Det medisinske fakultet

Basic ultrasound, echocardiography and Doppler for clinicians


by Asbjørn Støylen, dr. med.

Contact address: asbjorn.stoylen@ntnu.no
This section updated:  November 2010

This web page is intended as an introduction to basic ultrasound physics and technology for clinicians without technical  or mathematical background. A basic knowledge of the physical principles underlying ultrasound, will give a better understanding of the practical limitations in ultrasound, and the technical solutions used to solve the problems. This will give a clearer picture of the reasons for the problems and artifacts. Technical or mathematical background is not necessary, explanations are intended to be intuitive and graphic, rather than mathematical. This section is important for the understanding of the basic principles described in detail in the section on measurements of strain rate by ultrasound. Especially in order to understand the fundamental principles that limits the methods.The priciples will also be useful to gain a basic understanding of echocardiography in general, and may b read separately even if deformation imaging is not interesting.

The page is part of the website on Strain rate imaging.

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Ultrasound

Ultrasound is simply sound waves, like audible sound. Although some physical properties is dependent on the frequency, the basic principles are the same. Sound consists of waves of compression and decompression of the transmitting medium (e.g. air or water), traveling at a fixed velocity. Sound is an example of a longitudinal wave oscillating back and forth in the direction the sound wave travels, thus consisting of successive zones of compression and rarefaction. Transverse waves are oscillations in the transverse direction of the propagation. (For instance surface waves on water or electromagnetic radiation.)


Fig. 1. Schematic illustration of  a longitudinal compression wave (top) and transverse wave (bottom). The bottom figure can also represent the pressure amplitude of the sound wave.

The audible sound frequencies  are below 15 000 to 20 000 Hz, while diagnostic ultrasound is in the range of 1 - 12 MHz. Audible sound travels around corners, we can hear sounds around a corner (sound diffraction). With higher frequencies the sound tend to move more in straight lines like electromagnetic beams, and will be reflected like light beams. They will be reflected by much smaller objects (because of sorter wavelengths), and does not propagate easily in gaseous media.

The wavelength  is inversely related to the frequency f by the sound velocity c:

 

Meaning that the velocity equals the wavelength times the number of oscillations per second, and thus:


The sound velocity i a given material is constant (at a given temperature), but varies in different materials (117):
Material
 Velocity ( m/s)
Air
 330
Water
 1497
Metal
 3000 - 6000
Fat
 1440
Blood
1570
Soft tissue
 1540

Ultrasound is generated by piezoelectric crystals that vibrates when compressed and decompressed by an alternating current applied across the crystal, the same crystals can act as receivers of reflected ultrasound, the vibrations induced by the ultrasound pulse.

What are the ultrasound data?

The ultrasound data can be sampled at different levels of complexity as shown below:

Basically, a reflected ultrasound pulse is a waveform. However, storing the full waveform, called RF data, is demanding in terms of storage, as each point on the curve would have to be represented in some way or other. However, if the full RF data are stored, the amplitude and frequency data could both be calculated in post processing.
The pulse has a certain amplitude. Just storing the amplitude is much les demanding (corresponding more or less to one number per pulse). This is the only data that are used in grey scale imaging, where the amplitude is displayed as brightness of the point correspåonding to the scatterer as in B-mode and M-mode.
 However, the reflected ultrasound pulse has a frequency (or a spectrum of frequencies), and this can be represented as a numerical value per image pixel as well, as described in Doppler imaging. Still, the amount of data is far less than the RF data.


Imaging by ultrasound

Basically, all ultrasound imaging is performed by emitting a pulse, which is partly reflected from a boundary between two tissue structures, and partially transmitted (fig. 2). The reflection depends on the difference in impedance of the two tissues. Basic imaging by ultrasound does only use the amplitude information in the reflected signal. One pulse is emitted, the reflected signal, however, is sampled more or less continuously (actually multiple times). As the velocity of sound in tissue is fairly constant, the time between the emmision of a pulse and the reception of a reflected signal is dependent on the distance; i.e. the depth of the reflecting structure. The reflected pulses are thus sampled at multiple time intervals (multiple range gating), corresponding to multiple depths, and displayed in the image as depth.

Different structures will reflect different amount of the emitted energy, and thus the reflected signal from different depths will have different ampluitudes as shown below. The time before a new pulse is sent out, is dependent of the maximum desired depth that is desired to image.




Fig. 2.  Schematic illustration of  the reflection of an ultrasound pulse emitted from the probe P, being reflected at a, b and c.  Part of the pulse energy is transmitted from the scatterer a, the rest is transmitted, part from b and the rest from c. When the pulse returns to P, the reflected pulse gives information of two measurements: The amplitude of the reflected signal, and the time it takes returning, which is dependent on the distance from the probe(twice the time the sound uses to travel the distance between the transmitter and the reflector, as the sound travels back and forth). The amount of energy  being reflected  from each point  is given in  the diagram as the amplitude. When this is measured, the scatterer is displayed with amplitude and position. Thus, the incoming pulse a a is the full amplitude of P. At b, the incoming (incident) pulse is the pulse transmitted  through a. At c, the incident pulse is the transmitted pulse from b. (In bot cases minus further attenuation in the interval.)

The ratio of the amplitude (energy) of the reflected pulse and the incident is called the reflection coefficient. The ratio of the amplitude of the incident pulse and the transmitted pulse is called the transmission coefficient. Both are dependent on the differences in acoustic impedance of the two materials. The acoustic impedance of a medium  is the speed of sound in the material × the density:
Z = c ×

The reflecting structures does not only reflect directly back to the transmitter, but scatters the ultrasound in more directions. Thus, the reflecting structures are usually termed scatterers.


It's important to realise that the actual amount of energy that is reflected back to the probe; i.e. the amplitude of the reflected signal, is not only dependent on the reflection coefficient. The direction of the reflected signal is also important.
Thus:
    - An irregular scatterer will reflect only a portion back to the probe.
    - A more regular scatterer will reflect more if the reflecting surfaces are perpendicular to the ultrasound beam.
Effect of size and direction of the reflecting surface.  The two images on the left shows a perfect reflecting surface. Most of the energy (but not all, as the wavefront is not flat), will reflect back to the transducer resulting in a high amplitude echo, when the surface is perpendicular to the ultrasound beam. On the other hand, if this surface is tilted 45º, almost all energy will be reflected away from the surface, resulting in a very low amplitude echo.  The two images to the right shows a scatterer with a more curved surface, resulting in more energy being spread out in different directions, this will give a lower amplitude signal when oriented toward the beam, but reflects more energy back towards the probe if it is tilted. The effect of the direction of the reflecting surface in a  long axix image of the left ventricle.  The echo resulting from the  septum-blood interface (arrows) is  far  stronger in the regions where the surfaces are perpendicular to the ultrasound beamns (blue arrows), compared to the region between where the surface is slanted compared to the ultrasound beams.

Cyclic variations in the amplitude in reflected ultrasound (integrated backscatter) with heart cycle. This reflect the variations in reflexivity, but not myocardial density, as the myocardium is incompressible. Thus, most of the amplitude variations must be due to changes in fibre directions.


Thus, the apparent density of the tissue on the ultrasound image is as dependent on the fibre direction. A part of the heart where the fibres run mainly in a direction across the ultrasound beams, will look much denser. Variations in amplitude (brightness of the reflected signal) do not necessarily mean differences in density, but may also mean variations in reflectivity due to variation in the direction of the reflections. Thus, integrated backscatter can be used for studying of cyclicity, but it is not useful for tissue characterisation.




The time lag, , between emitting and receiving a pulse is the time it takes for sound to travel the distance to the scatterer and back, i.e. twice the range, r, to the scatterer at the speed of sound, c, in the tissue. Thus:

The pulse is thus emitted, and the system is set to await the reflected signals, calculating the depth of the scatterer on the basis of the time from emission to reception of the signal. The total time for awaiting the reflected ultrasound is determined by the preset depth desired in the image.

The received energy at a certain time, i.e. from a certain depth, can be displayed as energy amplitude, A-mode. The amplitude can also be displayed as the brightness of the certain point representing the scatterer, in a B.mode plot. And if some of the scatterers are moving, the motion curve can be traced by letting the B-mode image sweep across a screen or paper as illustrated in fig. 3. This is called the M-mode (Motion).






Fig. 3a.  The ultrasound  image is built up as a line of echoes based on the time lag and  amplitude of the reflected signals.
3b. The reflected signals can be displayed in three different modes. A-mode (Amplitude) shows the depth and the reflected energy from each scatterer.  B-mode (Brightness) shows the energy or signal amplitude as the brightness (in this case the higher energy is shown darker, against a light background) of the point. The bottom scatterer is moving. If the depth is shown in a time plot, the motion is seen as a curve, (and horizontal lines for the non moving scatterers) in a M-mode plot (Motion).

 

Attenuation

It follows that the ultrasound waves are attenuated as some of the energy is reflected or scattered. Thus, in passing through tissue, the energy is attenuated due to the reflection that is necessary to build an image. This is about 10% of the total energy loss. In addition, the ultrasound waves are diffracted, resulting in further diffusion of the waves out into the surrounding tissue and loss in the energy available for reflection (imaging). However, the most important factor is the the ultrasound energy is attenuated due to absorption in the tissue, this absorption process generates heating of the tissue. Thus, the total energy in the ultrasound beam decreases with depth, this is called attenuation. ( And the reflections are further attenuated in passing back toward the probe).

Absorption is important for two reasons:
The absorption is dependent on many factors (117):
  1. The density of the tissue. The higher the density, the more absorption. Thus the attenuation is fluid < fat < muscle < fibrous tissue < calcifications and bone.
  2. The frequency of the ultrasound beam. The higher the frequency, the more absorption. In human tissue, a general approximation is that the attenuation is 1 dB/cm MHz. (however, that is for one way, in imaging the distance is 2* the depth). Thus, the desired depth to be imaged, sets the limit for how high frequency that can be used. AS can be seen, penetration might be increased by increasing the transmitted energy, but this would increase the total absorbed energy as well, which has to stay below the safety limits.
It will also have effect on the image in other ways, as shown blow.

Attenuation. Imaging of a homogeneous tissue, f.i. liver will change the apparent density behind structures with different attenuation.
Behind a structure with high reflexivity (e.g. a calcification), there will be high attenuation, (white; left). Hence, the sector behind receives less energy, and appears less dense (darker), the area behind may even be a full shadow.
Behind a strcture with low reflexivity (e.g. a fluid) there is little attenuation (black; right), the tissue receives more energy and appears denser (brighter - "colouring") than the surrounding tissue. 
 Liver with a gallbladder in front, containing gallstones. The gallstones are dense, with a shadow behind. The rest of the gallbladdeer is fluid filled, thus the sector behind the fluid appears denser than the neighbouring tissue due to "colouring".


Attenuation can be dealt with by gain, increasing gain amplifies the reflected signal in post processing. However, increased gain increases signal and noise in the same manner.
Low amplitude signals can be filtered away, resulting in filtering out cavity noise, however at the price of risking to loose low amplitude signals (e.g. from valves.) by the reject function.
Finally, the grey scale can be compressed, resulting in a steeper saturation curve. This means that the picture goes to full saturation (pure white) at a lower amplitude, while the brightness of low amplitude signals are reduced.

It is important to realise that all these are post-processing functions that manipulates the image on the screen, without improving the signal quality itself, or the fundamental signal to noise ratio.


 








Principle of gain, reject and compress.  All curves display brightness of the display in relation to the amplitude of the rejected signal. An ordinary gain curve is shown in black, using a linear brightness scale, displays the full range of amplitudes. Increasing gain (red curve), will increase all signals, including the weakest, as in the noise. The disadvantage, in addition to increasing noise, is that the strongest signals will be saturated, so details may disappear. Compress is shown as the blue curve. This results in a steeper brightness curve, resulting in less brightness of the weakest echoes, and more brightness of the strongest. Thus, weak echoes may disappear together with background noise, while strong echoes will be saturated, resulting in loss of detail.  Finally reject is shown by the light grey zone, siply displaying all signals below a certain amplitude as black. (The black brightnes curve drops abruptly to zero at the reject limit (dark grey line).  A combination of high gain and reject will give an effect fairly similar to the compress function.

Image with default gain, reject and compress settings
 Same image with high gain (top) showing increased denity of the endocardium, but loss of detail due to birghtness saturation and a corresponding increase in cavity noise and low gain (bottom), showing reduction in cavity noise, but loss of detail (see endocardium in lateral wall).
 Same image with increased reject (top) showing reduction in cavity noise, but also with slight loss of detail (endocardium in lateral wall) and compress function with less detail in the myocardium due to increased brightness.



All commercial equipment today has a time gain compensation (TGC), increasing the gain of the reflected signals with increasing time from the transmitted pulse. This is equivalent to increasing the gain with increasing depth. However, this is not a perfect solution, as the noise is constant with depth, while the reflected signals become weaker, and with TGC, the noise will be gained as well as the signal, and the signal-to-noise ratio will decrease, thus the resulting signal will end up as a grey blur at a certain depth. This effect can be seen below. Before harmonic imaging, the TGC was adjustable, relying on the operator to optimise the visibility. AS the greater part of cavity noise is removed by the harmonic imaging, most modern equipment has automated TGC, but retains the possibility of manual adjustment.

For practical medical purposes, the penetration for good imaging is about 10 - 20 cm at 3.5 MHz (adult cardiac), 5 - 10 cm at about 5 MHZ (pediatric cardiac), 2-5 cm at 7.5 MHz, 1-4 cm at 10 MHZ, the last two frequencies being in the vascular domain. However, one method to bypass some of the attenuation problem is by harmonic imaging. Thus the beam is transmitted at a certain frequency, and the received signal is analysed at twice that frequency (Fourier analysis). This increases the signal to noise ratio of the reflected signal, especially at the deepest parts of the image, without a similar loss of resolution.

M-mode

The M-mode was the first ultrasound modality to record display moving echoes from the heart (118), and thus the motion could be interpreted in terms of myocardial and valvular function. The M-modes were originally recorded without access to 2-dimensional images.
a

 b

 c
Fig.  4.  Typical M-mode images. a from left ventricle, b from the mitral valve and c from the aortic valve as indicated on the 2D long axis image above. Here the amplitude is displayed in white on dark background.


Depth resolution. Bandwidth:

The depth resolution of the ultrasound beam, is the resolution along the beam. This is dependent on the length of the transmitted ultrasound pulse. In a blood / tissue interface, the dividing line can be seen as a bright line, which does not reflect a tissue stricture (typically NOT the intima, being far too thin to be seen with ultrasound at the present frequences), but the pulse length. This is the reason for the ASE convention, where depths are measured from leading - to - leading edge of the echoes, as this will neutralise the pulse length form measurements.


Ideally, the pulse length in imaging (B- and M-mode) should be as short as possible, but this is dependent on the physical properties of the probe. Most probes will ring in the resonance frequency for a few oscillations, and thus produce a pulse with a length of several oscillations. By Fourier analysis, the frequency content of the pulse will be less dispersed, the longer the pulse is. Thus, the pulse length is inversely proportional with  the spread of the frequency, i.e. the bandwidth of the pulse, as shown below. This will have consequences for Doppler imaging, where frequencies, and not amplitudes are analysed.


Two different pulses with the same frequency, but different duration (pulse length), i.e. Number of oscillations. The shortest pulse has a wider dispersion of frequencies, i.e. a greater bandwidth. After Angelsen (117).



Higher frequencies will result in shorter pulses for the same number of oscillations, i.e. reduce pulse length without increasing bandwidth to the same degree.  Thus, for imaging, the ideal pulse would be highest possible frequency (depending on the required depth penetration) and the shortest possible pulse length. However, as noise is unevenly distributed in different frequency domains, harmonic imaging, which analyses at half the frequency, will result in less noise. Harmonic imaging thus doubles the pulse length for a given frequency, and results in thicker echoes.





Halving the frequency results in half the number of oscillations per time unit, or longer time (= pulse length) for the same number of oscillations. Thus halving the frequency, as in second harmonic analysis, will result in longer pulse length. However, the bandwidth is far less affected.
Second harmonic (1.7/3.5 MHz) left and fundmental (3.5 MHz) right images of LV septum, showing how the echo from the blood/septum interface (arrows) is thicker in harmonic imaging, due to the reduction in frequency. Observe, however, how cavity noise is much reduced in harmonic imaging, resulting in a far more favorable signal-to-noise ratio.
 The thickness of the surtface echoes is dependent n the pulse length, and thus also on the frequency.  This picture of the septum illustrates how the leading-to-leading ASE convention shown in red, eliminates the pulse length in measurement (as the echo blooms in both directions), while the Penn convention will result in increasing overestimation of the thickness by increasing pulse length as it incorporates the interface on both sides.

The most important point is that the echo from an interface reflects the pulse length, and is NOT a picture of the endothelium.

2-dimensional imaging:

A 2-dimensional image is built up by firing a beam vertically, waiting for the return echoes, maintaining the information and then firing a new line from a neighboring transducer along a tambourine line in a sequence of  B-mode lines. In a linear array of ultrasound crystals, the electronic phased array shoot parallel beams in sequence, creating a field that is as wide as the probe length (footprint). A curvilinear array has a curved surface, creating a field in the depth that is wider than the footprint of the probe, making it possible to create a  smaller footprint for easier access through small windows. This will result in a wider field in depth, but at the cost of reduced lateral resolution as the scan lines diverge.

A pulse is sent out, ultrasound is reflected, and the B-mode line is built up  from the reflected signals.
Linear array.
Curvilinear array
The linear array gives a large probe surface (footprint) and near field, and a narrow sector. A curvilinear array will also give a large footprint and near field, but with a wide sector.

 But in order to achieve a footprint sufficiently small to get access to the heart between the ribs, and with a sufficiently wide far field, the beams has to diverge from virtually the same point. This means that the image has to be generated by a single beam originating from the same point, being deflected in different angles to build a sector image (cf. figs. 6 and 7).

This can be achieved by a single transducer or array sending a single beam that is stepwise rotated, either mechanically or electronically.
A very small footprint can be achieved by a mechanical probe, sending only one beam, but being mechanically rotated by a motor. Finally with a slightly larger footprint, a phased array with electronic focusing and steering, can generate a beam sweeping at an angle similar to the mechanical probe. Beamforming by phased array, also enables focusing of the ultrasound beam as shown. Focusing can also be performed in a mechanical probe, by a concentric arrangement of several ring shaped transducers, an annular array. This will focus the beam in both transverse directions at the same time, as indicated in fig. 7A.


The next line in the image is then formed by a slight angular rotation , making the beam sweep across a sector:


By making the ultrasound beam sweep over a sector, the image can be made to build up an image, consisting of multiple B-mode lines.
 		c. In principle, the image is built up line by line, by emitting the pulse, waiting for the reflected echoes before tilting the beam and emitting the next pulse. Resulting in an image being built up with a whole frame taking the time for emitting the total number of pulses corresponding to the total number of lines in the image.

This means that as a pulse is sent out, the transducer has to wait for the returning echoes, before a new pulse can be sent out, generating the next line in the image.
2D echocardiography. A line is sent out, and as all echoes along the beam are received, the picture along the beam is retained, and a new beam is sent out in the neighboring region. building up the next line in the image.  one full sweep of the beam will then build up a complete image; i.e one frame. A cine-loop is then a sequence of frames; i.e. a movie.

The present technology is sufficient to  build up a picture wit sufficient depth and resolution with about 50 frames per second (FPS), which gives a good temporal resolution for 2D visualisation of normal heart action ( about 70 beats per min.). However, the eye has a resolution of about 25 frames per second, so there may seem to be excess information. But off-line replay may be done at reduced frame rate, thus enabling the eye to utilise a higher temporal resolution.

Beamforming


Fig. 7A. Mechanical transducer. The sector is formed by rotating a single transducer or array of transducers mechanically, firing one pulse in each direction and then  waiting for the return pulse before rotating the transducer one step. In this beam there is electronic focusing as well, by an annular array.
 B. Electronic transducer in a phased array. By stimulating the transducers in a rapid sequence , the ultrasound will be sent out in an interference pattern. According to Huygens principle, the wavefront will behave as a single beam, thus the beam is formed by all transducers in the array, and the direction is determined by the time sequence of the pulses sent to the array. Thus, the beam can be electronically steeredand will then sweep stepwise over the sector in the same way as the mechanical transducer in A, sending a beam in one direction at a time.
 



Beam focusing:


Dynamic focusing. The same principle of phase steering can be applied to make a concave wavefront, resulting in focusing  of the beam with its narrowest part  a distance from the probe. Combining  the steering in B and C will result in a focussed beam that sweeps across the sector, as in the moving image above.
 Resulting Ultrasound beam as shown by a computer simulation, focusing due to the concave wavefront created by the dynamic focusing. The wavelength is exaggerated for illustration purposes. Image Courtesy of Hans Torp.

Focusing is illustrated above. In a mechanical probe, there may be several transducers, arranged in a circular array, focusing the beam in a manner analogous to that shown in fig. 7c. In a circular array, however, the focusing can be done in all directions transverse to the beam direction, i.e. in the imaging plane and transverse to the plane, while a linear array can only focus in one direction, in the imaging plane.

Annular focusing in all directions both in plane and transverse to the plane.
Linear focusing in the imaging plane only.

A matrix array, can focus in both directions at the same plane.

The focusing increases the concentration of the energy at the depths where the beam is focussed, so the energy in each part of the tissue has to be calculated according to both wavelength, transmission and focusing to ensure that the absorbed energy stays within safe limits.

Lateral resolution


The apparent width of the scatterer in the image is more or less given by the lateral resolution of the beam. (The thickness in the axial direction is determined by the depth resolution, i.e. the pulse length as discussed above). In addition, two echoes within one beam, will only be separated by the difference in depth.







The lateral resolution of a beam is dependent on the focal depth, the wavelength and  probe diameter (aperture) of the ultrasound probe.
(Reproduced from Hans Torp by permission)		 Thus a small scatterer will appear to be "smeared out", and the apparent size in the image is determined by the beam width and pulse length.  As the pulse length is less than the beam width, the object will
 be "smeared out" most in the lateral direction.
 Two scatterers at the same depth, separated laterally by less than the beam width, will appear as one.
 Two scatterers at different depths will appear separate  if separated by more than the pulse length.
But, if separated both laterally and in depth, they will appear as being in the same line, if lateral separation is within the beam.


Thus:

Line density

The width of the echo will be determined by the beam width, and thus the distance between the beams (most ultrasound scanners today will intrapolate between beams if the distance between the beams is greater than the beam width). Ideally, the distance between the beam width should be the same as the beam width at the focal depth, for maximal resolution, thus lateral resolution of a beam determining the line density. This means that the line density would be suited to the beam width. This, however, holds only for a linear array.

However, as the beam width also increases at depths greater than the focal depth, the ideal line density for a sector probe is the one where  beam distances are equal to the beam width at the focal depth. This will give the best lateral resolution. A line density that is so high as to make lines overlap, will not result in increased lateral resolution. A line density that leaves gaps between the lines, will have less than optimal lateral resolution as determined by the probe aperture and focal depth.

But as the time it takes to build each line in the image for any given depth that is desired, the number of beams in an image limits the frame rate. And if a greater sector width is desired without reducing the frame rate, the line density is reduced (same number of lines over a wider angle).

Thus,  the line density itself is limited by other factors as well:
Due to these factors, the line density often falls below the theoretically desirable described above, and the line density, not the probe size and wavelength becomes the limiting factor for the lateral resolution.
Two points in a sector that is to be scanned.
The ultrasound scan will smear the points out according to the lateral resolution in each beam.
Two different lateral resolutions, the speckles can be seen to be "smeared". In this case the loss of resolution in the right image is due to lower line density . By rights the image should appear as split in different lines as indicated in the middle, as each beam is separated, line density being less than optimal relative to the beam width. Instead the image is interpolated beween lines. This reduction in line density is done to achieve a higher frame rate, as illustrated below.

So a distinction should be made between the lateral beam resolution, given by the fundamental properties of the system, and the image resolution that is a compromise between the requirements of frame rate, angle width and depth.

Thus, the line density can be increased by
  1. Reducing the sector width (gives higher line density by spreading the lines over a smaller angle)
  2. Reducing frame rate (enables time for builing more lines between frames)
  3. Reducing depth (enables a higher line density for a given frame rate, as the shorter lines takes shorter time to build).
This is discussed in detail below:

Temporal resolution (frame rate):

To imagine moving objects, structures such as blood and heart, the frame rate is important, related to the motion speed of the object. The eye generally can only see 25 FPS (video frame rate), giving a temporal resolution of about 40 ms. However, a higher frame rate and new equipment offers the possibility of replay at lower rate, f.i. 50 FPS played at 25 FPS, which will in fact double the effective resolution of the eye.

In quantitative measurement, whether based on the Doppler effect or 2D B-mode data, sufficient frame rate is important to avoid undersampling.  In Doppler, the frame rate is also important in the  Nykvist phenomenon.

The temporal resolution  is limited by the sweep speed of the beam. And the sweep speed is limited by the speed of sound, as the echo from the deepest part of the image has to return before the next pulse is sent out ad a different angle in the neighboring beam.

Depth
If the desired depth is reduced, the time from sending to receiving the pulse is reduced, and the next pulse (for the next beam) can be sent out earlier, thus increasing sweep speed and frame rate, as shown below.



As the depth  of the sector determines the time before next pulse can be sent out, higher depth results in longer time for building each line, and thus longer time for building the sector from a given number of lines, i.e. lower frame rate.
 Thus reducing the desired depth of the sector results in shorter time between pulses, and thus shorter time for building each line, shorter time for building the same number of lines, i.e. higher frame rate. In this case, the depth has been halved, and the time for building a line is also halved.

The important point of this, is that reducing the depth to the field of interest will give a higher frame rate, that can either be used for higher temporal resolution, or for increased spatial resolution or sector width (see later). In practice, when studying the ventricles, the atria should be excluded.

The field of view should be limited to the field of interest.
In this case, in the image to the left, the depth has been halved, reducing the time for building each line to half, thus also halving the time for building the full sector, doubling the frame rate.

Number of beams: Sector width and line density.

The sweep speed can also be increased by reducing the number of beams  for a full sector. Reducing the number of lines in the image will reduce the time for building up the whole image. This can be achieved by either decreasing the sector angle (width), but keeping the line density, i.e. reducing the field of view but keeping lateral resolution. Decreasing the line density, but keeping the same sector angle  will achieve the  same increase in frame rate, but  reduce lateral resolution.  

Fig.7a. A sector with a given depth, sector width and line density determines the frame rate.
 b. Reducing sector width, but maintaining the  line density, gives unchanged lateral resolution but higher frame rate, at the cost of field of view.
 c. Reducing the  line density instead and maintaining sector width, results in lower number of lines, i.e. lateral resolution, and gives the same increase in frame rate.


As can be seen there has always to be a trade off. Frame rate is a compromise between sector size (width and depth), resolution (line density) and frame rate. Line density is a trade off with frame rate and sector width. The fundamental limitation is the speed of sound.

This is shown in the example below:



Ultrasound acquisitions of the same ventricle at frame rate 34 ( left), 56 (middle) and 112 (right), all other setting being equal. Increased frame rate is achieved by reducing the number of lines; i.e. the line density. This can be seen as an increasing width of the speckles in the image with increasing frame rate, resulting in a lateral blurring of the image. The first step from 34 to 56 seems to retain an acceptable image quality, indicating that the line density was  redundant  at the lowest frame rate.  ( In fact, it may seem that the image in the middle has the best quality, as the left image seems more grainy. But the graininess is the real appearance of the echoes, while the more homogeneous appearance in the middle and the left is due to smearing). However, as line density decreases toward the bottom of the sector (by the divergence of the lines), the effect is mos clearly seen here, i.e. in the atrial walls, the mitral ring and valve.  In the image to the right, the endocardial definition is lost.  As it is the echoes that are smeared, the effect will result in an apparent decreased cavity size. 

These images also illustrates the drawback of time gain compensation, all three images has the same TGC, showing about the same brightness of the walls from base to apex, (the attenuation being offset by the TGC), but with increasing cavity noise.

Multiple line acquisition (MLA)
However, a method for increasing the frame rate for a given sector and line density, is to fire a wide transmit beam, and listen on more narrow receiver beams (crystals) simultaneously. This is called multiple line acquisition (MLA), and is illustrated below:

In this example, a wide beam is fired, and for each of the four transmit beams, there are four receiver beams (4MLA). thus, the frame rate is increased fourfold for the same number of lines.
However, simultaneous reception from neighboring lines will result in some contamination of the signals from lateral beams, in addition it will increase the effect of side lobes. At present, 4MLA is used in colour tissue Doppler, but only 2MLA in B-mode imaging. The limitation ion B-mode is due to the point that there will be interference beween the simultaneous receiver beams, creating a line pattern. Smoothing this in the image will result in "semaring", and thus give much of the effect seen by reducing line density, and thus there is no gain in the mLA technique..

artifacts:

Shadows



Both air and bone will attenuate the ultrasound beam almost totally, thus creating a shadow. However, a shadow has different effects depending on the distance from the probe. A distant shadow will simply create a drop out behind the shadowing object. On the other hand, a shadow close to the probe will simply reduce the effective aperture, thus not creating a drop out, but instead reducing the lateral resolution. The principle is illustrated below.
Illustration of effects of shadows on an ultrasound beam. Left: no shadow. Middle, a shadow distant from the beam (e.g. a calcification or the lung seen at a distance), resulting in a shadow with no image below it. Left a shadow close to the transducer surface (e.g. lung edge or rib) will result in a narrow beam (reduced apparent aperture) which will not be seen as a shadow in the picture, but rather a reduced lateral resolution.  (Simulation image courtesy of Hans Torp.) The effect of the depth of the origin of the shadows in the images is shown below, indicated by the green arrows.
If the near shadow is in the centre of the probe, the result may be that the beam is split in two, resulting in two apparent apertures. The effect on the image is shown below.

Left, shadow originating at a depth of ca 3 cm, as can seen by the visible structures of the chest wall closer to the probe.  The shadow is probably due to the edge of the lung.  Right; a small repositioning of the probe solves the problem. Left shadow originating close to the chest wall (< 1 cm), probably the edge of a costa.  It can be seen as a shadow, but the main effect is loss of lateral resolution in the shadow, and again a small repositioning of the probe solves the problem as seen to the right. Split image due to two virtual appertures, caused by a near shadow in the middle of the probe footprint.









Drop out of the anterior wall in this 2-chamber view due to a lung shadow distant from the probe. However, the lateral resolution may be seen to be reduced at the basal part of the  border between the picture and the shadow.
 Reduced lateral resolution due to costal shadow. The effects of both costae and shadows will vary, according to the distance from the probe. In this case the patient was extremely thin, thus there was virtually no distance between the probe and the costa. In this case, no localised shadow can be seen, the costa was to the left in the image, where resolution is poorest.


Stationary reverberations:

Stationary reverberations are caused by stationary structures, usually in the chest wall, causing the ultrasound to bounce back and forth between the skin and the structure, increasing the time before the echo returns and giving rise to an apparent stationary structure deeper down.


Left, the principle of stationary reverberations, showing how the ultrasound pulse bounces twice between a stationary structure in the body wall and the surface (thick line), causing the reflected pulse to arrive the transducer three times as late as if it hadn't bounced (dotted, thick line). This is then interpreted as the reflector being located three times it's true depth. Right, the appearance  in grey scale image, see the stationary echo in the lateral wall (arrows).

Reverberations needs not necessarily be totally stationary, if the reflecting surface that gives rise to the echo moves, the reverberations will move as well.

Stationary echoes and noise is also referred to as "clutter". This noise may also result in a more random pattern (shadowy reverberations), resulting in a more blurred picture as shown below.

Side Lobes

Each beam is not solely concentrated in the main beam as illustrated above. In addition, some of the energy is dispersed in side lobes originating among other things from interference as illustrated below.



Simulated beam with focusing, showing interference pattern dispersing some of the beam to the sides. (image courtesy of Hans Torp).
 Side lobes from a single focussed ultrasound beam. These side lobes will also generate echoes from a scatterer hit by the ultrasound energy in the side lobes, i.e. outside the main beam.


As echoes  from a scatterer in the side lobe pathway is perceived coming from the main beam, this will result in a false echo. AS the beam sweeps back and forth, this again will result in the echo being smeared out across the sector.




This results in a false echo in the main beam
 AS the beam with side lobes sweeps across the sector, each echo from the scatterer in both the main beam and the side lobes will generate the false echo in the position of the main beam
 Resulting in a smeared out echo across a large part of the sector.





Side lobes originating from the fusion line of the aortic cusps, seen to extend into both the LV cavity and the aortic root cavity (arrows).
As opposed to reverberations, the side lobes moves with the structure, and may change with time (in this case the echo intensity of the fusion line decreases as the valve opens, and thus the intensity of the side lobes too) .
 An image riddled with side lobes. Again they can be seen to move in the axial direction with the structures they are originating from.  (The echo in the middle of the right ventricle cavity is an ICD lead, and not an artifact.)

As opposed to reverberations, the side lobes will move as well as increase and decrease in intensity in parallel with the source of the echo as shown in the illustrations above.

Foreshortening

For correct display of the left ventricle, the imaging plane has to transect the apex. This is ensured by finding the apex beat by palpation. However, the apex does not necessarily offer the optimal window for imaging, and the intercostal space above may give a better view. However, this may lead to a geometrical distortion as illustrated below:
Apical position of the probe. The two orthogonal planes can be seen to bisect the apex.
Foreshortening by erroneous placement of the probe. The two orthogonal planes can be seen to bisect the wall, not the apex.
In one plane this will not be evident, as the intersecting wall still shows an ellipsoid shape.

This is shown in the images below:

A: LV in correct 4-chamber plane. Probe in the 5th intercostal space. Image is less than optimal in quality.
 B: Foreshortened image in 4 chamber plane, by placing the probe in the 4th intercostal space. Wall definition as well as detail become clearer.
C: Rotating the probe in position B to the two chamber plane, shows the apex to be out of sector, thus the four chamber image is foreshortened.

The foreshortened image in four chamber view may seem to be better, at least for wall motion assessment, but the consequences are:
  • The apex is not imaged, and any apical abnormalities will be missed.
  • Volumes and LV length may be underestimated (I've seen diagnosed bi atrial enlargement due to foreshortening simply because the normal atria looked bigger in relation to the foreshortened ventricle)
  • The anterior wall is partly apical and partly circumferential. Thus longitudinal shortening, especially by speckle tracking may be circumferential rather than longitudinal. This may also be related to the curvature dependency of strain as measured by speckle tracking.

Non linear wave propagation and harmonic imaging.

Non linear propagation of the signal in the body, leads to distortion of the waves in the signal. But this again leads to a dispersion of the wavelength content in the signal, as assessed by Fourier analysis in the received signal. By Fourier analysis it is thus possible to send at half the frequency (typical 1.7 MHz as opposed to 3.4 MHz in native imaging), but receive at the same frequency (the second harmonic frequency: Twice the frequency is one octave higher). Thus, it improves penetration, which is important especially in obese subjects, while it retains the resolution (almost).

The noise from clutter and aberrations is mainly in the primary frequency, so the use of second harmonic will suppress noise, improving the noise-to-signal ratio. Also, the echoes from the side lobes are mainly in the primary frequency and will be reduced in second harmonic imaging.

Thus second harmonic imaging leads to:

1: Reduced noise and side lobe artifacts
2: Improved depth penetration.

However, due to the increase in pulse length with higher frequency, harmonic imaging also leads to:

3: Thicker echoes from speckles as discussed above.






Non-linear propagation. The upper panel shows the waveform of a pulse as originally transmitted, and after 6 mm transmission through tissue. The lower panels shows how the energy distribution is shifted to a more evenly distribution between more frequencies. (image courtesy of Hans Torp). Fourier analysis of the resulting signal in native frequency (left) and second harmonic mode (left) shows that the native signal contains much more energy at all depth, while the harmonic signal contains most of the energy at a certain depth, in this case at the level of the septum, showing a much better signal-to-noise ratio.(image courtesy of Hans Torp). Energy distribution of the signal from cavity (lower curve) and septum (upper curve), showing the same phenomenon as the middle picture. The difference between cavity signal (being mostly clutter) and tissue is small in the native frequency domain (1.7 MHz), but there is little clutter at the harmonic frequency (3.4 MHz). Thus, filtering the native signal will reduce clutter, as shown below. (image courtesy of Hans Torp).



Examples of the effect of harmonic imaging can be seen below.



The same image in  harmonic (left) and fundamental (right) mode, showing the improved signal-to-noise ratio in harmonic imaging, especially in rducing noise from the cavity.  (Thanks to Eirik Nestaas for correcting my left-right confusion in this iamge text)
Stationary reverberation in harmonic (left) and fundamental (right) imaging, showing the effect of harmonic imaging on clutter.
 Fundamental (left) and harmonic (right) images of the left ventricle at the level of the chorda tendineae. The echo of the chorda (blue arrow) can be seen to be thicker in harmonic imaging, due to the longer pulse length. The echo generates a side lobe that can be seen to the right of the chorda. The side lobe is more prominent in fundamental than harmonic imaging. Note also the reduction in cavity noise from the right and left ventricle in the harmonic image.



However, halving the transmit frequency will also halve the Nykvist limit, and thus is less suited to Doppler imaging as will be discussed below.

Speckle formation:

The gray scale image is seen to consist of a speckled pattern. The pattern is not the actual image of  the scatterers in the tissue itself, but the interference pattern generated by the reflected ultrasound:
Interference pattern. Here is simulated two wave sources or scatterers at the far field (white points). The emitted or reflected waves are seen to generate a speckle pattern (oval dots) as the amplitude is increased where wave crests cross each other, while the waves are neutralised where a wave crest crosses a though. This can be seen by throwing two stones simultaneously in still water . The speckle pattern can be seen in front of the scatterers, towards the probe.
 Irregular interference pattern. This is generated by more scatterers somewhat randomly distributed. The speckle pattern is thus random too.  Again there may be a considerable distance between the speckles and the scatterers generating the pattern.


Speckle tracking
Humpback whale diving. Each humpback has an unique (speckle) pattern on the underside of the tail (and flukes). Thus each individual can be identified by its speckle pattern. Photographs at different times and places can thus track the wandering of each individual all over the area it wanders, without recourse to anything else than the pattern.  - Speckle tracking! This is thus a method with low frame rate, giving mainly the extent of wandering over a long time period (the sampling intwerval). To measure swimming velocity, a Doppler sonar would have been useful.

The speckle pattern can be used to track myocardial motion due to two facts about the speckle pattern.

1. The randomness of the speckle pattern ensures that each region of the myocardium has its own unique speckle pattern: that can differentiate a region from other regions, just as in the whales.



Principle of speckle tracking. Demonstrating the difference between two different regions of the myocardium by their different random speckle pattern. The two enlarged areas show completely different speckle patterns, which is due to the randomness of the interference. This creates an unique pattern for any selected region that can identify this region and hence, the displacement of the region in the next  frame.

2. The speckle pattern remains reasonably stable, and the speckles follow the myocardial motion. This can be demonstrated by M-mode:
An M-mode along the septum demonstrates how the speckles is shown as motion curves. It is evident that many speckles are only visible during part of the heart cycle, but if the speckle pattern is compared from frame to frame, the changes will be small. The grainy texture of the lines is due to the limited frame rate as the M-mode on the right is reconstructed from the 2D image at the left. When the speckle pattern is followed by an M-mode in the wall, the alternating bright and dark points are seen as alternating bright and dark lines. The lines remaining to a large degree unbroken, shows the pattern to be relatively stable, the speckles moving along with the true myocardial motion, and thus myocardial motion can be tracked by the speckles.

By this, defining a region (kernel) in one frame, this kernel can be identified as region in the next frame with the same size and shape with the most similar speckle pattern, and the motion of the kernel can be tracked from frame to frame.




Speckle tracking. Real time M-mode demonstrates how the speckle pattern follows the myocardial motion.
Defining a kernel in the myocardium will define a speckle pattern within (red). Within a defined search area ( blue), the new position of the kernel in the next frame (green) can be recognised by finding the same speckle pattern in a new position. The movement of the kernel  (thick blue arrow) can then be measured.
 Speckle tracking search algorithm. The kernel is defined in the original frame at t=0 (red square). In the next frame, at t=t, the algorithm defines a search area (white square), and the search is conducted in all directions for the matching kernel.

The algorithm for this seas is simple, it simply searches for the area with the smallest difference in the total sum of pixel values, the smallest sum of absolute differences (SAD). This has been shown to be as effective as cross correlation (26, 27). However, the speckle pattern will not repeat perfectly. This is due to both true out of plane motion (rotation and torsion relative to apical planes and longitudinal deformation relative to short axis planes) and to small changes in the interference pattern. But the frame to frame change is small, and the approach to recognition is statistical. This means, however, that the search should be done from frame to frame, the changes over longer time intervals will be to great.


More about the mathematics can be found here.
 
In principle, pure speckle tracking  is direction independent, and can track crosswise. However, lateral resolution is important in delineating the speckles in the lateral direction. If the lateral resolution is low,  the interpolation will result in a "smeared" picture, with speckles that are nor so easily tracked in the lateral direction. In addition the lateral resolution decreases in depth with sector probes.
LOngitudinal speckle tracking in apical 4 chamber view. The resulting tracking of the kernels shown in motion. As can be seen, with a drop out apicolateral, this ROI tracks less than perfect, giving too low strain both in LA and MA segments. Speckle tracking can be applied crosswise. In this parasternal long axis view, the myocardial motion is tracked both in axial and transverse (longitudinal) direction. It is evident that the tracking is far poorer in the inferior wall, due to the poor lateral resolution at greater depth.

However, drop outs and reverberations will affect the tracking, and in the lateral direction, low lateral resolution will "smear" the speckles in the lateral direction, making tracing less perfect, as can be seen in the parasternal long axis image above.

The speckle pattern will not repeat perfectly. This is due to both true out of plane motion (rotation and torsion relative to apical planes and longitudinal deformation relative to short axis planes) and to small changes in the interference pattern. But the frame to frame change is small, and the approach to recognition is statistical, the basic algorithms are shown here. This means, however, that the with lower frame rate, the changes from frame to frame are greater, resulting in poorer tracking. Higher heart rate (f.i. in stress) will result in the same, as the number of frames per cycle will be less, i.e. lower relative frame rates.

Thus: speckle tracking is frame rate sensitive:
  1. Too low frame rate will result in too great changes from frame to frame, resulting in poor tracking. This may also limit the use in high heart rates, as the motion and thus frame to frame change increases relative to the frame rate.
  2. Too high frame rate is obtained by reduced lateral resolution, and thus resulting in poorer tracking at least in the transverse direction.
Thus, both too high and too low frame rate may affect speckle tracking adversely. With the present equipment, the optimal frame rate seems to be between 40 - 70 if image quality is good, slightly higher with poorer image quality.

As speckle tracking can track in both transverse and axial directions, with a sufficient number of kernels, deformation can in principle be measured in two dimensions, as as discussed in the section about a new application called "2D strain".

Within its limitations, however, speckle tracking can be used for measuring displacement, velocity, strain and strain rate as described below.









The Doppler effect
Christian Andreas Doppler
(1803 - 1853)
My cat Doppler
(2004 -     

The Doppler effect was discovered by Christian Andreas Doppler (1803 - 1853), and shows how the frequency of an emitted  wave changes with the velocity of the emitter or observer. The theory was presented in the royal Bohemian society of Science in 25th of May1842 (5 listeners at the occasion!), and published in 1843 (119). The premises for his theoretical work was faulty, as he built his theory on the work of James Bradley who erroneously attributed the apparent motion of stars against the background (parallax effect) to the velocity of the earth in its orbit (instead of the effect of Earth's position in orbit on the angle of observation). Further, Doppler attributed the differences in colour of different stars to be due to the Doppler effect, assuming all stars to be white.  Finally, he theoretised over the effect of the motion  of double stars that rotate around each other, assuming a  Doppler effect  from the motion.  The changes in wavelength from the Doppler effect, however, is too small to be observed.

However, Doppler did a theoretical derivation of the effect of the motion of the source or observer on the perceived wavelength from the premises of a constant propagation velocity of the waves in the medium, and this is entirely correct, valid both for sound waves and electromagnetic radiation of all kinds. The basis for the Doppler effect is that the propagation velocity of the waves in a medium is constant, so the waves propagates with the same velocity in all directions, and thus there is no addition of the velocity of the waves and the velocity of the source. Thus, as the source moves in the direction of the propagation of the waves, this does not increase the propagation velocity of the waves, but instead increases the frequency.The original derivation of the Doppler principle as well as the extension to reflected waves is explained in more detail here. As a work of theoretical physics, it is thus extremely important. In addition, it has become of practical importance, as  the basis for the astronomical measurement of the velocity of galaxies by the red shift of the spectral lines, in Doppler radar, Doppler laser and Doppler ultrasound.

The theory was experimentally validated by the Dutchman Christoph Hendrik Diderik Buys Ballot (120), with the Doppler effect on sound waves, who placed musicians along a railway line and on a flatbed truck, all blowing the same note, and observed by subjects with absolute pitch, who observed the tones being a half note higher when the train was approaching as compared to the stationary musicians, and a half note lower as the train receded.

(This can be observed in everyday phenomena such as the sound of f.i. an ambulance siren, the pitch (frequency) is higher when the ambulance is coming towards the observer, hanging as it passes,  and lower as it goes away.

This is illustrated below:

 

The Doppler effect. As the velocity of sound in air (or any other medium ) is constant, the sound wave will propagate outwards in all directions with the same velocity, with the center at the point where it was emitted. As the engine moves, the next sound wave is emitted from a point further forward, i.e. with the center a little further forward. Thus the distance between the wave crests is decreased in the direction of the motion, and increased in the opposite direction. As the distance between the wave crests is equal to the wavelength, wavelength decreases (i.e. sound frequency increases) in front of the engine, and increases (sound frequency decreases) behind it. This effect can be heard, as the pitch of the train whistle  is higher coming towards a listener than moving away, changing as it passes. The effect on the pitch of the train whistle was published directly, but later than Doppler and Buys Ballot.


If the sound source is stationary, the effect on  moving observer is similar. The train will meet the wave crest with shorter intervals, as the train moves into the incoming sound. In ultrasound, the wave is sent from a stationary transducer, the moving blood or muscle is  firstly moving towards the transducer and then following the reflected wave towards the transducer, thus the Doppler shift is approximately twice as great. In the case of reflected ultrasound, the Doppler shift is:

where is the angle between the direction of the motion and the ultrasound beam (insonation angle).

Thus, in the case of reflected ultrasound, the velocity of blood or tissue can be measured by the Doppler shift of the reflected ultrasound:
Basically, the Doppler effect can be used to measure blood and tissue velocities from the Doppler shift of reflected ultrasound:

where v is the blood or tissue velocity, c is the sound velocity in tissue, f0 is the transmitted frequency, fD is the Doppler shift of reflected ultrasound and is the insonation angle, between the ultrasound beam and the direction of motion (velocity vector).

Spectral analysis:

Analysis of the returned Doppler signal resolves the Doppler shift into the different frequencies, as well as the amplitude for each spectrum. As shown above, the frequency is related to the different velocities.


Spectral analysis. The Doppler frequencies are distributed according to this frequency - amplitude diagram. The Tissue echoes has high amplitude of the reflected signal, but low velocities (resulting in low Doppler frequencies). Blood has higher velocities 8with a wider distribution), but lower amplitude. The signal is filtered. For Doppler flow, a high pass filter (low velocity reject) is applied to suppress the tissue echoes. The filter is variable, and thus can be applied to select for v very high velocities in insufficiency jets. A low pass filter (high velocity reject) can be applied to suppress noise above the velocity range. In tissue Doppler, the high pass filter can be removed, or at least partially, to allow the low velocities from the tissue (usually on the order of 1/10 of flow). It can partially be maintained to suppress absolutely stationary echoes, among other from the reverberations. The blood signal can be removed both by reducing the gain, and by applying a low pas filter. For detailed explanation see below.  Applied from Angelsen BA Ultrasound  imaging (117)

In the heart, the velocities are variable with time. Thus the velocity, or frequency shift is a function of time. At the same time, the amplitude is a function of the velocities, as the velocities will be  concentrated in a narrow band.  To display this, an approach similar to the M-mode can be applied. Amplitude is displayed as brightness, in a manners similar to B.mode, while Frequency / Velocity are displayed on the y - axis and time on the x - axis. This results in the typical Doppler flow velocity curves:
Typical spectral flow curves. Left: Flow in the Left ventricular outflow, right in the mitral annulus. Velocities toward the probe (positive Doppler shift) are shown as positive velocities, velocities away from the probe (negative Doppler shift) are shown as negative velocities. The velocities are sampled at a certain depth by pulsed Doppler (see below). Thus, the velocity distribution is limited to a fairly narrow band. Note the absence of velocities near zero, due to the high pass filter.


Width of the spectrum

As can be seen from the illustrations above, the velocity spectrum has a certain width. In the blood, this partly reflects the spread of velocities between the multiple scatterers in the blood, as shown below:



Spectrum width. Left mitral flow, showing a fairly narrow spectrum band, indicating a relatively homogeneous velocity distribution.
Right pulmonary venous flow, showing a wide distribution of velocities.


However, this is not the only explanation. As the bandwidth is a function of the pulse length as illustrated above, spectral analysis will yield a spectrum that minimum is as wide as the bandwidth. Ideally, the pulse length in Doppler should be long, in order to increase velocity resolution. However, this will reduce the spatial (axial) resolution.

Thus, if the dispersion of velocities is larger than the bandwidth, as in flow measurements, this is the most important. On the other hand, if there is little dispersion of velocities, as in tissue Doppler, the width of the spectrum reflects the bandwidth.

But the insonation angle also has an influence, not only in the velocity measurement according to the Doppler equation. but also in the bandwidth as shown below.


As can be seen, the direction of the motion  of the scatterer in relation to the direction of the pulse, may influence the number of oscillations that are actually used for measurement. Thus, a high  insonation angle is equivalent to a virtual shortening of the pulse length, and results in a wider bandwidth (spectrum).

Finally, stationary reverberations, creating artificially stationary echoes will result in widening of the spectrum in tissue Doppler. In flow, this will basically be removed by the high pass filter.

Pulsed and continuous wave Doppler:

Doppler pulses can either be used as a pulsed Doppler, where a pulse is sent out, and the frequency shift in the reflected pulse is measured after a certain time. This will correspond to a certain depth (range gating), i.e. velocity is measured at a specific depth, which can be adjusted. The width is the same as the beam width, and the length of the sample volume is equal to the length of the pulse. The same transducer is used both for transmitting and receiving.

A problem in pulsed Doppler is that the Doppler shift is very small compared to the ultrasound frequency. This makes it problematic to estimate the Doppler shift from a single pulse, without increasing the pulse length too far. A velocity of 100 cm/s with a ultrasound frequency of 3.5 MHz results in a maximum Doppler shift of  2.3 KHz. The solution to this problem is shooting multiple pulses in the same direction and produce a new signal with one sample from each pulse, the Doppler curve from this signal will be a new curve with the frequency equal to the Doppler shift. (This means that a full package of pulses is considered one pulse in the sampling frequency sense).

The pulsed modus results in a practical limit on the maximum velocity that can be measured. In order to measure velocity at a certain depth, the next pulse cannot be sent out before the signal is returned.  The Doppler shift is thus sampled once for every pulse that is transmitted, and the sampling frequency is thus equal to the pulse repetition frequency (PRF). Frequency aliasing occurs at a Doppler shift that is equal to half of the PRF.
fD = ½ * PRF
This is illustrated below with an analogy:

The Nykvist phenomenon.

The Nykvist phenomenon (121) is an effect of the relation between the sampling frequency and the observed velocity. If you sample at a certain frequency, the direction of the motion becomes ambiguous, more frequent sampling will give the correct direction, less frequent  sampling results in an apparent motion in the opposite direction. This can be observed with a stroboscopic light, for instance illuminating the flow of water, or with old fashioned wagon wheels in old moves which often seem to revolve slowly backwards when the wagon moves forwards.


This is illustrated below.

Constant rotation velocity, decreasing sampling frequency:

The easiest is to show how reducing the sampling frequency affects the apparent motion. All circles rotate with the same rotation velocity clockwise. The sampling frequency is reduced from left to right. It can be seen that the red dots is at the same positions when they are seen to move.




a:      8:1
8 samples per rotation, the red point is seen in eight positions during the rotation.
 b:     4:1
4 samples per rotation, the red point is seen to rotate just as fast, but is only seen in four positions
 c:    2:1
2 samples per rotation, i.e. the sampling frequency is exactly half the rotation frequency. Here, the red dot is only seen in two positions, (but it is evident that it is in the same positions at the same time as in a and b). However,  it is impossible to decide which way it is rotating. This is the Nykvist limit; sampling rate = 1/2 rotation rate.
 d:    1.5:1
1.5 samples per rotation,or  one sample per three quarter rotation, making it seem that the red dot is rotating counter clockwise. Again, the dot is in the same position at the same time as in a and b.



Constant sampling frequency, increasing rotation velocity

The same principle applies when there is a fixed sampling frequency, but increasing rotational velocity. In the images below, the frames are seen to shift simultaneously, but the positions of the red dots are different due to the different rotational velocity.




a:    1:8
One rotation per 8 samples. The sampling catches the red dot in 8 positions during one rotation.
 		b:    1:4
Rotation velocity twice that i a; one rotation per four samples, the sampling catches the red dot only in four positions during one rotation.
 		c:   1:2
Rotation velocity four times a; one rotation per two samples, this catches the red dot in only two positions, giving directional ambiguity as above.
 		d:   1:1,5
Rotation velocity six times a; one rotation per 1,5 samples, or 3/4 rotation per sample, giving an apparent counter clockwise rotation.

Sampling from increasing depth will  increase the time for the pulse returning, thus increasing the sampling interval and decrease the sampling  frequency.  The Nykvist limit thus decreases with depth. This means that pulsed Doppler has depth resolution, but this leads to a limit to the velocities that can be measured.

The Nyquist limit is dependent on the depth (D) of the sampling volume. The larger distance from the probe, the  longer time for the pulse to pass to the desired depth and back, and the  lower the Nyquist limit. The time for the return of the pulse is:

T = 2D/c and the maximum PRF as PRF = c/2D

Thus, the maximum Doppler shift that can be measured is:
fD = c/4D (from: fD = ½ * PRF) and from the Doppler equation ( v = fD * c / 2f0) the maximum velocity (Nyquist velocity vN):
vN = c2/8Df0


Aliasing is no problem in pulsed tissue Doppler, as tissue velocities are far below the Nykvist limit. However, in colour tissue Doppler, harmonic imaging is halving the effective frequency, and leads to aliasing as shown below.

Continuous Doppler,

on the other hand, will measure all velocities along the ultrasound beam: The beam is transmitted continuously, and the received echoes are sampled continuously with no range gating. Thus, there is no information about the time interval from the signal to the reflection, and, hence, no information about the depth of the received signal; the signal may come from any depth. The continuous Doppler has no Nykvist limit, and can measure maximal velocities. It is used for measuring high velocities.

This means that both methods has limitations: pulsed Doppler has velocity ambiguity at high velocities, and continuous wave Doppler has depth or range ambiguity. Thus, for continuous Doppler the pulse length can be long, as there is no depth resolution, while in pulsed Doppler it has to be shorter in order to achieve a sufficient depth resolution.



Examples of continuous wave versus pulsed wave, and the Nykvist effect. Left: Aortic insufficiency shown by cw Doppler. It van be seen that there are a fair distribution of velocities in the whole spectrum. However, There are far more velocities blow 2 m/s. In this case, the low pass filter is only set to suppress tissue velocities. If the point is to get a clear visualisation of the maximal velocities in the jet, at 4 - 6 m/s, the filter should be set higher. Left, the same patient by pulsed Doppler of the LVOT. The outflow can be seen as a narrow band, within the velocity range, while the regurgitant jet has velocities outside the Nykvist range, and there is total velocity ambiguity.

High Pulse Repetition Frequency (HPRF)

A way around the problem, is to use high pulse repetition frequency pulsed Doppler. This means that one or more new pulses are sent out before the echo from the desired depth of the first is received. This will increase the pulse repetition frequency, and thus increase the Nykvist limit as the fN = ½ * PRF. On the other hand it will be impossible to determine which pulse is the origin of the echo, and thus it will result in a partial depth ambiguity.


The principle of HPRF.  Pulses are transmitted with three times the frequency that is necessary to allow the echo from the furthest depth to return. Thus, the echo of pulse 1 will return from level 3 at the same time as the echo of pulse 2 from level 2 and and of pulse 3 from level 1, and there is no way to determine whether a signal is from level 1, 2 or 3. HPRF pulsed Doppler recording (right). with one sample volume in mid ventricle and one in the mitral ostium. The recording shows a systolic dynamic gradient (due to inotropic stimulation with dobutamine), as well as an ordinary mitral inflow curve.  There is no way in the pulsed recording to determine which velocities that originate from which sample volume (except from á priori knowledge, of course). 


Colour Doppler mode (CFM)

The colour Dppler mode is based on analysing the phase shift of the reflected pulse, which is equivalent to the Doppler shift. Shooting at least two (or more) pulses in rapid sequence, (NOT to be confused by sampling one pulse at two timepoints as illustrated above) results in the possibility to analyse the Doppler shift in terms of the phase shift between the pulses. The phase shift analysis is based on the principle that when pulse 2 hits a moving scatterer, the scatterer will have moved a little away from, or towards the probe, and the return pulse 2 will then be in  a different phase from pulse 1. The distance the scatterer has moved, is of course a function of the velocity of the scatterer and the time between pulses given by the pulse repetition frequency, thus:
d = v * t = v * 1/PRF.

In order to avoid aliasing, the distance the scatterer moves should be less than half a wavelength, i.e. the Nykvist limit. This is thus given by the pulse repetition frequency, exactly as in pulsed Doppler. In this case, the PRF is 1/t, the time delay between the two pulses in one package. In order to obtain an acceptable Nykvist limit, the PRF need to be at least 1 KHz which will result in a Nykvist limit of 19.8 at a depth of 15 cm. 





Two pulses sent toward a scatterer with a time delay  t2 - t1 = 1/PRF. Given that the scatterer has a velocity, it will have moved a distance, d, that is a function of the velocity and the time (d = v x t).  Thus, pulse 2 travels a longer (or shorter) distance equal to d with the speed of sound, c, before it is reflected.  During the time pulse2 has travelled the distance d to the new position of the scatterer and back to the point of the reflection of pulse 1, i.e. a distance 2d  pulse 1 has travelled the same distance away from the reflection point. (The scatterer will have travelled further, but this is not relevant).  Thus the diasplacement of the waveform of pulse 2 relative to pulse 1, is 2d. By sampling the two pulses simultaneously at two timepoints, as shown in the previous illustration, the phase of each pulse can be determined. The phase analysis  of the relative positions of all four points is done by autocorrelation, a quick (and dirty?) method that allows online computation.


The phase analusis is done by treating the waveform as a sine curve, where the phase of a point is represented as an angle. This is described in more detail in the mathemathics section.

The two pulses represent a pulse package. The time between the two pulses represent 1/PRF; determining the Nykvist limit. The time between two packages is 1/FR. The time between packages can be used for sampling B-mode data for a B-mode image in a composite image.

The method will give the velocity at a certain depth, depending on the range gating as in pulsed Doppler. By gating multiple sampling times from the same pulse package in post processing, the Doppler shift can be measured at multiple depths along the line. As opposed to HPRF, only one pulse package can be sent out along one line in order to await the return signal down to maximum depth (range), in order to avoid range ambiguity. This means that the method is more similar to B-mode than HPRF, and the velocity ambiguity is similar to pulsed Doppler. In fact, colour Doppler can be seen as a B-mode, where the phase (or frequency) shift is analysed instead of the amplitude, stored as numerical values in each pixel (as amplitude is in B-mode), but displayed as colour instead of brightness.

This again means that in order to build a sector, the next package has to be sent along next line, and a new package can only be sent out along the first line when a complete set of scan lines has been built up. Thus the time between packages in one line, and hence, the temporal resolution, is dependent on the frame rate, just as in B-mode as described above. The temporal resolution can be increased by reducing sector depth, sector width and line density. In addition, sampling at longer gating intervals, due to one package being two pulses, reduces radial (depth) resolution. However, as the region of flow is only the cavity, and the region of interest usually being only part of this, reducing sector size in order to obtain an acceptable frame rate is a feasible solution as shown below.





Principle of package acquisition. Time depth diagram of the position of a moving scatterer. Each dot represent one pulse. Packages of two pulses are sent to the scatterer with intervals. The time between the pulses in one package is given by 1/PRF, and decides the Nykvist limit. The time between packages is the time it takes to build a full sector of lines in colour flow mode (CFM) and is given by the frame rate (FR), the time interval being 1/FR. This decides the temporal resolution of the CFM.
CFM sector superposed on a B-mode sector. By reducing sector size, line density and sampling frequency, the CFM image can achieve an acceptable frame rate.  This is feasible because the region of interest for the flow is usually only a part of the ROI for The B-mode, ,flow being intracavitary as shown below.

In principle, two pulses are sufficient for phase analysis with perfect signals. However, more than two pulses can be used in order to make the analysis more robust, in the autocorrelation method. This results in better accuracy (reliability) of the velocity estimate. This, however increases the duration of each package (as the time between two pulses is 1/PRF, a package of N pulses will have a duration of N-1/PRF), and thus it will affect frame rate.

A theoretical frame rate of 1 KHz will make the use of packets unnecessary (as then PRF = FR, and phase analysis can be done from one puls to the next, with the same Nykvist frequency). This is technically feasible (215)



Also, the possibility to display the full velocity information in each pixel is limited. In order to display multiple velocities in real time over a sector, the numerical values are usually displayed only semi quantitatively as color. Power Doppler shows the amplitude of the Doppler shifted signal, i.e. the blood flow.
Power Doppler image of the renal circulation. The amplitude is a function of the number of scatterers, i.e. the number of blood cells with a Doppler shift. This is shown as the brightness (hue) of the signal. In addition, direction of flow can be imaged by different colours (red - positive flow - towards probe, blue - negative colours - away from probe), and still the brightness may show the amplitude.
Colour flow showing a large mitral regurgitation. Velocities away from the probe is shown in blue (converting to red where there is aliasing), towards the probe is red. In this image, the green colour is used to show the spread (variance) of velocities. This will also reflect areas of high velocities (high variance due to aliasing). The sector with colour flow is seen to be far smaller than the B-mode sector. The image displays the direction, extent and timing of the jet.





2D colour flow gives mainly information on the direction of velocities, as well as colour M-mode giving the direction - timing information. However, the information is numerical, and can be extracted as is done in colour tissue Doppler, but this gives far less accurate values than pulsed and cw Doppler, as well as a reduced frame rate.







Recording from a patient with apical hypertrophic cardiomyopathy. Ejection can be seen in blue, and there is a delayed, separate ejection from the apex due to delayed relaxation. There is an ordinary mitral inflow (red), but no filling of the apex in the early phase (E-wave), while the late phase (A-wave) can be seen to fill the apex.  Left,  a combined image in HPRF and  colour M-mode.  The PRF is adjusted to place two samples at thr mitral annulus and in the mid ventricle just at the outlet of the apex. The mitral filling  is shown by the green arrows,  and the late filling of the apex is marked by the blue arrow.  In addition, theere is a dynamic mid ventricular gradient shown by the red arrow, with aliasing in the ejection signal in colur Doppler. The delayed ejection from the apex is marked by the yellow arrow (the case is described in (87).  The utility of the different methods is evident: HPRF (or cw Doppler) for timing and velocity measurement, but with depth ambiguity, colour M-mode for timing and location of the different jets, direction being displayed by the colour. 
 

The phase analysis is often done by the process known as autocorrelation. This will result in a values that does not reflect the spectrum, but only mean values in the spectrum. But if there is clutter in the region (stationary echoes), this will be incorporated in the mean, resulting ion lower values. In Doppler flow, this can be filered by the high pass filter, and thus will represent a small problem. In tissue Doppler, this may be a more significant problem, as the velocities are only about 1/10 of the flow values, and thus clutter may be more difficult to separate from true velocities. Thus, a substantial amunt of clutter may reduce autocorrelation values for tissue Doppler more than pulsed Doppler as discussed below. In addition, it is customary to analyse the tissue Doppler values in native, rather than harmonic imaging, due to the Nykvist limitation. Thus, there is a greater amount of clutter than if harmonic imaging had been used, as shown in B-mode images.

For optimal colour flow, it is important to realise that there may, in some scanners, be an inverse relation between the gain of colour Doppler and B-mode. (In some scanners it is possible to adjust the priority, or to adjust the gain settings separately). This, however, is an acquisition finction, and not image adjustment, and thus cannot be compensated afterwards. This is illustrated below:


Effect on B-mode gain on colour Doppler imaging. Left pulmonary venous flow by pwDoppler, showing a systolic flow component, although low velocities. Middle, colour M-mode of the same patient. Only the diastolic flow component can be seen. Right, reducing B-mode gain increases the gain of colour flow, and the systolic pulmonary venous flow can be seen.




Tissue Doppler.

The Doppler principle can be used both for blood flow and Tissue velocities. Tissue Doppler was first decribed in 1989 (53). It is simply a question of different filtering of the Doppler signals. The main principle is that blood has high velocity (Typically above 50 cm/s, although also all velocities down to zero), but low density, resulting in low intensity (amplitude) reflected signals. Tissue has high density, resulting in high intensity signals, but low velocity (typically below 20 cm/s). The difference in the applications used for the two sets of signals is mainly differences in filtering, applying a high pass filter in Doppler flow, and low pass filter in tissue Doppler (Although the latter is not absolutely necessary).




The diagram to the left shows the placement of flow and tissue signals on this intensity (amplitude) / velocity diagram. Velocity given as the height ogf the bars, intensdity showb both by the placement on the x axis, as well as the darkness of the bars, black being the highest intensity. The flow signals are low intensity but mostly high velocity, while the tissue is exclusively low velocity, high intensity. The heart valves, however, are solid structures which moves with the velocity of the passing blood, resulting in high intensity signals giving a saturation of the Doppler spectrum. A typical flow curve from the right ventricular outflow tract is shown to the left, with the valve click.


Application of a high pass filter (low velocity reject) shown schematically to the left and in practice applied to a mitral flow curve to the right. Velocities lower than the limits of the green bar (showing the range of the filter) are removed seen in the dark zone in the middle of the spectrum. The setting rejects velocities at blood intensities below 15 - 20 cm/s, which is too high for normal flow velocities as in this instance, although may often be useful  in continuous wave Doppler recordings of high velocities in jets.

The filter is adjustable and is here reduced to 10 cm/s


Further reduction in the filter below 10 cm/s results in high intensity signals becoming visible, especially in early diastole. This is tissue signals from the mitral ring.


Fully removing the filter results in a dense band of high intensity tissue signals around the baseline. The signal is difficult to analyse, as it has so high amplitude that the display is saturated.


Decreasing the scale and gain (shown as all signals being illustrated in lighter colour, but with the same relative placement on the x axis), and placing the sample volume near the mitral ring, discloses the tissue velocity curve of the ring, still taken with an ordinary Doppler. The flow signal, having a much lower amplitude, is removed simply by reducing the gain.


All modern ultrasound machines today has separate applications for tissue Doppler which optimises the signal for this purpose, among other things by applying a low pass filter that removes most of the flow velocities. This results in a cleaner signal.

Colour Tissue Doppler

The basic principles of colour tissue Doppler are the same as for colour flow mode (222). The difference is the same as in spectral Doppler flow versus colur Doppler flow, except for the differences in filtering as shown above. colour However, the region of interest is the same as in B-mode, being the myocardium. Thus the solution of a small sector within the B-mode sector is unfeasible. However, as velocities are measured only along the ultrasound beams, and cavity signals are filtered by low gain / low pass filter, the line density need not be high. In addition, as the data are for numerical analysis, not imaging, artifacts from MLA are unimportant. Thus, high frame rate in a full B-mode sector is achieved with a very low line density, and a higher MLA (typically 4). Thus the B-mode and colour Doppler images are displayed superposed, but the acquisition is interleaved, recording a multiple of colour Doppler images between each B-mode frame. (For instance 16 lines acquired by 4 MLA in colour Doppler interleaved with one B-mode image acquired with 64 lines at 2MLA for every fourth Doppler image will result in a Doppler frame rate of 160 for a B-mode frame rate of 40).

Both velocity and strain rate information can be displayed on the B-mode image, just as in colour flow. However, as seen below, in this case it does not display location of jets, as most motion and deformation is fairly more uniform. In addition, shifts are too quick to visualise entirely.

Thus, in colour tissue Doppler, the main function is to get data for post processing, either for semi quantitative analysis in curved anatomical M-mode as shown below, or to extract quantitative data  as velocity or deformation curves.  The advantage over pulsed Doppler, is the near simultaneity of the data over the sector, being important in comparing regional motion from different segments of the wall. In colour flow, simultaneity is less important, and so quantitative information is rather acquired by pulsed or cw Doppler, giving far higher temporal resolution.


Curved anatomical M-mode

This method, developed by Lars Åke Brodin and Bjørn Olstad shows the whole time sequence in one wall at a time. (18). By this method, a line is drawn in the wall, and tissue velocity data are sampled for the whole time interval (e.g. one heart cycle) and displayed in colour along a line in a time plot, as shown below. This has the advantage of displaying the whole sequence in a still picture, giving a temporal resolution like the frame rate of the 2D tissue Doppler.
Velocity and strain rate imaging of the same (normal) left ventricle.  The colour sector can bee seen to be equal to the B-mode sector.Velocity is red in systole when all parts of the heart muscle moves toward the probe (apex) and blue in diastole. The changes are too quick to observe entirely, to make full use of the information the image has to be stopped and scrolled. Curved anatomical M-mode (CAMM). A line is drawn from apex to base, and velocity data over time are sampled along the line and displayed in colour along a straight line. The numbers on the curve and the M-mode are included for reference and corresponds to the numbers on the B-mode image.  This example shows the septum from the apex to base along one axis, and one heart cycle along the other, in a two - dimensional space - time plot. S: systole, E: early relaxation, A: atrail contraction. Extracted velocity curves from three points in the septum. As in colour flow, the M-mode gives the depth - time - direction information, while the curves give the quantitative information. 


Thus: 2D images show the whole sector image at one point in time, velocity or strain (rate) traces shows the whole time sequence (f.i. a heart cycle) at one point in space, while CAMM shows the time sequence as well as the length of the line, but only semi quantitative motion / deformation information.

Measurement of peak values in relation to the Doppler spectrum

The width of the spectrum in Doppler flow is mainly determined by the dispersion of velocities. However, the main use of pulsed wave tissue Doppler is for measuring annulus velocities, and the annulus is stiff, with propapility of dispersion of velocities. Also, in pulsed tissue Doppler, the insonation angle is small. Despite this, the spectrum has a certain width, indicating a spectrum of velocities:


Spectrum width in tissue Doppler.  Image courtesy of H Dalen.


Thus, the main determinant of the width of the spectrum is determined by the bandwidth. Ideally, the most representative value of the spectrum is the modal velocity (the velocity in the middle of the spectrum), not the maximal value (at the top of the spectrum). This will also be the values that are most similar to the tissue Doppler values obtained by autocorrelation (colour tissue Doppler). Thus, most of the differences reported between pw Tissue Doppler (145, 165) and colour tissue Doppler are due to the width of the spectrum.



Left: spectral tissue Doppler, illustrating the width of the spectrum. The modal (average) velocity curve is indicated in black.  Right, colour tissue
Doppler from the same location in the same patient, obtained by autocorrelation, giving average values directly. Peak values correspond fairly.

In addition, the width of the spectrum is dependent on the gain setting. Increased gain setting increases the peak values (145) as illustrated below:



Same tissue Doppler recording with two different gain settings. We see that peak systolic velocity differs by 2 cm/s, and the lowest gain setting
is closest to the modal velocity.

It is evident that pw tissue Doppler should be obtained and analysed by the lowest readable gain setting.
On the other hand, filtering of the colour tissue Doppler reduces peak values, and will also increase the  difference between the methods (145). Autocorrelation with high filtering or low frame rate will thus underestimate the true mean value.


However: Modal velocity may be influenced by stationary reverberations. Thus, spectral values will incorporate pixels with zero velocity, and the spectrum will broaden, with a reduction in average velocity due to artifacts, as illustrated below.



A patient with a stationary reverberation in the basal lateral wall (left), with the sampling point for both pw and colour Doppler is indicated. .  The pw spectral Doppler (middle) can be seen to be broadened, comprising all velocities between peak and zero. This is due to stationary pixels due to the artifact, but a sufficient number of pixels with normal velocity (and a number of pixels in between).  Averaging the spectrum will result in too low values, as the lowest values are due to artifact.  The autocorrelation values (right) are likewise average values, thus too low.

Thus, maximum values are more robust in relation to artifacts than both taking the middle of the spectrum and colour tissue Doppler. For the present, the best compromise seems to be:
pw Tissue Doppler should be measured as maximal values, but with the lowest possible gain setting.


Strain rate imaging by tissue Doppler

The simultaneous measurement of velocities in the whole sector, enables the measurement of velocity differences. This is the prerequisite for measuring the velocity gradient or strain rate by tissue Doppler.


Thus, longitudinal velocity gradient, is a measure of longitudinal strain rate. However, it can be shown that this is equal to the velocity gradient over a fixed distance. Strain rate by tissue Doppler measures the velocity gradient of two points over a segment with a fixed distance (In the latest scanner software, the velocity gradient is in fact calculated by linear regression of all pixel velocities within the delta x, to reduce noise.):

This is a different algorithm from the velocity gradient, but it can be proved (here) that the two formulas result in the same ratio. The distance  is called the offset distance or strain length.


a

b
a, longitudinal velocity gradient, where v1 and v2 are two different velocities measured at points 1 and 2, and L the instantaneous length of the segment between those points. The velocity gradient is given in the formula.  b, strain rate measured by tissue Doppler, as the velocities of a segment with fixed length as shown by the formula. It can be proven that these two approaches yield the same result.

Thus:        


Strain rate by linear regression

Instead of measuring just the velocities at the ends of the offset distance;  or  respectively, the velocity gradient / strain rate can be calculated as the slope of the regression line of all velocities along the offset distance as described originally (14). With perfect data, the values will be identical, both formulas defining the slope. With imperfect data, this method will tend to make the method less sensitive to errors in velocity measurements, as the value is an average of more measurements.



Strain rate calculated over an offset  (strain length) of  12 mm (L). "True" strain rate at the end points are v1 = 0 and v2 = 1.2 cm s-1 giving a strain rate of -1.0 s-1 (blue squares), the strain rate is actually the slope of the line between the points, being equal to (v2 - v1)/L. Due to random variability of the measurements, the measured values deviate from the slope. Here velocities are sampled for each 0.5 mm along the strain length (red points), and are seen to be dispersed around the true strain rate line. The regression line through the points (red line) is fairly close to the true strain line, and results in a strain rate measurement of -1.14 s-1. This makes the measurement far less vulnerable to measurement variability than simply measuring the two velocities at the end of the strain length (points in the green open squares), and compute SR = (v2 - v1)/L shown by the green line, yielding a strain rate of -1.63 s-1.

Strain by speckle tracking

Motion of one kernel can thus be measured throughout one heart cycle. Velocity can be derived from the motion curve or calculated by the motion divided by the frame interval. With two kernels, the relative displacement per distance can be derived. This is equal to strain. Likewise the relative velocity per distance (velocity gradient / strain rate) can be calculated or derived from the strain curve.

Strain by speckle tracking has been validated by ultrasonomicrometry in the longitudinal direction (124) as well as for rotation (125).

 One way of using this approach, is to place defined ROI in the myocardium at the segmental borders and measure segmental strain and strain rate directly by changes in segment length. This is true segmental strain rate, and angle independent measures, eliminating the insonation angle problems discussed elsewhere..





Kernel displacement  Displacement curve obtained by tracking through a whole heart cycle shown to the right, derived velocity curve shown below. From two different kernels, the relative displacement and hence, strain as well as strain rate can be derived. With kernels at all segmental borders, segmental motion and deformation can be tracked, as shown to the right.
 And the length variations of the segments between the kernels kan be followed through the heart cycle.

From strain, the strain rate can be derived by temporal derivation.

The advantage of this method is that it tracks in two dimensions, along the direction of the wall, not along the ultrasound beam, and thus is angle independent. This means true longitudinal strain. The disadvantage is that if the algorithm does not track one kernel correctly, the strain values will be wrong for the segments on both sides of the kernel. This is evident in areas of drop outs or reverberations as shown here. This can be overcome by increasing the number of kernels, or manually avoiding placing a kernel in an area of drop out or reverberation.


It is important to be aware that strain measured by speckle tracking  is Lagrangian strain directly. Derivation yields Lagrangian strain rate, while tissue Doppler yields Eulerian strain rate. Commercially, it has been customary to display the Eulerian or natural strain rate (velocity gradient) derived directly from tissue velocity data, but to apply a correction when integrating to strain rate and display Lagrangian strain. For Strain rate from speckle tracking, a reverse correction has to be applied, either calculating velocity from displacement and then strain rate from velocity, or calculate strain rate from strain and then apply a conversion. Thus, if Lagrangian strain is derived directly from the segment length change, the Eulerian strain rate can be derived by the simple conversion:
 

This, however, is not as simple as it looks. As can be seen from the equation, the formula converts the strain rate at a specific time from Lagrangian to Eulerian strain rate, but as shown above, the peak strain rate is not simultaneous by the two calculations. Thus, peak Lagrangian strain rate (which is lower than Eulerian), will be converted to Eulerian strain at the time of peak Lagrangian strain, not the peak Eulerian strain itself, so the conversion is practically useless. This is in contrast to the conversion between strains, as bot Eulerian and Lagrangian strain are cumulated maximal strain, which is at end systole and simultaneous.




Speckle tracking can also be combined with tissue Doppler in various combinations for more efficient tracking as described below.


Strain rate is displayeed as yellow to orange in systole (shortening) and cyan to blue in the two diastolic phases early and late filling (lengthening), but green in periods of no deformation. The changes are too quick to observe entirely, to make full use of the information the image has to be stopped and scrolled. Combined strain rate image with one systolic and one diastolic frame displaued in B.mode, below the CAMM from the septum and below that the strain rate (yellow) and strain curves from one point in septum.

Relation between velocity, displacement, strain and strain rate

AS seen below, velocity information may be extracted from the colour image, and strain rate may be derived, displacement integrated and strain may be derived/integrated from velocity data.

Derived measurements



Integration
of velocity
Velocity traces may be considered the raw data. All other modalities are integrated or derived from this. Here normal function is shown at normal basal velocity (6 cm/s), as well as normal decrease from base to apex.  This decrease is evident by visual assessment alone, as the distance between the curves. The distance between the curves then is a direct visual assessment of strain rate.  Strain rate curves can be obtained by spatial derivation of velocity:


Displacement curves obtained by integration of velocity. Temporal integration reduces velocity to motion. In principle, strain could be obtained by spatial derivation of displacement, although not used:







Integration
of strain  rate
The strain rate curves are the spatial derivative of velocity, showing the time course of the velocity gradient. This is equivalent to the local deformation rate. The curves, however, are noisy, shown in this unfiltered image, the increase in random noise is a consequence of derivation.

Strain is regional deformation. This can be obtained both by spatial derivation of strain (not used at present) and temporal integration of strain rate (used). 



Harmonic tissue Doppler

Harmonic imaging in tissue Doppler leads to aliasing, as seen by this colour M-mode. However, Strain rate imaging, using the velocity differences, will neutralise this, as usually both velocities in the equation (V1 - V2)/L are aliased, and thus the difference remains the same. So in strain rate imaging, aliasing is effectively unwrapped, as shown previously (167).



Colour M-mode (CAMM) of tissue velocities in fundamental (above) and harmonic (below) imaging. Slight aliasing can be seen in native
imaging in the e' wave at the base. In harmonic imaging, there is aliasing both in the S' wave, and the e' wave (double).
Colour tissue Doppler curved M-mode in harmonic imaging, velocity plot (above), strain rate (below). As can be seen there is heavy aliasing in the
velocity plot, but no aliasing in strain rate imaging.

However, this would imply that separate recordings had to be taken for strain rate and velocity analysis, instead of post processing strain rate from TDI recordings.



Tracking by combined speckle tracking and tissue Doppler

Tracking tissue motion by speckle tracking is described above. Modern ultrasound equipment has the capability of acquiring second harmonic greyscale images with an acceptable frame rate of 40 - 50 FPS and good lateral resolution simultaneously with tissue Doppler data. This opens the possibility of tracking along the ultrasound beam by tissue Doppler, simply by calculating the displacement from one frame to next from the velocity and frame interval, while tracking transverse to the ultrasound beam can be done by speckle tracking (124).

Image showing the speckle tracking from both modalities. The kernels are shown as the small, round, yellow circles.  The longitudinal search area along the ultrasound beam by tissue Doppler is shown in red. The lateral search area by speckle tracking is shown in white.

This has several advantages:
  1. It increases computational speed, as the SAD is calculated over a far smaller area, especially as the longitudinal motion is greater than the transverse.
  2. If the method is used to compute longitudinal velocities or strain rate, the longitudinal tracking is done with the high sampling frequency of tissue Doppler, thus reducing the possibility of undersampling.
  3. It utilises the full dataset inherent in the combined image. It is not reasonable to assume that discarding part of the information in the acquisition results in better data.
  4. Using tissue Doppler in this way, eliminates the angle dependency.
This method can be used in different ways to analyse strain rate imaging (127) as described here and has even been shown to be clinically useful in stress echo (128).




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Editor: Asbjorn Støylen, Contact address: asbjorn.stoylen@ntnu.no, Updated: 2010