Ultrasound in Med. & Biol., Vol. 31, No. 5, pp. 663–671, 2005Copyright © 2005 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/05/$–see front matter

doi:10.1016/j.ultrasmedbio.2005.01.011

● Original Contribution

EFFECT OF VESSEL CURVATURE ON DOPPLER DERIVED VELOCITYPROFILES AND FLUID FLOW

R. KRAMS,* G. BAMBI,† F. GUIDI,† F. HELDERMAN,* A.F.W. VAN DER STEEN*and P. TORTOLI

†

*Department of Biomedical Engineering, Erasmus MC, Rotterdam, The Netherlands and †Dipartimento diElettronica e Telecomunicazioni, Universita degli Studi di Firenze, Firenze, Italy

(Received 9 August 2004; revised 20 January 2005; in final form 27 January 2005)

Abstract—Side-branches and curvatures in the arterial tree yield deviations from the axial oriented velocity. Velocityor volume flow estimates based on the assumption that flow is axially oriented are of limited value at these sites. Thisarticle evaluates information obtainable by using a multigate Doppler ultrasound (US) instrument used with curvedphantoms, which resemble the human coronary arteries. The comparison of experimental velocity data with dataprovided by an accurate computational fluid dynamics (CFD) method shows differences in the range of 4 to 11% forfour curvatures with different radii. Multigate data are also used to estimate the volume flow in the curved segmentsat different experimental conditions. An error lower than 15% is obtained, to be compared with a 24% error obtainedby assuming a parabolic velocity profile. In particular, it is shown that the residual error is not related to the smalldeviation of the velocity vectors from the axial direction due to the presence of secondary velocity components, whichare found to be of magnitude less than 10% with respect to the axial velocity component. (E-mail:r.krams@erasmusmc.nl) © 2005 World Federation for Ultrasound in Medicine & Biology.

Key Words: Doppler, Velocity profiles, CFD, Secondary velocity.

INTRODUCTION

Atherosclerosis has a distinctive distribution pattern, i.e.,near curvatures or side-branches, which cannot be ex-plained by the current risk factors (Brand et al. 1989;Honda et al. 2001; Krams et al. 1997). Several labora-tories have provided evidence that deviations of thenormal velocity field occur at these sites (Brand et al.1989, Friedman et al. 1986; Gnasso et al. 1997; Honda etal. 2001; Ku et al. 1985; Shaaban and Duerinckx 2000;Wentzel et al. 2001b; Zarins et al. 1983).

Kinetics of fluid elements in curved segments isquite complex, due to the secondary velocity componentsinduced by centrifugal forces (Chang and Tarbell 1988;Dash et al. 1999; Kang and Tarbell 1983; Mann andTarbell 1990; Perktold et al. 1991; Sabbah et al. 1986;Weston and Tarbell 1997). As the secondary componentsare large in the symmetry plane of the curved segmentand as these components are directed outward, the max-imum velocity shifts to the outer curvature (Chang and

Address correspondence to: R. Krams, M.D., Ph.D., Departmentof Biomedical Engineering Erasmus University Rotterdam, P.O. Box

1738 3000 DR Rotterdam, The Netherlands. E-mail: r.krams@erasmusmc.nl

663

Tarbell 1988; Mann and Tarbell 1990; Perktold et al.1991; Weston and Tarbell 1997). The possible presenceof atherosclerotic plaques yields further deviations fromthe ideal parabolic velocity distribution (Wentzel et al.2001a; Wentzel et al. 2001b; Wentzel et al. 2000).

Several methods have been developed for measur-ing velocity patterns near atherosclerotic plaques, includ-ing MRI-, echo- and Doppler-derived fluid flow. (Fried-man et al. 1986; Gnasso et al. 1997). MRI-based methodsare non invasive but offer a rather low resolution (Glor etal. 2003; Long et al. 2003; Thomas et al. 2003; Wu et al.2004). Fluid flow measurements based upon correlationof subsequent IVUS images needs a catheter to be lo-cated inside the artery, thereby affecting the velocityprofile and the shear stress pattern (Chandran et al. 1996;Cheng et al. 2003; Krams et al. 1999b; Liu et al. 2001).Doppler-based techniques (Evans et al., 2000) are par-ticularly promising. The earlier intra vascular Dopplermethods measure maximum velocity over a cross-sectionand calculate flow and shear stress after assuming aparabolic velocity profile (Doucette et al. 1992). How-ever, as stated above, such assumption is not realistic andneither this method nor other methods measuring veloc-

ity in a single point (single-gate Doppler) provide con-

664 Ultrasound in Medicine and Biology Volume 31, Number 5, 2005

sistent fluid flow estimates (Doucette et al. 1992). An-other limitation common to all Doppler approaches isthat, to convert the measured frequencies to velocities,the beam-to-flow angle can only be estimated on thebasis of geometrical considerations, by assuming that, atany measurement point, the velocity vector is orientedalong the local vessel axis. The unavoidable presence of(transversely oriented) secondary velocity components incurved vessel segments makes such an assumption afurther possible source of measurement error.

The limitations of single point measurements arepartially overcome by real-time multigate instrumentsthat have been recently developed (Tortoli et al. 2002;Tortoli et al. 1997; Tortoli et al. 1999) for measuring theactual distribution of Doppler frequencies along the USbeam axis. This paper investigates the effects of vesselcurvature on Doppler-derived velocity profiles and onrelated fluid flow estimates. CFD-derived velocity pro-files are used as a reference for evaluating the accuracyof experimental velocity profiles measured with the mul-tigate system in curved phantoms resembling the humanarterial coronary system. Fluid flow estimates based ondifferent Doppler (single-gate and multigate) techniquesare then evaluated. The effect of secondary velocitycomponents on these estimates is also discussed.

MATERIALS AND METHODS

Ultrasound systemA piezoelectric unfocused 16 MHz transducer

(manufactured at the Institute of Fundamental Research,PAN, Warsaw) is excited at 16 kHz pulse repetitionfrequency (PRF) to generate bursts of US energy. TheUS beam is narrower (�6 dB) than 1 mm over a range ofabout 15 mm, as measured through an advanced systemdedicated to US probe characterization (Esaote S.p.A.,Florence, Italy). Circuits for electrical excitation of thepiezoelectric element and for amplification of receivedechoes are included in the radio-frequency front-end.Phase-quadrature demodulation, allowing forward andreverse flows to be distinguished, provides two base-band signals.

Acquisition-processing systemThe multigate processing system consists of a PCI-

bus plug-in card (to be housed in a PC), where threesections can be identified. The first section performsanalog conditioning of input signals provided by the USsystem; the second carries out acquisition by means oftwo analog-to-digital converters (ADCs); and the thirdperforms data management and processing in a highspeed digital signal processor (DSP) associated withSDRAM and PCI channel.

For each pulse transmitted at PRF rate, the system

digitizes 64 complex samples with 14-bit resolution. Theoperator can change the time distance between these sam-ples, corresponding to the spacing between the range cells,to match the analyzed range to the actual region-of-interest.Processing is performed by the TMS320C6202 (Texas In-struments Incorporated, Dallas, Texas, USA) that computesthe true spectra of the Doppler signals related to the subse-quent range cells in real time. The computed power spectraldensities are sent to the host PC for real-time display.During real-time operation, it is possible to store all dataacquired over a time interval several seconds long into afile, for possible post processing.

Software measurement toolsAll real-time and postprocessing operations are un-

der user control by means of a bundled software, namedGASP (global acquisition and signal processing), whichconsists of an integrated shell for Windows™ (MicrosoftCorp., Redmond, Washington, USA) family operatingsystem. It includes applications for postprocessing car-ried out in LabVIEW™ (National Instruments Corp.,Austin, Texas, USA) and a Visual C�� (MicrosoftCorp., Redmond, Washington, USA) coded console forreal-time data acquisition and visualization (Bambi et al.2004).

By using the post processing modules, it is possibleto evaluate different features of each acquisition. Up to200 subsequent spectral profiles are ensemble-averagedto reduce the spectral variance (Evans et al. 2000). Foreach of the 64 averaged spectra, related to the 64 corre-sponding depths, the peak frequency is estimated throughthe threshold method (Marasek et al. 1994). A polyno-mial fit of the 64 experimental points is carried outthrough the LabVIEW™ “HouseHolder” function.

Frequencies have been converted to velocities ac-cording to the Doppler equation, corrected to take intoconsideration the intrinsic spectral broadening effect:

V �fmax

2

��cos� �1

2k sin�� (1)

where V is the flow velocity, fmax is the estimated peakfrequency, � is the US wavelength, � is the Dopplerangle and k is a factor depending on the transducer(Tortoli et al., 1997). For the 16 MHz transducer used inour experiments, k � 0.2.

Flow phantomA closed-loop hydraulic circuit was used, where

changing the height difference between an upper and alower reservoir controls fluid velocity. The correspond-ing flow rate (ml/s) was measured through a variable area

flow-meter (e50-2600, ASA SpA, Milano, Italy). The

Effect of curvature on Doppler flow ● R. KRAMS et al. 665

test fluid was a water suspension of polyamide backscat-tering particles (Orgasol 2001D NAT 1, ELF Atochem,Paris, France) with mean diameter of 10 �m. Viscosityof the fluid was 1 cP.

The measurement section includes a polyethylenetube with 3 mm inner diameter (ID) and 4 mm outerdiameter (OD). This tube was chosen because it is suf-ficiently rigid for flow studies and the wall only supportslongitudinal ultrasound propagation. The acousticalproperties of the material have been measured by themethod described by (Selfridge 1985): the longitudinalsound speed was 2100 m/s and the critical angle ofincidence in water was 47° � 2°. The polyethylene tubewas fixed to a Plexiglas template in such a way as topresent five possible different measurement sites of Dop-pler-derived velocity profile. There was one straight tubeand four bends with the following curvature radii: A) 47mm, B) 23 mm, C) 43 mm, D) 27 mm (Fig. 1). The inletlength of the tube before the measurement site was 40 cmin length. The distance between curves A and B was 10cm (the same between curves C and D), while curve Cwas 30 cm far from curve B. The tube is immersed inwater to facilitate the interrogation by the US probe. Anin-house-developed device allowed accurate positioningof the probe in the plane of symmetry of the curved tube.

The volume flow was set at two flow rates (0.5 and1.0 mL/s). These values were chosen to generate Reyn-olds numbers of 68 and 247, respectively, values in therange found in the human coronary arteries (Krams et al.1997; Wentzel et al. 2001b). The corresponding Deannumbers (i.e., � � Re��(a/R), where a is the tube

Fig. 1. Experimental set-up. (a) Housing of the template in thewater tank. (b) Details of bends A and C: template curvature 45mm, US beam directed from outside to inside (bend A) and

from inside to outside (bend C).

diameter, R is the curvature radius and Re is the Reyn-

olds number) were 17 and 90, respectively. For eachspeed, the Doppler transducer was positioned in twodifferent ways for the four curvatures. For curves B andC, the beam was directed from the inner wall to the outerwall, while, for curves A and D, the beam was directedfrom the outer to the inner wall (Fig. 1).

CFD-derived velocity profilesA 3D-mesh, equivalent to the geometry of the hydrau-

lic phantom, was generated in a well-validated computa-tional fluid dynamics package (Sepran, Sepra, Delft, TheNetherlands). The mesh consisted of approximately 2600elements distributed over 80 cross-sections that formed atube with a diameter of 3 mm. The tube described an arc ofa circle with radii of curvature of A) 47 mm, B) 23 mm, C)43 mm, and D) 27 mm. The cross-sectional resolution ofthe mesh was 0.15 mm2 in the middle of the tube, increas-ing to 0.075 mm2 at the wall. As the number of cross-sections was constant, the axial resolution changed withcurvature and varied between 1.3 mm to 2.34 mm, for acurvature of 23 mm to 47 mm, respectively.

Each element of the 3D-mesh contained 27 nodes,leading to approximately 27000 nodes per simulation. Ineach node, the full incompressible Navier-Stokes equationsare implemented. The non linear convective term in theNavier-Stokes equations was linearized with a Newton-Raphson approach. To obtain the pressure unknowns fromthe discrete Navier-Stokes equations, a penalty functionapproach was used. Eliminating the pressure unknownswith the penalty method and linearization of the convectiveterm result in a set of linear equations with velocity un-knowns. These equations were solved by a numerical solverof a well-validated commercial available finite elementpackage (Sepran, Sepra, Leiden, The Netherlands), whichwas implemented on a workstation (HP 715/80, HewlettPackard Nederland, Utrecht). Details and elaborate valida-tion of the solver have been presented before by differentlaboratories under different test conditions (Gijsen et al.1996; Gijsen et al. 1998; Krams et al. 1998; Van de Vosseet al. 1990; Wentzel et al. 2001a). We have defined thefollowing boundary conditions: parabolic, steady inflow of0.5 and 1.0 mL/s at the entrance of the vessel, constantstress gradient at the outlet and no-slip conditions at themesh wall. The fluid was modeled as a Newtonian fluidwith a density of 1050 kg/m3 and a viscosity of 1 cP. Thesevalues were chosen as they reflect those used in the flowphantom. CFD-derived velocity profiles were computedalong the same directions as those established by the ex-perimental conditions.

Data analysis: CFD- and Doppler-derived velocity pro-files accuracy

The CFD accuracy corresponding to the chosen

mesh density and shear stress gradients has been prelim-

666 Ultrasound in Medicine and Biology Volume 31, Number 5, 2005

inarily evaluated by comparing the CFD-derived velocityprofiles, Vcfd, in a straight tube with the analytical solu-tion, Va. The analytical solution was obtained from thesettings of the hydraulic system, the diameter of thevessel and the assumption of parabolic velocity profile.The average of the normalized difference, (Vcfd � Va)/Va, was used as a parameter of the accuracy. Subse-quently, both mesh density and numerical error wererepeatedly doubled, until a difference lower than 1%between the CFD-derived velocity field and the analyti-cal solution was obtained. The final mesh density wasthen used to derive all other CFD data.

To compare Doppler- with CFD-derived velocity pro-files, we calculated the projection of the CFD-derived 3 Dvelocity vector on the direction of the US beam. The mag-nitude of the projected velocity vector (Vcfd) was comparedwith the Doppler-derived velocity (VDoppler) by calculatingthe normalized difference [(Vcfd � VDoppler)/VDoppler*100].The difference between Doppler and CFD (�Doppler�cfd)velocities was defined as the average of this normalizeddifference (in %).

Data analysis: volume flowFluid flow was calculated from the experimental

data in two ways. The first approach was based on asingle-gate Doppler technique, where only the maximumvelocity was taken into consideration. This approachfollows the method most often used in (clinical) Dopplermeasurements. By assuming a parabolic velocity profile,the average velocity was estimated as 0.5 � maximumvelocity. Average velocity multiplied by vessel area pro-duced fluid flow (“fluid flow I”). In a second approach,the multigate Doppler technique was used to generate a3 D velocity profile from the 1-D Doppler-derived ve-locity profile scaled in the direction perpendicular to thedirection of Doppler measurements. Note that the 3 Dvelocity profile produced in this way is symmetricalaround the plane of curvature. Flow (in ml/s) was calcu-lated as the double integral of the local flux, which wasdefined as the product of local velocity and local area(“fluid flow II”). These two methods were compared tothe absolute flow, which was known from the setting ofthe flow phantom.

Local fluid flow was also calculated from the 3 DCFD-derived velocity field, similar as for the Dopplermeasurements, with the exception that now the entirevelocity field was available. These calculations wereperformed on the basis of the axial projected velocityvector (“fluid flow corrected for secondary velocity”)and of the total velocity vector (“fluid flow uncorrectedfor secondary velocity”). The effect of secondary veloc-ity upon fluid flow was determined by the ratio of these

fluid flow estimates (in %).

RESULTS

Figure 2 shows an example of the spectral profileobtained in bend C (Fig. 1). Power spectral densityobtained after averaging 100 spectral frames is coded ingrey levels, as function of frequency (kHz) on the hori-zontal axis and of depth (mm) on the vertical axis.Maximum frequencies are superimposed together withthe polynomial fit obtained through the LabVIEW™

“Householder” function. Figure 3 reports the distributionof secondary and axial velocities obtained with the CFDfor the cross- section located in the center of a bend with47 mm curvature.

Accuracy of the CFD and Doppler method in straighttubes

CFD-derived profiles showed an accuracy of 0.2 %[Fig. 4, (a) and (b)], for entrance flows of 0.5 and 1.0mL/s. Differences between the Doppler-derived velocityprofile and the analytical solution were 6.9% and 4.3%,for 0.5 and 1.0 mL/s entrance flows, respectively [Fig. 4,(c) and (d)]. Changing mesh density in the curved seg-ment changed the velocity profiles by 0.1%, i.e., by lessthan the error presented above. These data indicated thatthe CFD-derived velocity profiles could be used as ref-erence for the Doppler-derived measurements.

Velocity profiles in curved phantomsIn general, the Doppler-derived velocity profiles

Fig. 2. Example of the spectral profile obtained on the bend C(Fig. 1), averaging 100 spectral frames. Power spectral densityis coded in grey levels, as function of frequency (kHz) in thehorizontal axis and depth (mm) in vertical axis. Maximumfrequencies are displayed, together with the polynomial fitobtained through the LabVIEW™ “Householder” function.

behaved similarly to the CFD-derived velocity profiles,

Effect of curvature on Doppler flow ● R. KRAMS et al. 667

with skewness increasing with entrance velocity andcurvature (compare Figs. 5 and 6). The differences be-tween Doppler-derived and CFD-derived velocity pro-files (�Doppler�cfd) were small, ranging in the order of 5

Fig. 3. Example of CFD calculation, including the form andsize of the mesh. The cross-section used for all calculations waslocated in the middle of the mesh, to reduce entrance and outleteffects. The secondary velocity components (upper panel) andthe axial velocity components (lower panel) are indicated onthe right side of the figure. Note that the secondary componentsare less than 10% of the axial components.

Fig. 4. Accuracy of CFD measurements determined bycorresponding analytical solution a) and d). The Doppanalytical solution in c) and d). a) and c) of the right coobtained at 1.0 mL/s. The velocity components projected

angle of 60°). Note that, because of this choice, the values diff

to 12%, despite the large variations of hemodynamicconditions (Figs. 5, 6 and Table 1). The range of differ-ences between CFD- and Doppler-derived velocity pro-files was independent of entrance velocity and curvature(Table 1).

Fluid flowFluid flow calculated by assuming a parabolic pro-

file differed from real flow by between 2% and 25%,with a 12% average difference (Table 2). This differencedecreased to 1 to 14% (average: 6%) when the multigate-derived velocity profiles were considered (Table 2).

To evaluate the role of secondary velocity compo-nents on these estimates, fluid flow was computed byconsidering both the total velocity vector (hence includ-ing the contribution of transverse components) and theaxial-projected velocity vector. The results differed byquantities ranging between 0.3% and 2% (Table 2, lastcolumn).

DISCUSSION AND CONCLUSION

Doppler techniques have been used for decades tomeasure blood velocity and fluid flow in the humancardiovascular system (Doucette et al. 1992; Tortoli et al.2002; Tortoli et al. 1997; Tortoli et al. 2001). The first

rison of the CFD-derived velocity profile (O) with theived velocity profiles are displayed, together with theis obtained at 0.5 mL/s. b) and d) at the left column ishe ultrasound beam are displayed in this figure (Doppler

compaler-derlumnupon t

er by a factor of two with the axial velocity component.

ppler a

668 Ultrasound in Medicine and Biology Volume 31, Number 5, 2005

approaches estimated maximum velocity in a single sam-ple volume spanning the entire cross-section. After as-suming a parabolic velocity profile fluid flow was esti-mated. (Doucette et al. 1992) showed this approach to befeasible if vessel segments were not too curved.

Fig. 5. Comparison between Doppler- and CFD-derived vto D corresponds to bends A to bend D. For reference to tupon the ultrasound beam are displayed in this figure (Do

Fig. 6. Comparison between Doppler and CFD-derived velto D correspond to bends A to bend D. For reference to th

upon the ultrasound beam are displayed in this figure. (Doppler a

Both axially oriented forces and centrifugal forcesinfluence the kinetics of fluid elements moving in curvedsegments (Doucette et al. 1992; Friedman et al. 1986;Naruse and Tanishita 1996). While the axially orientedforces induce fluid flow, the centrifugal forces induce

profiles for bends A to D. Entrance conditions: 1 mL/s Arent bends see Fig. 1. The velocity components projectedngles: 105° for bends A and B, 75° for bends C and D).

rofiles for bends A to D. Entrance conditions: 0.5 mL/s. Arent bends, see Fig. 1. The velocity components projected

elocityhe diffe

ocity pe diffe

ngles: 105° for bends A and B, 75° for bends C and D).

Effect of curvature on Doppler flow ● R. KRAMS et al. 669

secondary (i.e., in-plane) velocity components not di-rectly contributing to volume flow. Furthermore, thedirection of the velocity vector is not parallel to thecenter line of the vessel segment under study and max-imum velocity shifts toward the outer wall (Gijsen et al.1999; Kang and Tarbell 1983)

The present study allowed evaluating the contribu-tion of secondary velocity components and the deviation

Table 1. Doppler-derived and CFD-derived maximumvelocity values and their difference in straight and curved

vessel segments

Dopplercm/s

CFDcm/s �Doppler-cfd %

Entrance flow 1.0 ml/s

Straight 14.2 14.2 4.3Bend A 5.6 5.9 9.0Bend B 6.4 6.6 12.4Bend C 6.8 6.8 7.9Bend D 7.7 5.8 10.1

Entrance flow 0.5 ml/s

Straight 7.0 7.1 6.9Bend A 3.2 2.9 7.5Bend B 3.2 3.6 5.1Bend C 4.4 3.8 8.4Bend D 3.1 3.0 11.1

The maximal of the velocity profiles in the direction of the US beamis reported. Doppler maximum value of the Doppler-derived velocityprofile (cm/s); CFD: maximum value of the CFD-derived velocityprofile (cm/s). �Doppler-cfd: average difference between CFD- and Dop-pler-derived velocity profiles (%). Note: the difference is obtained byaveraging all data points, while the maximum is based on a single datapoint. For description of bends A to D, see Fig. 1.

Table 2. Fluid flow calculated from Doppler-derive

Maximum velocitycm/s

Average veloccm/s

Entrance

Straight 28.3 14.2Bend A 21.7 15.5Bend B 24.6 17.3Bend C 26.3 17.0Bend D 27.7 17.4

Entrance

Straight 14.0 7.1Bend A 12.5 8.4Bend B 12.4 8.4Bend C 16.8 11.4Bend D 12.0 8.7

The maximal and the averages of the Doppler-derived velocity proficalculated from either half of the maximum velocity (“Parabolic profile

The effect of secondary velocity components was evaluated by comparing theon the axial velocity (�tf-af in %). For description of bends A to D, see Fig.

of the velocity profile from a parabolic profile on acurved phantom resembling the human coronary system.The nonparabolic velocity profiles were measured by amultigate technique (Tortoli et al., 1997). The accuracyof this method was first tested against an accurate CFD-derived velocity profile. Doppler-derived velocity pro-files exhibited the same shape the CFD-derived velocityprofiles and differed by quantities of less than 12%,despite large variations of hemodynamic conditions.

The multigate data were also used for fluid flowevaluation. To improve volume flow estimates on thebasis of Doppler profiles, we generated a 3-D velocityfield before calculation of fluid flow. This step wasnecessary because the velocity profiles were non sym-metric in the plane of curvature and classical conversiontechniques based upon symmetry did not work. To gen-erate the 3-D velocity field, we assumed a parabolicprofile perpendicular to the Doppler measurements. As aconsequence, flow was considered to have a symmetryperpendicular to the direction of the Doppler measure-ments. The latter assumption holds only if curvature is inplane. With this method, an error lower than 15% wasobtained, while, with the single-gate Doppler-basedmethod, the error was � 24%.

The role of the secondary velocity components onthe residual errors appeared to be of minor importance(see Table 2). This lack of importance was related to thesmall magnitude of the secondary component with re-spect to the axial component (i.e., 10%), caused by thelow curvatures of the phantom. These curvatures werechosen based upon physiological values measured inhuman coronary blood vessels and femoral arteries

city indices in straight and curved vessel segments

Fluid flow Iml/s

Fluid flow IIml/s �tf-af%

1.0 ml/s

1.00 1.00 00.76 1.05 0.70.87 1.14 0.50.93 1.01 0.40.98 0.96 0.6

0.5 ml/s

0.50 0.50 00.44 0.50 2.00.41 0.53 0.70.62 0.55 0.40.42 0.53 0.3

he direction of the vessel center-line are reported. Fluid flow (ml/s) isflow I”), or from the Doppler-derived velocity profile (“fluid flow II”).

d velo

ity

Flow

Flow

les in t”, “fluid

estimates of fluid flow based on the total velocity vector to those based1.

670 Ultrasound in Medicine and Biology Volume 31, Number 5, 2005

(Chang and Tarbell 1988; Krams et al. 1999a; Wentzel etal. 2001b; Wentzel et al. 2000).

One of the major limitations of the present studywas due to the difficulty in fixing a precise position of theprobe for each cross-section in the flow phantom. Smalldifferences between the expected and the actual experi-mental position’s could produce noticeable differences inthe detected velocity profiles. In addition, steady flowand Newtonian fluid were assumed because of theirsimplicity both in experimental and numerical condi-tions. While pulsatile conditions might have increasedthe secondary components, non-Newtonian fluid proper-ties may have reduced these components (Gijsen et al.1999). The overall effect is presently unknown and ex-trapolation to non-Newtonian and pulsatile conditionsmust be done with caution. On the other hand, the use ofwater instead of plasma should have minimal effects onthe present findings, because the conditions for compar-ison were flow-controlled and not pressure-controlled.Hence, the present conditions resulted in similar velocityprofiles for water and plasma, with a smaller pressuregradient for water.

Acknowledgments—Dr. R. Krams is an established investigator of theDutch Heart foundation of the Netherlands. Dr. J.J. Wentzel is appre-ciated for helping in developing some of the algorithms. This work wassupported by the Italian Ministry of Education, University and Re-search (COFIN 2002). Special thanks are due to Prof. Andrzej Nowicki(IPPT-PAN, Warsaw, Poland) for providing transducer used in theexperimental work.

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