# Physics of ultrasound as it relates to echocardiography

By Aleksandr Rovner, MD

Sound is created by a mechanical vibration and transmits energy through a medium (usually elastic). As ultrasound is transmitted, there are parts of the wave that are compressed (increase in pressure or density) and parts that are rarefied (decrease in pressure or density). When used in diagnostic echocardiography, the frequency is usually above 20,000 Hz (20 kHz), and it is not audible to a human ear.

Fig. 1

There are several properties of ultrasound that are useful in clinical cardiology. Since ultrasound is a mechanical wave in a longitudinal direction, it is transmitted in a straight line and it can be focused. These waves obey laws of reflection and refraction. Since small objects in the human body will reflect ultrasound, it is possible to collect the reflected data and compose a picture of these objects to further characterize them.

Major drawback of ultrasound is the fact that it cannot be transmitted through a gaseous medium (like air or lung tissue), in clinical echo certain windows are used to image the heart and avoid the lungs. As ultrasound transverses tissue, its energy decreases. This is called attenuation and is more pronounced in tissue with less density (like lung). There are seven parameters that describe ultrasound waves.

Period of an ultrasound wave is the time that is required to capture one cycle, i.e., the time from the beginning of one cycle till the beginning of the next cycle. The units of period is time and typical values in echo is 0.1 to 0.5 microsecond. Period of ultrasound is determined by the source and cannot be changed by the sonographer.

Fig. 2

Frequency is the inverse of the period and is defined by a number of events that occur per unit time. The units of frequency is 1/sec or Hertz (Hz). Since f = 1/P, it is also determined by the source and cannot be changed.

Fig. 3

Amplitude is an important parameter and is concerned with the strength of the ultrasound beam. It is defined as the difference between the peak value and the average value of the waveform. It is expressed in decibels or dB, which is a logarithmic scale. It can be changed by a sonographer. Amplitude decreases as the ultrasound moves through tissue, this is called attenuation. Amplitude decreases usually by 1 dB per 1 MHz per 1 centimeter traveled. For example, if we have a 5 MHz probe and the target is located at 12 cm (24 cm total distance), then the amplitude attenuation will be 1 dB x 5 MHz x 24 cm = 120 dB which nearly 6000 fold decrease.

Fig. 4

Power of ultrasound is defined as the rate of energy transfer and is measured in Watts. It is determined by the sound source and it decreases as the beam propagated through the body. Intensity of the ultrasound beam is defined as the concentration of energy in the beam. Intensity = Power / beam area = (amplitude)^2 / beam area, thus it is measured in Watts per cm^2. It is the key variable in ultrasound safety. Intensity also decreases as the ultrasound propagates through tissue.

Fig. 5

Wavelength is defined as the length of a single cycle. It is measured in the units of length. It is determined by both the source and the medium. Wavelength cannot be changed by the sonographer. It influences the longitudinal image resolution and thus effect image quality. Typical values of wavelength are 0.1 – 0.8 mm. Wavelength (mm) = Propagation speed in tissue (mm/microsecond) / frequency (MHz). High frequency means short wavelength and vice versa.

Fig. 6

Propagation speed in human soft tissue is on average 1540 m/s. It is defines as to how fast the ultrasound can travel through that tissue. It is determined by the medium only and is related to the density and the stiffness of the tissue in question. Density of the medium is related to its weight and the stiffness of the medium is related to its “squishability”. As the medium becomes more dense, the slower is speed of ultrasound in that medium (inverse relationship). The stiffer the tissue, the faster will the ultrasound travel in that medium (direct relationship). There are tables where one can look up the velocity of sound in individual tissues. Range equation – since ultrasound systems measure the time of flight and the average speed of ultrasound in soft tissue is known (1540 m/s), then we can calculate the distance of the object location. Distance to boundary (mm) = go-return time (microsecond) x speed (mm/microsecond) / 2. So far we have defined the ultrasound variables and parameters. In the next section will talk more about pulsed ultrasound. Pulse Duration is defined as the time that the pulse is on. It is determined by the number of cycles and the period of each cycle. In clinical imaging, a pulse is comprised of 2-4 cycles and the pulse duration is usually between 0.5 to 3 microseconds. Pulse duration does not change with depth, thus it cannot be changed by the sonographer. Pulse Duration (msec) = # of cycles x period (msec). Since Wavelength (mm) = Propagation speed in tissue (mm/microsecond) / frequency (MHz), this can be rewritten as 1/frequency = wavelength / propagation speed. And since period = 1/frequency, then the Pulse Duration = (# of cycles x wavelength) / Propagation speed.

Fig. 7

Pulse Repetition Period or PRP is the time between the onset of one pulse till the onset of the next pulse. Aagain, it is measured in units of time. This parameter includes the time the pulse is “on” and the listening time when the ultrasound machine is “off”. It can be changed by the sonographer by varying the depth to which the signal is send. Since the Pulse Duration time is not changed, what is changed is the listening or the “dead time”. PRP = 13 microseconds x the depth of view (cm). It follows from this equation that the deeper is the target, the longer is the PRP. The typical values of PRP in clinical echo are form 100 microseconds to 1 millisecond.

Fig. 8

A related parameter to PRP is the Pulse Repetition Frequency or PRF. PRP and PRF are reciprocal to each other. PRF is the number of pulses that occur in 1 second. This parameter is not related to the frequency of ultrasound. PRF can be altered by changing the depth of imaging. It is measured in Hertz (Hz). PRF = 77,000 / depth of view (cm). As evident from the equation, as the location of the target gets further away, the PRF decreases. PRF is related to frame rate or sampling rate of the ultrasound. I would like to talk about Duty Factor (DF) here. This parameter is related to ultrasound bioeffects, but since it is also related to pulsed ultrasound it is reasonable to introduce it in this section. DF is defined as a percent of time that the ultrasound system is on while transmitting a pulse. DF = pulse duration (sec) / pulse repetition period (sec) x 100. It has units of % and ranges from 0 (the system is off) to 100 (the system is on continuously). Typical valued of DF in clinical imaging are 0.1% to 1% (usually closer to 0), thus the machine is mostly listening during clinical imaging. Another interesting point to note is the fact that since the sonographer changes the PRF by changing the depth, they indirectly change the duty factor. And lastly, one must realize that an anatomic image cannot be created with a continuous wave ultrasound. Since one must listen for the return signal to make an image, a clinical echo machine must use pulsed signal with DF between 0.1 and 1%.

Back to propertied of pulsed ultrasound, we need to discuss spatial pulse length. Up to now we introduced properties that were related to timing. Spatial Pulse Length is the distance that the pulse occupies in space, from the beginning of one pulse till the end of that same pulse. It is measured in units of distance with typical values from 0.1 to 1 mm. SPL (mm) = # cycles x wavelength (mm). Axial or longitudinal resolution (image quality) is related to SPL. Axial resolution = SPL/2 = (# cycles x wavelength)/2.

Fig. 9

We will now talk about interaction of ultrasound with tissue. As we discussed in the section of amplitude, the energy of ultrasound decreases (attenuation) as it travels through tissue. The stronger the initial intensity or amplitude of the beam, the faster it attenuates. Standard instrument output is ~ 65 dB. So for a 10 MHz transducer, the maximum penetration would be as follows: 1 dB/cm/MHz x 10 MHz x (2 x max depth) = 65 dB. Max depth = 65/20 = 3.25 cm. If we use a 3.5 MHz transducer and apply the same formula for max depth, will get Max depth = 65/7 = 9.3 cm. Attenuation of ultrasound in soft tissue depends on the initial frequency of the ultrasound and the distance it has to travel. As we saw in the example above, in soft tissue the greater the frequency the higher is the attenuation. So we can image deeper with lower frequency transducer. The further into the tissue the ultrasound travels, the higher the attenuation is, so it is ultimately the limiting factor as to how deep we can image clinically relevant structures.

Fig. 10

There are 3 components of interaction of ultrasound with the tissue medium: absorption, scattering, and reflection. Absorption of ultrasound by tissue implies loss of energy that is converted to heat. The highest attenuation (loss of energy) is seen in air, the lowest is seen in water.

Fig. 11

Reflection is the process were propagating ultrasound energy strikes a boundary between two media (i.e., the RV free wall in the parasternal long axis) and part of this energy returns to the transducer.

Fig. 12

If the reflector is very smooth and the ultrasound strikes it at 90 degree angle (perpendicular), then the reflection is strong and called specular.

Fig. 14

If the incidence is not 90 degree, then specular reflectors are not well seen. Another instance when specular reflection is produced is when the wavelength is much smaller than the irregularities of the media/media boundary. Diffuse or Backscatter reflections are produced when the ultrasound returning toward the transducer is disorganized. This occurs when the ultrasound wavelength is similar size to the irregularities of the media/media boundary. When the ultrasound wavelength is larger than the irregularities of the boundary, the ultrasound is chaotically redirected in all directions or scatters.

Fig. 15

If the reflector is much smaller than the wavelength of the ultrasound, the ultrasound is uniformly scattered in all directions and this is called Rayleigh scattering. Red blood cell would be an example of Rayleigh scatterer. Rayleigh scattering is related to wavelength to 4th power. Backscatter is what produces the relevant medical imaging.

Fig. 16

Let us talk about Impedance (Z). This is an important concept and it is related to reflection of ultrasound energy. It is calculated and is not measured directly. The higher the difference of the acoustic impedance between two media, the more significant is the reflection of the ultrasound. That is why we use coupling gel between the ultrasound transducer and the skin. By using the gel, we decrease the impedance and allow the ultrasound to penetrate into the tissue. Otherwise, the impedance between skin/transducer is so high that all the energy will be reflected and no image will be produced.

Fig. 17

More of on reflection – it occurs only when the acoustic impedance of one media is different from acoustic impedance of the second media at the boundary. If the ultrasound hits the reflector at 90 degrees (normal incidence), then depending on the impedances at the boundary the % reflection = ((Z2 - Z1) / (Z2 + Z1))^2. Then transmission is 1 - % reflection. Physics of oblique incidence is complex and reflection/transmission may or may not occur. We do know that the incident intensity is equal to the sum of the transmitted and reflected intensities.

Fig. 18

Refraction is simply transmission of the ultrasound with a bend. This occurs when we have an oblique incidence and different propagation speed from one media to the next. The physics of the refraction is described by Snell’s law. Sine (transmission angle)/sine (incident angle) = propagation speed 2/ propagation speed 1.

Fig. 19

More on image quality or resolution. We have touched upon axial resolution (ability to differentiate objects that are located along the imaging beam axis) when we discussed spatial pulse length. The smaller the axial resolution length, the better the system is and it can resolve structures that are closer together. Thus the shorter the pulse length, the better picture quality. Current transducers are designed with the minimum number of cycle per pulse to optimize image quality. The primary determinant of axial resolution is the transducer frequency. Axial resolution (mm) = 0.77 x # cycles / frequency (MHz). One must remember that attenuation is also dependent on the transducer frequency, thus a tradeoff must be reached.

Fig. 20

Lateral resolution is the minimum distance that can be imaged between two objects that are located side to side or perpendicular to the beam axis. Again, the smaller the number the more accurate is the image. Since the beam diameter varies with depth, the lateral resolution will vary with depth as well. The lateral resolution is best at the beam focus (near zone length) as will discuss later when will talk about the transducers. Lateral resolution is usually worse than axial resolution because the pulse length is usually smaller compared to the pulse width.

Fig. 21

Temporal resolution implies how fast the frame rate is. FR = 77000/(# cycles/sector x depth). Thus frame rate is limited by the frequency of ultrasound and the imaging depth. The larger the depth, the slower the FR is and worse temporal resolution. The higher the frequency is, the higher is the FR and the temporal resolution improves. Sonographer can do several things to improve the temporal resolution: images at shallow depth, decrease the #cycles by using multifocusing, decrease the sector size, lower the line density. However one can realize quickly that some of these manipulations will degrade image quality. And this is in fact correct: improving temporal resolution often degrades image quality. M-mode is still the highest temporal resolution modality within ultrasound imaging to date.

Before we talk about Doppler Effect, let us discuss the ultrasound transducer architecture and function. The current transducers became available after the discovery that some materials can change shape very quickly or vibrate with the application of direct current. As important is the fact that these materials can in turn produce electricity as they change shape from an external energy input (i.e., from the reflected ultrasound beam). This effect of vibration form an application of alternative current is called a piezoelectric effect (PZT).

Fig. 22

Many materials exist in nature that exhibit piezoelectric effect. Ccommercial transducers employ ceramics like barium titanate or lead zirconate titanate. The transducer usually consists of many PZT crystals that are arranged next to each other and are connected electronically. The frequency of the transducer depends on the thickness of these crystals, in medical imaging it ranges 2-8 MHz. An ultrasound pulse is created by applying alternative current to these crystals for a short time period. Afterwards, the system “listens” and generates voltage from the crystal vibrations that come from the returning ultrasound. An important part of the transducer is the backing material that is placed behind the PZT, it is designed to maximally shorten the time the PZT crystal vibrates after the current input is gone also known as ringing response. By decreasing the ringdown time, one decreases the pulse length and improves the axial resolution. In addition, the backing material decreases the amount of ultrasound energy that is directed backwards and laterally.

Fig. 23

In front of the PZT, several matching layers are placed to decrease the difference in the impedance between the PZT and the patient’s skin. This increases in efficiency of ultrasound transfer and decrease the amount of energy that is reflected from the patient. Let us talk about the shape of the ultrasound beam. Since there are many PZT crystals that are connected electronically, the beam shape can be adjusted to optimize image resolution. The beam is cylindrical in shape as it exits the transducer, eventually it diverges and becomes more conical. The cylindrical (or proximal) part of the beam is referred to as near filed or Freznel zone. The image quality and resolution is best at the focal depth that can be determined by Focal depth = (Transducer Diameter)^2 x frequency /4. When the ultrasound beam diverges, it is called the far field.

Fig. 24

One would state that the best images are acquired using a large diameter transducer with high frequency. However, as we have learned, high frequency transducers have significant attenuation issues. In addition, larger diameter transducers are impractical to use because the imaging windows are small. The way around these problems is electronic focusing with either an acoustic lens or by arranging the PZT crystals in a concave shape.

Fig. 25

In clinical imaging, the ultrasound beam is electronically focused as well as it is steered. This became possible after phased array technology was invented. By applying electrical current in a differential manner and adjusting the timing of individual PZT excitation, the beam can travel in an arch producing a two-dimensional image. If one applies electricity in a differential manner from outside inward to the center of the transducer, differential focusing can be produced resulting in a dynamic transmit focusing process.

Fig. 26

Briefly, I would like to touch upon real time 3D imaging. In order to accomplish this, the PZT elements need to be arranged in a 2D matrix. Each PZT element represents a scan line, by combining all the data, a 3D set is reconstructed. For example, if we have a matrix of 128 by 128 PZT elements, one can generate over 16 thousand scan lines. With careful timing for individual excitation, a pyramidal volumetric data set is created. When imaged several times per minute (>20), a real time image is achieved.

Fig. 27

Image production is a complex process. Echo instrumentation must generate and transmit the ultrasound and receive the data. Then the data needs to be amplified, filtered and processed. Eventually the final result needs to be displayed for the clinician to view the ultrasound information. As the first step in data processing, the returning ultrasound signals need to be converted to voltage. Since their amplitude is usually low, they need to be amplified. The ultrasound signal usually is out of phase so it needs to be realigned in time. At this point one has the raw frequency (RF) data, which is usually high frequency with larger variability in amplitudes and it has background noise. The next step is filtering and mathematical manipulations (logarithmic compression, etc) to render this data for further processing. At this stage one has sinusoidal data in polar coordinates with distance and an angle attached to each data point. This information needs to be converted to Cartesian coordinate data using fast Fourier transform functions. Once at this stage, the ultrasound data can be converted to analog signal for video display and interpretation. Image display has evolved substantially in clinical ultrasound. Currently, 2D and real time 3D display of ultrasound date is utilized. Without going into complexities of physics that are involved in translating RF data into what we see every day when one reads echo, the following section will provide the basic knowledge of image display. If one can imagine a rod that is imaged and displayed on an oscilloscope, it would look like a bright spot. Displaying it as a function of amplitude (how high is the return signal) is called A-mode. If one converts the amplitude signal into brightness (the higher the amplitude the brighter the dot is), then this imaging display is called B-mode.

Fig. 28

Using B mode data, once can scan the rod multiple times and then display the intensity and the location of the rod with respect to time. This is called M-mode display. Using B-mode scanning in a sector created a 2D representation of anatomical structures in motion.

Fig. 28 (All 3 modes of display are depicted: A, B, and M)

Fig. 29

Second Harmonic is an important concept that is used today for image production. The basis for this is that fact that as ultrasound travels through tissue, it has a non-linear behavior and some of its energy is converted to frequency that is doubled (or second harmonic) from the initial frequency that is used (or fundamental frequency).

Fig. 30

There are several parameters that make second harmonic imaging preferential. Since it is produced by the tissue, the deeper the target the more second harmonic frequency is returned. As the ultrasound beam travels through tissue, new frequencies appear that can be interrogated. Second harmonic data gets less distortion, thus it produces better picture. Also, the second harmonic is strongest in the center of the beam, thus it has less side lobe artifacts. At the chest wall the fundamental frequency gets the worst hit due to issues that we have discussed (reflection, attenuation) – if one can eliminate the fundamental frequency data then these artifacts will not be processed. One concept of eliminating fundamental frequency data is called pulse inversion technology. The transducer sends out 2 fundamental frequency pulses of the same amplitude but of different phase. As these pulses are reflected back to the transducer, because of the different phase they cancel each other out (destructive interference) and what is left is the second harmonic frequency data which is selectively amplified and used to generate an image.

Fig. 31

Doppler Effect is change in frequency of sound as a result of motion between the source of ultrasound and the receiver. Greater velocity creates a larger shift in ultrasound frequency.

Fig. 32

An example of a moving object in cardiac ultrasound is red blood cells. Typical values for Doppler shift is 20 Hz to 20 kHz, thus comparing to the fundamental frequency, the Doppler shift is small. Since it “rides” on top of the much larger frequency (i.e., 5 MHz), the process of extracting this data is termed demodulation. Doppler shift = (2 x reflector speed x incident frequency x cosine (angle)) / propagation speed. There are two important concepts that must be emphasized. First, the Doppler shift is highly angle dependent. Since cosine (90) = 0 and cosine (0) = 1, then the most true velocity will be measured when the ultrasound beam is parallel to the axis of motion of the reflector. At perpendicular axis, the measured shift should be 0, however usually some velocity would be measured since not all red blood cells would be moving at 90 degree angle.

Fig. 33

The other concept is the direction of the motion of the reflector. When the reflector is moving away from the source of the ultrasound, the shift is negative, and when the reflector is moving towards the source of ultrasound the shift is positive. Continuous wave (CW) Doppler required 2 separate crystals, one that constantly transmits, and one that constantly receives data. There is no damping using this mode of imaging. One can measure very high velocities (i.e., velocities of aortic stenosis or mitral regurgitation). The advantage of CW is high sensitivity and ease of detecting very small Doppler shifts. The disadvantage of CW is the fact that echos arise from the entire length of the beam and they overlap between transmit and receive beams. Thus one cannot determine where in the body the highest velocity is coming from – range ambiguity.

Fig. 33

Pulsed wave (PW) Doppler requires only one crystal. It alternates between transmitting and receiving data. The transducer “listens” for the data at a certain time only, since the sampling volume is coming from the location that is selected by the sonographer (i.e., the velocity at the LVOT or at the tips of the mitral valve). This is called range resolution. The major disadvantage of PW Doppler is aliasing. In PW mode, the transducer has to sample a certain frequency at least twice to resolve it with certainty. This put a limit on the max velocity that it can resolve with accuracy. 2 x Doppler frequency (Nyquist) = PRF. If the velocity is greater than the sampling rate / 2, aliasing is produced. The following maneuvers can be performed to eliminate aliasing: change the Nyquist limit (change the scale), select a lower frequency transducer, select a view with a shallower sample volume.

Fig. 34

Imaging and PW Doppler can be achieved with a single crystal transducer (both are created using pulsed ultrasound). With 2D imaging, one uses high frequencies and the incidence is usually at 90 degrees. With PW Doppler, one uses lower frequency and the incidence is usually at 0 degrees for optimal data.

Color Flow Doppler uses pulsed Doppler technique. The velocity data is encoded in color, and it reports mean velocities. Since it is a pulsed Doppler technique, it is subject to range resolution and aliasing. Color data is extremely complex and consumes significant computational resources, thus several assumptions are made to speed up this process. Returned echo frequencies are compared to a predetermined threshold to decide whether this is a 2D image vs Doppler shift. Once the computer decides that the frequency is low enough to be a Doppler shift data, repetitive sampling determines the mean velocity and variance. Then a color is assigned using a color look-up table rather than doing a discrete Fourier transform for each data point. Velocities that move toward the transducer are encoded in red, velocities that move away are encoded in blue. One must remember that the color jets on echo are not equal to the regurgitant flow for a number of reasons. The regurgitant flow is a three dimensional structure with jet momentum being the primary determinant of jet size. This parameter is effected by the jet velocity as well as flow rate. Blood pressure will affect the velocity and thus the regurgitant flow. Chamber constraints will have an effect on the appearance of the color jet, especially eccentric jets. Lastly, the settings of the echo machine will have an effect on how the color flow jet appears on the screen.

## Reference:

• Feigenbaum's Echocardiography, 7th Edition
• Sidney K. Edelman, PhD. Lecture notes from 2005 ASCeXAM Review course