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ULTRASOUND
Ultrasound – like sound – is the propagation of vibrations; the main difference is that while the audible frequency range is 20 Hz – 20 kHz, the operating frequency of the devices used in medical diagnostics is often 5 MHz or higher. The operation of an ultrasound device is based on the emission of ultrasound and on the measurement of the time it takes for the sound reflected off a certain surface to return. Bats find flying mosquitoes in the same way and radars operate on a similar principle as well, only they emit radio waves.
On average, the velocity of propagation of sound is c=1540 m/s within the body. From the reflection time of the waves the distance of reflection is {EQUATION(size="80")}z=c\Delta t/2{EQUATION}.
It is important to divide {EQUATION(size="80")}\Delta c{EQUATION} by 2, because the sound covers the distance back and forth.
The reason why ultrasound is used in the device is that the theoretically achievable resolution is proportional to the applied wavelength (0.2 mm – 2 mm in practice). The sketch of an ultrasound device is depicted in Figure 1.
{IMG(fileId="2659",thumb="y",max="400",align="center",desc="Figure 1.")}{IMG}
The signal generator generates an electronic signal {EQUATION(size="80")}P(t){EQUATION}, which usually is a sine wave that is amplitude-modulated by Gaussian distribution; approximately 3 waves can be considered to have amplitudes different from 0. The signal gets into the transducer through a switch.
The transducer converts the electronic signal into mechanical vibrations. The conversion is carried out based on the piezoelectric effect, i.e. the expansion or contraction of certain crystals (e.g. the quartz crystal) in specific directions when a voltage is applied to them. The vibration generated in this way leaves the transducer in the form of ultrasound.
Without discussing the details of the wave theory, it can be said that since the outlet diameter of the transducer is several times larger than the wavelength of the generated ultrasound, the outgoing wave propagates as a plane wave. The advantage of this is that diffraction is little and the ultrasound propagates practically perpendicular to the emitting surface. Reflection can be expected from this direction. The disruptive effects of multiple reflection are reduced by the fact that the energy of the reflected wave is only a small fraction of the original energy of the wave; thus, the intensity of the signal reflected multiple times is significantly lower than that of the wave that is reflected only once.
The ratio of the energy of the reflected wave to that of the original wave is called reflexivity. Reflexivity in an {EQUATION(size="80")}(x,y,z){EQUATION} point in space is denoted by {EQUATION(size="80")}R(x,y,z){EQUATION}. Reflexivity can be direction-dependent (specular) or omnidirectional (diffuse). As far as medical imaging is concerned, the latter is more advantageous. Specular reflection is usually avoided by the setting the transducer. Diffuse reflection is produced when the reflective surface is rough or small compared to the wavelength.
In order to avoid strong reflection off the skin, an ointment designed specifically for this purpose is applied on the skin.
The reflected signal gets into the transducer, where it is converted into an electronic signal due to the inverse piezoelectric effect. The signal gets from the transducer into the signal processing unit through the switch, which has been switched over in the meantime. Let the output signal be denoted by {EQUATION(size="80")}e(t){EQUATION}.
If the signal {EQUATION(size="80")}P(t){EQUATION} were reflected perfectly off the emitting surface of the transducer, and it did not sustain a loss or become distorted, then the time needed to cover the distance would be {EQUATION(size="80")}e(t)=p(t-2z/c){EQUATION}. Instead of P, p is written in the formula because the transducer and the signal processing unit may distort the original signal.
Let us assume that the emitting surface of the transducer is in the plane {EQUATION(size="80")}z=0{EQUATION}, and its centre is in the origin. In this case, the characteristics of the emitting surface is {EQUATION(size="80")}S(x,y){EQUATION}which expresses the degree to which an {EQUATION(size="80")}(x,y){EQUATION} point of this plane lets through the signal; in the points of the emitting surface {EQUATION(size="80")}S(x,y)\approx1{EQUATION}, otherwise it is 0. Since both the original and the reflected signals have to pass through the emitting surface, a reflected signal is obtained only in those points where {EQUATION(size="80")}S(x,y)\approx1{EQUATION}. The reflected signal can be expressed by {EQUATION(size="80")}S^{2}(x,y)p(t-2z/c){EQUATION}.
If reflection is not perfect, then {EQUATION(size="80")}R(x,y,z)S^{2}(x,y)p(t-2z/c){EQUATION}.
However, the signal is attenuated as it passes through the body; attenuation per unit length is {EQUATION(size="80")}$e^{-\alpha}{EQUATION}. The signal covers a distance of 2z within the body (Magyar!), thus attenuation in this case is {EQUATION(size="80")}$e^{-2\alpha z}{EQUATION}. Further attenuation is caused by the fact that the signal does not propagate as a plane wave after reflection; instead, the point of reflection behaves as a point source. This means that the intensity of the reflected signal is inversely proportional to the square of the distance travelled. Therefore, the total attenuation is {EQUATION(size="80")}$e^{-2\alpha z}/z^{2}{EQUATION} .
If we want to take into account the effect of the signal reflected from all points in space, then we have to integrate over the whole space; thus, the magnitude of the measured signal:
{EQUATION()}e(t)=\left | \int \int \int R(x,y,z)S^{2}(x,y)p( t-\frac{2z}{c})\: dxdydz \right |{EQUATION}
If {EQUATION(size="80")}P(t){EQUATION} is “short”, i.e. it differs from 0 only in the case of small t values, then {EQUATION(size="80")}p(t){EQUATION}is short as well, which means that it is not 0 only if {EQUATION(size="80")}t-2z/c\approx 0{EQUATION}. In this short interval {EQUATION(size="80")}z\approx ct/2{EQUATION}, and {EQUATION(size="100")} \frac{e^{-2\alpha z}}{z^{2}}\approx \frac{e^{-\alpha ct}}{(ct/2)^{2}}{EQUATION}. Since the expression on the right is not dependent on the integration variables, it can be brought out in front of the integral. Dividing by this factor, the reflected signal corrected by distance (time) is obtained:
{EQUATION()}e_c(t)=\left | \int \int \int \frac{e^{-2\alpha z}}{z^{2}} R(x,y,z)S^{2}(x,y)p( t-\frac{2z}{c})\: dxdydz \right |{EQUATION}
Since the emitting surface of the transducer is small (i.e {EQUATION(size="80")}S(x,y){EQUATION} differs from 0 only if {EQUATION(size="80")}x,y \approx 0{EQUATION} , and since {EQUATION(size="80")}p(t){EQUATION} differs from 0 only if {EQUATION(size="80")}z\approx ct/2{EQUATION} ), the value of {EQUATION(size="80")}e_{c}(t){EQUATION} is affected only by the value reflexivity takes on in a small environment of the point, thus it can be written that
{EQUATION(size="100")}R(0,0,z)=R(0,0,ct/2)\approx e_c(t)K=e(t)K(ct/2)^{2}e^{\alpha ct}{EQUATION},
where {EQUATION(size="80")}K{EQUATION} is a constant dependent on the device.
Therefore, the value of {EQUATION(size="80")}$R(0,0,z){EQUATION} is obtained in the following way: the device measures the value of {EQUATION(size="80")}e(t){EQUATION} and it multiplies the measured value by its current value dependent on {EQUATION(size="80")}K(ct/2)^{2}e^{\alpha ct}{EQUATION}.
By relocating the transducer, the value of reflexivity can be determined in arbitrary points.