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Effects of beam steering in pulsed-wave ultrasound velocity estimation
Ultrasound in Med. & Biol., Vol. 31, No. 8, pp. 1073–1082, 2005 Copyright © 2005 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/05/$–see front matter
doi:10.1016/j.ultrasmedbio.2005.04.005
● Original Contribution EFFECTS OF BEAM STEERING IN PULSED-WAVE ULTRASOUND VELOCITY ESTIMATION AARON H. STEINMAN,* ALFRED C. H. YU,† K. WAYNE JOHNSTON,†‡ and RICHARD S. C. COBBOLD† *Vivosonic Inc., Toronto, ON, Canada; †Institute of Biomaterials and Biomedical Engineering and ‡Department of Surgery, University of Toronto, Toronto, ON, Canada (Received 15 November 2004, revised 29 March 2005, in final form 7 April 2005)
Abstract—Experimental and computer simulation methods have been used to investigate the significance of beam steering as a potential source of error in pulsed-wave flow velocity estimation. By simulating a typical linear-array transducer system as used for spectral flow estimation, it is shown that beam steering can cause an angle offset resulting in a change in the effective beam-flow angle. This offset primarily depends on the F-number and the nominal steering angle. For example, at an F-number of 3 and a beam-flow angle of 70°, the velocity error changed from ⴚ5% to ⴙ 5% when the steering angle changed from ⴚ20° to ⴙ 20°. Much higher errors can occur at higher beam-flow angles, with smaller F-numbers and greater steering. Our experimental study used a clinical ultrasound system, a tissue-mimicking phantom and a pulsatile waveform to determine peak flow velocity errors for various steering and beam-flow angles. These errors were found to be consistent with our simulation results. (E-mail: [email protected]) © 2005 World Federation for Ultrasound in Medicine & Biology. Key Words: Ultrasound array, Flow velocity errors, Beam steering, Simulations, Pulsatile flow, Flow phantom experiments.
INTRODUCTION
source of error associated with the US system, namely, that associated with the use of beam steering. Geometric and technical considerations in the study of a patient have made the use of electronic US beam steering of considerable importance for color flow imaging and spectral flow velocity estimation. Studies have directly and indirectly investigated beam steering as a potential source of error in the estimation of the maximum velocity. For instance, Daigle et al. (1990) used a string phantom and a sinusoidal waveform to estimate this error for a linear array. A comparison of their results for steered and unsteered beams indicated that there was no noticeable difference. In fact, for cases in which the aperture is partially truncated, their steered results appear to perform better than the corresponding unsteered results. Winkler and Wu (1995), using a steady-flow model, also reported that the transverse Doppler equation (Newhouse et al. 1987, 1994) was not consistent with their experimental results for beam steering angles of ⱖ 20°. They indicated that this inconsistency could be caused by experimental errors. Subsequently, Steinman et al. (2001) performed steady-flow measurements with a linear array. They reported that, for a given beam-flow angle, the variations in maximum velocity estimates obtained with different beam-steering angles were gener-
Measurements of the peak velocity using ultrasound (US) spectral flow velocity estimation (also called spectral Doppler) is commonly used for determining the severity of a stenosis in the carotid, renal and peripheral arteries and for detecting stenoses in bypass grafts. In some institutions, it is the only preoperative diagnostic technique that is used before carotid endarterectomy and reoperative surgery on bypass grafts. However, a number of studies have shown that estimations of the peak velocities may be significantly in error (Daigle et al. 1990) and that these errors can range from ⫺4% up to 47% (Hoskins 1996). Several different potential sources of error have been suggested (Steinman 2004; Steinman et al. 2004a) and these can be classified as those associated with the US system itself, those associated with the technologist and the corresponding examination technique and those arising from the geometry of the vessel, such as its tortuosity. This paper examines one possible
Address correspondence to: Professor Richard S. C. Cobbold, Institute of Biomaterials and Biomedical Engineering, University of Toronto, Rosebrugh Bld., Toronto, Ontario M5S 3G9 Canada. E-mail: [email protected] 1073
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ally within experimental error. As a result, it was concluded that beam steering is not a significant source of error for steady-flow conditions. This finding was apparently supported by preliminary beam-steered sample volume (SV) simulation results (Steinman et al. 2004), which indicated that, at the same beam-flow angle, the characteristics of the beam-steered SV were not significantly different from that with no steering. However, as will be shown, a more detailed examination of the earlier results, together with the results of new experiments and simulations, have revealed the presence of important differences that can make significant contributions to the peak-velocity estimation error. Our purpose in this paper is to provide the results of a detailed study of the effects of beam steering on peak velocity estimation. Presented and analyzed are the results from a computer-simulation model together with experimental results obtained using a tissue-mimicking phantom under pulsatile flow conditions. Theory and definitions The estimated maximum velocity can be expressed in terms of the slow-time maximum frequency as (Jensen 1996):
max ⫽
c f max , 2f 0 cosF
(1)
where c is the speed of sound within the SV, fmax is the estimated maximum frequency obtained from the slowtime power spectrum, f0 is the frequency of the gated received signal (generally taken to be the transmit source center frequency) and F is the beam-flow angle. Four different transducer orientations with respect to a scatterer are illustrated in the scaled drawing of Fig. 1. The scaling corresponds closely to that used both in the simulations and experiments. Note that beam steering has been applied to transducers A and C. As such, the beams of transducers A, B and C are all focused at the same point with the same beam-flow angle (⬇ 60°). Based on this geometry, the transducer-flow angle (the physical angle) P is defined as the angle made by the normal of the transducer surface to the velocity vector; this value is different for all three transducer positions. We define the nominal steering angle N S as the difference between the nominal (assumed) beam-flow angle N F and N the transducer-flow angle (i.e., N ⫽ ⫺ ). It should S F P also be noted that, if the beam is steered using a constant aperture length, then the F-number (ratio of the focal distance to the effective length of the active aperture) decreases. This reduction occurs as a result of the effective aperture being reduced by a factor of cos(N S ) and the focal length being increased by a factor of 1/cos(N S ). For example, if N ⫽ 20°, these two changes cause the S
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Fig. 1. Scaled drawing to illustrate beam steering in the y-z plane with a linear array. Hatched or shaded areas indicate the active elements of the transducer aperture. Three different transducer orientations with the same focal point are shown as A, B and C; these correspond to the same nominal beam-flow angle of 60° and three different transducer-flow angles, P of 40, 60 and 80°, respectively. By definition, transducer A has positive nominal steering angle; for transducers C and D, angles are negative. Nonsteered lateral focal depth fl is the distance from center of the array to focal point (i.e., center of sample volume). Steered lateral focal depth is distance from center of active aperture to focal point (—) and is given by fls ⫽ fl/cos(N S ). Scaling corresponds to that used in simulations and experiments.
F-number to be reduced by nearly 12%. Thus, to maintain the same F-number with steering, the aperture length must be increased. For the purpose of analysis, we shall define two more quantities. The first is the offset angle that may arise because the true beam direction may differ from its nominal value. This quantity is defined as: ⌬ ⫽ FT ⫺ FN ⫽ ST ⫺ SN ,
(2)
where the superscript T stands for true and N for nominal. The second is the estimated maximum velocity percentage error, which is defined as: max ⫽ E%
max max N ⫺ T ⫻ 100% , max T
(3)
max where Tmax is the true maximum velocity and N is the nominal maximum velocity as obtained experimentally or by simulation. Note that a positive value of Emax % corresponds to an overestimation error and a negative value corresponds to an underestimation error.
SIMULATION METHODS Our simulations were based on software that was written, tested and evaluated over the past seven years. The code was originally developed for simulating Bmode imaging using 1-D and 2-D linear and curved arrays and was subsequently extended for use in pulsedwave velocity estimation (Steinman 2004; Steinman et al. 2004b). The simulator assumed propagation and scattering to be linear processes, even though it is known that
Beam steering errors ● A. H. STEINMAN et al.
Table 1. Transducer and propagation medium parameters Parameter
Values
Element width (mm) Element height (mm) Kerf (mm) Elevation lens model: n of subelements Elevation lens focus (mm) Lateral F-number, Center frequency, f0 (MHz) Apodization Attenuation coefficient (dB/(cm · MHz) Speed of sound (m/s)
0.27 4.0 0.03 70 11.25 2, 3 or 4 5.0 none 0.7 1540
the peak negative pressures in some commercial Doppler US systems can be sufficient to cause nonlinear propagation. However, Li et al. (1993), using commercial systems, were unable to detect any nonlinear effects on the slow-time power spectrum or the maximum frequency. Based on this finding, the assumption of linear propagation seems to be a reasonably good approximation. Beam steering and focusing are achieved by applying appropriate time delays to each transducer element within the chosen aperture for both the transmit and receive functions. The transmit and receive delays are equivalent if the transmit and receive focal points and their apodization functions are identical. These values were calculated using the scheme described by von Ramm and Smith (1983). The assumed simulation parameters closely match those of a commercial US system (Acuson 128XP/10; Mountain View, CA, USA) with an L7 transducer that incorporates a fixed lens in the elevation direction with a focal length of 11.25 mm. Although details of the assumed transducer properties have been previously reported (Steinman et al. 2004b), for convenience they are presented in Table 1 along with the properties of the transmission medium. To calculate the response produced by a given transmit waveform, the impulse-response method was used with a sampling frequency of 0.1 GHz. In the elevation direction, each transducer element was divided into 70 rectangular subelements. The trapezoidal method (Jensen and Svendsen 1992) was used to calculate the impulse response for each subelement. The SV axial length is primarily determined by the transmit pulse and receive gate durations. For the transmit pulses, two sinusoidal pulses were used, whose front and back edges were cosine-tapered over a single cycle. The parameters and characteristics of these pulses are summarized in Table 2. For the receive gate, a rectangular time gate was used. The gate timings correspond to the times at which the edges of the time gate bisect the signals received from scatterers located at the desired
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axial limits of the SV. Such a gate has been called the “compromise gate” (Steinman et al. 2004b). Strategy for sample volume comparisons To determine the SV shape on the y-z plane (see Fig. 1), a 2-D grid of scatterers spaced by 0.05 mm was used. The gated received signal energy from each individual scatterer was determined and, by combining the results for all scatterers, the SV shape was calculated. To provide quantitative comparisons of the effects of beam steering on the SV shape, careful attention was paid to certain issues concerning details of the methodology. In the present study, to compare the SV energy distributions using a color display, the steered SV energies were normalized to the corresponding nonsteered values and a display threshold of ⫺30 dB was used. This comparison approach should be contrasted with the method used in our previous study (Steinman et al. 2004b), in which self-normalization and a threshold of ⫺20 dB were used. With this new comparison approach, the true beam-flow angle was determined for each SV by simultaneously rotating the self-normalized ⫺6, ⫺12 and ⫺18 dB contours of the SV until they were judged to be best matched to the corresponding nonsteered SV. It was estimated that the accuracy of this method was within ⫾ 0.5°. The centers of the two desired SV axial lengths (ᐉP ⫽ 1.5 and 5 mm) were chosen to coincide with the two lateral focal depths selected for our studies. These two depths corresponded to points close (15 mm) to and well away (35 mm) from the elevation focus. In addition to the above variables, two F-numbers, F2 and F4 were studied, leading to eight permutations in all. Each of the eight SVs was simulated for five nominal steering angles: 0°, ⫾ 20° and ⫾ 30°. In our experimental studies, when a beam steering of ⫾ 20° was used at a focal depth of 35 mm, aperture truncation effects caused the F-number to be increased from 2 to 3. Consequently, in addition to the above permutations, simulations were also performed for F3 at a lateral focal depth of 35 mm with ᐉP ⫽ 1.5 mm using the same steering angles as listed above. All simulations were performed using an attenuation of 0.7 dB/(cm · MHz) although, for some of the SVs, they were repeated without attenuation. The relatively narrow bandwidth of the transmit pulses (Table 2) en-
Table 2. Properties of transmitted pulses Desired axial length, ᐉP
1.5 mm
5 mm
Center frequency, f0 (MHz) Number of cycles Duration of transmit pulse, P (s) Length of transmit pulse (mm) Fractional bandwidth
5 9 1.8 1.39 0.107
5 15 3.0 2.31 0.061
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N Fig. 2. Effects of beam steering angle (S) on SV energy profiles in axial-lateral plane. (a) N S ⫽ 0°, (b) S ⫽ 20° and N (c) S ⫽ 30°. The origin (0,0) is at the lateral steered focus in flow coordinate system. Parameters used: F2, desired axial length ⫽ 1.5 mm, fl ⫽ 15 mm and FN ⫽ 60°. Note that (b) and (c) are normalized to maximum value in (a).
abled the effects of frequency-dependent dispersion to be ignored. To extend the lateral region over which the SV can be positioned, a reduced aperture size may be used when the center of the aperture approaches either end of the transducer array. Using a small hydrophone, Hoskins et al. (1999) measured the aperture size of two commercial systems, one of which was the Acuson 128XP/10 with an L7 transducer. They observed significant aperture truncation for the L7 transducer when a large aperture setting was selected. As appropriate, this truncation effect was included in the simulation modeling. Moreover, for the smaller apertures used in this study, a nearly constant F-number with beam steering was maintained, in accord with that used in the Acuson system. This was achieved by increasing the aperture to compensate for the changes in the focal length and effective aperture with beam steering, as described in the previous section. Single-scatterer power spectrum To study the power spectrum, the simulations assumed a single scatterer moving at 35 cm/s through the lateral steered focus and a pulse-repetition interval of 80 s. Each fast time-gated signal was demodulated, enabling
the slow time signal to be obtained and then analyzed in terms of its slow time-power spectrum. Each of the eight SV permutations was simulated at a nominal beam-flow angle of 60° for five nominal steering angles: these were 0°, ⫾ 20° and ⫾ 30° (i.e., transducer-flow angles of 30°, 40°, 60°, 80° and 90°). SIMULATION RESULTS AND DISCUSSION Sample volume results Typical SV energy distribution results on the plane x ⫽ 0 are shown in Fig. 2. These plots clearly indicate that the energy within the contours decreases with increasing beam steering. This decreasing trend was seen for both lateral focal depths (15 and 35 mm), even when the simulations were performed without attenuation and without elevation lens focusing. For the distributions shown, the change from N S ⫽ 0° to 30° was approximately ⫺7 dB. To determine whether or not the steered and unsteered SVs are in exact alignment, self-normalized contour graphs were obtained, as shown in Fig. 3. It is evident from these plots that SV misalignments were
Fig. 3. Energy contours for comparing the effects of beam steering on the SV profile. (a) ⫺12 dB contours for N S ⫽ ⫺30°, (dashed), 0° (solid) and 30° (dotted); (b) ⫺6 dB contours for N S ⫽ ⫺30° (dashed), 0° (solid) and 30° (dotted); and (c) ⫺6 dB contours for N S ⫽ 0° (solid), 20° (dotted) and 30° (dashed). The contours are for the same conditions as listed in Fig. 2, except they are self-normalized.
Beam steering errors ● A. H. STEINMAN et al.
present when beam steering was used. The misalignment consisted of: 1. a small change in the position of the SV, 2. a small change in the shape in the contours, and 3. a significant change in the true beam-flow angle. The first two changes, from a practical standpoint, are not significant but, as will be seen, the change in the true angle is important. It seems likely that our new technique (see Simulation Methods section) for comparing the effects of beam steering on the SV helped us to reveal the sources of discrepancy in the true beam-flow angle when beam steering is used. Table 3 shows that the true beam-flow angle (TF ⫽ ⌬ ⫹ N F ) changes as the nominal steering angle increases. The results indicate the importance of the Fnumber in determining the offset angle. In particular, for a fixed focal depth, the offset magnitude reduces as the aperture size is decreased (F-number increased). This reduction can be seen by comparing data columns 1 and 3 for a focal depth of 15 mm and by comparing data columns 4, 5 and 6 for a lateral focal depth of 35 mm. It should be noted that virtually identical results were obtained for the two SV lengths (data columns 1 and 2) at the same F-number. The table also shows that, as the focal depth is increased from 15 to 35 mm for a constant F-number (1st and 4th data columns), there appears to be a small reduction in the magnitude of the offset angle. The notion of the offset angle as defined by eqn (2) and its impact on the true beam-flow angle can be better understood by examining the qualitative sketches in the panels of Fig. 4. They provide an intuitive description showing that the offset angle can affect the velocity estimation accuracy, depending on the sign of the beam-steering angle and whether the physical angle is greater or less than 90°. By examining the ratio between the nominal velocity and the true velocity, a more quantitative description of this phenomenon can be provided. Based on eqns (1), (2) and (3), this ratio is given by:
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N cos(FN ⫹ ⌬) , ⫽ T cos(FN)
(4a)
which enables the percentage velocity error to be expressed as: max E% ⫽
冋
册
cos(FN ⫹ ⌬) ⫺ 1 ⫻ 100% . cos(FN)
(4b)
This expression, together with the assumption that the simulated values for the steering offset angles given in the last two columns of Table 3 are correct, enabled the graphs shown in Fig. 5 to be obtained. In this figure, the percentage velocity errors are shown as a function of the nominal beam-flow angle for two nominal beam steering angles and two F-numbers. The asymmetry of the results about the 90° axis should especially be noted. From another perspective, there is symmetry in the results between positive steering below 90° and negative steering above 90°. This symmetry can be understood by examining the physical location of the active aperture, such as transducers A and D shown in Fig. 1 (e.g., the active aperture for N F ⫽ 60° generated from ⫹ 20° steering is identical to the one for N F ⫽ 120° generated from ⫺20° steering). Under the same argument, symmetry also exists in the results between negative steering below 90° and positive steering above 90°. Single-scatterer power spectra A typical set of power spectra from the single scatterer moving at 35 cm/s through the center of the wide aperture SV at a beam-flow angle of 60° is shown in Fig. 6 for five beam-steering angles. In the absence of any spectral broadening, the ideal slow time-frequency generated by a single scatterer moving at 35 cm/s for N F ⫽ 60° when f0 ⫽ 5 MHz can be calculated from eqn (1) as 1136 Hz. It can be seen that, as the steering increases, the peak amplitudes decrease and the mean frequency changes in different directions depending on the sign of
Table 3. Offset angles (⌬) for various steering angles Sample volume characteristics Axial length, mm Unsteered focal depth, mm F-number Unsteered aperture, mm Nominal steering/angle, N S* ⫺30° ⫺20° 0° 20° 30° * Offset angle, ⌬ (⫾ 0.5°)
1.5 15 2.0 7.5 3° 2° 0° ⫺2° ⫺3°
5 15 2.0 7.5 3° 2° 0° ⫺2° ⫺3°
1.5 15 4.0 3.75
1.5 35 2.0 17.5
1.5° 1° 0° ⫺1° ⫺1.5°
2.5° 1.5° 0° ⫺1.5° ⫺2.5°
1.5 35 3.0 13.2 2° 1° 0° ⫺1° ⫺2°
1.5 35 4.0 8.8 1° 0.5° 0° ⫺0.5° ⫺1°
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Fig. 4. The effect of (a) and (b) negative and (c) and (d) positive beam steering on velocity estimation accuracy for transducerflow angles of ⬍ and ⬎ 90°. ( · · · · ) Actual beam direction, (—) beam direction in absence of any angle offsets caused by beam steering. Arrow ⫽ scatterer velocity direction. (a) and (d) correspond to negative velocity error and (b) and (c) to a positive error.
the steering angle. The shifts of the steered spectra away from the nonsteered peak indicate the possibility of changes in the true beam-flow angle. Moreover, the asymmetry of the steered power spectra suggests a different scatterer path through the SVs. We also found that, for a smaller aperture (F4) with its wider SV dimension (Steinman et al. 2004a), the power spectra had a smaller band width with a mean frequency close to that of the nonsteered beam. Moreover, the trends found in this series of studies were independent of attenuation, lateral focal depth and elevation focusing although, as expected, the amplitudes differ.
Fig. 5. Percentage velocity error for nominal beam steering angles of 20° and ⫺20° for F-numbers 3 and 4. Simulation results are based on the last two columns of Table 3.
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Fig. 6. Typical slow-time power spectra generated by a single scatterer moving at 35 cm/s through the center of a 1.5 mm SV at a depth of 15 mm for the wide aperture (F2), FN ⫽ 60° and five beam steering angles. Solid line, S ⫽ 0°; black dotted line, N N S ⫽ ⫺20°; grey dotted line, S ⫽ 20°; black dashed line, N ⫽ ⫺30°; grey dashed line, N S S ⫽ 30°.
EXPERIMENTAL METHODS Experimental system The pulsatile flow system shown in Fig. 7a is similar in design to a previous design used for steady-flow experiments (Steinman et al. 2001). To ensure accurate positioning of the transducer, a custom-designed mechanical system with three degrees of freedom was used. The flow phantom, which was designed to model a nonstenosed artery, used a 7.9-mm inner diameter latex tube with a wall thickness of 1.6 mm, embedded at a depth of 15 mm in a tissue-mimicking medium (R.G. Shelley Inc., Toronto, ON, Canada) (Rickey et al. 1995). The bloodmimicking fluid, whose properties are listed in Table 4, was based on that described by Ramnarine et al. (1998). Nylon beads (Orgasol™ ELF Autochem, Paris, France) with a mean diameter of 5 m were used as scatterers at a concentration corresponding to a hematocrit of 5%. Although this hematocrit value is considerably less than that of normal blood, it can effectively circumvent nonNewtonian viscous effects and settling of scatterers in the flow tube and, thus, there are fewer experimental difficulties that complicate the estimation of peak flow velocity. Furthermore, it is reasonable to expect that, at the 5% hematocrit level, the characteristics of the ensemble-averaged slow time-power spectra would be the same as those for a higher hematocrit under the same flow conditions. The flow waveform shown in Fig. 7b is similar to that of the carotid artery (Shehada et al. 1993), but has reduced flow in the diastolic region. To extend the duration of the peak quasistationary flow region, a frequency of 40 flow cycles per min was selected. The Womersley number for the fundamental frequency was
Beam steering errors ● A. H. STEINMAN et al.
Fig. 7. System used for pulsatile flow experiments. (a) The flow bypass and gate valves determine flow through the phantom. The ballast, combined with the filter, was designed to minimize the amount of air bubbles pumped through the system (Lubbers 1999). All experiments used an Acuson 128XP/10 system (7-MHz B-mode, 5-MHz spectral Doppler) with an L7 transducer. (b) Flow waveform.
4.3 and the time-averaged Reynold’s number was 305. The values and errors associated with the various parameters of our flow system are listed in Table 4. It should be noted that the changes in the speed of sound between tissue and fluid (and vice versa) were accounted for in calculating the nominal beam-flow angle. Signal acquisition and processing For the waveform used, the peak flow can be considered as quasistationary over a time interval of ⬇20 ms. To achieve a good signal-to-noise ratio for the spectral estimation over this relatively short time interval, ensemble averaging was used. However, because there were fluctuations in the flow waveform period and no accurate synchronizing signal was available, a waveform alignment technique was used to ensure that the averaging was being performed on the same signal segments. The alignment method consisted of cross-correlating the flowmeter signal to determine the time interval between successive pulses. The resulting alignment error was estimated to be ⫾ 5 ms, corresponding to half of the flowmeter sampling period. Using a 20-ms Tukey window for each flow pulse, the power spectrum was obtained using a standard fast Fourier transform method. This procedure was repeated for each waveform and the ensemble averaged power spectrum was determined by using a minimum of 40 cycles. The maximum frequency was estimated from the integrated power spectrum using the signal-noise slope intersection method proposed by Steinman et al. (2001). In this method, the integrated power spectrum is modeled with a transition (knee) region separating the signal and noise regions. The slope in the transition region can be considered to be equal to the sum of the contribution from the signal region and the complementary contribution from the noise region. The estimated maximum
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frequency is defined to be in the transition region for a specific fractional signal contribution. Consequently, the maximum frequency is not taken as the limit of no signal (at the noise end of the transition region). For these experiments, a fractional signal contribution of 0.3 was selected to optimize the estimation performance at a beam-flow angle of 60°. Tests have shown that, under these conditions, the signal-noise slope intersection method is approximately independent of the signal-tonoise ratio (Steinman 2004). The true maximum center-line flow velocity was calculated from the digitized volume flow waveform. Specifically, this value was determined by using the Womersley equation and a Fourier analysis of the volume flow waveform (Evans and McDicken 2000). A maximum velocity was computed for each individual flow cycle and subsequently averaged over all of the flow pulses, yielding our reference value of Tmax ⫽ 98 ⫾ 6 cm/s. Protocol The experiments were conducted with the lateral transducer focal depth set to 35 mm under nonsteered conditions: the corresponding steered focal depth was 37.2 mm (35/cos 20°). At this focal depth, the nonsteered beam measurements for two different aperture sizes were available and used, 17.4 mm and 8.7 mm, corresponding to F2 and F4, respectively. For the smaller aperture, the steered aperture was identical to the unsteered value, so that the F-numbers were the same (i.e., F4). However, when the beam was steered by N X ⫽ ⫾ 20° at the larger aperture setting, aperture truncation was present, causing the F-number to be increased from the no-steering value of F2 to F3. The experiments were conducted in 5° increments over a range of transducer-flow angles (P) from 50° to 130° and, at each angle, readings were taken for nominal steered angles (N S ) of ⫺20°, 0° and 20°. Note that the spectral gain was set just below saturation for the strongest signal case and kept constant for the
Table 4. Experimental values and associated errors Property
Value ⫾ error
Units
Speed of sound in phantom Speed of sound in fluid Fluid density Fluid dynamic viscosity Fluid attenuation Transducer positioning Transducer positioning angle Flow measurement error (reading ⫹ calibration ⫹ drift) (%) Beam/flow angle Beam/flow angle Center-line maximum flow velocity Flow signal alignment
1540 ⫾ 10 1547 ⫾ 3 1042 ⫾ 1 3.74 ⫾ 0.1 0.06 ⫾ 0.01 ⫾ 0.5 ⫾ 0.5
m/s m/s kg/m3 cP dB/(cm · MHz) mm degrees
⫾5 30 ⫾ 2.6 60 ⫾ 0.9 98 ⫾ 6 ⫾5
L/min degrees degrees cm/s ms
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Fig. 8. Comparison of velocity estimation errors for nominal beam steering angles of 0°, ⫺20°, 20° and FN ⫽ 60°. (a) F2 (unsteered), F3 (steered); (b) F4 (both steered and unsteered). To improve clarity, the SD of data in the horizontal (angle) direction is shown as bars at top of each graph.
experiments. Also, the center of the SV was kept at the same location at the center of the vessel when beam steering was initiated. In this region, the velocity profile is fairly constant around peak systole. EXPERIMENTAL RESULTS AND DISCUSSION Beam-flow angle-dependence The maximum velocity error is plotted in Fig. 8 for both steered and unsteered beams at various beam-flow angles and using the aperture settings previously noted. For Fig. 8a, the nonsteered F-number was 2 and the steered was 3 but, for both the steered and nonsteered results in Fig. 8b, it was 4. It should be noted that physical restrictions on the transducer placement prevented measurements from being made for certain ranges of beam-flow angles and, hence, discontinuity gaps are seen in the experimental results. For the nonsteered case, our pulsatile flow results
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are quite similar to those previously reported by Hoskins (1996) using a string phantom and to those reported by Steinman et al. (2001) using a steady-flow tissue-mimicking phantom. The effects of changes in aperture size, as seen by comparing the results of Fig. 8a and b, are also consistent with prior observations. Specifically, for a reduced aperture (larger F-number), Fig. 8b shows that the velocity estimation error is decreased relative to Fig. 8a. Unlike the simulation results for determining the offset angle error, the experimental maximum velocity error involves additional error contributions from components that may not be associated with beam steering. Some of the primary factors that contribute to the experimental errors have been summarized in the first table of the study by Christopher et al. (1995). Among these factors, intrinsic spectral broadening is well established as a major error source for both steered and nonsteered conditions. It is especially relevant to our measurements, because they were made at peak flow, during which the velocity profile is fairly constant over the SV, thereby making the effects of extrinsic spectral broadening insignificant. For the nonsteered results, it should be noted that the error rapidly increases as the beam-flow angle increases. The error increase can be explained by noting that the amplitude modulation resulting from the finite scatterer transit time through the SV concomitantly decreases as the beam-flow angle increases and, thus, the extent of intrinsic spectral broadening is more prominent at larger beam-flow angles (Newhouse et al. 1987). Spectral broadening may also arise in the form of frequency modulation caused by deviations in the beam-flow angle as scatterers move through the SV (Ata and Fish 1991; Guidi et al. 2000). Hoskins et al. (1999) showed that these changes in the subtended angle can be significant for a phased linear array and can, therefore, contribute to the velocity estimation error. Effects of beam steering Figure 8a shows that, as the beam-flow angle approaches 90° from the left-hand side, the percentage velocity error for a negative beam steering angle (N S ⫽ ⫺20°) is less than that forN ⫽ 0°. On the other hand, as S 90° is approached from the right-hand side of the graph, the error for negative steering is greater than that for N S ⫽ 0°. The reverse is true when the beam steering angle is positive (N S ⫽ 20°). The asymmetry in the measured velocity estimation errors can also be seen in Fig. 8b, although to a lesser extent. These results indicate the presence of an underlying process that affects the velocity errors.
Beam steering errors ● A. H. STEINMAN et al.
COMPARISON OF EXPERIMENTAL AND SIMULATION RESULTS The conditions under which the nonsteered experimental results of Fig. 8a were obtained are equivalent to those of the fourth data column in Table 3. Conversely, for the steered results, because aperture truncation caused the F-number to be increased to 3, the experimental conditions correspond to the fifth data column in Table 3. As previously noted, the experimental results in the absence of steering contain systematic contributions to the velocity error from several sources. In the presence of steering, it is reasonable to assume that these systematic sources are also present together with the errors caused by the offset angle. Hence, the experimental velocity errors shown in Fig. 8 are larger in magnitude than the theoretical errors shown in Fig. 5, and the asymmetry of the errors about 90° is identical in form for both figures. We also qualitatively compared the intensity of the received signals by observing the recorded image displays. Such a comparison was performed for a fixed beam-flow angle for a constant spectral gain. It was found that the intensity was less when steering was used, thereby providing some experimental support for the SV results of Fig. 2 and the single-scatterer results of Fig. 6. DISCUSSION AND CONCLUSIONS This paper set out to establish whether or not electronic beam steering is a significant source of error in linear-array pulsed-wave US velocity-estimation systems. The experimental results and the corresponding conclusions are qualitatively consistent with the SV simulations, the single-scatterer simulations and the pulsatile-flow phantom experiments. For both experimental and simulation results, one or more of several factors could cause the velocity error. One factor is related to the delays used to achieve the required beam steering on both transmit and receive. For the purpose of calculating the delays, each transducer element in the active aperture was generally assumed to be a point source/receiver located at the center of each element. However, the transducer elements are 2-D structures, and the cylindrical lens used for focusing in the elevation direction is 3-D in nature (even though it is commonly simulated as a 2-D structure). The assumption of point-like transducer elements may, thus, no longer be valid when the 2-D aperture dimensions of the transducer become comparable with the lateral focal depth. Such an assumption might be a possible cause of the angle offsets, but further simulation investigations are needed. Understanding the source of the offset angle error would enable a correction table to be deduced
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either to correct the delays or to provide a corrected beam-flow angle. Four primary conclusions can be derived from the results and discussions. 1. The amplitude of the gated received signal (the energy in the SV) decreases when steering is increased. 2. A greater degree of beam steering leads to a greater beam-flow angle offset that, in turn, may lead to further errors in maximum velocity estimation. 3. The degree of beam steering offset is not dependent on SV axial length but, rather, on the F-number. Also, a lateral focal depth that is farther away from the elevation focus slightly reduces this offset. 4. The maximum velocity error curves for beam steering exhibit asymmetry in their characteristics as a function of the beam-flow angle. From a clinical perspective, beam steering is typically used at beam-flow angles of 60° or less. In this region, the errors associated with beam steering, although they exist, are probably not very significant compared with other errors, such as those caused by spectral broadening. In addition, at these typical beamflow angles, the beam is steered away from 90°, which will actually cause a reduction in the overestimation error of the maximum velocity. However, at higher beam-flow angles, larger beam steering angles (⬎ 20°) and steering toward 90°, beam steering errors can become large. Acknowledgements—The authors thank the Canadian Institutes of Health Research (K.W. Johnston and R.S.C. Cobbold) and the R. Fraser Elliott Chair (K.W. Johnston) for partial funding support. In addition, they are grateful to Elaine Lui for her helpful insights, Dr. G. Meng for assistance in the preparation of the blood-mimicking fluid and Dr. J. Myers for help with the experimental design. They are also pleased to acknowledge the assistance of Acuson (Siemens) for providing information on their L7 transducer.
REFERENCES Ata OW, Fish PJ. Effect of deviation from plane wave conditions on the Doppler spectrum from an ultrasonic blood flow detector. Ultrasonics 1991;29:395– 403. Christopher DA, Burns PN, Hunt JW, Foster FS. The effect of refraction and assumed speeds of sound in tissue and blood on Doppler ultrasound blood velocity measurements Ultrasound Med Biol 1995;21:187–201. Daigle RJ, Stavros AT, Lee RM. Overestimation of velocity and frequency values by multielement linear array Dopplers. J Vasc Technol 1990;14:206 –213. Evans DH, McDicken WN. Doppler ultrasound: physics, instrumentation and signal processing, 2nd ed. West Sussex, UK: Wiley, 2000. pp. 5–26. Guidi G, Licciardello C, Falteri S. Intrinsic spectral broadening (ISB) in ultrasound Doppler as a combination of transit time and local geometrical broadening. Ultrasound Med Biol 2000;26:853– 862. Hoskins PR. Accuracy of maximum velocity estimates made using Doppler ultrasound systems. Br J Radiol 1996;69:172–177. Hoskins PR, Fish PJ, Pye D, Anderson T. Finite beam-width ray model for geometric spectral broadening. Ultrasound Med Biol 1999;25: 391– 404.
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Jensen JA. Estimation of blood velocities using ultrasound: A signal processing approach. Cambridge, UK: Cambridge University Press, 1996. pp. 86 –110. Jensen JA, Svendsen NB. Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers. IEEE Trans Ultrason Ferroelec Freq Control 1992;39:262–267. Li S, McDicken WN, Hoskins PR. Nonlinear propagation in Doppler ultrasound. Ultrasound Med Biol 1993;19:359 –364. Lubbers J. Application of a new blood-mimicking fluid in a flow Doppler test object. Eur J Ultrasound 1999;9:267–276. Newhouse VL, Censor D, Vontz T, Cisneros JA, Goldberg BB. Ultrasound Doppler probing of flows traverse with respect to beam axis. IEEE Trans Biomed Eng 1987;34:779 –789. Newhouse VL, Faure P, Cathignol D, Chapelon J-Y. The transverse Doppler spectrum for focused transducers with rectangular apertures. J Acoust Soc Am 1994;95:2091–2098. Ramnarine KV, Nassiri DK, Hoskins PR, Lubbers J. Validation of a new blood- mimicking fluid for use in Doppler flow test objects. Ultrasound Med Biol 1998;24:451– 459. Rickey DW, Picot PA, Christopher DA, Fenster A. A wall-less vessel phantom for Doppler ultrasound studies. Ultrasound Med Biol 1995;21:1163–1176.
Volume 31, Number 8, 2005 Shehada RE, Cobbold RSC, Johnston KW, Aarnink R. Three-dimensional display of calculated velocity profiles for physiological flow waveforms. J Vasc Surg 1993;17:656 – 660. Steinman AH. Errors in phased array pulse-wave ultrasound velocity estimation systems. Ph.D. thesis. University of Toronto, 2004. Steinman AH, Lui EYL, Johnston KW, Cobbold RSC. Beam steering in pulsed Doppler ultrasound velocity estimation. IEEE Ultrason Symp Proc 2004a;3:1745–1748. Steinman AH, Lui EYL, Johnston KW, Cobbold RSC. Sample volume shape for pulsed flow velocity estimation using a linear array. Ultrasound Med Biol 2004b;30:1409 –1418. Steinman AH, Tavakkoli J, Myers JG Jr, Cobbold RSC, Johnston KW. Sources of error in maximum velocity estimation using linear phased-array Doppler systems with steady flow. Ultrasound Med Biol 2001;27:655– 664. von Ramm OT, Smith SW. Beam steering with linear arrays. IEEE Trans Biomed Eng 1983;30:438 – 452. Winkler AJ, Wu J. Correction of intrinsic spectral broadening errors in Doppler peak velocity measurements made with phased sector and linear array transducers. Ultrasound Med Biol 1995;21: 1029 –1035.
doi:10.1016/j.ultrasmedbio.2005.04.005
● Original Contribution EFFECTS OF BEAM STEERING IN PULSED-WAVE ULTRASOUND VELOCITY ESTIMATION AARON H. STEINMAN,* ALFRED C. H. YU,† K. WAYNE JOHNSTON,†‡ and RICHARD S. C. COBBOLD† *Vivosonic Inc., Toronto, ON, Canada; †Institute of Biomaterials and Biomedical Engineering and ‡Department of Surgery, University of Toronto, Toronto, ON, Canada (Received 15 November 2004, revised 29 March 2005, in final form 7 April 2005)
Abstract—Experimental and computer simulation methods have been used to investigate the significance of beam steering as a potential source of error in pulsed-wave flow velocity estimation. By simulating a typical linear-array transducer system as used for spectral flow estimation, it is shown that beam steering can cause an angle offset resulting in a change in the effective beam-flow angle. This offset primarily depends on the F-number and the nominal steering angle. For example, at an F-number of 3 and a beam-flow angle of 70°, the velocity error changed from ⴚ5% to ⴙ 5% when the steering angle changed from ⴚ20° to ⴙ 20°. Much higher errors can occur at higher beam-flow angles, with smaller F-numbers and greater steering. Our experimental study used a clinical ultrasound system, a tissue-mimicking phantom and a pulsatile waveform to determine peak flow velocity errors for various steering and beam-flow angles. These errors were found to be consistent with our simulation results. (E-mail: [email protected]) © 2005 World Federation for Ultrasound in Medicine & Biology. Key Words: Ultrasound array, Flow velocity errors, Beam steering, Simulations, Pulsatile flow, Flow phantom experiments.
INTRODUCTION
source of error associated with the US system, namely, that associated with the use of beam steering. Geometric and technical considerations in the study of a patient have made the use of electronic US beam steering of considerable importance for color flow imaging and spectral flow velocity estimation. Studies have directly and indirectly investigated beam steering as a potential source of error in the estimation of the maximum velocity. For instance, Daigle et al. (1990) used a string phantom and a sinusoidal waveform to estimate this error for a linear array. A comparison of their results for steered and unsteered beams indicated that there was no noticeable difference. In fact, for cases in which the aperture is partially truncated, their steered results appear to perform better than the corresponding unsteered results. Winkler and Wu (1995), using a steady-flow model, also reported that the transverse Doppler equation (Newhouse et al. 1987, 1994) was not consistent with their experimental results for beam steering angles of ⱖ 20°. They indicated that this inconsistency could be caused by experimental errors. Subsequently, Steinman et al. (2001) performed steady-flow measurements with a linear array. They reported that, for a given beam-flow angle, the variations in maximum velocity estimates obtained with different beam-steering angles were gener-
Measurements of the peak velocity using ultrasound (US) spectral flow velocity estimation (also called spectral Doppler) is commonly used for determining the severity of a stenosis in the carotid, renal and peripheral arteries and for detecting stenoses in bypass grafts. In some institutions, it is the only preoperative diagnostic technique that is used before carotid endarterectomy and reoperative surgery on bypass grafts. However, a number of studies have shown that estimations of the peak velocities may be significantly in error (Daigle et al. 1990) and that these errors can range from ⫺4% up to 47% (Hoskins 1996). Several different potential sources of error have been suggested (Steinman 2004; Steinman et al. 2004a) and these can be classified as those associated with the US system itself, those associated with the technologist and the corresponding examination technique and those arising from the geometry of the vessel, such as its tortuosity. This paper examines one possible
Address correspondence to: Professor Richard S. C. Cobbold, Institute of Biomaterials and Biomedical Engineering, University of Toronto, Rosebrugh Bld., Toronto, Ontario M5S 3G9 Canada. E-mail: [email protected] 1073
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ally within experimental error. As a result, it was concluded that beam steering is not a significant source of error for steady-flow conditions. This finding was apparently supported by preliminary beam-steered sample volume (SV) simulation results (Steinman et al. 2004), which indicated that, at the same beam-flow angle, the characteristics of the beam-steered SV were not significantly different from that with no steering. However, as will be shown, a more detailed examination of the earlier results, together with the results of new experiments and simulations, have revealed the presence of important differences that can make significant contributions to the peak-velocity estimation error. Our purpose in this paper is to provide the results of a detailed study of the effects of beam steering on peak velocity estimation. Presented and analyzed are the results from a computer-simulation model together with experimental results obtained using a tissue-mimicking phantom under pulsatile flow conditions. Theory and definitions The estimated maximum velocity can be expressed in terms of the slow-time maximum frequency as (Jensen 1996):
max ⫽
c f max , 2f 0 cosF
(1)
where c is the speed of sound within the SV, fmax is the estimated maximum frequency obtained from the slowtime power spectrum, f0 is the frequency of the gated received signal (generally taken to be the transmit source center frequency) and F is the beam-flow angle. Four different transducer orientations with respect to a scatterer are illustrated in the scaled drawing of Fig. 1. The scaling corresponds closely to that used both in the simulations and experiments. Note that beam steering has been applied to transducers A and C. As such, the beams of transducers A, B and C are all focused at the same point with the same beam-flow angle (⬇ 60°). Based on this geometry, the transducer-flow angle (the physical angle) P is defined as the angle made by the normal of the transducer surface to the velocity vector; this value is different for all three transducer positions. We define the nominal steering angle N S as the difference between the nominal (assumed) beam-flow angle N F and N the transducer-flow angle (i.e., N ⫽ ⫺ ). It should S F P also be noted that, if the beam is steered using a constant aperture length, then the F-number (ratio of the focal distance to the effective length of the active aperture) decreases. This reduction occurs as a result of the effective aperture being reduced by a factor of cos(N S ) and the focal length being increased by a factor of 1/cos(N S ). For example, if N ⫽ 20°, these two changes cause the S
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Fig. 1. Scaled drawing to illustrate beam steering in the y-z plane with a linear array. Hatched or shaded areas indicate the active elements of the transducer aperture. Three different transducer orientations with the same focal point are shown as A, B and C; these correspond to the same nominal beam-flow angle of 60° and three different transducer-flow angles, P of 40, 60 and 80°, respectively. By definition, transducer A has positive nominal steering angle; for transducers C and D, angles are negative. Nonsteered lateral focal depth fl is the distance from center of the array to focal point (i.e., center of sample volume). Steered lateral focal depth is distance from center of active aperture to focal point (—) and is given by fls ⫽ fl/cos(N S ). Scaling corresponds to that used in simulations and experiments.
F-number to be reduced by nearly 12%. Thus, to maintain the same F-number with steering, the aperture length must be increased. For the purpose of analysis, we shall define two more quantities. The first is the offset angle that may arise because the true beam direction may differ from its nominal value. This quantity is defined as: ⌬ ⫽ FT ⫺ FN ⫽ ST ⫺ SN ,
(2)
where the superscript T stands for true and N for nominal. The second is the estimated maximum velocity percentage error, which is defined as: max ⫽ E%
max max N ⫺ T ⫻ 100% , max T
(3)
max where Tmax is the true maximum velocity and N is the nominal maximum velocity as obtained experimentally or by simulation. Note that a positive value of Emax % corresponds to an overestimation error and a negative value corresponds to an underestimation error.
SIMULATION METHODS Our simulations were based on software that was written, tested and evaluated over the past seven years. The code was originally developed for simulating Bmode imaging using 1-D and 2-D linear and curved arrays and was subsequently extended for use in pulsedwave velocity estimation (Steinman 2004; Steinman et al. 2004b). The simulator assumed propagation and scattering to be linear processes, even though it is known that
Beam steering errors ● A. H. STEINMAN et al.
Table 1. Transducer and propagation medium parameters Parameter
Values
Element width (mm) Element height (mm) Kerf (mm) Elevation lens model: n of subelements Elevation lens focus (mm) Lateral F-number, Center frequency, f0 (MHz) Apodization Attenuation coefficient (dB/(cm · MHz) Speed of sound (m/s)
0.27 4.0 0.03 70 11.25 2, 3 or 4 5.0 none 0.7 1540
the peak negative pressures in some commercial Doppler US systems can be sufficient to cause nonlinear propagation. However, Li et al. (1993), using commercial systems, were unable to detect any nonlinear effects on the slow-time power spectrum or the maximum frequency. Based on this finding, the assumption of linear propagation seems to be a reasonably good approximation. Beam steering and focusing are achieved by applying appropriate time delays to each transducer element within the chosen aperture for both the transmit and receive functions. The transmit and receive delays are equivalent if the transmit and receive focal points and their apodization functions are identical. These values were calculated using the scheme described by von Ramm and Smith (1983). The assumed simulation parameters closely match those of a commercial US system (Acuson 128XP/10; Mountain View, CA, USA) with an L7 transducer that incorporates a fixed lens in the elevation direction with a focal length of 11.25 mm. Although details of the assumed transducer properties have been previously reported (Steinman et al. 2004b), for convenience they are presented in Table 1 along with the properties of the transmission medium. To calculate the response produced by a given transmit waveform, the impulse-response method was used with a sampling frequency of 0.1 GHz. In the elevation direction, each transducer element was divided into 70 rectangular subelements. The trapezoidal method (Jensen and Svendsen 1992) was used to calculate the impulse response for each subelement. The SV axial length is primarily determined by the transmit pulse and receive gate durations. For the transmit pulses, two sinusoidal pulses were used, whose front and back edges were cosine-tapered over a single cycle. The parameters and characteristics of these pulses are summarized in Table 2. For the receive gate, a rectangular time gate was used. The gate timings correspond to the times at which the edges of the time gate bisect the signals received from scatterers located at the desired
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axial limits of the SV. Such a gate has been called the “compromise gate” (Steinman et al. 2004b). Strategy for sample volume comparisons To determine the SV shape on the y-z plane (see Fig. 1), a 2-D grid of scatterers spaced by 0.05 mm was used. The gated received signal energy from each individual scatterer was determined and, by combining the results for all scatterers, the SV shape was calculated. To provide quantitative comparisons of the effects of beam steering on the SV shape, careful attention was paid to certain issues concerning details of the methodology. In the present study, to compare the SV energy distributions using a color display, the steered SV energies were normalized to the corresponding nonsteered values and a display threshold of ⫺30 dB was used. This comparison approach should be contrasted with the method used in our previous study (Steinman et al. 2004b), in which self-normalization and a threshold of ⫺20 dB were used. With this new comparison approach, the true beam-flow angle was determined for each SV by simultaneously rotating the self-normalized ⫺6, ⫺12 and ⫺18 dB contours of the SV until they were judged to be best matched to the corresponding nonsteered SV. It was estimated that the accuracy of this method was within ⫾ 0.5°. The centers of the two desired SV axial lengths (ᐉP ⫽ 1.5 and 5 mm) were chosen to coincide with the two lateral focal depths selected for our studies. These two depths corresponded to points close (15 mm) to and well away (35 mm) from the elevation focus. In addition to the above variables, two F-numbers, F2 and F4 were studied, leading to eight permutations in all. Each of the eight SVs was simulated for five nominal steering angles: 0°, ⫾ 20° and ⫾ 30°. In our experimental studies, when a beam steering of ⫾ 20° was used at a focal depth of 35 mm, aperture truncation effects caused the F-number to be increased from 2 to 3. Consequently, in addition to the above permutations, simulations were also performed for F3 at a lateral focal depth of 35 mm with ᐉP ⫽ 1.5 mm using the same steering angles as listed above. All simulations were performed using an attenuation of 0.7 dB/(cm · MHz) although, for some of the SVs, they were repeated without attenuation. The relatively narrow bandwidth of the transmit pulses (Table 2) en-
Table 2. Properties of transmitted pulses Desired axial length, ᐉP
1.5 mm
5 mm
Center frequency, f0 (MHz) Number of cycles Duration of transmit pulse, P (s) Length of transmit pulse (mm) Fractional bandwidth
5 9 1.8 1.39 0.107
5 15 3.0 2.31 0.061
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N Fig. 2. Effects of beam steering angle (S) on SV energy profiles in axial-lateral plane. (a) N S ⫽ 0°, (b) S ⫽ 20° and N (c) S ⫽ 30°. The origin (0,0) is at the lateral steered focus in flow coordinate system. Parameters used: F2, desired axial length ⫽ 1.5 mm, fl ⫽ 15 mm and FN ⫽ 60°. Note that (b) and (c) are normalized to maximum value in (a).
abled the effects of frequency-dependent dispersion to be ignored. To extend the lateral region over which the SV can be positioned, a reduced aperture size may be used when the center of the aperture approaches either end of the transducer array. Using a small hydrophone, Hoskins et al. (1999) measured the aperture size of two commercial systems, one of which was the Acuson 128XP/10 with an L7 transducer. They observed significant aperture truncation for the L7 transducer when a large aperture setting was selected. As appropriate, this truncation effect was included in the simulation modeling. Moreover, for the smaller apertures used in this study, a nearly constant F-number with beam steering was maintained, in accord with that used in the Acuson system. This was achieved by increasing the aperture to compensate for the changes in the focal length and effective aperture with beam steering, as described in the previous section. Single-scatterer power spectrum To study the power spectrum, the simulations assumed a single scatterer moving at 35 cm/s through the lateral steered focus and a pulse-repetition interval of 80 s. Each fast time-gated signal was demodulated, enabling
the slow time signal to be obtained and then analyzed in terms of its slow time-power spectrum. Each of the eight SV permutations was simulated at a nominal beam-flow angle of 60° for five nominal steering angles: these were 0°, ⫾ 20° and ⫾ 30° (i.e., transducer-flow angles of 30°, 40°, 60°, 80° and 90°). SIMULATION RESULTS AND DISCUSSION Sample volume results Typical SV energy distribution results on the plane x ⫽ 0 are shown in Fig. 2. These plots clearly indicate that the energy within the contours decreases with increasing beam steering. This decreasing trend was seen for both lateral focal depths (15 and 35 mm), even when the simulations were performed without attenuation and without elevation lens focusing. For the distributions shown, the change from N S ⫽ 0° to 30° was approximately ⫺7 dB. To determine whether or not the steered and unsteered SVs are in exact alignment, self-normalized contour graphs were obtained, as shown in Fig. 3. It is evident from these plots that SV misalignments were
Fig. 3. Energy contours for comparing the effects of beam steering on the SV profile. (a) ⫺12 dB contours for N S ⫽ ⫺30°, (dashed), 0° (solid) and 30° (dotted); (b) ⫺6 dB contours for N S ⫽ ⫺30° (dashed), 0° (solid) and 30° (dotted); and (c) ⫺6 dB contours for N S ⫽ 0° (solid), 20° (dotted) and 30° (dashed). The contours are for the same conditions as listed in Fig. 2, except they are self-normalized.
Beam steering errors ● A. H. STEINMAN et al.
present when beam steering was used. The misalignment consisted of: 1. a small change in the position of the SV, 2. a small change in the shape in the contours, and 3. a significant change in the true beam-flow angle. The first two changes, from a practical standpoint, are not significant but, as will be seen, the change in the true angle is important. It seems likely that our new technique (see Simulation Methods section) for comparing the effects of beam steering on the SV helped us to reveal the sources of discrepancy in the true beam-flow angle when beam steering is used. Table 3 shows that the true beam-flow angle (TF ⫽ ⌬ ⫹ N F ) changes as the nominal steering angle increases. The results indicate the importance of the Fnumber in determining the offset angle. In particular, for a fixed focal depth, the offset magnitude reduces as the aperture size is decreased (F-number increased). This reduction can be seen by comparing data columns 1 and 3 for a focal depth of 15 mm and by comparing data columns 4, 5 and 6 for a lateral focal depth of 35 mm. It should be noted that virtually identical results were obtained for the two SV lengths (data columns 1 and 2) at the same F-number. The table also shows that, as the focal depth is increased from 15 to 35 mm for a constant F-number (1st and 4th data columns), there appears to be a small reduction in the magnitude of the offset angle. The notion of the offset angle as defined by eqn (2) and its impact on the true beam-flow angle can be better understood by examining the qualitative sketches in the panels of Fig. 4. They provide an intuitive description showing that the offset angle can affect the velocity estimation accuracy, depending on the sign of the beam-steering angle and whether the physical angle is greater or less than 90°. By examining the ratio between the nominal velocity and the true velocity, a more quantitative description of this phenomenon can be provided. Based on eqns (1), (2) and (3), this ratio is given by:
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N cos(FN ⫹ ⌬) , ⫽ T cos(FN)
(4a)
which enables the percentage velocity error to be expressed as: max E% ⫽
冋
册
cos(FN ⫹ ⌬) ⫺ 1 ⫻ 100% . cos(FN)
(4b)
This expression, together with the assumption that the simulated values for the steering offset angles given in the last two columns of Table 3 are correct, enabled the graphs shown in Fig. 5 to be obtained. In this figure, the percentage velocity errors are shown as a function of the nominal beam-flow angle for two nominal beam steering angles and two F-numbers. The asymmetry of the results about the 90° axis should especially be noted. From another perspective, there is symmetry in the results between positive steering below 90° and negative steering above 90°. This symmetry can be understood by examining the physical location of the active aperture, such as transducers A and D shown in Fig. 1 (e.g., the active aperture for N F ⫽ 60° generated from ⫹ 20° steering is identical to the one for N F ⫽ 120° generated from ⫺20° steering). Under the same argument, symmetry also exists in the results between negative steering below 90° and positive steering above 90°. Single-scatterer power spectra A typical set of power spectra from the single scatterer moving at 35 cm/s through the center of the wide aperture SV at a beam-flow angle of 60° is shown in Fig. 6 for five beam-steering angles. In the absence of any spectral broadening, the ideal slow time-frequency generated by a single scatterer moving at 35 cm/s for N F ⫽ 60° when f0 ⫽ 5 MHz can be calculated from eqn (1) as 1136 Hz. It can be seen that, as the steering increases, the peak amplitudes decrease and the mean frequency changes in different directions depending on the sign of
Table 3. Offset angles (⌬) for various steering angles Sample volume characteristics Axial length, mm Unsteered focal depth, mm F-number Unsteered aperture, mm Nominal steering/angle, N S* ⫺30° ⫺20° 0° 20° 30° * Offset angle, ⌬ (⫾ 0.5°)
1.5 15 2.0 7.5 3° 2° 0° ⫺2° ⫺3°
5 15 2.0 7.5 3° 2° 0° ⫺2° ⫺3°
1.5 15 4.0 3.75
1.5 35 2.0 17.5
1.5° 1° 0° ⫺1° ⫺1.5°
2.5° 1.5° 0° ⫺1.5° ⫺2.5°
1.5 35 3.0 13.2 2° 1° 0° ⫺1° ⫺2°
1.5 35 4.0 8.8 1° 0.5° 0° ⫺0.5° ⫺1°
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Fig. 4. The effect of (a) and (b) negative and (c) and (d) positive beam steering on velocity estimation accuracy for transducerflow angles of ⬍ and ⬎ 90°. ( · · · · ) Actual beam direction, (—) beam direction in absence of any angle offsets caused by beam steering. Arrow ⫽ scatterer velocity direction. (a) and (d) correspond to negative velocity error and (b) and (c) to a positive error.
the steering angle. The shifts of the steered spectra away from the nonsteered peak indicate the possibility of changes in the true beam-flow angle. Moreover, the asymmetry of the steered power spectra suggests a different scatterer path through the SVs. We also found that, for a smaller aperture (F4) with its wider SV dimension (Steinman et al. 2004a), the power spectra had a smaller band width with a mean frequency close to that of the nonsteered beam. Moreover, the trends found in this series of studies were independent of attenuation, lateral focal depth and elevation focusing although, as expected, the amplitudes differ.
Fig. 5. Percentage velocity error for nominal beam steering angles of 20° and ⫺20° for F-numbers 3 and 4. Simulation results are based on the last two columns of Table 3.
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Fig. 6. Typical slow-time power spectra generated by a single scatterer moving at 35 cm/s through the center of a 1.5 mm SV at a depth of 15 mm for the wide aperture (F2), FN ⫽ 60° and five beam steering angles. Solid line, S ⫽ 0°; black dotted line, N N S ⫽ ⫺20°; grey dotted line, S ⫽ 20°; black dashed line, N ⫽ ⫺30°; grey dashed line, N S S ⫽ 30°.
EXPERIMENTAL METHODS Experimental system The pulsatile flow system shown in Fig. 7a is similar in design to a previous design used for steady-flow experiments (Steinman et al. 2001). To ensure accurate positioning of the transducer, a custom-designed mechanical system with three degrees of freedom was used. The flow phantom, which was designed to model a nonstenosed artery, used a 7.9-mm inner diameter latex tube with a wall thickness of 1.6 mm, embedded at a depth of 15 mm in a tissue-mimicking medium (R.G. Shelley Inc., Toronto, ON, Canada) (Rickey et al. 1995). The bloodmimicking fluid, whose properties are listed in Table 4, was based on that described by Ramnarine et al. (1998). Nylon beads (Orgasol™ ELF Autochem, Paris, France) with a mean diameter of 5 m were used as scatterers at a concentration corresponding to a hematocrit of 5%. Although this hematocrit value is considerably less than that of normal blood, it can effectively circumvent nonNewtonian viscous effects and settling of scatterers in the flow tube and, thus, there are fewer experimental difficulties that complicate the estimation of peak flow velocity. Furthermore, it is reasonable to expect that, at the 5% hematocrit level, the characteristics of the ensemble-averaged slow time-power spectra would be the same as those for a higher hematocrit under the same flow conditions. The flow waveform shown in Fig. 7b is similar to that of the carotid artery (Shehada et al. 1993), but has reduced flow in the diastolic region. To extend the duration of the peak quasistationary flow region, a frequency of 40 flow cycles per min was selected. The Womersley number for the fundamental frequency was
Beam steering errors ● A. H. STEINMAN et al.
Fig. 7. System used for pulsatile flow experiments. (a) The flow bypass and gate valves determine flow through the phantom. The ballast, combined with the filter, was designed to minimize the amount of air bubbles pumped through the system (Lubbers 1999). All experiments used an Acuson 128XP/10 system (7-MHz B-mode, 5-MHz spectral Doppler) with an L7 transducer. (b) Flow waveform.
4.3 and the time-averaged Reynold’s number was 305. The values and errors associated with the various parameters of our flow system are listed in Table 4. It should be noted that the changes in the speed of sound between tissue and fluid (and vice versa) were accounted for in calculating the nominal beam-flow angle. Signal acquisition and processing For the waveform used, the peak flow can be considered as quasistationary over a time interval of ⬇20 ms. To achieve a good signal-to-noise ratio for the spectral estimation over this relatively short time interval, ensemble averaging was used. However, because there were fluctuations in the flow waveform period and no accurate synchronizing signal was available, a waveform alignment technique was used to ensure that the averaging was being performed on the same signal segments. The alignment method consisted of cross-correlating the flowmeter signal to determine the time interval between successive pulses. The resulting alignment error was estimated to be ⫾ 5 ms, corresponding to half of the flowmeter sampling period. Using a 20-ms Tukey window for each flow pulse, the power spectrum was obtained using a standard fast Fourier transform method. This procedure was repeated for each waveform and the ensemble averaged power spectrum was determined by using a minimum of 40 cycles. The maximum frequency was estimated from the integrated power spectrum using the signal-noise slope intersection method proposed by Steinman et al. (2001). In this method, the integrated power spectrum is modeled with a transition (knee) region separating the signal and noise regions. The slope in the transition region can be considered to be equal to the sum of the contribution from the signal region and the complementary contribution from the noise region. The estimated maximum
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frequency is defined to be in the transition region for a specific fractional signal contribution. Consequently, the maximum frequency is not taken as the limit of no signal (at the noise end of the transition region). For these experiments, a fractional signal contribution of 0.3 was selected to optimize the estimation performance at a beam-flow angle of 60°. Tests have shown that, under these conditions, the signal-noise slope intersection method is approximately independent of the signal-tonoise ratio (Steinman 2004). The true maximum center-line flow velocity was calculated from the digitized volume flow waveform. Specifically, this value was determined by using the Womersley equation and a Fourier analysis of the volume flow waveform (Evans and McDicken 2000). A maximum velocity was computed for each individual flow cycle and subsequently averaged over all of the flow pulses, yielding our reference value of Tmax ⫽ 98 ⫾ 6 cm/s. Protocol The experiments were conducted with the lateral transducer focal depth set to 35 mm under nonsteered conditions: the corresponding steered focal depth was 37.2 mm (35/cos 20°). At this focal depth, the nonsteered beam measurements for two different aperture sizes were available and used, 17.4 mm and 8.7 mm, corresponding to F2 and F4, respectively. For the smaller aperture, the steered aperture was identical to the unsteered value, so that the F-numbers were the same (i.e., F4). However, when the beam was steered by N X ⫽ ⫾ 20° at the larger aperture setting, aperture truncation was present, causing the F-number to be increased from the no-steering value of F2 to F3. The experiments were conducted in 5° increments over a range of transducer-flow angles (P) from 50° to 130° and, at each angle, readings were taken for nominal steered angles (N S ) of ⫺20°, 0° and 20°. Note that the spectral gain was set just below saturation for the strongest signal case and kept constant for the
Table 4. Experimental values and associated errors Property
Value ⫾ error
Units
Speed of sound in phantom Speed of sound in fluid Fluid density Fluid dynamic viscosity Fluid attenuation Transducer positioning Transducer positioning angle Flow measurement error (reading ⫹ calibration ⫹ drift) (%) Beam/flow angle Beam/flow angle Center-line maximum flow velocity Flow signal alignment
1540 ⫾ 10 1547 ⫾ 3 1042 ⫾ 1 3.74 ⫾ 0.1 0.06 ⫾ 0.01 ⫾ 0.5 ⫾ 0.5
m/s m/s kg/m3 cP dB/(cm · MHz) mm degrees
⫾5 30 ⫾ 2.6 60 ⫾ 0.9 98 ⫾ 6 ⫾5
L/min degrees degrees cm/s ms
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Fig. 8. Comparison of velocity estimation errors for nominal beam steering angles of 0°, ⫺20°, 20° and FN ⫽ 60°. (a) F2 (unsteered), F3 (steered); (b) F4 (both steered and unsteered). To improve clarity, the SD of data in the horizontal (angle) direction is shown as bars at top of each graph.
experiments. Also, the center of the SV was kept at the same location at the center of the vessel when beam steering was initiated. In this region, the velocity profile is fairly constant around peak systole. EXPERIMENTAL RESULTS AND DISCUSSION Beam-flow angle-dependence The maximum velocity error is plotted in Fig. 8 for both steered and unsteered beams at various beam-flow angles and using the aperture settings previously noted. For Fig. 8a, the nonsteered F-number was 2 and the steered was 3 but, for both the steered and nonsteered results in Fig. 8b, it was 4. It should be noted that physical restrictions on the transducer placement prevented measurements from being made for certain ranges of beam-flow angles and, hence, discontinuity gaps are seen in the experimental results. For the nonsteered case, our pulsatile flow results
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are quite similar to those previously reported by Hoskins (1996) using a string phantom and to those reported by Steinman et al. (2001) using a steady-flow tissue-mimicking phantom. The effects of changes in aperture size, as seen by comparing the results of Fig. 8a and b, are also consistent with prior observations. Specifically, for a reduced aperture (larger F-number), Fig. 8b shows that the velocity estimation error is decreased relative to Fig. 8a. Unlike the simulation results for determining the offset angle error, the experimental maximum velocity error involves additional error contributions from components that may not be associated with beam steering. Some of the primary factors that contribute to the experimental errors have been summarized in the first table of the study by Christopher et al. (1995). Among these factors, intrinsic spectral broadening is well established as a major error source for both steered and nonsteered conditions. It is especially relevant to our measurements, because they were made at peak flow, during which the velocity profile is fairly constant over the SV, thereby making the effects of extrinsic spectral broadening insignificant. For the nonsteered results, it should be noted that the error rapidly increases as the beam-flow angle increases. The error increase can be explained by noting that the amplitude modulation resulting from the finite scatterer transit time through the SV concomitantly decreases as the beam-flow angle increases and, thus, the extent of intrinsic spectral broadening is more prominent at larger beam-flow angles (Newhouse et al. 1987). Spectral broadening may also arise in the form of frequency modulation caused by deviations in the beam-flow angle as scatterers move through the SV (Ata and Fish 1991; Guidi et al. 2000). Hoskins et al. (1999) showed that these changes in the subtended angle can be significant for a phased linear array and can, therefore, contribute to the velocity estimation error. Effects of beam steering Figure 8a shows that, as the beam-flow angle approaches 90° from the left-hand side, the percentage velocity error for a negative beam steering angle (N S ⫽ ⫺20°) is less than that forN ⫽ 0°. On the other hand, as S 90° is approached from the right-hand side of the graph, the error for negative steering is greater than that for N S ⫽ 0°. The reverse is true when the beam steering angle is positive (N S ⫽ 20°). The asymmetry in the measured velocity estimation errors can also be seen in Fig. 8b, although to a lesser extent. These results indicate the presence of an underlying process that affects the velocity errors.
Beam steering errors ● A. H. STEINMAN et al.
COMPARISON OF EXPERIMENTAL AND SIMULATION RESULTS The conditions under which the nonsteered experimental results of Fig. 8a were obtained are equivalent to those of the fourth data column in Table 3. Conversely, for the steered results, because aperture truncation caused the F-number to be increased to 3, the experimental conditions correspond to the fifth data column in Table 3. As previously noted, the experimental results in the absence of steering contain systematic contributions to the velocity error from several sources. In the presence of steering, it is reasonable to assume that these systematic sources are also present together with the errors caused by the offset angle. Hence, the experimental velocity errors shown in Fig. 8 are larger in magnitude than the theoretical errors shown in Fig. 5, and the asymmetry of the errors about 90° is identical in form for both figures. We also qualitatively compared the intensity of the received signals by observing the recorded image displays. Such a comparison was performed for a fixed beam-flow angle for a constant spectral gain. It was found that the intensity was less when steering was used, thereby providing some experimental support for the SV results of Fig. 2 and the single-scatterer results of Fig. 6. DISCUSSION AND CONCLUSIONS This paper set out to establish whether or not electronic beam steering is a significant source of error in linear-array pulsed-wave US velocity-estimation systems. The experimental results and the corresponding conclusions are qualitatively consistent with the SV simulations, the single-scatterer simulations and the pulsatile-flow phantom experiments. For both experimental and simulation results, one or more of several factors could cause the velocity error. One factor is related to the delays used to achieve the required beam steering on both transmit and receive. For the purpose of calculating the delays, each transducer element in the active aperture was generally assumed to be a point source/receiver located at the center of each element. However, the transducer elements are 2-D structures, and the cylindrical lens used for focusing in the elevation direction is 3-D in nature (even though it is commonly simulated as a 2-D structure). The assumption of point-like transducer elements may, thus, no longer be valid when the 2-D aperture dimensions of the transducer become comparable with the lateral focal depth. Such an assumption might be a possible cause of the angle offsets, but further simulation investigations are needed. Understanding the source of the offset angle error would enable a correction table to be deduced
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either to correct the delays or to provide a corrected beam-flow angle. Four primary conclusions can be derived from the results and discussions. 1. The amplitude of the gated received signal (the energy in the SV) decreases when steering is increased. 2. A greater degree of beam steering leads to a greater beam-flow angle offset that, in turn, may lead to further errors in maximum velocity estimation. 3. The degree of beam steering offset is not dependent on SV axial length but, rather, on the F-number. Also, a lateral focal depth that is farther away from the elevation focus slightly reduces this offset. 4. The maximum velocity error curves for beam steering exhibit asymmetry in their characteristics as a function of the beam-flow angle. From a clinical perspective, beam steering is typically used at beam-flow angles of 60° or less. In this region, the errors associated with beam steering, although they exist, are probably not very significant compared with other errors, such as those caused by spectral broadening. In addition, at these typical beamflow angles, the beam is steered away from 90°, which will actually cause a reduction in the overestimation error of the maximum velocity. However, at higher beam-flow angles, larger beam steering angles (⬎ 20°) and steering toward 90°, beam steering errors can become large. Acknowledgements—The authors thank the Canadian Institutes of Health Research (K.W. Johnston and R.S.C. Cobbold) and the R. Fraser Elliott Chair (K.W. Johnston) for partial funding support. In addition, they are grateful to Elaine Lui for her helpful insights, Dr. G. Meng for assistance in the preparation of the blood-mimicking fluid and Dr. J. Myers for help with the experimental design. They are also pleased to acknowledge the assistance of Acuson (Siemens) for providing information on their L7 transducer.
REFERENCES Ata OW, Fish PJ. Effect of deviation from plane wave conditions on the Doppler spectrum from an ultrasonic blood flow detector. Ultrasonics 1991;29:395– 403. Christopher DA, Burns PN, Hunt JW, Foster FS. The effect of refraction and assumed speeds of sound in tissue and blood on Doppler ultrasound blood velocity measurements Ultrasound Med Biol 1995;21:187–201. Daigle RJ, Stavros AT, Lee RM. Overestimation of velocity and frequency values by multielement linear array Dopplers. J Vasc Technol 1990;14:206 –213. Evans DH, McDicken WN. Doppler ultrasound: physics, instrumentation and signal processing, 2nd ed. West Sussex, UK: Wiley, 2000. pp. 5–26. Guidi G, Licciardello C, Falteri S. Intrinsic spectral broadening (ISB) in ultrasound Doppler as a combination of transit time and local geometrical broadening. Ultrasound Med Biol 2000;26:853– 862. Hoskins PR. Accuracy of maximum velocity estimates made using Doppler ultrasound systems. Br J Radiol 1996;69:172–177. Hoskins PR, Fish PJ, Pye D, Anderson T. Finite beam-width ray model for geometric spectral broadening. Ultrasound Med Biol 1999;25: 391– 404.
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Jensen JA. Estimation of blood velocities using ultrasound: A signal processing approach. Cambridge, UK: Cambridge University Press, 1996. pp. 86 –110. Jensen JA, Svendsen NB. Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers. IEEE Trans Ultrason Ferroelec Freq Control 1992;39:262–267. Li S, McDicken WN, Hoskins PR. Nonlinear propagation in Doppler ultrasound. Ultrasound Med Biol 1993;19:359 –364. Lubbers J. Application of a new blood-mimicking fluid in a flow Doppler test object. Eur J Ultrasound 1999;9:267–276. Newhouse VL, Censor D, Vontz T, Cisneros JA, Goldberg BB. Ultrasound Doppler probing of flows traverse with respect to beam axis. IEEE Trans Biomed Eng 1987;34:779 –789. Newhouse VL, Faure P, Cathignol D, Chapelon J-Y. The transverse Doppler spectrum for focused transducers with rectangular apertures. J Acoust Soc Am 1994;95:2091–2098. Ramnarine KV, Nassiri DK, Hoskins PR, Lubbers J. Validation of a new blood- mimicking fluid for use in Doppler flow test objects. Ultrasound Med Biol 1998;24:451– 459. Rickey DW, Picot PA, Christopher DA, Fenster A. A wall-less vessel phantom for Doppler ultrasound studies. Ultrasound Med Biol 1995;21:1163–1176.
Volume 31, Number 8, 2005 Shehada RE, Cobbold RSC, Johnston KW, Aarnink R. Three-dimensional display of calculated velocity profiles for physiological flow waveforms. J Vasc Surg 1993;17:656 – 660. Steinman AH. Errors in phased array pulse-wave ultrasound velocity estimation systems. Ph.D. thesis. University of Toronto, 2004. Steinman AH, Lui EYL, Johnston KW, Cobbold RSC. Beam steering in pulsed Doppler ultrasound velocity estimation. IEEE Ultrason Symp Proc 2004a;3:1745–1748. Steinman AH, Lui EYL, Johnston KW, Cobbold RSC. Sample volume shape for pulsed flow velocity estimation using a linear array. Ultrasound Med Biol 2004b;30:1409 –1418. Steinman AH, Tavakkoli J, Myers JG Jr, Cobbold RSC, Johnston KW. Sources of error in maximum velocity estimation using linear phased-array Doppler systems with steady flow. Ultrasound Med Biol 2001;27:655– 664. von Ramm OT, Smith SW. Beam steering with linear arrays. IEEE Trans Biomed Eng 1983;30:438 – 452. Winkler AJ, Wu J. Correction of intrinsic spectral broadening errors in Doppler peak velocity measurements made with phased sector and linear array transducers. Ultrasound Med Biol 1995;21: 1029 –1035.