Institute of Fundamental Technological Research

The nonlinear acoustics equation for a dissipative medium is analytically solved. Continuous wave stimulation and an axisymmetric Gaussian spatial profile of the boundary conditions were assumed. The approximation of the D’Alambert operator by the wave diffusion operator was applied and justified. In this approximation and assuming classical absorption (dispersion), the equation to be solved is presented by the Khokhlov-Zabolotskaya-Kuznetsov model. A sequence of functions describing the spatial distribution of the harmonic components of the disturbance was determined. They are the form of spatially modulated Gauss functions for harmonic wave numbers (frequencies). For a lossless medium a universal numerical sequence describing non-linear interactions and harmonic generations was determined. In other cases, the description of the cooperation of dispersion and non-linear interactions in the harmonic generation process is given by a sequence of functions dependent on the dispersion coefficient and with boundary values given by the universal sequence mentioned above. It was unexpectedly discovered that the influence of geometrical parameters of the beam on nonlinear interactions depends on dispersion, and component of the dispersion, absorption may strengthen harmonic generation. In general, dispersion spatially modulates the amplitude and phase of nonlinear interactions. This is not against the law of conservation of energy. The energy exchange between the fundamental (initiating) component and other harmonics is described. The analytical solution was compared with the numerical one. The numerical solution was obtained in the scheme implementing the full Helmholtz operator (no axial - wave diffusion- approximation).

nonlinear propagation, analytical solution, Gaussian beam, nonlinear interactions, dispersion

The objective of this paper is an experimental study of the most crucial parameters of the received acoustic signals (e.g. signal-to-noise ratio (SNR), side-lobes level (SLL), axial resolution) obtained as a result of si-multaneous emission of mutually orthogonal Golay complementary sequences (MOGCS) to demonstrate their feasibility of being used in ultrasound diagnostics. Application of the MOGCS in ultrasound measurements allows the image reconstruction time to be shortened without decreasing the resulting quality of econstructed images in comparison with regular complementary Golay coded sequences (CGCS). In this paper two sets of 16-bits long MOGCS were implemented in the Verasonics VantageTM (Verasonics Inc., Kirkland, WA, USA) scanner. Ultrasound data were generated using a perfect reflector, a custom-made nylon wire phantom and tissue mimicking phantom. Parameters of the detected MOGCS echoes like SNR, SLL and axial resolution were determined and compared to that of the standard CGCS and the short two-sine cycles pulse. It was evidenced that applying MOGCS did not compromise the parameters of the separated and compressed echoes in compa-rison to the other types of transmitted signal – the CGCS and the short pulse. Concretely, both the MOGCS and CGCS yield similar SNR increase in comparison to the short pulse. Almost similar values of the axial reso-lution estimated at the full width at the half maximum level for all types of the transmitted signals were also obtained. At the same time, using the MOGCS the data acquisition speed can be increased twice in comparison with the CGCS signal.

coded excitation; mutually orthogonal Golay codes; synthetic aperture; ultrasound imaging

Acoustical Driving Forces (ADF), induced by propagating waves in a homogeneous and inhomogeneous lossy fluid (suspension), are determined and compared depending on the concentration of suspended particles. Using integral equations of the scattering theory, the single particle (inclusion) ADF was calculated as the integral of the flux of the momentum density tensor components over the heterogeneity surface. The possibility of negative ADF was indicated. Originally derived, the total ADF acting on inclusions only, stochastically distributed in ambient fluid, was determined as a function of its concentration. The formula for the relative increase in ADF, resulting from increased concentration was derived. Numerical ADF calculations are presented. In experiments the streaming velocities in a blood-mimicking starch suspension (2 μm radius) in water and Bracco BR14 contrast agent (SF6 gas capsules, 1 μm radius) were measured as the function of different inclusions concentration. The source of the streaming and ADF was a plane 2 mm diameter 20 MHz ultrasonic transducer. Velocity was estimated from the averaged Doppler spectrum obtained from originally developed pulsed Doppler flowmeter. Numerical calculations of the theoretically derived formula showed very good agreement with the experimental results.

streaming suspension, scattering, acoustical driving force, Doppler measurements, contrast agents

Quantitative ultrasound has been widely used for tissue characterization. In this paper we propose a new approach for tissue compression assessment. The proposed method employs the relation between the tissue scatterers' local spatial distribution and the resulting frequency power spectrum of the backscat- tered ultrasonic signal. We show that due to spatial distribution of the scatterers, the power spectrum exhibits characteristic variations. These variations can be extracted using the empirical mode decomposition and analyzed. Validation of our approach is performed by simulations and in-vitro experiments using a tissue sample under compression. The scatterers in the compressed tissue sample approach each other and consequently, the power spectrum of the backscattered signal is modified. We present how to assess this phenomenon with our method. The proposed in this paper approach is general and may provide useful information on tissue scattering properties.

tissue characterization, tissue compression, quantitative ultrasound, empirical mode decomposition, signal anaysis

Equations of nonlinear acoustic wave motion in a non-classical lossy medium are used to derive generalised formulas describing the phenomena of reflection and transmission. Integral, non-local operators that are caused by the nonlinear effects in wave propagation and occur in reflection and transmission formulas are given in a form in which classical linear reflection and transmission coefficients are explicitly separated. Numerical calculations are performed for a simplified, one-dimensional wave travelling in a lossless medium. These simplifications reveal the pure effect of the impact of nonlinearities on the reflection and transmission phenomena. We consider adjacent media with different properties to illustrate various aspects of the problem. In particular, even if two media have the same linear impedance and the same material modules of the third order, we observe an explicit effect of the nonlinearity on the reflection phenomenon. The theoretical predictions are confirmed qualitatively by numerical calculations based on the finite difference time domain method.

non-linear sound wave, non-linear reflection, non-classical absorption, soft tissues

A method of suppression of grating lobes is presented, analyzed, and verified. The method is based on creating a Virtual Receiving Subaperture (VRS) by adding virtual transducer elements not existing in the physical layout of the receiver. The VRS channels are filled with data based on signals from real channels. The analytical model of the synthetic aperture imaging system's impulse response is presented to describe the properties of the VRS. The model shows a reduction of the receiving grating lobes' amplitude (with a comparison to the main lobe’s amplitude) by a magnitude equal to the number of receiving transducer elements. It is shown that effective properties of the entire system with a VRS are similar to a system with a pitch in the receiving aperture that is twice as small. The numerical calculations of the impulse response show a doubling of the signal to noise ratio, which results in a reduction of the receiving grating lobes. For experimental validation, the generalized Plane Wave Imaging with and without the VRS is compared with a basic synthetic transmit aperture (STA) imaging. The experiment confirmed that the use of a VRS allows for visualizat ion of the objects in a medium in which they are not imaged without a VRS or are visualized with a lower contrast. The reduction of grating lobes attained using the proposed method is at the level of 15dB in the visualization of the superficial cyst.

grating lobes, image quality, synthetic aperture, virtual subaperture

Streaming velocity depends on intensity and absorption of ultrasound in the media. In some cases, such as ultrasound scattered on blood cells at high frequencies, or the presence of ultrasound contrast agents, scattering affects the streaming speed. The velocities of acoustic streaming in a blood-mimicking starch suspension in water and Bracco BR14 contrast agent were measured. The source of the streaming was a plane 20MHz ultrasonic transducer. Velocity was estimated from the averaged Doppler spectrum. The single particle driving force was calculated as the integral of the momentum density tensor components. For different starch concentrations, the streaming velocity increased from 8.9 to 12.5mm/s. This corresponds to a constant 14% velocity increase for a 1 g/l increase in starch concentration. For BR14, the streaming velocity remained constant at 7.2mm/s and was independent of the microbubbles concentration. The velocity was less than in reference, within 0.5mm/s measurement error. Theoretical calculations showed a 16% increase in streaming velocity for 1 g/l starch concentration rise, very similar to the experimental results. The theory has also shown the ability to reduce the streaming velocity by low-density scatterers, as was experimentally proved using the BR14 contrast agent.

ultrasound, radiation force, starch, contrast agent, blood, thrombolysis

The aim of this work was to find the differences between random media by analyzing the properties of the ultrasound signals backscattered from the inhomogeneities. A numerical model is used to generate two types of random media. The first has the randomness in scatterers’ positions and the second has the randomness in the size and acoustical properties of scatterers. The numerical model of wave scattering has been used to simulate the RF (radio frequency) signals caused by the incident pulse traveling as a plane wave. The markers of randomness type differences between the scattering media were obtained with the help of the spectral and wavelet analysis. The effect of differences in randomness type is more spectacular when the wavelet analysis is performed.

spectrogram, scalogram, wavelets, random scattering structure

Quantitative ultrasound methods are widely investigated as a promising tool for tissue characterization. In this paper, a novel quantitative method is developed which can be used to assess scattering properties of tissues. The proposed method is based on analysis of oscillations of the backscattered echo power spectrum. It is shown that these oscillations of the power spectrum are connected with the distances between scatterers within the medium. Two techniques are proposed to assess the scatterer’s distribution. First, we show that the inverse Fourier transform of the backscattered echo power spectrum corresponds to a histogram of the distances between scatterers. Second, the Hilbert-Huang transform is used to directly extract the power spectrum oscillations. Both methods are examined by means of a numerical experiment. A cellular gas model of a biological medium is considered. Results are presented and discussed. Both methods can be used to evaluate the scatterer’s distribution by means of the power spectrum oscillations.

quantitative ultrasound, signal analysis, wave scattering

The main aim of this study was to verify the suitability of the hydrogel sonographic model of the left ventricle (LV) in the computed tomography (CT) environment and echocardiography and compare the radial strain calculations obtained by two different techniques: the speckle tracking ultrasonography and the multislices computed tomography (MSCT). The measurement setup consists of the LV model immersed in a cylindrical tank filled with water, hydraulic pump, the ultrasound scanner, hydraulic pump controller, pressure measurement system of water inside the LV model, and iMac workstation. The phantom was scanned using a 3.5 MHz Artida Toshiba ultrasound scanner unit at two angle positions: 0° and 25°. In this work a new method of assessment of RF speckles’ tracking. LV phantom was also examined using the CT 750 HD 64-slice MSCT machine (GE Healthcare). The results showed that the radial strain (RS) was independent on the insonifying angle or the pump rate. The results showed a very good agreement, at the level of 0.9%, in the radial strain assessment between the ultrasound M-mode technique and multislice CT examination. The study indicates the usefulness of the ultrasonographic LV model in the CT technique. The presented ultrasonographic LV phantom may be used to analyze left ventricle wall strains in physiological as well as pathological conditions. CT, ultrasound M-mode techniques, and author's speckle tracking algorithm, can be used as reference methods in conducting comparative studies using ultrasound scanners of various manufacturers.

Computed tomography, Echocardiography, Left ventricle, Speckles tracking, Strain, Ultrasound phantoms

The study shows the direct relationship between the temperature field and the parallel changes that are taking place in backscattered ultrasonic signals from the breast tissue in vivo when heated to the temperature of approximately 42 o C. The non-uniform temperature field inside the heating tissue was determined by the numerical model using FEM. It is shown that the spatial distribution of intensities of the backscattered signals coincides with the temperature distribution field predicted by the numerical model in some areas. The result indicates the possibility of the indirect measurement of the temperature rise in the breast tissue in vivo by measuring the intensity variations of the ultrasound echo.

temperature field, backscattered ultrasonic signals, breast tissue in vivo, FEM model

The comparison between theoretical and numerical solutions of the reflection/transmission problem for the acoustic plane wave normally incident on the discontinuity surface between two nonlinear lossy media was presented. Numerical calculations made under the assumption that the two media have the same impedance, allow to single out the effect of nonlinearities in the description of the reflection and transmission phenomena, so they agreed with theoretical predictions. It was shown that theoretically obtained and numerically calculated results mutually confirmed themselves

Nonliear reflection, transmission, Riccati equation

The aim of this work is to present preliminary results of the layer thickness assessment method based on optimization approach. The developed method is based on a multilayer model structure. The measured acoustic signal reflected from the layer is compared with a simulated signal on the basis of a multilayer model. The cost function is defined as the difference between the reflected signal measured using pulse echo approach and the simulated signal. The thickness of the solid layer is the parameter which minimizes the cost function yielding desired solution. Minimization of the cost function is performed with the simulated annealing algorithm. The results obtained with the developed method using measurement data of a custom design model are compared with the reference value and the accuracy of the method is checked. The relative error of the thickness estimation is 1.44 %.

Bone quantitative ultrasound, Human cortical bone, Human cancellous bone, Ultrasound attenuation, Layered media

Nowadays there are new modalities in ultrasound imaging allowing better characterization of tissue regions with different stiffness. We are proposing a novel approach based on compression and rarefaction of tissue simultaneously with imaging. The propagating wave is a combination of two pulses. A low frequency pulse is expected to change the local scattering properties of the tissue due to compression/rarefaction while a high frequency pulse is used for imaging. Two transmissions are performed for each scanning line. First, with the imaging pulse that propagates on maximum compression caused by a low frequency wave. Next, the low frequency wave is inverted and the imaging pulse propagates over the maximum rarefaction. After the processing of the subtracted echoes from subsequent transmissions including wavelet transform and band-pass filtering, differential images were reconstructed. The low frequency wave has a visible impact on the scattering properties of the tissue which can be observed on a differential image.

The present work concerns the sonographic model of the left ventricle (LV) examined in the Computed Tomography (CT) environment and compare radial strain calculations obtained by two different techniques: the speckle tracking ultrasonography and the Multislices Computed Tomography (MSCT). The Left Ventricular (LF) phantom was fabricated from 10% solution of the poly(vinyl alcohol) (PVA). Our model of the LV was driven by the computer- controlled hydraulic piston Super -Pump (Vivitro Inc., Canada) with adjustable fluid volumes. The stroke volume was set at of 24ml. The fluid pressure was changed within range of 0- 60 mmHg, and the pulse rate was of 60 cycles/per minute. The relationships between computer controlled left ventricular wall deformations and its visual izations of the echocardiographic and CT imaging, both in the normal and pathological conditions were examined. The difference of assessment the Radial Strain between two methods was not exceeding 1.1%.

Two different tissue phantoms of the left ventricle to imitate a beating left ventricle were developed: first was prepared using a sponge material and second phantom was constructed using a polyvinyl alcohol material modeled into a homogeneous hollow cylinder: approximately 10 cm and 12 cm in length for the first and second phantom, respectively. Both phantoms were 5 cm in diameter, with a wall thickness of 1.0 cm. Additionally, a small part of the wall of the second phantom was processed to simulate the stiffness of myocardial infarction. The phantoms were connected at the end to an adjustable external pump. The pulse volume inside the cylinder was set between 12 to 50 ml at rates of 40, 60, 100, 120 beats/minute. The phantoms were immersed in water for ultrasound scanning with two different insonation angles (90 and 65 degrees). Strain and strain rate were measured with different combinations of angles and pulse rates. The main aim of this work was to develop the new method for validation of the human infarct wall strain calculation procedures using the speckles tracking.

soft tissue, phantom, ultrasound

We explain, motivation behind this work and briefly describe foundation of new method which we have developed for efficient solution in PC environment of the nonlinear propagation equation with the boundary conditions applied for both circular and not circular transducers (like array). Comparison between new and old method will be presented for strongly nonlinear disturbance. At the end we will demonstrate the results of the numerical calculations of the nonlinear field propagating from the array.

Nonliear propagation, Envelope waves, Fast calculations

The main objective of this study is to develop an echocardiographic model of the left ventricular and numerical modeling of the speckles- markers tracking in the ultrasound (ultrasonographic) imaging of the left ventricle. The work is aimed at the creation of controlled and mobile environment that enables to examine the relationships between left ventricular wall deformations and visualizations of these states in the form of echocardiographic imaging and relations between the dynamically changing distributions of tissue markers of studied structures.

left ventricle, echocardiography, speckle modeling, ultrasound phantoms, strain, strain rate

An analytical model of imaging using synthetic aperture (SA) methods is presented. This model takes into account: fundamental features of an environment, of an electric transmission/reception path and a description of SA structure - possible schemes of transmission, reception and image formation. Then two schemes are analyzed: a proposed cumulative synthetic transmit aperture (CSTA) and for comparison of the standard STA schemes. For both methods identical basic parameters - equal sequences of transmit and receive transducers were applied. The distinctive feature of CSTA is gathering (summing up) echoes of subsequent transmissions in one acquisition mat ix sufficient for image reconstruction. In traditionally applied STA methods a separate acquisition matrix for each transmission is created. Therefore there are a dozen to several dozen more matrices and the time of image reconstruction is at least several times longer than in CSTA. The presented experimental results obtained using wire and tissue mimicking phantoms have shown the comparable imaging quality in both methods.

ultrasound imaging, synthetic aperture

The integral equations that describe scattering in the media with step-rise changing parameters have been numerically solved for the trabecular bone model. The model consists of several hundred discrete randomly distributed elements. The spectral distribution of scattering coefficients in subse- quent orders of scattering has been presented. Calculations were carried on for the ultrasonic frequency ranging from 0.5 to 3 MHz. Evaluation of the contribution of the first, second, and higher scattering orders to total scattering of the ultrasounds in trabecular bone was done. Contrary to the approaches that use the lCT images of trabecular structure to modeling of the ultrasonic wave propagation condition, the 3D numerical model consisting of cylindrical elements mimicking the spatial matrix of trabeculae, was applied. The scattering, due to interconnections between thick trabeculae, usually neglected in trabecular bone models, has been included in calculations when the structure backscatter was evaluated. Influence of the absorption in subsequent orders of scattering is also addressed. Results show that up to 1.5 MHz, the influence of higher scattering orders on the total scattered field characteristic can be neglected while for the higher frequencies, the relatively high amplitude interference peaks in higher scattering orders clearly occur.

Multiple scattering, Complex media, Bone

The paper describes the investigations intended to compare the results of experimental measurements f backscattering properties of the trabecular bone with the results of computer simulations. Ultrasonic RF echoes were collected using two bone scanners operating at 0.58 and 1.3 MHz. The simulations of the backscattered RF echoes were performed using the scattering model of the trabecular bone that consisted of cylindrical and spherical elements uniformly distributed in waterlike medium. For each measured or simulated RF backscatter the statistical properties of the signal envelope were determined. Experimental results suggest deviations of the backscattering properties from the Rayleigh distribution. The results of simulation suggest that deviation from Rayleigh distribution depends on the variation of trabeculae diameters and the number of thin trabeculae. Experimentally determined deviations corresponded well to the deviations calculated from simulated echoes assuming trabeculae thickness variation equaled to the earlier published histomorphometric study results.

trabecular bone, scattering statistics, bone model

Recently a new technology of piezoelectric transducers based on PZT thick film has been developed as a response to a call for devices working at higher frequencies suitable for production in large numbers at low cost. Eight PZT thick film based focused transducers with resonant frequency close to 40 MHz were fabricated and experimentally investigated. The PZT thick films were deposited on acoustically engineered ceramic substrates by pad printing. Considering high frequency and nonlinear propagation it has been decided to evaluate the axial pressure field emitted (and reflected by thick metal plate) by each of concave transducer differing in radius of curvature – 11 mm, 12 mm, 15 mm, 16 mm.
All transducers were activated using AVTEC AVG-3A-PS transmitter and Ritec diplexer connected directly to Agilent 54641D oscilloscope. As anticipated, in all cases the focal distance was up to 10% closer to the transducer face than the one related to the curvature radius. Axial pressure distributions were also compared to the calculated ones (with the experimentally determined boundary conditions) using the angular spectrum method including nonlinear propagation in water. The computed results are in a very good agreement with the experimental ones. The transducers were excited with Golay coded sequences at 35–40 MHz. Introducing the coded excitation allowed replacing the short-burst transmission at 20 MHz with the same peak amplitude pressure, but with almost double center frequency, resulting in considerably better axial resolution. The thick films exhibited at least 30% bandwidth broadening comparing to the standard PZ 27 transducer, resulting in an increase in matching filtering output by a factor of 1.4–1.5 and finally resulting in a SNR gain of the same order.

transducers, thick film, high frequency ultrasound, pulse compression, Golay codes

Some important details of the Backscatter Effective Cross-Sections (BECS) obtained for random scattering structures (like trabecular bone) are explain by comparison with the results obtained by means of the simplified theoretical model. The simplified model was (establish) and justified on the basis of the structural analysis of the results obtained for exact model of the field scattering on complex structures. The simplified model is commonly used in description of the scattering on the regular structures like crystal. Comparison with experimental results for the trabecular bone is also presented. The results allowed to conclude that crystallographic methods could be potentially useful for extracting characteristic features of trabecular bone.

Multi scattering, Random structures

The use of therapeutic ultrasound continues to grow. A focused ultrasonic wave can increase the tissue temperature locally for the non-invasive cancer treatment or other medical applications. The authors have designed a seven-element annular array transducer operating at 2.4 MHz. Each element was excited by sine burst supplied by a linear amplifier and FPGA control circuits. The acoustic field, generated by a transducer was initially numerically simulated in a computer and next compared to water tank hydrophone measurements performed at 20, 40 and 60 mm focal depth. The results showed good agreement of the measurements with theory and the possibility to focus the ultrasound in the preselected area. The total acoustic power radiated by the annular array was equal to 2.4W.

ultrasonic therapy, annular array transducer, ultrasonic field

A trabecular bone consists of trabeculae whose mechanical properties differ significantly from the surrounding marrow, therefore an ultrasonic wave propagating within the bone structure is strongly scattered. The aim of this paper was to evaluate the contribution of the first, second and higher order scattering (multiple scattering) into the total scattering of ultrasound in a trabecular bone. The scattering due to the interconnections between thick trabeculae, usually neglected in trabecular bone models, has been also studied. The basic element in our model of the trabecular bone was an elastic cylinder with a various finite-length and diameter as well as orientation. The applied model was taking into account variation of both, elements size and their spatial configuration. The field scattered on the bone model was evaluated by solving numerically the integral form of the generalized Sturm-Liouville equation describing a scalar wave in inhomogeneous and lossy media. For the scattered fields calculated numerically the effective cross-sections were determined. The influence of absorption on the scattering coefficients was demonstrated. The results allowed to conclude that within the frequency range from 0.5 to 1.5 MHz contribution of the second order scattering to the effective backscattering cross-section is at least 500 times lower than that due to the first order scattering. It was noticed that for a frequency higher than 1.5 MHz fast growth of the backscattering (reflection) coefficients, calculated for the second order scattering, occurs.

multi-scattering, random complex structures, trabecular bone model

Beneficial biological effects in soft tissues can be induced by focused ultrasound of low intensity (LIFU). For example, increasing of cells immunity to stress can be accomplished through the enhanced heat shock proteins (Hsp) expression induced by the low intensity focused ultrasound. The possibility to control the Hsp expression enhancement in soft tissues in vivo can be the potential new therapeutic approach to neurodegenerative diseases that utilizes the known feature of cells to increase their immunity to stresses through the Hsp expression enhancement. The controlling of the Hsp expression enhancement by adjusting the level of exposure to ultrasound energy would allow evaluating of ultrasound-mediated treatment efficiency. Our objective was to develop the numerical model capable of predicting in space and time temperature fields induced in multilayer nonlinear attenuating media by a circular focused transducer generating pulsed acoustic waves and to compare the results calculated for two-layer configuration of media: water - fresh rat liver with the experimental data. The measurements of temperature variations versus time at 5 points on the acoustic beam axis within the tissue sample were performed using 0.2-mm diameter thermocouples. Temperature fields were induced by the transducer with 15-mm diameter, 25-mm focal length and 2-MHz centre frequency generating tone bursts with the intensity ISPTA varied between 0.45 W/cm2 and 1.7 W/cm2 and duration varied between 20 and 500 cycles at the same 20-% duty cycle and 20-min exposure time. Quantitative analysis of the obtained results allowed to show that, for example, for the acoustic beam with intensity ISPTA = 1.13 W/cm2 exposure time to ultrasound should not be longer than 10 min to avoid cells necrosis following the 43-oC temperature threshold exceeding.

low intensity focused ultrasound, soft tissues, temperature fields, ultrasonic regimes, therapy efficiency

The paper describes the computer simulations allowing investigating the properties of the ultrasound pulse-echo signal, as it is received on the transducer surface after scattering in trabecular bone. A novel computer simulation model provides better understanding of ultrasonic scattering in porous bone structure and it can be also used to yield an ideal environment in which, the effects of various parameters (scatterer mechanical and geometrical properties, scatterer’ concentration), the shape of incident wave and experimental conditions influencing the scattering of ultrasonic waves in trabecular bone structure can be examined individually. The results proved that the computer simulation has a particular relevance in studying scattering in cancellous bone which may be approximated as a collection of two populations of scatterers, cylindrical and spherical that imitate thick and thin trabeculae respectively.

bone modeling, scattering simulation, osteoporosis, trabecular bone

In this paper a new method utilizing nonlinear properties of tissues to improve contrast-to-noise ratio is presented. In our novel method the focused circular transducer is excited with two-tone bursts (including the 2.2 MHz fundamental and 4.4 MHz second harmonic frequencies) with specially coded polarization of each tone. This new approach was named Multitone Nonlinear Coding (MNC) because the choice of both tones polarization and amplitude law, allowing optimization of the probe receiving properties, depends on nonlinear properties of tissue. The numerical simulations of nonlinear fields in water and in tissue-like medium with absorption coefficient of 7 Np/(m•MHz) are performed. The comparison between the proposed method and the Pulse Inverse (PI) method is presented. The concept of the virtual fields was introduced to explain properties of both the Pulse Inversion and MNC methods and to compare their abilities. It was shown that for the same on-source pressure an application of the MNC method allows to decrease the mechanical index about 40 %, to improve lateral resolution from 10 to 30 % and to gain the signal-to-noise ratio up to 8 times with respect to the PI method.

harmonic imaging, ultrasonography, nonlinear propagation

Integral equations that describe scattering on the structure with step like abruptly changing physical parameters, have been numerically solved on example of the trabecular bone model. The model consists of several hundred elements with randomly selected parameters. The spectral distribution of scatter coefficients in subsequent orders - first second and third- of scattering has been presented.

Multiple scattering in absorbing medium, Complex random structure, Back scatter coefficients

Ultrasound is safe, convinient and inexpensive modality which may be useful for soft tissues treatment. A range of beneficial biological effects induced by ultrasound depends on the exposure level used during treatment. At high intensities instantaneous tissue necrosis is desired, whereas at lower intensities remedial reversible cellular effects may be produced. For example, increasing of cell immunity against stress can be obtained through the heat shock proteins (Hsp) expression enhancement. The possibility of the Hsp expression enhancement in soft tissues in vivo by means of controlled exposure to ultrasound would allow to evaluate the treatment efficiency. Ultrasonic regimes can be controlled by adjusting the ultrasound intensity, frequency, pulse duration, duty-cycle and exposure time. The goal of this work was to develop the numerical model capable of predicting in space and time the temperature fields induced by circular focused transducer generating tone bursts in multilayer nonlinear attenuating media, which is intended for the Hsp expression enhancement therapeutic applications. The acoustic pulsed pressure field generated by the transducer was calculated using our original 3D numerical solver [1]. For prediction of the temperature distributions in multilayer biological media the Pennes bio-heat transfer equation was solved numerically. The 3D thermal fields induced in a rat liver in vitro by a 2 MHz transducer of 15-mm diameter and 25-mm focal length during ultrasonic Hsp expression enhancement treatment using various acoustic beam intensities and exposure time was predicted.

multilayer biological media, ultrasound exposure parameters, local thermal fields, numerical prediction

The primary goal of this work was to verify experimentally the applicability of the recently introduced time-averaged wave envelope (TAWE) method 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 (20mm x 20 mm) and rectangular (10mm x 25 mm) 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. 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. 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.

rectangular focused apertures, pulsed acoustic fields, nonlinear distortion, numerical modelling and experiments

The aim of this paper is to present the new method for relative changes measurements of intima-media thickness in the common carotid artery wall. The numerical solver was created for calculation of the fields of ultrasonic beams and scattered fields under different boundary conditions and different angles of penetration of ultrasonic beams with respect to the position of the arterial wall. The cylindrical model of the artery was changing the radius and thickness of the wall under cyclic variation of blood pressure. The presented method was verified on a pipe made of latex. The paper describes also the initial results of examinations of the intima-media thickness. The good agreement for the angle dependence and the perpendicular ultrasonic beam displacement from the longitudinal axis of the artery segment between the numerical calculation and experimental results was obtained for different artery diameters.

The aim of this paper is to compare the results of the mathematical modeling and experimental results of the ultrasonic waves scattering in the inhomogeneous dissipative medium. The research was carried out for an artery model (a pipe made of a latex), with internal diameter of 5 mm and wall thickness of 1.25 mm. The numerical solver was created for calculation of the fields of ultrasonic beams and scattered fields under different boundary conditions, different angles and transversal displacement of ultrasonic beams with respect to the position of the arterial wall. The investigations employed the VED ultrasonic apparatus. The good agreement between the numerical calculation and experimental results was obtained.

Model of the artery, scattering, numerical solver

The goal of this work was to verify experimentally the applicability of the recently developed Time-Averaged Wave Envelope (TAWE) method [1] as a tool for fast prediction of pulsed nonlinear pressure fields from focused nonaxisymmetric 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 25 mm) 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. Comparison of the experimental and numerical calculations results 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.

rectangular focused apertures, pulsed acoustic fields, nonlinear distortion, numerical modelling

The aim of this paper is to compare the results of examinations of the intima-media thickness (IMT) in the wall of the carotid artery model by means of zero-crossing and correlation methods. The research was carried out on the elastic artery model (a pipe made of latex), with the internal diameter of 3, 5 and 8 mm and the wall thickness of 0.75, 1.25 and 2 mm. A numerical solver was created for the purpose of calculating the fields of ultrasonic beams and scattered fields under different boundary conditions, different angles and transversal displacements of ultrasonic beams in respect of the position of the arterial wall. A VED ultrasonic apparatus was used during the investigations. The frequency of the transmitted ultrasound was 6.75 MHz. The numerical solver was used for the creation of ultrasonic RF reference signals. A good conformity was obtained between changing the numerical reference of the IMT and the results of determining the IMT by both the zero-crossing method and the correlation method.

ultrasound, carotid artery, intima-media thickness, numerical model

The aim of this paper is to compare the results of the mathematical modeling and experimental results of the ultrasonic waves scattering in the inhomogeneous dissipative medium. The research was carried out for an artery model (a pipe made of a latex), with internal diameter of 3, 5 and 8 mm and wall thickness of 0.75, 1.25 and 2 mm. The numerical solver was created for calculation of the fields of ultrasonic beams and scattered fields under different boundary conditions, different angles and transversal displacement of ultrasonic beams with respect to the position of the arterial wall. The investigations employed the VED ultrasonic apparatus. The frequency of the transmitted ultrasound was 6.75 MHz. The good agreement between the numerical calculation and experimental results was obtained. The numerical solver is used for verified proposed methods for determining of the IMT in the artery walls.

ultrasound, scatering, artery model, numerical calculations, eksperimental results, Comparison

We present a new method that uses nonlinear properties of tissue to improve contrast-to-noise ratio.
In our novel method, the acoustic source is activated with two tone-bursts (2.2 and 4.4 MHz), with specially designed polarization of the individual tone-burst. This new approach is called multitone nonlinear coding (MNC) because the choice of polarization of both tones (and their amplitudes), allowing optimization of the receiving properties, depends on the nonlinear properties of tissue. The calculations were done for two tone-bursts propagating in the tissue-like glossy medium with absorption of 7 Np/m · MHz. The method was experimentally verified by scanning the incident pulses propagating in soft tissue and by scanning the thread phantom immersed in water. The concept of the virtual fields was introduced to explain abilities and properties of pulse inversion and MNC and to compare the two methods. Comparison of the spatial field distribution obtained using MNC with the conventional harmonic imaging approach, in which the second harmonic is used to reconstruct the image, is presented. It was shown that, for the same peak pressure amplitude, the resulting mechanical index was about 40% lower for MNC, lateral resolution was 10% to 30% better and, what seems to be the most encouraging, the signal gain was up to eight times higher than pulse inversion.

harmonic imaging, ultrasonography, nonlinear propagation

The developed solver of acoustic field was used for simulation of the artery wall thickness examination. It is capable of describing spatial and time-dependent distribution of an ultrasonic beam, that is emitted by a piezoelectric ring transducer and then backscattered on cylindrical surfaces of the walls in artery models. The electrical signal received corresponds closely with the actual RF signal that is obtained during measurements at the output of the ultrasonicVED apparatus. The theoretical model of the artery for creating the ultrasonic reflected echoes was used. The internal radius of the artery model was 3mm for the diastolic pressure and 3.3mm for the systolic pressure. The intima- media thicknes (IMT) of the artery wall was changed from 0.48mm to 0.44mm respectively. The echoes-tracking solver based on the zero-crossing and correlation methods was used for detecting changes of the IMT.

ultrasound, common carotid artery, elasticity, intima-media thickness, numerical solver

The aim of the study was the elaboration of a mathematical model to describe the process of acoustic wave propagation, generated by an ultrasonic probe in a inhomogenous loosing medium. Numerical calculations make it possible to define waveforms for electric signals that are generated when ultrasonic waves, being reflected and backscattered by an artery model, are then received by the ultrasonic probe. It is the signal that pretty well corresponds with the actual RF signal that is obtained during measurements at the output of anultrasonic apparatus. The developed solver of acoustic field was used for simulation of the artery wall tickness examination. The theoretical model of the atrery for the creating the simulated ultrasonic reflected echoes was used. The internal radiusof the artery model was 3mm for the diastolic pressure and and 3.3mm for the systolic pressure. The intima-media thicknes (IMT) of the artery wall was changed from o.48 to 0.44 respectively. The solver based on zero-crossing method was used for detecting changes of the IMT.

ultrasound, mathematical model, scatering, inhomogenous loosing medium, numerical simulations, intima-media changes

A novel, free from paraxial approximation and computationally efficient numerical algorithm capable of predicting 4D acoustic fields in lossy and nonlinear media from arbitrary shaped sources (relevant to probes used in medical ultrasonic imaging and therapeutic systems) is described. The new WE (wave envelopes) approach to nonlinear propagation modeling is based on the solution of the second order nonlinear differential wave equation reported in [J. Wojcik, J. Acoust. Soc. Am. 104 (1998) 2654-2663; V.P. Kuznetsov, Akust. Zh. 16 (1970) 548-553]. An incremental stepping scheme allows for forward wave propagation. The operator-splitting method accounts independently for the effects of full diffraction, absorption and nonlinear interactions of harmonics. The WE method represents the propagating pulsed acoustic wave as a superposition of wavelet-like sinusoidal pulses with carrier frequencies being the harmonics of the boundary tone burst disturbance. The model is valid for lossy media, arbitrarily shaped plane and focused sources, accounts for the effects of diffraction and can be applied to continuous as well as to pulsed waves. Depending on the source geometry, level of nonlinearity and frequency bandwidth, in comparison with the conventional approach the Time-Averaged Wave Envelopes (TAWE) method shortens computational time of the full 4D nonlinear field calculation by at least an order of magnitude; thus, predictions of nonlinear beam propagation from complex sources (such as phased arrays) can be available within 30-60 min using only a standard PC. The approximateratio between the computational time costs obtained by using the TAWE method and the conventional approach in calculations of the nonlinear interactions is proportional to (1/N)**2, and in memory consumption to 1/N where N is the average bandwidth of the individual wavelets. Numerical computations comparing the spatial field distributions obtained by using both the TAWE method and the conventional approach (based on a Fourier series representation of the propagating wave) are given for circular source geometry, which represents the most challenging case from the computational time point of view. For two cases, short (2 cycle) and long (8 cycle) 2 MHz bursts, the computational times were 10 min and 15 min versus 2 h and 8 h for the TAWE method versus the conventional method, respectively.

Nonliear propagation, Envelope waves, Fast calculations

The study was aimed at elaboration of a mathematical model to describe the process of acoustic wave propagation in an inhomogeneous and absorbing medium, whereas the wave is generated by an ultrasonic probe. The modelling proces covered the phenomenon of ultrasonic wave backscattering on an elastic pipe with dimensions similar to the artery section. Later on the numerical codes were determined in order to calculate the fields of ultrasonic waves, as well as backscattered fields for various boundary conditions. Numerical calculations make it possible to definethe waveforms for electric signals that are produced when ultrasonic waves, being reflected and backsvattered by an artery model, are then received by the ultrasonic probe. It is the signalwhich pretty well corresponds with the actual RF signal that is obtained during measurements at the output of anultrasonic apparatus.

ultrasound, backscattering, artery, numerical model

The nonlinear pulsed acoustic pressure field from a focused square aperture is considered. Experimental measurements in water of a 4D sound field radiated from a 2.8 MHz focused square transducer of a 20 mm side and a 80 mm focal distance for excitation level producing an average acoustic pressure P0 = 0.14 MPa at its surface are presented. The obtained results are compared with the numerical calculation results for the same boundary conditions. The novel, free from paraxial approximation and computationally efficient numerical algorithm was used to simulate the 4D nonlinear pulsed pressure field from the nonaxisymmetric acoustic source. Our theoretical model was based on the Time-Averaged Pressure Envelope (TAPE) method recently developed that enable to represent the propagated pulsed disturbance as a superposition of sinusoidal wavelets with carrier frequencies being the harmonics of the initial tone burst and with envelopes determined by the TAPE method. The novel approach to the solution of the nonlinear wave equation enabled to simulate full 4D nonlinear field for given boundary conditions in a dozen or so minutes utilizing the computational power of the standard PC.

square spherically focused transducer, pulsed waves, nonlinear propagation in water, sound fields, numerical modelling

Imaging of living cells or tissues at a microscopic resolution, where GHz frequencies are used, provides a foundation for many new biological applications. The possible temperature increase causing a destructive influence on the living cells should be then avoided. However, there is no information on possible local temperature increases at these very high frequencies where, due to strongly focused ultrasonic beams, nonlinear propagation effects occur. Acoustic parameters of living cells were assumed to be close to those of water; therefore, the power density of heat sources in a water medium was determined as a basic quantity. Hence, the numerical solution of temperature distributions at the frequency of 1 GHz was computed for high and low powers generated by the transducer equal to 0.32 W and 0.002 W. In the first case, typical nonlinear propagation effects were demonstrated and, in the second one, propagation was almost linear. The focal temperature increase obtained in water equaled 14°C for the highest possible theoretical repetition frequency of fr = 10 MHz and for the thermal insulation at the sapphire lens-water boundary. Simultaneously, the scanning velocity of the tested object was assumed to be incomparably low in respect to the acoustic beam velocity. The maximum temperature increase in water occurred exactly at this boundary, being equal there to 20°C. It was shown that, first of all, the very high absorption of water was significant for the temperature distribution in the investigated region, suppressing the focal temperature peaks. Because the temperature increases are proportional to the repetition frequency, so for example, at its practical value of fr = 0.1 MHz, all temperature increases will be 100 times lower than listed above. For the low transducer power of 0.002 W, the corresponding temperature increases were about 140 times lower than those for the high power of 0.32 W. The presented solutions are devoted mainly to the reflection pulse mode; however, they can be also applied for the transmitting (continuous-wave) mode, as shown in an example. Pressure distributions were computed for the acoustic field of the microscope for the first and higher harmonics. Hence, at the frequency of 1 GHz, the effective focal radius in water measured as the −6-dB amplitude pressure drop was found to be 1,1 μm, and 0.7 μm for the second harmonic, independently of the assumed transducer power. So the width of the beam, scanning the living cells in the focal region, was equal to 2.2 μm at the fundamental frequency of 1 GHz.

Temperature, Acoustic microscopy, Living cells, Temperature increase, Pressure

The influence of finite aperture and frequency response of piezoelectric ultrasonic hydrophone probes on the Thermal and Mechanical Indices was investigated using a comprehensive acoustic wave propagation model. The experimental verification of the model was obtained using a commercially available, 8 MHz, dynamically focused linear array and a single element, 5 MHz, focused rectangular source. The pressure–time waveforms were recorded using piezoelectric polymer hydrophone probes of different active element diameters and bandwidths. The nominal diameters of the probes ranged from 50 to 500 μm and their usable bandwidths varied between 55 and 100 MHz. The Pulse Intensity Integral (PII), used to calculate the Thermal Index (TI), was found to increase with increasing bandwidth and decreasing effective aperture of the probes. The Mechanical Index (MI), another safety indicator, was also affected, but to a lesser extent. The corrections needed were predicted using the model and successfully reduced the discrepancy as large as 30% in the determination of PII. The results of this work indicate that by accounting for hydrophones' finite aperture and correcting the value of PII, all intensities derived from the PII can be corrected for spatial averaging error. The results also point out that a caution should be exercised when comparing acoustic output data. In particular, hydrophone's frequency characteristics of the effective diameter and sensitivity are needed to correctly determine the MI, TI, and the total acoustic output power produced by an imaging transducer.

Ultrasound imaging, Nonlinear propagation, Spatial averaging, Safety indices

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 μm. 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.

Time gating frequency analysis (TGFA), Time delay spectrometry (TDS), High frequency hydrophone calibration, Nonlinear hydrophone calibration, High frequency ultrasound, Ultrasonic metrology

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 model’s 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.

Nonlinear propagation modeling, Nonlinear propagation, JW model

Nonlinear propagation effects produced by focused pulses in blood were measured over a 20-cm range, being inspired by diagnostic applications in cardiology. The initial and maximum pressures applied during measurements in blood were equal to 0.40 MPapp and 0.76 MPapp, while the pressure estimated at the patient body surface equalled 0.70 MPapp. Measurements of the frequency characteristic and the linearity of the ultrasonic probe used in experiments were performed in water. A numerical procedure developed previously was applied in blood to calculate the pressure distribution of its first and second harmonics along the beam axis. The comparison of numerical and measured distributions in blood at a temperature of 37°C showed rather good agreement. Using numerical methods, a proportional growth of the second harmonic with the increased applied initial pressure was first observed, and finally the maximum limiting effect was found. In this way, much higher level of harmonics could be obtained. However, there arise the questions of the transmitting system construction and of the nonuniform resolution in the case of harmonic imaging when increasing the applied initial pressure.

Ultrasound, Pulses, Nonlinear propagation, Blood, Cardiology

Measurements in the very near field of piezoelectric transducers are fundamental for many ultrasonic problems. In such cases also the transducer vibrations should be known to perform mathematical models of radiated beams. Acoustic pressure measurements near to the transducer surface can give the necessary information. The pressure of the radiated wave at the transducer surface corresponds to its normal vibration velocity multiplied by the [ampersand]rho;c value of the medium. However, this is valid only for the central wave, when the edge wave of the transducer can be ignored. On the other hand, pressure measurements on and very near to the transducer surface are not possible because of the voltage leakage between the electronic transmitter and the PVDF hydrophone used in such measurements. By means of a numerical model, central and edge waves were found for a plane PZT transducer 7.5mm in radius, with the applied 2.7MHz voltage pulse composed of 3 cycles. Two types of boundary conditions of Dirichlet and Neumann were considered showing a negligible difference in the case of short pulses. Basing on numerical and experimental results, practical conditions were determined which make it possible to carry out pressure measurements in the very near field of the transducer, and hence to determine the transducer vibrations which are important for modeling ultrasonic pulse beams.

Anisotropy is inherently related to microstructural arrangement within a representative volume of material. The microstructure can be represented by a second order tensor whose eigenvectors specify the orientation of the axes of material symmetry. In this paper, failure criteria for geomaterials are formulated in terms of the stress state and a microstructure tensor. The classical criteria for isotropic materials are generalized for the case of orthotropy as well as transverse isotropy. The proposed approach is illustrated by a simple example demonstrating the sensitivity of the uniaxial strength of the material to the orientation of the sample relative to the loading direction.

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 abdomiinal 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 μs, 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 ISAPA= 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°C for nonlinear and 1.84°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°C and the maximum fetal elevation was 1.65°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.

Ultrasound, Nonlinear propagation, Temperature, Obstetrics

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 μs 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 ISPPA= 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.

Ultrasound, Pulses, Nonlinear propagation, Diagnostics

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 — dominate, the power (energy) of the disturbance is conserved. Just as in the linear notation, the only reason why the total power (energy) changes is linear absorption, but that one which occurs under the conditions of nonlinear propagation. In consequence, the equations of power (and energy) balancing the disturbance have the same formal shape in nonlinear and linear descriptions. The equations provide a theoretical basis for different, easier, and more accurate methods than those used previously for determination (numerical and experimental) of, e.g., the power density of heat sources generated by sound. The function of the nonlinear gain of absorption and the function of effective absorption were also introduced. On the basis of quasi-dispersion equations the phenomenon of overtone generation (not harmonics) is shortly discussed.

Acoustic absorption, Acoustics, Nonlinear acoustics, Absorption coefficient, Acoustic analysis

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.

The pressure gradient along the ultrasonic beam results in medium streaming. Following Nyborg's 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 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 μW to 6 mW, the burst width was 0.5 μs 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.

Ultrasound, Streaming, Nonlinear ultrasound effects, Doppler

The theory of wave reflection from spherical obstacles was applied for determination of the cause of the shadow created by plane wave pulses incident on rigid, steel, gaseous spheres and on spheres made of kidney stones. The spheres were immersed in water which was assumed to be a tissuelike medium. Acoustic pressure distributions behind the spheres with the radii of 1 mm, 2.5 mm and 3.5 mm were determined at the frequency of 5 MHz. The use of the exact wave theory enabled us to take into account the diffraction effects. The computed pressure distributions were verified experimentally at the frequency of 5 MHz for a steel sphere with a 2.5-mm radius. The experimental and theoretical pulses were composed of about three ultrasonic frequency periods. Acoustic pressure distributions in the shadow zone of all spheres were shown in the amplitude axonometric projection, in the grey scale and also as acoustic isobar patterns. Our analysis confirmed existing simpler descriptions of the shadow from the point of view of reflection and refraction effects; however, our approach is more general, also including diffraction effects and assuming the pulse mode. The analysis has shown that gaseous spherical inclusions caused shadows with very high dynamics of acoustic pressures that were about 15 dB higher in relation to all the other spheres. The shadow length, determined as the length at which one observes a 6-dB drop of the acoustic pressure, followed the relation r−6dB = 3.7a2λ with the accuracy of about 20% independent of the sphere type. λ denotes the wavelength and a the sphere radius. Thus, a theoretical possibility of differentiating between gaseous and other inclusions and of estimation of the inclusion size in the millimeter range from the shadow was shown. The influence of the frequency-dependent attenuation on the shadow will be considered in the next study.

Shadow, Pulses, Spheres, Ultrasonography

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.

Temperature, Ultrasonography, Time, Hazard

Transient solutions of the thermal conductivity equation for the two-dimensional case of an elongated cylíndrical 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°C as a function of the insonation time and the beam radius which exclude the possibility of any hazardous effect caused by temperature elevation.

Lithotripsy, Obstetrics, Ultrasonography, Temperature, Hazard

In this paper we investigate how to assess the ultrasound nonlinearity coefficient using plane wave imaging. We employ the technique based on excitation of the medium with ultrasonic pulses of increasing amplitude level. As the pulse pressure is increased, due to medium nonlinearity, higher fraction of energy is transferred from the fundamental to higher harmonics during the propagation. In this case the amplitude of the backscattered echo is not linear in respect to the initial pulse amplitude at source. This phenomenon can be used for the nonlinearity coefficient assessment and show its implementation for the plane wave imaging. The method was validated experimentally using a wire phantom immersed in water and scanned using the Verasonics scanner. We discuss the usefulness of the proposed technique and its shortcomings. In comparison to other nonlinearity coefficient assessment methods, the presented technique works in the pulse-echo mode and it doesn't require information on second harmonic or using a special wide-band transducer. The method can be implemented directly into a medical scanner.

Plane wave imaging, nonlinear ultrasound, quantitative ultrasound, coefficient of nonlinearity

Streaming velocity mainly depends on the intensity and absorption of ultrasound in the media. For high frequencies exceeding 20 MHz the speed of streaming in blood is also affected by scattering effects on the blood cells and contrast agent microbubbles. According to theoretical calculations, 12.2 % increase in streaming velocity for 1 g/l starch concentration rise should be expected. The theory has also shown the reduction of the streaming velocity by low-density scatterers, estimated decrease was -9.7 % for BR14 contrast agent. The experimental measurements of streaming velocities were done in blood mimicking fluid and Bracco BR14 microbubbles dissolved in water. The streaming was generated by a plane 20 MHz ultrasonic transducer driven by a pulsed Doppler flowmeter. For starch concentration changing from 0.01g/l (reference fluid) up to 1g/l the streaming velocity increased by 13%, very close to the theoretical prediction. For BR14, the measured velocity was 9% less than in reference fluid and remained independent on the microbubbles concentration.

streaming, radiation force, starch, contrast agent, blood

A new toolbox for the simulation of acoustic wave fields is described. The toolbox, USim, is designed to make an acoustic modeling of ultrasound propagation in tissues reliable and fast. The forward simulations of the wave field are based on the Born-Neumann single scattering approximation of the solution of Sturm-Liouville equation. The toolbox allows simulating the ultrasound wave propagation in non-homogeneous media containing finite size scatterers similar to those existing in a real tissues. The approach of calculations conducted in Fourier space increases the efficiency and allows taking into account the absorption and density phenomena in a simple and correct way.

acoustic propagation, numerical simulations, fourier domain

Nowadays there are new modalities in ultrasound imaging allowing better characterization of tissue regions with different stiffness. We are proposing an approach based on simultaneous propagation of two waves being a combination of two pulses differing in pressure and frequency: a low frequency pulse is expected to change the local scattering properties of the tissue due to compression/rarefaction while a high frequency pulse is used for imaging. Two transmissions are performed for each scanning line. First, with the imaging pulse that propagates on maximum compression caused by a low frequency wave. Next, the low frequency wave is inverted and the imaging pulse propagates over the maximum rarefaction. After the processing of the subtracted echoes from subsequent transmissions including wavelet transform and band-pass filtering, differential images were reconstructed. The low frequency wave has a visible impact on the scattering properties of the tissue which can be observed on a differential image.

The main goal of this study was to verify the suitability of sonographic model of the left ventricle (LV) in Computed Tomography (CT) environment and compare radial strain calculations obtained by two different techniques: speckle tracking ultrasonography and Multislices Computed Tomography (MSCT). The Left Ventricular (LF) phantom was fabricate from 10% solution of the poly(vinyl alcohol) (PVA). Our model of the LV was driven by the computer-controlled hydraulic piston Super-Pump (Vivitro Inc., Canada) with adjustable fluid volumes. During cycle of the pump, the Stroke Volume (SV) of water was pumped into the LV phantom and returned to the pump, resulting in changing the inner and outer diameters of the phantom. The stroke volume was set at of 24ml. The fluid pressure was changed within range of 0-60 mmHg, and the pulse rate was equal 60 cycles/per minute. The relationships between computer controlled left ventricular wall deformations and its visualizations of the echocardiographic and CT imaging, both in the normal and pathological conditions were examined. The difference of assessment the Radial Strain between two methods was not exceeding 1.1%.

The Synthetic Aperture (SA) method provides a new solution in ultrasound diagnostics. It has particular importance in applications where frame rate and image resolution are crucial. Our new approach named Cumulative Synthetic Transmit Aperture (CSTA) allows optimizing SA in terms of memory size and computational power. The proposed CSTA algorithm requires 25 times less memory than a reference STA method for 64 elements transducer. This makes feasible implementation of CSTA on a low-power embedded GPU.

ultrasonic imaging, synthetic aperture, GPU

Scattering-based ultrasonic methods potentially enable assessment of microscopic structure of bone. In our previous study we have shown how to account the higher orders of scattering in the backscatter evaluation. Here, the role of multiple scattering in bone backscatter is assessed for the varying bone porosity. The set of integral equations corresponding to the singular Sturm–Liouville equation was solved numerically allowing for the calculation of the field scattered forward and backward in the bone model. The results indicate that for the porosities below 85% the approximation of the first order scattering is not valid for all considered frequencies. The Laue’s equations were applied to explain considerable increase of the scattering at some frequencies. It was shown that the deformation of trabecular mesh results in flattening of the scattering peaks

trabecular bone, ultrasounds scattering, multiple scattering

The main aim of this work was numerical modeling of temperature fields induced in soft tissues in vivo by pulsed focused ultrasound during neurodegenerative disease treatment and experimental verification of the proposed model for a rat liver. The new therapeutic approach to neurodegenerative diseases consists of stimulation of enhanced expression of the Heat Shock Proteins (HSP) which are responsible for immunity of cells to stress. During therapy the temperature rise in tissues in vivo should not exceed 6 oC above level of the thermal norm (37 oC). First, the 3D acoustic pressure field, and the rate of heat production per unit volume due to that field, were calculated using our 3D numerical solver capable of predicting nonlinear propagation of pulsed high intensity waves generated from circular focused acoustic sources in multilayer configuration of attenuating media. The two-layer configuration of media (water - rat liver) assumed in calculations fairly well approximated both the real anatomic dimensions of rat liver and the geometric scheme of our experimental set-up. A numerical solution of the Pennes bio-heat transfer equation which accounted for the effects of heat diffusion, blood perfusion and metabolism rates, was employed to calculate the temperature fields induced in the rat liver by the ultrasonic beam. The numerical simulation results were verified experimentally using a thermocouple inserted in the liver of a rat under anesthesia at the beam focus. The quantitative analysis of the obtained results enabled estimation of the effects of several acoustic and thermal parameters of the rat liver in vivo on the temperature rise, as well as determination of exposure time for ultrasonic beams with varied acoustic power generated by a 2-MHz circular transducer of 15-mm diameter and 25-mm focal length, in order to avoid the tissue overheating that leads to cells necrosis, which would be unacceptable in neurodegenerative disease treatment.

low power ultrasound, tissue heating, HSP expression enhancement, cell immunity, degenerative deceases treatment

Recent research has shown that beneficial therapeutic effects in soft tissues can be induced by the low power ultrasound (LPUS). For example, increasing of cells immunity to stress can be accomplished through the heat shock proteins (Hsp) expression enhancement induced by the low intensity ultrasound. The possibility to control the Hsp expression enhancement in soft tissues in vivo by pulsed focused low power ultrasound may be the potential new therapeutic approach to the neuro-degenerative diseases that utilizes the known feature of cells to increase their immunity to stresses through the Hsp expression enhancement. The controlling of the Hsp expression enhancement by adjusting the level and time of exposure to ultrasound energy would allow evaluating of the ultrasound-mediated treatment efficiency. Ultrasonic regimes are controlled by adjusting the pulsed ultrasound waves intensity, frequency, duration, duty cycle and exposure time. Our objective was to develop the numerical model capable of predicting in space and time temperature fields induced by a circular focused transducer generating tone bursts in multilayer nonlinear attenuating media and to compare the numerically calculated results with the experimental data obtained for soft tissues in vitro. The acoustic pressure field in multilayer biological media was calculated using our original numerical solver. For prediction of temperature fields the Pennes’ bio-heat transfer equation was employed. The measurements of temperature distributions along and across the acoustic beam axis were performed using the experimental facility containing the temperature-controlled 1% saline solution tank, the tested tissue sample immersed in the tank and the USB-TEMP module connected with 8 thermocouples positioned within the tissue sample along and across the acoustic axis. Temperature measurements in vitro were carried out in a fresh rat liver using the 15 mm diameter, 25 mm focal length and 2 MHz central frequency transducer generating tone bursts with the spatial peak temporal average acoustic intensity varied between 0.325 and 1.95 W/cm2, duration varied from 20 to 500 cycles at the same 20 % duty cycle and the exposure time varied up to 20 minutes. The measurement data were compared with numerical simulation results obtained under experimental boundary conditions. Good agreement between the theoretical and measurement results for all cases considered has verified the validity and accuracy of our numerical model. The accuracy of the proposed model most of all depends on the accuracy of the source pressure measurements, estimation of tissues attenuation coefficients and their frequency-dependence law as well as determination of the tissue thermal conductivity. The measurement results confirmed that for pulses considered the temperature rise induced in tissue depends on the duty cycle, not on the tone burst duration. Quantitative analysis of the obtained results allowed to show that, for example, for the acoustic intensity ISPTA = 1.1 W/cm2 (P0 = 0.184 MPa) exposure time of the fresh rat liver should not exceed 10 min to avoid cells necrosis (T > 43 oC).

low intensity focused ultrasound, soft tissues, temperature fields, ultrasonic regimes, therapy efficiency

The presented investigations were intended to compare the experimental results obtained from the heel bones in vivo with the results of simulations. Ultrasonic RF echoes were collected using two bone scanners operating at 0.58 MHz and 1.3 MHz. The simulations of the backscattered RF echoes were performed using the scattering model of the trabecular bone that consisted of cylindrical and spherical elements uniformly distributed in water. For each measured or simulated RF backscatter statistical properties of the signal envelope was determined. Experimental results suggest deviations from the Rayleigh distribution. Simulation results suggest that deviations from Rayleigh distribution depend on the variation of trabeculae diameters and the number of thin trabeculae. Experimentally determined deviations corresponded well with the deviations calculated from simulated echoes assuming trabeculae thickness variation equal to published histomorphometric study results.

trabecular bone, scattering, statistical properties

Trabecular bone consists of trabeculae which mechanical properties differ significantly from the surrounding marrow and therefore the ultrasonic wave is strongly scattered within the bone structure. The aim of the presented paper was the evaluation of the contribution of the first, second and higher order scattering (multiple scattering) into total scattering of ultrasounds in the trabecular bone. The scattering due to interconnections between thick trabeculae, usually neglected in trabecular bone models, has been also studied. The basic element in our model of trabecular bone was an elastic cylinder with finite-length and varying diameter and orientation. The applied model was taking into account variation of elements size and spatial configuration. The field scattered on the bone model was evaluated by solving numerically the integral form of the Sturm-Liouville equation that describes scalar wave in inhomogeneous media. For the calculated scattered fields the effective cross-sections as well as the Broadband Ultrasonic Backscatter (BUB) were determined. The influence of the absorption on scattering coefficients was demonstrate. The results allowed to conclude that within the frequency range from 0.5 to 1.5 MHz the contribution of the second order scattering to the effective backscattering cross-section is at least 500 times lower than the one due to the first order scattering. BUB, calculated under the same assumptions, is 20 times lower. Above the 1.5 MHz the fast growth of the BUB, calculated for the second order scattering, occurs.

Complex media, Modeling, Multiple scattering, Trabecular bone,Ultrasound

The new therapeutic approach to neurodegenerative diseases consisting in stimulation of enhanced expression of the Heat Shock Proteins (HSPs) responsible for immunity of cells to stress is proposed. During such therapy local temperature rises in tissues should not exceed 6 oC above the thermal norm (37 oC). Tissue heating at temperatures higher than 43 oC may lead to an inability of cells to divide or to their death. The main aim of this work is numerical modeling of temperature rises induced locally in soft tissues in vivo by pulsed focused ultrasound during neurodegenerative diseases therapy and experimental verification of correctness and accuracy of the proposed model in a rat liver. Both the pulsed acoustic fields and power density of heat sources produced locally in tissues due to those fields were calculated using our 3D numerical solver capable of predicting nonlinear propagation of pulsed finite-amplitude acoustic waves generated from circular focused sources in multilayer configuration of attenuating media with arbitrary attenuation law. The two-layer configuration of media (water - rat liver) assumed in calculations fairly well approximated both the real anatomic dimensions of the rat liver and geometric scheme of our experimental set-up. The Pennes bio-heat transfer equation accounting for the effects of heat diffusion, blood perfusion and metabolism rates was employed to calculate the temperature fields. The numerical simulation results were verified experimentally using a thermocouple positioned in the rat liver at the acoustic beam focus. The quantitative analysis of the obtained results enabled estimation of the effects of the source acoustic power, blood perfusion and metabolism rates on the temperature rises in the rat liver as well as determination of the level and time of exposure to ultrasound in order to avoid the tissue overheating above 43 oC that leads to cells necrosis. The ultrasonic 20-cycle tone bursts with 20 % duty cycle and initial acoustic pressure amplitude varied from 0.131 to 0.226 MPa were generated from a 2-MHz circular transducer with a 15-mm diameter and 25-mm focal length during 10-min exposure time. It was shown that for the 0.226-MPa source pressure the exposure time should not exceed 5 minutes in order to avoid the rat liver overheating. The thermal conductivity coefficient for the rat liver in vivo was determined to be 0.47.

pulsed focused ultrasound, tissues heating, HSPs expression enhancement, cells immunity, neurodegenerative diseases treatment

The semi-empirical scattering models of trabecular bone were developed and examined for their abilities to mimic the frequency dependent backscattering coefficient measured in the cancellous bone. In the simulation of the bone RF echoes the real properties of the bone and experimental conditions were taken into account. Three types of trabeculae mimicking scatterers were considered. First, the bone consisted of cylinders with varying thickness (Gamma distributed) within the population, was assumed. The next two cases accounted for the contribution of thick and thin trabeculae to the total backscattered signal. The second model assumed existence of two populations of the cylindrical scatterers significantly differing in the average value of Gamma distributed diameters. Finally, the mixed model composed of thick and thin trabeculae modeled respectively by cylindrical and spherical scatterers was examined. The last selection resulted from the similarity found between scattering on small sphere and finite cylinder. Calculated echoes demonstrated the usefulness of the mixed model. Frequency dependence of backscattering coefficient agreed well with the experimentally determined dependences. The study showed also that the amplitude histograms calculated using demodulated RF echoes deviate from the Rayleigh distribution when the variation of scatterers’ diameters increases.

trabecular bone structure, modeling, backscatter

In our previous study we have developed the simulation technique that enables determination of the ultrasound signal received at the pulse-echo transducer surface after interrogation of cancellous bone. The simulation can be applied for different scattering models of a trabecular structure. In this study we examined newly developed scattering models of the trabecular bone for their abilities to mimic the frequency dependent backscattering coefficient measured in the cancellous bone. Three types of trabeculae mimicking scatterers were considered. First the bone consisted of cylinders with varying thickness (Gamma distributed) within the population, was assumed. The next two cases accounted for the contribution of thick and thin trabeculae to the total backscattered signal. The second model assumed existence of two populations of the cylindrical scatterers significantly differing in the average value of Gamma distributed diameters. Finally, the mixed model composed of thick and thin trabeculae modeled respectively by cylindrical and spherical scatterers was examined. The last selection resulted from the similarity found between scattering on small sphere and finite cylinder. Calculated echoes demonstrated the usefulness of the mixed model. Frequency dependence of backscattering coefficient agreed well with the experimentally determined dependences.

trabecular bone, scattering, bone model

Recently a new technology of piezoelectric transducers based on PZT thick film has been developed as a response to a call for devices working at higher frequencies suitable for production in large numbers at low cost. Eight PZT thick film based focused transducers with resonant frequency close to 40 MHz were fabricated and experimentally investigated. The PZT thick films were deposited on acoustically engineered ceramic substrates by pad printing. Considering high frequency and nonlinear propagation it has been decided to evaluate the axial pressure field emitted (and reflected by thick metal plate) by each of concave transducer differing in radius of curvature - 11 mm, 12 mm, 15 mm, 16 mm.
All transducers were activated using AVTEC AVG-3A-PS transmitter and Ritec diplexer connected directly to Agilent 54641D oscilloscope. As anticipated, in all cases the focal distance was up to 10% closer to the transducer face than the one related to the curvature radius. Axial pressure distributions were also compared to the calculated ones (with the experimentally determined boundary conditions) using the angular spectrum method including nonlinear propagation in water. The computed results are in a very good agreement with the experimental ones. The transducers were excited with Golay coded sequences at 35-40 MHz. Introducing the coded excitation allowed replacing the short-burst transmission at 20 MHz with the same peak amplitude pressure, but with almost double center frequency, resulting in considerably better axial resolution. The thick films exhibited at least 30% bandwidth broadening comparing to the standard PZ 27 transducer, resulting in an increase in matching filtering output by a factor of 1.4-1.5 and finally resulting in a SNR gain of the same order. Examples of skin scans obtained with the new thick-film transducers are presented.

transducers, thick film, high frequency ultrasound, pulse compression, Golay code

The theoretical and experimental studies of the finite amplitude acoustic waves propagation in attenuating media from nonaxisymmetric sources rather rarely can be found in literature in spite of the fact that probes of the rectangular geometry (such as linear phased arrays) are commonly used in clinical practice for medical ultrasonic imaging purposes. The main reason of such situation is a lack in simpler theoretical models and in computationally efficient numerical algorithms that are able to predict accurately the nonlinear effects in 4D ultrasound fields from pulsed, arbitrarily shaped sources (plane and focused) in biological media with arbitrary frequency-dependent absorption. In recent years the only study describing the computationally efficient numerical model that is able to simulate accurately the 4D nonlinear ultrasound field in water and in biological tissues from pulsed nonaxisymmetric sources was developed by Zemp et al. [1]. Their model is based on the second order operator-splitting method, proposed by Tavakkoli et al., with the modified fractional step scheme whereby the combined effects of diffraction and absorption are accounted for over half-steps and the effects of nonlinear harmonic interactions over full incremental steps. The computation of diffraction and absorption sub-steps was based on the angular spectrum technique with modified sampling method (to obtain computational savings due to larger axial propagation steps) while the computation of nonlinear steps was based on the time-domain solution to Burgers' equation. There are not reports yet describing an experimental confirmation of an agreement between the simulated nonlinear acoustic pulsed fields in water or in soft tissues from nonaxisymmetric focused sources (obtained by using the numerical model proposed) and nonlinear field from realistic probes.

Apertures, Numerical simulation, Acoustic pulses, Absorption, Ultrasonic imaging, Computational modeling, Acoustic propagation, Focusing, Phased arrays, Nonlinear acoustics

Tissue Harmonic Imaging (THI) was introduced in 1997 [1] and is today routinely used in diagnostic ultrasound. The benefits of harmonic imaging in many clinical situations have been proved. However, it has to be stressed that it is away from optimal because only half of the available transducer bandwidth is used for image formation - lower half for transmission and upper half during reception.

Ultrasonic imaging, Image resolution, Polarization, Absorption, Image coding, Pulse inverters, Dynamic range, Nonlinear equations, Boundary conditions, Propagation losses

Harmonic imaging at frequencies up to 15 MHz is now routinely used in clinical practice and frequencies well beyond 20 MHz are considered for diagnostic ultrasound imaging applications. However, currently available measurement tools are not fully adequate to characterize such high frequency systems, primarily due to the combined effects of limited frequency responses and spatial averaging effects. To alleviate this problems, a comprehensive wave propagation model has been developed and tested. The model can predict the linear and nonlinear acoustic wave propagation generated by differently shaped acoustic radiators at virtually any point in the field and takes into account spatial averaging effects introduced by hydrophone probes and their associated frequency responses. The applicability of the model in hydrophone probe calibration up to 100 MHz is demonstrated. Also, a novel calibration technique termed Time-Gating Frequency Analysis (TGFA) is briefly described and calibration results in the frequency range up to 60 MHz for hydrophones having effective diameters between 150 and 500 /spl mu/m are presented. Also presented are the results of the investigation that determined the effect of using hydrophone probes of different diameters and bandwidth on Spatial-Peak Pulse-Average Intensity (I/sub SPPA/). It was found that the values of I/sub SPPA/ increased with decreasing effective aperture of the hydrophone probe and its bandwidth.

Ultrasonic imaging, Frequency, Sonar equipment, Probes, Calibration, Acoustic propagation, Nonlinear acoustics, Bandwidth, Acoustic measurements, Current measurement

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 theoretical calculations were verified using a purpose-designed 32 MHz pulsed Doppler unit. The applied average acoustic power was changed from 1 /spl mu/W to 6 mW. The experiments were done on 4 mm diameter flat and focused transducers. The streaming velocity was measured along the ultrasonic beam from O to 20 mm. 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 grams to 40 grams of starch in 1 litre of distilled water. For applied acoustic powers, the streaming velocity changed linearly from 0.2 to 40 mm/s. Theoretical solutions for both plane and focused waves agreed with experimental results.

Acoustic beams, Poisson equations, Acoustic transducers, Ultrasonic transducers, Differential equations, Ultrasonic variables measurement, Acoustic waves, Impedance, Navier-Stokes equations, Boundary conditions

Acoustical Driving Forces (ADF), induced by propagating waves in a homogeneous and inhomogeneous lossy fluid (suspension), are determined and compared depending on the concentration of suspended particles. Using integral equations of the scattering theory, the single particle (inclusion) ADF was calculated as the integral of the flux of the momentum density tensor components over the heterogeneity surface. The possibility of negative ADF was indicated. Originally derived, the total ADF acting on inclusions only, stochastically distributed in ambient fluid, was determined as a function of its concentration. The formula for the relative increase in ADF, resulting from increased concentration was derived. Numerical ADF calculations are presented. In experiments the streaming velocities in a blood-mimicking starch suspension (2 mu m radius) in water and Bracco BR14 contrast agent (SF6 gas capsules, 1 mu m radius) were measured as the function of different inclusions concentration. The source of the streaming and ADF was a plane 2 mm diameter 20 MHz ultrasonic transducer. Velocity was estimated from the averaged Doppler spectrum obtained from originally developed pulsed Doppler flowmeter. Numerical calculations of the theoretically derived formula showed very good agreement with the experimental results.

Numerical analysis of the reflection/transmission problem for a non-linear acoustic wave is studied. The wave is a plane wave and it is incident normally on the plane discontinuity surface between two lossy media. Numerical calculations are proceeded with the help of self written software (in Fortran). The influence of different propagation parameters (properties of two different media) on the reflected and transmitted wave fields are discussed. Particularly, it is shown that although two media have the same impedance, the effect of the propagation nonlinearities is still existing in the reflected and transmitted fields. The performed analysis qualitatively confirmed theoretical predictions quite well.

Nonliear reflection, transmission, Riccati equation, Numerical caslculations