IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 6, JUNE 1996 589

Measurement of Volumetric Flow with No Angle Correction Using Multiplanar

d Doppler Ultrasound Jens K. Poulsen"

Abstract-In this paper we show that by scanning at points on the surface of a sphere that the normal angle correction used in pulsed Doppler flow measurements is no longer necessary. Thus, it is possible to measure three-dimensional (3-D) flow using multiplanar ultrasound even though we only get one-dimensional (1-D) velocity information from pulsed Doppler ultrasound. The technique handles the three basic problems in flow measurements using ultrasound Doppler: The variations of the cross-sectional area, the time dependent changes in the velocity field, and the dependency of the angle of insonation. The technique is tested in a flow phantom using different angles of insonation to validate the angle independence of this new technique. Using six different angles of insonation in the range 0" to 69" with flowrates in the range of 0-170 mYs a linear dependence was found to be: measured (color Doppler) = 0.98 real flow (reference) +1.36 mYs, with a 95% confidence interval of f13.9 mYs.

I. THEORETICAL CONSIDERATIONS

HE general equation for flow out of a surface of an T incompressible fluid is found from conservation of matter [l] (by dividing the mass flowrate by the density of the fluid) to be

F = / / ? -dA (1) Surface

i.e., from a double integration over the velocity field of interest on a given surface. F is equal to the volumetric flowrate through the given_surface, is a three-dimensional (3-D) vector and so is dA. ? is aligned in the same direction as the velocity vector in a particular point on the surface and has a magnitude equal to the velocity at this point. The magnitude of the vector d A is equal to the area of a small surface element, and its direction is along the normal.

Generally speaking,+we are unable to find the scalar prod- uct between ? and dA since all three velocity components (U,, vy , v,) at a given point are not easily measured by current ultrasonic methods. While this may change in the future using echo correlation [2], [3], speckle tracking [4], methods derived from SAR [5]-[7] or some other velocity estimation method

Manuscript received March 23, 1995; revised February 26, 1996. This work was supported by Grants from The Danish Heart Foundation, Copenhagen, The Karen Elise Jensen Foundation, Copenhagen, and Aarhus University Research Foundation, Aarhus. Asterisk indicates corresponding author.

*J. K. Poulsen is with the Institute of Experimental Clinical Research, Skejby Sygehus, 8200 Aarhus N, Denmark (e-mail: skejjkp@aau.dk).

W. Y. Kim is with the Department of Thoracic and Cardiovascular Surgery T, Aarhus University Hospital, Skejby Hospital, Denmark.

Publisher Item Identifier S 0018-9294(96)03986-9.

and W. Yong Kim

[8], a standard B-mode Doppler scan only provides velocity information in the direction of the ultrasonic beam.

we decide the area over which we want to perform the ce integration to be a surface with a constant distance to

the ultrasonic transducer, the situation is as shown in Fig. 1. The multiplanar transducer works like an ordinary B-mode transducer but has the ability to rotate the scanplane around the central axis. It should be noted that the multiplanar transducer is placed outside the tube. The area vector, d 2 , is now aligned in the same direction as the ultrasonic beam and this means that we now measure the scalar product of a general velocity vector and a unit-size vector aligned in the same direction as the area vector. As a result we do not have to know all three velocity components to calculate the scalar product, and thus the flow-the geometry is such that no correction has to be made.

Strictly, this is of course only true for infinitesimal sample volumes, with these placed on closely spaced points equidis- tant from the transducer. For practical purposes, we have to perform a limited number of B-mode scans, so that we have enough information to accurately describe our chosen surface (Fig. 1).

The sampling density on the sphere's surface has to be kept as high as possible, although the finite size of the sample volume, and the demand for a short measurement time, force us to make some approximations. However, numerical integration is an operation that is usually easy to perform with good results. Small individual errors in the velocity estimates tend to cancel out and numerous scans only have to be made for flow fields with an irregular structure, provided the velocity estimator used is consistent.

The limitations of this basic theory depend on the accuracy of the ultrasonic equipment only, and not on the flow charac- teristics. If the flow is characterized by the presence of highly irregular and disordered variation of the velocity with time, then we can only get a statistical estimate of the mean flow, since the standard pulsed ultrasonic method can measure the velocity in only one sample volume at a time.

The first reference to ultrasonic scanning along a spherical surface seems to be Hemp, who proposes the concept in conjunction with transit-time flow meters [9]. Recently an application of spherical scanning has been described using a spherically shaped transducer to generate a spherical ultrasonic field by itself [lo]. However, the method is limited to intra- vascular applications and is also somewhat sensitive to flow

0018-9294/96$05.00 0 1996 IEEE

590 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 6, JUNE 1996

Multiplanar ultrasonic transducer

Fig. 1. Multiplanar ultrasonic transducer scanning at points on the surface of a sphere.

nli::ide the vessel, a stable attenuation in the media, and a more or less critical orientation of the transducer inside the vessel. Since the experimental basis for the intravascular single-scan transducer concept is still not fully investigated, a thorough validation will have to show if this is a feasible way to measure volumetric flow.

The method described in the present study does not suffer from these limitations since the flow is measured from an integration of discrete units of velocity times area, and much technical research has already been devoted to obtaining the small sample volumes needed. This method employs standard B-mode scans to obtain the information and is therefore completely noninvasive.

A. Calculation of 3 - 0 Flow with no Angle Correction Factor

beam (from the transducer to the sample volume). That is, the measured velocity at a point is the real velocity projected onto the direction of the ultrasonic beam. The minus sign occurs since velocities are registered positive when an object is moving toward the transducer. The direction of the surface area vector d 2 is normal to the surface. The normal vector to the surface of a sphere at a particular location has the same direction as a vector pointing from the center of the sphere to this particular point on the surface. This means we can write

B&m dA = d 2 (3)

and thus the flow out from a small area dA is

d . d 2 = d . (Bezm dA) = ( d . Bezm) dA

The basic theory can be expressed in mathematical terms as = -V, dA. (4) follows: The measured ultrasonic velocity can be found from

V, = - d . Beam (2)

where V, is the measured velocity determined by the ultra- sonic equipment, d is the vector with direction and magfitude

It should be noted that there is no angle correction factor any longer since both V, and dA are scalars. If we are interested in the flow into a surface, the sign is reversed and by using (1) we get

+

equal to the velocity at the sample volume, and Beam is Flowin = / J v, d ~ . ( 5 ) the unit-size vector pointing in the direction of the ultrasonic Surface

POULSEN AND KIM MEASUREMENT OF VOLUMETRIC FLOW WITH NO ANGLE CORRECTION 591

Rotation 4 k=4

Rotation5 k=5

otauon L k=2

Rotation 1 k=l

A Z

(c )

Fig. 2. The area of integration for the technique. Six scanning planes are shown though other choices could have been made. Each surface area, AA, is placed centered around each sample volume as shown in (a) and (c). If more than one vessel is present inside the scan-sectors, the flow of interest should be chosen by proper choice of angles O , + : (a) top view, (b) 3-D view, and (c) sideview.

This double integral can be approximated by the double sum

Flow;, M CVm AA AA-iO

where AA is an area on the spherical surface with outer boundary as defined by the two pair of angles 4 and I9 [see Fig. 2(a) and (c)].

The mathematical conditions that have to be satisfied in order to make this limit equal to the true integral are that the velocity field is continuous over the surface and that the boundary of the surface is not too complicated. These are true for all practical purposes. A full proof requires advanced calculus, e.g., [ll]. The double sum can be expressed in

spherical coordinates

Flow;, = Vm(+, 0, R) AA (7) AA+O

where R is the radius of the sphere and + , O are angles that define a particular point; see Fig. 2. For pulsatile flow or flow that is best described with stochastic methods, we can calculate the mean value of the flow using N time divisions

592 IEEE TRANSACTIONS ON BIOM

the number of scan 1 “rotations” chosen. Fr rd “rotation” will be used to describe the the ultrasonic trans-

axis, see Figs. 1 and 2(a) for a sphere sampled in six rotations. The angle increment factors &,As are found from

[Fig. 2(a) and (c)]

OM - 01 a0 = ___

4N - 41 A4=-

M - 1 and

P - l

The last equation set is valid only if all rotations are equally Fig 3. The overflow sys

own in the center of a scanline in Fig. 2(c). made when approximating a dou

In theory we should let an accurate estimation of th

pulsatile flow we can let the repetition period, to mi

to a multiple Of

or when calculating A. Laminar FZow Stu

From surface integration we can find the area of a small section of a sphere to be (see Appendix for the details)

AA A$R2 I COS 0, - COS On+, I.

collection time is U

flow can be found measurement of the

From this we can deduce the general flow formula to be

~ A$R2 Flow = ___ y Vm(k n, R, iaT>

z = 1 n=l Ic= l N

(12) collection method the ultrasonic

Latex tubin nction of the scan sector k

acoustic properties B. Determination of Area

velocity in a sample vol

POULSEN AND KIM. MEASUREMENT OF VOLUMETRIC FLOW WITH NO ANGLE CORRECTION 593

I Receiver

Fig. 4. timer.

Close view of the infrared (IR) transmitter and receiver used in the

the transmitted and received pulses so that the highly sensitive receive section in the scanner was not overloaded.

Most of the measurements were performed using the closed flow phantom shown in Fig. 3. For small angles of insonation (515O) it was neccessary to change the measurement setup because it was difficult to make undistorted ultrasonic scans through the latex wall when scanning at small angles. Thus, an open system was used (not shown) inasmuch as the latex tube was cut and the ultrasonic transducer was placed in front of the outlet latex tube. In this case a second throttle was added to the exit of the test section to be able to adjust the fluid level to a fixed height inside the test system.

B. Pulsatile Flow Study

The pulsatile system is composed of a steerable piston “Su- perDupr” and a generator (Vivitro Systems, Victoria, Canada), which was set to give the FDA recommended flow curve €or testing of heart valves. Two artificial heart valves were used to control the direction of flow while the same test section as used for the steady state flow measurements was used as an acoustic window, see Fig. 5. The pulse rate was set to 80 beats per minute and the flow was varied between 1.0 and 4.1 l/min. Because of pressure variations during each beat, the latex tube showed large variation in area with time (up to 30%) and provided a good opportunity to test if the color flow modality could accurately follow the changing cross-sectional area of the tube. The fluid was collected manually using a fixed number of beats, typically 12. The error in the reference flow measurement using the timed collection method again was relatively small if the start and end of each collection was set in the part of the flow cycle where almost no fluid was flowing.

C. Timer

The timed collection method requires a measure of the weight of the collected fluid, found using a precision scale, and

Large reservoir

Steady level

Test section \ Latex

Fig. 5. The pulsatile flow system. Latex is used as acoustic window

the collection time, found using a timer that is running while the fluid is being collected. A light beam is broken during the collection of fluid-while the light beam is broken, the timer counts up (see Fig. 3).

An optical transmitter sends 2000 light pulses per second, these are received and amplified (see Fig. 4 for the optical setup). When a transmitted optical pulse is not detected at the receiver, the timer counts up. The counter value is divided by two to give a basic measurement accuracy of 1 mS. The biggest problem involved in the time measurement is to ensure the stream of fluid and the light beam are broken simultaneously by the collection bucket. Furthermore, one has to ensure the flowrate is stable after changing it (this was checked by always measuring the flowrate at least twice whenever it had been changed).

The timer was tested in conjunction with the overflow system. It was possible to reproduce measured values within 0.1% when measuring the flow over a 1-min collection time.

D. Doppler Measurements

The Doppler measurements were done using a mechanical annular phased array transducer operated by a Vingmed Sound CFM-800A system (Vingmed Sound, Horten, Norway). The transmission frequency was 6.0 MHz. A total of six rotations of the multiplanar transducer were used. The frame rate was five and data from one second was captured in the laminar study in each rotation, four consecutive seconds of flowdata were captured in the pulsatile study (this was done using an ECG-simulator to enable the scanner to capture data for such a long time by using an artificial low heartrate of around 30, since we could only capture data from two heartbeats). In each B-mode scan 60 radial scan lines were captured. Of these approximately two-thirds had velocities different from zero, depending upon the geometry. Thus 1800 V AA values (six rotations times 60 velocity values times five frames) were captured for each flow estimate in the laminar flow study and 7200 values for the pulsatile study. Of these only 1200 and 4800 (the velocity values inside the tube) were used for the flow estimates. The velocity estimates using color Doppler were found with the “maximum quality”, this corresponds to 34 pulse packets for each velocity estimate. We used a

594 E E E

f 0.05-0.08 m / s depending on the

of the digital velocity estimator. A gain setting which was too low would result in an underestimation or no registered velocities at all, and a setting which was too high would result in a large overestimation (probably due to clipping). This can be kept from be

not done during the measurements, n expect much larger variances in the Jim1 result. e, the individual steps on our GAIN-control were

could not therefore, expect our flow estimates to be m r. The velocity was checked during one rotation only, using pulsed ultrasound Doppler with Fourier analysis spectral output as a reference. A point in the middle of the tube was used and the GAIN was adjusted so

oppler was normalized was also used in all other the same problems with the metric flow in vivo [12].

asound using asyn- chronous sampling. By this term we mean that the start of each frame using the ultrasonic scanner is not synchronized in

le. The time between in a single rotation is actually equally spaced. e start of the frames is random with respect to

the source. This means that individual measurements in a rably, thus we have to en we measure a given

can find the average value as stated by the Weak Law of Large Numbers [l?].

the sum of a stocastical

disease. The disadvantage,

This sampling scheme was

POULSEN AND KIM: MEASUREMENT OF VOLUMETRIC FL.OW WITH NO ANGLE CORRECTION 595

TABLE I

Center angle 34 38 60 69 0 37 51

System type Closed Closed Closed Closed Open Open Closed Flow range I 3 16-139 13-128 20-160 17-171 9-130 16-143 17-69

Correction (r) 0.991 0.997 0.997 0.996 0.993 0.995 0.979 Slope: value 0.961 0.889 1.020 1.001 1.032 0.928 0.950

95% limits (low) 0.856 0.831 0.955 0.930 0.930 0.854 0.788 (high) 1.067 0.947 1.084 1.072 1.135 1.002 1.112

p-value (offset = 1.0 0.423 0.002 0.500 0.982 0.488 0.056 0.494 Offset: value 0.99 1.01 6.70 0.28 -5.47 4.62 4.61

95% limits (low) -6.87 -3.58 0.57 -6.41 -13.08 -1.64 -2.11 (high) 11.22 5.59 12.84 6.96 2.14 10.88 11.34

p-value (offset = 0.0 0.593 0.626 0.036 0.926 0.136 0.127 0.152

Reynolds number based on viscosity equal to 4.0 cP, value for pulsatile flow vary through each beat, an approximate mean value is shown (area variations are not included). Center angle is given in degrees, offset in ml/s

Reynolds number 268-2333 22 1-2 142 332-2685 285-2861 146-2181 184-2396 286-1159

that a two-dimensional spatial averaging of the color Doppler data is used by the machine for better visual appearance.

The measured flow data was analyzed using linear regres- sion analysis (this type of analysis was the only one done on the pulsatile flow study since it was performed using only one angle of insonation). This is appropriate since we have a well defined and precise reference (namely the timed collection method). To further validate the angle independence of this technique we performed a multiple regression analysis with 48 degrees of freedom (six slopes, six offsets, input data-60 points-is the reference flow, output data is the color Doppler flow). A Students t-test was performed on each of the six different lines [Fig. 6(a)-(f)] to test the following two hypothesis:

1) that the slope of the lines is equal to one and 2 ) that the offset of the lines is equal to zero.

The analysis was performed on the laminar flow data under

Model 1) That all six lines had a different offset and slope

The test was considered significant if p < 0.05.

the following different assumptions:

yz = a, + ,&xz + E

where xi is the reference flow, yz is the measured flow by color Doppler (six lines, i = 1. . .6 ) , a; and pi is the offset and slope for the individual lines, and E is an Gaussian distributed error term. That all six lines had a common offset but different slope dependent upon the angle

Model 2)

y; = a + pix; + E

where x, is the reference flow, yyz is the measured flow by color Doppler, a is the offset, pa is the individual slopes, and t is an Gaussian distributed error term. That all six lines had a common slope but different distinct offset dependent upon the angle

Model 3)

Global model)

where x is the reference flow, y; is the measured flow by color Doppler, ai and /3 is the individual offset and the common slope, and t is an Gaussian distributed error term. A global error model that assumed the same slope for all lines but with a Gaussian distributed offset and error term

y = a +p. + E

where x is the reference flow, y is the mea- sured flow by color Doppler, a is an Gauss- ian distributed offset (changes for each new angle setup), and E is an Gaussian distributed error term (changes for each measurement) with zero mean, thus, the total error depends upon the sum of a and E .

IV. RESULTS The angle of insonation between the ultrasonic probe and the

main stream axis was varied between 0' and 69". Fig. 6(a)-(d) shows the regression lines for the closed tube phantom study. It is seen that there is little dependence of the mean angle of insonation with respect to slope of the lines and offset on the ordinate [see Table 1 for results of the analysis using Model l)]. All of the measurements yielded regression lines with a slope near unity and a relatively small offset at the ordinate. A strong correlation between the reference data and the spherical scanning method was found.

It is seen from Table I that the first t-test was passed for all angles except when the angle is equal to 3 8 O , which had a p- value = 0.002-thus we will have to reject the hypothesis that the slope is equal to one in this case. The second test (namely that the offset is equal to zero) were passed for all angles except 60°, where it had a p-value marginally lower than 0.05 (namely p = 0.036). This discrepancy could be due to errors in estimation of velocity, area and wall motion filter setting. An F-test gives no statistical significance (p = O.O56)--thus we can accept the hypothesis that the slope is the same for the six lines using this model.

596 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 6, JUNE 1996

I60

140

120

100

80

60

40

20

0 0 20 40 60 80 100 120 140 160

(a)

140

120

100

80

60

40

20

n Y

0 20 40 60 80 100 120 140 0 50

(b)

140 I I I I I I

120 -

100 -

80 -

0 50 100 150 200 0 20 40 60 80 100 120 140

(d) (e)

100 150 200

(C)

0 20 40 60 80 100 120 140 160 0 10 20 30 40 50 60 70 80

(0 (€9

Fig. 6. Regression lines for flow measured using ultrasound versus the timed collection method. Angle = (a) 34", (b) 38'. (c), 60°, and (d) 69'. Regression lines for the open flow system, Angle = (e) O', (f) 37'. (8) Regression line for the pulsatile flow system. Angle = 51". (a)-(f) x axis: Flow rate (color Doppler) (mVs); y axis: Steady state flow rate (timed collection method) (mVs). (g) x axis: Pulsatil flow rate (timed collection method) (mVs).

The result of analysis using Model 2) can be seen in Table 11. It is seen that a slope equal to one can be accepted for all lines except the angle 38" and 60". The reason there are now two lines, where we cannot accept a slope equal to one is that the offset is now fixed (close to zero). An F-test on the offset was statistically insignificant ( p = 0.28), thus we can accept the hypothesis that the offset is equal to zero in this model.

The result of analysis using Model 3) can be seen in Table 111. An F-test gives significance ( p = 0.0001)-thus we have

to reject that the offset is the same if we simultaneously accept the same slope for the six lines.

The global model) gave the following equation:

y = LY + 0.982 + E

with the mean of a equal to 1.36 ml and a standard deviation of 4.96 ml/s, t has a standard deviation of 4.88 mVs. Thus we can find twice the total standard deviation of the color Doppler estimates compared to our reference as twice the square root

POULSEN AND KIM MEASUREMENT OF VOLUMETRIC FLOW WITH NO ANGLE CORRECTION 591

Coefficient Estimated value

SEE

PI Pa P3 P4 PS P6 0.970 0.885 1.069 0.990 0.953 0.961 0.022 0.024 0.020 0.020 0.025 0.022

TABLE III Coefficient

Estimated value SEE

a1 a2 a3 a4 a5 0 6 1.209 10.551 2.464 -4.960 -1.791 1.325 1.885 2.184 2.191 2.189 2.183 2.187

where SEE is the standard error estimate of the value.

(Reference - color Doppler) flow $

to appear on the tissue images then turned just one step down. The angle-independent flow measurement method was originally developed for use with the color Doppler method. Since this modality is semiquantitative and requires calibration (in vitro done by adjusting the ultrasonic receiver gain until correspondence with Fourier processed data in a single point is obtained), it should be interesting to use the Fourier transform method directly on all sample points on the chosen sphere. This should yield a more consistent estimator although the frame rate will suffer from this approach, since the color Doppler method typically uses 15-40 ultrasonic pulses for each velocity estimate, while the Fourier transform pulsed Doppler based method typically uses 128-256 pulses for each velocity estimate. Thus this approach should be somewhat slower but also yield superior average flow values. By using phased array multi-pulse processing the frame rate can be improved a lot as the latest scanner generation with very high frame rates has shown, so this may not be a serious obstacle.

The velocity used in each sample volume can be found from the modal frequency (i.e.. the Doppler shift frequency corresponding to the bin at which the energy content is the largest) obtained from the power specb” found in Fourier processing. Alternately one could use the first moment function and thus get this expression for the mean frequency shift

-20 I I I I I 0 50 100 150 200

Reference flow $ Fig. 7. Difference plot of all points from laminary flow. The 95% confidence intervals is marked as f 2 0.

of the sum of the variances, we find 2u = 13.9 mVs. In Fig. 7 a difference plot is been shown including all points from the laminary study, the mean difference is -0.46 ml/s and twice the standard deviation of the differences is 13.8 ml/s (this is slightly different from the global model because the slope is implicit equal to one).

V. DISCUSSION The in vitro study has validated the basic theory of this

new method and shown an agreement between the timed collection method and the spherical scanning method. Even though a relatively coarse velocity estimator was used (namely color Doppler), the linear regression analysis showed good results with slopes near one and a relatively small offset. The results were obtained by checking the color Doppler modality in a single point to assure that the sensitive gain- control was set correctly. Future ultrasonic equipment could either do this automatically or use some other and more stable modality to obtain the actual velocity estimates. We did not have the same problems with the gain control in vivo [12], in this case it was turned up until noise began

where f is the frequency, P ( f ) is the power density, and F, is the sample frequency (usually equal to the pulse repetition frequency). By using this formula for calculating the Doppler frequency shift and thereby obtain the individual velocities in the basic flow summation formula [(12)] we get an expression for the flow.

It is possible to use this expression to advantage in future- generation ultrasonic scanners. It is not clear whether the modal frequency approach or first moment method will per- form best. While the first moment method will offer some statistical advantages, the modal frequency approach is more insensitive to noise since we only focus on the part of the spectrum where the energy is largest (however, for very low SNR’ s this will eventually fail completely). Autoregressive processing of the Doppler signal is also possible, the main advantage as compared to Fourier processing is the possibility to combine the angle-independent method with continous flow monitoring and not only just obtain mean flow estimates, since the autoregressive methods typically uses 30-50 pulses for each velocity estimate. These methods do, however, rely on assumptions about the Doppler shift signal and perform best on modest to high SNRs.

The measured flow rate only showed small variation for different angles of insonation. This shows that even though the actual ultrasonic equipment suffers from basic limitations such as finite sidelobes, wall motion filtering effects, finite size of the sample volume and some errors in the velocity estimates (so that the restrictions imposed on the theory are violated to

598 IEEE TRANSACTIONS ON BIOMEDI

scheme and thus only beiween the two were necce consistent results (data and a standard d

m one second in An in vivo measure

ments are needed before size sample volume se the volumetric flow

estimates and a shorte have infinite small sample

our in vitro on the B-mode

tent of this phenomenon was dependent on the

It has been shown is possible even thou Provides I-* velocity

sidelobes of the ultrasonic terally no attenuation in water

on the assumption of an i sample volumes, a full

measurement using st

It requires heavy processing but this could be a minor o

Our own in vivo vali

complicated apparatus, th the advancement of

comparison to thermodilution

estimate and the wall-motio

POULSEN AND KIM. MEASUREMENT OF VOLUMETRIC PLOW WITH NO ANGLE CORRECTION 599

and by symmetry

We find the surface area of a sphere bounded by a horizontal plane, which is placed parallel to the x-v plane, so that the maximum value of the x- or y-coordinate inside the surface of interest is p [see Fig. 2(c)]

By a transformation to polar coordinates we get

[4] L. N. Bohs and G. E. Trahey, “A novel method for angle independent ultrasonic imaging of blood flow and tissue motion,” IEEE Trans. Biomed. Eng., vol. 38, no. 3. pp. 280-286, 1991.

[5] U. Moser, M. Anlier, and P. M. Schumacher, “Ultrasonic synthetic aperture imaging used to measure 2-D velocity fields in real time,’’ in Proc. IEEE Int. Symp. Circuits, Syst., 1991, pp. 738-741.

[6] U. Moser, A. Vieli, P. M. Schumacher, P. Pinter, and M. Anlier, “Ein Doppler-ultraschall-Gerat zur Bestimmung des blut-volumenflusses,” Ultraschall in der Medizin, vol. 13, no. 2, pp. 71-79, 1992.

[7] U. Moser, P. M. Schumacher, and M. Anliker, “Benefits and limitations of the C-mode Doppler procedure,” Acoust. Imag., vol. 21, pp. 509-522, 1995.

[8] V. L. Newhouse, D. Cathignol, and J.-Y. Chapelon, “Study of vector flow estimation with transverse Doppler,” in Proc. ZEEE Ultrason. Symp., 1991, pp. 1259-1263.

[9] J. Hemp, “Theory of transit time ultrasonic flowmeters,” J. Sound, Vibration, vol. 84, no. 1, pp, 133-147, 1982.

[lo] W. G. R. Gibson, R. C. S. Cobbold, and K. W. Johnston, “Principles and design feasibility of a Doppler ultrasound intravascular volumetric flowmeter,” IEEE Trans. Biomed. Eng., vol. 41, no. 9, pp. 898-908, Sept. 1994.

[ l l ] T. M. Apostol, Mathematical Analysis. Reading, MA: Addison- Wesley, 1958, pp. 267, theorem 10.24.

[12] W. Y. Kim, J. K. Poulsen, K. Terp, and N.-H. Staalsen, “A new method for measurement of volumetric flow: In vivo validation,” J. Am. Coll. Cardiol., pp. 182-192, Jan. 1996.

[13] P. G. Hoel, S. C. Port, and C. J. Stone, Introduction to Probability Theory. Boston, MA: Houghton Mifflin, 1971, ch. 4.6, theorem 8.

[14] N. E. Haites, D. H. R. Mowat, F. M. McLennan, and J. M. Rawles, “How far is the cardiac output?,” Lancet, no. 2, pp. 1025-1027, 1984.

[15] S. R. Underwood and D. N. Firmin, Magnetic Resonance of the Cardio- vascular System.

[16] G. H. Mostbeck, G. R. Caputo, and C. B. Higgins, “MR measurement of blood flow in the cardiovascular system,” Am. J. Radiol, vol. 159, pp. 453-461, 1992.

[17] J. M. Bland and D. G. Altman, “Statistical Methods for Assessing Agreement Between Two Methods of Clinical Measurement,” Lancet,

Oxford, U.K.: Blackwell Scientific, 1991.

v;. 1, pp. 307-310, 1986.

Reading, MA: Addison-Wesley, 1992, p. 1012. From this we now find an area bounded by angles On, 8,+1 and with a slice width of A$. Because

[IS] G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry.

pn = RsinO, and by symmetry [Fig. 2(a), (c)]

AA = &+I - A,_

= A ~ R I 4 ~ 2 - ~2 sin2 en - J R ~ - ~2 sin2 en+ll

= A+R~I~=- JXl = A ~ R ~ 1 COS en - COS e,+, 1. (22)

The absolute sign is needed for general values of On, 8,+1 to make the area positive, the last simplification takes care of problems that occur when the angles On and O,+l are placed on opposite sides of 5.

Jens Kristian Ponlsen was bom in Thisted, Den- mark, on January 30, 1966. He received the MSc. degree in electrical engineering from The Technical University of Denmark in 1992. He is currently working on a Ph.D. degree at Aarhus University dealing with high resolution image analysis for tissue characterization using scanning acoustic mi- croscopy.

Since 1992 he has been working on realtime anal- ysis of Doppler signals, blood-flow measurements, and signal processing technology.

ACKNOWLEDGMENT

The authors thank N. T. Andersen, C . Djurhuus, J. C . Djurhuus, A. Takahashi, and M. Vaeth for advice and technical assistance.

REFERENCES

[I] B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals ofFluid Mechanics. New York Wiley, 1990, p. 225.

[21 D. Dotti, R. Lombardi, and P. Piazzi, “Vectorial measurement of blood velocity by means of ultrasound,” Med. Biol. Eng. Comp., vol. 30, pp. 219-225, 1992.

[3] S. Matsumoto, A. Ohya, and M. Nakajima, “Two-dimensional velocity distribution measurement by ultrasonic two-receiver interferometry,” in Proc. ZEEE Ultrason. Symp., 1991, pp. 1255-1258.

Won Yong Kim was born April 11, 1965 in Seoul, Korea. He received the MD degree in February, 1992 from the University of Aarhus, Denmark. From February, 1992 to October, 1995 he studied for the Ph.D. degree at the Department of Cardio- thoracic and Vascular Surgery T, Aarhus University Hospital, Skejby Hospital. The Ph.D. thesis is enti- tled “Blood Flow Measurements by Pulsed Doppler Ultrasound and Velocity Encoding Magnetic Reso- nance Imaging, with special Reference to the Mitral

His main research areas involve investigation of hemodynamics in the cardiovascular system with special emphasis on evaluation of cardiac per- formance using noninvasive methods. He is currently working as a medical doctor to fulfil his internship.

Valve”.