ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 50, no. 1, january 2003 89 Correspondence Velocity Fluctuation Reduction in Vector Doppler Ultrasound Using a Hybrid Single/Dual-Beam Algorithm Robin Steel, Peter J. Fish, Senior Member, IEEE, Kumar V. Ramnarine, Aline Criton, Helen F. Routh, and Peter R. Hoskins Abstract—In order to reduce the fluctuations in the ve- locity magnitude estimate, we propose a modification to the standard algorithm for reconstructing the (two com- ponent) vector velocity from the measured Doppler shifts in two directions. This uses the standard dual-beam algo- rithm, combined with temporal smoothing, to find only the velocity angle, then uses the single-beam algorithm to esti- mate the velocity magnitude. We present initial data show- ing the significant reduction in velocity estimate fluctuation that this hybrid method achieves compared to the standard algorithm. I. Introduction I n blood velocity measurement applications in which the velocity direction is not necessarily parallel to the vessel walls, such as in flows close to arterial stenoses, velocity estimates made with conventional single-beam Doppler ul- trasound can have large errors [1], [2]. Many techniques have been proposed in an attempt to improve estimation performance in such situations, including spatial quadra- ture [3]–[6], cross correlation [7], [8], bandwidth estimation [9], [10], and cross-beam Doppler systems [2], [11], [12], the latter using velocity component information from more than one direction. We previously demonstrated [13], prin- cipally using simulations, that the fluctuation in both the velocity magnitude and angle estimate in small interbeam angle dual-beam systems using the conventional algorithm is usually significantly larger than that in a single-beam system operating at the same beam-vessel angle. Several authors have pointed out [14]–[18] that one method to re- duce these unwanted effects is to increase the interbeam angle. However, except in intraoperative systems, this ap- proach usually is not practicable because of the limited scanhead footprint made necessary by clinical access con- straints. In the work reported here, which expands on our preliminary findings using the standard dual-beam al- gorithm [13], we propose an alternative hybrid approach Manuscript received February 26, 2002; accepted July 17, 2002. This work was funded by the EPSRC. R. Steel and P. Fish are with the School of Informatics, University of North Wales, Bangor, LL57 1UT, UK (e-mail: robin@informatics.bangor.ac.uk). K. Ramnarine and P. Hoskins are with the Department of Medical Physics and Medical Engineering, University of Edinburgh, Edin- burgh, UK. A. Criton and H. Routh are with Philips Medical Systems. to the reduction of fluctuation in dual-beam velocity es- timates that is applicable even to the case of small in- terbeam angle. It uses the conventional dual-beam esti- mator to determine just the velocity direction, which is then temporally smoothed, using the assumption that it is slowly varying in time. The velocity magnitude then is estimated, using the standard single-beam algorithm ap- plied to whichever of the two beams has a lower Doppler angle to this smoothed velocity direction. We show that this approach has potential advantages over the conven- tional algorithm, and we present simulation and experi- mental data from a steady flow phantom and initial in vivo data to illustrate these. II. Theory A. Standard Dual-Beam Algorithm and Its Velocity Fluctuation The conventional dual-beam velocity estimator was re- viewed in [2], so we present only a brief summary here. Left and right virtual beam vectors L and R are defined to lie along the bisectors of the relevant transmit and receive beams, with magnitude equal to the cosine of half the an- gle between these two beams [13]. We apply the standard Doppler equation to the Doppler frequencies f L and f R for each virtual beam to give the velocity components, v L and v R , respectively, from which the velocity magnitude is calculated as: v =  v 2 L + v 2 R − 2v L v R cos(θ) sin(θ) , (1) and orientation as: θ bv = arctan  v R − v L v R + v L  cot  θ 2  , (2) where θ is the interbeam angle between the left and right virtual beams and θ bv is the angle between the velocity vec- tor and the bisector of L and R . In practice, v L and v R will fluctuate about their mean values. In the case of streamline flow through the sample volumes at high signal-to-noise ratio, the fluctuation results from the random distribution of blood cells passing through the sample volumes and the effect on the Doppler signal spectrum and the mean fre- quency estimates from the windowed Doppler signal can be quantified easily. The effect of these fluctuations on the velocity estimate in (1) and (2) was considered in [13] in which the following results, valid when f L and f R are the spectral means, were derived. The standard deviation, ε v , in the velocity magnitude estimate, (1), is given approxi- mately by: 0885–3010/$10.00 c  2003 IEEE