Receive / Transmit Aperture Selection for 3D Ultrasound Imaging with a 2D Matrix Transducer

: Recently, we realized a prototype matrix transducer consisting of 48 rows of 80 elements on top of a tiled set of Application Speciﬁc Integrated Circuits (ASICs) implementing a row-level control connecting one transmit / receive channel to an arbitrary subset of elements per row. A fully sampled array data acquisition is implemented by a column-by-column (CBC) imaging scheme (80 transmit-receive shots) which achieves 250 volumes / second (V / s) at a pulse repetition frequency of 20 kHz. However, for several clinical applications such as carotid pulse wave imaging (CPWI), a volume rate of 1000 per second is needed. This allows only 20 transmit-receive shots per 3D image. In this study, we propose a shifting aperture scheme and investigate the e ﬀ ects of receive / transmit aperture size and aperture shifting step in the elevation direction. The row-level circuit is used to interconnect elements of a receive aperture in the elevation (row) direction. An angular weighting method is used to suppress the grating lobes caused by the enlargement of the e ﬀ ective elevation pitch of the array, as a result of element interconnection in the elevation direction. The e ﬀ ective aperture size, level of grating lobes, and resolution / sidelobes are used to select suitable reception / transmission parameters. Based on our assessment, the proposed imaging sequence is a full transmission (all 80 elements excited at the same time), a receive aperture size of 5 and an aperture shifting step of 3. Numerical results obtained at depths of 10, 15, and 20 mm show that, compared to the fully sampled array, the 1000 V / s is achieved at the expense of, on average, about two times wider point spread function and 4 dB higher clutter level. The resulting grating lobes were at − 27 dB. The proposed imaging sequence can be used for carotid pulse wave imaging to generate an informative 3D arterial sti ﬀ ness map, for cardiovascular disease assessment.


Introduction
Arteriosclerosis is a very common cause of death worldwide, caused by deposition of lipids in the vessel wall in the form of plaques [1]. The relation between arterial stiffness and cardiovascular risk has been investigated in several studies as advanced arteriosclerosis can occlude arteries and involve changes of the vessel wall that reduce its elasticity and flexibility [2][3][4]. Estimation of the arterial stiffness second can be achieved. However, for CPWI, 1000 V/s are needed [8,35,36]. Therefore, we are obligated to limit the number of receive/transmit shots to 20. To do so, in this paper, we propose to interconnect elements into non-delayed apertures which can be shifted in the elevation direction. We investigate the effects of receive and transmit aperture sizes and aperture shifting step in numerical simulations and in an in vitro experimental setup. An angular weighting (AW) method (based on the directivity pattern of the receive apertures) is used in the image reconstruction procedure to reduce the effects of grating lobes (caused by the increased effective pitch in elevation direction). The goal is to find a trade-off to form good-quality B-mode images, providing suitable inputs for motion estimation methods used in wall velocity measurement [8,13].

Imaging Schemes
For CPWI, the lateral direction in our 2D assessment (the elevation direction in 3D imaging) lies perpendicular to the carotid artery. For people of age over 60 years, the carotid diameter is 7.4 ± 0.7 mm and 8.2 ± 0.8 mm and over 70, it is 7.6 ± 0.8 mm and 9 ± 0.7 mm, for females and males, respectively [37]. The effective covered aperture size is calculated as (Nr + 19Ns) × Pitch, where Nr is the receive aperture size, Ns is the shift of the receiving aperture between sequential shots, and Nt is transmit aperture size. With Ns 1 and 2 and with a small Nr, the obtained effective aperture is much less than what is needed to cover the whole carotid artery. As we aim to have an effective covered aperture size big enough for all ages, we neglect Ns of 1 and 2 in our study.
A tile of 12 × 40 elements (rows × columns) requires 12 channels for transmission and 12 channels for data reception (Figure 1a). The row-level circuitry allows us to put two tiles head-to-head in a row to form an array of 12 × 80 elements without increasing the channel count [33]. In this study, the array is fabricated using 4 × 2 (rows × columns) tiles providing 48 × 80 (azimuth × elevation) elements ( Figure 1b; the carotid has a diameter of 6-9 mm [37]; the array is 14 × 12 mm). A single plane wave imaging with full data acquisition can be implemented using a column-by-column (CBC) data acquisition approach which requires 80 transmit/receive shots. In this approach, all the 80 elements are connected to the transmit bus, and only a single element in the elevation direction is connected to the receive bus. For CPWI, we need 1000 V/s which restricts the number of transmit-receive shots to 20 for one volume. These 20 shots can be implemented in different manners; an example is shown in Figure 2b where the transmit aperture of 50 elements is shifted over three elements for each step, co-aligned with the receive aperture of 10 elements.
Applying dynamic focusing to the received data having an effective element pitch larger than half of a wave length introduces grating lobes [24]. In our study, the 2D matrix transducer has an elevation pitch of 150 µm, and the operating frequency ranges from 6 to 9 MHz, i.e., the shortest wavelength is 170 µm in tissue. Therefore, grating lobes in the elevation direction will appear.
The current ASIC architecture can accommodate 20 arbitrary transmit/receive patterns in the provided memories, which can be used to define sparse patterns (out of the scope of this paper) [34]. In addition to these element level memories, elements can be grouped into a set of transmit/receive apertures with an arbitrary size and shifted along elevation direction with one element per clock cycle [34]. In this study, we use this feature to define regular aperture patterns (see Figure 2b for example). The elements contributing in a receive aperture are interconnected (switched in parallel without further inter-element delays) in practice, and one time series is achieved for the receive aperture.

Numerical Studies on the Beam Profile
As element interconnection only happens in the elevation direction, our numerical evaluation was only conducted in the elevation direction. Simulations were conducted in Field II [38,39] to investigate the beam profile of the different transmit/receive sequences. The image reconstruction in azimuthal direction is unaffected and therefore neglected. We varied the settings provided in Table 1. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 17 The proposed transducer consisting of 4 × 2 tiles with respect to the carotid artery (carotid has a diameter of 6-9 mm; the array is 12 × 14 mm).

Numerical Studies on the Beam Profile
As element interconnection only happens in the elevation direction, our numerical evaluation was only conducted in the elevation direction. Simulations were conducted in Field II [38,39] to investigate the beam profile of the different transmit/receive sequences. The image reconstruction in azimuthal direction is unaffected and therefore neglected. We varied the settings provided in Table  1. The impulse response consisted of a Gaussian-modulated sinusoidal waveform (central frequency of 7.5 MHz, bandwidth of 80%). Normalization and log-compression were used to form final images. The response of receive aperture size (Nr), the shift of the receiving aperture between sequential shots (Ns) and the transmit aperture size (Nt) were evaluated using point scatterers positioned at depths of 10, 15, and 20 mm, as we expect the carotid artery in this range [8]. Figure 2b illustrates the way that the Nr, Nt, and Ns parameters were defined and implemented in 20 transmit-receive shots. The receive and transmit apertures were concentric and shifted over the same distance.
The level of grating lobes, lateral width of point spread function (PSF) at −6 dB called full-widthhalf-maximum (FWHM) and level of sidelobes were used to evaluate the image quality as result of the different settings [24]. The finalized imaging sequence was evaluated with a numerical tissue mimicking phantom in which an anechoic cyst having a radius of 3 mm was positioned at the depth of 15 mm. The phantom contained scatterers from −15 to 15 mm in the lateral direction to visualize the off beam energy including the grating lobes. Multi-scatterer and multi-cyst phantoms were also imaged. We use column-by-column imaging scheme as the ground truth (GT) in our study.

Experimental Study
The 2D matrix array was mounted in a box with an acoustically transparent window (25 µm thick polyimide) and the whole setup was submerged in a tank filled with water (see Figure 3a). The transducer was excited by a custom-programmed experimental ultrasound system (Vantage 256, Verasonics, Kirkland, WA, USA). A picture of the two-needle phantom used in our experiments is provided in Figure 3d. The axial distance between the plastic foam, that acts as base to the needles, and the surface of our 2D transducer was about 26 mm. Matlab 2018b and MeVisLab (MeVis Medical Solutions AG, Fraunhofer MEVIS, Bremen, Germany) software was used to process the data and show 3D images, respectively.  The impulse response consisted of a Gaussian-modulated sinusoidal waveform (central frequency of 7.5 MHz, bandwidth of 80%). Normalization and log-compression were used to form final images. The response of receive aperture size (Nr), the shift of the receiving aperture between sequential shots (Ns) and the transmit aperture size (Nt) were evaluated using point scatterers positioned at depths of 10, 15, and 20 mm, as we expect the carotid artery in this range [8]. Figure 2b illustrates the way that the Nr, Nt, and Ns parameters were defined and implemented in 20 transmit-receive shots. The receive and transmit apertures were concentric and shifted over the same distance.
The level of grating lobes, lateral width of point spread function (PSF) at −6 dB called full-width-half-maximum (FWHM) and level of sidelobes were used to evaluate the image quality as result of the different settings [24]. The finalized imaging sequence was evaluated with a numerical tissue mimicking phantom in which an anechoic cyst having a radius of 3 mm was positioned at the depth of 15 mm. The phantom contained scatterers from −15 to 15 mm in the lateral direction to visualize the off beam energy including the grating lobes. Multi-scatterer and multi-cyst phantoms were also imaged. We use column-by-column imaging scheme as the ground truth (GT) in our study.

Experimental Study
The 2D matrix array was mounted in a box with an acoustically transparent window (25 µm thick polyimide) and the whole setup was submerged in a tank filled with water (see Figure 3a). The transducer was excited by a custom-programmed experimental ultrasound system (Vantage 256, Verasonics, Kirkland, WA, USA). A picture of the two-needle phantom used in our experiments is provided in Figure 3d. The axial distance between the plastic foam, that acts as base to the needles, and the surface of our 2D transducer was about 26 mm. Matlab 2018b and MeVisLab (MeVis Medical Solutions AG, Fraunhofer MEVIS, Bremen, Germany) software was used to process the data and show 3D images, respectively. (e) The imaging phantom: a foam pad with five needles, positioned in different depths (from 10 to 20 mm). The distance between the foam and the transducer surface is about 28 mm. The numbers next to the needles are used for target localization in Section 3. used (Nt = 80) and 20 shots are given. For a receive aperture equal to one element (Nr = 1) and step size of three elements (Ns = 3), the total covered aperture is 8.7 mm resulting in a main beam width of about 2 mm at 10 mm distance. The grating lobe level is −20 dB as expected from the 1.5 cycle excitation and 20 shots and positioned at a lateral distance of −5 mm caused by the large step (Ns = 3, effective pitch = 450 µm, λ = 200 µm). The mean beam width remains about the same for larger Nr up to 5 (see Figure 4e) since two opposite effects play a role. The first one is the slowly increasing covered aperture (e.g., for Nr = 5, the covered aperture is 9.3 mm). The opposite one is the increasing directivity of the receiving apertures (for Nr = 5 the directivity is about 10 • ), which makes the contribution of the outer elements less. For larger receive apertures, Nr = 10 the main beam width is increased to about 3 mm and it further increases to 4.2 mm for Nr = 20. For Nr = 20, the covered aperture is admittedly the largest, but very directive (2 • ), and therefore only the middle part of the covered aperture is contributing in image reconstruction. For Ns = 4 (see Figure 4b), the grating becomes stronger, making the grating lobes appear closer to the mainlobe, at −4 mm lateral distance (see the rectangles in Figure 4a      Since the effective aperture can be very large and directive, angular weighting, as described in [40], can be applied in reception. This results in reduced grating lobes (compare the first and second rows in Figure 4) and about 16 and 13 dB lower clutter level for Ns of 3 and 4, respectively (see Figure 5a where the pixel intensities within the red rectangular regions shown in Figure 4a were used for statistical analysis). The level of grating lobes (for different Nr) obtained with the AW technique is presented in Figure 5b. The AW technique hardly works for Nr of 1 up to 5 due to the large angular view of the receive aperture. Figure 6 shows the FWHM as function of aperture settings. It indicates that the smaller the Nr, the lower the FWHM (i.e., better lateral resolution). This is mainly because we use the dynamic focusing technique in reception, and increment in Nr results in a narrower directivity in each receive aperture, which reduces the effective number of elements (i.e., aperture size) contributing in the reconstruction of a given pixel (voxel in 3D).

Numerical Results: Point Scatterer
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 17 Figure 6 shows the FWHM as function of aperture settings. It indicates that the smaller the Nr, the lower the FWHM (i.e., better lateral resolution). This is mainly because we use the dynamic focusing technique in reception, and increment in Nr results in a narrower directivity in each receive aperture, which reduces the effective number of elements (i.e., aperture size) contributing in the reconstruction of a given pixel (voxel in 3D). An analysis of Figures 4 and 6 shows that the AW technique helps with suppression of the grating lobes at the expense of resolution. With an AW, a higher contrast can be expected [40], helping the motion estimation methods used to estimate the wall velocities in CPWI [8]. Thus, we use AW technique in the reconstruction through rest of the paper.
As seen in Figure 4c,d, grating lobes are not well-suppressed for Nr = 1 (see the red circles). For Ns = 3, these effects reduce about 6 dB from Nr = 1 to Nr = 5 and then reduce with a lower rate along with oscillation (see Figures 5b and 4c). Almost the same pattern happens for Ns = 4 from Nr = 1 to An analysis of Figures 4 and 6 shows that the AW technique helps with suppression of the grating lobes at the expense of resolution. With an AW, a higher contrast can be expected [40], helping the motion estimation methods used to estimate the wall velocities in CPWI [8]. Thus, we use AW technique in the reconstruction through rest of the paper.
As seen in Figure 4c,d, grating lobes are not well-suppressed for Nr = 1 (see the red circles). For Ns = 3, these effects reduce about 6 dB from Nr = 1 to Nr = 5 and then reduce with a lower rate along with oscillation (see Figures 4c and 5b). Almost the same pattern happens for Ns = 4 from Nr = 1 to Nr = 5. We also noticed the same trend happening at depths of 15 and 20 mm (depths of interest for CPWI [8])-results are not provided here. Based on the lower grating lobes for Ns = 3 than Ns = 4 (Figure 5b), the reduction of grating lobes from Nr of 1 to 5 for Ns = 3 (Figure 5b), and the same resolution in Nr = 5 for Ns of 3 and 4 (see zoomed version in Figure 6), we proceed with Ns = 3 and Nr = 5. It is true that Nr = 5 increases the FWHM more than a smaller Nr (see Figure 6), but lower grating lobes are more beneficial/needed in CPWI than a higher resolution due to motion estimation algorithms used in CPWI [8,13]. Although even lower grating lobes can be achieved by a Nr larger than 5 (Figure 5b), the rate of FWHM increment is much larger than the rate of grating lobes decrement (see Figure 6) when Nr increases. It is expected to have 7 dB higher signal-to-noise ratio (SNR) with Nr = 5 than Nr = 1. Moreover, with Ns = 3 and Nr = 5, an effective covered aperture of 9.3 mm can be obtained, which is large enough to cover the carotid artery in all ages [37].
We conducted the same assessment for scatterers positioned at depths of 15 and 20 mm (results not shown). For different Ns, almost the same difference in the mean of the level of grating lobes (red lines in Figure 6a) is achieved. The same effects indicated by the red circles Figure 4c,d occur, but with lower power. Following the same procedure Ns = 3 and Nr = 5 can be selected for receive aperture in these depths as well.

Transmit Aperture
So far, we used a full-aperture transmission. In this section, we evaluate the effects of the transmit aperture size (Nt) on the PSFs obtained with the selected reception parameters (Nr = 5 and Ns = 3). Figure 7a-c shows the PSFs obtained with different Nt while a scatterer is positioned at the depths of 10, 15, and 20 mm, respectively. A small Nt leads to strong artifacts caused by the grating lobes. This is due to the omnidirectional pressure field in transmission which causes excitation in the direction of the grating lobes with the same energy as that of main lobe. As the Nt gets larger, the transmission pressure field gets more directional toward the front side of the array and becomes like what is shown in Figure 2a. In higher depths, the effects of grating lobes are lower since a lower pressure is transmitted in the direction of the grating lobes (see Figure 2a).
The effects of grating lobes are lower when Nt is larger than 20 (see Figure 7a-c). From Nt of 1 to 50 the FWHM decreases, and reaches an asymptotic minimum for Nt larger than 50. However, around a Nt of 50, the effects of sidelobes are still more powerful than larger Nt (compare the regions shown by the red and green dashed rectangles in Figure 7a-c). Since the larger the Nt, the lower the effects of sidelobes, we proceed with a full-aperture transmission (Nt = 80).

Lateral Shift Variance
In Figure 7d,e, point scatterers positioned at the depths of 10 and 20 mm, respectively, are laterally shifted from −6.5 to 6.5 mm. This image is normalized to its maximum to fairly track the effects of grating lobes (see the green arrows) [41]. Figure 7d,e indicates that if a scatterer is positioned in the direction of the grating lobes, its effect is much lower than that of a scattering target in the middle of the array. This is mainly because of the AW technique in reception and the fact that the direct transmitted pressure field excites targets in the lateral center of the array with a higher power, compared to those in the grating lobes angles (see Figure 2a). The best resolution is also obtained when the scatterer is at the lateral center.
The effects of grating lobes are lower when Nt is larger than 20 (see Figure 7a-c). From Nt of 1 to 50 the FWHM decreases, and reaches an asymptotic minimum for Nt larger than 50. However, around a Nt of 50, the effects of sidelobes are still more powerful than larger Nt (compare the regions shown by the red and green dashed rectangles in Figure 7a-c). Since the larger the Nt, the lower the effects of sidelobes, we proceed with a full-aperture transmission (Nt = 80).

Comparison with Ground Truth
A comparison between the reconstructed images by the proposed imaging sequence (full-aperture transmission, Nr = 5 and Ns = 3) and the CBC imaging sequence (ground truth, GT) at three depths of interest is provided in Figure 8a,b, along with their corresponding PSFs in Figure 8c. The AW is applied to the GT to have a fair comparison. By the proposed imaging sequence, the FWHM is almost doubled due to the larger receiving aperture, and the mean clutter level, on average, is 4 dB higher than GT, with a level of grating lobes of about −24, −27, and −29 dB, at the depths of 10 mm, 15 mm, and 20 mm, respectively. See the Supplementary Materials ( Figure S1) for the results obtained using the multi-scatterer phantom.

Numerical Results: Cysts Phantom
The results obtained with the cyst phantom are shown in Figure 9. The images are laterally limited due to the narrow transmission beam profile (see Figure 2a). The proposed scheme (Figure 9a) is more laterally limited because of the smaller effective aperture size than the CBC GT (Figure 9b). The red arrows in Figure 9a show the effects of grating lobes. To statistically compare these images, pixel intensities within two regions of interest (ROI1 and ROI2, the red and yellow circles in Figure 9b) are used. For the proposed imaging sequence, the mean of ROI1 is 6 dB higher than GT (see Figure 9c) due to the effects of the grating lobes and sidelobes (see Figure 8c). The 3 dB higher mean in ROI2 provided by the proposed imaging sequence (Figure 9c) is also due to the larger FWHM and more constructive interferences of the point scatterers. See the Supplementary Materials ( Figure S2) for the results obtained using the multi-cyst phantom. three depths of interest is provided in Figure 8a,b, along with their corresponding PSFs in Figure  8c. The AW is applied to the GT to have a fair comparison. By the proposed imaging sequence, the FWHM is almost doubled due to the larger receiving aperture, and the mean clutter level, on average, is 4 dB higher than GT, with a level of grating lobes of about −24, −27, and −29 dB, at the depths of 10, 15, and 20 mm, respectively. See the Supplementary Materials ( Figure S1) for the results obtained using the multi-scatterer phantom.

Numerical Results: Cysts Phantom
The results obtained with the cyst phantom are shown in Figure 9. The images are laterally limited due to the narrow transmission beam profile (see Figure 2a). The proposed scheme (Figure 9a) is more laterally limited because of the smaller effective aperture size than the CBC GT (Figure 9b). The red arrows in Figure 9a show the effects of grating lobes. To statistically compare these images, pixel intensities within two regions of interest (ROI1 and ROI2, the red and yellow circles in Figure 9b) are used. For the proposed imaging sequence, the mean of ROI1 is 6 dB higher than GT (see Figure 9c) due to the effects of the grating lobes and sidelobes (see Figure 8c). The 3 dB higher mean in ROI2 provided by the proposed imaging sequence ( Figure  9c) is also due to the larger FWHM and more constructive interferences of the point scatterers. See the Supplementary Materials ( Figure S2) for the results obtained using the multi-cyst phantom.  Figure 10 presents the experimental results of the 3D reconstructed images using the GT and the proposed imaging sequence (full-aperture transmission, Nr = 5 and Ns = 3). The AW is applied on the GT to have a fair comparison. The red, yellow, green, and blue arrows show the foam, the two needles, the grating lobes, and the clutter in the azimuth direction, respectively. The smaller size of foam in Figure 10b,e is due to a smaller effective aperture size in the elevation direction. A better visualization on the needles is achieved in Figure 10e compared to Figure 10d due to the AW technique.  Figure 10 presents the experimental results of the 3D reconstructed images using the GT and the proposed imaging sequence (full-aperture transmission, Nr = 5 and Ns = 3). The AW is applied on the GT to have a fair comparison. The red, yellow, green, and blue arrows show the foam, the two needles, the grating lobes, and the clutter in the azimuth direction, respectively. The smaller size of foam in Figure 10b,e is due to a smaller effective aperture size in the elevation direction. A better visualization on the needles is achieved in Figure 10e   The maximum intensity projection (MIP) over a depth of 1-23 mm of the 3D experimental images obtained with the two-needle phantom is shown in Figure 11a-c. The elevation FWHM obtained with the proposed imaging sequence at the depth of about 10 mm is doubled, in accordance with the results provided in Figure 8a,b. This is the reason that the targets look stretched in the elevation direction, as seen in Figures 11a and 10b. The azimuth FWHM for different receive parameters stays the same (about 620 m) as the proposed imaging sequence does not significantly affect the performance of the 2D transducer in the azimuth direction. The same clutter patterns for different Ns, but the same Nr, also prove this matter (see Figure 11d). The clutter level in azimuth direction is higher than elevation direction due to the larger pitch (300 μm) and lower number of elements (48). The maximum intensity projection (MIP) over a depth of 1-23 mm of the 3D experimental images obtained with the two-needle phantom is shown in Figure 11a-c. The elevation FWHM obtained with the proposed imaging sequence at the depth of about 10 mm is doubled, in accordance with the results provided in Figure 8a,b. This is the reason that the targets look stretched in the elevation direction, as seen in Figures 11a and 10b. The azimuth FWHM for different receive parameters stays the same (about 620 µm) as the proposed imaging sequence does not significantly affect the performance of the 2D transducer in the azimuth direction. The same clutter patterns for different Ns, but the same Nr, also prove this matter (see Figure 11d). The clutter level in azimuth direction is higher than elevation direction due to the larger pitch (300 µm) and lower number of elements (48).

Discussion
The aim of this study was to find a set of imaging parameters that provides a good image quality using an aperture-shifting matrix array, within the boundary condition of a limited number of transmissions (20), such that a volume frame rate of 1000 per second could be reached. We aimed for such high volume rate to facilitate CPWI (seeing the 4.5 m/s pulse wave pass laterally within a few volumes [8]). Depths of evaluation were 10, 15, and 20 mm as the anterior and posterior walls are expected at these depths [8]. The effects of receive aperture size (Nr), the aperture shifting distance per pitch (Ns), and transmit aperture size (Nt) were studied.
Improving the quality of B-mode images helps in CPWI as it facilitates the motion estimation algorithms to track the axial displacement of the arterial walls [8,13]. This is essential for accurate local pulse wave velocity estimation required to generate an elasticity map. Our assessment and parameters selection priority were as follows: (1) the size of effective aperture; as it indicates whether we can have a comprehensive view on the carotid artery or not [37], (2) suppression of grating lobes; the lower the level of grating lobes, the higher the contrast, which is important for motion estimation algorithms to better estimate the axial displacement of carotid walls [8], and (3) resolution/sidelobes. Both Ns of 3 and 4 could provide sufficiently large effective aperture. Due to the lower grating lobes achieved with Ns = 3, compared to Ns = 4, Ns = 3 was selected. Selection of Nr imposed a trade-off between FWHM and grating lobes. The rate of decrement of grating lobes and increment of FWHM in Ns = 3 led to a Nr of 5. Almost the same trends happened in all the three depths of interest for CPWI [8], which led us to Ns = 3 and Nr = 5. The finalized imaging sequence, with a boundary condition of 20 transmit-receive shots, consists of a full-aperture transmission (Nt = 80), Nr = 5, Ns = 3 and using the angular weighting (AW) technique in the image reconstruction procedure.
Signal-to-noise ratio (SNR) is also an important criterion in the performance of motion estimation techniques. However, it is not included in our study since in previous studies, an acceptable SNR in CPWI was achievable with both 1D or 2D arrays [8,13], and it is expected that this is the case with our 2D matrix array as well [34]. If SNR is not good enough (either for CPWI or other applications which need the same effective aperture size) to perform a reliable motion

Discussion
The aim of this study was to find a set of imaging parameters that provides a good image quality using an aperture-shifting matrix array, within the boundary condition of a limited number of transmissions (20), such that a volume frame rate of 1000 per second could be reached. We aimed for such high volume rate to facilitate CPWI (seeing the 4.5 m/s pulse wave pass laterally within a few volumes [8]). Depths of evaluation were 10, 15, and 20 mm as the anterior and posterior walls are expected at these depths [8]. The effects of receive aperture size (Nr), the aperture shifting distance per pitch (Ns), and transmit aperture size (Nt) were studied.
Improving the quality of B-mode images helps in CPWI as it facilitates the motion estimation algorithms to track the axial displacement of the arterial walls [8,13]. This is essential for accurate local pulse wave velocity estimation required to generate an elasticity map. Our assessment and parameters selection priority were as follows: (1) the size of effective aperture; as it indicates whether we can have a comprehensive view on the carotid artery or not [37], (2) suppression of grating lobes; the lower the level of grating lobes, the higher the contrast, which is important for motion estimation algorithms to better estimate the axial displacement of carotid walls [8], and (3) resolution/sidelobes. Both Ns of 3 and 4 could provide sufficiently large effective aperture. Due to the lower grating lobes achieved with Ns = 3, compared to Ns = 4, Ns = 3 was selected. Selection of Nr imposed a trade-off between FWHM and grating lobes. The rate of decrement of grating lobes and increment of FWHM in Ns = 3 led to a Nr of 5. Almost the same trends happened in all the three depths of interest for CPWI [8], which led us to Ns = 3 and Nr = 5. The finalized imaging sequence, with a boundary condition of 20 transmit-receive shots, consists of a full-aperture transmission (Nt = 80), Nr = 5, Ns = 3 and using the angular weighting (AW) technique in the image reconstruction procedure.
Signal-to-noise ratio (SNR) is also an important criterion in the performance of motion estimation techniques. However, it is not included in our study since in previous studies, an acceptable SNR in CPWI was achievable with both 1D or 2D arrays [8,13], and it is expected that this is the case with our 2D matrix array as well [34]. If SNR is not good enough (either for CPWI or other applications which need the same effective aperture size) to perform a reliable motion estimation, a larger Nr can be used. However, it would cause a larger FWHM as a consequence of relatively more direct receive apertures and applying the dynamic focusing in reception. Therefore, in the case of using a large Nr (i.e., 20), a more sophisticated image reconstruction method, which takes the narrow directivity of a large Nr into account, should be used in the elevation direction to form a 3D volume [42].
The final version of our 2D transducer will have 10 tiles in the azimuth (row) direction and two tiles in the elevation (column) direction, making 120 × 80 (row × column) elements available to form a 3D image. Having an azimuth pitch of 300 µm, the size of our 2D transducer in azimuth direction will be 36 mm, thus providing an aperture size big enough to track the carotid pulse wave velocity (estimated to be around 4.5 m/s [8]). All the evaluations and decisions in this paper were conducted using static imaging phantoms while in CPWI, there are several types of motion, i.e., carotid wall movement caused by pulse wave propagation, overall movement by breathing, and patient/transducer movement. Motion of the object might lead to unwanted decorrelation of the signals recorded by the sequentially switched apertures, leading to lower image quality. To calculate the risk of such decorrelation, we performed the following calculation. The wall velocity is, estimated from the interframe axial displacement, reported to be about 5 mm/s [8,13]. Considering the 1 ms needed for data acquisition by the proposed imaging sequence for a 3D volume, the target moves 5 µm in the axial direction. In this scenario, the PSF obtained with a dynamic scatterer, moving 5 µm in 20 transmit-receive shots, well overlaps with a static scatterer, which is mainly because the motion is lower than 10% of the wavelength (200 µm) during the time needed to acquire the data of a 3D volume [43]. Therefore, we can assume that the coherent compounding from the 20 acquisitions is not negatively affected. In addition, the global movement is slow, will not interfere with the coherent image reconstruction, and can be removed by subtraction of displacement of the tissues and that of carotid artery [8].
With 20 kHz pulse repetition frequency, reflection artifacts come from depths higher than 37.5 mm. In this study, these reflection artifacts are ignored due to tissue attenuation (≈0.4-0.6 dB/MHz/cm) as a scatterer at a higher depth (e.g., 50 mm) is attenuated by 28 dB compared to the reflection coming from 12.5 mm depth. A more advanced beamforming algorithm might be helpful with suppression of these artifacts as well [44][45][46][47], taking the change of speckle patterns into account [48].
Finally, although the reported results are linked to the specifications of the matrix transducer reported in our previous publication (see [34]), we would like to stress that the procedure taken in this study to select receive/transmit parameters can be used for other array transducers (either 1D or 2D) to obtain a higher frame/volume rate at a given number of data acquisition channels.
As a future study, we will use the proposed imaging sequence to conduct clinical 3D carotid pulse wave imaging and generate 3D elasticity maps.

Conclusions
In this paper, we evaluated the effects of receive/transmit parameters on the image quality provided by a linear array in which elements are mutually interconnected, which mimics the elevation direction of our prototype 2D matrix transducer. The parameter selection procedure was based on the effective covered aperture size, level of grating lobes, and resolution/sidelobes. To achieve 1000 volumes/second, only 20 transmit-receive shots were available, considering a pulse repetition frequency of 20 kHz and imaging depth of 37.5 mm. To suppress the effects of grating lobes, an angular weighting technique was used. The proposed imaging sequence consisted of a full-aperture transmission (all 80 elements), receive aperture size of 5, and aperture shifting distance of three elements. For depths of 10-20 mm, the results show that, on average, about two times wider point spread function at −6 dB and marginally (4 dB) higher clutter level were obtained, compared to the ground truth, and grating lobes level of about −27 dB lower than the main lobe. Having a proper imaging sequence based on 1000 V/s could be beneficial to generate 3D maps of carotid artery stiffness, providing valuable information regarding the cardiovascular risk.