## Basic Doppler ultrasound for clinicians

Christian Andreas Doppler
(1803 - 1853)
The Doppler effect
My cat Doppler
(2004 - 2019)
The page is part of the website on Strain rate imaging

This section updated:  November 2018

This section is the Doppler part of the former "Basic ultrasound, echocardiography and Doppler for clinicians". Due to the size and number of illustrations, the page tended to load very slowly. It has now been split into one section on Basic ultrasound, Doppler, and this one on Doppler, including tissue Doppler. In addition, the background paragraphs from the previous mathematics section on the derivation of the Doppler equation and  the phase analysis have been included here (but can be bypassed, of course, for those not interested).

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 principles will also be useful to gain a basic understanding of Doppler echocardiography in general, and may be read separately, even if deformation imaging is not interesting.

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## The Doppler effect

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:

#### Derivation of the Doppler equation

The following is the original derivation of the equation by CA Doppler (119).

The Doppler effect is the effect of the velocity of the
observer (A) or the wave source (B) on the perceived wavelength,. The basic fact is that the velocity of a wave, c, is constant in a given medium, and equal to the number of oscillations per second, times the wavelength (length of one oscillation):

The time of one oscillation, i.e. the time it takes for the wave to move one wavelength is then:

 A B The Doppler effect for a stationary wave source and a moving observer. For the time the wave has moved the distance , the observer has moved the distance closer to the source (determined by the velocity v of the observer), corresponding tho the difference between the real wavelength and the apparent wavelength . This results in an apparent shortening of wavelength (increase in frequency) as shown by Doppler (119) and reproduced below this image. . The Doppler effect for a moving source and a stationary observer. In the time between waves (which is 1/f0), the source has moved the distance closer to the source, (determined by the velocity v of the source), corresponding tho the difference between the real wavelength and the apparent wavelength . This results in an apparent shortening of wavelength (increase in frequency) as shown by Doppler (119) and reproduced below this image. Thanks to Hon Chen Eng of University of Toledo who pointed out an inconsistency in the original illustration and showed a better way of illustrating the Doppler effect in this image. An observer (blue) moving towards a stationary wavesource with the velocity: will meet the wave as the wave have moved a distance , which is the perceived wavelength. The observer has moved the distance: The motion of the wave and the motion of the observer happen during the same time interval: The motion of the observer thus shortens the original wavelength  by , i.e: Thus: The change in frequency, the Doppler shift is: If the source moves toward a stationary observer with the velocity:              In the time the wave moves one wavelength, the source moves the distance:                                                                             The motion of the wave and the motion of the source happen during the same time interval: The distance from the next wave emitted from the new position of the source (small dotted red circle) to the observer (blue) is shortened by  in the direction of the motion, so the new wavelength representing the distance between the first and second waves is Thus: The change in frequency, the Doppler shift is: If  v << c, then: and:

For reflected ultrasound, the effect is twice as great. A reflector moving towards the source will shorten the incoming wavelength in the same way as an observer moving towards the source, and the reflected ultrasound wavelength will be further shortened in the same way as a moving source following the reflected ultrasound, Thus, the effect is:

(The approximation in case B is small, the velocity of ultrasound in tissue is 1540 m/s, while the velocity of blood is between 0,2 and 6 m/s, and tissue between 0.05 and 0.2 m/s, giving a v/c of maximum 0.004, i.e the approximation in B is maximum 0.4% and in reflected ultrasound 0.2%). In all cases, if the velocity vector has an angle with the direction of the observation, the effective velocity is as discussed below, in the case of reflected ultrasound, the angle is the angle between the emitted ultrasound beam and the  velocity vector. Thus the full Doppler equation for reflected ultrasound is:

In all cases it is evident that velocities at right angle to the ultrasound beam will result in no Doppler shift (Cos 90º = 0), and if the reflector moves away from the ultrasound source, there is a negative Doppler shift (cos 180º = -1).

For motion measured in the same direction as the sound propagation.

#### Angle deviation in Doppler:

As described in the basic ultrasound section, distances (e.g. wall thickness) and motion by M-mode) increases with increasing angle:

 As a reflector moves from a to b in the direction 1, the true motion (displacement) is L1. If the ultrasound beam deviates from the direction of the motion by the angle , the apparent length along the ultrasound beam will be L2, which is the hypothenuse of the triangle, and thus L2 = L1 / cos (). Thus angle deviation of M-mode measures will always over estimate the real motion (as opposed to Doppler measurements). The angle error in displacement measurement demonstrated in a reconstructed M-mode.  Increasing angle between M-mode line and direction of motion increases the overestimation of the MAPSE.

Ultrasound, being high frequency, behaves more like beams, as explained in the beginning, there may be an angle between the direction of the motion and the ultrasound beam. This means that  the angle deviation will result in a change in the measured velocity, but in this care the angle deviation results in an underestimation of velocity by the cosine of the angle deviation.

Left: distance along the axis x, imagined along the ultrasound beam y where is the angle between the direction of the motion and the ultrasound beam (insonation angle).. The angle effect is:

This means that motion along an M-mode line will increase by the cosine of the angle deviation. This is explained above. The angle deviation gives an increase in wavelength, similar to the increase in distance:

Frequency of ultrasound is the inverse of wavelength, so:

thus:

and:

Thus, in the case of reflected ultrasound, the complete Doppler equation is:

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).Thus:

Vascular colour ultrasound. Left: Near 90° insonation angle, resulting in a weak signal and velocities close to zero by the colour scale. Left, angulation of insonation signals, resulting in higher velocities (compare the colour bar top, right) that are not removed by the clutter filter,. Image courtesy of Ingvild Kinn Ekroll.

 Basically, the measured velocities decrease by the cosine function, being 0 at 90° insonation: Angle distortion in Doppler. The image on the left has applied angle correction, and then adjusted to scale.

But this also means that if motion is derived from integration of velocities, it will decrease with the angle, while tracking by B- or M- mode results in an increase with the angle as discussed in the basic ultrasound section.

#### Angle correction:

If we know the true direction of flow, it is possible to correct the velocities by applying the cosine function. This means that the velocity will be multiplied by the cosine of the angle for all values. This is feasible, only if one knows the true flow velocity direction, as in vascular ultrasound. Here, there will always be an insonation angle, while the true direction can be assumed being parallel to the direction of the vessel. However, with increasing angle correction, the corrected angle will rise rapidly, and thus increase the possibility for errors in the correction:

Using angle correction, the coorrected velocities will increase by 1/cos(a), meaning that there will be extreme corrections at angles > 45°, and errors in the correction will increase too.

In intracardiac jets on the other hand, the jet direction is in general not known. As regurgitant or stenotic areas are small, compared to the chamber they enter, and the valves in addition being diseased (and hence, assymetric), the jets may in practice have any direction withinn a range of 180°. This means that the angle is unknown, and no angle correction can be estimated.

In intracardiac jets, the basic technique will be to use the probe to align with the flow, the best insonation will be the direction giving the highest jet velocities. Angle correction should not be applied!

#### Phase analysis

In practice the analysis is done in terms of the phase shift: A wave can be described as a sine wave, and thus, any point on the wave can be described by which phase of the wave the point is. The phase of a point is represented as an angle. as illustrated below:

 Phase analysis.  If the waveform is treated as a sine curve,  every point on the curve corresponds to an angle, and the phase of the point in the curve can be described by this angle; the phase angle .  From the diagram, it's also evident that a full wavelength, , is equivalent to 2, and for every point the corresponding fraction of a wavelength is equivalent to an angle which is the fraction of 2. However, from the diagram at the top, it is evident that by sampling the waveform only once, the phase is ambiguous, it is not possible to separate the phase of point a from point b.  The two points are separated by a quarter of a wavelength, or 90° (). In order to determine the phase of the points unambiguously, the pulse has to be sampled at to points separated by less than a quarter wavelength. Then it can be seen that point a is in increasing phase from a1 to a2,  corresponding to a phase angle of  0 - /2 while b is in a decreasing phase corresponding to an angle of /2 - .

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.
d = v * t = v * 1/PRF.

 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 displacement of the waveform of pulse 2 relative to pulse 1, is 2d. This corresponds to a phase shift from pulse 1 to pulse 2 of  . By sampling the two pulses simultaneously at two timepoints, as shown in the previous illustration, the phase of each pulse can be determined as shown below.

Thus, the phase analysis is based on the phase shift from one pulse to next, instead of the apparent shortening of the wavelength by the motion of the source or observer (or reflector, which amounts to the combination of both. However, the phase shift from one pulse to the next is completely equivalent to the change in wavelength in one pulse, so the two are equivalent for all practical purposes.

The motion of the scatterer, d, is the same as the motion in the derivation of the Doppler equation, and the  displacement between the pulses, 2d is the same as the combined Doppler equation for reflected ultrasound: the displacement of sorce/observer as a fraction of the wavelength.

The time between the two pulses represent 1/PRF. The Doppler shift is very small compared to the ultrasound frequency. 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 This results in 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.

Thus, as a wavelength equals 2, the displacement (2d) of pulse 2 in relation to pulse 1 relative to equals the phase shift relative to 2:    2d/= /2. As d = v*t, t = 1/PRF and = c/f, it follows that:

= 4  *f0 / v*PRF

 A series of pulses shot successively. It is also evident the even without motion, there is a phase shift between pulses, but this is equal for transmitted and reflected ultrasound Measuring the velocity of an object by phase analysis. The velocity of the scatterer is shown by the dotted red line, showing the phase in each pulse, and then the phase shift through the pulse package is illustrated by the full sinusoid red line shown to the left, where the troughs and peaks of the red line represents the scatterer's position at the peaks and troughs of each pulse, i.e. the phase in trelation to the twop pulses. In principle, the phase shift can be sampled between each pulse pair. The sinusoid curve is the phase shift curve, and the frequency is equal to the Doppler frequency.

The low frequency sinusoid curve represent the Doppler shift frequency, which again is proportional to the velocity,  and the sampling frequency is thus equal to the pulse repetition frequency PRF.

#### Clutter (high pass) filter

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 by me from Angelsen BA Ultrasound  imaging (117)

Typically, tissue has high amplitude and low velocity, while blood has low amplitude, but a high velocity. As B-mode is a pure amplitude imaging, this means that tissue is bright, blood is black.

 Amplitude imaging (B-mode). Tissue echoes have a high amplitude, blood a low amplitude (due to low reflexivity). Thus tissue is visualised , while the blood is not visible in the present gain setting. As seen from the B-mode image to the left, the tissue is not stationary, but the velocities are low, compared to blood. Too see blood flow velocities, it is possible to increase gain, but that will also increase the low velocity signals from the tissue to saturation. They are generally considered clutter nois, when dealing with blood flow, although it is not true clutter in the reverberation sense. It is possible to filter the low velocities by a "high pass filter", that allows high velocities to pass, while removing low velocity signals independent of amplitude. This will remove both tissue signals and reverberation noise. The low velocities around the baseline have been removed, the width of the filter is indicated by the green band to the left. This is also called "clutter filter" or "low velocity reject".

In principle the Doppler signal can be sampled at the same sampling rate as M-mode. around 1000 samples per second, and the frequency shift could be determined between any two pulses. However, this would yield only one velocity value per pulse. In reality, the blood has multiple velocities, and to display all, the Doppler method uses spectral frequency analysis.

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 below 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.

### Spectral analysis:

The motion of the heart muscle and blood has components with many different velocities, i.e. different Doppler frequencies.

 As described above, the reflected signal contains multiple Doppler frequencies, because the blood flow has multiple velocities. This is shown above, left exemplified by three velocities. Different amounts of blood have different velocities, as indicated by the line thickknesses.Below is shown the compound cirve resulting from the three frequencies, which is the signal received by the probe. Fourier analysis can resolve the frequencies into different sine curves with different frequencies again. IN this case the method is called fas Fourier transform. The different amount of blood with each velocity will result in different amplitude of the signal in the different frequencies as shown below.

Fourier 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.

In the heart, the velocities are variable with time. Thus the velocity, or frequency shift is a function of time. The Fourier analysis thus uses a time window. At the outset, the time window decides the effective display rate, thus a window of 100 ms at a sampling frequency of 1000 Hz, would give a display rate of 33. However, the time window used for analysis can use a sliding window, that means that windows overlap, which will increase the effective display rate. And the samples within the window can be weighted (f.i. by a Gaussian algorithm), so not the whole window has equal weight. Thus the effective display rate will be higher, but usually in the vicinity of 90 - 120, depending on depth, probe frequency, sample size etc.

### Pulsed and continuous Doppler

Continuous vs pulsed wave Doppler. In continuous Doppler, one half of the aperture is dedicated sending, and one half receiving the reflected signal. Both signals are continuous, and all frequencies can be sampled continuously between each pulse from the overlap region. But the frequencies are received from all depths simultaneously, so there is no information about which depth the received signal originates. There is range ambiguity. The modality is best suited to measure peak velocities where they are high. Pulsed wave Doppler on the other hand, as shown below, uses the whole aperture for both send and transmit. A pulse package, consisting of at least two pulses is sent out, and the receive signal is sampled at a certain time, corresponding to a certain depth. The method shows the depth where the velocity corresponds to the spectrum. However, the pulsed modality results in a practical limitation in how high velocities that can be sampled, limited by the Nykvist phenomenon (aliasing) as explained below. Thus, pw Doppler shows velocity ambiguity above the Nykvist limit.

The width is the same as the beam width, and the length of the sample volume is equal to the length of the pulse.

Thus, in cw Doppler the sampling frequency equals the pulse repetition frequency. Continuous Doppler 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. Cw Doppler will then sample all velocities in the sensitive region, irrespectively of depth, there is range ambiguity. The continuous Doppler has no Nykvist limit, and can measure maximal velocities. It is used for measuring high velocities.

As cw Doppler emits with one part of the transducer and receives with another, the direct analysis of the reflected frequency is feasible, and the velocity can be calculated directly from the Doppler shift:

cw Doppler. The emitted frequency is shown in blue, hitting a reflector. If the reflector moves towards the wave source, the frequency in the reflected signal will  be higher (orange), if the reflector moves away from the source, the frequency will be lower (grey).

Still, there will be more than one frequency in the reflected signal, and thjis will be analysed by Fast Fourier transform.

Pw Doppler will only sample velocities at one depth. In pulsed Doppler, the pulse repetition frequency PRF is given by the sampling depth. A pulse cannot be sent out before the previous pulse has returned, so the pulese interval equals  the time for a pulse to be sent to the sampling depth and back, with the velocity of sound. The Doppler shift is thus sampled once for every pulse  that is transmitted. Thus, the pulse interval interval (PI) is given by the distance s = 2x depth, and the speed of sound c = 1540 m/s in soft tissue.
PI = 2d / c, and PRF = 1 / SI = c / 2d

This gives the maximum PRF at different depth:

 Depth SI PRF 1 cm 0.000013s 77000 Hz 5 cm 0.00006s 15400 Hz 10 cm 0.00013s 7700 Hz 15 cm 0.00019s 5133 Hz 20 cm 0.00026 s 3850 Hz

 Phase shift analysis. The distance between the pulses represent the pulse interval, or 1/pulse repetition frequency (1/PRF) This means that the phase shift curve can be sampled only with a frequency equal to the PRF. Halving the pulse repetition frequency doubles the sampling interval. This results in the Doppler shift curve being sampled at half the number of pulses. This may result in velocity ambiguity as described below. A more rapidly moving scatterer will then result in a higher frequency of the phase shift, i.e. a higher Doppler frequency. Thus the frequency is proportional to the velocity. It also means that the phase shift curve is sampled fewer times per oscillation, giving an equivalent effect as reducing the PRF..

The limitations of Pw Doppler is the limited sampling rate, that allows only velocity measurement under a certain velocity, the Nykvist limit.

#### The Nykvist limit.

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

 Cw Doppler, sampling the phase shift curve (Dopplwer frequency) once per pulse. The curve is very well reproduced. Pw Doppler samples the curve with much lower sampling frequency (PRF), but still sufficient so the curve can be reproduced, both the value and direction of the velocity can be measured. Pw Doppler where sampling frequency (PRF) is 4 × the Doppler frequency. The curve will still reproduce the troughs and peaks of the curve, and the information of the direction, and doesn't fit the alternate curve (same frequency, but out of phase (corresponding the the same velocity in the opposite direction. Sampling at 2 × the Doppler frequency (i.e.) twice per oscillation, PRF = 2 × fd, the curve cannot be reproduced, i.e. the Doppler frequency cannot be measured, the samples fit both curves equally well, and the velocity direction is ambiguous. - and the samples fit equally well the double Doppler frequency, i.e. twice the velocity. Finally when PRF is < 2 × Fd, the sampling will fit other velocities as well, in this case 1.5 × velocity.
The phenomenon where velocities becomes ambiguaous above Nykvist limet, is called aliasing.

It is often observed with old fashioned wagon wheels in old moves which often seem to revolve slowly backwards when the wagon moves forwards.

This is illustrated below as an example.

### 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.

Frequency aliasing occurs at a Doppler shift that is equal to half of the PRF. fD = ½ × PRF, i.e. two samples per wavelength, as described above. As

fDmax = ½ × PRF
vmax = c × PRF / 4 f0 cos()

Thus the maximum velocity (Nykvist limit) depends on the transmit frequecy, while the PRF depends on the depth. As the maximum PRF decreases with depth, so does the Nykvist limit. PRF = c / 2d. Thus:

vmax = c2 / 8 d f0 cos()
 Depth Maximum (Nykvist) velocity Transmit frequency (f0) 2 MHz 5 MHz 10 MHz 1 cm 1480 cm/s 590 cm/s 295 cm/s 5 cm 295 cm/s 120 cm/s 60 cm/s 10 cm 150 cm/s 60 cm/s 30 cm/s 15 cm 100 cm/s 40 cm/s 15 cm/s 20 cm 75 cm/s 30 cm/s 15 cm/s

 Aorta flow velocity curve. Aorta flow velocity curve sampled at too low PRF. Aliasing is evident. both positive and negative velocities are present. Aorta flow velocity curve sampled at too low PRF. There mat be a real limit, due to the sampling depth. In that case, by baseline adjustment, the limit for aliasing can be adjusted to 2× Nykvist, (but at the cost of total aliasing in the other direction.

If the velocities are much higher than the Nykvist, aliasing will occur at many multiples of the Doppler freqency:

Aliasing at 1/2, 1 and 2 × sampling frequency

This is typical in high velocity jets of valve insufficiencies, and aortic stenosis.

 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 below 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. 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 far outside the Nykvist range, and there is total velocity ambiguity.

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.

#### 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, a dynamic gradient like this is usually mid ventricular, and the mitral inflow in the annulus is easily recognised).

Thus, HPRF can be uses to get higher velocity range, at the cost of semi-selectivity for depth. The differentiation of the different velocity curves will be dependent on recognition from prior knowledge. .

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.

At the same time, the amplitude is a function of the velocities, as the velocities will be  concentrated in a narrow band, in pulsed Doppler, corresponding to the dispersion of the velocities in the sample volume.  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.

Spectral analysis can also be done in Pw Doppler. Using pulse packages of two pulses, only one velocity per pulse can be measured. Using more pulses per package, a spectral analysis can be resolved showing the distribution of velocities within the range window. Typically, this will be a spectral band of velocities, while cw Doppler , showing all velocities at all depths will show the velocities "smeared out" over the whole range.

 Cw Doppler signal from LVOT. The velocities can be seen to be present in all ranges, although the peak velocities are close to the curve shown by Pw Doppler to the left. . In addition, a part of the mitral flow can be seen as well, showing that the overlap area of the two cw secctors is less well focussed. Pw Doppler signal from the same LVOT. Most of the velocities can be seen to be collected in a narrow band, roughly corresponding to the peak velocities of the cw Doppler. Also, there is less contamination from the mitral flow, as the sample volume (range gate) is in the focussed part of the beam.

Using pulsed Doppler, the interval between the pulses can be used for updating the B-mode. However, this means that the B-mode pulses will interfere with the phase analysis at the start and end of the Doppler pulse, and reduce the signal quality of Doppler. In cw Doppler this will be even more pronounced. Thus, the signal will detorierate if B-mode is active, and the rule is to freeze B-mode when Doppler is acquired.

Signal quality with B-mode active and frozen. B-mode is frozen during Doppler acquisition, at the time of the white arrow, as evidenced by the break in the Doppler curve. The improvement in Doppler quality is evident.

#### 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, i.e. the dispersion of velocities within the sample volume  as shown below:

 Mitral flow, showing a fairly narrow spectrum band, indicating a relatively homogeneous velocity distribution within the sample volume, which is placed between the tip of the mitral cusps during filling, where the inflow jet is most narrow.. Pulmonary venous flow in the same subject, showing a wide distribution of velocities, within the sample volume placed in the right upper pulmonary vein. The sample volume is the same size.Venous velocities are much lower, but also varies from 0 to 0.5 m/s simultaneously.

The width of the spectrum will also be influenced by gain.

#### Gain in Doppler

 Pw Doppler recording from aorta descendens. Gain is set so high that the thermal noise is very visible. The main (modal) velocities is shown to be in a saturated band. Same recording at low gain. The modal velocities is still visible, while the less intense velocities above that are not visible, and the band is narrower, thus the peak velocities will be slightly lower. Mitral flow in high gain. Around the main spectrum is seen some noise spikes. Same recording in low gain, removes noise spikes.

Which amount of gain to use is a question of the data that are interesting. In cw Doppler, the main issue is usually peak velocities (high velocities), so gain should be high enough to show these. This means gain should be high enough, not to loose the highest values, in practice the thermal noise should be visible. In pulsed Doppler that might be the case in vascular Doppler, where the peak velocities (in the centre of the flow), may have lower amplitude that the main bulk of the blood flow.

In cardiac ultrasound, this may differ. Often in flow through orifices, the Doppler is used to calculate volume flow, which means that the most interesting is the modal velocity, the main spectral band. As the width of this band may vary with gain as seen above, this means that gain should be as low as possible. Also this will reduce noise spikes, that may interfere with peak values. The width of the spectrum is especially important in tissue Doppler. There, the width of the spectrum is not due to velocity dispersion, but to bandwith, as explained in the tissue Doppler section. Thus, the middle of the spectrum is the representative value, and the width of the spectrum should be as low as possible. And as tissue velocities are about 1/10th of blood velocities, the error induced by the bandwidth is far bigger, relatively:

In this case, differences in gain leads to a difference of 3 cm/s in systolic peak values, and 3.5 cm/s in early diastolic peak values.

As said above, velocity dispersion is not the only explanation for the width of the spectrum.

As the bandwidth is a function of the pulse length as described in the ultrasound section, bandwith is a function of the pulse length. Spectral analysis will yield a spectrum that minimum is as wide as the bandwidth.

 As described above, a pulse has a certein bandwidth, describing the frequency content of the pulse. In spectral analysis, this will give a spectrum of a certain width, corresponding to the velocity distribution of flow velocities. In phase analysis, this will correspont to a certain distribution of phase angles as illustrated. Autocorrelation, however, will only result in the average phase angle. In the case of stationary noise (clutter) as f.i. reverberations, the autocorrelation will result in an average phase angle that is in between the signal and the noise. The clutter noise will have to be removed by a low velocity filter in order to avoid severe underestimation of flow velocities.

Ideally, the pulse length in Doppler should be long, in order to increase velocity resolution. However, this will reduce the spatial (axial) resolution and the PRF.

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 velocities, 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).

Vascular ultrasound, illustrating how the beam to flow angle affects the spectral width, even when angle correction is applied, in line with the above discussion. This spectral broadening also affect the values of the peak velocities. Flow direction is evident in vascular ultrasound, being in line with the vessel, not so evident in intracardiac flow. Image courtesy of Ingvild Kinn Ekroll.

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.

#### Colour Doppler mode (CFM)

Basically, colour Doppler is pulsed Doppler. 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. A pulse package is trensmitted, but instead of only one range (time) gate, there are multiple gating of the return pulse (the opposite of HPRF). This results in there being no depth ambiguity (except in reverberation artefacts). This means that the method is similar to B-mode. 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.It also means that the PRF is limited by the maximum depth of the colour sector, i.e. the farthest range sample, and the Nykvist limit for maximum velocity is determined by that.

Secondly, the method is used for building a sector, just as in B-mode, where the next pulse is sent out with a small deviation, building a new line in the sector, as in B-mode. However, in order to use phase analysis, more than one pulse (at least two) must be sent along each line, before the next line can be sent out. This places a limit on the maximum frame rate, which then is given by the PRF as well as the number of pulses in a package.

The time between packages can be used for sampling B-mode data for a B-mode image in a composite image. In the Doppler signal, tissue echoes is removed by clutter filtering, but an amplitude filtering may also be applied.

 In colour Doppler one pulse package is sent out as in Pw Doppler, but the return signal is sampled multiple times as in B-mode. Since there is only one transmit pulse (package) at a time, there is no range ambiguity, each return sample corresponds to one specific deph, as in B-mode.. Relation between PRF and frame rate. The diagram illustrates a scatterer moving in a Doppler field. In order to do phase analysis, at least two pulses (a pulse package) need to be sent out along one line, the time between them corresponding to the PRF, which again is limited by the maximum depth of the colour sector. When the Doppler shifts have been sampled along one line by a pulse package, a new pulse package is sent out along the neighboring line, building a sector image analogous to B-mode. Thus, the position of the scatterer can be seen to be sampled only with the frame rate, which is lower than the PRF, depending on the depth, width and resolution of the colour sector. 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 colour Doppler, there is not spectral analysis. As seen by the colour display, there is only one velocity value per pixel.
The phase analysis  of the relative positions of all four points is done by autocorrelation, a quick (and dirty?) method that allows online computation. This gives the mean Doppler frequency per pulse package per pixel.

CFM sector superposed on a B-mode sector. Blue: Negative velocities (away from the probe), Red: Positive velocites. The display is semi quantitative, but the underlying data are quantitative. Both B-mode and colour Doppler is acquired at the same time, but with different pulses, beamfoming and line density. The sector with colour flow is seen to be smaller than the B-mode sector. The image displays the direction, extent and timing of the jets.

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. The PRF can be increased by reducing depth, reducing line density in the Doppler window, or reducing sector width.

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 (272), and has been implemented in a novel application; Ultra high frame rate tissue Doppler (UFR-TDI) (
215 ,268).

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).

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.

### Relation between flow velocity and flow

Doppler gives flow as a velocity. This means the velocity shows how rapidly the blood volume moves along the path, the distance per time v = d/t and is given in m/s. The velocity says nothing about the amount of blood, however. Flow is the volume rate: volume per time Q = V/t and is usually given in litres per minute. But given constant orifice area, any changes in velocity will be proportional to the change in blod volume flow, showing that there is a fundamental relation between cross sectional area of the flow, and the velocity:

Relation between velocity and flow with constant flow velocity. In this case, the velocity is distance / time v = d / t. During this time interval the distance times cross sectional area defines a volume V = A × d, which is the volume circumscribed by the motion with the velocity v. Flow (volume rate, volume per time) during the same interval t is Q = V / t, so Q = A × d / t = A × v.

#### Variable flow velocity - pulsatile flow

This, however, holds only for constant flow velocity. Blood flow is pulsatile, but the fundamental equations of motion still hold:

With varying velocity, velocity is the derivative of distance per time, and distance is the integral of the velocity. Integrating the velocity over a certain time, then defines the length of a cylinder travelling with that varying distance. Multipålyong by area gives the vloume of a cylinder, that equals the volume flow during that time.

This relation holds when veloctiy is varable, but cyclic, i.e. pulsatile flow.

In the heart this can be used to calculate the stroke volume (437, 438):

Stroke volume by Doppler. The cross sectional area is given by the diameter, under the assumption of a circular orifice. Tracing the velocity curve over one heart cycle, gives the integral of the velocity by the area under the curve (VTI). This represents the stroke distance, i.e. the distance traveled by something (blood) travelling by the varying velocity given by the Doppler curve. The stroke volume is then given by the area × the velocity integral.

#### Variable area - flow across stenoses:

In a continuous channel, flow is contiguous, and must be the same across every cross section, i.e. independent of cross sectional area:

 As the area A1 is larger than A2, in order to push the same amount of blood through A2, the velocity v2 must be higher than v1. As the flow is the same, and given by A×v for continuous, and A×VTI for pulsatile flow, the ratio of velocities / velocity time integrals is the inverse of the ratio of areas. This is the continuity equation. Using the continuity equation, as the LVOT diameter (and area) is known, tracing the VTI of the LVOT flow (pw Doppler to do it in the correct level) as well as the VTI through the valve (cw Doppler). The VTI equals the stroke length, and the stroke length times the atra, equals the stroke volume. As the stroke volume is constant, the two cylinders have equal volume, and thus,  the valve stenosis area (AVA) can be calculated by AVA = LVOT area × VTILVOT / VTIAO

The continuity equation for aortic valve area has been validated against the Gorlin formula (439):

### Relation between flow velocity and pressure:

Fundamentally, both velocity and pressure represents energy. The potential energy in a fluid under pressure, is given by E = P × V, while the kinetic energy is E = ½ m v2. But this means that when velocity increases, this kinetic energy has to be recruited from somewhere, which is the pressure energy. Thus, as velocity increases, pressure has to drop:

As velocity increases for the same volume that passes point 1, also must pass point 2, the increase in kinetic energy pressure is recruited from the pressure energy. Thus, there is a pressure drop from 1 to 2. The full equation for the acceleration of the fluid is given in the Bernoully equation. However, it has been shown that both the flow acceleration part and the friction parts are so much smaller than the first part (which is the basic energy difference), and may be ignored. Thus, the modified Bernoully equation relates pressure differences to the square of velocity differences. And if v2 is much smaller than v1, the modified equation may be simplified even more. And of course, the simplest form, is still an acceptable approximation if P2 is much lower than P1.

The simplified Bernoully equation has been shown to be valid for pressure gradients across mitral stenosis (440, 441), aortic stenosis (442), and estimation of RV pressure from tricuspid stenosis (443).

#### Pressure recovery

As discussed above, as flow converges towards a stenosis, there is little friction, meaning that there is laminar flow towards the stenosis. When flow has passed the stenosis, and into a receiving chamber where there is larger area, the velocity will decrease again. However, this may have different consequences. If there is perfectly laminar flow after the stenosis, the friction element is still so small, that the kinetic energy reverts to pressure: there is pressure recovery, and the pressure rises to the pre stenotic level. Doppler measurement will still measure the maximum pressure drop, and so will a manometer placed at the narrow part, but manometers before and after the stenosis will not register pressure drop.

On the other hand, if the flow is not perfectly laminar, there will be turbulence after the stenosis, resulting in frictional energy loss, and the velocity will decrease without restoring pressure energy, i.e. the energy is lost, and there is not pressure increase after the stenosis. In that case, there the Doppler gradient may be a true measure of pressure drop.

Full, partial and zero pressure recovery. The pressure drop corresponding to the velocity increase, is Pmax, the maximum pressure drop, given by P1 - P2. The net pressure drop through the stenosis, Pnet, however, is given by P1 - P3. Cw Doppler measures Pmax, (and so will a manometer placed directly into the stenosis, and may thus over estimate the effect of the stenosis.
 Left is perfectly laminar flow through the stenosis. In this case, the post stenotic velocity decelerates without energy loss, and the kinetic energy is converted back into pressure again. Here, there is no net pressure drop through the stenosis. Driving pressure at P1 does not have to be increased to maintain pressure at P3, and the pressure drop at P2 is temporary. It must be remarked, however, that pressure recovery cannot be more than the initial pressure. In the middle is partial pressure recovery. Some of the pressure energy converted to kinetic energy through the stenosis is lost when the flow velocity decelerates after the stenosis, in the form of turbulence resulting in friction. But some of the energy is recovered to pressure energy again. Thus, there is a net gradient over the stenosis, but this is less than the maximum gradient. The maximum gradient by Doppler will over estimate the net gradient. The red line represents the situation if the driving pressure is constant, thus the post stenotic pressure drops. The blue line represent the situation if P3 is regulated (as in the aorta). Then P1 has to be increased corresponding to Pnet in order to maintain P3. This represents the extrta work or load induced by the stenosis. Right, there is total energy loss through the stenosis, all kinetic energy due to the velocity increase through the stenosis is lost in turbulence and friction. Thus, Pmax =  Pnet and the maximum gradient is a measure of increased work (load) in order to maintain P3.

As discussed above, if pressure measured across an aortastenosis is higher than normal intraventricular pressure, there has to be a net gradient, as the maximum pressure drop has to be positive.

## Tissue Doppler.

As mentioned above, the tissue velocities are present i, and with far higher n the Doppler spectrum, and with far higher amplitude than the blood flow. In blood flow Doppler, the tissue signals are usually removed by the clutter filter, in order to get cleaner blood flow signals.

Spectral Doppler from the LVOT. Both tissue velocities and blood flow velocities can be seen, and by reducing gain below saturation of the tissue signal, the tissue velocities can be seen as distinct, and typical curves, showing both S', e' and a' peaks. The image demonstrates both that flow velocities are aboput ten times tissue velocities, but also that the Blood flow Dopple rhas a much lower amplitude.

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).

#### Spectral tissue Doppler

It is again important to realise that pulsed tissue Doppler has a high sampling rate (up to 1000), but a low temporal resolusion (effective frame rate usually below 100 FPS) due to the Fourier analysis over a long window as explained above.

 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, intensity shown 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 velocity signals. Intensity may be seen to be higher.  A typical flow curve from the LVOT 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 the LVOT 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 below 15 cm/s.  Wall velocities are generally lower, and is filtered. The filter is adjustable, and is here reduced below 10 cm/s. This results in high intensity signals becoming visible, especially in early diastole. This is tissue signals from the LVOT. Further reduction esults in high intensity tissue signals around the baseline. The signal is difficult to analyse, as it has so high amplitude that the display is saturated. Fully decreasing the filter, and decreasing the gain, (shown as all signals being illustrated in lighter colour, but with the same relative placement on the x axis), discloses the tissue velocity curve, while the flow signal, having a much lower amplitude, is much less visible. Reducing the scale, increases the resolution of the tissue velocities, that are still taken with ordinary Doppler. 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 colour 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. 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 higher temporal resolution.

Thus, tissue velocities can be extracted to numerical traceses, to enable measurements of
However, both the primary velocity data, as well as the calculated date can be displayed as parametric (colour) images images, of which the curved M-mode is the most useful so far. Mainly for assessment of timing.

#### 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.

#### Velocity traces

The information coded in the colour images, is fundamentally numerical for all varieties of colour Doppler. Thus, the velocity time traces can be extracted fom any point in the image as shown below.

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.

As the apex is stationary, while the base moves, the displacement and velocity has to increase from the apex to base as shown below.

 As the apex is stationary, while the base moves toward the apex in systole, away from the apex in diastole, the ventricle has to show differential motion, between zero at the apex and  maximum at the base. As motion decreases from apex to base, velocities has to as well. This is seen very well in this plot of pwTissue Doppler recordings showing decreasing velocities toward apex. Thus, there is a velocity gradient from apex to base

The simultaneous measurement of velocities by colour Doppler in the whole sector, enables the measurement of instantaneous velocity differences.

The velocity gradient is the velocity difference between two material points, divided by the instantaneous difference between them:

 Longitudinal velocity gradient, where v1 and v2 are two different velocities measured at points 1 and 2, and L the length of the segment between those points. Spatial distribution of systolic velocities as extracted by autocorrelation. This kind of plot is caled a V-plot (247). It shows velocities as near straight lines, and thus, a constant velocity gradient, which is the slope of the curve from base to apex. .

The distance L changes with time, if v1 and v2 are different. The unit of the velocity gradient is cm/s/cm, which is equal to s-1.

Longitudinal strain rate, on the other hand, was originally measured as the instantaneous velocity difderence between two fixed points in space, divided by the distance between them.

This is a different algorithm from the velocity gradient, but it can be proved that the two formulas result in the same ratio. The distance  is called the offset distance or strain length.
 The strain rate can be described by the instantaneous velocity gradient, in this case between two material points, but divided by the instantaneous distance between them. In this description, it is the relation to the instantaneous length, that is the clue to the Eulerian reference. train rate is calculated as the velocity gradient between two spatial points. As there is deformation, new material points will move into the two spatial points at each point in time. Thus, the strain that results from integrating the velocity gradient, is the Eulerian strain. In this view, the relation to the spatial, rather than material reference is very evident.

The two methods shown above are not identical, But it can be shown that they are equivalent.
In the material velocity gradient the L changes in the spatial it is constant. Thus, they are not equal, except at the point in time when x equals L, then v(x) = v2 and v(x+x) = v1. However, Usually, however, L will differ from x, for most frames and objects, and the velocities will hence differ too. Under the assumption that the strain is equally distributed over the length of the object (spatially constant), SR will still be equal to the velocity gradient, i e the values of the two ratios will be the same. Strain being spatially constant means that the velocity increases linearly along the length as shown in the diagram:

For any L that is different from x, v2 – v1 will be greater or smaller than v(x) – v(x + x) by the same ratio. In the figure, this is evident, as the slope of the curve is the same wherever it is measured. As v1 and v2 are the velocities of the end points of L, the ratios SR and VG will be the same, and thus the expressions are equivalent: SR = VG and the strain rate by tissue Doppler (SR) equals Eulerian strain rate. Thus:

#### Velocity gradient 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.

#### 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 little probability 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. The bandwith is determined by the pulse length and pulse frequency, and represents the frequency spectrum in the received signal. As this frequency is due to the frequency distribution in the transmitted signal, as well as the uncertainty in frequency measurement, the frequency distribution is statistical, and more or less normally distributed around the mean frequency (or velocity).

Left: spectral tissue Doppler, illustrating the width of the spectrum. The modal (or mean) 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.

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) as shown below.

 Recordings from basal septal mitral ring in a subject without substantial clutter. Spectral Doppler shows the dispersion of velocities, although this is probably an effect of bandwidth. The colour Doppler recording is superposed and aligned with both vertical and horizontal scale. In this instance can be seen to give values close to the middle of the spectrum (modal velocity).

However, historically the peak value (at the top of the spectrum), has been used in pulsed 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. But if so, this would give a good correlation between methods, and a more or less constant offset (165, 267).

Id addition the width of the spectrum is sensitive to gain settings. As the frequencies are normally distributed around the mean, this means that the intensity in the periphery of the band is lowest, and will disappear earlier when gain is reduced. Increased gain setting increases the peak values (145) as illustrated below:

In this case, differences in gain leads to a difference of 3 cm/s in systolic peak values, and 3.5 cm/s in early diastolic peak values.

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. However, the modal velocity itself, remains unchanged by the gain setting.

Spectral Doppler reconstructed from IQ data. Candidates for measuring peak systolic velocity by the PW tissue Doppler spectrum. RED circle: peak of the spectrum at normal gain, GREEN circle: upper edge of
the strongest part (the part visualised at minimal gain), BLUE circle: middle of the strongest part. MAGENTA circle and line: autocorrelation. As seen, imn this example the autocorrelation corresponds to the middle of the spectrum. (Figure
courtesy of Svein Arne Aase, modified from (266))

We tested this in a preliminary study (266), using spectral Doppler reconstructed from IQ data.

For a reference method, Peak slope of systolic M-mode in the same time point was chosen:

Reference method. A: pw Doppler from the mitral ring (reconstructed from RF data). Peak velocity of the ring displacement can be identified. B: This corresponds to the maximal slope of the M-mode line at the same time point. C: The M-mode in the same time window from the RF data. This gives a far better resolution in space and slope.  D: In the RF M-mode the steepest sloe was identified automatically. This will be a reference for the maximal velocity. (Figure courtesy of Svein Arne Aase, modified from (266))

In this study of 9 healthy volunteers, we found that the middle of the spectrum was the cloasest value (although statstically significant under estimation, but that peak value of the strongest part would be close enough (266), with just a slightly higher over estimation.

As this figure shows, the peak spectrum results in a substatial over estimation. reducing the gain improves the over estimation, while the modal velocity is closest to the reference. Autocorrelation on the other hand results in significant under estimation, due to the presence of clutter. Only four subjects showed almost totall correspondence between autocorrelation and modasl velocity from spectral doppler. (Figure courtesy of Svein Arne Aase, modified from (266))

In perfect image quality, modal velocity should be equal to autocorrelation, but in the case there is clutter, autocorrelation would not give similar values, as clutter would be incorporated into the mean. In the study above, this was the case with most subjectys, resulting in a significant difference between modal velocity from spectral Doppler and autocorrelation. This is further discussed below.

It is evident that the modal velocity is closest to the "true" velocity, and thus if peak values are used, they 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.

#### Reverberations (clutter) in tissue Doppler

However: Autocorrelation velocities may be influenced by reverberations. The mechanism for the formation of reverberations are explained in the basic ultrasound section.

The principle of clutter effect on spectral and autocorrelation velocities. In this simulation, an ordinary spectral celocity cirve is seen. The peak velocities depend on the width of the spectrum as shown above. The modal velocity is the middle of the spectrum. A stationary clutter band ois added, with zero velocities, and with nearly the same amplitude and spectrum width. The modal velocity is the zero line. The autocorrelation in this case will then be the average of both, far removed from the true modal velocity curve.

 Image from another subject in the study shown above (266). In this subjech there is some clutter from reverberations, as seen by the band in systole close to the zero line. In this case the peak velocity by autocorrelation is lower than the modal velocity of the main spectral band, which still was the one closest to the RF M-mode reference. (Figure courtesy of Svein Arne Aase, modified from (266)) Clutter filtering may reduce the problem, as seen here. There is a band of clutter close to zero velocities, but as seen here, the spectral modality makes it very easy to separate the true and clutter velocities. However, the clutter affects the autocorrelation velocity (red line), giving lower velocities, but with clutter filter this effect is removed (red line) , and the peak value is substantially higher. Image modified from (268).

In some cases, the mean velocity may be very far from the "true" modal velocity:

In this case, the spectral Doppler is "smeared out, all the way down to the baseline. The mean of all frequencies is shown on the spectrum in black. It is far from the peak values. To the right is the values from autocorrelation, which are similar to the mean values.

Reverberations may not be entirely stationary. It the reflecting surface that gives rise to the reverberation moves, the reverberation bands will move also, as seen in the simulations above, as well as the real acquisition below

.
Normal tissue Doppler curve from the mitral ring. Peak systolic velocity around 8 cm/s. However, a band with approximately the same shape, but systolic amplitudes of around 2 m/s van be seen as well, probaby a reverberation from the apical part.

Reverberations are often nearly stationary echoes, meaning that the echoes will incorporate zero velocities (clutter). Most of the reverberation echoes are in the fundamental imaging frequency. However, harmonic imaging halves the frequency, and thus the Nykvist limit. This means that there will be aliasing at half the velocity in harmonic imaging, compared to fundamental. And this is within the range of tissue velocities. Thus harmonic imaging is unfeasible in tissue Doppler, and the harmonic acquisition makes the method more vulnerable to reverberations than B-mode, as explained below.

 Shadowy reverberations covering the anterior wall in this 2-chamber image. It is differentiated from drop out, as we can se a "fog" of structures covering the anterior wall. The structures are stationary. On the other hand, this is not distinct reverberations shadows, but incoherent clutter. Recordings from the basal anterior ring in a subject with substantial clutter. The tyrue signal is clearly visible as a normal curve, and can be seen separately from the clutter band, which is the horizontal spectral band along the baseline. The colour Doppler recording is superposed and aligned with both vertical and horizontal scale. The colour Doppler, using the autocorrelation algorithm, results in mean velocities that incorporate both signal and clutter, giving a severe underestimation of velocities.

This example might be supposed to be mostly drop out, but pw tissue Doppler curves from the same examination show normal peak velocities in the anterior wall as shown below. Once again spectral Doppler is able to overcome the clutter problem, showing true peak mitral ring systolic velocities of 8.5 cm/s, compared to the peak values of 2 cm/s seen by colour Doppler above.

 Basal spectral tissue Doppler curve in the anterior wall. Peak systolic velocity ca 8.5 cm/s. Midwall spectral tissue Doppler curve in the anterior wall. Peak systolic velocity ca 6.5 cm/s. Apcal spectral tissue Doppler curve in the anterior wall. Peak systolic velocity ca 5 cm/s.

From this, it would seem that clutter is solely non-random noise, but in fact they also increases (relative) random noise.

Basically clutter is stationary echoes resulting in zero velocities as described more in detail in the basic ultrasound section. Thus, cutter is basically systematic noise, not random. However, in colour tissue Doppler clutter will also lead to increase in random noise. This is again due to the autocorrelation algorithm. The velocity estimate in each pixel will be an average of the amount of clutter and of moving echoes. The final velocity estimate will vary according to the relative amplitude of the clutter and the moving echo (weighted) (284), and this varies according to the speckle pattern as described here. Thus, in areas with much clutter, the velocity variations are larger than in high quality recordings:

 IColour M-mode from the image shown above. The curved M-mode shows a fairly homogenous and normal signal in the inferior wall (top), but more or less random noise in the anterior wall (bottom), where the noise is seen as vertical stripes of alternating colours. Velocity curves from the anterior wall, showing noise, and not much more, but at low level (within ± 0.3 cm/s). The principle of the effect of clutter. V-plot with clutter showing how the mean velocities are reduced, compared to the mormal expected values (red line). But in additions the variation of the velocity estimates from pixel to pixel is much higher, resulting in increased noise, but with reduced mean values. Combined pulsed Doppler (yellow bands) and colour Doppler green Aligned horizontally and vertically. The noise level can be seen to be b´very low, compared to the peak velocities shown in the pulsed Doppler recording. The clutter is the horizontal band around the baseline, and the width of the spectrum in this case is the noise.

Further examples are shown below:

 Reverberations in the septum of a normal ventricle. The colour bands are  stationary In pulsed wave tissue Doppler, the clutter will show up as a high amplitude band of zero velocities, but the true velocity curves can be seen as entirely separate from the clutter line, and thus peak velocities can still be measured.

Colour Doppler, calculating average velocities, will average in the clutter. Looking at the same example as above, we see the difference in peak values:

Velocity curves from the reverberations shown in the video above. Left colour Doppler from three sites showing peak systolic values of about 3, 2 and 1 cm/s, respectively. The pulsed Doppler recordings from the same sites separates the clutter from the velocity signal, and thus we find peak systolic velocities of about 12, 11 and 8 cm/s, respectively. Normally the systematic difference between the two methods is only about 1.5 - 2 cm/s as shown in the HUNT study.

Thus, maximum values are more robust in relation to artifacts than both taking the middle of the spectrum and than colour tissue Doppler, but the spectrum width will lead to som overestimation of peak values:

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.

### Ultra high frame rate tissue Doppler (UFR-TDI)

Ultra high frame rate tissue Doppler is done by combining more principles:

• Using very little focussing (planar beams). This is feasible as tissue Doppler doesn't use harmonic imaging
• Planar beams allows a high MLA factor. Again, this is feasible as tissue Doppler is about acquiring numerical values, not pretty images.
• Broad beams and high MLA factor allows the whole of one wall to be covered by one transmit beam.
• Designing the software so there is only sent one transmit beam along each wall per frame, i.e. only two pulses per frame. This is the extreme example of exchanging spatial for temporal resolution.

By this method, using two broad, unfocussed (planar) beams, each covering one wall, as well as 16 MLA and sparse interleaved B-mode imaging, it has proved possible to increase the TDI frame rate substatially (172, 268). it has been possible to increase frame rate to 1200 FPS in 2D imaging.

Few beams give high frame rate. Image courtesy of Svein Arne Aase, modified from (172).

Already this has shown new information about both the pre ejection and post ejection dynamics.

With this method, it is possible to acquire IQ (RF) data with FR > 1000. This makes it possible to process restrospective tissue Doppler from the whole field (i.e. that covered by the two transmit beams), simultaneously from one heart cycle. as in colour Doppler.

Retrospective spectral Doppler curves from base, midwall and apex, all acquired from the same heart cycle, showing the decreasing velocities from base (1) to apex (3). The simultaneity of Doppler data from the whole field, allows the velocity gradient to be imaged below, as in colour Doppler. This gradient is taken from the window in mid systole shown in the top left. Image courtesy of Lars Christian Naterstad Lervik.

This allows thequalitative  assessment of strain rate from Doppler curves, construction of a V-plot, and all relatively unaffected by clutter, as described in principle above and in the pitfalls section.

### 3-dimensional tissue Doppler

Tissue Doppler is still limited to one velocity direction only. This means that the term "3-dimensional" refers to a three dimensional distribution of tissue velocities only, not velocity  vectors in a three dimensional coordinate system. However, data from the whole ventricle can be put together on a surface model of the left ventricle.

 3D tissue Doppler is basically a grid of numerical values on a ventrricular surface. Triplane tissue Doppler, showing three standard planes, with the assumption that the angle between them is 60°, the rest of the data between the planes are than interpolated. This gives a circumferential resolution of 60°.

As tissue data as about acquiring (and displaying, f.i. by colour) numerical data, the method do not have the same limitation as 3D B-mode. One method is to combine information from three standard planes, and then interpolating the data between the planes by for instance spline. The method has been explained elsewhere. This has been done both by combining sequentially acquired standard planes. It could also be done as a simultaneous triplane acquisition, but at the cost of a substantially reduced frame rate. Thus, freehand scanning has been preferred.

Systolic (left) and early diastolic (right) frame showing the 3D surface of the left ventricle, where longitudinal tissue Doppler values are displayed with colour. Red shows velocities toward the apex, blue velocities away from the apex.  (mark the colour gradient from base to apex, reflecting the velocity gradient).

This version of three dimensional tissue Doppler may be used for display, but also for area measurement, as the data are distributed over a representation of the (approximate) real ventricular area.

With the Ultra high frame rate tissue Doppler method, it is also possible to acquire three dimensional tissue Doppler in real time.

Using a 3D matrix probe, sending an array of 3x3 broad unfocussed or planar beams, and using a 4x4 matrix of receive beams for each transmit beam, giving a 16 (or 4x4) MLA, we have been able to achieve a volume rate of about 500 VPS (280), i.e. Ultra high frame rate 3D tissue Doppler.

 Principle of beam formation, showing a matrix of 3x3 wide transmit beams (brown circles) and for each beam an array of 4x4 receive beams, i.e. a 16 MLA. (After 280). Distribution of the transmit beams in relation to a cross section of the ventricle, endocardial and epicardial surfaces marked with black lines and arrows. the energu distribution of the beams is shown by the colour hue. The transverse plane shown to the right is marked by the thick line. (After 280). Distribution of the transmit beams in an apical plane, the level of the cross section to the left is marked by the thick line. As evident from the illustration, the transmit beams do not cover the whole sector, but will cover most of the walls. (After 280).

By this method, t is possible to achieve a high circumferential resolution through the MLA technique at the same time as a high temporal resolution. The result can be displaued as a 3D figure, as with reconstructed 3D, and both curved M-modes and tivelocity curves can be extracted from this matrix:

 3D surface with tissue velocity display. The ring represents a line for extraction of the curved M-mode shown top, left. (After 280). Data display from the 3D velocity figure to the right. Top: curved M-mode, showing the time variation of apically directed velocities in a ring around the mid ventricle.  Bottom, velocity curves from the basolateral part, red from UFR 3D TVI, extracted from the 3D data to the left, blue velocity from the same point in the same subject, but acquired bt conventional colur TDI (i.e. a different heartbeat).

#### 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, but might improve the reverberation sensitivity of strain rate imaging by tissue Doppler.

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