J. Wójcik

Conservation of energy and absorption in acoustic fields for linear and nonlinear propagation

In the present paper, the energy effects accompanying a strong sound disturbance of a medium are analyzed. The waves may be, in time, periodic — continuous or pulsed — or have the form of single pulses. The description is based on equations which are commonly applied in nonlinear acoustics. The Fourier analysis, elements of the theory of linear operators, and analytical functions are applied. A general method is given for the construction of the absorption operator in the domain of space–time coordinates (x,t), to which the small-signal absorption coefficient corresponds. By analogy to linear equations and the corresponding dispersions equations, the quasi-dispersion equations in the case of nonlinear description are introduced. Simplification of the “classical” equation of nonlinear acoustics was performed. The relations between absorption operators in the space and time domains are shown. It is demonstrated that in nonlinear interactions, where terms of such type — nonlinear function of pressure — domin...

Calibration of ultrasonic hydrophone probes up to 100 MHz using time gating frequency analysis and finite amplitude waves.

A number of ultrasound imaging systems employs harmonic imaging to optimize the trade off between resolution and penetration depth and center frequencies as high as 15 MHz are now used in clinical practice. However, currently available measurement tools are not fully adequate to characterize the acoustic output of such nonlinear systems primarily due to the limited knowledge of the frequency responses beyond 20 MHz of the available piezoelectric hydrophone probes. In addition, ultrasound hydrophone probes need to be calibrated to eight times the center frequency of the imaging transducer. Time delay spectrometry (TDS) is capable of providing transduction factor of the probes beyond 20 MHz, however its use is in practice limited to 40 MHz. This paper describes a novel approach termed time gating frequency analysis (TGFA) that provides the transduction factor of the hydrophone probes in the frequency domain and significantly extends the quasi-continuous calibration of the probes up to 60 MHz. The verification of the TGFA data was performed using TDS calibration technique (up to 40 MHz) and a nonlinear calibration method (up to 100 MHz). The nonlinear technique was based on a novel wave propagation model capable of predicting the true pressure-time waveforms at virtually any point in the field. The spatial averaging effects introduced by the finite aperture hydrophones were also accounted for. TGFA calibration results were obtained for different PVDF probes, including needle and membrane designs with nominal diameters from 50 to 500 micro m. The results were compared with discrete calibration data obtained from an independent national laboratory and the overall uncertainty was determined to be +/-1.5 dB in the frequency range 40-60 MHz and less than +/-1 dB below 40 MHz.

Estimation of acoustical streaming: theoretical model, Doppler measurements and optical visualisation

An approximate solution for the streaming velocity generated by flat and weakly focused transducers was derived by directly solving the Dirichlet boundary conditions for the Poisson equation, the solution of the Navier-Stokes equation for the axial components of the streaming velocity. The theoretical model was verified experimentally using a 32 MHz pulsed Doppler unit. The experimental acoustical fields were produced by three different 4 mm diameter flat and focused transducers driven by the transmitter generating the average acoustic power within the range from 1 microW to 6 mW. The streaming velocity was measured along the ultrasonic beam from 0 to 2 cm. Streaming was induced in a solution of water and corn starch. The experimental results showed that for a given acoustic power the streaming velocity was independent of the starch density in water, changed from 0.3 to 40 grams of starch in 1 l of distilled water. For applied acoustic powers, the streaming velocity changed linearly from 0.2 to 40 mm/s. Both, the theoretical solutions for plane and focused waves and the experimental results were in good agreement. The streaming velocity field was also visualised using the particle image velocimetry (PIV) and two different evaluation methods. The first based on the FFT-based cross-correlation analysis between small sections for each pair of images and the second employing the algorithm of searching for local displacements between several images.

Nonlinear and linear propagation of diagnostic ultrasound pulses

The effect of nonlinear propagation in fluid followed by soft tissue was studied both theoretically and experimentally for a most crucial case in obstetrical ultrasonography. For this purpose, short pressure pulses, with the duration time of 1.3 micros and a carrier frequency of 3 MHz, radiated by a concave transducer into water, with maximum intensities up to the value of 18 W/cm2, were computed and measured. The ultrasonic beam had the physical focus at the distance of 6.5 cm, where the highest focal intensity of I(SPPA) = 242 W/cm2 was obtained. In front of the transducer, at a distance of 7 cm, artificial tissue samples prepared on the basis of ground porcine kidney, with a thickness of 0.5, 1.5 and 3 cm, were placed in water. Pressure pulses and their spectral components were produced numerically and measured by means of a PVDF hydrophone in water before and after penetrating the tissue samples. The theoretical analysis and measurements were carried out, in every case, for two signal levels: for a high level assuring nonlinear propagation and for a low one where conditions of linear propagation were fulfilled. In this way, it was possible to compare directly the effects of nonlinear and linear propagation, in every case showing a good conformity of theoretical values with measured ones. A method of determination of the effective frequency response of the hydrophone was elaborated to enable quantitative comparisons of numerical and experimental results. The theoretical part of our study was based on a paper of Wójcik (1998), enabling us to compute the characteristic function of nonlinear increase of absorption. An agreement of up to 10% was obtained when comparing theoretical and measured values of these functions in the investigated beam in water and behind tissue samples. The results obtained showed that the recently given theory of nonlinear absorption, based on the spectral analysis and the elaborated numerical procedures, may be useful in various practical ultrasonic medical problems and also in technological applications.

Acoustic streaming: Comparison of low-amplitude linear model with streaming velocities measured by 32-MHz Doppler

The pressure gradient along the ultrasonic beam results in medium streaming. Following Nyborgs analysis of the Navier-Stokes equation, Wu and Du developed an approximate solution for the streaming velocity generated by flat and weakly focused transducers. We have modified their solution of the Poisson equation by directly deriving the Dirichlet boundary conditions to be applied for this type of equation. Our numerical results (for the linear case) were about one half smaller for flat and weakly focused on Gaussian beam transducers compared to the results by Wu and Du. The theoretical calculations were verified using a purpose-designed 32-MHz pulsed Doppler unit. The applied average acoustic power was changed from 1 microW to 6 mW, the burst width was 0.5 microseconds and the pulse repetition frequency was 32 kHz. The experiments were done on 4-mm-diameter flat and focused (focal distance = 8 and 12 mm) transducers. The streaming was measured along the ultrasonic beam from 0-20 mm; at all positions, the maximum Doppler frequency was estimated from the recorded spectra. Streaming was induced in a solution of water and corn starch. The experimental results showed that, for a given acoustic power, the streaming velocity was independent of the starch density in water changed from 0.3-40 g of starch in 1 l of distilled water. For applied acoustic powers, the streaming velocity changed linearly from 0.2-40 mm/s. Both the theoretical solutions for plane and focused waves and the experimental results were in good agreement.

Nonlinear propagation model for ultrasound hydrophones calibration in the frequency range up to 100 MHz

To facilitate the implementation and verification of the new ultrasound hydrophone calibration techniques described in the companion paper (somewhere in this issue) a nonlinear propagation model was developed. A brief outline of the theoretical considerations is presented and the models advantages and disadvantages are discussed. The results of simulations yielding spatial and temporal acoustic pressure amplitude are also presented and compared with those obtained using KZK and Field II models. Excellent agreement between all models is evidenced. The applicability of the model in discrete wideband calibration of hydrophones is documented in the companion paper somewhere in this volume.

Fast prediction of pulsed nonlinear acoustic fields from clinically relevant sources using Time-Averaged Wave Envelope approach: comparison of numerical simulations and experimental results

The primary goal of this work was to verify experimentally the applicability of the recently introduced time-averaged wave envelope (TAWE) method [J. Wójcik, A. Nowicki, P.A. Lewin, P.E. Bloomfield, T. Kujawska, L. Filipczyński, Wave envelopes method for description of nonlinear acoustic wave propagation, Ultrasonics 44 (2006) 310-329.] as a tool for fast prediction of four dimensional (4D) pulsed nonlinear pressure fields from arbitrarily shaped acoustic sources in attenuating media. The experiments were performed in water at the fundamental frequency of 2.8 MHz for spherically focused (focal length F=80 mm) square (20 x 20 mm) and rectangular (10 x 25mm) sources similar to those used in the design of 1D linear arrays operating with ultrasonic imaging systems. The experimental results obtained with 10-cycle tone bursts at three different excitation levels corresponding to linear, moderately nonlinear and highly nonlinear propagation conditions (0.045, 0.225 and 0.45 MPa on-source pressure amplitude, respectively) were compared with those yielded using the TAWE approach [J. Wójcik, A. Nowicki, P.A. Lewin, P.E. Bloomfield, T. Kujawska, L. Filipczyński, Wave envelopes method for description of nonlinear acoustic wave propagation, Ultrasonics 44 (2006) 310-329.]. The comparison of the experimental results and numerical simulations has shown that the TAWE approach is well suited to predict (to within+/-1 dB) both the spatial-temporal and spatial-spectral pressure variations in the pulsed nonlinear acoustic beams. The obtained results indicated that implementation of the TAWE approach enabled shortening of computation time in comparison with the time needed for prediction of the full 4D pulsed nonlinear acoustic fields using a conventional (Fourier-series) approach [P.T. Christopher, K.J. Parker, New approaches to nonlinear diffractive field propagation, J. Acoust. Soc. Am. 90 (1) (1991) 488-499.]. The reduction in computation time depends on several parameters, including the source geometry, dimensions, fundamental resonance frequency, excitation level as well as the strength of the medium nonlinearity. For the non-axisymmetric focused transducers mentioned above and excited by a tone burst corresponding to moderately nonlinear and highly nonlinear conditions the execution time of computations was 3 and 12h, respectively, when using a 1.5 GHz clock frequency, 32-bit processor PC laptop with 2 GB RAM memory, only. Such prediction of the full 4D pulsed field is not possible when using conventional, Fourier-series scheme as it would require increasing the RAM memory by at least 2 orders of magnitude.

Temperature elevations computed for three-layer and four-layer obstetrical tissue models in nonlinear and linear ultrasonic propagation cases

The authors computed temperature elevations in a three-layer and a four-layer tissue model, assuming the crucial obstetrical case when the ultrasonic pulse propagating through the abdominal wall and the fluid-filled bladder penetrates into soft fetal tissues. To consider nonlinear propagation, the authors applied a new theory of nonlinear increase of absorption recently developed by the first author. Computations were carried out for pulses with a carrier frequency of 3 MHz, duration time of 1.33 micros, and pulse repetition frequency of 3.3 kHz. Similar computations were carried out for a four-layer tissue model corresponding to the third trimester of gestation. The ceramic piezoelectric transducer 2 cm in diameter radiated the ultrasonic beam focused at a distance of 6.5 cm. The intensities at the radiating transducer (at the source) were I(SAPA) = 10 and 5 W/cm2. Temperature elevations and distributions were determined numerically for various values of low-amplitude absorption coefficients assumed to be the same as attenuation coefficients. It was shown in the three-layer tissue model that the maximum temperature elevation can be about 50% higher for nonlinear than for linear propagation. The maximum fetal temperature elevation in this case was 2.36 degrees C for nonlinear and 1.84 degrees C for linear propagation. The temperature elevation in the abdominal wall was lower than those temperatures when the attenuation of the abdominal wall was assumed to be a low value of 0.05 Np/cm.MHz (0.45 dB/cm.MHz). However, when it was increased to 0.16 Np/cm.MHz (1.4 dB/cm.MHz), the temperature elevation of the abdominal wall reached 3.2 degrees C and the maximum fetal elevation was 1.65 degrees C. In such cases, the abdominal wall became the principal source of heat production. In this case, the difference between fetal temperature elevations for nonlinear and linear propagation was only about 10%. The results obtained in the four-layer tissue model, in which the uterus tissue also was represented, show that temperature elevations in this case are about 3.6 times lower than in the three-layer tissue model, with comparable attenuation of the abdominal wall. Differences between nonlinear and linear propagation in the four-layer tissue model are negligible. The temperature elevations obtained were proportional to the pulse repetition frequency, without changing temperature distributions in the ultrasonic beam. In this manner, fetal temperature elevations can be reduced by reducing the repetition frequency.

Estimation of transient temperature elevation in lithotripsy and in ultrasonography

Transient solutions of the thermal conductivity equation for the two-dimensional case of an elongated cylindrical focus in the ultrasonic beam were derived and applied for lithotripsy and obstetrical ultrasonography. Assuming uniform and Gaussian distributions in the focus of the beam cross section, it was possible to estimate the temperature elevation arising in lithotripsy for various repetition frequencies of shock-wave pulses and for various radii of the beam. In obstetrical ultrasonography where the blood perfusion is difficult to determine, the authors suggested that the insonation time be used as the decisive factor for the temperature determination. Values of focal intensities were found necessary to increase the tissue temperature by 1 degree C as a function of the insonation time and the beam radius which exclude the possibility of any hazardous effect caused by temperature elevation.

Temperature elevation in focused Gaussian ultrasonic beams at various insonation times

Transient solution of the thermal conductivity equation for the three-dimensional case of the Gaussian ultrasonic focused beam was derived and applied for cases relevant to medical ultrasonography. Quantitative results for the case of a homogeneous medium with constant values of thermal coefficients and constant absorption as well as for the two-layer tissue model used in obstetrics were presented for various diagnostic probes used in ultrasonography. The possible effects of perfusion and nonlinear propagation were neglected. The results obtained are in agreement with results of other authors when considering the steady-state and the infinitely short insonation time. The computations show the influence of the insonation time on the temperature elevation, thus making it possible to introduce its value as a factor in limiting the possible harmful effects in ultrasonography. This has been shown in diagrams presenting the temperature distribution along the beam axis of 6 different diagnostic probes for various insonation times and demonstrating the corresponding temperature decrease when limiting the insonation time to 5 and 1 min. For instance, the highest temperature elevation (for probe number 1, see Table 1) decreases 2.6 and 5 times with respect to the steady-state temperature when the insonation time equals 5 and 1 min, respectively.