Singleton Focusing Ultrasound Emitters

Single-element focusing ultrasound emitters are devices that create focused ultrasound beams and are made in the form of a single piezoelectric emitting element , the surface of which in most cases is given a spherical or cylindrical shape ^[1] . The most widely used are the so-called spherical focusing emitters, which are in shape a spherical segment in the form of a cup, whose diameter is much greater than the ultrasound wavelength ^[1] . In such systems, the wavefront converging into focus has an initially spherical shape, which leads to the concentration of ultrasound energy in the focal region. The diameter of the focal region is much smaller than the diameter of the emitter and is comparable in order of magnitude with the wavelength of ultrasound. Due to this feature, the intensity of the ultrasound in focus significantly exceeds the intensity on the surface of the source. Along with single-element emitters, focused beams can be created more complex in arrangement and control of multi-element emitters ( phased antenna arrays ), which are not considered here.

The single-element focusing ultrasound emitters are most widely used in clinical and experimental medicine ^[2] . Typically, the intensity on the surface of piezoceramic ultrasound emitters during continuous operation does not exceed 10 W / cm ² , and with good cooling of the emitter - 20-40 W / cm ² . Record intensity values obtained on the surface of piezoelectric plates in the continuous radiation mode reach 300 W / cm ² ^[3] . At the same time, when using modern focusing systems, including single-element emitters, it is not difficult to obtain in the focus hundreds and thousands of times higher ultrasound intensities and reach levels on the order of thousands and tens of thousands of watts per 1 cm ² ^[1] ^[4] . This allows using certain parameters of ultrasonic exposure to induce non-invasively in the deep tissues of the body a variety of stimulating and therapeutic effects, as well as create destruction of a predetermined size without damaging the surrounding tissues, which is extremely important for medicine ^[2] ^[4] .

History

Focusing emitters made of quartz , whose surfaces were concave, were first proposed in the mid-1930s ^[5] ^[1] . In 1942, one of the first such emitters was used in experiments on liver samples and when exposed to focused brain structures of animals through the skull with focused ultrasound ^[6] . Such emitters were not only expensive and difficult to manufacture, but also did not allow to create the correct in-phase wavefront due to the direction of the piezoelectric properties of quartz. Since the electrical axis of the crystal forms different angles with the normal at different points on the spherical surface, the radiated energy is unevenly distributed over its surface ^[1] . Therefore, it is impossible to make a focusing emitter from quartz with a large surface curvature ^[7] .

Focusing emitters for use in medicine, manufactured in the late 1950s, were made on the basis of flat quartz emitters with focusing lenses made of plastic ^[8] ^[9] ^[10] . In the laboratory of prof. W. Fry ( Eng. W. Fry ), USA, a design consisting of four focusing emitters was used, the relative position of which was adjusted so that the focal regions of all emitters coincided with each other ^[9] ^[11] .

A significant drawback of such focusing systems is not only the design complexity, but also that up to 40% of the emitted acoustic energy is absorbed in the focusing lens ^[9] . In addition, due to the difference in the acoustic impedance of the lens and the propagation medium, part of the wave energy is reflected from the interfaces and is not focused properly. Overheating of the lenses and, as a result, their damage is especially pronounced at high frequencies and high intensities of ultrasound. Despite these drawbacks, focusing with lenses is actively used in modern focusing devices. So, in the ultrasonic sensors used in medical diagnostics, cylindrical lenses are used to specify the scanning area as a thin layer ^[12] ^[4] . Lens focusing is used in one of the designs of electromagnetic shockwave lithotripter ^[13] ^[14] . With the advent of three-dimensional printing, the manufacture of acoustic lenses has become easier and therefore their further use can be expected.

Another method for creating focused acoustic fields is a method based on the reflection of plane or spherically diverging waves from concave surfaces. Historically, such a focusing method has been widely used in extracorporeal shock wave lithotripsy. So, in electro-hydraulic lithotripters, an electric discharge in water is used as a source of a shock wave ^[13] ^[14] ^[15] . To focus this pulse on a kidney stone, a metal reflector is used, the surface of which is made in the form of a semi-ellipsoid of revolution. An electric discharge is produced in one of the focal points of the ellipsoid, and the target (kidney stone) is installed in another focus. A spherically diverging powerful acoustic pulse excited by an electric discharge turns into a focused wave that converges in the second focus of the ellipsoid ^[13] ^[14] . Another type of reflector is used in the construction of an “electromagnetic” lithotripter, in which a cylindrical membrane pulses a cylindrically diverging wave pulsed by a magnetic field. This wave is directed to the reflector with a profile formed when the parabola rotates around an axis passing through the focus and perpendicular to the parabola axis ^[16] . With this shape of the reflecting surface, the cylindrical wave converges at the focus of the parabola, which is aimed at the kidney stone; a similar approach is used in the construction of some therapeutic ultrasonic applicators ^[17] .

Appearance of a typical single-element piezoceramic emitter

The disadvantages of using reflectors for focusing ultrasound are the bulkiness of the structure and the losses associated with non-ideal reflection.

The disadvantages inherent in acoustic lenses and reflectors, the principle of operation of which actually copies the classical optical approaches, have largely eliminated acoustical-specific spherical emitters based on concave plates made of piezoceramics, which they began to use for medical purposes in the late 1960s ^[1] ^[18] ^[19] ^[20] ^[7] . Since then, the use of concave piezoceramic plates as a radiating element has become a generally accepted way of constructing single-element focusing ultrasound emitters. In addition to the obvious advantages in cost and manufacturing technology of such emitters, they are more preferable, since the direction of the piezoelectric axes created by the polarization at each point coincides with the direction to the center of curvature.

Acoustic Field Calculation Methods and Basic Relationships

The theory of sound focusing systems has been dealt with by a number of foreign researchers ^[21] ^[22] ^[23] . A large role in the development of the theory and methods of calculating focusing systems was played by the books of prof. L. D. Rosenberg ^[1] ^[24] , as well as the work of his students (I. N. Kanevsky, K. A. Nugolnykh, E. V. Romanenko, M. G. Sirotyuk). They defined criteria that allow a rational choice of focusing systems, studied the properties of the focal region, studied the structure of the acoustic field, etc. In the monographs of L. D. Rosenberg ^[1] ^[24] and I. N. Kanevsky ^[25] summed up the results of their own research, as well as summarizing the results of previous theoretical and experimental works of domestic and foreign authors, dealing with the problems of focusing ultrasonic waves. The use of focusing ultrasound emitters in medicine and physiology has been discussed in a number of books and reviews ^[26] ^[7] ^[27] .

To calculate the acoustic fields of focusing systems, including spherical emitters, a method based on the use of the Rayleigh integral ^[21] is often used. The essence of this method is that the radiating surface is considered as a set of elementary radiators of infinitely small size, emitting diverging spherical waves. Then the total complex sound pressure of the emitter at each point in the field is determined as the sum of the contributions made by each elementary emitter. In the calculations, the Rayleigh integral is represented approximately in the form of the sum of the contributions from individual elements of finite size. As such elements, for example, small square emitters ^[28] ^[29] , or elements in the form of rings of equal area ^[30] into which the radiating surface is divided, are most often chosen. As a result, the complex amplitude of the acoustic pressure $p$ ${\ displaystyle p}$ focusing emitter, the surface of which harmoniously oscillates at a frequency according to the law $\sim \exp(-i2\pi ft)$ ${\ displaystyle \ sim \ exp (-i2 \ pi ft)}$ , can be found according to the expression ^[29] :

p=-i{\frac {\rho ck}{2\pi }}\sum _{n}{\frac {v_{n}e^{(-\alpha +ik)R_{n}}}{R_{n}}}\Delta {}A_{n},

{\ displaystyle p = -i {\ frac {\ rho ck} {2 \ pi}} \ sum _ {n} {\ frac {v_ {n} e ^ {(- \ alpha + ik) R_ {n}} } {R_ {n}}} \ Delta {} A_ {n},}

Where $i$ ${\ displaystyle i}$ - imaginary unit $\rho$ ${\ displaystyle \ rho}$ - tissue density $c$ ${\ displaystyle c}$ Is the speed of sound in the tissue, $k=2\pi f/c$ ${\ displaystyle k = 2 \ pi f / c}$ Is the wave number $v_{n}$ ${\ displaystyle v_ {n}}$ - the complex amplitude of the normal component of the vibrational velocity on the surface $n$ ${\ displaystyle n}$ elementary emitter $\Delta {}A_{n}$ ${\ displaystyle \ Delta {} A_ {n}}$ Is the area of this emitter, $\alpha$ ${\ displaystyle \ alpha}$ - attenuation coefficient in the fabric and $R_{n}$ ${\ displaystyle R_ {n}}$ - the distance from the center of the elementary emitter to the point where the field is calculated.

Geometrical parameters of a single-element focusing emitter:

F

{\ displaystyle F}

- focal length,

a

{\ displaystyle a}

Is the radius of the emitter,

h

{\ displaystyle h}

- depth

\alpha _{m}

{\ displaystyle \ alpha _ {m}}

- half the opening angle,

r_{0}

{\ displaystyle r_ {0}}

Is the radius of the focal region,

l

{\ displaystyle l}

- the length of the focal region

Animation of a focused acoustic field created by a singleton source. The color shows the acoustic pressure values normalized to the amplitude of the wave in focus

In some cases, the Rayleigh integral can be used as an analytical method for calculating the acoustic fields of emitters. Such an analysis, for example, can be carried out for a practically important emitter, which in shape is a part of a spherical bowl uniformly oscillating in thickness ^[1] . The main geometric characteristics in this case are the aperture radius $a$ ${\ displaystyle a}$ and focal length $F$ ${\ displaystyle F}$ , as well as the depth of the bowl depending on them $h=F(1-\cos \alpha _{m})$ ${\ displaystyle h = F (1- \ cos \ alpha _ {m})}$ and half opening angle $\alpha _{m}=\mathrm {arcsin} (a/F)$ ${\ displaystyle \ alpha _ {m} = \ mathrm {arcsin} (a / F)}$ . For such a source, the Rayleigh integral gives an exact expression for the complex amplitude of acoustic pressure along the axis of symmetry ^[21] :

p(r=0,z)=p_{0}{\frac {e^{ikz}-e^{ikR_{\mathrm {max} }}}{1-z/F}},

{\ displaystyle p (r = 0, z) = p_ {0} {\ frac {e ^ {ikz} -e ^ {ikR _ {\ mathrm {max}}} {1-z / F}},}

Where $r$ ${\ displaystyle r}$ - the transverse coordinate, counted from the axis, $z$ ${\ displaystyle z}$ - the distance along the axis from the center of the emitter, $p_{0}=\rho cv_{0}$ ${\ displaystyle p_ {0} = \ rho cv_ {0}}$ - the characteristic amplitude of the wave at the source, $v_{0}$ ${\ displaystyle v_ {0}}$ - the amplitude of the vibrational velocity of the surface of the emitter, $R_{\mathrm {max} }=F{\sqrt {1+(1-z/F)^{2}-2(1-z/F)\cos \alpha _{m}}}$ ${\ displaystyle R _ {\ mathrm {max}} = F {\ sqrt {1+ (1-z / F) ^ {2} -2 (1-z / F) \ cos \ alpha _ {m}}}}$ - the distance from the observation point to the edge of the emitter. From here, in particular, follows the expression for the gain $K_{p}=p_{F}/p_{0}=kF(1-\cos \alpha _{m})=kh$ ${\ displaystyle K_ {p} = p_ {F} / p_ {0} = kF (1- \ cos \ alpha _ {m}) = kh}$ where $p_{F}=p(0,F)$ ${\ displaystyle p_ {F} = p (0, F)}$ - the amplitude of the wave in focus.

Studies of the acoustic fields of focusing emitters show that through the focal region within the main diffraction maximum at small angles $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ there is an almost flat wave. Therefore, when calculating the intensity of the sound field $I$ ${\ displaystyle I}$ in the focal region, the relation for a plane wave is usually used ^[1] ^[31] : $I=|p|^{2}/(2\rho c)$ ${\ displaystyle I = | p | ^ {2} / (2 \ rho c)}$ . The following approximate expression for the axial dependence of the wave intensity in the focal region follows from the expression written above for the pressure amplitude:

I(r=0,z)=I_{F}\left({\frac {F}{z}}\right)^{2}\mathrm {sinc} ^{2}\left[{\frac {K_{p}}{2}}\left({\frac {F}{z}}-1\right)\right],

{\ displaystyle I (r = 0, z) = I_ {F} \ left ({\ frac {F} {z}} \ right) ^ {2} \ mathrm {sinc} ^ {2} \ left [{\ frac {K_ {p}} {2}} \ left ({\ frac {F} {z}} - 1 \ right) \ right],}

Where $\mathrm {sinc} (x)=\sin x/x$ ${\ displaystyle \ mathrm {sinc} (x) = \ sin x / x}$ , $I_{F}=K_{p}^{2}I_{0}$ ${\ displaystyle I_ {F} = K_ {p} ^ {2} I_ {0}}$ - intensity in focus, $I_{0}=p_{0}^{2}/(2\rho c)$ ${\ displaystyle I_ {0} = p_ {0} ^ {2} / (2 \ rho c)}$ - characteristic intensity on the surface of the emitter. Also in the focal plane $z=F$ ${\ displaystyle z = F}$ with a good approximation, the transverse intensity profile is also expressed

I(r,z=F)=I_{F}\left[{\frac {2J_{1}(kar/F)}{kar/F}}\right]^{2},

{\ displaystyle I (r, z = F) = I_ {F} \ left [{\ frac {2J_ {1} (kar / F)} {kar / F}} \ right] ^ {2},}

Where $J_{1}$ ${\ displaystyle J_ {1}}$ - Bessel function of the first kind of the first order. Such a transverse intensity distribution, which has the form of a circular spot with diffraction rings surrounding it, is known in optics as the Airy disk .

The characteristic distribution of the amplitude of the acoustic pressure in the field of the focusing source in the plane passing through the axis of symmetry (above) and in the focal plane (below). The amplitude increases when the color changes from black, through red and yellow, to white

The maximum intensity in the center of the focal region at not very large angles $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ ( $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ <45 °) can be expressed in terms of the area ratio of the source $\pi a^{2}$ ${\ displaystyle \ pi a ^ {2}}$ and cross-sectional areas of the focal region $\pi r_{0}^{2}$ ${\ displaystyle \ pi r_ {0} ^ {2}}$ ^[one]

I_{F}=3.7I_{0}{\frac {\pi a^{2}}{\pi r_{0}^{2}}}.

{\ displaystyle I_ {F} = 3.7I_ {0} {\ frac {\ pi a ^ {2}} {\ pi r_ {0} ^ {2}}}.}

The factor 3.7 indicates that the intensity in the center of the focal region is greater than the average intensity over the entire focal plane, and also takes into account that only 84% of the focused energy passes through the focal spot, and 16% falls on the fraction of secondary maxima ^[1] . If the opening angles are not too small, it must be taken into account that the gain in vibrational velocity $K_{v}$ ${\ displaystyle K_ {v}}$ slightly different from the pressure gain:

K_{v}=K_{p}\cos ^{2}{\frac {\alpha _{m}}{2}},

{\ displaystyle K_ {v} = K_ {p} \ cos ^ {2} {\ frac {\ alpha _ {m}} {2}},}

due to which the gain in intensity $K_{I}$ ${\ displaystyle K_ {I}}$ also different from $K_{p}^{2}$ ${\ displaystyle K_ {p} ^ {2}}$ :

K_{I}=K_{p}^{2}\cos ^{2}{\frac {\alpha _{m}}{2}}.

{\ displaystyle K_ {I} = K_ {p} ^ {2} \ cos ^ {2} {\ frac {\ alpha _ {m}} {2}}.}

From the obtained formulas for the intensity distribution, important simple relations follow for the sizes of the focal region: the radius of the focal region $r_{0}=0.61\lambda F/a$ ${\ displaystyle r_ {0} = 0.61 \ lambda F / a}$ and its length $l=2\lambda /(1-\cos \alpha _{m})$ ${\ displaystyle l = 2 \ lambda / (1- \ cos \ alpha _ {m})}$ where $\lambda =c/f$ ${\ displaystyle \ lambda = c / f}$ - wavelength of ultrasound. Both of these parameters are determined by the zeros of intensity closest to the focus. As an example, for a radiator with a resonant frequency of 1 MHz, with a radius and focal length of 42.5 and 70 mm, respectively, and an opening angle $2\alpha _{m}$ ${\ displaystyle 2 \ alpha _ {m}}$ = 75 °, diameter $2r_{0}$ ${\ displaystyle 2r_ {0}}$ and focal length $l$ ${\ displaystyle l}$ are respectively 3 and 15 mm, and pressure and intensity gains $K_{p}$ ${\ displaystyle K_ {p}}$ = 60 and $K_{I}$ ${\ displaystyle K_ {I}}$ = 3255, respectively ^[7] .

The simple relations described above make it possible, with practical accuracy, to determine the sizes of the focal region and the amplification factors of single-element focusing emitters. In most medical applications of focused ultrasound, when it is used to actively influence the medium, emitters are used whose diameter is approximately equal to the radius of curvature of the radiating surface, i.e., the angle $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ is approximately 30 ^about . The length of the focal region is approximately 5-6 times greater than its diameter. If the angle $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ is a smaller value, then the ratio of the diameter of the focal region to its length decreases and thereby the locality of the impact on the irradiated object worsens ^[7] .

When using sources of large wave sizes, the question arises of the applicability of the Rayleigh integral for calculating the fields generated by focusing emitters. Another problem is related to the role of the assumption of uniform distribution of vibrational velocity on the surface of focusing emitters, since this condition is almost never satisfied when using real emitters made of piezoceramics. A series of papers ^[32] ^[33] ^[34] ^[35] ^{[36] is} devoted to the study of these questions. Briefly, the results of these studies can be formulated as follows ^[36] . The acoustic field of concave piezoelectric ceramic sources of large wave sizes is incorrectly predicted by the widely used theoretical model based on the assumption of a uniform distribution of the velocity of the radiating surface. The main reason for the indicated discrepancy between theory and experiment is the inhomogeneous character of the oscillation surface velocity of the emitter due to the appearance of Lamb waves at the edge of the piezoelectric plate. They propagate from the edge to the center of the plate and lead to a change in the amplitude of the oscillation velocity by more than 10% (sometimes much more) compared to the amplitude of the thickness mode of oscillations of the piezoelectric plate. These errors are absent in the case of piezocomposite sources.

Nevertheless, the Rayleigh integral, in spite of its approximate nature in the case of a non-planar emitting surface, allows one to predict with high accuracy the radiation of a concave source of large wave sizes and therefore can be used to calculate the fields of focusing sources at moderate focus angles. The value of the diffraction correction to the Rayleigh integral can be calculated on the basis of a developed numerical algorithm ^[34] ^[35] .

More complicated for theoretical analysis is the case when the focused wave has such a high intensity that the effects of acoustic nonlinearity begin to appear. Nonlinear modes are typical of many modern focused ultrasound applications in therapy. Acoustic Pressure Linear Gain Mentioned Above $K_{p}$ ${\ displaystyle K_ {p}}$ can reach several tens and higher, which leads to the fact that in some systems of ultrasound surgery and lithotripsy the peak acoustic pressure at the focus reaches several tens of MPa, and the intensity reaches levels up to 10000-30000 W / cm ² ^[37] ^[38] . At such high levels of intensity, an acoustic wave begins to change the properties of the medium and therefore propagates differently than waves of small amplitude. In particular, the initial sinusoidal profile begins to distort, and at some distance the wave can even become a shock ^[39] ^[40] ^[41] . In spectral language, such a distortion means the generation of high-frequency harmonics, which, on the one hand, are more intensively absorbed, and on the other, they are better focused. Because of this, with increasing wave amplitude at the source, the gain in intensity first increases, and then begins to decrease. With a further increase in the wave amplitude at the source, the intensity at the focus ceases to increase, i.e., saturation occurs. Peak Pressure Saturation Level $p_{sat}$ ${\ displaystyle p_ {sat}}$ can be approximately expressed analytically and has the order $p_{sat}\sim \rho c^{2}/\beta$ ${\ displaystyle p_ {sat} \ sim \ rho c ^ {2} / \ beta}$ where $\beta$ ${\ displaystyle \ beta}$ Is the parameter of the acoustic nonlinearity of the medium ^[41] . A specific assessment of the saturation level differs somewhat in the cases of pulsed and harmonic sources ^[42] ^[43] . A more accurate analysis, which allows you to refine analytical estimates and describe all the features of focusing (distortion of the waveform, the appearance of shock fronts, difference in gain for peak positive and negative pressures, etc.), can be performed using numerical simulation ^[44] .

Constructions

Focusing emitters designed and manufactured at the Acoustic Institute in the 1970s and 80s.

Here is a description of the designs of spherical focusing emitters developed in the 1970s and 80s. at the Acoustic Institute of the Academy of Sciences of the USSR (AKIN) for use in medicine and physiology ^[7] ^[27] . As experience has shown, the use of emitters (and the generators supplying them) with the smallest possible dimensions and weight in each particular case is essential and sometimes crucial for medical applications of focused ultrasound. These factors play a particularly important role in the clinical use of focusing emitters.

As a radiating element of the focusing transducers, as a rule, piezoceramic plates were used, which were part of a spherical shell in shape. Brief technical characteristics of typical focusing emitters based on concave piezoceramic plates are as follows: plate diameter 20-85 mm; focal length 15-70 mm; angle $\alpha _{m}$ ${\ displaystyle \ alpha _ {m}}$ 20-36 °; resonant frequency from the range of 0.5-3 MHz; plate thickness 0.8-4 mm, depending on frequency; plate area 3-55 cm ² ; the diameter of the focal region is 1-6 mm, and its length is 5-23 mm, depending on the frequency. The maximum acoustic power on a plate with a diameter of 85 mm was 120 W in continuous mode and 800 W in pulse mode. The weight of the emitters ranged from 150 to 400 g, which made it possible to use the micromanipulator of a standard stereotactic apparatus for their controlled movement in three mutually perpendicular directions ^[7] ^[27] . Replaceable cones of various heights were put on the case of emitters, a thin sound-transparent plastic film was stretched on the outlet opening of which. A removable focus indicator was provided, the tip of which is aligned with the center of the focal region. The internal volume of the cone between the piezoceramic plate and the film was filled with degassed water.

In most emitters, the distance between the slice of the cone and the center of the focal region was unchanged and was determined by the conditions of the experiment. In a number of designs of focusing emitters, this distance could be varied within the required limits using a mechanical device mounted in the emitter body and moving the piezoceramic plate relative to the outlet of the cone ^[7] ^[27] .

In cases where emitters with a large active surface were required, which were difficult to make from one piezoceramic plate, the so-called “Mosaic” emitters, which are a set of single elements glued to a metal (eg, aluminum) half-wave shell in the form of a part of a sphere ^[1] ^[7] .

Until the mid-1990s As the material from which the active elements of the focusing emitters are made, various modifications of piezoceramics that work well on radiation (for example, lead zirconate-titanate , etc.) were used. However, then the significant advantages of using piezocomposite materials for this purpose were demonstrated ^[45] ^[33] ^[46] ^[47] ^[48] . So, a widespread composite material with a connectivity type of 1-3 consists of small rods of lead zirconate titanate in a low density polymer. The volume concentration of piezoceramics is from 20 to 70% ^[49] , and the acoustic impedance is approximately the same fraction of the impedance of lead zirconate titanate. It is believed that this material will become dominant in the development of medical converters in the 21st century ^[50] Among its advantages are not only reduced impedance, which allows for better matching with the tissue, but also relatively weak vibrations of the material in the transverse direction.

Here are the parameters of extracorporeal (that is, installed outside the human or animal body) focusing emitters used in various foreign research centers involved in the use of focused ultrasound in medicine. Institute of Cancer Research , Royal Marsden Hospital, Sutton, UK Institute of Cancer Research , The Royal Marsden Hospital , UK (prof. G. ter Haar et al.) The most commonly used prototype focusing emitter for clinical use ^[51] . The emitter is made on the basis of piezoelectric ceramics with a fundamental frequency of 0.57 MHz; work is carried out at the third harmonic, that is, at a frequency of 1.7 MHz. Focal length is 150 mm; the total diameter is 100 mm, and the active part of the plate is 84 mm. The dimensions of the focal region at half the maximum intensity in focus are as follows: length 19 mm, diameter 1.64 mm.

A group of French researchers ( INSERM, Lyon, France ; Prof. Katignolles, Dr. Chaplon and others) uses a variety of spherical emitters, in particular, emitters with a radius of 100 mm, an aperture of 100 mm, a frequency of about 1 MHz, made of piezoceramic ( P1-60, Quartz et Silice, Nemours, France ), as well as from piezocomposite 1-3 ( Imasonic Besancon, France ) ^[33] .

At the Boston Therapy Ultrasound Laboratory at Harvard Medical School , Brigham and Women's Hospital , Boston, USA various spherical emitters are also used, in particular with a diameter of 100 mm, a focal length of 80 or 100 mm and a frequency of 1.5 MHz, designed to destroy cancer tumors under MRI control ^[30] .

In the laboratory of prof. C. Kane University of Michigan , USA used a focusing emitter with a diameter of 63.5 mm and with the same focal length and with an opening for a diagnostic sensor with a diameter of 13 mm ^[52] . The ultrasound frequency was 1.44 MHz, and the maximum electric power at a matched load of 120 W, which made it possible to achieve a peak intensity in the focal region of 2000 W / cm ² .

In the last decade, focusing systems developed in China by HAIFU Technology Company, Chongqing University of Medical Sciences, have been widely used in ultrasound surgery using focused ultrasound , Chongqing, China. The technical characteristics of these systems based on a single-element transducer are as follows: frequency from 0.8 to 2.4 MHz, aperture 12-15 cm, focal length varies from 9 to 15 cm through the use of six interchangeable aluminum lenses, the peak value of the intensity in focus, measured in water under conditions free field, is from 5 to 15 kW / cm ² ^[53] . In the center of the emitter there was an opening for placing a diagnostic transducer for imaging tumors and monitoring the surgical operation in real time.

Along with extracorporeal emitters, intracavitary focusing systems for surgical treatment of the prostate have also found clinical application. The ultrasonic method for this purpose is based on the use of a transrectally single-element focusing transducer with a fixed focal length, mechanically moved parallel to the rectum wall. The greatest successes in the development and clinical use of this method are currently achieved by two research groups - in the USA and France. The first of them ( Focal Surgery Inc., Milpitas, Calif., USA ) developed the Sonablate device ^[54] for the destruction of prostate tissue using several interchangeable, mechanically moved up to 45 mm, single-element emitters with a frequency of 4 MHz and with different focal lengths (30, 35 and 40 mm). The second group ( TechnoMed, France ) created the Ablatherm device, in which a single-element focusing emitter with a diameter of 35 mm had a focal length of 35 mm and was excited at a frequency of 2.25 MHz ^[55] .

In medical applications of focused ultrasound, designs of focusing emitters with an aperture on the axis have become increasingly used in order to install an instrument sensor in it for ultrasonic imaging of the medium. The consequence of this is a decrease in the maximum intensity in focus, as well as a slight narrowing of the width of the region at half the maximum intensity and an extension of the same region in the direction of the acoustic axis. These issues are discussed at a quantitative level, for example, in ^[23] ^[56] .

Applications

The main field of application of focusing ultrasound emitters is medicine. Hundreds of articles and several books have been devoted to the medical applications of focused ultrasound ^[4] ^[2] ^[7] ^[13] ^[27] , see also High Intensity Focused Ultrasound in Medicine .

Although high-intensity ultrasound is widely used in industry, primarily for ultrasonic cleaning ^[57] ^[58] ^[59] , focusing emitters are used little in industry, probably because in this case it is not often necessary to carry out local exposure to a small volume, in advance specific area of the environment. Nevertheless, focusing emitters have found useful applications for spraying liquids, humidifying air, and creating aerosols ^[59] ^[60] . Focusing ultrasound emitters of much lower intensity are used in sound imaging, medical diagnostics and in ultrasonic non-destructive testing of materials ^[12] to increase sound pressure and improve resolution in the transverse direction.

Strengths and weaknesses

Существенным достоинством одноэлементных фокусирующих преобразователей с поверхностью в виде части сферической оболочки является относительная простота их конструкции, изготовления и практического использования. Однако существенным недостатком подобных фокусирующих систем является их фиксированное фокусное расстояние. Так как объём фокальной области излучателя обычно значительно меньше того объёма среды, на который требуется воздействовать, то должны быть предусмотрены средства для удобного механического перемещения излучателя относительно объекта. Для этой цели могут быть использованы современные автоматизированные механические системы (позиционеры). Однако и здесь имеются свои сложности. Если размеры области ультразвукового воздействия достаточно велики, то использование излучателей с фиксированным фокусным расстоянием не всегда является наиболее удачным выбором, даже если для их перемещения используются автоматизированные механические системы. Значительно более широкие возможности здесь, безусловно, имеют ультразвуковые фазированные решётки ^[2] .

Notes

↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ Розенберг, Л. Д. Фокусирующие излучатели ультразвука // В кн.: Физика и техника мощного ультразвука / Под ред. Л. Д. Розенберга. Prince 1. Источники мощного ультразвука. — М.: Наука, 1967. — C. 149—206.
↑ ¹ ² ³ ⁴ Гаврилов, Л. Р. Фокусированный ультразвук высокой интенсивности в медицине. — М.: Фазис, 2013. −656 c. — ISBN 978-5-7036-0131-2
↑ Буров, А. К. Получение больших интенсивностей ультразвука в жидкости // Акустический журнал. — 1958. — Т. 4, № 4. — С. 315—320.
↑ ¹ ² ³ ⁴ Ультразвук в медицине. Физические основы применения Под ред. К.Хилла, Дж. Бэмбера, Г. тер Хаар. Пер с англ. under the editorship of Л. Р. Гаврилова, В. А. Хохловой, О. А. Сапожникова. — М.: Физматлит, 2008, 544 с., с.67.
↑ Greutzmacher, J. Piezoelektrishe Kristall mit Ultrashall konvergenz // Zh. Phys. −1935. — V. 96. — 342.
↑ Lynn, YG, Zwemer, RL, Chick, AJ, Miller, AE A new method for the generation and use of focused ultrasound in experimental biology // Journ. Gener. Physiol. — 1942. -V. 26. — P. 179—193.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ Гаврилов, Л. Р., Цирульников, Е. М. Фокусированный ультразвук в физиологии и медицине. — Л.: Наука, 1980. — 199 с.
↑ Fry, FJ Precision high intensity focusing ultrasonic machines for surgery // Amer. J. Phys. Med. — 1958. — V. 37, № 3. — P. 152—156.
↑ ¹ ² ³ Aström, К.E., Bell, E., Ballantine, Н. Т., Heidensleben, E. An experimental neuropathological study of the effects of high-frequency focused ultrasound on the brain of the cat // J. Neuropathol. Exp. Neurol. — 1961. — V. 20, № 4. — P. 484—520.
↑ Lele, PP Production of deep focal lesions by focused ultrasound — current status // Ultrasonics. — 1967. — V. 5. — P. 105—112.
↑ Fry, FJ, Ades, HW, Fry WJ Production of reversible changes in the central nervous system by ultrasound // Science. — 1958. — V. 127, № 3289. — P. 83-84.
↑ ¹ ² Szabo, TL Diagnostic Ultrasound Imaging: Inside Out. 2nd Edition. — Oxford, UK, Academic Press (Elsevier), 2014. — p. 130.
↑ ¹ ² ³ ⁴ Бэйли, М. Р., Хохлова, В. А., Сапожников, О. А., Каргл, С. Г., Крам Л. А. Физические механизмы воздействия терапевтического ультразвука на биологическую ткань (Обзор) // «Акустический журнал» — 2003. — Т. 49, № 4. — C. 437—464.
↑ ¹ ² ³ Leighton, TG, Cleveland, RO Lithotripsy // Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine. −2010. -V. 224, № 2. — P. 317—342.
↑ Наугольных, К. А., Рой, Н. А. Электрические разряды в воде. - M .: Science. — 1971. — 155 с.
↑ Rassweiler, J., Henkel, TO, Kohrmann, KU, Potempa, D., Junemann, KP, Alken, P. Lithotripter technology: present and future // Journal of Endourology. — 1992. — V. 6, № 1. -P. 1-13.
↑ Синельников, Е. Д., Филд, Т., Сапожников, О. А. Закономерности формирования зоны термического разрушения при лечении фибрилляции предсердий катетерным методом ультразвуковой абляции // Акустический журнал. — 2009. — Т. 55, № 4-5. — С. 641—652.
↑ Warwick, R., Pond, J. Trackless lesions in nervous tissues produced by high intensity focused ultrasound (high-frequency mechanical waves) // J. Anat. — 1968. — V. 102, № 3. — P. 387—405.
↑ Fry, FJ, Heimburger, RF, Gibbons, LV, Eggleton RC Ultrasound for visualization and modification of brain tissue // IEEE Trans. on Sonics and Ultrasonics. — 1970. — V. SU-17, № 3. — P. 165—169.
↑ Авиром, В. М., Адрианов, О. С., Выходцева, Н. И., Гаврилов, Л. Р., Меринг, Т. А., Сиротюк, М. Г. Разрушение глубоких структур мозга с помощью фокусированного ультразвука // Журн. higher нервн. деят. — 1971. — Т. 21, № 5. — С. 1110—1113.
↑ ¹ ² ³ O'Neil, HT Theory of focusing radiators // J. Acoust. Soc. Am. — 1949. -V. 21, № 5. — P. 516—526.
↑ Kossoff, G. Analysis of focusing action of spherically curved transducers // Ultrasound in Med. and Biol. — 1979. — V. 5, № 4. — P. 359—365.
↑ ¹ ² Clarke, RL Modification of intensity distribution from large aperture ultrasound sources // Ultrasound in Med. and Biol. — 1995. — V. 21, № 3. — P. 353—363
↑ ¹ ² Розенберг,Л. Д. Звуковые фокусирующие системы. — М.: АН СССР, 1949. — 112 с.
↑ Каневский, И. Н. Фокусирование звуковых и ультразвуковых волн. — М.: Наука, 1977. — 336 с.
↑ Fry, FJ Intense focused ultrasound: its production, effects and utilization // In: Ultrasound: its applications in medicine and biology / FJ Fry ed. New York: Elsevier, 1978. Pt. 2. — P. 689—736.
↑ ¹ ² ³ ⁴ ⁵ Вартанян, И. А., Гаврилов, Л. Р., Гершуни, Г. В., Розенблюм, А. С., Цирульников, Е. М. Сенсорное восприятие. Опыт исследования с помощью фокусированного ультразвука. — Л.: Наука, 1985. — 189 с.
↑ Ocheltree, K., Frizzell, L. Sound field calculations for rectangular sources // IEEE Trans. Ultrason. Ferroelectr. Freq. Ctrl. — 1989. — V. 36, № 2. — P. 242—248.
↑ ¹ ² Goss SA, Frizell LA, Kouzmanoff JT, Barich JM, Yang JM Sparse random ultrasound phased array for focal surgery // IEEE Trans. Ultras. Ferroelectr. Freq. Ctrl. 1996. V. 43. № 6. P. 1111—1121.
↑ ¹ ² Cline, HE, Hynynen, K., Watkins, RD, Adams, WJ, Schenck, JF, Ettinger, RH, Freund, WR, Vetro, JP, Jolesz, FA Focused US system for MR imaging-guided tumor ablation // Radiology. — 1995. — V. 194, № 3. — P. 731—737.
↑ Bergmann, L. Der Ultraschall und seine Anwendung in Wissenschaft und Technik / Zurich, 1954. (Пер на русск. яз: Бергман Л. Ультразвук и его применение в науке и технике / М.: ИЛ. 1956. — 726 с.)
↑ Cathignol, D., Sapozhnikov, OA, Zhang, J. Lamb waves in piezoelectric focused radiator as a reason for discrepancy between O'Neil's formula and experiment // J. Acoust. Soc. Am. — 1997. — V. 101, № 3. — P. 1286—1297.
↑ ¹ ² ³ Cathignol, D., Sapozhnikov, OA, Theillere, Y. Comparison of acoustic fields radiated from piezoceramic and piezocomposite focused radiator // J. Acoust. Soc. Am. — 1999. — V. 105, № 5. — P. 2612—2617.
↑ ¹ ² Катиньоль, Д., Сапожников, О. А. О применимости интеграла Рэлея к расчёту поля вогнутого фокусирующего излучателя // Акустический журнал — 1999. — Т. 45, № 6. — С. 816—824.
↑ ¹ ² Сапожников, О. А., Синило, Т. В. Акустическое поле вогнутой излучающей поверхности при учёте дифракции на ней // Акустический журнал. — 2002. — Т. 48, № 6. — С. 813—821.
↑ ¹ ² Сапожников, О. А. Мощные ультразвуковые пучки: диагностика источников, самовоздействие ударных волн и воздействие на среду при литотрипсии. Дис.: докт. физ.-мат. n — М. 2008. −296 с.
↑ Wu, F., Wang, ZB, Chen, WZ, Zou, JZ, Bai, J., Zhu, H., Li, KQ, Xie, FL, Jin, CB, Su, HB, and Gao, GW Extracorporeal focused ultrasound surgery for treatment of human solid carcinomas: early Chinese clinical experience // Ultrasound Med. Biol. — 2004. -V. 30, № 2. -P. 245—260.
↑ Kreider, W., Yuldashev, PV, Sapozhnikov, OA, Farr, N., Partanen, A., Bailey, MR, Khokhlova, VA Characterization of a multi-element clinical HIFU system using acoustic holography and nonlinear modeling // IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control. −2013. -V. 60, № 8. — P. 1683—1698.
↑ Зарембо, Л. К., Красильников, В. А. Введение в нелинейную акустику (Звуковые и ультразвуковые волны большой интенсивности). — М.: Наука, 1966, — 519 с.
↑ Руденко, О. В., Солуян, С. И. Теоретические основы нелинейной акустики. — М.: Наука, 1975. — 287 с.
↑ ¹ ² Sapozhnikov, OA High-intensity ultrasonic waves in fluids: Nonlinear propagation and effects. / In: Power Ultrasonics. Applications of High-intensity Ultrasound, ed. by Gallego-Juarez, JA, and Graff, KF Chapter II. Woodhead Publishing Series in Electronic and Optical Materials, № 66. — Cambridge: Elsevier, 2015. — P. 9-35.
↑ Bacon, DR Finite amplitude distortion of the pulsed fields used in diagnostic ultrasound // Ultrasound Med. Biol. −1984. — V. 10, № 2. -P. 189—195.
↑ Руденко, О. В., Сапожников, О. А. Явления самовоздействия пучков волн, содержащих ударные фронты // Успехи физических наук. — 2004. — Т. 174, № 9. -С. 973—989.
↑ Карзова, М. М., Аверьянов, М. В., Сапожников, О. А., Хохлова, В. А. Механизмы насыщения нелинейных импульсных и периодических сигналов в фокусированных акустических пучках // Акустический журнал. — 2012. -Т. 58, № 1. — С. 93-102.
↑ Kluiwstra, JU, McGough, RJ, Cain, CA Therapeutic ultrasound phased arrays: practical consideration and design strategies // IEEE Ultrason. Symp. Proc. — 1996. — P. 1277—1280.
↑ Chapelon, JY, Cathignol, D., Cain, C., Ebbini, E., Kluiwstra, JU, Sapozhnikov, OA, Fleury, G., Berriet, R., Chupin, L., Guey, JL New piezoelectric transducers for therapeutic ultrasound // Ultrasound in Med. & Biol. — 2000. — V. 26, № 1. — P. 153—159.
↑ Fleury, G., Berriet, R., Le Baron, O., Huguenin, B. New piezocomposite transducers for therapeutic ultrasound / 2nd International Symposium on Therapeutic Ultrasound. Seattle — 31/07 — 02/08/2002.
↑ Cathignol, D. High intensity piezoelectric sources for medical applications: technical aspects // In Nonlinear Acoustics at the Beginning of the 21st Century / Ed. by Rudenko OV and Sapozhnikov OA (Proc. of 16th ISNA, Moscow, 2002). — 2002. — V. 1. — Р. 371—378.
↑ Shung, KK, Zipparo, M. Ultrasonic transducers and arrays // IEEE Engineering in Med. and Biol. — 1996. Nov/Dec. — P. 20-30.
↑ Foster, FS Transducer materials and probe construction // Ultrasound in Med. and Biol. 2000. — V. 26, Suppl. 1. — P. S2-S5.
↑ Rivens, IH, Clarke, RL, ter Haar, GR Design of focused ultrasound surgery transducers // IEEE Trans. Ultrason. Ferroelectr. Freq. Ctrl. — 1996. — V.43, № 6. — P. 1023—1031.
↑ Kluiwstra, J.-UA, Tokano, T., Davis, J., Strickberger, SA, Cain, CA Real time image guided high intensity focused ultrasound for myocardial ablation: In vivo study / In Proc. IEEE Ultrason. Symp. — 1997. — P. 1327—1330.
↑ Kennedy, JE, Wu, F., ter Haar, GR, Gleeson, FV, Phillips, RR, Middleton, MR, Cranston, D. High-intensity focused ultrasound for the treatment of liver tumours // Ultrasonics. — 2004. — V. 42, № 1-9. — P. 931—935.
↑ Foster, RS, Bihrle, R., Sanghvi, NT, Fry, FJ, Donohue, JP High-intensity focused ultrasound in the treatment of prostatic disease // Eur. Urol. — 1993. — V. 23, Suppl. 1. — P. 29-33.
↑ Gelet, A., Chapelon, JY, Margomari, J., Theillere, Y., Gorry, F., Souchon, R., Bouvier, R. High-intensity focused ultrasound experimentation on human benign prostatic hypertrophy // Eur. Urol. — 1993. — V. 23, Suppl. 1. — P. 44-47.
↑ Beissner, K. Some basic relations for ultrasonic fields from circular transducers with a central hole // J. Acoust. Soc. Am. — 2012. — V. 131, № 1. — P. 620—627.
↑ Физика и техника мощного ультразвука / Под ред. Л. Д. Розенберга. Prince 3. Физические основы ультразвуковой технологии. — М.: Наука, 1970. — 682 с.
↑ Harvey, G., Gachagan, A., Mutasa, T. Review of high-power ultrasound — industrial applications and measurement methods // IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. — 2014. — V. 61, № 3. — P. 481—495.
↑ ¹ ² Power Ultrasonics. Applications of High-intensity Ultrasound, ed. by Gallego-Juarez, JA, and Graff, KF Cambridge: Elsevier, Woodhead Publishing. — 2015. — 1167 p., ISBN 978-1-78242-028-6
↑ Акопян, В. Б., Ершов, Ю. А. Основы взаимодействия ультразвука с биологическими объектами / М.: МГТУ им. Н. Э. Баумана, 2005. — 223 с.

Literature

Розенберг Л. Д. Фокусирующие излучатели ультразвука // В кн.: Физика и техника мощного ультразвука / Под ред. Л. Д. Розенберга. Prince 1. Источники мощного ультразвука. — М. : «Наука», 1967. — С. 149−206 .
Гаврилов Л. Р. Фокусированный ультразвук высокой интенсивности в медицине. — М. : «Фазис», 2013. — 656 с. — ISBN 978-5-7036-0131-2 .
Гаврилов Л. Р., Цирульников Е. М. Фокусированный ультразвук в физиологии и медицине. — Л. : «Наука», 1980. — 199 с.
Вартанян, И. А., Гаврилов, Л. Р., Гершуни, Г. В., Розенблюм, А. С., Цирульников, Е. М. Сенсорное восприятие. Опыт исследования с помощью фокусированного ультразвука. — Л.: Наука, 1985. — 189 с.
Хилл К. Р., Бэмбер Дж., тер Хаар Г. ред. Ультразвук в медицине. Физические основы применения. / Пер. from English — М. : «Физматлит», 2008. — 544 с. — ISBN 978-5-9221-0894-2 .
Szabo, TL Diagnostic Ultrasound Imaging: Inside Out. 2nd Edition. — Oxford, UK, Academic Press (Elsevier), 2014, 572 p.
Бэйли, М. Р., Хохлова, В. А., Сапожников, О. А., Каргл, С. Г., Крам, Л. А. Физические механизмы воздействия терапевтического ультразвука на биологическую ткань (обзор). — Акуст.ж., 2003, т.49, № 4, с.437-464.
Акопян, В. Б., Ершов, Ю. А. Основы взаимодействия ультразвука с биологическими объектами / М.: МГТУ им. Н. Э. Баумана, 2005. — 223 с.

Links

[Розен-1] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ Розенберг, Л. Д. Фокусирующие излучатели ультразвука // В кн.: Физика и техника мощного ультразвука / Под ред. Л. Д. Розенберга. Prince 1. Источники мощного ультразвука. — М.: Наука, 1967. — C. 149—206.

[ЛГ-2] ¹ ² ³ ⁴ Гаврилов, Л. Р. Фокусированный ультразвук высокой интенсивности в медицине. — М.: Фазис, 2013. −656 c. — ISBN 978-5-7036-0131-2

[3] Буров, А. К. Получение больших интенсивностей ультразвука в жидкости // Акустический журнал. — 1958. — Т. 4, № 4. — С. 315—320.

[УЗ_в_мед-4] ¹ ² ³ ⁴ Ультразвук в медицине. Физические основы применения Под ред. К.Хилла, Дж. Бэмбера, Г. тер Хаар. Пер с англ. under the editorship of Л. Р. Гаврилова, В. А. Хохловой, О. А. Сапожникова. — М.: Физматлит, 2008, 544 с., с.67.

[5] Greutzmacher, J. Piezoelektrishe Kristall mit Ultrashall konvergenz // Zh. Phys. −1935. — V. 96. — 342.

[6] Lynn, YG, Zwemer, RL, Chick, AJ, Miller, AE A new method for the generation and use of focused ultrasound in experimental biology // Journ. Gener. Physiol. — 1942. -V. 26. — P. 179—193.

[ЛГ_ЕМЦ-7] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ Гаврилов, Л. Р., Цирульников, Е. М. Фокусированный ультразвук в физиологии и медицине. — Л.: Наука, 1980. — 199 с.

[8] Fry, FJ Precision high intensity focusing ultrasonic machines for surgery // Amer. J. Phys. Med. — 1958. — V. 37, № 3. — P. 152—156.

[Astrom-9] ¹ ² ³ Aström, К.E., Bell, E., Ballantine, Н. Т., Heidensleben, E. An experimental neuropathological study of the effects of high-frequency focused ultrasound on the brain of the cat // J. Neuropathol. Exp. Neurol. — 1961. — V. 20, № 4. — P. 484—520.

[10] Lele, PP Production of deep focal lesions by focused ultrasound — current status // Ultrasonics. — 1967. — V. 5. — P. 105—112.

[11] Fry, FJ, Ades, HW, Fry WJ Production of reversible changes in the central nervous system by ultrasound // Science. — 1958. — V. 127, № 3289. — P. 83-84.

[Szabo-12] ¹ ² Szabo, TL Diagnostic Ultrasound Imaging: Inside Out. 2nd Edition. — Oxford, UK, Academic Press (Elsevier), 2014. — p. 130.

[Бэйли-13] ¹ ² ³ ⁴ Бэйли, М. Р., Хохлова, В. А., Сапожников, О. А., Каргл, С. Г., Крам Л. А. Физические механизмы воздействия терапевтического ультразвука на биологическую ткань (Обзор) // «Акустический журнал» — 2003. — Т. 49, № 4. — C. 437—464.

[Leighton-14] ¹ ² ³ Leighton, TG, Cleveland, RO Lithotripsy // Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine. −2010. -V. 224, № 2. — P. 317—342.

[15] Наугольных, К. А., Рой, Н. А. Электрические разряды в воде. - M .: Science. — 1971. — 155 с.

[16] Rassweiler, J., Henkel, TO, Kohrmann, KU, Potempa, D., Junemann, KP, Alken, P. Lithotripter technology: present and future // Journal of Endourology. — 1992. — V. 6, № 1. -P. 1-13.

[17] Синельников, Е. Д., Филд, Т., Сапожников, О. А. Закономерности формирования зоны термического разрушения при лечении фибрилляции предсердий катетерным методом ультразвуковой абляции // Акустический журнал. — 2009. — Т. 55, № 4-5. — С. 641—652.

[18] Warwick, R., Pond, J. Trackless lesions in nervous tissues produced by high intensity focused ultrasound (high-frequency mechanical waves) // J. Anat. — 1968. — V. 102, № 3. — P. 387—405.

[19] Fry, FJ, Heimburger, RF, Gibbons, LV, Eggleton RC Ultrasound for visualization and modification of brain tissue // IEEE Trans. on Sonics and Ultrasonics. — 1970. — V. SU-17, № 3. — P. 165—169.

[20] Авиром, В. М., Адрианов, О. С., Выходцева, Н. И., Гаврилов, Л. Р., Меринг, Т. А., Сиротюк, М. Г. Разрушение глубоких структур мозга с помощью фокусированного ультразвука // Журн. higher нервн. деят. — 1971. — Т. 21, № 5. — С. 1110—1113.

[Neil-21] ¹ ² ³ O'Neil, HT Theory of focusing radiators // J. Acoust. Soc. Am. — 1949. -V. 21, № 5. — P. 516—526.

[22] Kossoff, G. Analysis of focusing action of spherically curved transducers // Ultrasound in Med. and Biol. — 1979. — V. 5, № 4. — P. 359—365.

[Clarke-23] ¹ ² Clarke, RL Modification of intensity distribution from large aperture ultrasound sources // Ultrasound in Med. and Biol. — 1995. — V. 21, № 3. — P. 353—363

[LD-24] ¹ ² Розенберг,Л. Д. Звуковые фокусирующие системы. — М.: АН СССР, 1949. — 112 с.

[25] Каневский, И. Н. Фокусирование звуковых и ультразвуковых волн. — М.: Наука, 1977. — 336 с.

[26] Fry, FJ Intense focused ultrasound: its production, effects and utilization // In: Ultrasound: its applications in medicine and biology / FJ Fry ed. New York: Elsevier, 1978. Pt. 2. — P. 689—736.

[Варт-27] ¹ ² ³ ⁴ ⁵ Вартанян, И. А., Гаврилов, Л. Р., Гершуни, Г. В., Розенблюм, А. С., Цирульников, Е. М. Сенсорное восприятие. Опыт исследования с помощью фокусированного ультразвука. — Л.: Наука, 1985. — 189 с.

[28] Ocheltree, K., Frizzell, L. Sound field calculations for rectangular sources // IEEE Trans. Ultrason. Ferroelectr. Freq. Ctrl. — 1989. — V. 36, № 2. — P. 242—248.

[Goss-29] ¹ ² Goss SA, Frizell LA, Kouzmanoff JT, Barich JM, Yang JM Sparse random ultrasound phased array for focal surgery // IEEE Trans. Ultras. Ferroelectr. Freq. Ctrl. 1996. V. 43. № 6. P. 1111—1121.

[Cline-30] ¹ ² Cline, HE, Hynynen, K., Watkins, RD, Adams, WJ, Schenck, JF, Ettinger, RH, Freund, WR, Vetro, JP, Jolesz, FA Focused US system for MR imaging-guided tumor ablation // Radiology. — 1995. — V. 194, № 3. — P. 731—737.

[31] Bergmann, L. Der Ultraschall und seine Anwendung in Wissenschaft und Technik / Zurich, 1954. (Пер на русск. яз: Бергман Л. Ультразвук и его применение в науке и технике / М.: ИЛ. 1956. — 726 с.)

[32] Cathignol, D., Sapozhnikov, OA, Zhang, J. Lamb waves in piezoelectric focused radiator as a reason for discrepancy between O'Neil's formula and experiment // J. Acoust. Soc. Am. — 1997. — V. 101, № 3. — P. 1286—1297.

[Cath-33] ¹ ² ³ Cathignol, D., Sapozhnikov, OA, Theillere, Y. Comparison of acoustic fields radiated from piezoceramic and piezocomposite focused radiator // J. Acoust. Soc. Am. — 1999. — V. 105, № 5. — P. 2612—2617.

[Катин-34] ¹ ² Катиньоль, Д., Сапожников, О. А. О применимости интеграла Рэлея к расчёту поля вогнутого фокусирующего излучателя // Акустический журнал — 1999. — Т. 45, № 6. — С. 816—824.

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