Institute of Fundamental Technological Research

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

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

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 propagation of heat was analyzed in the lens of an acoustic microscope used for testing of living cells at the frequency of 1~GHz. Information concerning the propagation of heat is necessary for determination of thermal boundary conditions which influence the temperature increase in the tested samples representing acoustical properties of water. The time of temperature propagation from water, heated due to high absorption, to the sapphire body of the lens was estimated to be 0.77 ms. To carry out these calculations the derivations of Carslow and Jaeger [2] and of Tautz [6] were adjusted. On the other hand the propagation time of the acoustic wave in the sapphire body equalled 0.0093 [ampersand]mu;s only. The time of image formation in the microscope is rather long being equal from one to several seconds due to mechanical inertia of the support vibrating together with the tested sample. The heat capacities of the water volume and the sapphire body were found to be comparable. However, if the heat capacity of the water volume would be many time smaller then the time of the finally attained temperature would be elongated. This effect can be neglected since the time of image formation is 3 orders of magnitude longer than the time of penetration of the sapphire body by the heat supplied by water. As the result a temperature equilibrium will be obtained with the average boundary temperature of water. In such a case no heat flux will penetrate the boundary water - sapphire and the condition of the thermal insulation at the boundary will be fulfilled. This thermal boundary condition makes it possible to determine the real temperature increase in biological specimens.

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.

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

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

Examination of the carotid artery stenosis is very important in the diag-nosis of cerebrovascular diseases. New possibilities in the diagnosis of stenotic lesions are provided by ultrasonic Doppler angiography. The aim of this paper is to present a Doppler imaging system developed by the authors for the exami¬nation of blood flowing in carotid arteries. The system is based upon a 5 MHz bi-directional c.tv. Doppler flowmeter with a separate output for anterograde and retrograde flows. A special bank of filters converts signals into various levels of the gray-scale display which correspond to the value of the blood flow velocity. The ultrasonic probe is held by the scanning arm. The position of the probe on the skin of the patient is electronically sensed by the position-sensing circuity which causes the bright spot on the image display to move according to the position of the probe. The Doppler image from the artery is stored in a digital memory system. The clinical results obtained by means of this system showed good agreement with X-ray arteriography for obstructions occluding more than 50 per cent of the arterial diameter.

The reflection of a plane ultrasonic wave from a gas bubble in the blood was considered quantitatively in the range ka < 1 (a is the radius of the bubble, k = 27s/A and is the wavelength). Taking as an example the elbow vein (vena basilica), the losses in a signal caused by electroacoustical transducing, attenua¬tion of the wave in tissues, reflection of the wave from the bubble and diver¬gence of the reflected wave were evaluated using a typical ultrasonic Doppler device with a frequency of 8 MHz.
It was shown that a single gas bubble with the radius 1.6 tim already gives a signal which is received by the device; this signal, however, is masked by the signal caused by the scattering of the wave by blood cells and, in addi¬tion, this bubble is instable. Determination of the level of signals scattered in blood indicated that in the case investigated it is possible to detect in the vein a single gas bubble with its radius greater than 16 vm or gas bubbles with a radius of 11 m lying at the stability limit provided that their density exceeds 35 cm-3.
The present calculation procedure permits the determination in a specific anatomic case of the level of signals scattered by blood, the level of electronic noise and also the determination of the detectability of gas bubbles in blood caused, for example, by the decompression of divers or by the caisson disease. In the case of the pulmonary artery, using a frequency of 5 MHz, minimum radii of detectable gas bubbles greater than 70 pm were obtained.

In order to analyse the possibility of the determination of the properties of soft tissues using ultrasonic (noninvasive) methods, a review was made of the basic physical properties characterizing acoustic wave propagation in these tissues (propagation velocity, attenuation, scattering). In a discussion of the results of "in vitro" investigations, it was shown that investigations of backscattering can be very significant for cardiological applications; the heart muscle with scars caused by an infarct is characterized by an increased pro-portion of collagen-rich connective tissue which has the value of the back-scattering coefficient greater by a factor of some dozens than that of normal muscle. This is related to the higher echographic visualizability of collagen than other soft tissues and suggests the practical possibility of noninvasive distinguishing of the regions of the muscle with scars caused by the infarct from normal muscle. This possibility was confirmed by "in vivo" investiga-tions performed on dogs by the method of grey level histograms obtained from ultrasonograms of dog's hearts.
This paper has the character of a review.

The detectability of a small, hypothetical cylinder-shaped blood vessel with a diameter of 0.1 mm has been considered analytically using the ultrasonic echo method. Soft tissues surrounding the vessel have been taken as homoge¬neous and not causing reflections of ultrasonic waves. They have been ascribed both with bulk and shear elasticity. A beam of longitudinal ultrasonic waves incident on the vessel has been taken in the form of a focused beam at the focus of which the blood vessel has been placed. It has been assumed that the reflection of ultrasonic waves from the blood vessel is caused by the difference between the velocity of waves in the tissue surrounding the vessel and that in blood.
Assuming a frequency of ultrasonic waves of 2.6 MHz, a diameter of the transmit-receive piezoelectric transducer of 2 cm, a focal length of this transducer of 10 or 8 cm, voltage of the transmitter of 250 V and sensitivity of the receiver of 10-5V, the conditions of detectability have been determined.
It has been shown that the signal of an echo from the blood vessel assumed is potentially detectable. Its magnitude depends critically on the distance betwe¬en the vessel and the surface of the body, resulting from the attenuation of waves in tissues penetrated.

The detectability of a small blood vessel of a radius 0.1 mm by the pulse echo ultrasonic method using a frequency of 2.5 MHz was estimated. It was assu¬med that the vessel was surrounded by a homogeneous soft tissue (i.e. causing no reflection) and was in the near region of the far field radiated in a continuous manner by a plane transducer of a diameter of 2 cm. The soft tissue and the walls of the vessel were assumed to have the same elastic properties as those of a li¬quid.
The measurements were carried out in a plane polar coordinate system, where the incident wave, the reflected wave and the wave penetrating into the vessel were expressed in terms of Bessel and HankeI functions. The boundary conditions were assumed in the form of the equality of the acoustic pressures and the normal components of acoustic velocitits on each side of the surface of the vessel. Thence the magnitude of the reflected wave was determi¬ned.
The losses of the signal due to the reflection of the wave, its divergence and absorption, are shown in the form of a graph from which it can be seen that the signal from the vessel considered is essentially detectible, although it lies near the noise level, and is critically dependent on the distance from the transducer due to attenuation in the tissues penetrated.
The detectability of the plane boundaries of soft tissues was also de-termined, indicating that at a distance of 20 cm from the transducer a difference in the characteristic acoustic impedance of the tissues of 0.2% is sufficient to give a detectable echo.

This paper describes the wave phenomena occurring in a needle used for puncture of body organs, under the simplifying assumption that this needle is an ideal elastic cylinder immersed in an ideal liquid.
Investigations carried out by the author using an echo method showed' that the velocity of the wave propagating in the needle immersed in water is close to the velocity of the wave propagating in water.
The author has analysed the propagation of waves along a cylindrical rod of infinite length immersed in a liquid, solving the wave equations for displacement potentials in the rod surrounding liquid, taking into consideration the boundary conditions on the rod surface. It was found that it is possible for the velocity of the propagating wave to be lower than the velocity of the wave in water, with the wave being guided by the rod and the surrounding liquid layer. The characteristic equation obtained was solved numerically for a 1.5 mm diameter steel rod immersed in water at wave frequencies of 3 and 5 MHz. Stress distributions, acoustic pressure and the propagating wave dis¬placements were determined. It can be concluded from the character of the wave that it is a surface wave.
The results obtained can be used as the first approximation to the problem of wave propagation along a needle in the case where the needle wall thickness and the frequency are adequately large.

Temperature increases on the body surface from the use of two different ultrasonic Doppler blood flow meters were determined experimentally. Using a thermographic method it was shown that the temperature increase can be significant. Two continuous wave ultrasonic Doppler blood flow meters induced temperature increases on the body surface of 12.5°C and 2.3°C respectively, after an insonation time of 100 sec. Theoretical estimates and experimental results have shown an approximate agreement in the values of the temperature increase.

The thermal effect arising from the sonification of soft tissues by an ultrasonic beam, cylindrical in shape, with the assumption of an even distri¬bution of heat sources in the beam, has been considered.
The analysis is based on the solution of the thermal conductivity equation, using the Laplace transformation as in the author's paper [1]. The formulae obtained permit determination of the rise in and distribution of temperature inside and outside the ultrasonic beam for sonification times longer than 20 s.
The formulae have been applied to estimate the temperature changes encountered in ultrasonic continuous wave Doppler methods used in medical diagnosis. For example, with a cylindrical ultrasonic beam of radius 2.2 mm, frequency 5 MHz, mean spatial intensity of 0.1 W/cm2 and sonification time of 100 s, the estimated value of the temperature increase at the centre of the beam was 1.8 °C.
The values obtained are overestimated since they do not consider the transfer of heat by the circulating blood or the thermal conductivity along the ultrasonic beam, which is particularly evident for higher frequencies.

The authors applied two absolute methods for the determination of the intensity of ultrasonic focused beams of 2·5 MHz frequency used in ultrasonograph UG-4 for abdominal visualization: the electrodynamic and the capacitance methods. For the first method a special electrodynamic transducer was constructed which made it possible to measure the acoustic (particle) velocity in the focal point of the ultrasonic beam radiating into water. Hence the focal intensity could be determined.

The two methods gave almost the same intensity values equal to 23 Wcm−2 in peak, (∼ 23 mWcm−2 time average value) although the mean surface value near to the radiating transducer was equal only to 0·59Wcm−2 in peak (∼0·59 MWcm−2 time average value).

Next, the loss of the ultrasonic signal was measured in pregnant women in vivo using for this purpose steel balls as reflecting targets placed in the gravid uterus. The echo amplitudes obtained in the uterus were compared with echoes received from the same steel balls placed in (nonabsorbing) water in the same position relative to the radiating transducer.

In this manner it could be found that intensity levels inside of the uterus were 6–14 dB lower than at the body surface, depending on the anatomical characteristics of the patient.

Thus the maximum (focal) intensities which are to be expected in the gravid uterus in early pregnancy will be between 0·9 and 5·5 Wcm−2 in peak (0·9–5·5 mWcm−2 time average values) when applying the ultrasonograph UG-4.

The authors present results of quantitative measurements of blood in vivo in the carotid artery of man. The Doppler pulse technique was used after being previously verified for steady state flows in tubes and for pulsating flows in a canine aorta where the electromagnetic method was also used for comparison.

An ultrasonic probe with two transducers was adapted for determination of the angle of the ultrasonic beam in relation to the vessel allowing the measurement of the vessel diameter which was also determined by means of the ultrasonographic B-mode technique.

By means of the Doppler pulse method profiles of the blood velocity in the carotid artery were determined as a function of time.

The continous wave Doppler technique together with the zero-crossing system and spectral analysis were also used for making measurements.

The flow velocity and the shape of the flow curve with time obtained with the above techniques showed good agreement. The measured flow rate in the carotid artery amounted to QM = 1.61/min (maximum instantaneous value) and Q0 - 0.531/min (mean time value).