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November 1996

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>Hey Gang,
>I need some opinions, (ya Terry, I know I'm ugly).
>How important are flow volume measurements in the peripheral vascular world?
(snip)
we believe volume flow estimates add useful info in infrainguinal vein
graft surveillance (see our paper in video journal of color flow imaging
1993;3:81-93.), possibly helpful in axillofemoral, but no help in
prosthetic aortofemoral or femoropopliteal grafts
we occasionally find volume flow etimates of value in dialysis graft
problems and to estimate volume flow in (often very high volume) native
brachial artery to cephalic vein fistulae, both before and after banding
them
nearly all systems etimate time averaged velocity (TAV), which is
presumably weighted by relative contributions in the velocity (frequency)
domain, as is done in our acuson and atl machines, and multiply this by
estimated cross sectional area- h-p systems in the past estimated time
average peak velocity, presumably due to their echocardiography heritage,
our current h-p system gives time averaged mean (again, i think weighted in
the velocity domain)
philips approach is different, probably more accurate (see paper from u of
vermont about a year ago comparing several commercial systems)
the following is the text for a talk i gave at a meeting last spring:

INTRODUCTION
Noninvasive estimation of volume flow of blood has become possible in both
peripheral vascular and cardiac testing with the development of duplex
devices.  These devices allow real-time measurement of blood vessel
dimension and Doppler estimated velocity of blood flow.  Ideally, the
product of the cross sectional area of the vessel and instantaneous blood
flow velocity (integrated over the vessel cross section and over velocity
itself) should provide the instantaneous value of volume flow.  Integration
of this product over time should provide average volume flow.
Unfortunately, basic physics and current engineering realities prevent the
practical achievement of a system that provides these exact quantities.
Nevertheless, some reasonable assumptions can be made that allow some
estimation of volume flow.  I will attempt in this chapter to review some
of the basic physics, limitations, and practice of volume flow estimation
on current commercially available duplex systems.  The following material
has in large part been presented previously.1

Measurement of volume flow in native arteries and bypass grafts has been a
subject of great interest in Vascular and Cardiac Surgery for many years.
Although highly accurate in vitro measurements can be made using an open
circuit and timed collections of shed blood, a closed circuit is necessary
to provide even a rough approximation of normal arterial hemodynamics.
Although both in-line and coaxial electromagnetic flow probes have been
available for many years, they are somewhat cumbersome, require calibration
prior to each use if accurate measurements are to be obtained, are not
practical for use in awake humans, and are clearly of no use in the
surveillance of lower extremity bypass grafts.  The more recently available
implantable ultrasonic flow probes are more easily used, but they are also
impractical for use in awake humans and in graft surveillance programs.
Although measurement of Doppler shift has been possible since the first use
of the ultrasonic techniques to detect blood flow more than thirty years
ago and the Doppler equation is relatively straightforward, accurate
measurement of the Doppler angle and estimation of flow velocity in vivo
was not practical prior to the development of the duplex scanner some
twenty years ago.2,3  Even when estimation of velocity was possible,
further refinements of basic physiologic knowledge, hardware, and software
were necessary before reproducible and reasonably accurate estimates of
volume flow were possible.  Once such volume flow estimates were available
and reasonably automated on grey scale scanners, a number of investigators
explored the possible uses of these flow estimates both in native vessels
and in graft surveillance protocols.4-7
This article is not meant to be a comprehensive treatise on physics and
engineering of duplex estimation of velocity and flow.  However, there are
many assumptions and potential sources of error in the ultrasonic
estimation of velocity and flow and a basic understanding of these
potential sources of error gives a more realistic perspective on the
accuracy and reliability of velocity and volume flow estimation.
The noninvasive measurement of velocity and flow has tremendous potential
to assist in understanding blood flow in health and disease and in clinical
practice as well as in the basic research laboratory.  Although the role of
duplex volume flow estimation remains controversial and some equipment
manufacturers seem to be backing away from volume flow estimation, such
measurements are considered an important tool in many modern vascular
laboratories, particularly for routine infrainguinal bypass graft
surveillance.
DUPLEX ESTIMATION OF VELOCITY AND VOLUME FLOW
Basic mathematical considerations
Although the detected Doppler shift is directly proportional to blood flow
velocity, the hand held Doppler has been used primarily as a qualitative
tool with analog recordings of velocities serving to allow inspection of
wave form shape and measurement of rapidity of early systolic upstroke.
However, as described in many basic textbooks of noninvasive vascular
testing, it can be shown that flow velocity can be estimated using the
equation:
1)      v       =       Df*c
                        2*f0*cos(q)
where v is velocity, Df is the estimated Doppler shift in frequency of
sound reflected from the moving blood, c is the speed of sound in the
medium (approximately 1.54 * 105 cm/sec in human tissue other than bone),
f0 is the carrier frequency of the Doppler, and q is the angle between the
Doppler "beam" and the direction of blood flow.8  Peak systolic velocity
(PSV) is frequently of great interest to clinicians, particularly in the
surveillance of lower extremity bypass grafts.7,9-14  Modern duplex
scanners provide each of the necessary parameters to estimate the
instantaneous value of velocity, vinst and this use of the Doppler shift is
the most common approach to velocity estimation and volume flow estimation.
Furthermore, it is also easily shown that instantaneous volume flow
(Qinst) can be estimated as:
2)      Qinst   =       vinst*A
where A is cross sectional area.  If we assume that blood vessels or bypass
grafts are tubular and circular in cross section, then area is given by the
well known formula A = pr2 where p ‰ 3.1416 and r is the radius of the
(round) vessel.  Duplex scanners can also be used to estimate r, although
in practice, diameter (d = 2*r) is measured.
Unfortunately, even at this still theoretical stage, we have already
encountered several potential sources of error in duplex measurement of
flow.  Even in the absence of turbulence, red cells do not all move at the
same velocity and velocity varies significantly across the cross section of
the vessel.  There are several common approaches to dealing with this
problem.15,16  The first is based on the knowledge that under laminar flow
conditions, theoretical and empirical data suggest that average velocity
over the vessel cross section is a predictable fraction of the velocity in
the center of the vessel, where velocity is expected to be highest (Figure
1).  This fraction appears to be about 0.5.  One simply measures the mean
velocity (weighted average across the velocity spectrum using Fourier
analysis) with a small sample volume placed in the center of the vessel
(Figure 1a) and assumes that this velocity is about twice the average
velocity across the cross section of the vessel.  A second, and more
commonly used approach, is to use a large pulsed Doppler sample volume
which encompasses the entire vessel diameter (Figure 1b).  Velocity is then
averaged (integrated) over both frequency and the diameter of the vessel
assuming that this approximates the average velocity over the cross section
of the vessel.  This approach depends more heavily on the quality of the
spectrum analyzer that is part of all modern duplex scanners but probably
gains from the fact the Doppler "beam" and the sample gate have a finite
width, which may approach 2 mm (at -3dB compared to center intensity),
roughly half the diameter of the typical infrainguinal vein bypass graft,
and the contribution of velocities lateral to the center line of the
Doppler beam contribute proportionately less than those along the beam.16
Phased array or mechanical probe systems that use the same transducer
crystal for imaging and Doppler may have narrow "beams" and may tend to
overestimate mean velocity.  Systems with separate imaging and Doppler
transducer crystals may actually be designed with very narrow image "beams"
to enhance spatial resolution and wider Doppler "beams" to allow "even
insonation" of the entire lumen.  Transit time broadening in pulsed Doppler
systems and tissue attenuation of ultrasound signals also cause imprecision
in velocity estimation, although they tend to be offsetting in practice.16
Finally, the assumption of laminar flow and the implication that all flow
vectors are parallel to the vessel axis is a reasonable approximation in
long straight segments, but this assumption may not be valid in areas where
there are significant changes in direction, diameter, or vessel branch
points.  For example, it is well known that flow patterns in the carotid
bifurcation are extremely complex and even estimating center stream PSV may
be very unreliable in this area since the true relationship between the
Doppler beam and flow vectors is not known with currently available
instruments.  Thus, it may be argued that velocity should never be
estimated from Doppler data; rather Doppler shift should be reported.17
Furthermore, while instantaneous PSV is of great interest, instantaneous
flow is generally of less interest than average flow.  Flow varies
significantly in magnitude and often in direction over the cardiac cycle.
Thus, flow is more often calculated using the velocity averaged over the
velocity spectrum, vessel cross section, and time, yielding the quantity
most often called time averaged velocity or TAV, a quantity available on
most modern duplex scanners.  Substituting d÷2 for r in the area
calculation, volume flow is estimated:
3)      Q       =       TAV*A   =       TAV*pd2
                                        4
All measurements (Df, q, TAV, and d) are made with operator direction by
the duplex scanner and Q is usually automatically calculated by the
scanner's computer hardware and software.16-19  TAV is calculated on some
currently available scanners as peak velocity averaged over time rather
than the velocity averaged over both frequency and time.  A correspondingly
higher estimate of volume flow results from this approach.
Although the basic approach outlined above is the starting point for duplex
estimation of volume flow, many theoretical and practical limitations of
this most simple approach threaten to confound accurate estimation of flow.
Thus, many investigators and equipment manufacturers have experimented
with different approaches to velocity and volume flow estimation.
Individual manufacturers are often reluctant to discuss specifics of
hardware and software for obvious competitive reasons.  As a result of
this, each scanner model should be tested extensively both in vitro and in
vivo over a wide range of flow conditions to calibrate the estimates of
velocity and volume flow.  Although some in vitro and in vivo calibrations
have been published 5,20,21 we will be more confident about this technique
when these results are confirmed in other laboratories.
Basic technical considerations
The basic hardware in all modern duplex instruments is adequate to perform
flow estimation.  Software is available for nearly all scanners, although
as noted above, some manufacturers seem to be moving away from flow
estimation.  At least one manufacturer, who supported volume flow
estimation on its previous model, does not offer volume flow estimation
software on their current model.  Assuming that one has access to one of
the other scanner models, the software is available, although some
manufacturers remain curiously resistant to installing it.
In practice, one identifies a relatively straight segment of bypass graft
or native vessel that has a uniform diameter, appears round on cross
section, appears on a brief pulsed Doppler or color flow interrogation to
have relatively uniform velocities over the length of the segment imaged,
and is remote from anastomoses or bifurcations (including areas of
suspected retained arteriovenous fistulae of in-situ saphenous vein
grafts).  Typically, a center stream PSV with a minimized sample volume is
measured first.  Next, a diameter is measured.  When imaging the vessel or
graft of interest in longitudinal orientation, one attempts to measure the
diameter through the center of the vessel by moving the probe slightly from
side to side and holding the probe at the point of maximum apparent
diameter.  Measurements on either side of the center will underestimate
diameter, the effect of which is squared in the area equation and flow
estimate.  One should take care not to compress the graft during diameter
measurement or collection of spectra since this will lead to both
underestimation of diameter and perturbation of laminar flow.  This is of
particular concern when scanning in-situ saphenous vein or other
subcutaneous grafts.  Some software packages allow measurement from the
transverse orientation to be saved and used later after TAV measurement
(presumably from the same segment of artery or graft) to reduce the chance
of underestimating diameter.  Next, on most scanners, the pulsed Doppler
sample volume is placed over the vessel and opened to encompass slightly
more than the entire vessel diameter.  The Doppler angle is adjusted as
appropriate but one should attempt to achieve proper orientation with an
optimal Doppler angle q of 60° to reduce the error in velocity estimation.
Doppler angles less than but near 60° probably do not produce significant
errors in velocity estimation.  As Doppler angle increases above 60°,
precision of the velocity estimate deteriorates rapidly and Doppler angles
greater than 60° should be avoided.16,18,22  A Doppler shift (velocity)
spectrum is then collected over as many cardiac cycles as possible.  A
minimum of 4 is ideal, but as few as 2 are probably acceptable with a good
quality spectrum and when the patient has a regular heart rhythm.  The
spectrum is then marked from a characteristic point (most commonly end
diastole) to the same point of a subsequent cardiac cycle.  The spectrum
analyzer is then used to calculate TAV and the derived values are used to
calculate Q from Equation 3 above.
In an effort to improve accuracy, some equipment and software use more
complicated methods to calculate cross sectional area.  For instance, some
models ask the operator to approximate the vessel circumference with an
oval in cross section and the area of this ellipse is used for A in
equation 3.  Unfortunately, this method requires an image plane that is a
true perpendicular to the axis of the vessel, probably more difficult to
achieve than a true diameter in longitudinal orientation.  Edge detector
technology can be applied to measure instantaneous (dynamic) diameter
throughout the cardiac cycle.19  This method requires instantaneous volume
flow (instead of velocity) to be averaged over time.
Sources of error
Despite the seemingly simple nature of diameter measurement, this is a
source of considerable error in estimation of volume flow.  This is due in
part to the fact that any error in measurement is magnified since radius
(diameter) is squared in the area calculation.4  Inaccurate diameter
measurement was probably the most important source of error with previous
generations of scanners.  For instance, many of our PSV and volume flow
estimates have been performed with an older grey scale scanner.  The B-mode
image of this generation of machine, although improved with each successive
hardware and software upgrade, is probably not equal to most current
generation color scanners.  Thus, there was always a degree of uncertainty
with respect to placement of calipers to measure diameter and to the
assumption that the diameter measured in longitudinal orientation was the
actual diameter and not a (shorter) chord.
In an effort to assess the magnitude of diameter measurement errors, we
reviewed the results of duplex examinations of ePTFE axillofemoral and
femorofemoral grafts.  Although it is difficult to know the true diameter
of vein grafts, ePTFE graft diameter is determined by the size of the
mandril over which the graft is extruded and ePTFE grafts are quite
inelastic so that diameter change is negligible over the cardiac cycle.
Assuming no enlargement after implantation and no significant accumulation
of thrombus or other material within the graft, the measured diameter in
vivo should be nearly equal to the manufacturer's nominal diameter.  Our
own observation with axillofemoral and femorofemoral graft limbs of known
internal diameter is that measured diameter varies considerably from known
diameter (Figure 2).  Furthermore, there appears to be as much likelihood
of overestimating as underestimating diameter, making accumulation of
thrombus an unlikely cause of this error.  This appears to be due at least
in part to ambiguity in deciding what part of the image represents the
luminal surface of the graft.  Therefore, magnification and finer caliper
"vernier" precision would offer only a partial solution to the problem.
Furthermore, in contrast to ePTFE grafts, vein graft and even Dacron graft
diameters may change substantially during the cardiac cycle, thus further
confounding accurate diameter measurement.  Measurement of diameter from
reduced size B-mode images displayed with concurrent Doppler spectra is
possible on some scanners.  We believe this is likely to result in
decreased precision of diameter measurement and should be avoided.
Inspection of B-mode images and the effect of slight movements of the
electronic calipers are quite revealing in this regard.  The rather
surprising inability to measure diameter accurately with the duplex scanner
would seem to correlate with the findings of Grigg et al, who noted good
interobserver agreement when measuring PSV but less encouraging
interobserver agreement when estimating diameter and volume flow.4
As image resolution improves with newer equipment, diameter measurement is
likely to become much more secure.  Although the assumption that vessels or
grafts are tubular with a uniform diameter may also be a source of error,
there is a paucity of data to support the purported improved results with
oval approximations, manual tracing, or automatic tracing of vessel/graft
area in cross section.  Advances in ultrasonic tissue characterization and
edge detection may improve the automated measurement of vessel area.
A common technical error is to place caliper marks along the Doppler beam
indicator line (M-line) instead of placing them directly opposite each
other along a line perpendicular to the long axis of the vessel.  This has
the effect of overestimating diameter and, therefore, significantly
overestimating area and volume flow.  The sensitivity of this method to
errors in diameter measurement cannot be overemphasized and the operator is
cautioned to take great care when measuring diameter.
Several of the potential sources of error in Doppler estimation of velocity
are listed above.  Some of these deserve special emphasis when applied to
clinical use.  The Doppler angle must be carefully measured by proper
alignment of the Doppler angle indicator and the B-mode image of the vessel
examined.  The angle should be as near to 60° as possible, although as
noted above, even the assumption that flow vectors are parallel to the
vessel or graft axis may not be correct.17  Doppler angles greater than 60°
are associated with rapidly increasing imprecision in velocity estimation
due to the behavior of the cosine function as q approaches 90°.16,22  It is
essential that TAV be estimated over as many complete cardiac cycles as
possible to reduce the risk of over- or underestimation.  As noted above,
use of at least 4 cardiac cycles is ideal but as few as 2 may be
satisfactory when the heart rhythm is regular.  A common error we have seen
as technologists begin to use volume flow estimation is the tendency to
simply mark both ends of the spectrum without regard for where the ends lie
with respect to the cardiac cycle.  We believe that end diastole is the
least ambiguous point in the cardiac cycle and we encourage our
technologists to mark as many cycles as are possible to reduce error.
Estimation of velocity is sensitive to cardiac dysrhythmiae, particularly
atrial fibrillation, and averaging over more cardiac cycles (time) reduces
the likely error produced by dysrhythmiae.  Some scanners place the B-mode
image next to the spectrum in "single screen" format.  The latter format
allows the spectrum from only a few cardiac cycles to be displayed on the
screen and full width spectra may require the use of "dual screen" format
on these machines.  Alternatively, the spectral sweep rate may be reduced
on machines that use this format so that enough spectral data may be
obtained, although this results in some loss of spectral visual detail.
Some scanners display the image above and the spectrum below and,
therefore, allow full screen width spectra on the same screen as the image.
Finally, many technologists neglect to open the Doppler sample gate to
encompass a region larger than the vessel diameter.  Since center stream
velocity is greater than the average velocity over the entire cross
section, this has the effect of overestimating TAV and, therefore, flow.
This is easily forgotten since a graft surveillance duplex exam may include
both a center stream PSV measurement (with the sample volume minimized) and
a volume flow estimate.
Although velocity estimation is much more automated than diameter
measurement on modern duplex scanners, this places the operator more at the
mercy of the equipment.  There are many theoretical and practical sources
of error in the use of Doppler ultrasound to measure blood flow velocity
which may or may not be addressed by hardware and software engineers for
the various scanners.  Some investigators have pointed out the limitations
of the above approach to velocity estimation and have proposed potentially
more accurate methods.23,24  A unique approach to velocity measurement
called color velocity imaging (CVI) has recently been described.  This
method involves measurement of change in sound travel time rather than
Doppler shift and is theoretically appealing, although a critical aspect of
this methodology, recognition of a unique group of red cells by
"cross-correlation" 24, remains intellectually challenging to the author.
We have only minimal experience with this technique.  Nevertheless, at
least two reports have described testing of both the above described
conventional as well as the CVI (Time Domain) approach and both reports
described excellent correlation between CVI method flow estimates and in
vivo or in vitro standards. 20,21  At the time of this writing, CVI is to
my knowledge available on only one brand of commercially available duplex
scanner.  We look forward to further work in this area.
Variation in serial estimates of velocity and flow
Despite the various assumptions that are involved in estimating velocity
averaged over cross section or TAV, we noted surprisingly similar estimates
of PSV when ePTFE axillofemoral and femorofemoral grafts were compared over
several months of follow-up (Figure 3).  The relationship between the first
and second volume flow estimated in these grafts was statistically quite
significant (p < 0.001, Figure 4) but not as strong as that for PSV, almost
certainly due at least in part to the error in diameter measurement noted
above.
The relationships between PSV measured at the first compared to the second
(Figure 5a) and the second compared to the third (Figure 5b) postoperative
graft surveillance study following autogenous vein bypass to infrapopliteal
targets for limb threatening ischemia were not as striking as that for the
ePTFE grafts, but were nevertheless quite significant (p < 0.001).
Similarly, although there is little published information regarding the
accuracy of volume flow estimated on the various systems and most duplex
scanner manufacturers readily admit that the absolute measurements obtained
are not very accurate, we and others have noted that volume flow estimates
taken from similar regions of a patient's graft with the same scanner are
reasonably stable over time.6  However, the correlation of serial volume
flow estimates is not as close as that for PSV (Figure 6).
Different scanner models may yield volume flow estimates that differ by a
factor of two or more because of differences in spectrum analyzer design
and algorithms for computing velocity.  All the data contributing to
Figures 2-6 were obtained with grey scale scanners over a seven year
period.  Our subsequent observations with several models of color scanners
have suggested significant variations between estimated volume flow and
even PSV when studying the same graft.  However, PSV measurements appear
reproducible and volume flow estimates appear to be relatively reproducible
using the same machine over time.  The practical implication of this is
that serial exams of an individual graft should be performed with the same
scanner.
CONCLUSIONS
The promise of noninvasive reliable and accurate estimation of volume flow
remains incompletely fulfilled.  Some limitations are probably not soluble,
but others may yield with further advances in engineering.  Continued
improvements in hardware and software and new approaches to velocity
measurement may significantly improve our ability to estimate volume flow
in the clinical and research laboratories.
 References
1.      Schneider JR, Size GP, Laubach M, et al. Estimation of velocity and
volume flow in native arteries and bypass grafts using the duplex scanner:
current status and limitations. Video J Color Flow Imaging 1993;3:81-93.
2.      Fish P, Walters D. Beam/vessel angle problem in Doppler flow
measurement. In Taylor DEM, Whamond D, eds. Noninvasive clinical
measurement. Turnbridge-Wells UK, Pitman Publishing, pp. 105-13, 1977.
3.      Strandness DE. Duplex scanning: Past, present, and future. Semin
Vasc Surg 1988;1:2-8.
4.      Grigg MJ, Wolfe JHN, Tovar A, Nicolaides AN. The reliability of
duplex derived haemodynamic measurements in the assessment of femoro-distal
grafts. Eur J Vasc Surg 1988;2:171-81.
5.      Field JP, Musson AM, Zwolak RM, McDaniel MD, Walsh DB, Cronenwett
JL. Duplex arterial flow measurements in normal lower extremities. J Vasc
Technol 1989;13:13-9.
6.      Kupinski AM, Stone MP, DePalma H, et al. Is reactive hyperemia a
reliable indicator of impending bypass failure? J Vasc Technol
1990;14:163-165.
7.      Zwolak RM, Cronenwett JL, McDaniel MD, et al. Vascular laboratory
detection of failing lower extremity bypass grafts. Presented at the 45th
Annual Meeting of the Society for Vascular Surgery, June 4, 1991, Boston,
Massachusetts.
8.      Zagzebski JA. Physics and instrumentation in Doppler and B-mode
ultrasonography. In Zwiebel WJ (ed): Introduction to vascular
ultrasonography, 3rd ed. Philadelphia, W.B. Saunders, 1992, pp. 19-43.
9.      Bandyk DF, Cato RF, Towne JB. A low flow velocity predicts failure
of femoropopliteal and femorotibial bypass grafts. Surgery 1985;98:799-809.
9.      Bandyk DF, Kaebnick HW, Stewart GW, Towne JB. Durability of the in
situ saphenous vein arterial bypass. A comparison of primary and secondary
patency. J Vasc Surg 1987;5:256-68.
11.     Bandyk DF, Schmitt DD, Seabrook GR, Adams MB, Towne JB. Monitoring
functional patency of in situ saphenous vein bypasses: The impact of a
surveillance protocol and elective revision. J Vasc Surg 1989;9:286-96.
12.     Mills JL, Harris EJ, Taylor LM Jr, Beckett WC, Porter JM. The
importance of routine surveillance of distal bypass grafts with duplex
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13.     Taylor PR, Wolfe JHN, Tyrrell MR, Mansfield AO, Nicolaides AN,
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Surg 1990;77:1125-1128.
14.     Bandyk DF. Monitoring after distal reconstruction. In Bernstein EF
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15.     Burns PN. Principles of deep Doppler ultrasonography. In Bernstein
EF, ed: Vascular Diagnosis. St. Louis, CV Mosby, pp. 258-60, 1993.
16.     Burns PN, Jaffe CC. Quantitative flow measurement with Doppler
ultrasound: Techniques, accuracy, and limitations. Radiol Clin North Am
1985;23:641-57.
17.     Phillips DJ, Beach KW, Primozich J, Strandness DE Jr. Should
results of ultrasound Doppler studies be reported in units of frequency or
velocity. Ultrasound Med Biol 1989;15:205-12.
18.     Taylor KJW, Holland S. Doppler US: Part I. Basic principles,
instrumentation, and pitfalls. Radiology 1990;174:297-307.
19.     Zwiebel WJ. Spectrum analysis in Doppler vascular diagnosis. In
Zwiebel WJ (ed): Introduction to vascular ultrasonography, 3rd ed.
Philadelphia, W.B. Saunders, 1992, pp. 45-65.
20.     Maulik D, Kadado T, Downing G, Phillips C. In vitro and in vivo
validation of time domain velocity and flow measurement technique. J
Ultrasound Med 1995;14:939-47.
21.     Winkler AJ, Wu J, Case T, Ricci MA. An experimental study of the
accuracy of volume flow measurements using commercial ultrasound systems. J
Vasc Technol 1995;19:175-80.
22.     Rizzo RJ, Sandager G, Astleford P, et al. Mesenteric flow velocity
variations as a function of angle of insonation. J Vasc Surg
1990;11:688-94.
23.     Kitney RI. Spectral estimation, image processing, and
three-dimensional solid modeling in the study of arterial disease. In
Bernstein EF (ed). Recent advances in noninvasive diagnostic techniques in
vascular disease. St. Louis, Mosby, 1990, pp. 325-41.
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Imag 1986;8:73-85.
(figures not included because they're not text)

have a ball

joe


Joseph R Schneider    Web Page http://pubweb.acns.nwu.edu/~jschneid/JRS.HTML
Politically correct means always having to say you're sorry.



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