Vol. 1, No. 1,1979
ULTRASONIC IMAGING
BLOOD FLOW MEASUREMENT USING THE ATTENUATIONGOMPENSATED VOLUME FLOWMETER Charles F. Hottinger' and James D. Meindl Department of Electrical Engineering Stanford University Stanford, California 94305
Measurement of blood volume flow using Doppler ultrasound has traditionally required determination of the Doppler angle, vessel size, and shape of the velocity profile. In transcutaneous measurements of blood flow, this problem has been a serious drawback in clinical usage. Volume flow is formally defined as the area integral of fluid flux normal to an arbitrary sample plane. Based on this definition, a pulsed Doppler volume flowmeter has been developed which measures flow normal to a thin uniformly illuminated sample volume. Calibration of the flowmeter is described, and implications for clinical use are discussed. Key words: blood flow; blood flow simulation; Doppler; first moment detection; transducer; uniform illumination; volume flow. INTRODUCTION For nearly two decades, Doppler ultrasound has been recognized as a potential modality for noninvasive quantitation of blood flow [ 11. Such a capability would find widespread clinieal application in measuring flows through a number of vessels, including the aorta, as well as the pulmonary, carotid, and femoral arteries, among others [2]. Certain physical aspects of the measurement process inherent to Doppler ultrasound, however, early appeared as serious drawbacks in estimating blood volume flow (in cm3/s). Chief among these apparent problems was the need to determine the Doppler angle in order t o estimate true blood velocity [3]. This is illustrated with reference to the cw (continuous wave) transducer configuration shown in figure 1. If it is assumed that the incident ultrasound beam and the reflected beam scattered by the moving red blood cells within the vessel are roughly anti-parallel, the apparent Doppler shift detected by the flowmeter is given by [ 11
where X is the ultrasound wavelength and 0 is the "Doppler angle" describing the angle between the velocity vector T a n d the sound beam. Thus, only the velocity component parallel t o the beam axis i s measured in figure 1, and the true velocity v cannot be estimated unless 0 is also known.
1.
Present address: Unirad Corporation, Denver, Colorado. Address correspondence regarding this paper to: C. Hottinger, Research Manager, Unirad Corporation, P.O. Box 39002, Denver, Colorado 80239
0161-7346/79/010001-15$02.00/0
1
Copyright @ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
ULTRASONIC IMAGING
Vol. 1, No.1, 1979
TRANSMITTING TRANSDUCER
VESSEL
RECEIVING BEAM
Fig. 1
Transcutaneous detection of blood flow using continuous-wave (cw) ultrasound. I f the transmitted and received beams are antiparallel, the detected Doppler shift indicates the velocity component along with beam axis.
Since volume flow, stated in simple terms, is a product of velocity v and area A
h [cm3/s1 = v x A
(2)
it was thought that v must be known in order to estimate h. Several groups have employed triangulation techniques in order to measure actual blood velocity [3,41.
Other problems, besides estimation of Doppler angle, remained in estimating volume flow. In eq. (2). as can be seen, the vessel area must be determined. This is particularly difficult if the lumen diameter is changing appreciably during the cardiac cycle. For this reason, various imaging schemes have been proposed to determine lumen area A. Still further complications arise i f the blood velocity varies with position across the lumen. To overcome this difficulty, pulsed Doppler velocity meters have been proposed to sample the velocity profile by measuring the Doppler shifts at a number of points along the beam path. Such an approach leads to considerable difficulty in transcutaneous flow quantitation. Several articles by Baker carefully analyzed the problems encountered using Doppler ultrasound to estimate blood flow and promised little prospect that these problems could be overcome [3,51.
It now appears these problems in estimating flow using Doppler ultrasound may be avoided by recognition of the general nature of flow as based on flux normal to an arbitrary sample surface [6,71. Of three generic configurations proposed, one in particular, as described in this paper, measures flow normal to an arbitrary surface in a particularly direct manner. Ascan beseen, the modality of pulsed Doppler ultrasound can therefore be demonstrated to be much more applicable to volume flow measurement than had been previously recognized. VOLUME FLOW DEFINED The general definition of volume flow can be considered with reference to figure 2 [ 8 ] .As indicated, a flux of incompressible fluid passes through a one-sided measurement surface shown here as a plane. A t any instant of time, volume flow
s,
2
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
U
Fig. 2
Volume flow defined in terms of flux normal to a measurement surface. (a) Volume flow of an incompressible fluid measured with respect to a one-sided measurement syrface (b) Volume flow through incremental area 6 A during time 6 t (6 06 t) defined by%fa A6 t.
s.
through any incremental area of this surface is defined as illustrated in figure 2(b). Through an area 6A, the displacement of fluid between times t and t + 6 t i s indicated by the parallelepiped whose bases are each of area 6A. The volume of fluid contained within this prism is
where
6h = instantaneous rate of volume flow through 6A 3=instantaneous local velocity o f the fluid through 6A O = angle between the velocity vector 9 and a unit vector 7 i normal t o 6A
Because the volume of the prism is determined by the product of the base area (6A) and the height (v6t cos O), the right rectangular prism in figure 2(b) (dotted lines) has an equivalent volume. As a result, only the component of velocity normal t o 6 A and defined by cos O =%$is relevant t o the determination of flow through 6A. The instantaneous rate of volume flow through 6 A is therefore
66 =
lim 6 t 4
s&it
=ITcose6A = vn6A 6t
(4)
whereT-xis expressed as Vn. The total instantaneous flow through S is the sum of the incremental flows
h=
6hi=
vni6Ai
i
i
3
(5)
ULTRASONIC IMAGING
Vol. 1, No. 1,1979
and, thus, the area density of flow is given by lim 6h (6) =vn 6A-0 6A Based on this area density, the total volume flow through Scan be expressed in integral form as
-
WhereTspecifies location and da is an incremental area. Because this integral is the most general definition of instantaneous volume flow through a sample surface, it is valid without regard to the spatial and temporal variations of the local fluid velocity-vector f i e l d J K t ) . It will be recognized that eq. (7) is directly analogous to the definition of electric current,
=
fs
Jnda
with respect to measurement surface S where normal current density Jn is the area density of electric current [91.
VOLUME FLOWMETER SYSTEM In this section, we give a brief description of a pulsed Doppler system that measures volume flow through an arbitrary sample surface in a measurement process based on eq. (7). As will be shown, the flow estimate is thereby independent of the actual Doppler angle and vessel size, as well as variations in the velocity profile across the lumen [6,7, 101. Figure 3 shows the transducer configuration and block diagram for the Attenuation-Compensated Volume Flowmeter. As shown in figure 3(a), a two-element pulsed Doppler transducer uniformly illuminates a vessel lying within the "near field" of the array (where the beam is approximated by the geometric projection of the transducer) with repeated short bursts of ultrasound. The signals returning to both etements are simultaneously range-gated and then processed by the circuitry in the block diagram in figure 3(b) tocreate two thin sample volumes. Volume 1, as shown, constitutes a large uniformly illuminated sample "plane" covering the entire lumen of the vessel; the projection of the lumen onto this plane is indicated by the dotted . 2 is a small disc of area A, which is a part of the line and has area A p ~ o j Volume larger plane forming Volume 1. As illustrated, A, must lie totally within the lumen. In a manner common to most pulsed Doppler systems 111-141, two Doppler audio signals [with power spectra (S, (f)and S, (f) in W/Hz) are produced by the circuitry indicated in figure 3(b) (15, 161. When the bandpass filter (BPF) properly compensates for sampling effects arising from the pulsed nature of the waveforms, the following relations can be shown to be valid [171,
jl
(f) df=T(z) 9 1, (2) A ~ R O J
4
(94
ULTRASONIC IMAGING
Vol. 1, No. 1,1979
&p
VOLUME 2
VOLUME 1
(includer Vol. 2)
TWO- ELEMENT TRANSDUCER
Fig. 3
AttenuationCompensated Volume Flowmeter (ACVF). (a) Transducer configuration. (b) Block diagram.
By virtue of Parseval's theorem [18], eqs. (9a) and (9b) indicate that the power levels in each channel are determined by the following quantities: T(z) [dimensionless] : round-trip transmission efficiency representing the effects of attenuation caused by the intervening tissue between the transducer and vessel at range z. q [cm-' ] : the volumetric scattering coefficient representing the scattering of
blood [191. I, (z) [W/cm] : the response of beam 1 t o a volumetric distribution of small random scatteren, 1 cm on each edge, having an effective TI of 1 cm-', located at range z in the absence of attenuation [ 191 This parameter indicates transducer sensitivity.
.
I, (z) [W/cm] : indicates transducer sensitivity of beam 2, analogous t o I, (z) for beam 1.
As can be seen from eq. (9a), the power level S S , (f) df is proportional to ApRoJ, the projection of the vessel lumen onto the sample surface created by gat-
5
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
ing beam 1. Changes in Ap OJ during the cardiac cycle or respiration will therefore be indicated by changes in JS, (f) df. In eq. (9b) for SS, (f) df, however, no such variation occurs since A, determined by the transducer design [as is I, (211, lies totally within the vessel. The power level SS, (f) df given by eden (t) in figure 3(b), therefore, provides an indirect measurement of the product .T (z) r), representing the combined effects of attenuation and target scattering efficiency. By referring to figure 3(b), it can be seen that a voltage is produced proportional to the first moment of S, (f) because the combined f' filter and power detector act as a first-moment detector. The voltage level enum (t) applied to the numerator input of the divided circuit in figure 3(b) can therefore be stated as K SfS, (f) df, where J?cis the gain of the adjustable gain amplifier. In considering the physical significance of the first moment of S, (f), the sample plane corresponding to Volume 1 in figure 3(a) may be divided into incremental areas of size 6Ai, as in figure 2(a), such that
Each incremental area within the lumen gives rise to an increment of Doppler signal power.fSi, (f) df defined by (1l a ) where
j S i l (f) df = T (z) TI I ,
( 2 ) 6Ai
Within each 6A,, the normal velocity is expressed as v,,. Si, (f) is therefore given as
(1Ib)
The first moment of each
2 l f S i l (f) df = T (z) TI I, (z) -v,,6Ai
x
(12)
where Doppler frequency (indicated here by f) corresponds to velocity normal to the sample surface in figure 3(a) 2 f =-Vn
(13)
x
But from eq. (5) the incremental flow through SA, is given by (from 5)
6 Qi = vni6 A,
where
d = c 6di =
vni6Ai
i
i Therefore, from eqs. (101, (11). and (12).
6
(from 5)
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
The voltage applied to the numerator input of the divider circuit in figure 3(b) can therefore be expressed as enum (t)= K
J
2
fS,(f) df = KT (zJ 1) I, (z) -Q A
(15)
Thus, the first moment of S, (f) is proportional to the volume flow 6 through the sample surface at range z, but the constant of proportionality includes the effects of attenuation by the media and target scattering efficiency. The ratio estimated by the divider in figure 3(b) compensates for these effects, however. From eqs. (9b) and (151,
where K is chosen so that K
(1:
A :),
=
The output voltage in eq. (16) therefore provides an estimate of volume flow Q by means of compensation for the combined effects of scattering and attenuation (giving rise to the name Attenuation-Compensated Volume Flowmeter). The proportionality factor given by (I, (z)/l, (z) A,) is determined only by the transducer design and the vessel range z and may be calculated a priori or measured empirically in a calibration procedure. The variable gain amplifier with adjustable gain f l compensates for this factor. Thus, for the simplest case, where I, (z) = I, (z), and where within the near field of the transducer A, is constant, then
and is constant with respect to range z. Because the measurement process, as expressed in eq. (161, is based on the general definition of flow [expressed in eq. (7)] in a particularly direct manner, the estimation is not affected by a number of factors affecting other Doppler ultrasound flowmeters [3]. As an example, the Doppler angle in eq. (1) need not be known so as to estimate true velocity v because only the velocity component measured vn is relevant to the estimation of Q. In this sense, the measurement process is independent of the Doppler angle. In addition, the precise vessel size need not be known in order to estimate & if the vessel lumen lies entirely within the large sample plane shown in figure 3(a). Even variations in lumen diameter during the cardiac cycle are thereby accommodated. As a final example, it should be noted that spatial variations in the velocity profile do not affect the flow estimate. The demodulator of the type shown in figure 3(b) can process a wideband Doppler power spectrum. With a suitable directional version of the Doppler circuitry, even flow involving simultaneous forward and reverse motion across the sample plane can be correctly estimated 1201.
DEMONSTRATION OF ANGLE-AREA INDEPENDENCE
Results from a set of in vitro experiments demonstrating independence of the flow estimate with respect to Doppler angle and vessel size are illustrated in figures 4 thru 6 [17, 211. The 5.7 MHz LTZ-2 transducer used in this particular test was 25 mm on a side, with a 5 mm central zone, following the configuration in figure 3(a). All data were taken at one range (25 mm) and with two vessel sizes (12 and
7
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
1.00-
o.50-
0.00
Fig. 4
7.310~ cos8 V W N E FLOW = 37 ml/s D = 16mm
I
I
I
I
I
Estimated volume flow as a function of Doppler angle for a given vessel size and flow rate. The curve proportional to the cosine of the Doppler angle is plotted for cornparison. Best zero-order fit is shown [22]
.
2.00r
FLOWMETER OUTPUT (Volts) 8=46.7O D=12mm D=16mm A
1.50-
1.00 -
r 2 = 0.99699
0.00 0
Fig. 5
1
10
I
20 VOLUME FLOW (m 1/51
I
30
I 40
Estimated volume flow as a function of actual flow for two vessel sizes to illustrate the independence of flow estimate with respect t o lumen diameter. Best first-order fit is shown 1221.
16 mm). Simulation of the vessels was accomplished with dialysis tubing, and a water suspension of polystyrene particles was used in place of blood. To demonstrate angle independence, a set of points corresponding to one vessel size and flow rate is plotted as a function of Doppler angle in figure 4. A straight line of zero slope is fitted to these points and the correlation coefficient is indicated. For comparison, a curve proportional to the cosine of the Doppler angle is also plotted. Dependence of flow estimates on angle has been reduced by one order of magnitude from that exhibited by a cosine curve.
8
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
FLOWMETER OUTPUT [ VOltS)
42.a0
ULTRASONIC IMAGING
BLOOD FLOW MEASUREMENT USING THE ATTENUATIONGOMPENSATED VOLUME FLOWMETER Charles F. Hottinger' and James D. Meindl Department of Electrical Engineering Stanford University Stanford, California 94305
Measurement of blood volume flow using Doppler ultrasound has traditionally required determination of the Doppler angle, vessel size, and shape of the velocity profile. In transcutaneous measurements of blood flow, this problem has been a serious drawback in clinical usage. Volume flow is formally defined as the area integral of fluid flux normal to an arbitrary sample plane. Based on this definition, a pulsed Doppler volume flowmeter has been developed which measures flow normal to a thin uniformly illuminated sample volume. Calibration of the flowmeter is described, and implications for clinical use are discussed. Key words: blood flow; blood flow simulation; Doppler; first moment detection; transducer; uniform illumination; volume flow. INTRODUCTION For nearly two decades, Doppler ultrasound has been recognized as a potential modality for noninvasive quantitation of blood flow [ 11. Such a capability would find widespread clinieal application in measuring flows through a number of vessels, including the aorta, as well as the pulmonary, carotid, and femoral arteries, among others [2]. Certain physical aspects of the measurement process inherent to Doppler ultrasound, however, early appeared as serious drawbacks in estimating blood volume flow (in cm3/s). Chief among these apparent problems was the need to determine the Doppler angle in order t o estimate true blood velocity [3]. This is illustrated with reference to the cw (continuous wave) transducer configuration shown in figure 1. If it is assumed that the incident ultrasound beam and the reflected beam scattered by the moving red blood cells within the vessel are roughly anti-parallel, the apparent Doppler shift detected by the flowmeter is given by [ 11
where X is the ultrasound wavelength and 0 is the "Doppler angle" describing the angle between the velocity vector T a n d the sound beam. Thus, only the velocity component parallel t o the beam axis i s measured in figure 1, and the true velocity v cannot be estimated unless 0 is also known.
1.
Present address: Unirad Corporation, Denver, Colorado. Address correspondence regarding this paper to: C. Hottinger, Research Manager, Unirad Corporation, P.O. Box 39002, Denver, Colorado 80239
0161-7346/79/010001-15$02.00/0
1
Copyright @ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
ULTRASONIC IMAGING
Vol. 1, No.1, 1979
TRANSMITTING TRANSDUCER
VESSEL
RECEIVING BEAM
Fig. 1
Transcutaneous detection of blood flow using continuous-wave (cw) ultrasound. I f the transmitted and received beams are antiparallel, the detected Doppler shift indicates the velocity component along with beam axis.
Since volume flow, stated in simple terms, is a product of velocity v and area A
h [cm3/s1 = v x A
(2)
it was thought that v must be known in order to estimate h. Several groups have employed triangulation techniques in order to measure actual blood velocity [3,41.
Other problems, besides estimation of Doppler angle, remained in estimating volume flow. In eq. (2). as can be seen, the vessel area must be determined. This is particularly difficult if the lumen diameter is changing appreciably during the cardiac cycle. For this reason, various imaging schemes have been proposed to determine lumen area A. Still further complications arise i f the blood velocity varies with position across the lumen. To overcome this difficulty, pulsed Doppler velocity meters have been proposed to sample the velocity profile by measuring the Doppler shifts at a number of points along the beam path. Such an approach leads to considerable difficulty in transcutaneous flow quantitation. Several articles by Baker carefully analyzed the problems encountered using Doppler ultrasound to estimate blood flow and promised little prospect that these problems could be overcome [3,51.
It now appears these problems in estimating flow using Doppler ultrasound may be avoided by recognition of the general nature of flow as based on flux normal to an arbitrary sample surface [6,71. Of three generic configurations proposed, one in particular, as described in this paper, measures flow normal to an arbitrary surface in a particularly direct manner. Ascan beseen, the modality of pulsed Doppler ultrasound can therefore be demonstrated to be much more applicable to volume flow measurement than had been previously recognized. VOLUME FLOW DEFINED The general definition of volume flow can be considered with reference to figure 2 [ 8 ] .As indicated, a flux of incompressible fluid passes through a one-sided measurement surface shown here as a plane. A t any instant of time, volume flow
s,
2
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
U
Fig. 2
Volume flow defined in terms of flux normal to a measurement surface. (a) Volume flow of an incompressible fluid measured with respect to a one-sided measurement syrface (b) Volume flow through incremental area 6 A during time 6 t (6 06 t) defined by%fa A6 t.
s.
through any incremental area of this surface is defined as illustrated in figure 2(b). Through an area 6A, the displacement of fluid between times t and t + 6 t i s indicated by the parallelepiped whose bases are each of area 6A. The volume of fluid contained within this prism is
where
6h = instantaneous rate of volume flow through 6A 3=instantaneous local velocity o f the fluid through 6A O = angle between the velocity vector 9 and a unit vector 7 i normal t o 6A
Because the volume of the prism is determined by the product of the base area (6A) and the height (v6t cos O), the right rectangular prism in figure 2(b) (dotted lines) has an equivalent volume. As a result, only the component of velocity normal t o 6 A and defined by cos O =%$is relevant t o the determination of flow through 6A. The instantaneous rate of volume flow through 6 A is therefore
66 =
lim 6 t 4
s&it
=ITcose6A = vn6A 6t
(4)
whereT-xis expressed as Vn. The total instantaneous flow through S is the sum of the incremental flows
h=
6hi=
vni6Ai
i
i
3
(5)
ULTRASONIC IMAGING
Vol. 1, No. 1,1979
and, thus, the area density of flow is given by lim 6h (6) =vn 6A-0 6A Based on this area density, the total volume flow through Scan be expressed in integral form as
-
WhereTspecifies location and da is an incremental area. Because this integral is the most general definition of instantaneous volume flow through a sample surface, it is valid without regard to the spatial and temporal variations of the local fluid velocity-vector f i e l d J K t ) . It will be recognized that eq. (7) is directly analogous to the definition of electric current,
=
fs
Jnda
with respect to measurement surface S where normal current density Jn is the area density of electric current [91.
VOLUME FLOWMETER SYSTEM In this section, we give a brief description of a pulsed Doppler system that measures volume flow through an arbitrary sample surface in a measurement process based on eq. (7). As will be shown, the flow estimate is thereby independent of the actual Doppler angle and vessel size, as well as variations in the velocity profile across the lumen [6,7, 101. Figure 3 shows the transducer configuration and block diagram for the Attenuation-Compensated Volume Flowmeter. As shown in figure 3(a), a two-element pulsed Doppler transducer uniformly illuminates a vessel lying within the "near field" of the array (where the beam is approximated by the geometric projection of the transducer) with repeated short bursts of ultrasound. The signals returning to both etements are simultaneously range-gated and then processed by the circuitry in the block diagram in figure 3(b) tocreate two thin sample volumes. Volume 1, as shown, constitutes a large uniformly illuminated sample "plane" covering the entire lumen of the vessel; the projection of the lumen onto this plane is indicated by the dotted . 2 is a small disc of area A, which is a part of the line and has area A p ~ o j Volume larger plane forming Volume 1. As illustrated, A, must lie totally within the lumen. In a manner common to most pulsed Doppler systems 111-141, two Doppler audio signals [with power spectra (S, (f)and S, (f) in W/Hz) are produced by the circuitry indicated in figure 3(b) (15, 161. When the bandpass filter (BPF) properly compensates for sampling effects arising from the pulsed nature of the waveforms, the following relations can be shown to be valid [171,
jl
(f) df=T(z) 9 1, (2) A ~ R O J
4
(94
ULTRASONIC IMAGING
Vol. 1, No. 1,1979
&p
VOLUME 2
VOLUME 1
(includer Vol. 2)
TWO- ELEMENT TRANSDUCER
Fig. 3
AttenuationCompensated Volume Flowmeter (ACVF). (a) Transducer configuration. (b) Block diagram.
By virtue of Parseval's theorem [18], eqs. (9a) and (9b) indicate that the power levels in each channel are determined by the following quantities: T(z) [dimensionless] : round-trip transmission efficiency representing the effects of attenuation caused by the intervening tissue between the transducer and vessel at range z. q [cm-' ] : the volumetric scattering coefficient representing the scattering of
blood [191. I, (z) [W/cm] : the response of beam 1 t o a volumetric distribution of small random scatteren, 1 cm on each edge, having an effective TI of 1 cm-', located at range z in the absence of attenuation [ 191 This parameter indicates transducer sensitivity.
.
I, (z) [W/cm] : indicates transducer sensitivity of beam 2, analogous t o I, (z) for beam 1.
As can be seen from eq. (9a), the power level S S , (f) df is proportional to ApRoJ, the projection of the vessel lumen onto the sample surface created by gat-
5
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
ing beam 1. Changes in Ap OJ during the cardiac cycle or respiration will therefore be indicated by changes in JS, (f) df. In eq. (9b) for SS, (f) df, however, no such variation occurs since A, determined by the transducer design [as is I, (211, lies totally within the vessel. The power level SS, (f) df given by eden (t) in figure 3(b), therefore, provides an indirect measurement of the product .T (z) r), representing the combined effects of attenuation and target scattering efficiency. By referring to figure 3(b), it can be seen that a voltage is produced proportional to the first moment of S, (f) because the combined f' filter and power detector act as a first-moment detector. The voltage level enum (t) applied to the numerator input of the divided circuit in figure 3(b) can therefore be stated as K SfS, (f) df, where J?cis the gain of the adjustable gain amplifier. In considering the physical significance of the first moment of S, (f), the sample plane corresponding to Volume 1 in figure 3(a) may be divided into incremental areas of size 6Ai, as in figure 2(a), such that
Each incremental area within the lumen gives rise to an increment of Doppler signal power.fSi, (f) df defined by (1l a ) where
j S i l (f) df = T (z) TI I ,
( 2 ) 6Ai
Within each 6A,, the normal velocity is expressed as v,,. Si, (f) is therefore given as
(1Ib)
The first moment of each
2 l f S i l (f) df = T (z) TI I, (z) -v,,6Ai
x
(12)
where Doppler frequency (indicated here by f) corresponds to velocity normal to the sample surface in figure 3(a) 2 f =-Vn
(13)
x
But from eq. (5) the incremental flow through SA, is given by (from 5)
6 Qi = vni6 A,
where
d = c 6di =
vni6Ai
i
i Therefore, from eqs. (101, (11). and (12).
6
(from 5)
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
The voltage applied to the numerator input of the divider circuit in figure 3(b) can therefore be expressed as enum (t)= K
J
2
fS,(f) df = KT (zJ 1) I, (z) -Q A
(15)
Thus, the first moment of S, (f) is proportional to the volume flow 6 through the sample surface at range z, but the constant of proportionality includes the effects of attenuation by the media and target scattering efficiency. The ratio estimated by the divider in figure 3(b) compensates for these effects, however. From eqs. (9b) and (151,
where K is chosen so that K
(1:
A :),
=
The output voltage in eq. (16) therefore provides an estimate of volume flow Q by means of compensation for the combined effects of scattering and attenuation (giving rise to the name Attenuation-Compensated Volume Flowmeter). The proportionality factor given by (I, (z)/l, (z) A,) is determined only by the transducer design and the vessel range z and may be calculated a priori or measured empirically in a calibration procedure. The variable gain amplifier with adjustable gain f l compensates for this factor. Thus, for the simplest case, where I, (z) = I, (z), and where within the near field of the transducer A, is constant, then
and is constant with respect to range z. Because the measurement process, as expressed in eq. (161, is based on the general definition of flow [expressed in eq. (7)] in a particularly direct manner, the estimation is not affected by a number of factors affecting other Doppler ultrasound flowmeters [3]. As an example, the Doppler angle in eq. (1) need not be known so as to estimate true velocity v because only the velocity component measured vn is relevant to the estimation of Q. In this sense, the measurement process is independent of the Doppler angle. In addition, the precise vessel size need not be known in order to estimate & if the vessel lumen lies entirely within the large sample plane shown in figure 3(a). Even variations in lumen diameter during the cardiac cycle are thereby accommodated. As a final example, it should be noted that spatial variations in the velocity profile do not affect the flow estimate. The demodulator of the type shown in figure 3(b) can process a wideband Doppler power spectrum. With a suitable directional version of the Doppler circuitry, even flow involving simultaneous forward and reverse motion across the sample plane can be correctly estimated 1201.
DEMONSTRATION OF ANGLE-AREA INDEPENDENCE
Results from a set of in vitro experiments demonstrating independence of the flow estimate with respect to Doppler angle and vessel size are illustrated in figures 4 thru 6 [17, 211. The 5.7 MHz LTZ-2 transducer used in this particular test was 25 mm on a side, with a 5 mm central zone, following the configuration in figure 3(a). All data were taken at one range (25 mm) and with two vessel sizes (12 and
7
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
1.00-
o.50-
0.00
Fig. 4
7.310~ cos8 V W N E FLOW = 37 ml/s D = 16mm
I
I
I
I
I
Estimated volume flow as a function of Doppler angle for a given vessel size and flow rate. The curve proportional to the cosine of the Doppler angle is plotted for cornparison. Best zero-order fit is shown [22]
.
2.00r
FLOWMETER OUTPUT (Volts) 8=46.7O D=12mm D=16mm A
1.50-
1.00 -
r 2 = 0.99699
0.00 0
Fig. 5
1
10
I
20 VOLUME FLOW (m 1/51
I
30
I 40
Estimated volume flow as a function of actual flow for two vessel sizes to illustrate the independence of flow estimate with respect t o lumen diameter. Best first-order fit is shown 1221.
16 mm). Simulation of the vessels was accomplished with dialysis tubing, and a water suspension of polystyrene particles was used in place of blood. To demonstrate angle independence, a set of points corresponding to one vessel size and flow rate is plotted as a function of Doppler angle in figure 4. A straight line of zero slope is fitted to these points and the correlation coefficient is indicated. For comparison, a curve proportional to the cosine of the Doppler angle is also plotted. Dependence of flow estimates on angle has been reduced by one order of magnitude from that exhibited by a cosine curve.
8
Vol. 1, No. 1,1979
ULTRASONIC IMAGING
FLOWMETER OUTPUT [ VOltS)
42.a0
