High-Contrast and -Resolution 3-D Ultrasonography with a Clinical Linear Transducer Array Scanned in a Rotate-Translate Geometry

Featured Application

This single-side access 3-D synthetic aperture scanning method can be readily applied with existing 1-D transducer arrays to more reliably monitor response of lesions over the course of therapy.

Abstract

We proposed a novel solution for volumetric ultrasound imaging using single-side access 3-D synthetic aperture scanning of a clinical linear array. This solution is based on an advanced scanning geometry and a software-based ultrasound platform. The rotate-translate scanning scheme increases the elevation angular aperture by pivoting the array (−45° to 45°) around its array axis (axis along the row of its elements) and then scans the imaged object for each pivoted angle by translating the array perpendicularly to the rotation axis. A theoretical basis is presented so that the angular and translational scan sampling periods can be best adjusted for any linear transducer array. We experimentally implemented scanning with a 5-MHz array. In vitro characterization was performed with phantoms designed to test resolution and contrast. Spatial resolution assessed based on the full-width half-maximum of images from isolated microspheres was increased by a factor of 3 along the translational direction from a simple translation scan of the array. Moreover, the resolution was uniform over a cross-sectional area of 4.5 cm². Angular sampling periods were optimized and tapered to decrease the scan duration while maintaining image contrast (contrast at the center of a 5-mm cyst on the order of −26 dB for 4° angular period and a scan duration of 10 s for a 9-cm³ volume). We demonstrated that superior 3-D ultrasound imaging can be obtained with a clinical array using our scanning strategy. This technique offers a promising and flexible alternative to development of costly matrix arrays toward the development of sensitive volumetric ultrasonography.

1. Introduction

Three-dimensional (3-D) ultrasound (US) imaging is increasingly prevalent in biomedical imaging [1]. Compared to conventional 2-D images, volumetric US display provides a detailed view of anatomical structures at various orientations. Thereby, visualization of structures and the associated measurements are less dependent on the skill and experience of the sonographer and measurements can be more readily and independently repeated on the same data set by other radiologists [2]. Additionally, volumetric US imaging facilitates fusion with images from other modalities for improved diagnosis or therapy planning. Furthermore, for longitudinal studies or for the monitoring of the progression or regression of pathology in response to therapy, 3-D US display facilitates the comparison of structures examined at different time points or by different practitioners.

Most current approaches developed to produce 3-D US images are based on volumetric reconstruction after acquiring successive 2-D images with an ultrasound 1-D array at different spatial positions and/or orientations [2]. For example, a 3-D ultrasound image of a fetus can be obtained using a mechanically wobbled linear array [3]. After acquisition of a series of 2-D images, the volume is reconstructed from the 2-D envelope-detected images positioned in a 3-D volume using specific spatial transformations [4]. These transformations can be retrieved during free-hand scanning with position sensing [5,6], free-hand scanning without position sensing, or mechanical scanning with a predefined motion [2]. Since these approaches use 2-D, envelope-detected images to produce a volumetric display, they can be implemented as external post-acquisition processes using images acquired with a conventional ultrasound machine. When a voxel has been viewed with different orientations of the array, incoherent spatial compounding can be applied to improve image quality [7]. However, the use of envelope-detected images limits the gain in terms of contrast or spatial resolution compared to what could be expected from coherent compounding [8]. To obtain the greater advantages provided by coherent compounding, ultrasound signals or images must be accessed prior to envelope detection.

At the cutting edge of ultrasound technology, planar arrays combined with programmable and fast acquisition electronics are emerging to achieve volumetric imaging rates in the kHz range. A variety of transducer spatial arrangements have been proposed including row-column arrays [9], matrix [10,11], and sunflower [12] arrays. The ultrasound beam is electronically controlled and received signals are combined with coherent compounding. Synthetic aperture imaging [13,14], which consists of sequentially emitting and/or receiving with different array-element subsets, can be used when the number of array elements is larger than the number of transmit/receive channels. Planar array technology has attracted a lot of attention because it enables high-rate imaging of whole volumes and thereby enables advanced imaging modes throughout the volumes such as elastography [11], ultrafast Doppler [15], and super-resolution ultrasound imaging [16]. However, the technology is expensive since transducer planar arrays are still prototyping and a large channel count may be needed to probe the array aperture at a high rate. Additionally, the development of small but highly sensitive transducer elements that can be arranged in an array to cover a significant planar aperture with good spatial sampling is a complex technological challenge. Currently, planar arrays providing small elements and good spatial sampling sacrifice sensitivity and an overall aperture extent. Planar arrays with better sensitivity typically use larger elements at the expense of an element directivity (limiting the effective aperture) and degraded spatial sampling. Therefore, volumetric image quality with this technology has yet to rival that provided with 2-D array transducers.

Three-dimensional synthetic aperture scanning provides an intermediate solution between the incoherent compounding of 2-D, single-plane images and volumetric imaging with planar arrays. It consists of a mechanical scan of a transducer using a system providing access to the data prior to envelope detection, followed by coherent compounding of the signals acquired at different scan positions to synthetize a larger aperture during the volumetric image reconstruction process. The method has been developed for simple scanning geometries of clinical ultrasound arrays designed for 2-D imaging using translation in the out-of-plane dimension [17], rotation around the median axis of the image [18], and a rocking motion [19]. The method advantageously uses already-developed transducer arrays that are technologically mature and highly sensitive and facilitate the extension of methods to the clinical setting. The time required for mechanical displacements prohibits high-rate scanning of the volume, but the method provides higher image quality than the previously mentioned approaches because coherent compounding can be coupled with a high number of ultrasound signals recorded with very sensitive transducer elements. To date, 3-D synthetic aperture scanning has only been investigated in proof-of-concept studies [17,18,19]. It has not yet been developed within a 3-D ultrasound scanner. One reason may be the relatively limited enhancement in spatial resolution that has been obtained with the simple scanning geometries that have been investigated thus far. Furthermore, ultrasound platforms that provide access to the raw channel or radiofrequency ultrasound data that are required for synthetic aperture scanning have only recently become widely available. Now is the time to reconsider 3-D synthetic aperture scanning with more advanced scanning geometries for applications that require high image quality but modest volume-scanning rates such as monitoring of a lesion’s progression or therapeutic response.

In this paper, we investigated the performance of a 3-D, synthetic aperture scanning scheme with a complex scanning geometry based on a rotate-translate scan with a linear ultrasound array. With the combined and predefined motions, a planar surface above the imaged volume was sequentially and efficiently populated with transducers having different spatial positions (array element and translated position) and orientations (angle of rotation about the normal to the surface). This combination of orientations provided a larger synthetic angular aperture in the elevation direction of the array, and thereby improved the resolution and contrast along the elevational axis. The proposed rotate-translate scanning approach was inspired from a scheme previously developed for photoacoustic (PA) tomography [20]. PA tomography is a hybrid imaging modality in which the ultrasound wavefield generated by laser excitation of optical absorbers is captured by transducer elements used only as detectors. The experimental implementation of the scanning scheme in this ultrasonic reception-only configuration resulted in high-quality, 3-D, PA images. In the present study, we demonstrated for the first time the applicability of a rotate-translate scanning scheme for pulse-echo US volumetric imaging and we showed experimentally that higher and more uniform 3-D resolution can be obtained than for existing synthetic aperture approaches. The original scheme proposed for photoacoustic imaging was limited due to the long scan duration. Herein, the scanning parameters were optimized through theoretical considerations and experimental implementation to dramatically reduce the number of scanning positions and thereby the acquisition time while maintaining high contrast and uniform spatial resolution. In particular, we studied the influence of the rotation step on the spatial resolution and image contrast. The system was characterized in vitro with imaging phantoms and the performance was compared to synthetic aperture scanning with a translation-only scan to highlight the superior image quality of this new, more complex scanning scheme.

2. Materials and Methods

2.1. Presentation of the Scanning Scheme

The rotate-translate scanning scheme is diagrammed in 2-D (in the axial-elevation plane) for one element of the linear US array (Figure 1). The extension to 3-D is direct when the lateral dimension of the array (i.e., axis along the row of active array elements) is taken into account. We considered here a linear array with a cylindrical lens, which is typical of clinical linear US arrays for 2-D ultrasonography. The focusing power of the lens is usually weak, which ensures a large depth of field but a relatively poor elevation resolution. These depth-of-field and resolution performances are directly linked to the small angular aperture 2.β of the element (Figure 1a) or the large f-number in the elevation direction. The f-number is equal to F/D where F and D are the focal distance and the width of the array element, respectively (Figure 1a). To benefit from the large depth of field but with a significant improvement in the elevation spatial resolution, we developed a synthetic aperture approach. The synthesis of the aperture increases the angular aperture for each array element over an extended imaged area thanks to a rotate-translate scanning scheme. The rotation increases the angular aperture and the 1-D translation ensures coverage of the imaged zone.

To obtain a single-side access, the rotation axis was set to pass through the center of the transducer face O_a (Figure 1a) and to be perpendicular to the axial direction of the transducer (w-axis) [20]. This axial direction is named transducer axis. The translation axis is perpendicular to both the rotation axis and the transducer axis when the rotation angle α_rot equals zero (Figure 1b). Translation across the imaged area is performed at a constant angle, α_rot, and then repeated for different angles of rotation (Figure 1c). All the translation scans had the same length L. To benefit from the full depth of field around the focus, the imaged area was centered on the focal distance F in the e_z-direction (Figure 1c). As a consequence, the translational scan at each angle, α_rot, was centered on the point l_{trans center}(α_rot) = −F × tan(α_rot) in the e_x-direction (Figure 1c). To have a constant step Δl between successive translational positions of the transducer axis, the translational scan at α_rot was divided into tomographic positions separated by Δl/cos(α_rot).

The ultrasound pulse-echo signals were recorded for each tomographic position of the transducer element. In this configuration and with an adapted Δl (see Section 2.2), each point of the imaged area was probed with all the different orientations of the transducer. Therefore, for each pixel of the sampled area, the ultrasound signals from the different positions could be combined coherently during the reconstruction process to build a larger synthetic aperture. This process also corresponds to spatial coherent compound imaging in the elevation direction.

2.2. Spatial Sampling of the Scan

Our scanning scheme combines motion of the US transducer in translation and in rotation. Therefore, both the linear and the angular spatial sampling periods need to be set. To improve upon the approach proposed for PA tomography in [20], we sought to determine the sampling periods offering the best compromise between image quality and acquisition speed. Indeed, increasing the number of tomographic positions usually leads to higher image quality (contrast, homogeneity) but will result in longer acquisition time.

The minimal sampling periods required to avoid grating lobes and other aliasing artifacts on the image depend both on the transducer geometrical parameters (Figure 1a) and the reconstruction method. The reconstruction algorithm for 3-D imaging is detailed in Appendix A. Our reconstruction algorithm is based on a basic delay-and-sum process. In the 2-D configuration, the transducer is assumed to be a point transducer located at O_a (Figure 1a), which emits circular waves (cylindrical in 3-D) and has an isotropic detection pattern. The spatial impulse response of the transducer is not modeled, but its directivity is taken into account with an amplitude mask. For a given position of the transducer, the pulse-echo signal only contributes to the image in a slice of width 2 × d_u (Figure 1b) around the transducer axis. The distance d_u is a distance to the transducer axis in the elevation direction. In this slice, an apodization window (20% cosine taper) was used to avoid discontinuities. The pulse-echo (two-way) −12 dB full width of the ultrasound beam is 1.2 × H [21] where H = λ_c × f-number, with λ_c the wavelength at the center frequency f_c of the transducer. So, to account for the focusing of the transducer, we set d_u = 0.6 × H.

The linear sampling period Δl is defined as the distance between the transducer axis for two successive translation positions at a constant angle α_rot (Figure 1c). The angular sampling period, Δα, corresponds to the angular change between two successive translations across the imaged area. Performing Nyquist–Shannon sampling in translation and in rotation [22], when considering point detectors in the reconstruction, would require a large number of positions and would lead to a prohibitive scan duration. To obtain more efficient scanning, we reduced the synthetic array to a rotation array: For each pixel in the imaged area, we coherently added signals corresponding a single position in translation for each rotation angle α_rot. Thereby, we avoided generating a synthetic array with the successive translation positions at a given α_rot. The selection of the translation positions was made based on the amplitude mask. We should have Δl ≥ 2 × d_u so that Δl ≥ 1.2 × H. On the other hand, an effective coverage of the imaged area would require Δl ≤ H, because H is the two-way full-width half-maximum (FWHM) of the ultrasound beam [23]. We chose to set Δl = H, as a compromise justified by the 20% cosine taper of the apodization mask. One can note that Δl and d_u are strongly linked because of the reduction of the synthetic array to a rotation array. For the translation-only scan used for comparison, a synthetic translation array was formed. Therefore, the volumes probed by successive translation positions should overlap and then Δl_{translation-only} ≤ 2 × d_u. We set Δl_{translation-only} = H/8 [24] (see also Appendix B) while d_u was still defined by d_u = 0.6 × H to account for the transducer directivity.

Angular sampling in our scanning geometry has not yet been studied to the best of our knowledge. From studies performed for other geometries, we found (Appendix B) that the minimal angular sampling period is Δα = β. Herein, we studied the influence of the spatial sampling on the imaged quality for three different nominal angular sampling periods:

Δα₁ = β, Δα₂ = β/2 and Δα₃ = β/4

(1)

with β = atan(D/(2 × F)).

(2)

We chose to perform our study with an angular range ${\alpha }_{rot}\in \left[-45°,45°\right]$, which provides a large angular aperture for ultrasound imaging and will strongly showcase the potential of our approach. However, side lobes are expected from a finite and uniformly sampled angular aperture. They can be reduced with amplitude apodization whose amplitude value depends on α_rot, but this solution leads to signal damping on the edge of the scanned interval and we considered this to be a sub-efficient use of the recorded signals. Therefore, we implemented spatial density tapering as a means to reduce both side lobes and the number of rotation positions. Because of its simplicity, we used the Doyle–Skolnik approach, as described in [25]. The angular positions of the synthetic array were determined considering a Hamming window function over the entire angular range [−45°, 45°] and the nominal angular sampling period Δα. The relative reduction in the number of rotation positions compared to a uniformly sampled array is approximately −46% (i.e., the area ratio between the Hamming and the rectangular windows). The translation endpoints for the different rotation positions are shown in Figure 2 for a translation range L = 30 mm and the angular sampling period Δα₂. Due to the spatial density taper, the effective angular sampling step was Δα₂ around α_rot = 0 and increased with the value of |α_rot|. Therefore, the bounds of the angular range [−45°, 45°] were not reached during the scan process. The effective bounds depend on the value of Δα. With an amplitude apodization and the Hamming window function, the recorded signals at α_rot = ±45° would have had a weight of 0.08 compared to a weight of 1 around α_rot = 0.

2.3. Experimental Implementation

We experimentally implemented the scanning scheme presented above with a clinical linear ultrasound array. This section presents the tomographic ultrasound scanner, the practical implementation of the scan, and the imaged samples.

2.3.1. Experimental Setup

The experimental setup presented in Figure 3 can be divided in three main parts: (1) the US acquisition system consisting of an US linear array driven by a programmable US platform, (2) the scanning system comprising two motorized stages and their motion controller, and (3) the synchronization system piloted with a programmable trigger generator. The acquisition process was fully automated.

The ultrasound transducer array was a 128-element linear array (L7–4, Linear Array, Philips Medical Systems, Seattle, WA, USA) with a 5-MHz center frequency and 298-µm inter-element spacing. In the elevation direction, the elements of the array had an aperture of D = 7.5 mm and were cylindrically focused to a focal distance of F = 25 mm, resulting in a f-number of 3.3. Therefore, as described in Equations (1) and (2), Δα₁ ≈ 8°, Δα₂ ≈ 4°, and Δα₃ ≈ 2°. Additionally, for this array, λ_c = 0.3 mm and H = 1 mm with a speed of sound of 1500 ms^-1. This linear array was chosen because it is widely used in the ultrasound community. Acoustic coupling between the sample and the ultrasound array was obtained by immersion in a water tank.

The array was driven by a programmable, 64-channel Vantage ultrasound system (Verasonics, Kirkland, WA, USA). For all the transmit events and all the receive events, only the 64 central elements of the array were used. Upon the arrival of each external trigger event, all these 64 array elements were fired to produce a “plane wave” emission (beam that is unfocused in the lateral direction of the array or the v-direction shown in Figure 1) [26]. Transmitted pulses were 1 cycle long at 5.2 MHz. For most acquisitions, all the elements in the transducer array were fired at the same time to produce an untilted plane wave (γ_st = 0) referred to as “straight transmit”. We also implemented an emission mode in which tilted plane waves were generated with different angles, γ_st, for successive trigger events. The steering angles were controlled electronically by introducing a small delay between the firing of adjacent elements according to the desired inclination angle. In this mode, named “steered transmit”, a cyclic emission sequence was used and, for each cycle, the tilted plane waves were emitted with increasing angular steering (

Table 1. Parameters of the different acquisition modes for a translation range L = 30 mm.

Scanning Mode	Rotate-Translate Scan				Translate-Only Scan
	Straight transmit			Steered transmit	Straight transmit
Mode #	1			2	3
Angular sampling period	Δα1 = β	Δα2 = β/2	Δα3 = β/4	Δα2 = β/2	0
Δα (°)	8	4	2	4	0
Number of angular positions 1	6	12	23	12	1
Steering angles γst (°)	0	0	0	−4; −2; 0; 2; 4	0
Acquisition rate (Hz) 2	50	50	50	250	50
Number of tomographic positions 2	267	508	947	2533	244
Acquisition duration (s)	5.3	10.2	19	10.1	4.9
Reconstruction duration (s) 3	11.2	20.9	39.7	95.2	14.2

¹ The number of angular positions results from the angular sampling period Δα and the density taper. ² The acquisition rate corresponds to the pulse repetition frequency (PRF) of the trigger generator, and the number of tomographic positions to the number of generated trigger pulses. ³ The reconstruction durations were measured for the largest imaged volume Δx × Δy × Δz = 30 × 19 × 30 mm³ and a pixel size of px × py × pz = 72 × 144 × 72 µm³, resulting in total of 417 × 133 × 417 ≈ 23 million voxels. The volumetric image reconstruction was performed using a Graphics Processing Unit (GPU) NVIDIA GeForce RTX 2060 SUPER. For the modes 1 and 2, some tomographic positions were discarded for the reconstruction (see Appendix A).

). The steered angles were chosen to ensure uncorrelated transmitted wave vectors. Given the active aperture of the array and the wavelength λ_c at its center frequency, the steered angles must theoretically be spaced by at least ~0.9° to have independent transmitted wave vectors [8]. We chose here an angular spacing of 2°. Moreover, the steered angles were chosen centered around zero to match the directionality of the array elements.

Pulse-echo signals were recorded at a sampling frequency of 20 MS/s, with a constant gain (no time gain compensation was applied) and for a time interval adapted to the distance range between 5 mm and 60 mm in front of the array. The transmitted pulse amplitude and the reception gains were adjusted to ensure a good amplitude digitalization of received signals without saturation. Ultrasound signals were stored in internal memories of the acquisition system.

We used two high-precision motorized stages. The rotation stage (M-061.PD, Physik instrumente, Karlsruhe, Germany) was mounted on the translation stage (L-511.44AD00, Physik instrumente, Karlsruhe, Germany). The two stages were controlled with a C-884 DC Motor Controller (Physik instrumente, Karlsruhe, Germany). Using integrated encoders, the positions of the stages were recorded with an accuracy of 0.001 degrees and 50 nm, respectively, at the time of arrival of an external trigger pulse to the controller. The positions were stored in an internal memory of the controller.

To synchronize the US plane wave emission and the recording of the stage positions, a trigger generator (BNC Model 577, Berkeley Nucleonics, San Rafael, CA, USA) was used to send external triggers simultaneously to both the programmable US system and the motion controller (Figure 3). The number of pulses and the pulse repetition frequency (PRF) depend on the acquisition parameters (Table 1) and were programmed in the pulse generator prior to each acquisition.

For each tomographic acquisition, the following sequence of events was used. First, the motors were positioned at the first tomographic position and the ultrasound system was initialized. Then, the ultrasound system and the motion controller were set to wait for a trigger pulse. The first trigger pulse started the motion of the stages following programmed trajectories (see Section 2.3.2). This trigger pulse and the next ones triggered the simultaneous recording of the stage positions and of the pulse-echo ultrasound signals. At the end of the scan, all the data from the ultrasound system and the motor controller were transferred to a computer.

2.3.2. Practical Implementation of the Scanning

Three different acquisition modes were employed. The first mode was a 3-D, rotate-translate tomographic scan with straight transmit events (γ_st = 0). The targeted endpoints of the motion were at l_{trans endpoints} (α_rot) = l_{trans center}(α_rot) ± L/2 and are illustrated in Figure 2 (black crosses). For a time-efficient scanning, we implemented continuous scanning rather than step-by-step scanning [27]. We chose to perform translation scans repetitively with a back-and-forth oscillatory motion. The rotation motion was performed in a single direction (increasing α_rot). To illustrate the stage motion, we notated A, B, C, and D as the first four endpoints of the scan in Figure 2. For two successive rotation angles, α_{rot 1} and α_{rot 2}, the translation endpoints were scanned in one translation direction for α_{rot 1} (from A to B for instance) and in the reverse direction for α_{rot 2} (from C to D)_. Thereby, we minimized the scanning time. The rotation motion starts from the second translation endpoint at α_{rot 1} (B in the example) and we aimed at reaching the second rotational position α_{rot 2} when the targeted first translational endpoint at α_{rot 2} was reached (C here). A translation motion was additionally needed between the two endpoint positions (B and C in the example). In the example, the first, positive-direction translation occurred from A to C. The rotational motion started when the translation stage reached B. When C was reached, the rotation motion and the positive-direction translation ended and the translation direction was reversed.

The trajectories of positioning stages had trapezoidal or triangular velocity profiles. Therefore, for each movement, the following parameters had to be set: the endpoints, the targeted velocity, and the acceleration. The deceleration was set equal to the acceleration. For the rotation stage, the endpoints α_{rot i} were defined by Δα and the spatial density tapper. The rotation velocity and acceleration were set to their maxima: 90 deg·s⁻¹ and 500 deg·s⁻², respectively. Most rotation moves had a triangular velocity profile. Their duration was then limited by the maximum rotation acceleration.

The translation velocity was set constant for each one-way translation at α_rot and equal to:

$$v\left({\alpha }_{\mathrm{rot}}\right)=\frac{\mathrm{PRF}}{N_{\mathrm{st}}}\frac{\Delta l}{\cos \left({\alpha }_{\mathrm{rot}} \right)},$$

(3)

where PRF corresponds to the trigger generator of the synchronization part and N_st is the number of electronic steering angles. N_st = 1 in this acquisition mode as only straight transmits were performed, and Δl was equal to H = 1 mm.

The translation acceleration and the PRF were determined empirically with the following criteria. We wanted that the rotation duration and the translation time between two successive endpoints at different α_{rot i} (such as B and C in Figure 2) were similar. The rotation duration was already reduced to its minimum; therefore, we had to adapt the translation parameters. We decided to adjust the translation acceleration. We calculated the maximum translation acceleration so that the translation part was covered only when at least 80% of the corresponding rotation distance was traveled. With this acceleration, we determined the maximum translation velocity so that, for a translation travel equal to L = 30 mm, 50% of the translation distance was traveled at a constant velocity (i.e., not in the acceleration or deceleration phase). The PRF was then deduced from Equation (3) by taking the maximum possible angle α_rot = 45°. With these criteria, we found the following set of parameters: a PRF of N_st × 50 Hz and, for the translation stage, an acceleration of 235 mm.s⁻² and a maximum velocity of 60 mm·s⁻¹ (maximum velocity from the manufacturer specifications: 90 mm·s⁻¹). With these parameters for L = 30 mm and Δα₂, 64% of the desired tomographic positions were acquired with an inter-position translation distance greater than 90% of Δl/cos(α_rot), the targeted translation sampling period.

The motion parameters (endpoints, direction, and velocity) of both stages were coded in a macro, stored in the controller memory. Motion parameters were automatically modified each time an endpoint was reached. In particular, the controller started the rotation motion when the translation coordinate of the corresponding endpoint was exceeded. This mode was implemented for the three angular sampling periods (Table 1).

The second acquisition mode was a 3-D, rotate-translate tomographic scan with steered transmit events. The parameters for the stages were the same as for the first acquisition mode. However, we used N_st = 5 steering angles and the PRF was thus increased to 250 Hz (Table 1). We can note that the translation velocities stayed the same but the targeted inter-position translation distance was Δl/(cos(α_rot) × N_st). Consequently, the number of tomographic positions was around five times higher for the same Δα. In this second acquisition mode, coherent compounding was performed both in the elevation direction of the array with the mechanical scanning and, additionally, in the lateral direction with electronic beam steering. We studied the effect of the additional compounding on the contrast and resolution of the 3-D image. This mode was implemented only for the median angular sampling period Δα₂.

The third acquisition mode was a translate-only scan with straight transmit events. The rotation stage was kept fixed at α_rot = 0, and a one-way translation scan was performed over the translation range L. The nominal translation sampling period Δl_{translation-only} was set to H/8 = 125 µm [24] (see also Appendix B). In continuous motion with a PRF of 50 Hz, the corresponding translation speed was set to v₀ = 6.25 mm·s⁻¹. This third mode was implemented to provide a benchmark for comparison with the proposed rotate-translate scan.

2.3.3. Imaging Phantoms

Four imaging phantoms were used to characterize our 3-D scanner. The phantoms were totally submerged (Figure 3), maintained fixed compared to the water tank, and centered in the imaged volume (Figure 1c).

The first phantom (Ph1) contained 200-µm-diameter polyethylene microspheres (BKPMS 180–212 um, Cospheric, Santa, Barbara, CA, USA) embedded in agar gel. The agar gel was prepared with agar powder 2% w/v (A1296, Sigma Aldrich, St. Louis, MO, USA) in water, and it was molded in 26.5-mm-diameter cylindrical mold. The microspheres had a significant acoustic contrast relative to the embedding medium (on the order of 40 dB). They were randomly, but sparely, distributed over the entire cross section of the cylinder, and could thus be identified individually. Ph1 was placed so that all microspheres had approximately the same y-coordinate, i.e., they all were within a plane perpendicular to the rotation axis. Ph1 was used to validate that the field of view can be adjusted by setting the translation range L (Figure 1c). Moreover, Ph1 enabled us to study the 3-D spatial resolution and its spatial homogeneity.

The second phantom (Ph2) was prepared with agar powder (2% w/v) and 1% w/v cellulose powder (Sigmacell cellulose Type 20) in water. Cellulose particles (20 µm) act as ultrasound scatterers to mimic the scattering properties of biological tissues for US imaging. The gel was molded in the cylindrical mold with three cylindrical solid inclusions of 5 mm in diameter and of the same length as the mold. The inclusions were removed when the gel was solidified. Thereby, Ph2 contained three cylindrical holes arranged along the vertices of an equilateral triangle centered on the phantom axis. The center-to-center distance between adjacent holes was about 9 mm. The holes were filled with water and Ph2 was placed with the axes of the cylinders aligned with the rotation axis (y-axis). The holes were far enough from each other to be considered as isolated anechoic regions embedded in a speckle-generating medium. They were used to measure the image contrast and its spatial homogeneity.

Two other phantoms were made of 0.36-mm-diameter polyester threads. The threads were verified to be acoustically scattering; however, the contrast with agar gel was found to be poor. To obtain a higher contrast, the threads were simply placed in water: a non-scattering medium for 5-MHz ultrasound waves.

The third phantom (Ph3) was composed of four threads parallel to each other and oriented along the rotation axis (y-axis). Three of the threads were arranged along the vertices of an equilateral triangle of 9-mm side length. The fourth thread was at the center of the triangle. Ph3 allowed us to obtain an estimate of line spread function at four positions with a large dynamic range and to derive a contrast metric, named the cystic resolution. The similar spatial arrangement between Ph2 and Ph3 allowed comparison of contrast measurements made with two suitable metrics (see Section 2.5).

For the last phantom (Ph4), a 3-D net was formed with knots between 12 threads. The net was a complex structure comprised of a thin, homogeneous scattering material. Because the threads were dyed in black, the ultrasound image of Ph4 can be easily compared with optical pictures. Ph4 allowed us to study the ability to image a complex shape with multiple orientations.

2.4. Image Reconstruction Algorithm

Image reconstruction was performed with a simple delay-and-sum beamforming algorithm. The two-way travel times between the US transducer element positions and each imaged voxel were computed. Then, the value of each voxel was computed by summing, over all the elements of the array and all the different tomographic positions, the signal values assessed at the voxel-associated travel times. Tomographic positions outside the translation limits of the theoretical scanning scheme (such as from B to C in Figure 2) were, however, discarded. A weighted sum was performed to implement several apodization windows and avoid side lobes and grating lobes, as detailed in Appendix A. Three-dimensional, envelope-detected images were obtained.

The 3-D display voxel grid was defined in the fixed coordinate system (O, e_x, e_y, e_z), presented in Figure 1b. Voxel dimensions were chosen equal to px × py × pz = 72 × 144 × 72 µm³, with an anisotropy reflecting the best-expected resolutions. The 3-D images were displayed using maximum amplitude projection (MAP) images along the axes of the coordinate system. For instance, the MAP image along the x-axis was obtained by projecting the voxels with the maximum amplitude along the x-axis in the yz-plane. For each (y,z) coordinate, the pixel value of the MAP image corresponded to the highest voxel value along the x-axis of the full volume. When the MAP images were displayed with a logarithmic gray scale, the logarithmic compression was performed after the maximum amplitude projection.

2.5. Characterization of the Imaging Performance

We assessed the image quality in terms of resolution, contrast, and spatial homogeneity using the phantoms presented in Section 2.3.3. The full-width at half-maximum (FWHM) criterion was used to assess the 3-D resolution of individual point-like objects. For the contrast, two criteria were evaluated and compared: a variant of the cystic resolution (CyR) and the simple contrast between an anechoic region and a speckle region. We studied the influence of both the angular sampling parameter Δα and the transmit mode (straight or electronically steered).

2.5.1. FWHM Resolutions

The 3-D resolution was assessed with phantom Ph1. To determine the FWHM resolution, the image of each microsphere was fit with a 3-D Gaussian blob using a nonlinear least-squares minimization. The three main axes of the blob were the axes of the Cartesian coordinate system, and the following equation was used:

$$g=A·exp\left\{-\left[\frac{{\left(x-x_0\right)}^2}{2.{\sigma }_x^2}+\frac{{\left(y-y_0\right)}^2}{2.{\sigma }_y^2}+\frac{{\left(z-z_0\right)}^2}{2.{\sigma }_z^2}\right]\right\},$$

(4)

where x₀, y₀, and z₀ are the coordinates of the center of the blob, A is the amplitude, and σ_x, σ_y, and σ_z are the Gaussian root mean square widths along the x, y, and z axes, respectively. The FWHM resolutions were computed using the equation: $FWH{M}_i=2\sqrt{2ln\left(2\right)}{\sigma }_i$ with i = x, y or z.

The expected FWHM resolution along the x-axis was on the order of λ_c, because of the large synthetic angular aperture. The wavelength λ_c = 0.3 mm was the wavelength at the center frequency of the array. Given the apodization along the y-axis (Appendix A), we expected a resolution on the order of 2 × λ_c. The resolution along the z-axis was limited by the ultrasound pulse width and was expected to be on the order of λ_c. Accordingly, the fits with the Gaussian blob were performed in volumes of 1.5 λ_c × 4 λ_c × 1.5 λ_c, each centered on an individual microsphere.

To compare the spatial resolution for different angular sampling periods, we used statistical quantifiers, such as the median and interquartile range (IQR), because of their robustness to outliers and reliability for various data distributions. The resolution comparisons are presented in boxplot displays.

2.5.2. The Cystic Resolution (CyR)

The CyR was introduced by Vilkomerson et al. [28] to evaluate the effects of side lobes and grating lobes of the point-spread function (PSF) on the contrast of ultrasound images. The CyR is expressed as the minimum detectable size of anechoic structures embedded in a speckle-generating background. Instead of evaluating the detection of anechoic cysts of various sizes, Ranganathan et al. [29] suggested using the image of the concentric PSF with a spherical void c. The CyR is then defined as the size of the spherical void that produces a contrast above a given threshold. In 2-D, the cyst contrast Cc is computed [29] from the ratio of PSF energy outside the void relative to the total PSF energy:

$$C{c}_{dB}=10lo{g}_{10}\left(\frac{{{\displaystyle \iint }}_{\left(x,z\right)\notin c}^{}{h}^2\left(x,z,x_0,z_0\right)dxdz}{{{\displaystyle \iint }}^{}{h}^2\left(x,z,x_0,z_0\right)dxdz}\right),$$

(5)

where $h\left(x,z,x_0,z_0\right)$ is the image of a point-like object centered at $\left(x_0,z_0\right)$, which stands for the PSF function.

For 3-D imaging, Rasmussen et al. [13] suggested to use the line spread function (LSF) instead of the PSF to access the CyR because the LSF is easier to measure experimentally and has a larger dynamic range. In our case, we were interested in measuring the effect of the scan parameters on the image quality. Therefore, we wanted to evaluate the CyR in xz-planes (perpendicularly to the rotation axis), using the LSF in the y-direction (along the rotation axis). Phantom Ph3 allowed us to compute a robust estimate of the LSF with little background noise. The Cc contrast was computed using the slice at y = 0 (center of the array) and the two closest uncorrelated xz-slices located at y = ±0.86mm. Each slice was a disk of 25 × λ_c diameter, which was a compromise between the need to encompass the entire spatial extent of the LSF and the need to avoid overlap between neighboring LSFs. To obtain a smooth Cc curve, the images were oversampled by a factor 2 in the xz-plane (px = pz = 36 µm). Because we expected anisotropy of the LSF, we extended the definition of the Cc and we proposed a sectorial cyst contrast.

We can then introduce a general definition of Cc as:

$$C{c}_{dB}^T=10lo{g}_{10}\left(\frac{E_T^{out}}{E_T^{tot}}\right),$$

(6)

$$E_T^{out}={\displaystyle \sum }_{ny=-1}^1\underset{\left(x,z\right)\in \left\{T\backslash c\right\}}{\overset{}{{\displaystyle \iint }}}{h}^2\left(x,y,z,x_0^y,z_0^y\right)dxdz,$$

(7)

$$E_T^{tot}={\displaystyle \sum }_{ny=-1}^1\underset{\left(x,z\right)\in T}{\overset{}{{\displaystyle \iint }}}{h}^2\left(x,y,z,x_0^y,z_0^y\right)dxdz,$$

(8)

where T is either the entire slice (Equation (6) is equivalent to (5) but for the LSF) or an angular sector of the slice (leading to sectorial contrast), c is the disk-shaped void, and T\c is the set of elements of T but not in c. The sectorial contrast was computed in angular sectors of π/10. The $h\left(x,z,y,x_0^y,z_0^y\right)$ is the image of the LSF, at a given y, and centered at $\left(x_0^y,z_0^y\right)$, the coordinates of maximum amplitude of this LSF image.

Similarly to Rasmussen et al. [29], we defined CyR as the cyst radius that produces a contrast $C{c}_{dB}$ of −20 dB. For the entire disk-shaped slice, this definition of CyR is called isotropic cystic resolution, iCyR, in the following. For a sector, the sectorial cyst resolution is called anisotropic cystic resolution, aCyR.

2.5.3. Image Contrast

Phantom Ph2 contained three anechoic structures in a speckle-generating medium that consisted of cylindrical 5-mm-diameter holes, invariant along the y-axis. To assess the detectability of this cyst size, we chose the simplest and most general contrast estimator used in ultrasound imaging [30]:

$$C{0}_{dB}=10lo{g}_{10}\frac{{\mu }_c}{{\mu }_s}=10lo{g}_{10}\frac{\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{c}_i^2}{\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{s}_i^2},$$

(9)

where ${\mu }_c$ is the mean signal power in the cyst area and ${\mu }_s$ is the mean signal power in the speckled background area. The cyst statistics and the background statistics are assessed on the same number of pixels N = 1000, equivalent to about 30 resolution cells to guarantee reliable statistics. The variables $c_i$ and $s_i$ denote the value of ith pixel of the cyst area and of the background area, respectively. The background area was chosen distant from the three holes. As previously, the contrast was computed using the slice at y = 0 and the two closest uncorrelated slices (xz images). Each slice was a disk with a diameter limited to 25 × λ_c.

As for estimation of the LSF based on Cc, a contrast curve was plotted by considering different and non-overlapping cyst areas of size N, all centered on the longitudinal axis of the cylindrical hole. The center-most area was a disk and the areas beyond this disk were concentric rings.

3. Results

3.1. Influence of the Translation Range on the Imaged Volume

The cross section of the volume probed at all orientations α_rot of the array and perpendicular to the rotation axis is presented in Figure 1c. This cross-sectional area was expected to have a diamond shape and its area should depend on the translation range L. We name DSCA this diamond-shaped, cross-sectional area. The cross section of the volume probed by none of the tomographic positions is named the blind area (Figure 1c). The regions outside of the DSCA (and the blind area) were probed with fewer orientations of the array. Therefore, we expected a degraded resolution and a lower amplitude for this part of the image.

To experimentally study the influence of L on the cross section of the imaged volume perpendicular to the rotation axis, we performed several tomographic acquisitions of Ph1 for three different translation ranges. The angular sampling was kept constant and equal to Δα₂. Ph1 contained 200-μm-diameter microspheres that were sparsely distributed over the cross section of its 26.5-mm-diameter cylindrical form (Figure 4a). We chose L values equal to 30 mm, 20 mm, and 10 mm, corresponding to 508, 396, and 280 tomographic positions and scanning durations of 10.2, 7.9, and 5.6 s, respectively. The ratio of the scanning duration relative to L became lower as L increased, while the sampling parameters and, in particular, the translation velocities computed from Equation (3) were identical for all scans. This is due to fact that the acceleration and deceleration phases of the translation motions had the same duration for all three scans, but these phases, during which the translation velocity was lower, were proportionally shorter as L increased. The plane of microspheres was placed around y = 0 mm. Each microsphere could be identified individually, allowing a qualitative evaluation of the effective imaged area and a comparison of the image quality inside and outside the DSCA. We reconstructed images of dimensions Δx × Δy × Δz = 30 × 2 × 30 mm³. Figure 4b–d presents the maximum amplitude projection (MAP) of the 3-D US image along the y-axis.

Figure 4b presents the US image for the largest translation range L = 30 mm. As illustrated by the circular markers, which are landmarks, the spatial distributions of the microspheres in the US image and in the optical pictures match (Figure 4a). This result demonstrates the ability of our system to perform US images in the xz-plane and to localize small scatterers.

For L = 30 mm, almost all microspheres were included in the DSCA. However, when the translation range was reduced, some microspheres entered the blind area on the side of the DSCA (along the x-direction), as illustrated by the microsphere marked with the yellow arrow in Figure 4, which is no longer visible for L = 10 mm. Red arrows in Figure 4d illustrate that microspheres were indeed reconstructed with better resolution and amplitude when they were in the vicinity or inside the DSCA, compared to when they were located in the vicinity of the blind area. The islets (Figure 4e–g) further show the image amplitude degradation with a zoom on one microsphere.

With the developed scanner, the operator can adjust the translation range to fit the region of interest within the DSCA. Thus, the scan duration can be reduced when the object to image is smaller and the translation range can be more strongly limited.

3.2. The FWHM Resolution

The resolution of isolated microspheres and their amplitude in the DSCA were evaluated for L = 30 mm, according to the method detailed in Section 2.5.1. Ph1 was comprised of about 100 microspheres (Figure 4a). The final data set excluded any pairs of microspheres positioned within 900 µm of each other and contained 52 microspheres distributed across the DSCA (Figure 5a). Ph1 was also imaged using the translate-only scanning mode (L = 30 mm). For this mode, the resolution along the x-axis was degraded. The fits with the Gaussian blobs were performed in volumes of 6 λ_c × 4 λ_c × 1.5 λ_c around each microsphere. Due to the small distance along x between neighboring microspheres, only 16 of them could be isolated in such volumes and included in the data set. The microspheres had a finite diameter of about 200 µm in diameter. Therefore, they could not exactly be considered as point-like, and the measured FWHM values were expected to be slightly larger than the actual resolution of the system. However, this diameter of microspheres ensured an efficient scattering of the ultrasound waves and a sufficient contrast to the supporting medium (agar gel).

Table 2. Full width at half maximum (FWHM) along the three directions of the coordinate system for microspheres of Ph1. Values are median ± interquartile range (IQR). Two acquisition modes are compared: a rotate-translate scan at Δα₂ (data set of 52 microspheres) and a translation-only scan (data set of 16 microspheres). The translation range was L = 30 mm for both acquisition modes.

FWHM	Axis of Interest	Rotate-Translate Scan	Translate-Only Scan
FWHMx	x-axis	371 µm ± 36 µm	1140 µm ± 222 µm
FWHMy	y-axis	756 µm ± 40 µm	709 µm ± 45 µm
FWHMz	z-axis	401 µm ± 23 µm	373 µm ± 26 µm

compares the median ± IQR values of the FWHM along the three directions of the coordinate system for the rotate-translate scan at Δα₂ and the translate-only scan. We noted that FWHMy and FWHMz are barely influenced by the rotation scan. Indeed, the resolution along the y-axis was mainly limited by the finite length of the active aperture of the linear array, and the resolution along the z-axis was mostly influenced by the duration of the pulsed echo. Figure 5c shows that FWHMy had a small deviation for z < 30 mm due to the fixed lateral f-number used in the reconstruction method (Appendix A). For z > 30 mm, the effective lateral f-number could no longer be maintained fixed by adjusting the apodized reception aperture of the array as the full length of the array was used. Therefore, the effective lateral f-number increased and the resolution degraded with the increasing depth z. The FWHMy values reported in Table 2 were computed for the full range of depths. Figure 5c illustrates that the estimated median value of FWHMy is robust to the outliers for z > 30 mm. The reported IQR values, which measure the statistical dispersion were, however, larger than they would have been if the microspheres at z > 30 mm had been discarded. Figure 5d shows the homogeneity of FWHMz with z. Unlike FWHMy and FWHMz, FWHMx was found to be 3.1 times smaller with the rotate-translate scan. The resolution along the x-axis was linked to the angular aperture along the elevation direction of the array. This angular aperture was increased from 2 × β = 16° in the translation-only mode to almost 90° (with a Hamming apodization) in the rotate-translate mode. Both FWHMy and FWHMx were limited by diffraction. However, Figure 5b demonstrates that FWHMx was noticeably more homogeneous, even for z > 30 mm, as opposed to FWHMy. Thus, our scanning scheme provided a homogeneous synthetic aperture in the DSCA. Figure 5e shows that the amplitude from scattering structures in the image decreased with depth for z > 20 mm. This result can be attributed to the attenuation of ultrasound waves in the medium. For z < 20 mm, the increase in the amplitude with z may be attributed to the spatial response of the array before the focal distance that was not modeled in the reconstruction algorithm.

The FWHM values were assessed for the four rotate-translate scans presented in Table 1. Figure 6 shows that the FWHMy and the FWHMz were stable for all the angular sampling periods Δα. The FWHMx was significantly larger for Δα₁ and was on the order of 360 µm (median) for the other angular samplings, with a similar IQR. This result indicates that the angular sampling period Δα₁ may be too loose to avoid strong side lobes in the vicinity of the main lobe. The steered transmit induced fewer outliers (red crosses), but did not have a significant influence on the median value of the FWHM.

3.3. The Cystic Resolution (CyR)

The cystic resolution further characterized the influence of the angular sampling on the image quality by quantifying the effect of the side lobes of the PSF on the image contrast. Figure 7a presents an image of Ph3 for the plane y = 0 with a logarithmic gray scale (Δα₂ and L = 30 mm). The side lobes of the LSF were visually similar for the four threads that were all included in the DSCA.

Figure 8a displays the cyst contrast Cc_dB of the image of Thread 2 for the four rotate-translate scans presented in Table 1. The iCyR corresponds to a radius that produces a contrast of −20 dB.

Table 3. Contrast parameters of the different acquisition modes. Translation range of L = 30 mm. The angular periods are Δα₁ ≈ 8°, Δα₂ ≈ 4°, and Δα₃ ≈ 2°.

Rotate-Translate Scan		Straight Transmit			Steered Transmit
Contrast Parameter	Location	Δα1	Δα2	Δα3	Δα2
iCyR (mm)	Thread 1	2.84	1.40	1.12	1.08
	Thread 2	1.48	1.30	0.97	0.90
	Thread 3	1.51	1.37	1.08	1.08
	Thread 4	1.48	1.40	1.04	0.97
	Median	1.49	1.38	1.06	1.02
C0 (dB) 1	Hole 1	−24.3	−25.8	−28.7	−31.5
	Hole 2	−24.4	−26.2	−28.9	−31.1
	Hole 3	−24.7	−25.3	−27.9	−31.5
	Median	−24.4	−25.8	−28.7	−31.5

¹ Contrast obtained at the center (i.e., for the smallest cyst radius r = 0.8 mm).

presents the iCyR values obtained for each thread. The spatial homogeneity of the iCyR can be noticed for a given angular sampling. One exception was observed for Thread 1 and Δα₁. In this case, the degraded FWHMx resolution for Δα₁ and the vicinity of Thread 4 biased the iCyR estimation. For straight transmits, the iCyR decreased with the decreasing value of Δα, which means that the spatial extension of the side lobes decreased and the contrast increased. The difference between Δα₂ and Δα₃ indicates that the latter angular sampling period should be preferred in case the image contrast is desired for the targeted application. Interestingly, the iCyR was close for the steered transmit at Δα₂ and for the straight transmit at Δα₃. The same result was observed for the Cc_dB (Figure 8a). Therefore, globally, the lobes of the LSF for Δα₂ could be mitigated by generating additional tilted planes waves and thereby reinforcing spatial averaging in the image during the reconstruction process.

Figure 8b compares the anisotropic sectorial cystic resolution aCyR for Thread 2 and the different angular sampling periods. While FWHMz and FWHMx had similar values (Figure 6) indicating an isotropic center peak for the LSF, we noted that the polar plot of the aCyR had the shape of a lemniscate (∞-symbol) for every angular sampling period. The aCyR was much smaller around the z-axis than along the x-axis. This anisotropy of the side lobes was linked to the finite synthetic angular aperture (around 90°) and its orientation. Therefore, the image contrast depended on the orientation of the structure in the xz-plane. Second, the angular sampling mostly influenced the aCyR along the x-axis. A left–right asymmetry of the aCyR can also be noticed. It may be due to the asymmetry of the scanning between the two translation directions in the practical implementation of the scanning scheme (Figure 2). On the right side of the polar plot ($\theta \in \left[-45°;45°\right]$), we observe smaller aCyR values for smaller Δα in the straight transmit mode. The steered transmit at Δα₂ led to aCyR value intermediate between those of the straight transmit at Δα₂ and Δα₃.

3.4. Image Contrast

The image contrast was also directly assessed on the three 5-mm-diameter anechoic holes of Ph2. Figure 7b shows an image of Ph2 for the plane y = 0. The holes are identified as well as the speckled background area used for the calculation of C0. The holes 1, 2, and 3 have similar positions to the threads 1, 2, and 3 of Ph3, respectively.

Table 3 presents the contrast C0 obtained for the four rotate-translate scans presented in Table 1 and for a cyst area defined by a disk of radius 0.8 mm centered on each hole. First, we noted that the cylindrical anechoic holes can all be detected with a contrast higher than 20 dB compared to the speckle background. This result is in agreement with the iCyR values determined in Section 3.3, because the radius of the holes was 2.5 mm and the largest median iCyR was 1.5 mm. The C0 values of Table 3 are similar for the three holes. The contrast can then be considered as spatially homogeneous in the DSCA. However, C0 decreases with the decreasing angular period Δα for the straight transmit. Moreover, the contrast was stronger for the steered transmit acquisition at Δα₂ than for the straight transmit acquisition at Δα₃. For these two acquisitions, the iCyRs were both on the order of 1 mm. We can note that the disk of the cyst area was at a distance larger than 1 mm from the edges of the hole. As a consequence, the influence of the LSF on C0 might be considered low, and the lower value of C0 for the steered transmit at Δα₂ was most likely linked to the higher number of tomographic positions in this mode (Table 1) and the resulting increased spatial averaging of the electronic noise during the reconstruction process. The C0 values of Table 3 can be considered as a measure of the contrast to noise of the different acquisition parameters. They all decreased with the number of tomographic positions.

The contrast was evaluated for non-overlapping concentric rings of the same area as the central disk. Figure 9a displays C0 as a function of the external radius of each ring for hole 1. As expected, C0 reached 0 dB for rings located outside of the hole. Inside and in the vicinity of the scattering edges of holes, C0_dB was influenced by the spread of the LSF. As for Cc shown in Figure 8a, the contrast C0 decreased with the decreasing Δα for the straight transmit. However, whereas the Cc values were similar for the straight transmit at Δα₃ and the steered transmit at Δα₂, C0 was lower for the steered transmit acquisition from a distance of about 1 mm from the edges.

Figure 9b–e illustrates the visibility of the anechoic hole 1 for the different rotate-translate acquisitions. For straight transmits, the center of the hole was indeed darker when Δα decreased. Additionally, the hole appeared larger in the lateral direction (along the x-axis). This result corresponds to the anisotropy of the LSF, identified by the aCyR (Figure 8b).

3.5. Complex 3-D Phantom

To focus on the xz-plane where the rotate-translate scanning scheme had the greatest impact. The scattering structures were confined in a plane for Ph1 and they were invariant along the y-axis for Ph2 and Ph3. Conversely, Ph4 was a 3-D scattering structure that was not confined to the plane y = 0 containing structures aligned along multiple orientations (Figure 10a). Threads were arranged in a net with knots and four-way junctions. The mesh was large to ensure that the branches could be resolved in the rotate-translate mode regardless of their orientation.

Figure 10a shows a photographic top view of the net and Figure 10b displays the MAP US image along the z-axis for a steered transmit at Δα₂. The two images correspond well, and both the knots and the branches can be identified in the MAP US image. The threads appeared continuous even if a local amplitude heterogeneity was visible along the threads with orientations having a significant x-component. This may be linked to randomly distributed microbubbles trapped on the thread. For the junctions, mostly oriented along the y-axis, the lower FWHMy may smooth the reconstruction. The continuity of the threads was also visible on the MAP images along the y-axis (Figure 10d) and the x-axis (Figure 10e). As for Figure 5, the amplitude decreased with the increasing z coordinate, but the amplitude did not strongly depend on the orientation of the thread.

Figure 10c,f presents MAP images corresponding to a translate-only acquisition performed for Ph4 just after the rotate translate acquisition. In the translation-only mode, the resolution along the x-axis was degraded throughout the entire imaged volume. This result was expected from Section 3.2, and it resulted in smearing and overlapping between the images of some threads (blue arrow in Figure 10c). The threads also appeared discontinuous with bright and dark spots in the translation mode (pink arrows in Figure 10c,f). Video S2 further demonstrates the higher 3-D image quality with the rotate-translate mode compared to the translate-only mode.

4. Discussion

We experimentally investigated the performance of a synthetic aperture rotate-translate approach for volumetric US imaging with a linear ultrasound array. In the elevation direction of the array, this approach demonstrated the synthesis of an angular aperture several times larger than the aperture provided by the native elevation focus of the array. The aperture was increased by a factor of 5 and a Hamming apodization was implemented. As a result, the elevation resolution was improved by a factor of 3 compared to a translate scanning approach. Moreover, the aperture synthesis in the elevation plane was obtained over a large area (the DSCA): An area of 4.5 cm² was covered here with a translation range of 3 cm. The elevation resolution was confirmed to be homogeneous in this area. Therefore, the rotate-translate scan was shown to yield superior image quality compared with a translation scan.

To the best of our knowledge, this study is the first demonstration that 3-D ultrasound imaging can be performed with a rotate-translate scanning scheme combined with coherent compounding. A similar scanning approach had previously been proposed for photoacoustic (PA) tomography [20]. However, the ultrasound sources in PA imaging are the illuminated optical absorbers. For PA imaging the transducer is scanned to a different orientation after each laser pulse to receive the ultrasound wavefield generated by absorbers in the medium. For US imaging, the ultrasound waves are both emitted and captured by the scanned US transducer array. As a consequence, the backscattered echoes from different orientations of the exciting US beam need to be recombined coherently to generate a high-quality image. This concept has been extensively used for plane-wave ultrasound imaging [8] and synthetic aperture imaging [31] with a fixed transducer array. However, in this case, the field of view and the accessible orientations are limited by the number and geometrical properties of the elements of the array. Coherent compounding was also recently shown for 2-D imaging using multiple transducer arrays with different orientations [32] or a rotation scan of an array [33]. Such approaches increase the angular aperture and the lateral resolution but the transducer arrays were always oriented within the same imaging plane. Similar results were obtained in 2-D PA imaging [34]. Here, we performed 3-D synthetic aperture scanning by applying coherent compounding from tomographic positions of the array with both different angular elevation orientations and different elevation spatial positions. Additionally, we combined the coherent summing of echoes obtained from different tomographic positions with the summing of echoes from beams electronically steered to different angles. For each steered angle, a data set with the same translation and angular sampling periods was recorded. The steered angles were chosen to ensure uncorrelated transmitted wave vectors. As a consequence, the coherent summation over all the steered angles during the reconstruction process reinforced coherent echoes and reduced incoherent noise, which led to image contrast enhancement.

The proposed rotate-translate scanning scheme is novel for 3-D US imaging. Furthermore, since both PA and US imaging use the same detector, acquisition parameters optimized for one imaging modality can be applied directly to the other. We benefited from previous innovations in rotate-translate PA tomography such as continuous scanning [27] that showed a reduction in the scan duration. We further reduced the scan duration by more fully assessing the needs in terms of angular sampling. Two main improvements were made to angular scanning. (1) The angular spacing was chosen based on the elevation focusing of the array and (2) a spatial density taper was applied. The smallest angular period considered here, Δα_3, is three times larger than the angular period in [20]: λ_c/F = 0.7°. Therefore, the optimization of the angular spacing reduced the number of tomographic positions by at least a factor of 3. Additionally, the spatial density taper reduced the number of tomographic positions by about 40% and reduced side lobes of the PSF in the elevation plane without any additional amplitude apodization. Indeed, strong side lobes were expected at ±45° with no apodization, but due to the spatial density taper used here they were not stronger at 45° than at 0° (Figure 8b). Finally, the total number of tomographic positions was thus reduced by at least by a factor of 5 (80% decrease) relative to the number previously reported in [20].

Three different angular periods, Δα₁ ≈ 8°, Δα₂ ≈ 4°, and Δα₃ ≈ 2°, were tested. The number of tomographic positions and the scan duration increased by a factor of 2 when the angular period was divided by 2 (Table 1). Metrics were computed to evaluate the impact of the angular sampling period on the image quality. We found that Δα₁ led to a significant degradation of the FWHM resolution along the x-axis compared to Δα₂ and Δα₃. Therefore, Δα₁ was determined to be too loose to prevent side lobes from degrading the main lobe of the PSF. The cystic resolution and the contrast on the edge of an anechoic region were also improved with the decreasing angular period. For the angular sampling, Δα₂, we performed acquisitions in straight transmit mode and in steered transmit mode. Interestingly, both contrast measurements improved with the addition of steered angles, while the angular sampling was the same. We can explain this result by the enhanced spatial averaging of the “diffraction noise” when steered transmits were performed. Moreover, the contrast parameters (Cc and C0) were found to be similar with the angular sampling Δα₂ in steered transmit and with Δα₃ in straight transmit. The steered transmit did not modify the acquisition duration but did substantially increase the size of the data set, while the scan duration at Δα₂ was half that at Δα₃. Therefore, a gain in the image quality can effectively be made without compromising the acquisition duration by the addition of a steered transmit. A systematic evaluation of the contrast for different numbers of steered angles was beyond the scope of this paper but is of great interest for our rotate-translate system and will be considered in future studies. The angular periods, Δα₂ and Δα_3, will also be implemented in PA tomography. As the tomographic system is implemented with a programmable ultrasound machine that can record US waves upon the arrival of a trigger pulse, PA tomography sequences with a pulsed laser could be implemented during scans. The interlacing of PA and US acquisitions will be investigated in future studies for simultaneous PA and US 3-D imaging of the same volume with the scanner. Additional strategies such as tissue harmonic coherent compound imaging using advanced US sequences to isolate non-linear echoes [35,36] and the use of contrast agents could be integrated within the tomographic system in the future to further improve US image contrast and specificity. Because of the scan duration, it is expected that contrast agents that could be retained within biological tissue would work best, such as targeted microbubbles [37] or nanodroplets converted to microbubbles upon optical [38,39] or ultrasound [40] activation. Feasibility has been shown, for example, to optically activate microbubbles from nanodroplets that are detectable with PA imaging so that, with the choice of appropriate dual-modality contrast agents, PA and US can be used together to characterize agent arrival and delivery [39].

The system was developed with a single-side access for an accessibility to various body parts. The scan duration is also a crucial point for in vivo application, clinical or preclinical. Long acquisitions may cause discomfort and tissue motion that can cause image blurring. As mentioned above, the number of tomographic positions was already reduced by adjusting the spatial sampling. But the acquisition duration is the number of tomographic positions multiplied by the acquisition rate. With the ultrasound machine used in this study, the acquisition rate could be in the kHz range. Currently, we limited the acquisition rate to 50 Hz because of the maximum acceleration of the rotation stage (see Section 2.3.2). The acquisition duration could then be greatly reduced with a faster rotation stage such as one with a magnetic motor. With our worm gear motor, we achieved a scan duration of 10 s for a 9-cm³ volume. This acquisition time is smaller compared to other rotate-translate scanning schemes that have been applied in vivo in PA tomography [41] and for incoherent spatial compounding of Doppler data [42,43]. Although these studies map functional information and not only pulse-echo US imaging, they give a reference for in vivo acquisition duration. Li et al. [41] performed multispectral PA tomographic acquisitions in 12 min. For 3-D Doppler tomography with incoherent compounding of Doppler images, reported acquisition times were of 15–20 min [42,43]. To avoid blurring due to respiratory motion, acquisition methods such as respiratory gating, i.e., acquisition only during the respiratory pause, can be considered. A weak oversampling can be applied to compensate for missing tomographic positions. With a current acquisition duration of 10 s for a 9-cm³ imaged volume (in the DSCA) in the steered transmit mode and a reconstruction duration of 95 s (Table 1), the scanning system can currently achieve a volumetric imaging rate of 0.57 volumes/min. The reconstruction duration could be further accelerated with a high-end graphics processing unit (GPU). However, even if the acquisition rate between tomographic positions was set in the kHz range, which is the maximum rate given the ultrasound propagation time and the ultrasound hardware, the full-volume acquisition duration will still be on the order of 1 second in the steered transmit mode. Therefore, real-time volumetric imaging (>5 volume/s [4]) would only be achievable with degraded image quality (rotate-translate scan in the straight transmit mode and with Δα ≥ Δα₂ or a translate-only scan). Such an approach could be used to position the volume of interest in the DSCA with a real-time display prior to a higher-quality image acquisition.

Finally, the developed scanning method is applicable to a broad range of already-available US arrays. The scan parameters best adapted for each array can be calculated based on its geometrical properties and the center frequency. Moreover, the one-side access design and the homemade (3-D printed) holder make the system easily adaptable to any kind of available probes. Although the feasibility was demonstrated with a 5-MHz ultrasound array, the method is of great interest for higher frequencies within the range of medical ultrasound (>15 MHz), in particular to improve the sensitivity to smaller tissue scatterers and the resolution compared to low frequencies. For applications and ultrasound acquisition modes that do not require high-frame-rate imaging, we believe that our 3-D synthetic aperture rotate-translate scanning method could greatly enhance the capacity of 3-D biomedical ultrasound imaging without the complex development of new transducer arrays. Among clinical applications for which 3-D ultrasonography has already been shown to be of high interest, our approach could specifically benefit those requiring a high contrast and resolution such as quantification of carotid atherosclerosis [44] and use of musculoskeletal ultrasonography to assess rheumatoid arthritis [4,45]. Additionally, we expect that our 3-D scanning method will help longitudinal monitoring of pathological tissues for which high-quality volumetric images are crucial for image processing and for accurate comparison of images acquired at different points in time.

5. Conclusions

We demonstrated here that a clinical ultrasound linear array designed for 2-D imaging can be used to obtain high-resolution and -contrast volumetric US images with a 3-D synthetic aperture scanning method. We developed and experimentally implemented an advanced scanning scheme combining rotation and translation motions of the array. The angular sampling period was adjusted and tapered to optimize the scan duration without compromising the image quality. An acquisition duration of 10s was achieved for a volume of 9 cm³. The image contrast was further enhanced by the addition of electronically steered exciting beams. The method is expected to benefit applications that require high image quality but a relatively low volumetric scanning rate such as longitudinal monitoring of pathological tissues.