2. DOPPLER FLOWMETERS

2. DOPPLER FLOWMETERS.

This chapter deals with both the fundamental and the practical aspects of Doppler ultrasound. Starting with a general description of the Doppler effect in Section 2.1., Section 2.2. proceeds to show how ultrasonic flowmeters can be used to interrogate the velocity of moving targets. Problems caused by beam geometry are also described before Section 2.3 moves on to the topic of range-discrimination. Section 2.4. deals with the important field of Doppler signal demodulation and includes descriptions of non-coherent, coherent, non-directional and directional methods of detecting the Doppler shifted returning echoes.

It is hoped that this chapter will not only provide the reader with a basic understanding of ultrasonic Doppler flowmeters but will also indicate some of the problems which can be encountered when using Doppler devices in the clinical situation.

2.1. THE DOPPLER EFFECT.

In its most basic form, the Doppler principle states that if a receiver (R) moves relative to a source (S) of sound waves (Fig.2.1.) then the frequency detected by the receiver is not the same as that transmitted by the source. To understand how this frequency shift occurs, consider the arrival times of, say, the peaks of the sound wave at the receiver. Suppose first of all that both the source and receiver are stationary as shown in Fig.2.1a. The rate as which the peaks are detected as the traveling sound wave hits the receiver is simply equal to the rate at which they were transmitted, that is, the source frequency ¦_s.

2.1a Moving Receiver.

If the receiver now begins to move in the direction of the source (Fig.2.1b) then the number of peaks received per unit time will correspond to the number transmitted plus the extra number of peaks intercepted (in the case four) by the receiver as it travels towards the source. The frequency as seen by the receiver has increased. Similarly, if the receive

moves away from the source as shown in Fig.2.1c, then the four waves which would have been detected if R had been stationary have not yet been received. The peak detection rate is therefore lower than normal and the received frequency ¦_R has decreased.

Mathematically, suppose the receiver R moves at a velocity v_R in the direction of the source S which emits ultrasound at a frequency ¦_s into a medium where the ultrasonic propagation velocity is c. The distance between successive peaks in the traveling wave is c/¦_s = l_s , the ultrasonic wavelength. In unit time the receiver moves a distance v_R and intercepts an extra number of peaks v / l_s. The received frequency ¦_R equals the total number of peaks detected per unit time and is therefore given by

or, since l =c/¦_s

The Doppler shift (or difference) frequency ¦_d is defined as the difference between the received frequency ¦_R and the transmitted frequency ¦_s , giving the conventional Doppler expression

2.1b Moving Source.

If it is the source that is moving then the Doppler effect can best be explained in a slightly different way by considering how this movement affects the distance between the peaks in the traveling wave. Figure 2.2a shows that when the source is stationary, the peak-to-peak distance is by definition the ultrasonic wavelength l_s. However, when the source is moving in the same direction as the wave (Fig.2.2b), successive peaks will be spaced closer by an amount equal to the distance Dl that the source has been able to move between the transmission of the two peaks. The stationary receiver therefore detects a frequency which appears to be higher than that fed to the source. Similarly, if the source moves backwards in the opposite direction to that of the wave as shown in Fig. 2.2c, then the spacing between peaks increases by Dl and the frequency at the receiver appears to have decreased.

Mathematically, if v_s, is the velocity of the source S in the direction of propagation, then in the time interval 1/¦_s between peaks the source will move a distance Dl given by

The wavelength l_R of the travelling wave (which is eventually detected at the receiver) is

which can be rewritten

The frequency ¦_n corresponding to this wavelength is given by rewriting Eqn (2.5a) in the form

or

which is an exact form of the Doppler equation for mechanical vibrations and shows how directly the Doppler shifted frequency is related to the transmitted frequency.

Dividing through by c in Eqn (2.7) gives

Since v/c is small, it is possible to use only the first order term in the expansion

to produce the simpler form of Eqn (2.8)

The Doppler shift frequency ¦_d ( = ¦_R - ¦_s) is then given by the familiar Doppler relation

2. 1c Reflection

The above analysis can be extended to describe reflection of ultrasound from a moving target simply by combining Eqns(2.3) and (2.11) which deal with receiver and source movement respectively. The target is considered first to be a moving receiver which detects Doppler shifted sound waves at a frequency ¦_R given by Eqn (2.2). This target then behaves as a travelling source, radiating waves at an already Doppler shifted frequency ¦_R which are then detected by a stationary receiver. The frequency ¦'_R seen by this receiver is, from Eqn (2.10), given by

Substituting for ¦_R from Eqn (2.2) then gives

Since |v_R| = |v_s| (= v), the target velocity, and because v << c so that terms in (v/c)² can be neglected, Eqn (2.13) can be rewritten as

and the Doppler difference frequency ¦_d, of the echo is

This equation could equally well have been derived by observing that, to a first approximation, it does not matter whether the source moves towards the receiver or the receiver moves towards the source: the Doppler shift frequency given by Eqns (2.3) and (2.1l) is the same in both cases. Thus it is the relative velocity between source and receiver which should be used to compute the Doppler shift.

2.1d Implications of the Basic Doppler Equation

Equation (2.15) describes how echoes reflected from a moving target are shifted from the transmitted frequency. However, the practical situation to which this expression relates is extremely restricting. Figure 2.3 illustrates a precise interpretation of the basic Doppler equation showing that a single-frequency Doppler difference waveform (¦) can be produced only if an infinitely wide plane target moves at a constant velocity through a monochromatic ultrasonic held which extends over an infinitely wide beamwidth. If any of these conditions (illustrated in Fig. 2 .3a) are not satisfied then the Doppler frequency spectrum cannot consist of the single spectral line shown in (g). For example, if either the beam or the target is finite (rather than infinite) in extent, then target movement will eventually cause variations in the amplitude of the reflected echo. This amplitude modulation immediately broadens the Doppler shift spectrum. Similarly if the target does not move at a constant velocity or if the transmitted ultrasound is not monochromatic, then the Doppler shift signal must once again contain more than one frequency component.

Thus the basic Doppler equation cannot be used to describe any of the practical situations encountered in medical applications of ultrasonic flowmeters. The next few sections will therefore examine more useful Doppler configurations by making use of the waveform and spectral plot approach similar to that introduced in Fig. 2.3.

2.2 THE CONTINUOUS-WAVE FLOWMETER

The most simple Doppler device is the continuous-wave (CW) flowmeter which was used by Satomura (1957) in the first successful attempt to monitor blood velocity non-invasively in a particular vessel. Since then, the continuous-wave flowmeter has been developed into an effective clinical tool and its many diagnostic applications are described in later chapters.

2.2a Basic Principles

The basic elements of a CW instrument are illustrated in Fig. 2.4. The master oscillator produces a sinusoidal waveform which is amplified and used to drive the transmitting transducer at its resonant frequency. This excitation creates an ultrasonic beam which occupies the continuous-wave diffraction field of the transmitting transducer. Targets within this beam reflect and backscatter echoes, some of which return eventually to the receiving transducer. (lt. should be mentioned that although in theory it should be possible to use a common transmitter-receiver, the practical problems associated with simultaneous transmission and reception of ultrasound using a single transducer usually make it necessary to operate separate transmitting and receiving elements. Thus in practice it is only those targets moving within the beam common to both transducers which will eventually contribute to the Doppler output.) The receiving preamplifier is a low-noise device which amplifies the weak returning echoes before they are detected in the demodulator. Doppler demodulation involves comparing the frequency of the received waveform with a replica of that transmitted which, in the case of Fig. 2.4, is derived directly from the master oscillator. The various techniques of Doppler demodulation are dealt with in more detail in Section 2.4. At this point it is sufficient to notice that the output from the demodulator is the Doppler difference waveform at frequency ¦_d.

2.2b Point Target

Figure 2.5 shows the ultrasonic waveforms and corresponding spectral plots when a single point target P moves through the ultrasonic beam of a CW flowmeter. (For those less familiar with frequency domain representation of signals the interpretation of spectral plots is described in Section 3.1.) The motion of the target can be resolved into component directions parallel and perpendicular to the beam axis. Velocity components along the beam produce Doppler shifted signals whereas movement across the beam causes amplitude modulation of the reflected echoes (d) as the target passes through the ultrasonic field. This variation in amplitude also appears on the Doppler difference signal, causing a broadening of the Doppler difference spectrum as shown in (g).

This "transit-time" broadening effect can be described by Fourier analysis which predicts that an amplitude-modulated sine wave must contain more than one frequency component. In fact it can be shown in general that if the amplitude of a sine wave varies in time then the modulating function (or waveform envelope) determines the shape of the frequency spectrum and, in particular, the more rapid the modulation the wider the frequency spread.

In the case of a CW flowmeter the amplitude modulation rate is determined by the transit time of the target. Thus for a constant velocity across the beam the Doppler frequency spread is inversely proportional to the beamwidth. This means that if, for

example, the beam were made narrower, then the transit time would decrease producing a faster amplitude modulation and widening the Doppler spectrum' Even a single particle

travelling at constant velocity through the beam of a CW flowmeter produces a complete spectrum of Doppler shift frequencies.