MAGNETIC RESONANCE IN MEDICINE
27,214-225 ( 1992)
Nuclear Magnetic Resonance Velocity Spectra of Pulsatile Flow in a Rigid Tube RICHARD E. WENDT III*’t’$ AND WAI-FANW0NC”t * Department of Radiology and t Magnelic Rcsonunce Center, Buylor College of Medicine, Houston, Texas 77030, and $Department of Radiology, The Methodist Hospital,Houston, Texas 77030 Received May 22, 1991; revised December 19, 1991; accepted January 7, 1992 Velocity spectra can be derived from velocity-encoded nuclear magnetic resonance (NMR) images. Velocity spectra are histograms showing the amounts of fluid flowing at different velocitiesin the sensitive volume ofthe measurement. Velocity spectra may prove to be useful in characterizing the flow of blood in small vessels, for example, in detecting the presence of stenoses and in evaluating their severity. NMR velocity spectra acquired in vivo are sufficientlycomplicated that a model system was designed and tested to investigate the velocity spectra of pulsatile flow. This study measured the NMR velocity spectra of pulsatile flow in a rigid tube and compared them to velocity spectra derived from Doppler ultrasound measurements and to velocity spectra inferred from a theoretical model driven by the measured pressure difference function. The experimental results from each technique agree. 0 1992 Academic Press, Inc. INTRODUCTION
Velocity-resolved nuclear magnetic resonance ( N M R ) imaging may be used to measure blood flow noninvasively. NMR imaging is not significantly affected by bones, by airspaces, or by depth-dependent attenuation in soft tissues and thus may be the only noninvasive means for evaluating the flow of blood in small, deeply buried vessels in the body. The NMR images of many small, but important vessels do not have adequate spatial resolution to construct detailed maps of the flow field across the lumen of the vessel. The vessel is represented by just a few pixels and the spatial dependence of velocity within the vessel lumen is hard to assess. The authors have investigated the use of velocity spectra ( I , 2) as a method for presenting the essential information in the velocity-resolved images in a manner that does not depend strongly on the spatial resolution of the original images. NMR velocity spectra of pulsatile flow in vivo are more complicated than those of steady flow ( 3 ) .The study reported here was undertaken to develop and test a physical model system for the evaluation of pulsatile flow by NMR velocity imaging. It tested the hypothesis that the NMR velocity spectra of pulsatile flow in a rigid tube are accurate. The results suggest that they are. The measurement of velocity spectra by NMR has a long history. The terms “velocity distribution” and “probability distribution” have been used by most investigators. They are used in the sense of a statistical distribution to mean the relative or absolute frequency of flow at particular velocities. Zhernovoi and Latyshev measured the velocity distribution of turbulent flow ( 4 ) . Grover and Singer ( 5 ) described a measurement 0740-3194192 $5.00 Copyright 0 1992 by Academic Press, h e . All rights of reproduction in any form reserved.
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FIG. I . Schematic diagram, not to scale, of the essential elements of the phantom setup: ( A ) fluid reservoir, (B) peristaltic pump, (C) damping element, (D) check valve, ( E ) respirator pump, (F) pulsation element, ( G ) tube in which flow was imaged (the pressure ports and the Doppler ultrasound transducer are not shown), and ( H ) return path for the fluid.
of the velocity distribution in the human finger. De Gennes (6) pointed out that the Fourier transform of the spin-echo signal would give an instantaneous velocity distribution of turbulent flow. Hayward et al. ( 7) calculated and measured the spin echo for plug and laminar flow along a magnetic field gradient and noted that the Fourier transform would yield the velocity distribution. Garroway measured the velocity distribution of laminar flow (8).Fukuda and Hirai ( 9 )measured the velocity distribution as a function of Reynolds number for turbulent flow. Feinberg and Jakab studied perfusion using NMR measurements of the velocity distribution (10). The authors prefer the term “velocity spectrum” to avoid confusion with the spatial velocity distribution v ( x , y , z , t ) discussed in fluid mechanics and occasionally used in the spatial sense in the NMR flow imaging literature (11, 12). What is more, the term “velocity spectrum” comes naturally to discussions of continuous-waveDoppler ultrasound since the Fourier transform of the Doppler ultrasound signal is the velocity distribution in the sense of a histogram or frequency distribution (13, 14). The effects of pulsatile flow on the spatial velocity distribution or, more properly, the velocity field, have been studied (15, 16) and the effects on the velocity spectrum have been alluded to ( 17). The pulse sequence used in this study relies on a stepped, bipolar gradient pulse for phase encoding of velocity ( 18-22), gradient waveform compensation for constant velocity (12, 23-25), and a spoiled, gradient echo image acquisition (26). The shape of the velocity spectrum is determined by the relative amounts of fluid moving at the different velocities o f flow within the vessel. In steady, fully developed flow, the authors have shown that laminar flow gives a uniform velocity spectrum while turbulent flowgives a peaked velocity spectrum (2). Velocity spectra of pulsatile flow in vivo have different shapes depending on the phase of the cardiac cycle. The present investigation was undertaken to determine how the velocity spectra of pulsatile flow should look. The theory of pulsatile flow has been described by McDonald ( I S ) and by Rodkiewicz ( 1 6 ) . Their derivations for tubes with rigid walls and circular cross sections are summarized in the Appendix and form the basis of the theoretical model used
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here. The theoretical model suggests that early in the pulsatile cycle when the fluid is being accelerated, the velocity profile is blunted and therefore the velocity spectrum should be peaked as in steady, turbulent flow. Later, when the fluid is moving well, the viscous forces will produce a more rounded velocity profile and the velocity spectrum should be flatter as in steady, laminar flow. Still later there may be retrograde flow in parts of the lumen of the vessel. This causes the velocity spectrum to have a significant amplitude at both positive and negative velocities. The three-dimensional set of velocity-resolved images produced by the pulse sequence is difficult to interpret. Especially in clinical data sets, most of the images have just a few bright spots that represent blood flow at nonzero velocities. Condensing these data sets into velocity spectra appears to preserve significant information about the nature of the flow in a vessel. This reduction in the complexity of the interpretation of velocity-resolveddata is even more important for pulsatile flow in which the velocities vary during the pulsatile cycle (e.g., the cardiac cycle) and the data are four-dimensional. METHODS
To investigate the NMR velocity spectra of pulsatile flow, a phantom was built so that the pulsatile flow velocities through it could be measured by NMR imaging and by range-gated, pulsed Doppler ultrasound, and could be inferred from pressure difference measurements. Contemporaneous measurements were made by the three methods and the data were processed and compared.
The Phantom The phantom contained two acrylic tubes 125 cm long with inside radii of 0.159 cm. The tubes were connected in series so that fluid flowed down one tube and then back up the other. All measurements were made on the second tube to ensure more complete magnetization of the fluid. The phantom was filled with a NiQ-doped glycerol-water solution with a viscosity of 2.7 cP. A small amount of cornstarch was added to provide reflective particles for the Doppler ultrasound to sense. The path of the fluid in the experimental apparatus is shown in Fig. 1. The fluid was driven by a peristaltic pump (Masterflex, Model CK-07520-00, Cole-Parmer, Chicago) with a size 16 (4.8-480 ml/min) pump head. A damping element was placed immediately downstream of the pump to minimize the pulsations introduced by the peristalsis of the pump. The outlet of the damping element was directed through a check valve to prevent backflow toward the pump. A small average flow rate of 0.532 ml/s was used. A pulsation element patterned after that described by Evans et af. (27) was constructed by enclosing a 50-ml rubber bulb in a sealed 1-liter jar. A water connection was provided through the closure of the jar to the bulb and an air connection was provided to the interior of the jar. The jar was pressurized through the air connection by an animal respirator (Dual Phase Control Respirator Pump, Model 613, Harvard Apparatus, South Natick, MA) using a period of 1.88 s (about 33 “breaths” per minute), a 50% “inspiration” (i.e., duty cycle), and a stroke volume setting of 400 ml. This gave a Womersley number ( 1 5 )of 1.77 for the fluid and acrylic tubing used. The respirator pump triggered the NMR system at a particular point in
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its “respiratory” cycle. The interior of the rubber bulb was connected to the flow system downstream of the check valve by a Y-connector. To measure the pressure drop along a length of the tubing, two pressure ports were made by drilling two holes 15 cm apart in the side of the acrylic tubing so that the ends of plastic Luer connectors from butterfly sets would fit snugly. The bores of the Luer connectors were cleared with a 0.04-cm drill. After the Luer connectors were glued in place with methylene chloride and then epoxy, the base of each Luer connector was removed by passing a 0.32-cm drill down the bore of the tubing. This necessitated cutting the tubing near the site of each pressure port. Each face of each cut was polished square and smooth and the tubing was rejoined with methylene chloride after the pressure ports were completely installed. The joints in the tubing were reinforced with beads of epoxy or with a short, telescoping collar of acrylic tubing. Several test joints were examined after construction. There were slight disturbances of the wall, but no protrusions of material into the bore of the tube. The ports were connected by 6 1-cm lengths of pressure tubing to physiologic pressure transducers (Model 290C, Hewlett-Packard, Palo Alto, CA). The pressure signals were transmitted through the radiofrequency screening of the NMR imager room using an FM telemetry system (Model OMP 720lC, Nihon-Kohden (America), Irvine CA) operating at 472 MHz. The pressure signals, along with the trigger pulse and the phasic Doppler ultrasound velocity signal, were digitized (Model AD- 1000,Real Time Devices, State College, PA) and stored in a personal computer using a data acquisition program written by the authors. The pressure gradient was determined by dividing the measured pressure difference by the distance between the ports and assuming that the system was rigid between the two ports. The pressure signal was calibrated using a sphygmomanometer attached to the manifold at the inlet of the pressure transducers. The velocity of fluid flow in the tube was also measured by range-gated Doppler ultrasound. A 45” notch was filed in the side of the tubing as deep as possible without breaking through to the bore of the tube. A bare 20-MHz ultrasonic transducer crystal with fine-wire leads was glued to one face of the notch using epoxy. The notch was filled in with epoxy. The fine leads were soldered to the conductors of a twisted pair cable that was attached to a range-gated Doppler ultrasound instrument built at Baylor College of Medicine (Craig J. Hartley, Cardiovascular Research Laboratory). The instrument was modified to permit computer control of the range gate depth. A series of velocity readings were made along the ultrasonic beam in i-mm increments.
NMR Data Acquisition and Processing The phantom was imaged in a 2.4-T, 40-cm-bore NMR imager and spectrometer (Biospec, Bruker Instruments, Rheinstetten, Germany) using a velocity-compensated, gradient echo, perpendicular velocity-measuring pulse sequence equivalent to that described previously ( I ) . This pulse sequence uses a stepped bipolar gradient pulse to encode motion. Sixteen gradient amplitude steps were used to encode velocity perpendicular to the transverse imaging plane. Seven time points were measured in an interleaved fashion at intervals of 200 ms beginning 80 ms after the trigger pulse. Thirty-degree radiofrequency pulses were used to reduce the partial saturation of slowly moving spins. One hundred twenty-eight spatial phase-encoding steps were used with
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a field of view of 15 cm. The data were reconstructed into sets of velocity- and timeresolved images on an offline computer using a three-dimensional Fourier transform program written by the authors. Velocity spectra from a region of interest covering the entire lumen of the tube were produced by summing the intensities of all pixels within the region and presenting those intensities as a function of velocity.
Doppler Ultrasound Data Acquisition und Processing A set of stepped, range-gated measurements was made at the beginning and end of each NMR data acquisition. The ultrasonic data were digitized using a 40-ms sampling period. Eight seconds of data were acquired at each range gate step from 1.O to 4.83 mm. The Doppler ultrasound signal near the far wall was of low amplitude and seemed to contain a small amount of reflection from other structures. Subsequent reconsideration of the data suggested that the far wall was not quite reached. The center of the vessel was assumed to be the point along the beam at which the velocity was the largest at “end systole.” The far wall appeared to be 2.84 mm from the center of the vessel along the ultrasonic beam. This suggests that the angle of the beam was 34” with respect to the direction of flow. It was assumed that the small amount of acrylic through which the beam passed in traversing the wall of the tube had a negligible effect on the reflection time and on the focus of the beam. The measurements were converted to units of velocity using the formula
where v is the measured velocity, A f is the Doppler frequency shift, c is the speed of sound in the medium, .fo is the Doppler ultrasound carrier frequency, and 0 is the angle of the beam with respect to the direction of flow. A value of 16 1,200 cm/s was chosen for c by linear interpolation between the speed of sound in water and the speed of sound in pure glycerol (28) for the 30%glycerol solution used. The recorded ultrasonic data were off by a factor of 2.5, which is one of the fixed gain increments of the preamplifier. It was assumed that the actual gain was incorrectly recorded and the data were multiplied by a factor of 2.5. The range-gated Doppler ultrasound data were initially available as a two-dimensional data set having time as one dimension and distance from the transducer as the other. A row of the dataset having a constant time is a velocity profile of the flow at that phase of the cycle of pulsation. The Doppler ultrasound instrument cannot measure accurately near the transducer, so that data from the center of the tube and on toward the far wall were used. The far half of the velocity profile was interpolated to 1024 points. A histogram was constructed by choosing the histogram bin by the value of the velocity of each point on the interpolated profile and adding to the bin the circumference of the circle with the radius of that point. The circumference is proportional to the area of a thin annulus with that radius and therefore to the volume of fluid in the lamina having that particular velocity. These histograms are thus Doppler ultrasound velocity spectra inferred from the measured velocity profile data.
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Theoretical Model Driven by Pressure Measurements Three computer programs were written to produce velocity spectra based on the theoretical model given in the Appendix and pressure difference measurements made during experiments. The first program takes a digitized pressure difference waveform and performs a discrete Fourier transform (DFT) to determine the parameters needed by the next program. The constant term of the Fourier transform of the pressure difference function was determined by calculating the steady pressure difference necessary to achieve the measured average flow rate. The second computer program was written to solve Eq. [ 71 for an arbitrary pressure gradient function that is input as a set of Fourier coefficients. The other inputs to the program are the viscosity of the fluid, the radius of the tube, and the fundamental frequency of the pulsatile driving force. The output of the program is a two-dimensional dataset that contains the velocity as a function of radius at a number of instants within the cycle of the driving force. The third program converts the velocity profile at each instant into a velocity spectrum at that instant by assuming that the velocity field within the tube is radially symmetric. Several methods were used to compare the velocity spectra from the three methods. The velocity spectra from the Doppler ultrasonic measurements and from the theoretical model were blurred so that they had the same velocity resolution as the NMR velocity spectra. The maximum velocity was defined as the highest velocity point at which the velocity spectrum was above the baseline. These points were identified by hand for each spectrum. The average velocity was computed by
The coherence function of two velocity spectra, S1(v), and S2(v),
was used as a measure of the correlation of pairs of velocity spectra. The summations are over the range of discrete velocities. The area of each spectrum was determined by summing the spectral amplitudes for each velocity and was normalized by dividing by the area averaged over the velocity spectra from a particular method at the trigger delays corresponding to the NMR measurements. RESULTS
Two experiments were performed on different days using the setup parameters described above. A typical tracing of the Doppler velocity signal at midstream in the tube, the trigger signal, the upstream and downstream pressure signals, and the difference of the two pressure tracings is shown in Fig. 2. The measurements were virtually the same in the two experiments. An example of the resulting Doppler ultrasound, NMR, and theoretical model data is shown in Fig. 3. The comparisons of the velocity spectra are shown in Fig. 4. DISCUSSION AND CONCLUSIONS
The purpose of the work reported here was to compare velocity measurements made by Doppler ultrasound and NMR, and inferred from pressure difference mea-
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surements with the intent of demonstrating the correctness of the NMR velocity measurements. There is reasonably good agreement among the three techniques in maximum velocity, average velocity, and coherence. The shapes of the NMR velocity spectra are very similar to those derived from the Doppler ultrasound velocity measurements and the pressure difference measurements. However, there are visual differences among the velocity spectra from the three techniques that may point to limitations of the experimental setup. The Doppler ultrasound velocity spectra are somewhat attenuated at zero velocity because the measurements did not quite extend to the far wall of the tube and thus slower flow was underrepresented in the measurement. The NMR velocity spectra appear to be attenuated at slower speeds. The angle of the Doppler ultrasound beam was inferred from the apparent distance from the center of the flow stream to the far wall. This gave reasonable agreement between the highest velocities measured by Doppler ultrasound and by NMR. Several factors in the Doppler ultrasound data are not precisely known, however. The distribution of reflective particles in the fluid was assumed to be uniform, yet it was observed that over the course of the experiment there was some settling of the cornstarch onto the bottom of the tubes. Despite agitation of the reservoir to maintain suspension of the particles in the fluid, this suggests a slight gradient in the concentration of particles as a function of distance along the ultrasonic beam in the data, since the Doppler ultrasound transducers pointed down with respect to the gravitational force vector. This gradient is probably negligible. The shape of the sensitive volume of the Doppler
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FIG.3. Velocity spectra determined by the three techniques. In the bottom row, the first panel from the left is a gray-scale depiction of the velocity profiles measured by range-gated, pulsed Doppler ultrasound. Time proceeds from bottom to top. The darker regions have the higher velocities. The very light regions have retrograde flow. The horizontal axis is position along the ultrasonic beam as determined by the range gate delay time. The transducer was located to the left of the panel. The data to the left of the vertical stripe artifact are unreliable because of ringing in the transducer. The second panel from the left is a gray-scale representation of the Doppler velocity spectra derived from the Doppler velocity profiles. The darker regions have more voxels of fluid moving at that velocity. The third panel from the left is Doppler velocity spectra plotted with height representing the number of voxels of fluid at a particular velocity. The vertical position of the baseline reflects the time delay after the start of the pulsatile cycle (denoted "R')). The times plotted correspond to the times at which the NMR velocity spectra shown in the fourth panel were taken. The top row is in the same format as the bottom row. It shows the velocity profiles and velocity spectra determined from Eq. [7] driven by the measured pressure difference function.
signal may be distorted by passage through the wall of the acrylic tube. The angle of mounting of the transducer crystal in the slot filed in the tube was affected by the height of the solder spot attaching the electrical lead to the underside of the crystal. The ultrasonic beam should have been at a greater angle with respect to the direction of flow if the transducer were truly at 45",because acrylic has a higher speed of sound than does a glycerol solution. The Doppler ultrasound instrument rejects Doppler frequency shifts below 200 Hz (=1.44 cm/s in this case). Thus, there was some attenuation of the Doppler ultrasound velocity spectra right at zero from the instrument itself. The NMR velocity measurements of slow flow may be attenuated by partial-saturation effects, even though the tip angle was set to 30" to reduce this effect. The plot of relative areas in Fig. 4 is consistent with partial saturation of slow flow. The areas of the NMR velocity spectra are too small at 880 and 1080 ms after the trigger, which is during the deceleration phase when the velocity spectra should be skewed toward zero velocity. There are several clinically significant implications of these results. Pulsatile flow in the circulation is more complicated than in long, straight, rigid tubes. Thus the interpretation of in vivo velocity spectra may involve additional factors such as the compliance of the vessel wall and the effects of a curved vessel and a noncircular cross section. In the event that these in vivo complications can be accounted for, it should
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be possible to solve the inverse problem, that is, to work backward from velocity spectra measured at several phases of the pulsatile cycle to an inference of the pressure gradient waveform (29, 30). The time-dependent pressure gradient function may be useful in characterizing the hemodynamic significance of a lesion in a vessel. In parts of the body where Doppler ultrasonic velocity measurements are difficult to make, NMR velocity imaging may be a useful substitute, despite its much longer data acquisition times. Experimentation with varying the parameters of the theoretical model has shown that the appearance of the theoretical velocity spectra is sensitive to the value of the
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constant term of the pressure gradient function. Measuring this constant term is notoriously difficult ( 1 5 ) , both in vivo and, as was found in these experiments, in phantoms without the use of a true differential pressure transducer. The shapes of the NMR velocity spectra, especially during the retrograde flow phase of the cycle, may provide a means for estimating the average pressure gradient even if an estimate of the entire pressure gradient function proves to be impossible. An important application of this technique may be in the measurement of blood flow in smaller vessels that occupy only a few pixels of an image, even one made at the highest possible resolution of an imaging system. The construction of the velocity spectrum by combining the velocity spectra of each pixel covering part of an image retains all of the velocity resolution of the original acquisition while approximating the effect of imaging an extremely small vessel. It should be noted that the “partial voluming” of surrounding stationary tissue with the lumen may cause the velocity spectrum of a small vessel to have a prominent peak at zero velocity. A topic for further research is the utility of velocity spectra describing the entire vessel when the spatial velocity distribution is not axially symmetric (i.e., when it has an azimuthal dependence) as may be the case for flow downstream of an asymmetric stenosis. The results reported here demonstrate that the velocity spectra derived from velocityresolved NMR imaging of pulsatile flow realistically represent the pulsatile motion of fluid in a model system. They imply that NMR velocity spectra measured in vivo should be good representations of the blood flow velocities. APPENDIX
The mathematical model of pulsatile flow of a Newtonian fluid in a long, straight tube of circular cross section is outlined below. The dimensionless parameter a, called the frequency parameter or the Womersley parameter, is given by the expression a, =
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where R is the radius of the vessel, f is the pulse rate in beats per second, n is a harmonic number of the fundamental pulse rate, and v is the kinematic viscosity. Assume laminar flow of a Newtonian fluid of constant viscosity and density in a rigid tube where the velocity distribution does not change along its length and is axial. The velocity w along the direction of flow z can be written as aw(p)-
at
+--1 a w )
1 ap+v& .(. p az
dr2
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where r is radial distance from the center of the tube, p is the density of the fluid, and p is the pressure. This equation says, in essence, that the acceleration of the fluid is
increased by the pressure drop or pressure gradient along the tube and decreased by the viscous coupling to the stationary walls of the tube. The pressure gradient is the driving function of this differential equation. Suppose that the pressure gradient is a periodic function that may be decomposed in a Fourier series
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where Po is the pressure gradient that would produce the average flow through the tube and Pcnand P,, are the coefficients of the nth cosine and sine terms of the Fourier expansion, respectively. Solving Eq. [ 5 ] for a pressure gradient expressed as a Fourier series,
where the functions A (r, a ) and B( Y, a ) incorporate the radial dependence of the velocity and are given by
+
A(r,a )=
ber a ber(ar/R) bei a bei(ar/R) ber2a + bei2a
181
B(r, a ) =
bei a ber(ar/R) - ber a! bei(ar/R) ber2a! bei2a
[91
+
W is the mean velocity and ber and bei are the real and imaginary parts of the Bessel function of the first kind of order one. ACKNOWLEDGMENTS The authors gratefully acknowledge support from a grant by the American Heart Association, Texas Affiliate. This work was also supported by the Herbert F. Frensley Endowment and the George and Cynthia Mitchell Fund of the Baylor Magnetic Resonance Center and by a grant from Siemens Medical Systems. The data were analyzed using the image processing and display program @adwritten by Nicholas J. Schneiders of Baylor College of Medicine. The authors thank Craig Hartley, Wolfgang Nitz, Roxann Rokey, Stephen Altobelli, David Feinberg, and Paul Murphy for helpful discussionsand Allison Marks for technical assistance. Thomas C. Grimes 111 modified the ultrasound instrument to facilitate computer control. REFERENCES 1. R. E. WENDT111, W. NITZ,P. H. MURPHY,AND R. N. BRYAN,Magn. Reson. Med. 10,71 ( 1989). 2. R. E. WENDT111, W. R. NITZ, AND P. H. MURPHY,Magn. Reson. Med. 15,90 ( 1990). 3. R. E. WENDT111, R. ROKEY,W.-F. WONG,AND A. MARKS,Invest. Radiol. 27,499 (1992). 4. A. I. ZHERNOVOI AND G. D. LATYSHEV,“Nuclear Magnetic Resonance in a Flowing Liquid,” Consultants Bureau, New York, 1965. 5 . T. GROVER AND J. R. SINGER,J.Appl. Phys. 42,938 ( 197 1 ). 6. P. DE GENNES,P h p . Lett. A 29, 20 ( 1969). 7. R. HAYWARD, K. PACKER,AND D. TOMLINSON, Mol. Phys. 23, 1083 ( 1972). 8. A. GARROWAY, J. Phys. D 7 , L159 (1974). 9. K. FUKCJDA AND A. HIRAI,J. Phys. SOC.Japan 47, 1999 (1979). 10. D. A. FEINBERC AND P. D. JAKAB,Mugn. Reson. Med. 16,280 ( 1990). 11. K. KOSE,K. SATOH,AND T. INOUYE,in “Book of Abstracts, Third Annual Meeting,” p. 433, Society of Magnetic Resonance in Medicine, Berkeley, CA, 1984. 12. A. CONSTANTINESC~, J. MALLET,A. BONMARTIN, c. LALWT,AND A. BRIGUET,Magn. Resun. Imaging 2,335 (1984).
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13. T. D. BROWN,R. H. GABEL,D. R. PEDERSEN,L. D. BELL,AND W. F. BLAIR,J. Biomech. 18, 927 (1985). 14. J. MORIN,K. JOHNSTON, AND Y. LAW,Ultrasound Med. Biol. 14, 175 ( 1988). 15. D. MCDONALD,“Blood Flow in Arteries,” Williams & Wilkins, Baltimore, MD, 1974. 16. C. M. RODKIEWICZ, in “Arteries and Arterial Blood Flow” (C. Rodkiewicz, Ed.), p. 327, SpringerVerlag, Vienna, 1983. 17. H. VAN As AND T. J. SCHAAFSMA,in “An Introductionto Biomedical Nuclear Magnetic Resonance” (S. B. Petersen, R. N. Muller, and P. A. Rinck, Eds.), p. 68, Thieme, New York, 1985. 18. P. R. MORAN,Magn. Reson. Imaging 1, 197 (1982). 19. D. NORRIS,in “Book of Abstracts, Third Annual Meeting,” p. 559, Society of Magnetic Resonance in Medicine, Berkeley, CA, 1984. 20. T . W. REDPATH,D. G. NORRIS,R. A. JONES,AND J. M. HUTCHISON, Phys. Med. Biol. 29,891 ( 1984). 21. M. MUERI,J. HENNIG,P. BRUNNER, AND H. FRIEDBURG, BrukerMedicalReport, pp. 29-31 (1986). 22. J. HENNIG,M. MORI, P. BRUNNER, AND H. FRIEDBURG, Radiology, 166,237 (1988). 23. E. HAACKEAND G. LENZ,Am. J. Roentgenol. 148, 1251 ( 1987). 24. P. M. PATTANY,J. J. PHILLIPS,L. c. CHIU,J. D.LIPCAMON,J. L. DUERK,J. M. MCNALLY,AND S. N. MOHAPATRA, J. Comput. Assisted Tomogr. 11, 369 (1987). 25. J. DUERKAND P. PATTANY,Magn. Reson. Imaging 6,321 (1988). 26. A. HAASE,J. FRAHM, D. M A ~ E IK., D. MERBOLDT,AND W. HANICKE, J. Magn. Reson. 67, 258 (1986). 27. A. J. EVANS,L. W. HEDLUND,AND R.J. HERFKENS,Invest. Radiol. 23, 579 ( 1988). 28. J. L. ROSEAND B. B. GOLDBERG,“Basic Physics in Diagnostic Ultrasound,” Wiley, New York, 1979. 29. D. A. FEINBERG, “Method and Apparatus for NMR Detection and Imaging of Flowing Fluid Nuclei,’’ U.S. Patent 4,602,641, 1983. 30. R. S. ADLER,T. L. CHENEVERT, J. B. FOWLKES,J. PIPE,AND J. M. RUBIN,J. Comput.Assisted Tomogr. 15,483 (1991).
27,214-225 ( 1992)
Nuclear Magnetic Resonance Velocity Spectra of Pulsatile Flow in a Rigid Tube RICHARD E. WENDT III*’t’$ AND WAI-FANW0NC”t * Department of Radiology and t Magnelic Rcsonunce Center, Buylor College of Medicine, Houston, Texas 77030, and $Department of Radiology, The Methodist Hospital,Houston, Texas 77030 Received May 22, 1991; revised December 19, 1991; accepted January 7, 1992 Velocity spectra can be derived from velocity-encoded nuclear magnetic resonance (NMR) images. Velocity spectra are histograms showing the amounts of fluid flowing at different velocitiesin the sensitive volume ofthe measurement. Velocity spectra may prove to be useful in characterizing the flow of blood in small vessels, for example, in detecting the presence of stenoses and in evaluating their severity. NMR velocity spectra acquired in vivo are sufficientlycomplicated that a model system was designed and tested to investigate the velocity spectra of pulsatile flow. This study measured the NMR velocity spectra of pulsatile flow in a rigid tube and compared them to velocity spectra derived from Doppler ultrasound measurements and to velocity spectra inferred from a theoretical model driven by the measured pressure difference function. The experimental results from each technique agree. 0 1992 Academic Press, Inc. INTRODUCTION
Velocity-resolved nuclear magnetic resonance ( N M R ) imaging may be used to measure blood flow noninvasively. NMR imaging is not significantly affected by bones, by airspaces, or by depth-dependent attenuation in soft tissues and thus may be the only noninvasive means for evaluating the flow of blood in small, deeply buried vessels in the body. The NMR images of many small, but important vessels do not have adequate spatial resolution to construct detailed maps of the flow field across the lumen of the vessel. The vessel is represented by just a few pixels and the spatial dependence of velocity within the vessel lumen is hard to assess. The authors have investigated the use of velocity spectra ( I , 2) as a method for presenting the essential information in the velocity-resolved images in a manner that does not depend strongly on the spatial resolution of the original images. NMR velocity spectra of pulsatile flow in vivo are more complicated than those of steady flow ( 3 ) .The study reported here was undertaken to develop and test a physical model system for the evaluation of pulsatile flow by NMR velocity imaging. It tested the hypothesis that the NMR velocity spectra of pulsatile flow in a rigid tube are accurate. The results suggest that they are. The measurement of velocity spectra by NMR has a long history. The terms “velocity distribution” and “probability distribution” have been used by most investigators. They are used in the sense of a statistical distribution to mean the relative or absolute frequency of flow at particular velocities. Zhernovoi and Latyshev measured the velocity distribution of turbulent flow ( 4 ) . Grover and Singer ( 5 ) described a measurement 0740-3194192 $5.00 Copyright 0 1992 by Academic Press, h e . All rights of reproduction in any form reserved.
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FIG. I . Schematic diagram, not to scale, of the essential elements of the phantom setup: ( A ) fluid reservoir, (B) peristaltic pump, (C) damping element, (D) check valve, ( E ) respirator pump, (F) pulsation element, ( G ) tube in which flow was imaged (the pressure ports and the Doppler ultrasound transducer are not shown), and ( H ) return path for the fluid.
of the velocity distribution in the human finger. De Gennes (6) pointed out that the Fourier transform of the spin-echo signal would give an instantaneous velocity distribution of turbulent flow. Hayward et al. ( 7) calculated and measured the spin echo for plug and laminar flow along a magnetic field gradient and noted that the Fourier transform would yield the velocity distribution. Garroway measured the velocity distribution of laminar flow (8).Fukuda and Hirai ( 9 )measured the velocity distribution as a function of Reynolds number for turbulent flow. Feinberg and Jakab studied perfusion using NMR measurements of the velocity distribution (10). The authors prefer the term “velocity spectrum” to avoid confusion with the spatial velocity distribution v ( x , y , z , t ) discussed in fluid mechanics and occasionally used in the spatial sense in the NMR flow imaging literature (11, 12). What is more, the term “velocity spectrum” comes naturally to discussions of continuous-waveDoppler ultrasound since the Fourier transform of the Doppler ultrasound signal is the velocity distribution in the sense of a histogram or frequency distribution (13, 14). The effects of pulsatile flow on the spatial velocity distribution or, more properly, the velocity field, have been studied (15, 16) and the effects on the velocity spectrum have been alluded to ( 17). The pulse sequence used in this study relies on a stepped, bipolar gradient pulse for phase encoding of velocity ( 18-22), gradient waveform compensation for constant velocity (12, 23-25), and a spoiled, gradient echo image acquisition (26). The shape of the velocity spectrum is determined by the relative amounts of fluid moving at the different velocities o f flow within the vessel. In steady, fully developed flow, the authors have shown that laminar flow gives a uniform velocity spectrum while turbulent flowgives a peaked velocity spectrum (2). Velocity spectra of pulsatile flow in vivo have different shapes depending on the phase of the cardiac cycle. The present investigation was undertaken to determine how the velocity spectra of pulsatile flow should look. The theory of pulsatile flow has been described by McDonald ( I S ) and by Rodkiewicz ( 1 6 ) . Their derivations for tubes with rigid walls and circular cross sections are summarized in the Appendix and form the basis of the theoretical model used
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here. The theoretical model suggests that early in the pulsatile cycle when the fluid is being accelerated, the velocity profile is blunted and therefore the velocity spectrum should be peaked as in steady, turbulent flow. Later, when the fluid is moving well, the viscous forces will produce a more rounded velocity profile and the velocity spectrum should be flatter as in steady, laminar flow. Still later there may be retrograde flow in parts of the lumen of the vessel. This causes the velocity spectrum to have a significant amplitude at both positive and negative velocities. The three-dimensional set of velocity-resolved images produced by the pulse sequence is difficult to interpret. Especially in clinical data sets, most of the images have just a few bright spots that represent blood flow at nonzero velocities. Condensing these data sets into velocity spectra appears to preserve significant information about the nature of the flow in a vessel. This reduction in the complexity of the interpretation of velocity-resolveddata is even more important for pulsatile flow in which the velocities vary during the pulsatile cycle (e.g., the cardiac cycle) and the data are four-dimensional. METHODS
To investigate the NMR velocity spectra of pulsatile flow, a phantom was built so that the pulsatile flow velocities through it could be measured by NMR imaging and by range-gated, pulsed Doppler ultrasound, and could be inferred from pressure difference measurements. Contemporaneous measurements were made by the three methods and the data were processed and compared.
The Phantom The phantom contained two acrylic tubes 125 cm long with inside radii of 0.159 cm. The tubes were connected in series so that fluid flowed down one tube and then back up the other. All measurements were made on the second tube to ensure more complete magnetization of the fluid. The phantom was filled with a NiQ-doped glycerol-water solution with a viscosity of 2.7 cP. A small amount of cornstarch was added to provide reflective particles for the Doppler ultrasound to sense. The path of the fluid in the experimental apparatus is shown in Fig. 1. The fluid was driven by a peristaltic pump (Masterflex, Model CK-07520-00, Cole-Parmer, Chicago) with a size 16 (4.8-480 ml/min) pump head. A damping element was placed immediately downstream of the pump to minimize the pulsations introduced by the peristalsis of the pump. The outlet of the damping element was directed through a check valve to prevent backflow toward the pump. A small average flow rate of 0.532 ml/s was used. A pulsation element patterned after that described by Evans et af. (27) was constructed by enclosing a 50-ml rubber bulb in a sealed 1-liter jar. A water connection was provided through the closure of the jar to the bulb and an air connection was provided to the interior of the jar. The jar was pressurized through the air connection by an animal respirator (Dual Phase Control Respirator Pump, Model 613, Harvard Apparatus, South Natick, MA) using a period of 1.88 s (about 33 “breaths” per minute), a 50% “inspiration” (i.e., duty cycle), and a stroke volume setting of 400 ml. This gave a Womersley number ( 1 5 )of 1.77 for the fluid and acrylic tubing used. The respirator pump triggered the NMR system at a particular point in
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its “respiratory” cycle. The interior of the rubber bulb was connected to the flow system downstream of the check valve by a Y-connector. To measure the pressure drop along a length of the tubing, two pressure ports were made by drilling two holes 15 cm apart in the side of the acrylic tubing so that the ends of plastic Luer connectors from butterfly sets would fit snugly. The bores of the Luer connectors were cleared with a 0.04-cm drill. After the Luer connectors were glued in place with methylene chloride and then epoxy, the base of each Luer connector was removed by passing a 0.32-cm drill down the bore of the tubing. This necessitated cutting the tubing near the site of each pressure port. Each face of each cut was polished square and smooth and the tubing was rejoined with methylene chloride after the pressure ports were completely installed. The joints in the tubing were reinforced with beads of epoxy or with a short, telescoping collar of acrylic tubing. Several test joints were examined after construction. There were slight disturbances of the wall, but no protrusions of material into the bore of the tube. The ports were connected by 6 1-cm lengths of pressure tubing to physiologic pressure transducers (Model 290C, Hewlett-Packard, Palo Alto, CA). The pressure signals were transmitted through the radiofrequency screening of the NMR imager room using an FM telemetry system (Model OMP 720lC, Nihon-Kohden (America), Irvine CA) operating at 472 MHz. The pressure signals, along with the trigger pulse and the phasic Doppler ultrasound velocity signal, were digitized (Model AD- 1000,Real Time Devices, State College, PA) and stored in a personal computer using a data acquisition program written by the authors. The pressure gradient was determined by dividing the measured pressure difference by the distance between the ports and assuming that the system was rigid between the two ports. The pressure signal was calibrated using a sphygmomanometer attached to the manifold at the inlet of the pressure transducers. The velocity of fluid flow in the tube was also measured by range-gated Doppler ultrasound. A 45” notch was filed in the side of the tubing as deep as possible without breaking through to the bore of the tube. A bare 20-MHz ultrasonic transducer crystal with fine-wire leads was glued to one face of the notch using epoxy. The notch was filled in with epoxy. The fine leads were soldered to the conductors of a twisted pair cable that was attached to a range-gated Doppler ultrasound instrument built at Baylor College of Medicine (Craig J. Hartley, Cardiovascular Research Laboratory). The instrument was modified to permit computer control of the range gate depth. A series of velocity readings were made along the ultrasonic beam in i-mm increments.
NMR Data Acquisition and Processing The phantom was imaged in a 2.4-T, 40-cm-bore NMR imager and spectrometer (Biospec, Bruker Instruments, Rheinstetten, Germany) using a velocity-compensated, gradient echo, perpendicular velocity-measuring pulse sequence equivalent to that described previously ( I ) . This pulse sequence uses a stepped bipolar gradient pulse to encode motion. Sixteen gradient amplitude steps were used to encode velocity perpendicular to the transverse imaging plane. Seven time points were measured in an interleaved fashion at intervals of 200 ms beginning 80 ms after the trigger pulse. Thirty-degree radiofrequency pulses were used to reduce the partial saturation of slowly moving spins. One hundred twenty-eight spatial phase-encoding steps were used with
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a field of view of 15 cm. The data were reconstructed into sets of velocity- and timeresolved images on an offline computer using a three-dimensional Fourier transform program written by the authors. Velocity spectra from a region of interest covering the entire lumen of the tube were produced by summing the intensities of all pixels within the region and presenting those intensities as a function of velocity.
Doppler Ultrasound Data Acquisition und Processing A set of stepped, range-gated measurements was made at the beginning and end of each NMR data acquisition. The ultrasonic data were digitized using a 40-ms sampling period. Eight seconds of data were acquired at each range gate step from 1.O to 4.83 mm. The Doppler ultrasound signal near the far wall was of low amplitude and seemed to contain a small amount of reflection from other structures. Subsequent reconsideration of the data suggested that the far wall was not quite reached. The center of the vessel was assumed to be the point along the beam at which the velocity was the largest at “end systole.” The far wall appeared to be 2.84 mm from the center of the vessel along the ultrasonic beam. This suggests that the angle of the beam was 34” with respect to the direction of flow. It was assumed that the small amount of acrylic through which the beam passed in traversing the wall of the tube had a negligible effect on the reflection time and on the focus of the beam. The measurements were converted to units of velocity using the formula
where v is the measured velocity, A f is the Doppler frequency shift, c is the speed of sound in the medium, .fo is the Doppler ultrasound carrier frequency, and 0 is the angle of the beam with respect to the direction of flow. A value of 16 1,200 cm/s was chosen for c by linear interpolation between the speed of sound in water and the speed of sound in pure glycerol (28) for the 30%glycerol solution used. The recorded ultrasonic data were off by a factor of 2.5, which is one of the fixed gain increments of the preamplifier. It was assumed that the actual gain was incorrectly recorded and the data were multiplied by a factor of 2.5. The range-gated Doppler ultrasound data were initially available as a two-dimensional data set having time as one dimension and distance from the transducer as the other. A row of the dataset having a constant time is a velocity profile of the flow at that phase of the cycle of pulsation. The Doppler ultrasound instrument cannot measure accurately near the transducer, so that data from the center of the tube and on toward the far wall were used. The far half of the velocity profile was interpolated to 1024 points. A histogram was constructed by choosing the histogram bin by the value of the velocity of each point on the interpolated profile and adding to the bin the circumference of the circle with the radius of that point. The circumference is proportional to the area of a thin annulus with that radius and therefore to the volume of fluid in the lamina having that particular velocity. These histograms are thus Doppler ultrasound velocity spectra inferred from the measured velocity profile data.
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Theoretical Model Driven by Pressure Measurements Three computer programs were written to produce velocity spectra based on the theoretical model given in the Appendix and pressure difference measurements made during experiments. The first program takes a digitized pressure difference waveform and performs a discrete Fourier transform (DFT) to determine the parameters needed by the next program. The constant term of the Fourier transform of the pressure difference function was determined by calculating the steady pressure difference necessary to achieve the measured average flow rate. The second computer program was written to solve Eq. [ 71 for an arbitrary pressure gradient function that is input as a set of Fourier coefficients. The other inputs to the program are the viscosity of the fluid, the radius of the tube, and the fundamental frequency of the pulsatile driving force. The output of the program is a two-dimensional dataset that contains the velocity as a function of radius at a number of instants within the cycle of the driving force. The third program converts the velocity profile at each instant into a velocity spectrum at that instant by assuming that the velocity field within the tube is radially symmetric. Several methods were used to compare the velocity spectra from the three methods. The velocity spectra from the Doppler ultrasonic measurements and from the theoretical model were blurred so that they had the same velocity resolution as the NMR velocity spectra. The maximum velocity was defined as the highest velocity point at which the velocity spectrum was above the baseline. These points were identified by hand for each spectrum. The average velocity was computed by
The coherence function of two velocity spectra, S1(v), and S2(v),
was used as a measure of the correlation of pairs of velocity spectra. The summations are over the range of discrete velocities. The area of each spectrum was determined by summing the spectral amplitudes for each velocity and was normalized by dividing by the area averaged over the velocity spectra from a particular method at the trigger delays corresponding to the NMR measurements. RESULTS
Two experiments were performed on different days using the setup parameters described above. A typical tracing of the Doppler velocity signal at midstream in the tube, the trigger signal, the upstream and downstream pressure signals, and the difference of the two pressure tracings is shown in Fig. 2. The measurements were virtually the same in the two experiments. An example of the resulting Doppler ultrasound, NMR, and theoretical model data is shown in Fig. 3. The comparisons of the velocity spectra are shown in Fig. 4. DISCUSSION AND CONCLUSIONS
The purpose of the work reported here was to compare velocity measurements made by Doppler ultrasound and NMR, and inferred from pressure difference mea-
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surements with the intent of demonstrating the correctness of the NMR velocity measurements. There is reasonably good agreement among the three techniques in maximum velocity, average velocity, and coherence. The shapes of the NMR velocity spectra are very similar to those derived from the Doppler ultrasound velocity measurements and the pressure difference measurements. However, there are visual differences among the velocity spectra from the three techniques that may point to limitations of the experimental setup. The Doppler ultrasound velocity spectra are somewhat attenuated at zero velocity because the measurements did not quite extend to the far wall of the tube and thus slower flow was underrepresented in the measurement. The NMR velocity spectra appear to be attenuated at slower speeds. The angle of the Doppler ultrasound beam was inferred from the apparent distance from the center of the flow stream to the far wall. This gave reasonable agreement between the highest velocities measured by Doppler ultrasound and by NMR. Several factors in the Doppler ultrasound data are not precisely known, however. The distribution of reflective particles in the fluid was assumed to be uniform, yet it was observed that over the course of the experiment there was some settling of the cornstarch onto the bottom of the tubes. Despite agitation of the reservoir to maintain suspension of the particles in the fluid, this suggests a slight gradient in the concentration of particles as a function of distance along the ultrasonic beam in the data, since the Doppler ultrasound transducers pointed down with respect to the gravitational force vector. This gradient is probably negligible. The shape of the sensitive volume of the Doppler
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FIG.3. Velocity spectra determined by the three techniques. In the bottom row, the first panel from the left is a gray-scale depiction of the velocity profiles measured by range-gated, pulsed Doppler ultrasound. Time proceeds from bottom to top. The darker regions have the higher velocities. The very light regions have retrograde flow. The horizontal axis is position along the ultrasonic beam as determined by the range gate delay time. The transducer was located to the left of the panel. The data to the left of the vertical stripe artifact are unreliable because of ringing in the transducer. The second panel from the left is a gray-scale representation of the Doppler velocity spectra derived from the Doppler velocity profiles. The darker regions have more voxels of fluid moving at that velocity. The third panel from the left is Doppler velocity spectra plotted with height representing the number of voxels of fluid at a particular velocity. The vertical position of the baseline reflects the time delay after the start of the pulsatile cycle (denoted "R')). The times plotted correspond to the times at which the NMR velocity spectra shown in the fourth panel were taken. The top row is in the same format as the bottom row. It shows the velocity profiles and velocity spectra determined from Eq. [7] driven by the measured pressure difference function.
signal may be distorted by passage through the wall of the acrylic tube. The angle of mounting of the transducer crystal in the slot filed in the tube was affected by the height of the solder spot attaching the electrical lead to the underside of the crystal. The ultrasonic beam should have been at a greater angle with respect to the direction of flow if the transducer were truly at 45",because acrylic has a higher speed of sound than does a glycerol solution. The Doppler ultrasound instrument rejects Doppler frequency shifts below 200 Hz (=1.44 cm/s in this case). Thus, there was some attenuation of the Doppler ultrasound velocity spectra right at zero from the instrument itself. The NMR velocity measurements of slow flow may be attenuated by partial-saturation effects, even though the tip angle was set to 30" to reduce this effect. The plot of relative areas in Fig. 4 is consistent with partial saturation of slow flow. The areas of the NMR velocity spectra are too small at 880 and 1080 ms after the trigger, which is during the deceleration phase when the velocity spectra should be skewed toward zero velocity. There are several clinically significant implications of these results. Pulsatile flow in the circulation is more complicated than in long, straight, rigid tubes. Thus the interpretation of in vivo velocity spectra may involve additional factors such as the compliance of the vessel wall and the effects of a curved vessel and a noncircular cross section. In the event that these in vivo complications can be accounted for, it should
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be possible to solve the inverse problem, that is, to work backward from velocity spectra measured at several phases of the pulsatile cycle to an inference of the pressure gradient waveform (29, 30). The time-dependent pressure gradient function may be useful in characterizing the hemodynamic significance of a lesion in a vessel. In parts of the body where Doppler ultrasonic velocity measurements are difficult to make, NMR velocity imaging may be a useful substitute, despite its much longer data acquisition times. Experimentation with varying the parameters of the theoretical model has shown that the appearance of the theoretical velocity spectra is sensitive to the value of the
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constant term of the pressure gradient function. Measuring this constant term is notoriously difficult ( 1 5 ) , both in vivo and, as was found in these experiments, in phantoms without the use of a true differential pressure transducer. The shapes of the NMR velocity spectra, especially during the retrograde flow phase of the cycle, may provide a means for estimating the average pressure gradient even if an estimate of the entire pressure gradient function proves to be impossible. An important application of this technique may be in the measurement of blood flow in smaller vessels that occupy only a few pixels of an image, even one made at the highest possible resolution of an imaging system. The construction of the velocity spectrum by combining the velocity spectra of each pixel covering part of an image retains all of the velocity resolution of the original acquisition while approximating the effect of imaging an extremely small vessel. It should be noted that the “partial voluming” of surrounding stationary tissue with the lumen may cause the velocity spectrum of a small vessel to have a prominent peak at zero velocity. A topic for further research is the utility of velocity spectra describing the entire vessel when the spatial velocity distribution is not axially symmetric (i.e., when it has an azimuthal dependence) as may be the case for flow downstream of an asymmetric stenosis. The results reported here demonstrate that the velocity spectra derived from velocityresolved NMR imaging of pulsatile flow realistically represent the pulsatile motion of fluid in a model system. They imply that NMR velocity spectra measured in vivo should be good representations of the blood flow velocities. APPENDIX
The mathematical model of pulsatile flow of a Newtonian fluid in a long, straight tube of circular cross section is outlined below. The dimensionless parameter a, called the frequency parameter or the Womersley parameter, is given by the expression a, =
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where R is the radius of the vessel, f is the pulse rate in beats per second, n is a harmonic number of the fundamental pulse rate, and v is the kinematic viscosity. Assume laminar flow of a Newtonian fluid of constant viscosity and density in a rigid tube where the velocity distribution does not change along its length and is axial. The velocity w along the direction of flow z can be written as aw(p)-
at
+--1 a w )
1 ap+v& .(. p az
dr2
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where r is radial distance from the center of the tube, p is the density of the fluid, and p is the pressure. This equation says, in essence, that the acceleration of the fluid is
increased by the pressure drop or pressure gradient along the tube and decreased by the viscous coupling to the stationary walls of the tube. The pressure gradient is the driving function of this differential equation. Suppose that the pressure gradient is a periodic function that may be decomposed in a Fourier series
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where Po is the pressure gradient that would produce the average flow through the tube and Pcnand P,, are the coefficients of the nth cosine and sine terms of the Fourier expansion, respectively. Solving Eq. [ 5 ] for a pressure gradient expressed as a Fourier series,
where the functions A (r, a ) and B( Y, a ) incorporate the radial dependence of the velocity and are given by
+
A(r,a )=
ber a ber(ar/R) bei a bei(ar/R) ber2a + bei2a
181
B(r, a ) =
bei a ber(ar/R) - ber a! bei(ar/R) ber2a! bei2a
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+
W is the mean velocity and ber and bei are the real and imaginary parts of the Bessel function of the first kind of order one. ACKNOWLEDGMENTS The authors gratefully acknowledge support from a grant by the American Heart Association, Texas Affiliate. This work was also supported by the Herbert F. Frensley Endowment and the George and Cynthia Mitchell Fund of the Baylor Magnetic Resonance Center and by a grant from Siemens Medical Systems. The data were analyzed using the image processing and display program @adwritten by Nicholas J. Schneiders of Baylor College of Medicine. The authors thank Craig Hartley, Wolfgang Nitz, Roxann Rokey, Stephen Altobelli, David Feinberg, and Paul Murphy for helpful discussionsand Allison Marks for technical assistance. Thomas C. Grimes 111 modified the ultrasound instrument to facilitate computer control. REFERENCES 1. R. E. WENDT111, W. NITZ,P. H. MURPHY,AND R. N. BRYAN,Magn. Reson. Med. 10,71 ( 1989). 2. R. E. WENDT111, W. R. NITZ, AND P. H. MURPHY,Magn. Reson. Med. 15,90 ( 1990). 3. R. E. WENDT111, R. ROKEY,W.-F. WONG,AND A. MARKS,Invest. Radiol. 27,499 (1992). 4. A. I. ZHERNOVOI AND G. D. LATYSHEV,“Nuclear Magnetic Resonance in a Flowing Liquid,” Consultants Bureau, New York, 1965. 5 . T. GROVER AND J. R. SINGER,J.Appl. Phys. 42,938 ( 197 1 ). 6. P. DE GENNES,P h p . Lett. A 29, 20 ( 1969). 7. R. HAYWARD, K. PACKER,AND D. TOMLINSON, Mol. Phys. 23, 1083 ( 1972). 8. A. GARROWAY, J. Phys. D 7 , L159 (1974). 9. K. FUKCJDA AND A. HIRAI,J. Phys. SOC.Japan 47, 1999 (1979). 10. D. A. FEINBERC AND P. D. JAKAB,Mugn. Reson. Med. 16,280 ( 1990). 11. K. KOSE,K. SATOH,AND T. INOUYE,in “Book of Abstracts, Third Annual Meeting,” p. 433, Society of Magnetic Resonance in Medicine, Berkeley, CA, 1984. 12. A. CONSTANTINESC~, J. MALLET,A. BONMARTIN, c. LALWT,AND A. BRIGUET,Magn. Resun. Imaging 2,335 (1984).
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