency, and distance. As this distortion progresses, energy shifts from the fundamental frequency f to higher harmonics. A value of σ = 1 is often taken to be the onset of nonlinearity, even though the change is gradual. In addition to being nonlinear, all materials have mechanical or acoustic loss. At ultrasonic frequencies, the most frequently occurring type of loss can be described by a frequency power law 10 α = α o f y (5)

3 J Ultrasound Med 18:33 41, 1999 SZABO ET AL 35 where y is a number usually between 1 and 2 and α o is a constant. The exchange of energy between nonlinear distortion and attenuation is complicated, because attenuation diminishes amplitude with increasing propagation distance and acts as a low pass filter that reduces the generation of higher harmonic energy. Haran and Cook 11 compared nonlinear propagation in a number of tissues and water through the use of a nonlinear model adapted to losses of the frequency power law type. A measure of how likely nonlinear distortion or attenuation will prevail is the Gol dberg number 8,12 Γ = σ/(αz) (6) When Γ = 1, nonlinear distortion or shock sets in. The larger Γ is, the more nonlinear distortion dominates, whereas for values of Γ < 1, attenuation is so strong that significant nonlinear distortion does not occur. A surprising result is given in Figure 1, where the Gol dberg number of water (266) is compared to those of tissue for a typical diagnostic pressure of 5 MPa and midrange frequency of 5 MHz, which are less than 14. The frequency loss power exponent for water is 2, and for most tissues it is approximately 1; therefore, the Γ for water increases with amplitude and decreases inversely with frequency, but for most tissues, it is nearly independent of frequency and changes more with amplitude. Clearly, water is very different from tissues in its ability to generate nonlinear distortion. The models described until now apply only to ideal plane wave sources; however, diagnostic imaging systems have focusing transducers that produce complicated fields which vary in amplitude and phase. Fortunately, nonlinear models that include focusing, attenuation, and moderate nonlinearity have been experimentally verified. 13,14 Most of the results reported to date have been for circularly symmetric transducers. Most diagnostic transducers in use are rectangular linear arrays and phased or curved linear arrays that employ a combination of electronic and static focusing. Baker and associates 15 experimentally verified a model for plane rectangular nonfocusing transducers; however, the model is so computationally intensive that a supercomputer was necessary. Typically a transducer array is capable of electronically focusing for many transmit focal zones in a number of modes, such as B-mode (two-dimensional), color flow imaging, and Doppler mode, and a range of drive levels. While models can be helpful for understanding nonlinear effects, modeling the large number of possibilities for each transducer is computationally prohibitive given the present state of the art. In summary, for a typical diagnostic ultrasound field from a focusing array, a complicated interplay exists among the evolving field and the nonlinearity and attenuation of the medium. Because nonlinearity is involved, these three effects are no longer simply multiplicative, and elaborate computer models are needed to predict the full complexity of the fields. EXPERIMENTAL METHODS Figure 1 Gol dberg numbers for tissue and water for a pressure of 5 MPa and a frequency of 5 MHz. To compare different measurement methods, a tissue mimicking material was chosen for reference. Watersaturated extra firm organic tofu was selected, and its acoustic properties were measured with a broadband attenuation technique. 10,16,17 The acoustic impedance was 1.61 MRayl, a midfrequency speed of sound of 1.52 mm/µs, an attenuation of 0.75 db/ [(MHz) y cm], and a frequency exponent y = 1.25 over a measurement range of 2 to 11 MHz. The attenuation coefficient varied up to 25% for samples from different containers so that each batch of samples was acoustically characterized before and after each acoustic output measurement session. The actual attenuation was used in most of the subsequent calculations, and a linear least squares fit to the attenuation characteristic, 1.36 db/(mhz cm), was used for a broadband derating factor discussed later. While β for tofu was not measured, an average midrange value for tissue, β = 5, was assumed, given that the range of β for tissues (muscle to fat) 6 is 4.4 to 6.

4 36 IN SITU VALUES OF ACOUSTIC OUTPUT PARAMETERS J Ultrasound Med 18:33 41, 1999 This value for tofu gives Γ = 3.9 for the conditions of Figure 1. Tofu has the advantage of being inexpensive, edible, biodegradable, and readily available; it has the disadvantage of a short (refrigerated) shelf life. Once a sealed package is opened, the useful life of tofu for consistent results is on the order of 2 days. We do not recommend that tofu be used as a substitute for water. Water is a more stable, universally reproducible measurement medium, so we recommend it over tissue mimicking materials and fluids, despite its strong nonlinearity. Our goal in this work was to relate water data to tissue data. For our experiments, tofu samples were placed in a number of Plexiglas cylinders of different lengths; each one was sealed at its ends with acoustically transparent plastic wrap (Cling Wrap, Glad, Union Carbide, Danbury, CT; thickness, 12.7 µm). Previous experiments with this window material showed negligible effects on acoustic parameters within experiemental error. The tofu samples became progressively shorter as our experiments proceeded. For each depth, the tofu sample occupied most of the acoustic path from transducer to hydrophone. Measurements were conducted in a standard measurement tank, with a 0.5 mm active spot diameter PVDF bilaminar membrane hydrophone (Model # Y , GEC-Marconi, Chelmsford, England) with digital waveform capture equipment. ELECTRIC ATTENUATION For our first experiment we adjusted the voltage drive level for water waveforms to match the observed distortion of a waveform from a tofu path. This method is somewhat different than the standard electric attenuation approach in which the drive level is decreased by an amount dictated by a derating factor. 3 For this equivalent nonlinearity approach, the voltage drive level applied to the transducer in water was varied to match the maximum rarefactional pressure p r of the waveform for the tofu propagation path. The ultrasound energy source was a circularly symmetric 5 MHz center frequency focusing transducer with a diameter of 10 mm and a radius of curvature of 10 cm. A typical set of waveforms (for 3.5 cm depth) is given in Figure 2, where the top waveform shows the extreme distortion in water at maximum drive level, the middle waveform has a milder distortion through tofu at the same drive level, and the bottom waveform matches well the one for tofu at a reduced voltage drive. Even though the tofu waveform has an apparently lower frequency content from attenuation compared to the water waveform at an equivalent voltage, the three output parameters of interest are close for these waveforms. Figure 3 compares the water matching method with tofu for the main acoustic output parameters p r, p c (peak compressional pressure), and I TA (time average intensity) as a function of depth. The matching worked well for the acoustic output parameters in this case. Unfortunately, we had a priori knowledge of what pressure level was needed for matching, information which would not be available generally. The results for drive voltage reduction using a simple derating factor were not as successful, as explained in the next section. BROADBAND ATTENUATION AND DERATING METHODS Compensating for overall tissue attenutaion for acoustic output measurements is most often done through the use of a derating factor. 1 The value used for derating purposes in the United States is 0.3 db/ (MHz cm), and it is intended as an overall conservative parameter, not as a realistic representation of tissue types. At first glance, it seems odd to multiply a time waveform by a constant derating factor containing a center frequency. Later we will derive the form of this factor (Equation 7), but first we consider a more straightforward approach that accounts for attenuation in a broadband sense. As proposed by Schafer, 2 broadband attenuation correction consists of the following steps: 1. Obtain complex spectrum of measured waveform using an FFT. 2. Multiply this spectrum by an appropriate complex attenuation factor at each frequency, e [α(f)z+i2πfz/ν(f)] 3. Take the inverse FFT of the product and derive usual output parameters. Here, the broadband attenuation factor contains phase velocity dispersion ν(f) as well as attenuation α(f). 10 The results for this process using the measured attenuation characteristic are shown in Figure 4 for the water waveform given previously in Figure 2. Compared to the tofu waveform, the signal corrected for attenuation appears different. This difference can be attributed to the fact that the linear attenuation correction of a water signal after it has undergone severe nonlinear distortion is not the same as the natural process in which attenuation and nonlinear effects interact.

5 J Ultrasound Med 18:33 41, 1999 SZABO ET AL 37 A simplified broadband derating factor can be derived based on the assumption of a Gaussianshaped measured spectrum with a 3 db fractional bandwidth in percent and a center frequency f c in megahertz. Assume to first order that the broadband attenuation is linear with frequency and dispersion is negligible. Follow the steps above but perform an analytic Fourier transform rather than a numerical FFT. The result is a Gaussian time pulse scaled by a constant. Setting the time to z/c o at the peak of the Gaussian envelope gives the following maximum amplitude: BBDR = exp ( α o f c z) exp [( α o f c z) 2 (BW) ] (7) Figure 2 Waveforms at a 3.5 cm depth: top, water path and maximum voltage; middle, tofu path and maximum voltage; bottom, water path with voltage adjusted to match p r in tofu waveform. where BBDR is the simplified broadband derating factor that can be adjusted for finite bandwidth BW, and it explains how a factor containing frequency constants is used to mutiply the amplitude of a time waveform. When the bandwidth is zero, this equation reduces to the monochromatic derating factor. 1 Note that the second exponent has a positive argument which diminishes the effect of the first exponent. This change can be explained by the fact that as bandwidth increases, a proportionately larger fraction of the bandwidth is lost to higher frequencies so that the effective center frequency is downshifted. For a 0.3 db/(mhz cm) attenuation, the contribution of the second exponential factor is typically only a few percent. Two attenuation estimation approaches are shown (for the case described in the last figure) in Figure 5, where the simple monochromatic derated (labeled H20der ) and broadband derated (labeled BBDR ) water data are compared to the tofu data. The broadband derating factor is closer to the tofu values for p r, overestimates p c, and closely tracks I TA over most depths. For p r and I TA, the monochromatic derating underestimates the tofu values. Compared to the equivalent voltage results of Figure 3, both broadband and monochromatic derating are less effective because the correction is done after nonlinear distortion has developed. DETERMINATION OF NONLINEAR CHARACTERISTICS Can the degree of nonlinear distortion be characterized by relatively simple indicators? The classic sigma nonlinearity parameter is not expected to work because it was derived for plane waves, and it, therefore, is not applicable to complicated focused fields (see section on Nonlinearity of Tissues). We introduce a field σ z that is similar to the classic plane wave σ but uses the value of mean pressure p m = (p c + p r )/2 at the measured field point: σ z = βp m (z)kz/(ρ o c o 2 ) (8) Finally, Bacon 18 derived a nonlinear propagation parameter σ m for focused Gaussian beams that has been extended to more general beams, σ m = σ z ln [(G 2 1) 1/2 + G]/(G 2 1) 1/2 (9) where G is a focal gain determined by 0.69 times the square root of the ratio of the transducer area to the 6 db focal area (or area at the depth of interest).

6 38 IN SITU VALUES OF ACOUSTIC OUTPUT PARAMETERS J Ultrasound Med 18:33 41, 1999 Figure 3 Output parameters as a function of axial depth for a tofu-filled path and maximum voltage and a water path with voltage adjusted to match tofu waveform p r. Figure 4 Waveforms at a 3.5 cm depth: top, water path and maximum voltage; middle, tofu path and maximum voltage; bottom, water path with maximum voltage and waveform corrected for broadband attenuation. For the tofu experiment discussed earlier, we recorded acoustic output parameters as a function of drive voltage for each depth. To determine p o for the classic plane wave σ, we measured the time average power as a function of voltage on a force balance, corrected for area and duty cycle, and used Equation 3. The output parameters were each plotted against the three σ types and compared to the waveforms at different values of σ s. The ranges turned out to be σ, 0.03 to 0.21; σ z, 0.28 to 2.9; and σ m, to Bacon 18 originally set up criteria for observable distortion for ranges of values of σ m, and these were incorporated into the Acoustic Output Measurement Standard 19 : σ m < 0.5, little distortion; 0.5 < σ m 1.5, significant distortion; and for σ m > 1.5, significant distortion and attenuation due to nonlinear effects. Of the three types of nonlinearity parameters, σ z best tracked the degree of nonlinearity observed; σ z < 1 corresponded to a quasi-linear region. For example, in Figure 2, top, σ z = 2.9, σ m = 1.7; and in Figure 2, bottom, σ z = 0.7, σ m = 0.4. More work is needed to determine how well σ z applies to a wider variety of conditions. Two other candidates for tracking the degree of nonlinearity were tried. The first was the ratio of p c /p r. We observed the waveforms in this study as a function of drive voltage. At the lowest voltages, p c = p r. As voltage increased, p c grew larger and p r decreased. Beyond some voltage, saturation set in, with little relative further change in peak pressures. These results are illustrated by Figure 6. A second nonlinearity indicator is the ratio of the maximum spectral value of the second harmonic (twice the fundamental frequency) b 2 to the peak spectral value of the fundamental frequency b 1. Again, as shown in Figure 6, a similar correspondence exists in the degree of distortion, with a gradual increase in the b 2 /b 1 ratio until a saturation level was reached. The dependence of an output parameter on drive level can be broken down roughly into three regions: linear, knee, and saturation.

7 J Ultrasound Med 18:33 41, 1999 SZABO ET AL 39 Figure 6 Nonlinearity indicators b 2 /b 1 and p c /p r as a function of normalized drive level σ z at a depth of 3.5 cm. Figure 5 Output parameters as a function of axial depth for a tofu-filled path and maximum voltage and a water path with maximum voltage with either monochromatic derating (H20der) or a broadband derating factor (BBDR). σ z, p c /p r, and b 2 /b 1 appear to have potential as nonlinearity indicators and for identifying these three regions. To determine the universality of these indicators we compared them to output parameter data from a curved linear array and a sector array. Similar relationships were found for these arrays as well as the same trends for the range of values for σ z and σ m. For example, these trends followed those in Figure 6, where σ z < 1 corresponded to a linear region, 1 σ z < 2 corresponded to a knee region, and σ z > 1 corresponded to a saturation region. IN SITU EXTRAPOLATION METHODS In general, the process of characterization is not well defined because of the variability of the nonlinear relationship between each acoustic output parameter and the voltage applied to the transducer. As mentioned earlier, a small signal linearization between the output variable and voltage is often a poor approximation. A better result can be obtained by using a least squares fit to output values up to voltage values corresponding to σ z ~ 1. As discussed in the last section, this range of σ z corresponds to a p c /p r ~ 1.4, indicating a slight nonlinearity. Results of applying this method are shown in Figure 7, where output data are shown against a fixed fraction of applied voltage. The first three points correspond to the σ z ~ 1.4 region. Note that for this case, the fits provide conservative estimates of acoustic output. CONCLUSIONS Gol dberg numbers provided an early indication that water as a nonlinear medium far surpassed any tissue in its ability to produce extremely distorted waveforms, even at relatively low drive levels. These numbers also show that for tissue, nonlinear effects prevail over attenuation at diagnostic drive levels. Further evidence of the dominance of nonlinear effects came through experiments in which tofu was a reference tissue mimicking material. Water waveforms, with their drive levels adjusted, could be made to closely match waveforms obtained at

8 40 IN SITU VALUES OF ACOUSTIC OUTPUT PARAMETERS J Ultrasound Med 18:33 41, 1999 maximum drive levels through a tofu propagation path in terms of the acoustic output parameters and degree of nonlinear distortion observed. In contrast, when water waveforms at maximum drive levels were corrected for broadband attenuation of a tofu path, the comparison to tofu waveforms was poorer. These effects are a consequence of the complicated evolution of focused fields in an attenuative, nonlinear medium. The interplay between attenuation and the nonlinearity of the medium develops along the propagation path and is not simply a multiplicative effect, 20 as normally assumed for linear media. In situ estimation methods that account for these effects in a linear way are doomed to be somewhat inaccurate. The challenge is to evaluate approximate estimation methods to determine those that are more accurate, consistent, and practical. Though accuracy is Figure 7 Extrapolation of output parameters using voltages up to 10 volts corresponding to a quasi-linear region up to σ z < 1.2 compared with output parameters as a function of scaled drive level. important, if the method cannot be reproduced and maintained consistently at different sites or if it is complicated, expensive, or unsafe, the method is not likely to gain acceptance. Most in situ estimation methods are based on imitating tissue equivalent attenuation. The simplest of these is the replacement of the water as a measurement medium with a tissue mimicking fluid. This fluid would have the benefits of combining attenuation with nonlinearity, but at present, the maintenance, calibration, temperature dependence, toxicity, chemical properties, compatibility, and, most importantly, long term stability of such fluid raise many other issues that need consideration. Until a suitable fluid can be found, water will remain the most universally acceptable measurement medium despite its untissuelike properties. A second alternative, not investigated here, is the use of plastic attenuators. 3,4 In this approach, disks of low density polyethylene of different thicknesses are combined and placed in a water path between the source transducer and hydrophone to simulate the loss of tissue occupying the full measurement path. While the concentrated attenuation and discrete spatial placements of the disks do not reproduce the actual distributed attenuation of tissue, this approach has its merits in not allowing the buildup of nonlinear distortion that occurs in a pure water path. More work needs to be done to develop a consistent protocol and to prove its efficacy in a variety of highly focused fields. Most derating schemes suffer from an ad hoc application of attenuation after nonlinear distortion in the water path has developed to an extreme degree. The standard monochromatic method in wide use simply scales or derates an already distorted water waveform. A broadband attenuation multiplication 2 is more accurate but does not recover the waveform shape measured through tissue mimicking material. A compromise solution is a constant broadband derating factor developed here, which accounts for differences in bandwidth among transducers. The correction for bandwidth is small for average tissues and even smaller for the standard 0.3 db in situ derating value. More accurate methods need to come to terms with the extent of nonlinear propagation effects. Part of the problem is the lack of a vocabulary for describing and communicating the characteristics of focused fields so that more consistent approaches can be developed. Because of the complicated interplay among nonlinearity, attenuation, and focusing, it is difficult to extract simple rules except in an empirical manner. Computer models have made

9 J Ultrasound Med 18:33 41, 1999 SZABO ET AL 41 great progress in simulating the combined interactions, but they are not at the point where predictions of the focused fields of rectangular arrays have been demonstrated at diagnostic levels. We have looked for nonlinearity indicators that could help identify the degree of nonlinear distortion occurring in data. Indicators can be useful in determining in which of three regions measurements fall. In this study, the classic plane wave σ nonlinearity parameter has not been found useful for focused fields. The parameter σ m derived for focused fields, in our experience, does not correspond with the extent of waveform distortion observed. These results have led us to use a field σ parameter, σ z, that gives a better correspondence for the cases we studied. This field σ parameter is somewhat easier to implement in that beam width measurements are not needed; more work is required to determine how universally applicable it is.two other indicators, the p c /p r ratio and the second harmonic ratio b 2 /b 1, have also shown potential for discriminating among the different regions described previously. These nonlinearity descriptors could play a role in the process of characterization and for identifying linear and quasilinear regions. By using a quasi-linear rather than a strictly small signal linear region, one can introduce mild nonlinearity into the characterization process to improve acoustic output prediction for a larger range of drive levels. This dash of nonlinearity, when added to the proposed broadband derating factor, provides a reasonable alternative for improved in situ estimation. REFERENCES 1. AIUM/NEMA: Standard for Real-time Display of Thermal and Mechanical Acoustic Output Indices on Diagnostic Ultrasound. Rockville, MD, American Institute of Ultrasound in Medicine and National Electrical Manufacturers Association, Schafer ME: Alternative approaches to in-situ intensity estimation. Proc IEEE 1990 Ultrasonics Symposium, p Preston RC, Shaw A, Zeqiri B: Prediction of in situ exposure to ultrasound: An acoustical attenuation method. Ultrasound Med Biol 17:317, Preston RC, Shaw A, Zeqiri B: Prediction of in situ exposure to ultrasound: A proposed standard experimental method. Ultrasound Med Biol 17:333, Christopher T, Carstensen EL: Finite amplitude distortion and its relationship to linear derating formulae for diagnostic ultrasound systems. Ultrasound Med Biol 22:1103, Duck FA: Physical Properties of Tissue. London, Academic Press, Blackstock DT: AIP Handbook. 3rd Ed. New York, Mc- Graw Hill, 1972, p Hamilton MF: Fundamentals and applications of nonlinear acoustics. In Wright TW (Ed): Nonlinear Wave Propagation in Mechanics AMD, Vol. 77. New York, American Society of Mechanical Engineers, 1986, p 1 9. Beyer RT: Nonlinear Acoustics. Woodbury, New York, Acoustical Society of America, Szabo TL: Causal theories and data for acoustic attenuation obeying a frequency power law. J Acoust Soc Am 97:14, Haran ME, Cook BD: Distortion of finite amplitude ultrasound in lossy media. J Acoust Soc Am 73:774, Gol dberg ZA: On the propagation of plane waves of finite amplitude. Sov Phys Acoust 3:340, Baker AC: Nonlinear pressure fields due to focused circular apertures. J Acoust Soc Am 91:713, Christopher T: Modeling the Dornier HM3 lithotripter. J Acoust Soc Am 95:3088, Baker AC, Berg AM, Sahin A, et al: The nonlinear pressure field of plane, rectangular transducers: Experimental and theoretical results. J Acoust Soc Am 97:3510, Zeqiri B: An intercomparison of discrete-frequency and broadband techniques for the determination of ultrasonic attenuation. In Evans DH, Martin K (Eds): Physics in Medical Ultrasound. The Institute of Physical Sciences in Medicine, York, UK, 1988, p Wu J: Determination of velocity and attenuation of shear waves using ultrasonic spectroscopy. J Acoust Soc Am 99:2871, Bacon DR: Finite amplitude distortion of the pulsed fields used in diagnostic ultrasound. Ultrasound Med Biol 10:189, NEMA: Acoustic Output Measurement Standard for Diagnostic Ultrasound Equipment. Washington, D.C., National Electrical Manufacturers Association, 1994, Appendix B 20. Duck FA, Perkins MA: Amplitude dependent losses in ultrasound exposure measurement. IEEE Trans UFFC 35:232, 1988