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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5814_Библиотеки_им_академика_М_И_Перельмана
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chapter six
Flow and displacement imaging
Ultrasonic B-mode real-time imaging can be combined with Doppler in
a scanner so that the scanner is capable of providing not only anatomical information, but also blood ow data. Both sets of information are
displayed simultaneously. A cursor line is typically superimposed on
the B-mode image to indicate the direction of the Doppler beam. A fast
Fourier transform (FFT) algorithm is used to compute the Doppler spectrum that is displayed in real time. This type of scanner is called duplex
scanner. More recently, electronic and computer speed is fast enough to
allow blood ow information superimposed on the B-mode image displayed in real time.
6.1 Color Doppler ow imaging
Color Doppler ow imaging systems are duplex scanners capable of displaying both B-mode and Doppler blood ow data simultaneously in real
time (Shung et al., 1992; Routh, 1996; Jensen, 1996; Ferrara and DeAngelis,
1997). The Doppler information is encoded in color. Conventionally the
color red is assigned to indicate ow toward the transducer, and the color
blue is assigned to indicate ow away from the transducer. The magnitude
of the velocity is represented by different shades of the color. Typically
the lighter the color, the higher the velocity. The color Doppler image is
superimposed on the gray-scale B-mode image. A color Doppler image of
carotid bifurcation in the neck is shown in Figure6.1.
The basic concept of the color Doppler is similar to that of the pulsed
Doppler instruments that extract the mean Doppler shift frequency
from a sample volume dened by the beam width and the gate width.
The only exception is that the color Doppler instruments are capable of
estimating the mean Doppler shifts of many sample volumes along a
scan line in a very short period of time, on the order of 30 to 50 ms. The
most straightforward way of achieving this is to compute the FFT from
each sample volume and then to calculate the mean frequency from the
Fourier spectrum. Unfortunately, current electronic and computer technology cannot yet do that. To be able to do so, fast algorithms have to be
developed.
159
160 Diagnostic ultrasound: imaging and blood ow measurements
t
2=π∂∂
t
mm
2
π
Figure 6.1 Color Doppler image of a carotid artery bifurcation. Blood ow is represented by the color image, whereas the gray-scale B-mode image delineates the
arterial anatomy. (Courtesy of Philips Medical Systems.)
An approach was derived from considering the phase of a wave. For
a plane wave given below,
(2 )
iftkz
pztpe
,
()
π−
=
0
(6.1)
where the phase of the wave φ = 2π ft-kz. The rst time derivative of φ
divided by 2π yields the frequency:
1
f
(6.2)
If the phase of a plane wave can be estimated, the frequency at a certain time may be approximated by the slope of the phase at that time, as
shown in Figure6.2, represented by the following equation:
1
−
1
f
≈
−
(6.3)
where φm and φ
denote the phases at two different times at mΔt and (m – 1)
m–1
Δt. For a complex function like the plane wave, p(z,t) = r(z,t) + ji(z,t), where
161Chapter six: Flow and displacement imaging
mt
mt
11
()
()
()
∫∫
−∞
−∞
()
φ
t
Δt
φ
m
φ
m–1
Figure 6.2 The frequency of a wave can be estimated approximately by the slope
of the phase at time t.
r and i represent the real and imaginary parts of the complex number. The
phase term is given by
izt
(,)
1=−
tan
(6.4)
rzt
(,)
Substituting Equation (6.3) into Equation (6.4), the phase term can be estimated from
,1
iz
,
1
f
=
2
t
π
izmt
()
−−
tan
,
rzmt
()
tan
−
rz
−
,( 1)
−
(6.5)
One such algorithm for estimating frequency from the phase of a
wave was based upon the well-known Wiener–Khinchine theorem, which
shows that the autocorrelation function H(τ) of a complex function p(t) is
the Fourier transform of the power spectrum P(ω) of p(t) (Kasai et al., 1985).
Mathematically, this is given by
∞
HptptdtPed
() () () ()
τ= −τ =ω ω
∞
ωτ
j
(6.6)
Alternatively, H(τ) can be written in the form of
τ
HH
()
τ
() |()| ()
τ= τ=τ
j
e
j
Ae
(6.7)
162 Diagnostic ultrasound: imaging and blood ow measurements
)(
ωω
HP
)(
−∞
Pd
−∞
−∞
where the magnitude and phase of H(τ) are, respectively, an even function
and an odd function. The symbol A is used to represent the magnitude of
H(τ) here.
From Equation (6.6),
∞
(0
(0
H
j
)=
d
∫
−∞
∞
)=∫ωωω
(6.9)
(6.8)
where the dot operation represents the rst derivative = ∂H(τ)/∂τ. Let <ω>
denote the mean of ω, and from the denition of mean angular frequency
and Equations (6.8) and (6.9),
∞
()
Pd
ωωω
<ω>=
∫
−∞
∞
()
Pd
∫
ωω
(0)
H
=
(6.10)
(0)
jH
This equation can be manipulated to become
(0)
<ω>=j
H
(6.11)
(0)
H
Further, the variance of angular frequency, σ2, is given by
The term <ω>2 can be calculated from Equation (6.10), and <ω2> by denition is
where the double dot operation denotes the second derivative = ∂2H(τ)/∂τ2.
22 2
σ=<ω >−<ω>
∞
ωωω
∫
2
<ω >=
−∞
∞
∫
2
()
Pd
()
Pd
ωω
(6.12)
(0)
H
−
=
(0)
H
Substituting Equation (6.13) into Equation (6.12),
2
H
(0)
2
σ= −
H
(0)
H
(0)
(6.14)
H
(0)
(6.13)
It can be further shown that for an ultrasonic imaging system trans-
()
T
T
0(
∂τ
==
() (0)()
T
T
T
T
](
2
2
mitting pulses with a pulse repetition frequency T,
163Chapter six: Flow and displacement imaging
<ω>=
(6.15)
2
σ
1
=−
2
H
T
(0)
(6.16)
HT
|()|
1
These are all simple arithmetic operations that require little time for
computation if the autocorrelation function H(T) can be estimated.
Equations (6.15) and (6.16) can be found by considering the fact that for
an even function, the rst derivative of the function at the origin = 0, and
for an odd function, the function = 0 at the origin, i.e.,
(0)
H
=
τ=
and
0
0) 0
(6.17)
A
()
∂τ
Therefore,
(0)(0)
(0)(0) (0)(0) (0)(0)
=+ =HAejAe jA
jj
(6.18)
From Equations (6.11) and (6.18),
<ω>= ≈
(0)
−
=
which is Equation (6.15). This expression says that the mean frequency of a
spectrum is equal to the slope of the phase of the autocorrelation function
at the origin that can be approximated by the difference in phase at the
origin and at one pulse repetition period T, assuming that the autocorrela-
tion function is sampled at internals of T. Similarly, it can be shown from
differentiating H(τ) twice that
2
=−
HA A(0)(0) [(0)
0)
(6.19)
Here A(τ) can be expanded into a Taylor series, ignoring third-order and
higher terms and assuming that τ is small,
τ
AA A() (0)
τ≈ +
(0)
(6.20)
164 Diagnostic ultrasound: imaging and blood ow measurements
2
)(
τ
H(0)
)
)
22
)
f(t – T)
Compute autocorrelation function and then average
Rearranging Equation (6.19),
AAA(0)
[(
=
2
0)]
τ− (6.21)
can be found by substituting Equation (6.21) into Equation (6.19).
Substituting H(0
, H(0
, and H(0) into Equation (6.14), Equation (6.16) is
obtained.
HT
A
2
2
σ≈τ−
()
τ
1
≈−
AT
(0)
2
()
1
H
(0)
This expression indicates that the variance of the frequency can be
estimated from the magnitude of the autocorrelation function at the origin and at T.
Figure 6.3 shows a version of the autocorrelation method that was
implemented in a commercial scanner a few years ago. Given a real-time
function f(t), its quadrature component g(t) can be found by shifting the
time function by 90°. A complex function z(t) = f(t) + jg(t) can be obtained.
The complex multiplier performs the operation
[f(t) + jg(t)] ∙ [f(t − T) − jg(t − T)]
The autocorrelation function is obtained by integrating the output of the
complex multiplier over a period of time, say nT, where n represents the successive pulses transmitted by a scanner to acquire the autocorrelation function.
f(t)
Delay T
Delay T
g(t)
Figure 6.3 A hardwired autocorrelator for estimating the autocorrelation function H(T) from a time signal f(t).
Complex
multiplier
g(t – T)
[ f (t) + jg (t)]•[ f (t – T) – jg(t – T)]
Integrator
Integrator
H
r
Hi(t)
(t
165Chapter six: Flow and displacement imaging
()
()
HT
HT
tm
1
50 15
⋅
Time
Voltage
The three unknowns in Equations (6.15) and (6.16) are readily attainable from
the following expressions:
() () () () tan
HT HT HT andT
22 1
=+ =
ri
i
−
r
where Hr and Hi are the real and imaginary parts of H.
It should be noted that H(T) is a function of time or is time dependent.
The accuracy of the estimated H(T) is ultimately determined by the time
duration in which the estimation is performed. The longer the time duration, the better the accuracy. This requirement must be comprised in realtime ultrasonic imaging. In the earliest color Doppler scanners, there were
50 scan lines with a frame rate of 15 per second. The dwelling time of the
ultrasound beam at any one direction is
d
=
s
1.33=
If the depth of view is 10 cm, the time needed for a pulse to make
a round-trip or time of ight is 0.13 ms assuming an ultrasound speed
of 1540 m/s. This means that 1.33/0.13 = 10 ultrasound pulses can be
transmitted in this time span, and that the autocorrelation function is
computed and averaged after 10 pulse transmissions. The autocorrelator
needs to compute the autocorrelation function for each pixel along a scan
line, as illustrated in Figure6.4, where the thin curve and the thick curve
H(T ) is computed from the sampled
data at these times
Voltage
T
Voltage
Figure 6.4 The autocorrelation function from which the mean and the variance
of the Doppler-shifted frequency are estimated is computed for each pixel along
a scan line in a color Doppler ow mapping system.
Pixel at z
n
z
z
Beam direction
z
166 Diagnostic ultrasound: imaging and blood ow measurements
Conventional Doppler
signal processor
Low pass filter
B-mode system
Figure 6.5 Block diagram of a color Doppler ow mapping system.
Auto-correlator
Velocity
calculator
Scan
converter
Display
represent, respectively, the pulse-echo waveform after each pulse transmission and the time variation of the echo at a certain pixel for which the
autocorrelation function is computed.
In a color Doppler system, the signal received by a probe is divided
into three paths, one for constructing the gray-scale B-mode image, one
for calculating the ow information from Doppler data using a hardwired
autocorrelator, and one for conventional Doppler measurements. This is
delineated in Figure6.5. Eight or more shades are used in these systems to
depict the magnitude of the velocity. The higher the velocity, the lighter the
shade. Since the basic principle of Doppler ow mapping is similar to that
of pulsed Doppler, the maximal Doppler frequency that can be detected
without aliasing is half of the pulse repetition frequency. Therefore, a
higher pulse repetition frequency is favored for avoiding aliasing and
increasing the accuracy of the autocorrelation. However, limited by the
frame rate and eld of view, the pulse repetition frequency in most color
Doppler systems is between 8 and 16 KHz, frequently resulting in aliasing
with color Doppler in cardiac imaging. To overcome these problems, the
image size may be reduced, or M-mode color Doppler where the beam is
xed in one direction may be used.
In the heart, the myocardium is in motion during a cardiac cycle, and
tissue color Doppler images of this motion can also be acquired with the
color Doppler methods previously described. The difference lies in that
myocardial motion is slower than blood ow and myocardial echoes are
stronger than blood. The spurious Doppler signals from blood in this case
can be eliminated by thresholding the echoes as illustrated in Figure6.6.
A tissue Doppler image of the heart where the color indicates the velocity
of myocardial motion is shown in Figure6.7.
Many clinical applications have been found for color Doppler ow
imaging, including diagnosing tiny shunts in the heart wall and valvular
167Chapter six: Flow and displacement imaging
Frequency
Amplitude
Tissue signal
Amplitude threshold
Blood signal
Figure 6.6 Tissue Doppler can be achieved by thresholding the Doppler signals
so as to suppress the Doppler signal from blood and retain only the Doppler
signals from tissues.
regurgitation and stenosis. It considerably reduces the examination time
in many diseases associated with ow disturbance. Problematic regions
can be quickly identied rst from the ow mapping. More quantitative
conventional Doppler measurements are then made on these areas.
Although color Doppler has now been widely used in a variety of
medical disciplines, it has several shortcomings. (1) Flow perpendicular
to the beam cannot be reliably detected. (2) Higher blood ow velocity
results in aliasing. (3) Its spatial resolution is poorer than B-mode grayscale imaging. (4) The mean velocity estimated is the average velocity
Figure 6.7 Tissue Doppler image of myocardial motion. Colored areas indicate
velocity of myocardium motion in the heart walls, and anechoic regions indicate
intracardiac blood pool. (Courtesy of Philips Medical Systems.)
168 Diagnostic ultrasound: imaging and blood ow measurements
)(
ωω
within a pixel or voxel. (5) Since the color Doppler image is overlaid over
the gray-scale B-mode, the overlay process is determined arbitrarily by
thresholding, which may result in vessel-wall overwrite obscuring the
slow blood ow signal near the wall. (6) Large echoes due to slow-moving
tissues can cause the “color ash artifact” because they overlap echoes
from owing blood. (7) The frame rate is reduced because separate ring
is needed to obtain a color Doppler image.
6.2 Color Doppler power imaging
Another way of displaying the color Doppler information, i.e., power
mode or energy mode imaging, has been introduced to minimize some
of the color Doppler problems (Rubin et al., 1994; Zagzebski, 1996). Instead
of the mean Doppler shift, the power contained in the Doppler signal is
displayed in this approach. There are several advantages to doing so. (1)
A threshold can be set to minimize the effect of noise. (2) The data can
be averaged to achieve a better signal-to-noise ratio. (3) The images are
less dependent upon the Doppler angle. Finally, (4) aliasing is no longer
a problem since only the power is detected. Because of these advantages,
signals from blood owing in much smaller vessels may be detected. The
images so produced have an appearance similar to that of x-ray angiography preferred by radiologists. The disadvantages of this approach are
that (1) it is more susceptible to motion artifacts due to frame averaging
and (2) the image contains no information on ow velocity and direction.
A color power Doppler image of a carotid artery bifurcation is shown in
Figure6.8. The orange region indicates that there is blood ow. The grayscale B-mode image delineates blood vessel wall and surrounding tissues.
Power Doppler imaging is in fact easier to implement than conventional color Doppler because Doppler power is readily available in conventional color Doppler systems. H(0) in Equation (6.11), which is needed
to calculate the mean Doppler frequency, is the power contained in the
Doppler spectrum. This becomes apparent when setting τ = 0 in Equation
(6.6), i.e.,
∞
(0)(
τ= ==
HptdtPd
2
∫
−∞
∞
∫
)
−∞
6.3 Time domain ow estimation
Blood ow velocity has been estimated directly from B-mode images,
termed speckle tracking, or from radio frequency (RF) echoes. These
alternatives accomplish ow blood measurements in the time domain.
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