A Transverse Velocity Spectral Estimation Method for Ultrafast Ultrasound Doppler Imaging

Abstract

A novel transverse velocity spectral estimation method is proposed to estimate the velocity component in the direction transverse to the beam axis for ultrafast imaging. The transverse oscillation was introduced by filtering the envelope data after the axial oscillation was removed. The complex transverse oscillated signal was then used to estimate the transverse velocity spectrum and mean velocity. In simulations, both steady flow with a parabolic flow profile and temporally-varying flow were simulated to investigate the performance of the proposed method. Next, the proposed approach was used to estimate the flow velocity in a phantom with pulsatile flow, and finally this method was applied in vivo in a small animal model. Results of the simulation study indicate that the proposed method provided an accurate velocity spectrogram for beam-to-flow angles from 45° to 90°, without significant performance degradation as the angle decreased. For the simulation of temporally-varying flow, the proposed method had a reduced bias (< −11.7% vs. 73.3%) and higher peak-to-background ratio (> 15.6 dB vs. 10.5 dB) compared to previous methods. Results in a vessel phantom show that the temporally-varying flow velocity can be estimated in the transverse direction obtained using the spectrogram produced by the proposed method operating on the envelope data. Finally, the proposed method was used to map the microvascular blood flow velocity in the mouse spinal cord, demonstrating estimation of pulsatile blood flow in both the axial and transverse directions in vivo over several cardiac cycles.

I. Introduction

Spectral analysis of the pulsed wave Doppler signal can be used to measure various temporal hemodynamic parameters such as the mean and peak blood flow velocities as well as the entire Doppler spectrum. Therefore, spectral Doppler has proven to be a useful tool in assessment and diagnosis of various cardiovascular diseases such as carotid artery diseases and deep venous thrombosis [1]. In contrast with the widely-adopted focused beam approach, ultrafast ultrasound imaging, which employs plane or diverging wave transmission, enables acquisition of anatomical images of the entire region of interest while also producing spectral Doppler measurements at all points in the image simultaneously [2, 3]. Ultrafast imaging combined with spectral Doppler, which is already available in commercial ultrasound scanners, is useful in various clinical scenarios including measuring carotid artery blood flow velocity [4] or evaluating the hemodynamics of the hepatic artery following liver transplantation [5]. Additionally, spectral analysis of the Doppler waveform has been used to map the cerebral vascular resistivity in neonates [6] and to extract blood flow Doppler indices in the left ventricle [7].

Despite being widely used in cardiovascular imaging, the conventional pulsed wave Doppler approach only provides the flow velocity along the axis of the ultrasound beam, i.e. towards or away from the transducer. When the beam-to-flow angle approaches 90° and the axial velocity component diminishes, it is challenging to estimate the blood flow velocity spectrum [8, 9].

In combination with ultrafast plane wave ultrasound imaging, vector Doppler imaging techniques were developed to estimate the blood flow velocity component along the lateral or transverse direction of ultrasound images [10-13]. By transmitting plane waves at two or more angles using an array transducer, the backscattered signal can be beamformed along specific angles, allowing the velocity components of the blood flow to be estimated at multiple angles. Therefore, the vector velocity of the blood flow can be obtained based on the multi-angle estimations. However, vector Doppler imaging techniques are not designed to provide the blood flow velocity spectrum along the lateral or transverse direction.

In order to obtain the flow velocity spectrum in the direction perpendicular to the beam axis, transverse oscillation can be introduced in the beamformed radio-frequency data. Transverse oscillation can be applied either by receive aperture apodization [14-16] or by filtering the beamformed frames [17, 18]. However, the major challenge of estimating the transverse flow velocity spectrum is separating the transverse oscillation from the axial oscillation in complex flow environments. In a previous study [8], it was proposed to use a fourth-order estimator to estimate the transverse flow velocity spectrum. However, the authors report that the transverse flow velocity spectrum was only usable when the beam-to-flow angle was larger than 60° [8], When the beam-to-flow angle was less than 60°, the spectrum widened significantly and became unreliable [8].

Estimating the blood flow velocity spectrum in the transverse direction remains a challenge. In particular, the flow angle relative to the beam may be unknown in vivo, especially for turbulent flow. In this study, we propose a new transverse velocity spectral estimation approach for use at varying beam-to-flow angles. This approach could be useful for quantifying the entire spectrum in large arteries with varying beam-to-flow angles, including in the hepatic artery following transplantation, cerebral vasculature in neonates, and spinal cord. In the proposed method, transverse oscillation is introduced to the envelope data, which is different from the conventional transverse oscillation method in which the transverse oscillation was introduced to the radio-frequency data.

The Field II simulation toolbox was used to validate the performance of this method for high-frequency ultrasound imaging of vessels. Temporally-varying flow conditions were also tested using a high-speed simulator, in which > 1000 frames of images of accelerating blood flow were simulated. The proposed method was compared with the conventional transverse oscillation-based fourth order transverse spectral velocity estimator [8]. Next, the method was tested to obtain the flow velocity spectrum in a 3.5 mm-diameter flow phantom. Finally, the proposed method was used to estimate the transverse flow velocity spectrum of the artery of the mouse spinal cord. The 2D flow velocity map can also be obtained by estimating the mean flow velocity based on the velocity spectra. By working with data containing oscillation in only the transverse direction, transverse velocity spectral estimation is demonstrated at varying beam-to-flow angles without loss of performance at smaller angles for the first time. In addition, performance of transverse spectral estimation is improved in terms of both reduced bias and increased peak-to-background ratio.

II. Methods

A. Ultrafast Data Acquisition and Reconstruction

In this study, the data were acquired using a coherent multiangle plane wave image acquisition approach. Specifically, 5 tilted plane waves were transmitted at −8°, −4°, 0°, 4°, 8° using the linear array (MS 400, VisualSonics). The received radiofrequency (RF) data were beam formed using delay-and-sum beamfonning. The beamformed RF data acquired at 5 angles were compounded coherently. Then, before envelope detection, the stationary clutter signal was removed using a singular value decomposition (SVD)-based filter. The default configuration of the image acquisition sequence is listed in Table I.

TABLE I.

Configuration of the high-frequency ultrasound imaging system in the Field II simulation study

Transducer array	MS 400 (VisualSonics)
Pitch	0.06 mm
Number of elements	256
Transmit frequency	23 MHz
Pulse length	3-cycle Gaussian pulse
RF signal sampling rate	92 MHz
Frame rate for different flow velocities	5000 (50 mm/s), 5000/8 (10 mm/s), 5000/41 (2 mm/s)
Number of frames	108 (50 mm/s), 67 (10 mm/s), 66 (2 mm/s)
Angles of plane waves	[−8°, −4°, 0°, 4°, 8°]
Pulse repetition frequency	25 kHz

B. Transverse Flow Velocity Spectrum Estimator

Transverse oscillation is created in order to estimate the flow velocity in the direction transverse to the beam. In order to generate the signal with transverse oscillation, the envelope of the multi-angle coherently compounded beamformed radiofrequency (RF) data was filtered frame-by-frame using a 2-D Fourier domain filter as proposed in [17]. It should be noted that the filtering was performed on the envelope data rather than the radiofrequency (RF) data containing axial oscillations at the transmitted frequency (f_c), which is different from the conventional transverse oscillation approach [17, 18], in which bi-directional oscillations are included (axial and transverse directions). This approach is also distinct from heterodyning [19-21] and the fourth order transverse spectral velocity estimator [8] to remove the axial oscillation, as the proposed approach operates on the envelope signal directly. Thus, the axial oscillation was first removed by taking the envelope of RF data, and the transverse oscillation can be directly used to estimate the transverse velocity alone. To our knowledge, this lias not been previously demonstrated. The wavelength of the transverse oscillation (λ_x) is determined by the 2-D Fourier domain filter. In this study, the transverse wavelength, λ_x, was set to be c/f_c (0.06 mm at 25 MHz), which is the wavelength of the transmitted ultrasound wave. The bandwidth of the transverse oscillation is determined by σ_x of the 2-D Fourier domain filter. In the simulation and phantom study, the σ_x equals to λ_x . In the animal model study, the σ_x was reduced to half of λ_x to ensure sufficient transverse resolution in microvascular image.

Following the filter in the transverse direction, the Hilbert transform of the filtered data was computed along the transverse direction to remove the negative frequency component of the data with transverse oscillation. The transverse flow velocity spectrum [ PSD_x(f) ] was obtained by estimating the power spectral density of the data along the slow-time dimension.

C.. Simulation Study

Acquisition with a high-frequency ultrasound imaging system was simulated using Field II [22] for imaging blood vessels. The configuration of the imaging system is described in Table I. The simulated RF data was beamformed off-line using delay-and-sum beamforming in Matlab (R2019a), with an F-number of 0.5. After beamforming and multi-angle compounding, a singular value decomposition (SVD)-based filter was used to remove the signal from the stationary tissue. This filter set the first two largest singular values to 0. Next, the flow velocity spectrum was obtained using the proposed method described in the previous section.

A phantom with a size of 6 × 1 × 4 mm³ (transverse × elevational × axial) was simulated. The diameter of the vessel in the phantom was 1 mm. To test the performance of the proposed method, vessels were simulated at four angles relative to the beam axis (90°, 75°, 60°, 45°). A parabolic flow pattern was simulated at three flow velocities (2, 10, and 50 mm/s at the center of the vessel). In the investigation of different flow angles, the default flow velocity is 50 mm/s. In the investigation of different flow velocities, the default vessel angle is 45°.

As the flow velocity decreases, a longer imaging duration is needed to ensure sufficient slow-time samples for spectral estimation. However, the long imaging duration of slower flow in Field II was extremely time-consuming due to the large number of scatterers simulated in each transmit-receive event to ensure at least 10 scatterers per resolution cell. Therefore, to reduce the computational burden, the imaging frame rate was decimated without violating the Nyquist sampling limit for velocities of 2 and 10 mm/s. The decimated frame rates at each flow velocity are shown in Table I. For each velocity and each flow angle, the simulation was repeated 5 times with different realizations of randomly-generated scatterers.

In evaluating the performance of the proposed approach, the bias of the velocity estimate is obtained by:

$$Bias=\frac{v_{estimated}-v_{true}}{v_{true}}\times 100\%$$ (5)

In addition to the constant flow rate, blood flow with temporally-varying flow velocity was also simulated to demonstrate the feasibility of the proposed method and allow comparison with the previous method [8]. To image continuously for a long duration at a high frame rate (5000 frames/s), the simulation was conducted using the high-speed, built-in simulator of the programmable Verasonics system, which significantly reduced computation time. The transmit frequency was 25 MHz based on the supported frequency of Verasonics system. A vessel with a diameter of 2 mm located at 4 mm in depth was simulated with a parabolic flow profile. No stationary tissue scatterers were included in this part of simulations due to the challenge of the computational time cost. The velocity in the center of the vessel linearly increased from 0 mm/s to 100 mm/s over 0.25 seconds. 1250 frames were simulated. For the simulation using Verasonics simulator, the RF data was beamformed offline using the built-in reconstructor of Verasonics system. The beamformed data were used for transverse spectral velocity estimation.

In addition to computing the bias of the velocity estimate, the performance of the proposed transverse spectral estimation method was also assessed by computing the peak-to-background ratio of the spectrogram. The peak-to-background ratio (PBR) is defined as the ratio between the peak of the power spectral density and the average power density over the entire spectrum excluding the peak power frequency. In the simulation, the flow spectrum was estimated at a specific flow velocity, which indicated that a single peak was expected in the spectrum at each time points. However, the broadening of the spectrum is usually unavoidable due to the finite bandwidth of the imaging system, and the estimator may also introduce additional artifacts, e.g., spectral leakage. Therefore, a high PBR value indicates reduced spectral broadening and leakage. The PBR is calculated at all time points and averaged across the entire period of the spectrogram of the accelerating blood flow.

In the simulation study, the proposed method was also compared with the conventional transverse oscillation method, which uses a fourth order spectral velocity estimator [8]. Specifically, the transverse oscillated signal was obtained by filtering the multiangle compounded beamformed RF signal frame-by-frame using a 2-D Fourier domain filter as proposed by Salles et.al [17]. Then, the power density spectrum for the transverse velocity component can be obtained using the fourth order spectral velocity estimator [8].

D. Phantom Study

To experimentally test the proposed transverse spectral estimation method, a gelatin phantom was fabricated using gelatin (6g/100 ml), graphite (2g/100 ml) and water (130 ml). A vessel with a diameter of 3.5 mm was constructed in the phantom. Corn starch was mixed with water to mimic ultrasound-scattering blood (concentration: ~10 g/L [23]). The centerline of the vessel is approximately 4 mm from the phantom surface. The angle of the vessel is approximately 90° relative to the beam axis. A peristaltic pump (DP-385, INTLLAB) was used to infuse the blood-mimicking fluid through the phantom. The mean flow rate was approximately 12.5 mL/min.

The MS 400 linear array transducer and a programmable ultrasound imaging system (Vantage 256, Verasonics) were used to acquire 1250 frames of images at 5000 frames/s. The transmit frequency was 25 MHz, which was determined based on the trade-off between image resolution and depth, the bandwidth (18-38 MHz) of the MS400 transducer and the supported transmit frequency of Verasonics system. For the clutter filter, the first 5 largest singular values were discarded to suppress the signal of the stationary vessel wall. To suppress the noise, the singular values smaller than the 800^th singular value were discarded as well. The flow velocity spectrum was estimated at the centerline of the vessel. The spectrogram was obtained using a temporal sliding window of 100 frames with a step size of 10 frames.

E. In vivo Imaging of Microvasculature in the Mouse Spinal Cord

The mouse spinal cord has dense microvasculature with blood flow in different directions, making it a good scenario for testing the proposed blood flow velocity estimation approach for all beam-to-flow angles, including smaller angles [8]. The ultrasound data were acquired in a mouse model according to a protocol approved by the Institutional Animal Care and Use Committee of Emory University. Before ultrasound imaging of the mouse spinal cord, the vertebrae from the thoracic and lumbar areas were removed to expose the dorsal surface of a 10 mm segment of the thoracolumbar spinal cord. The high-frequency linear array was then positioned ~ 2 mm above the dorsal surface. The gap between the transducer and the spinal cord was filled using ultrasound gel and mineral oil. The temperature and heart rate were monitored during anesthesia.

1250 frames of images were acquired at 5000 frames/s. The first 10 singular values were discarded to suppress stationary tissue. The singular values smaller than the 100^th singular value were discarded to suppress noise. After clutter filtering, the proposed method was used to estimate the transverse blood flow velocity spectrum in the anterior spinal artery. A series of velocity spectra were obtained at successive time points using a sliding temporal window with a length of 200 frames. The step size of the sliding window in the slow time dimension was 25 frames. The ensemble of 200 frames allowed measurement of a flow as slow as ~1.5 mm/s in the transverse direction and ~0.75 mm/s in the axial direction. Besides, the 2D mean flow velocity maps of both transverse and axial components were obtained by averaging the flow velocity estimation over the entire 0.25 s. The final velocity map was obtained by only retaining the pixels with signal intensity above −20 dB of the maximum signal intensity in the corresponding power Doppler image.

III. Results

A. Simulation Results

The point-spread functions simulated using Field II is shown in Fig. 1. The oscillation was mainly introduced on the transverse direction using the proposed method, in which the envelope data was fdtered using 2D Fourier domain fdter (Fig. 1b), while the oscillation can be seen on both axial and transverse directions using the conventional transverse oscillation approach, in which the RF data was filtered (Fig. 1c).

Fig. 1.

Field II-simulated point spread function obtained using (a) only delay-and-sum beamforming, (b) the proposed method, (c) the conventional transverse oscillation method. In the proposed method, the beamformed envelope data are filtered using a 2-D Fourier domain filter. In the conventional method, the beamformed RF data are filtered using a 2-D Fourier domain filter.

The B-mode images of the simulated blood vessels at 90°, 75°, 60°, and 45° are shown in Fig. 2a, 3a, 4a, and 5a, respectively. For the flow at 90°, the true flow velocity at the center of the vessel was set to be 50 mm/s in the simulation (Fig. 2b). The transverse (v_x) and axial (v_z) flow velocity spectra were estimated at the center of the vessel and averaged across a region having a size of one wavelength by one wavelength. The simulation was repeated five times. Therefore, five flow velocity spectra are shown in the grayscale spectrogram (Fig. 2b and 2c). The positive transverse flow velocity (v_x) can be visualized in the transverse flow velocity spectra. The axial flow velocity spectra are centered around 0 m/s, which is consistent with the simulation setup. The mean flow velocity (red circles in Fig. 2b) estimated from the spectrum appears to be lower than the flow velocity on the centerline.

Fig. 2.

Simulation of blood flow with flow direction of 90° relative to the beam axis. (a) B-mode image of a simulated vessel phantom. The flow velocity at the center of the vessel was 50 mm/s. The simulation was repeated 5 times. The flow velocity spectrum was obtained for each simulation. (b) Transverse velocity spectrum (v_x). (c) Axial velocity spectrum (v_z). The mean flow velocity estimated from the spectrum is indicated by the red circle. The ground truth of the flow velocity is indicated by the green line.

Fig. 3.

Simulation of blood flow with flow direction of 75° relative to the beam axis. (a) B-mode image of a simulated vessel phantom. The flow velocity at the center of the vessel was 50 mm/s. The simulation was repeated 5 times. The flow velocity spectrum was obtained for each simulation. (b) Transverse velocity spectrum (v_x). (c) Axial velocity spectrum (v_z). The mean flow velocity estimated from the spectrum is indicated by the red circle. The ground truth of the flow velocity is indicated by the green line.

Fig. 4.

Simulation of blood flow with flow direction of 60° relative to the beam axis. (a) B-mode image of a simulated vessel phantom. The flow velocity at the center of the vessel was 50 mm/s. The simulation was repeated 5 times. The flow velocity spectrum was obtained for each simulation. (b) Transverse velocity spectrum (v_x). (c) Axial velocity spectrum (v_z). The mean flow velocity estimated from the spectrum is indicated by the red circle. The ground truth of the flow velocity is indicated by the green line.

Fig. 5.

Simulation of blood flow with flow direction of 45° relative to the beam axis. (a) B-mode image of a simulated vessel phantom. The flow velocity at the center of the vessel was 50 mm/s. The simulation was repeated 5 times. The flow velocity spectrum was obtained for each simulation. (b) Transverse velocity spectrum (v_x). (c) Axial velocity spectrum (v_z). The mean flow velocity estimated from the spectrum is indicated by the red circle. The ground truth of the flow velocity is indicated by the green line.

Fig. 10.

The transverse (a, v_x) and axial (b, v_z) flow velocity maps of the mouse spinal cord microvasculature. The axes x and z denote the transverse and axial directions, respectively. The v_x and v_z spectral velocities of the artery, indicated by the green arrow in (a), are shown in (c) and (d). The entire segment of the artery is indicated by the green arrowheads in (b). The green curve is (c) and (d) are the mean velocity measured based on the spectrum.

The flow velocity spectra of a 75° vessel are shown in Fig. 3. The ground truth of the transverse component of the flow velocity was 48.3 mm/s (Fig. 3b). The ground truth axial flow velocity was −12.9 mm/s. The mean transverse flow velocity component measured using the transverse flow spectrum was 35.6±1.6 mm/s. The mean axial flow velocity component measured using the axial flow spectrum was −10.1±3.0 mm/s.

The flow velocity spectra of a 60° vessel are shown in Fig. 4. The ground truth of the transverse component of the flow velocity was 43.3 mm/s (Fig. 4b). The ground truth axial flow velocity was −25 irnn/s. The mean transverse flow velocity component measured using the transverse flow spectrum was 33.1±3.1 mm/s. The mean axial flow velocity component measured using the axial flow spectrum was −21.1±1.0 mm/s.

The flow velocity spectra of a 45° vessel are shown in Fig. 5. The ground truth of the transverse component of the flow velocity was 35.4 mm/s (Fig. 5b). The ground truth axial flow velocity was −35.4 mm/s. The mean transverse flow velocity component measured using the transverse flow spectrum was 29.0±5.0 mm/s. The mean axial flow velocity component measured using the axial flow spectrum was −29.3±3.3 mm/s.

The mean velocity measured using the flow spectra is shown in Fig. 6. The ground truth of the flow velocity is indicated by the green curve. The biases of the transverse flow velocity (v_x) estimated using the proposed method (blue curve in Fig. 6a) were −25%, −26%, −23%, and −18% at 90°, 75°, 60° and 45° respectively. In addition, v_x estimated using the fourth order estimator [8] are shown in Fig. 6 for comparison (red curve). The mean biases of the fourth order estimator were −39%, −34%, −35% and −33% at 90°, 75°, 60° and 45° respectively.

Fig. 6.

The mean flow velocity estimated using the spectra. (a) Transverse flow velocity v_x at varying beam-to-flow angles. (b) Axial flow velocity v_z at varying beam-to-flow angles. (c) Transverse flow velocity v_x at varying true flow velocities. (d) Axial flow velocity v_z at varying true flow velocities. The green curves indicate the true velocity. The blue curves indicate the mean flow velocity measured using the proposed method with error bars representing standard deviation. The red curve with error bars is the mean flow velocity measured using the fourth-order spectral estimator.

In addition, the mean velocity estimator was also tested for different flow velocities: 2 mm/s, 10 mm/s, 50 mm/s. Note that these are the centerline velocities, while the velocity components in the transverse and axial directions will be smaller values. The mean values of the transverse velocity component measured were 1.2±0.2 mm/s, 5.8±0.8 mm/s and 29.0±5.0 mm/s (Blue curve in Fig. 6c), while the ground truth values of the transverse flow velocity component were 1.4 mm/s, 7.1 mm/s and 35.4 mm/s respectively (green curve in Fig. 6c). The mean biases of the estimation of the transverse velocity using the proposed method were −16%, −18%, and −18%. For comparison, the mean transverse velocity components measured using the fourth order spectral estimator were 0.8±0.1 mm/s, 4.3±0.3 mm/s, 23.6±1.5 mm/s (red curve in Fig. 6c). The mean biases using the fourth order spectral estimator were −41%, −39%, and −33%. The mean axial velocity values measured were −1.2±0.1 mm/s, −5.8±0.6 mm/s and −29.3±3.3 mm/s, while the ground truth values of the axial flow velocity were −1.4 mm/s, −7.1 mm/s and −35.4 mm/s respectively (Fig. 6d).

The proposed method was also tested under conditions of temporally-varying flow velocity. The velocity spectrogram was obtained at the center of a 2 mm-diameter vessel while the flow was accelerating (Fig. 7). The proposed method (middle column in Fig. 7) was compared with the conventional transverse oscillation-based fourth order estimator (right column in Fig. 7) for transverse flow velocity estimation. It should be noted that the measurable velocity range of the fourth-order estimator is half of the range of the proposed method with the specified transverse wavelength. For this reason, the fourth-order flow spectra (Fig. 7c, 7f, 7i, 7l) beyond this measurable range is indicated with a white background.

Fig. 7.

Spectrogram of simulated accelerating blood flow. The left column is the axial velocity spectrogram (v_z). The middle column is the transverse velocity spectrogram obtained using the proposed method (V_x_proposed). The right column is the transverse velocity spectrogram obtained using the conventional transverse oscillation-based fourth order estimator (V_x_fourth-order). The beam-to-flow angle varies from 90° to 45° , as shown in the top row to the bottom row. The true flow velocity is indicated by the green line.

The mean bias of the velocity estimation was obtained over the entire period of acquisition. For the proposed method, the mean biases were −11.1 %, −11.7 %, −8.1 % and −1.9% at beam-to-flow angles of 90° (Fig. 7b), 75° (Fig. 7e), 60° (Fig. 7h) and 45° (Fig. 7k), respectively. For the conventional transverse oscillation-based fourth order estimator, the mean biases were −73.1 %, −73.3 %, −71.8 % and −71.6% at beam-to-flow angles of 90°, 75°, 60° and 45°, respectively. Furthermore, spectral artifacts were seen in the fourth order spectrogram (Fig. 7c, 7f, and 7i and 7l). The peak-to-background ratios were 10.0 dB, 9.6 dB, 10.5 dB and 10.0 dB for the fourth order spectrogram at 90°, 75°, 60° and 45°, respectively. For the proposed method, peak-to-background ratios were 16.3 dB, 15.6 dB, 15.6 dB and 15.7 dB, which were higher than those of the fourth order estimator.

The transverse spectra of the accelerating flow at specific time points are also shown in Fig. 8. As the fourth order estimator doubles the Doppler frequency shift [8, 19], The range of the frequency of the fourth-order spectra is scaled by a factor of 1/2 in Fig. 8 for comparison with the proposed method. As the spectra were normalized to the peak, the artifacts as well as the sidelobe levels can be directly compared between the proposed method and the conventional transverse oscillation-based fourth-order estimator. The proposed method exhibits lower artifacts and sidelobe levels than the fourth-order estimator, which is consistent with the peak-to-background ratios presented above.

Fig. 8.

The transverse flow spectra of the accelerating flow at different times (0.05, 0.1, 0.15 and 0.2 seconds). The beam-to-flow angle varies from 90° to 45° , as shown in the top row to the bottom row. The blue curves are spectra obtained using the proposed method. The red curves were obtained using the fourth order estimator. The ground-truth Doppler frequency of the flow is indicated by the vertical green line.

B. Results of Phantom Study

The B-mode image of the vessel phantom is shown in Fig. 9a with the flow direction indicated. The transverse flow velocity spectrogram (Fig. 9b) indicates that the blood-mimicking fluid flowed in a pulsatile manner, which was consistent with the experimental setup. The axial flow velocity spectrum (Fig. 9c) remained centered at 0 mm/s, as the flow in the axial direction remained approximately equal to zero due to the orientation of the vessel. The peak flow velocity at the center of the vessel was above 154 mm/s, which caused aliasing of the transverse spectrum in Fig. 9b.

Fig. 9.

Results of phantom study. (a) B-mode image of the vessel phantom. (b) Transverse velocity (v_x) spectrogram. (c) Axial velocity (v_z) spectrogram. The arrow in (a) indicates the direction of the flow. Arrowheads in (b) indicate the peak velocity of the pulsatile flow, which is aliasing in the spectrum.

C. In vivo imaging of mouse spinal cord

The pulsatile flow velocity of the spinal artery can be seen in the velocity spectrum (Fig. 10c and 10d). It should be noted that the diameter of the spinal artery is much smaller than the flow phantom (Fig. 9). The flow velocity of the spinal cord artery is also approximately 10 times lower than that of the flow phantom. Although the main objective of the proposed method is not estimating the mean flow velocity. The mean blood flow velocity maps can be obtained as shown in Fig. 10. To demonstrate the ability to identify axial and transverse velocities in a microvascular network with low flow velocity, the velocity maps are shown with a dynamic range of −5 to 5 mm/s. The overall diameter of the spinal cord is approximately 2 mm. The vessels having different flow directions can be identified in the velocity maps. The anterior spinal artery along the transverse (x) direction supplying the microvessels can be identified in both the v_x and v_z maps at a depth of 5 to 6 mm (indicated by the green arrow in Fig. 10a, and by the green arrowheads in Fig. 10b). This artery appears as two segments in the conventional axial flow velocity map (v_z. Fig. 10b) since the curvature of the artery caused the blood flow direction to be away from (blue-colored, v_z) <0 for x < −1 mm) and then towards the transducer (red-colored, v_z>0 for x> −1 mm). In the transverse velocity map (v_x. Fig. 10a), the artery was identified as one vessel with flow from left to right (v_x>0). The pulsation of the blood flow over several cardiac cycles can be visualized in transverse velocity spectrum (Fig. 10c) and the video of the dynamic velocity map (Supplemental Video 1).

IV. Discussion

As shown in the simulation results (Fig. 2 to Fig. 6), the estimated mean velocity is lower than the true velocity at the center of the vessel, i.e. there was a negative bias in the estimation. This might be due to the limited resolution in the elevation direction, which is visible in the B-mode image of the phantom. As shown in the B-mode images in Fig. 2 to Fig. 5, there is a significant number of stationary scatterers near and within the vessel, which is expected to be a dark hypoechoic regionfrom3.5 to 4.5 mm in depth. However, due to the finite thickness of the imaging plane in the elevation direction, the transducer still received echoes from the scatterers surrounding the vessel at depths of 3.5 to 4.5 mm. Although the stationary tissue was removed using an SVD filter, there were slowly flowing scatterers near the vessel wall, resulting in echoes being detected by the linear array transducer. This means that both the fastest-flowing scatterers near the centerline and the slowly flowing scatterers away from the centerline were recorded and displayed in the flow velocity spectrum. Therefore, the mean flow velocity measured using the spectra was lower than the true velocity of the scattererinthe centerline. Nevertheless, compared with the previous approach using the fourth order estimator, the proposed method showed reduced bias for estimation of the mean transverse velocity (v_x), as shown in Fig. 6.

In the simulation of a larger vessel (2 mm in diameter), the slowly flowing scatterers may also contribute to the spectral broadening as the velocity increases. However, the spectrograms obtained using the proposed method were mostly consistent with the true velocity in the vessel centerline at each angle, as indicated by Fig. 7. In the previous study of transverse spectral velocity estimation, the transverse flow velocity spectrogram obtained using the fourth-order estimator was only usable when the beam-to-flow angle was greater than 60° [8]. In our study, the fourth-order estimator exhibited bias of approximately −70%, while the proposed method exhibited a much lower bias (< −11.7%, Fig. 7). In addition. Fig. 7 demonstrates that by using the proposed method, one could reliably obtain the transverse velocity spectrogram when the beam-to-flow angle approaches 45°. The PBR of the proposed method was also higher than when using the fourth order estimator at each beam-to-flow angle, which was consistent with spectrogram results in Fig. 7 showing that the proposed method exhibited reduced spectral artifacts relative to the fourth order estimator. The advantages of the proposed method may be attributed to removing the axial oscillation before introducing the transverse oscillation using the envelope data, which is different from the conventional TO method that is applied on the RF data.

The proposed method was tested experimentally in a straight vessel phantom with a 90° beam-to-flow angle, as shown in Fig. 9. The results indicate that the proposed method could be used to track the temporally-varying flow velocity as shown in the transverse velocity spectrogram (v_x. Fig. 9b), which was barely visible in the axial velocity spectrogram obtained using the conventional method (v_z. Fig. 9c). As shown in the simulation, the main advantage of the proposed method is the capability to obtain the transverse velocity spectrogram at a small beam-to-flow angle, e.g. 45°. However, it is challenging to test the proposed approach in vessel phantoms at smaller beam-to-flow angles. To ensure a wide transverse spatial frequency bandwidth for generating transverse oscillation, the vessel needs to be placed close to the transducer surface, which made it unrealistic to build a phantom having a vessel at 45°. Therefore, the vessel in the phantom was built parallel to the transducer surface, similar to the previous work on transverse oscillation-based spectral velocity estimation [8].

In addition to the phantom study, the proposed method was used to image blood flow in the mouse spinal cord, which contains microvessels in various orientations. As shown in Fig. 10a and 10b, the transverse velocity map complements the conventional axial velocity map for identifying the arteries in the spinal cord. Both the transverse and axial flow velocity spectra can be obtained using the proposed method. However, the pulsatile flow was not as significant as that of the phantom study. This could be attributed to the lower flow rate and the much smaller diameter of the spinal artery compared with the phantom.

Previous studies [24, 25] indicate that spinal vasculature blood perfusion deficits may be a promising therapeutic target for rescuing injured nervous tissue and enhancing neuronal function. Various approaches have been developed and tested to improve blood perfusion after injury [26]. However, to validate the outcome of these approaches in preclinical animal models, it is necessary to assess the blood perfusion longitudinally. Various high frequency imaging methods have been developed to image the spinal cord blood perfusion with [27, 28] and without contrast agents [29]. However, only the 1D axial velocity component can be measured currently using conventional imaging methods, which makes it challenging to assess the perfusion quantitatively. The proposed method provides an advantage of quantifying both the axial and transverse velocity components of the blood flow, which makes it suitable for measuring the flow velocity of spinal vessels in the imaging plane. Therefore, the proposed method could potentially be used for monitoring spinal blood perfusion longitudinally in small animal models to guide development of treatments for spinal cord injury.

One of the limitations of the proposed method is the shallow imaging depth. It should be noted that the transverse resolution of the transverse velocity map depends on the selection of parameters for the Fourier domain filter, i.e. transverse oscillation wavelength λ_x , and the bandwidth σ_χ [17]. To ensure sufficient resolution, a small transverse wavelength was required, which required the spatial bandwidth of the image in the transverse direction to be sufficiently wide. Thus, a large receive aperture was used during beamforming to ensure sufficient bandwidth, which limited the presented approach to imaging of shallow vessels. Specifically, given the array size (maximum aperture size, 15.6 mm) and the small beamforming F-number (0.5), the maximum imaging depth was ~7.8 mm.

V. Conclusion

Based on high-frequency (>20 MHz) ultrafast ultrasound imaging (5000 frames/s), a transverse flow velocity spectral estimation method was proposed and validated for measuring the transverse blood flow velocity. The results obtained in the simulation study indicate that while the proposed method has some bias, mean velocity can be estimated for beam-to-flow angles of 90° to 45° without significant performance degradation at small angles, which is an advantage compared with the previous transverse spectral estimation method. The proposed method also has reduced bias in mean velocity estimation compared with the fourth order transverse spectral estimator. Furthermore, the proposed method provided a spectrogram with higher peak-to-background ratio (i.e. reduced spectral artifacts) compared with the fourth order estimator. The results obtained in the vessel phantom show that the temporally-varying flow velocity can be detected in the transverse velocity spectrogram obtained using the proposed approach, which cannot be detected using the conventional axial velocity spectral estimation method. The proposed method was demonstrated for obtaining the pulsatile blood flow spectrum on the transverse direction in a small animal model with a high-frequency ultrasound imaging system.

Appendix

To estimate the mean flow velocity in the transverse direction, the mean frequency $\left(f_{slow\_x}\right)$ of the slow-time signal was estimated according to:

$$f_{slow\_x}=\frac{\int {}_{-FR/2}^{FR/2}f\cdot PS{D}_x\left(f\right)df}{\int {}_{-FR/2}^{FR/2}PS{D}_x\left(f\right)df}$$ (1)

Then the mean transverse flow velocity $\left(v_x\right)$ was estimated by:

$$v_x=-f_{slow\_x}\cdot {\lambda }_x$$ (2)

FR is the frame rate, which is also the slow-time sampling rate of the blood flow.

In addition to the transverse velocity, the conventional axial velocity spectrum $\left[PS{D}_Z\left(f\right)\right]$ was also estimated using the data without transverse oscillation to estimate the axial velocity component $\left(v_z\right)$:

$$v_z=-f_{slow\_z}\cdot c/\left(2f_c\right)$$ (3)

In which f_c is the center frequency of the RF data, and c is the speed of sound. $f_{slow\_z}$ is given by [9, 30]:

$$f_{slow\_z}=\frac{\int {}_{-FR/2}^{FR/2}f\cdot PS{D}_z\left(f\right)df}{\int {}_{-FR/2}^{FR/2}PS{D}_z\left(f\right)df}$$ (4)