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Standard phase-domain pulsed Doppler techniques used in Colour Flow Mapping such as spectral Doppler or autocorrelation are monochromatic, focused on the analysis of the centre transmit frequency. As such all the algorithms using those approaches are limited: in terms of spatial Doppler resolution because of the long pulses typically used for transmission, in terms of frame rate because of the necessity to perform many Doppler lines repetitions and additional B-mode imaging transmissions, and in terms of accuracy which depends on the stability of the Doppler signal at the frequency considered. A velocimetry technique is presented which estimates the shifts between successive Doppler line segments using the phase information provided by the Fourier transform. Such an approach allows extraction of more information from the backscattered signal through the averaging of results from multiple frequencies inside the bandwidth, as well as the transmission of wide band-high resolution-pulses. The technique is tested on Doppler signals acquired with a research scanner in a straight latex pipe perfused with water and cellulose scatterers, and on an ultrasound contrast agent solution. The results are compared with the velocity estimates provided by standard spectral Doppler and autocorrelation methods. Results show that the proposed technique performs better than both other approaches, especially when few Doppler lines are processed. The technique is also shown to be compatible with contrast Doppler imaging. The proposed approach enables high frame rate, high resolution Doppler.

Cardiovascular diseases are a major modern health concern, responsible for one third of deaths worldwide. Assessing accurately blood perfusion and blood flow-rate are key elements for vascular diagnosis [1,2], and their observation at the locations of disease expression such as stenoses, or the measurement of the blood flow in small animal models, locations where the flow can be fast and has high frequency components, requires high frame rate, high resolution imaging modalities. To that end, Doppler ultrasound provides inexpensive, non-injurious, non-invasive and real-time flow measurements, and it is nowadays used routinely. However, the current implementations of ultrasound Doppler require the acquisition of several A-lines repeats to generate the local estimates, which severely slows down the image rate in the case of colour mapping [

According to Alam and Parker [

While many of the existing techniques can theoreticcally use down to a single repetition of a given data segment to perform local flow rate estimation, this is practically never the case and velocimetry techniques typically use several ultrasonic shots (see [7,12] p. 699) to provide an accurate and stable estimate.

A particular and quite fundamental limitation of the frequency/phase domain methods is their need for a demodulation step which is performed at a single frequency. In this aspect they require the transmission of long-narrowband-pulses [3,12,13], which reduces the Doppler resolution and is sometimes rendered ineffective because of the scattering dependant aspect of the radiofrequency signal generation.

This problem was noticed by Eriksson in 1995 [

In the present study, a basic but efficient velocimetry technique is proposed, which is based on the information contained in the phase of the Fourier transform of successive segments, and is as such a frequency/phase domain method. While no such straightforward technique has been found in the literature, such approach could be related to the methods reported by Atkinson and Woodcock [

The paper is organized as follows: the Materials and Methods part present the theory and the construction of the proposed velocity estimator, the in vitro experiments performed to gather radiofrequency data, and the processing performed to compute the velocity estimates of the proposed technique, which are compared with two fundamental methods of the same domain; then the Results are presented before Discussion of and Conclusion.

The fundamental pulse-echo Doppler processing consists is successive pulse-echo observations of the same spatial domain [_{PRF} seconds. Depth and time are considered to be equivalent according to the sound speed ratio c, assumed to be constant in biological tissues.

If the target from depth z_{1} and corresponding signal delay Dt_{z}^{+} has moved from its previous observation to depth z_{2} and corresponding time delay Dt_{z}, it can be written (1):

which express the fact that the signal has been slightly shifted between observation n and n + 1, and the acoustic signature observed at depth z_{1} will be observed at a different depth z_{2} at the next pulse repetition, assuming the amplitudes are preserved. If the target at depth z was moving at velocity component v_{z} along the Oz axis, the relation between the previous and the next positions delay is

where dz is the displacement of the local target between the two successive shots. Then Equation (1) turns into Equation (3)

which is the simple expression of the time delay between the corresponding acoustic signatures of the target between two pulses transmissions. Equation (3) can be then transferred to the frequency domain using the Fourier transform.

The discrete Fourier transform [

The transform from the time domain to the frequency domain is performed in Equation (4) where F {} denotes the Fourier transform and is the Fourier transform of signal s at repeat n+1 and depth Dt_{z}

where the time delay appears as a phase shift. Then the delay into the Fourier domain can be simply expressed through spectral division as provided by Equation (5)

and the velocity of the target is then given by Equation (6)

where Á {} denotes the imaginary part and Â {} the real part.

Equations (4) to (6) imply that the local velocity can be deduced from each frequency in the data segments. Thus each frequency is a potential estimator of the real local velocity component v_{z}, its final estimate being the mean value of for each available frequency w. is expressed in Equation (7), where the measured signal is assumed to be band-limited between B_{min} and B_{max}, and the estimates at each frequency are integrated and averaged over the bandwidth:

which gives Equation (8) after reordering the terms and introducing the incidence angle a between the ultrasound beam and the flow:

is the local velocity estimate which must be evaluated at each depth z. The bandwidth estimation is important as the phase information doesn’t have any meaning outside the bandwidth (i.e. in the noise signal), which is the standard issue in spectral division. However the technique is still valid for more complex “bandwidths”, which could well be defined over several sections of the spectrum. Equation (8) is the proposed Phase Fourier Doppler (PFD) estimator, which gives its strength to the proposed approach through combining the information at several frequencies to produce an improved estimate of the local velocity. Even with a small bandwidth, n available frequencies as much improve the accuracy of estimation and the need for Doppler lines repeats.

Assuming that the bandwidth definition is at half the amplitude (−6 dB) of the maximum spectrum peak, the integration domain is computed in the following study using (9)

where is the domain of definition of the integral of the estimator, which can be seen for evaluation purposes as logical map equal to 1 inside the bandwidth of, and zero otherwise, and the intersection domain of and.

The Fourier transform can unfortunately not be computed at each depth over a single sample, and an implementation of the proposed method must be performed, alike the other Doppler techniques, on successive segments of data, whose length depends on the desired Doppler resolution and the characteristics of the acquisition. The processing of successive segments with more or less overlapping [3,7,10] allows extracting a velocity profile. This is performed in the following sections after acquiring data using a flow phantom, and comparison is made with basic version of classic velocity estimators.

In order to test the proposed technique, RF ultrasound Doppler data were acquired on a phantom flow system set up in a water tank (^{−}^{1} to act as linear scatterers [^{−}^{1} (and corresponding maximum flow speed of ~20, ~26, ~35 and ~40 cm·s^{−}^{1}).

The Reynolds number at the highest flow rate was around Re = 1600 (See for calculation of the Reynolds number with: Mean velocity max: 20 cm·s^{−}^{1}; Characteristic diameter 0.8 cm; Fluid density 1 kg·L^{−}^{1}; Fluid viscosity ~1 cP). With Re < 2000, it is assumed that at the measurement site the flow was laminar and fully developed, with a parabolic velocity profile (while instabilities may start to appear in the flow at the higher flow rates,

as the flow gets increasingly disturbed with Re getting higher). Lower and higher flow rates were not achievable, as it was observed that cellulose was starting to decant at lower flow rates, disturbing the flow, and as higher flow rates led to a disturbed/turbulent flow and the corresponding flattened flow profile, useless for the study. This last point was likely due to the thinness of the pipe (200 µm), which is excellent for ultrasound measurements, but also very sensitive, over a great tube length, to flow disturbances and shear stress.

The ultrasound sequence was defined for a 4 cm acquisition depth for a focus at 2.5 cm, and the transmission was performed at f_{0} = 5 MHz. The L14-5/38 probe is said by Ultrasonix to have a centre frequency of 14 MHz with 5 MHz bandwidth, but it has been observed to transmit efficiently at 5 MHz. The excitation pulse shape on the Ultrasonix is a square wave controlled through a succession of “+”, “−” and “0”, a “plus” programming a positive rectangular excitation and a ‘minus’ a negative one. The amplitude of the excitation waves was 47 Volts (scanner’s specifications). The data were acquired using a single “+” of 100 ns for the transmission (pulse P1). For each frame, 128 ultrasound lines were acquired for B-mode observation, followed by 256 repetitions of the central line for Doppler processing. The pulse repetition frequency (PRF) was 13.56 kHz. The same acquisitions were performed using a longer excitation (pulse P2) “+−+−“, of total length of 400 ns, to enable comparing the results with a more standard-longer-frequency/phase domain Doppler pulse. Data were sampled at f_{s} = 40 MHz.

In order to allow testing the ability of the proposed method to work also in different frequency bands, after data acquisitions on the cellulose flow, ultrasound contrast agent (UCA) (Sonovue™, Bracco Research Inc., Switzerland) [

The angle a between the flow and the ultrasound beams was estimated on the B-mode image (see

The data were processed using the method presented earlier and results compared with one derived from the mean angular frequency of the spectral Doppler (SD) spectrum [5,6] and the autocorrelation (AC) [5,7,20] methods which also are fundamental frequency/phase domain techniques. The estimates are calculated for each algorithm for the same data segments and overlapping ratio, and the estimates compared with the fitted ideal parabolic flow profile. As elaborated versions of SD and AC methods exist, each method was computed in its basic form to account only on its fundamental ability to extract flow information from the data provided; equally, PFD is computed using the equations provided, without attempting to optimize any of the estimation steps. Furthermore, no wall filter was included in the different estimators to compare their natural ability to handle wall and near wall imaging. For the proposed Phase Fourier Doppler algorithm, the Doppler lines were taken by successive couples from a total Doppler lines number Nl, and the mean flow velocity profile and the standard deviation estimated from the successive estimations.

The AC and SD values were estimated in each data segment after quadrature demodulation at the centre transmit frequency (5 MHz) using a Matlab Blackman lowpass filtering, resulting in the generation of the complex Doppler signal. The AC values were derived from the phase of the cross-correlation of the local Doppler signal at lag 1 ([

The estimations were performed for all flows using different numbers of Doppler lines Nl, from 2 repeats, the minimum, up to 256, the total Doppler lines repeats acquired. The distance to the theoretical parabolic flow profile was computed for all techniques through the mean of the root of the mean square error (Mean RMS Error or M-RMS-E) between the mean flow profile and the theoretical profile for all flow rates. In other words, the final error parameter measured is the mean error for all flow rates, so that each technique accuracy is not assessed on a single profile estimation, but on its global performance over several flow profiles of different rates and function of the number of Doppler lines repeats provided to each algorithm.

In the case of contrast agent imaging, the data were band bass filtered around the harmonic of the 5 MHz transmitted frequency, that is between 8 MHz and 12 MHz, in order to process the nonlinear components of the acoustical response.

Examples of ultrasound B-mode image and of a set of Doppler lines are presented in _{s}/2.

The M-RMS-E estimation results are displayed in

It can be noticed that for an increasing number of Doppler lines processed, the SD method seems to require a minimum of Doppler lines repeats between 25 and 60 to provide stabilized estimates, and increasing the number of lines does not further improve the estimation significantly. The AC method performs slightly better for a small number of Doppler lines and only needs about ten lines to reach the SD method stabilized regions, however providing up to 50 lines still improves the estimation.

The instability and noisiness of the SD and AC results for small number of data lines and the impossibility of plotting clear error bars graphs in flow profile estimations led to the sole computation of their mean values over the total number of Doppler lines repeats Nl computed. This mean profile is the one used for error assessment in and accuracy comparison in

The PFD errors present the same trends, and processing more data allows to improve the estimations, however the initial 2 Doppler lines are sufficient (

techniques. A slight improvement of the PFD estimates can be observed when up to 10 Doppler lines are processed, and then increasing the number of lines does not seem to improve the estimations significantly.

Flow profiles for each flow rate are displayed in the best (in term of available information) Nl = 256 (

The standard deviations of the PFD estimates are not negligible and remain appreciatively the same across the flow profile. However it can be noticed that the AC and SD estimates oscillate inside or very close to the standard deviation domain of the PFD. A small irregularity - possibly a flattening-in the profiles can be observed for the estimates for the 600 ccm flow rate, at the top of the parabola.

In

The MRMS results for cellulose flow using pulse P2 are displayed in

Estimation results for the contrast flow are provided in Figures 8 and 9. Computation results of Mean RMS error to the theoretical parabolic profile as a function of the number of Doppler lines processed (

The SD and AC error profiles get very close when the number of Doppler lines processed is increased, but the AC errors tend decrease more rapidly. On another hand, when the number of Doppler lines processed is drastically reduced, the PFD error is close to the one obtained from SD and AC for a large number of Doppler lines.

This is further illustrated in

The smaller errors exhibited by PFD in

trend continuing at the greater flow rates, and coherently between all the estimation techniques, with the appearance of a flattened flow profile (not shown), which made the comparison with a theoretical profile impossible.

In

Demodulation of the same data on the same location at a different frequency, 4 MHz, which was present in the local spectrum, led to a restored accurate estimation of the local flow velocity (

estimate the flow rate properly at some locations if the frequency is not adapted to the local changes, and generate localized but significant errors.

Alam and Parker [

The drawback of this aspect is that the local amount of information used by the PFD is changing all the time, at least with the current implementation, so that each estimate is the result of an averaging over a different number of values, leading to inequalities in accuracy inside a given velocity profile. This could be avoided by limiting the number of values considered before averaging, but in the end would simply lead to worse estimates. The SD and AC method could also be corrected to avoid situations of divergence; for example the demodulation could be performed at different frequencies, in order to provide a mean estimate instead of a single one. However this would still lead to the processing of locally ill-defined frequencies, which would contribute to local divergence. Further stability could be achieved by performing a local estimation of the frequency content, but this would be equivalent to the minimum computation of a local FFT processing, which would be close to a PFD process. However the use of a longer pulse P2 (“+−+−”) did not significantly change the results obtained previously with the shorter pulse which can be transmitted by the scanner (

In the case of contrast imaging, the errors differences, as plotted in

The estimates in the wall in all estimations present (

However, the results demonstrate that the proposed technique can be used to compute the velocity profile in arbitrary frequency bands of the received signal, allowing it to combine with contrast imaging approaches. Further research in this matter should attempt to apply the PFD algorithm with non-linear imaging techniques such as Phase Inversion, Amplitude modulation or combined techniques [

An additional interesting aspect of the PFD is the possibility to obtain better Doppler resolution through the use of short pulses. Indeed, “standard” long pulses tend to spread and mix the responses of neighbouring scatterers, rendering each estimation window more likely to sense the velocities of the adjacent ones. The use of short pulses limits this spreading effect, making the flow estimates more local, which adds to the improved accuracy of the proposed technique.

But it should be noted that the present expression of the bandwidth selection (Equation (9) for the averaging of the velocity estimates does not take into account the fact that too high transmit frequency cannot be used if the local displacement is greater than half the corresponding wavelength. This is partly solved by the use of very high PRF, and it also works well with band limited pulses as the greater frequencies of the Fourier spectrum are supposed to be null. But in the hypothetical case of a perfect ultrasound impulse, even with an amplitude well defined, for a given PRF, the local velocity fixes the upper bound of the usable averaging domain, which is the ultimate limit of the PFD and of Doppler techniques in general.

Furthermore, the proposed implementation is appropriate for band limited pulses, which is suitable for ultrasound imaging, but cannot be applied straightforwardly to any kind of image, and would need to be adapted in different conditions of signal elaboration.

A shift appears in the calculation of the errors for all techniques, but the error of the PFD approach is lower. This is likely due to the fact that the PFD process averages the estimates through different frequencies, and suppresses the ill-defined ones, allowing to reduce the instabilities in the estimates, through this achieve a much lower mean error compared to the standard methods.

It should be noted that a single segment length, or gate size, (i.e. 64 samples segments) was tested in this study, which resulted from a compromise between spatial resolution and samples requirements for the calculation of the Fourier transform. The corresponding gate size is 1.6 µs, which can be compared with the 1 µs to 3 µs gates lengths range found in the Cobbold [

Finally, the proposed technique has been compared with basic implementations of the main estimator, SD and AC, while refined versions may prove more successful. However this permitted to account for their natural capacity to extract flow information from the RF data. Future implementations of the PFD may improve the proposed one, like SD and AC benefited from advanced or dedicated implementations. Future work should also consider comparison of the techniques in combination with a wall filtering, to account for their different abilities to include such step in their processing.

Standard pulsed Doppler techniques such as spectral Doppler or autocorrelation are limited by their single frequency approach and B-mode/Doppler spatial resolution due to the corresponding long pulses used for transmission. A Doppler method is presented that estimates the shifts between successive Doppler line segments using the phase information provided by the Fourier transform. In order to test and assess the performance of the proposed technique, signals were acquired with an Ultrasonix RP500 research scanner in a straight latex pipe perfused with water and cellulose scatterers. To compare the abilities of the technique with the existing phase domain Doppler approaches, the estimation results are computed along with the velocity estimates provided by standard-phase domain-spectral Doppler and autocorrelation methods, for different number of Doppler lines processed. The Mean RMS Error to the theoretical parabolic profile is computed for different flow rates properly developed using their mean estimates of each technique. Results show that the proposed method performs better than both spectral Doppler and autocorrelation techniques, especially when few Doppler lines are processed. This improved accuracy is due to extraction of more information from the backscattered signals, as the estimates are performed at multiple frequencies. If the use of a single frequency for estimation makes sense when continuous wave Doppler is performed, the standard pulsed Doppler approaches have wider band pulses while they rely on a single frequency only. This renders the velocity estimation dependent of the stability of that frequency in the backscattered signal, and requires increasing the number of Doppler lines repeats, which ultimately lowers the frame rate. It was shown that a phase domain Fourier approach can overcome that limitation by extracting multiple frequency information, through a combination of spectral division and averaging of the frequency dependent velocity estimates over the bandwidth. A possible drawback of the technique proposed here is that it does not produce a local Doppler spectrum detailing the range of velocities imaged in the Doppler resolution cell, and that it requires the estimation of the local bandwidth of the system. But regarding this latter point, results show that problems only occur when no signal is present, which is a typical weakness of Doppler techniques in general. As stated previously, one of the further strengths of the proposed technique over the other phase-domain ones is that it allows performing Doppler imaging with very wide-band pulses, which makes it able to gain in both B-mode and Doppler resolution, while providing more reliable velocity estimates. Through injection of Sonovue^{®} microbubbles in the system, it was demonstrated that this Doppler approach is compatible with nonlinear imaging techniques. Further research will attempt to combine it with the recently developed non-linear Doppler techniques, to handle the low signal-to-noise ratio situations, and of course to perform in vivo data processing. Despite the remaining limitations, Phase Fourier Doppler is a rare case of efficiency of spectral division and a significant step toward high resolution, high speed and high accuracy ultrasound Doppler, which demonstrates that ultrasound imaging continues to be a progressing modality full of promises.

The author thanks Professor Colin G. Caro and Dr. Adrien Desjardins for their support. This work has been originally funded by the Garfield Weston Foundation and the Henry Smith Charity (UK).