Three-dimensional passive acoustic mapping of high intensity focused ultrasound fields using sparse synthetic apertures from rotated one-dimensional linear arrays

Abstract

Accurate visualization of the focal spot in high-intensity focused ultrasound (HIFU) is essential for safe and effective therapeutic guidance. Passive acoustic mapping (PAM) provides a noninvasive method for reconstructing acoustic fields from scattered signals generated as HIFU pulses interact with tissue inhomogeneities. This study introduces a synthetic aperture strategy for three-dimensional beam reconstruction by mechanically rotating a one-dimensional linear ultrasound probe. Five synthetic configurations were assessed: 1D Linear, Cross Linear, Discrete Annular, Random Sparse, and Fibonacci Spiral. Each was formed by rotating a 64-element probe at predefined angles. A Full Aperture Array was included as a reference baseline. The Fibonacci Spiral Array achieved the most balanced performance among the synthetic configurations, combining high structural similarity (SSIM = 0.711 in X–Y) with strong sidelobe suppression (PSL = −12.538 dB in Y–Z), low mean squared error, and consistently high output quality across all evaluated planes. These results closely matched those of the Full Aperture Array, supporting similar resolution and localization accuracy. These findings indicate that spiral-based sampling provides a hardware-efficient, high-fidelity solution for volumetric PAM in therapeutic ultrasound, offering a practical alternative to conventional two-dimensional matrix arrays.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-42764-w.

Introduction

Passive acoustic mapping (PAM) is a non-invasive beamforming technique that reconstructs the spatial distribution of acoustic sources by passively receiving ultrasound emissions from a region of interest. In therapeutic ultrasound, particularly in high-intensity focused ultrasound (HIFU), PAM has demonstrated potential for real-time monitoring of energy deposition and acoustic field distribution^1,2. By mapping the focal zone of the HIFU beam, PAM enables spatial verification of treatment targeting, thereby enhancing the safety and efficacy of preclinical applications and showing promise for clinical translation³.

Ultrasound-based techniques for monitoring the HIFU focal zone include acoustic radiation force impulse (ARFI) imaging⁴, ultrasound elastography⁵, thermal strain imaging⁶, and echo decorrelation methods⁷. These approaches typically infer changes in tissue properties such as stiffness, temperature, or echo consistency induced by HIFU exposure. ARFI and elastographic techniques estimate tissue displacement or stiffness, thermal strain imaging tracks phase shifts due to localized heating, and echo decorrelation monitors speckle pattern disruption. While effective in specific contexts, these methods often require active transmission, rely on indirect thermal or mechanical tissue changes, or involve multiple acquisition cycles.

Although mechanically swept probes have long been employed for volumetric imaging, the present study extends this concept by introducing a flexible synthetic-aperture framework that enables software-defined control of array configuration and rotation strategy without hardware modification. This framework allows volumetric reconstruction using a single 1D probe, combining hardware simplicity with computational adaptability. In contrast, the PAM technique employed in this study utilizes backscattered signals generated by short HIFU pulses interacting with weak impedance inhomogeneities within the medium^8–10. This enables direct visualization of the acoustic field distribution without requiring temperature elevation, tissue deformation, or cavitation activity, making it particularly suitable for real-time, non-invasive monitoring of the HIFU focal zone under low acoustic energy exposure conditions.

Conventional passive acoustic mapping (PAM) relies on integrating acoustic emissions over time, which can result in reduced spatial resolution and elevated sidelobe levels due to accumulated off-axis signals. To overcome these limitations, spatiotemporal mapping techniques have been proposed, including instantaneous beamforming strategies¹¹, two-dimensional cavitation mapping integrated with B-mode imaging on clinical ultrasound platforms¹², and real-time volumetric PAM with optimized hardware configurations^13,14. These efforts have demonstrated improved spatial resolution, enhanced contrast, and real-time monitoring capabilities. Such advances are critical in therapeutic ultrasound applications, where accurate localization and quantification of acoustic activity, such as cavitation, is essential for effective and safe treatment delivery^15,16.

To bridge the gap between limited 1D arrays and high-cost 2D arrays, synthetic aperture techniques using mechanical probe rotation have been proposed. These methods emulate a 2D aperture by acquiring data at multiple orientations, enabling 3D field reconstruction without requiring a full matrix array. Recent studies have shown the feasibility of rotational synthetic apertures for volumetric mapping, demonstrating reduced hardware demands while maintaining imaging performance¹³. However, the spatial sampling pattern plays a pivotal role in image quality, especially under sparse acquisition conditions. Inadequate spatial coverage can lead to increased sidelobe levels, aliasing artifacts, and degraded localization accuracy^17–19.

To address the limitations of conventional array configurations, researchers have explored non-traditional spatial sampling strategies such as randomized and spiral distributions. Among these, the Fibonacci spiral has gained attention for its quasi-uniform angular coverage, low spatial redundancy, and deterministic geometry. Inspired by phyllotaxis patterns found in nature, the Fibonacci spiral has been adopted in applications such as antenna design and medical ultrasound imaging, where it has demonstrated favorable beamforming properties, including reduced sidelobe energy and improved isotropy^20–23. Its ability to achieve efficient spatial sampling with fewer acquisition points makes it a strong candidate for sparse 3D PAM reconstruction.

In this study, we investigate six representative synthetic aperture configurations that can be implemented by mechanically rotating a 1D linear ultrasound probe: 1D Linear Array (1D-LA), Cross Linear Array (C-LA), Discrete Annular Array (D-AA), Random Sparse Array (R-SA), Fibonacci Spiral Array (F-SA), and Full Aperture Arrays (F-AA), for 3D PAM of focused ultrasound fields. Each configuration was synthesized by rotating the probe around a fixed axis to simulate distinct spatial sampling patterns with varying degrees of regularity and aperture coverage, and to acquire volumetric data. This synthetic aperture strategy enables the exploration of these six array geometries without the need for complex hardware modifications. Among the tested configurations, we propose the Fibonacci Spiral as a hardware-efficient and high-resolution approach for visualizing the HIFU focal zone. Through numerical simulations, we evaluate spatial resolution, beam fidelity, sidelobe suppression, and localization accuracy across the six array geometries. The results demonstrate that the Fibonacci Spiral Array provides superior beam preservation and precise acoustic focus localization, supporting its potential for integration into practical therapeutic ultrasound monitoring systems.

Results

Synthetic array configurations and spatial distribution analysis

To characterize the spatial characteristics of the synthetic array configurations, we analyzed three representative 2D geometries: D-AA, R-SA, and F-SA. Additionally, the F-AA was included as a reference. The spatial layouts and inter-element connectivity were assessed using Delaunay triangulation (DT), as shown in Fig. 1, with summary metrics provided in Table 1. In each subplot, red lines indicate DT edges between array elements, which are marked by black dots. These triangulations provide insight into local density and geometric balance. The D-AA (Fig. 1a) exhibited a highly regular structure with four concentric rings (Coefficient of Variation, CV = 0.1245), producing nearly equilateral triangles across the aperture. The uniform DT mesh reflects consistent local spatial density and geometric symmetry. The R-SA (Fig. 1b) displayed irregularly spaced elements and nonuniform triangle size, which resulted in variable edge lengths and uneven spatial coverage (CV = 0.4199). This indicates that the random distribution leads to pronounced inhomogeneity in element spacing and triangulation mesh structure. The F-SA (Fig. 1c) showed a more organized radial distribution with spiral symmetry, resulting in more balanced triangulation compared to the random configuration (CV = 0.2567). The F-AA (Fig. 1d) served as a maximal sampling configuration, using 23,040 channels to achieve nearly uniform coverage across the domain. DT analysis for this configuration yielded the shortest mean edge length (0.0142) and the lowest standard deviation (0.0036), confirming extreme spatial regularity. Its CV (0.2547) was also very low, though the lowest CV was observed in the D-AA (0.1245). The 1D-LA and C-LAs were excluded from this DT-based spatial analysis due to their linear nature and inability to form valid 2D triangulation meshes.

Fig. 1.

Delaunay Triangulation (DT) of three array configurations within a normalized circular boundary (black solid line). Black dots indicate the positions of array elements, and red lines represent triangulated connections between neighboring elements. (a) Discrete Annular Array, (b) Random Sparse Array, (c) Fibonacci Spiral Array, (d) Full Aperture Array.

Table 1.

Delaunay triangulation (DT)-based spatial regularity metrics for each array type.

	1D-LA	C-LA	D-AA	R-SA	F-SA	F-AA
Mean length	0.4	0.8	0.2659	0.2263	0.2616	0.0142
Std. deviation	N/A	N/A	0.0331	0.0930	0.0672	0.0036
Coef. of var. (CV)	N/A	N/A	0.1245	0.4107	0.2567	0.2547

Mean edge length, standard deviation, and coefficient of variation (CV) were calculated to assess the uniformity of spatial distribution. CV was not calculated for 1D and Cross Linear arrays due to their non-planar, anisotropic geometry being unsuitable for DT analysis.

PAM reconstruction and performance comparison

Figure 2 compares the ground-truth beam pattern obtained from the full 3D simulation (Fig. 2a) with the reconstructed backscattered intensity distributions obtained by PAM using six synthetic array configurations (Figs. 2b–g). The simulation field represents forward (FWD) acoustic beam propagation, whereas the PAM results correspond to backward (BWD) propagation reconstructed from the received backscattered signals. This difference in propagation direction accounts for the minor variations in beam symmetry and focal extent observed between the simulated and reconstructed fields.

Fig. 2.

Comparison of beam profiles reconstructed using different array configurations for three-dimensional passive acoustic mapping (PAM). Reconstructed intensity maps are shown in three orthogonal planes: X–Y (axial-lateral), X–Z (axial-elevational), and Y–Z (lateral-elevational). All volumes were normalized and spatially aligned to the simulated focal point to allow direct comparison. The ground truth in (a), obtained from a full 3D simulation, exhibits a well-confined focal zone with minimal sidelobes, whereas sparse configurations display varying degrees of focal broadening and sidelobe artifacts depending on geometry and spatial sampling density. (a) Ground truth beam pattern from 3D simulation. (b) 1D-LA; only the X–Y plane reconstruction is available due to the absence of elevational sampling. (c) C-LA, (d) D-AA, (e) R-SA, (f) F-SA, (g) F-AA.

All PAM reconstructions were performed within the same spatial range as the simulation (X: 30–60 mm, Y and Z: −10–10 mm), and the focal region emerged within this coordinate domain as a result of beamforming, enabling direct comparison. Cross-sectional images are presented in three orthogonal planes (X–Y, X–Z, and Y–Z) to visualize focal structure and sidelobe characteristics. Across all configurations with elevational sampling, the focal region appears as a localized high-intensity area corresponding to the simulated focus. Variations among configurations primarily reflect differences in spatial sampling geometry, aperture coverage, and phase coherence.

The ground-truth field (Fig. 2a) exhibits an X-shaped intensity distribution in both the X–Y and X–Z planes, where the acoustic paths intersect near the geometric focus. In the Y–Z plane, the focal region appears well-focused and symmetric, with minimal sidelobe energy. The focal region is precisely located at the designed focus, exhibiting high spatial confinement and coherence in all directions.

The 1D-LA (Fig. 2b) reconstructs the focal region within the X–Y plane, providing only 2D backscattering information. Because elevational sampling along the Z-axis is absent, volumetric reconstruction cannot be achieved, and the corresponding X–Z and Y–Z cross-sections are unavailable. The reconstructed focal region appears elongated along the axial direction and exhibits high sidelobe levels, with noticeable artifacts extending anteriorly and posteriorly.

The C-LA (Fig. 2c) reconstructs the 3D focal region by combining two orthogonal 1D apertures. The focal region is formed near the intended focus but appears broader than the ground-truth pattern. In the X–Y and X–Z planes, strong wing-like background energy is observed adjacent to the main lobe. In the Y–Z plane, a cross-shaped intensity pattern with lateral and elevational sidelobes extends from the central focus.

The D-AA (Fig. 2d) reconstructs the 3D focal region using a concentric-ring configuration. The focal region is symmetrically formed, showing improved spatial confinement compared with C-LA. In the X–Y plane, the reconstructed focal region appears slightly elongated axially, and moderate sidelobes are distributed above and below the main lobe. In the X–Z plane, the focal region remains sharply defined with weak arc-shaped sidelobes. In the Y–Z plane, the focal region forms a nearly circular spot surrounded by a low-level diffuse background, and a weak ring-shaped sidelobe pattern is observed.

The R-SA (Fig. 2e) reconstructs the 3D focal region using the single, randomly distributed sparse array generated for this study. The focal region is discernible but exhibits irregular shape and reduced uniformity. In the X–Y and X–Z planes, the focal region appears elongated, surrounded by diffuse background energy and multiple asymmetric sidelobes. The intensity distribution lacks clear spatial symmetry. In the Y–Z plane, the focal region appears fragmented, and the background intensity is globally elevated, indicating diffuse energy spread across the plane.

The F-SA (Fig. 2f) reconstructs the 3D focal region using a Fibonacci spiral geometry. The focal region is sharply formed near the intended focus and appears compact and symmetric. In the X–Y and X–Z planes, the focal region exhibits strong intensity localization with relatively low background energy; however, high sidelobes are present along the axial direction. In the Y–Z plane, the focal region appears nearly circular with faint peripheral halos, well-suppressed sidelobes, and low background noise. The beam profile shows high isotropy and closely matches the simulated focus location.

The F-AA (Fig. 2g) reconstructs the 3D focal region using the full aperture sampling configuration. The reconstructed beam pattern exhibits a compact and symmetric focal region that most closely resembles the simulated reference. In the X–Y and X–Z planes, the focal intensity is sharply confined along the axial center, showing minimal broadening and well-suppressed sidelobes. The axial, lateral, and elevational intensity distributions are highly uniform. In the Y–Z plane, the focal region appears circular and compact, with negligible background energy or peripheral artifacts. The reconstructed field maintains consistent symmetry and localization across all planes, confirming that dense spatial sampling ensures accurate beam reproduction.

Quantitative evaluation of beam characteristics

Figure 3 shows the axial and radial intensity profiles reconstructed from all array configurations, normalized to their respective peak amplitudes. The black curve represents the reference profile obtained from the full 3D simulation. The F-AA exhibited the closest agreement with the reference, showing a narrow main lobe and minimal sidelobe energy. The F-SA achieved comparable focal confinement with an axial FWHM of 9.2 mm and a peak displacement magnitude of 0.86 mm, closely matching the F-AA (9.6 mm and 1.07 mm, respectively). In contrast, the C-LA showed a broad axial profile (9.9 mm), while the R-SA, despite a narrow FWHM (9.1 mm), exhibited a noisy profile with high, diffuse sidelobes (Fig. 3). Both arrays also showed larger displacements (> 0.9 mm), indicating reduced spatial confinement. The D-AA maintained stable focusing behavior with moderate sidelobe suppression, while the 1D-LA exhibited the widest beam profile due to the absence of elevational sampling. These results demonstrate that F-SA achieves focusing performance comparable to the full-aperture configuration while using a reduced number of sampling positions.

Fig. 3.

Line profile comparison of reconstructed beam patterns for different array configurations. (a) Axial intensity profiles along the focal axis (depth), normalized to the maximum intensity. (b) Radial intensity profiles across the lateral-elevational plane at the focal depth. The black curve represents the ground truth obtained from the full 3D simulation, while colored lines correspond to reconstructions from the 1D-LA, C-LA, D-AA, R-SA, F_SA, and F-AA.

Quantitative evaluation of beam quality

Table 3 summarizes the quantitative beam-quality metrics for each array configuration, including the structural similarity index (SSIM), peak signal-to-noise ratio (PSNR), mean squared error (MSE), and peak sidelobe level (PSL). These values were calculated for three orthogonal cross-sections relative to the reference simulation: axial-lateral (X–Y), axial-elevational (X–Z), and lateral-elevational (Y–Z) planes.

Table 3.

Quantitative comparison of beam quality across array configurations.

		1D-LA	C-LA	D-AA	R-SA	F-SA	F-AA
X–Y	MSE	0.006	0.008	0.003	0.005	0.004	0.006
	PSNR	22.338	21.112	24.628	23.146	23.920	22.519
	SSIM	0.655	0.642	0.686	0.678	0.711	0.572
	PSL(dB)	− 0.040	− 0.009	− 0.075	− 0.046	− 0.123	− 14.011
X–Z	MSE	0.111	0.007	0.004	0.004	0.005	0.006
	PSNR	9.563	21.503	23.777	24.180	23.155	22.155
	SSIM	0.538	0.646	0.687	0.657	0.689	0.629
	PSL(dB)	0.000	− 0.009	− 0.075	− 0.046	− 0.123	− 14.011
Y–Z	MSE	0.034	0.004	0.003	0.003	0.003	0.003
	PSNR	14.690	23.531	25.454	25.790	25.295	24.709
	SSIM	0.534	0.654	0.603	0.738	0.626	0.880
	PSL(dB)	0.000	− 10.455	− 12.858	− 5.702	− 12.538	− 22.351

The axial beam profile and radial beam cross section of each simulation are evaluated using mean squared error (MSE), peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) and peak sidelobe level (PSL) relative to the reference configuration.

The F-AA configuration, serving as the benchmark, provided a unique performance profile. While its SSIM (0.572–0.629) and PSNR (22.155–22.519) values in the X–Y and X–Z planes were surpassed by D-AA, R-SA, and F-SA, its clear superiority was in sidelobe suppression. It achieved remarkably low PSL values of −14.011 dB (in both X–Y and X–Z) and -22.351 dB (in Y–Z), substantially better than any other configuration. Furthermore, it achieved the highest SSIM by a wide margin in the Y–Z plane (0.880).

Among the sparse arrays, the D-AA, R-SA, and F-SA configurations all exhibited high PSNR values, often exceeding 23 dB and peaking at 25.790 dB for the R-SA in the Y–Z plane. The F-SA’s performance was notably dependent on the viewing plane: it achieved the highest SSIM of any sparse array in the X–Y plane (0.711), but its PSL was high in the X–Y and X–Z planes (−0.123 dB). However, its performance in the Y–Z plane was strong, with a competitive PSL of −12.538 dB.

The R-SA showed highly variable and asymmetric performance, as expected from a single random instance. Unlike the deterministic arrays, its metrics differed across all three planes. It achieved the highest PSNR (25.790 dB) and highest SSIM (0.738) of any sparse array in the Y–Z plane, but also had the worst Y–Z PSL (−5.702 dB) among the 3D-capable sparse arrays. In contrast, the C-LA and 1D-LA were consistently lower performers, with the 1D-LA showing significant degradation in all metrics for the X–Z and Y–Z planes (e.g., PSNR 9.563–14.690), as expected from its 2D-only nature.

Discussion

This study evaluated six synthetic aperture array configurations generated by the mechanical rotation of a 1D linear array for 3D PAM of focused ultrasound regions. The comparative analysis demonstrates that the array’s spatial geometry, as statistically quantified in Table 1 and visualized in Figs. 1 and 5, is a primary determinant of the reconstructed beam’s morphological structure (Fig. 2) and quantitative fidelity (Tables 2 and 3).

Fig. 5.

Normalized-scale distributions of 64 elements for five synthetic array configurations. Red rectangles represent individual piezoelectric elements, and the black solid circle indicates the normalized rotational boundary within which all elements are positioned. (a) Conventional 1D linear array (1D-LA, base form). (b) Cross Linear Array (C-LA) formed by orthogonal linear rotations. (c) Discrete Annular Array (D-AA) generated via discrete angular steps. (d) Random Sparse Array (R-SA) formed through irregular rotations. (e) Fibonacci Spiral Array (F-SA) created via quasi-uniform radial rotation. (f) Full Aperture Array (F-AA) synthesized by continuous rotations at 1° increments.

Table 2.

Quantitative evaluation of beam characteristics for different array configurations, including beam width (FWHM), peak intensity location, and peak displacement in three spatial dimensions (Unit: mm).

	FWHM			Peak location			Peak displacement			Magnitude
	X	Y	Z	X	Y	Z	X	Y	Z
Simulation	6.761	0.824	0.822	44.9	0.0	0.0	0.0	0.0	0.0	0.000
1D-LA	10.322	5.612	–	43.8	− 0.3	–	− 1.1	− 0.3	–	1.140
C-LA	9.874	7.348	7.115	44.1	− 0.3	− 0.3	− 0.8	− 0.3	− 0.3	0.906
D-AA	8.912	2.120	2.696	45.8	− 0.2	− 0.3	0.9	− 0.2	− 0.3	0.970
R-SA	6.945	2.795	3.805	45.7	− 0.5	− 0.4	0.8	− 0.5	− 0.4	1.025
F-SA	9.161	2.469	2.675	45.6	− 0.3	− 0.4	0.7	− 0.3	− 0.4	0.860
F-AA	9.640	3.376	3.712	45.8	− 0.3	− 0.5	0.9	− 0.3	− 0.5	1.072

The Delaunay triangulation (DT) method provided a quantitative framework to assess spatial regularity among array geometries²⁴. The coefficient of variation (CV) of edge lengths effectively captured differences in geometric uniformity²⁴. The D-AA exhibited the lowest CV, indicating highly regular spatial sampling due to its symmetric concentric ring structure^25,26. However, highly uniform arrangements often lead to prominent sidelobes and grating lobes, as previously reported in array design literature²⁷. In contrast, the R-SA showed the highest CV, reflecting significant spatial irregularity and clustering of elements, which can result in undesirable weighting effects during beamforming and elevated sidelobe levels²⁸. The F-SA achieved an intermediate CV, balancing spatial uniformity with a deterministic radial and angular distribution²⁸. This moderate CV directly contributed to its improved imaging performance, highlighting the advantages of carefully planned deterministic sampling patterns in synthetic aperture design²⁴.

A critical finding, enabled by the inclusion of the F-AA configuration, is that a full aperture is not universally superior^17,20. As shown in Fig. 5f (and Fig. S1), the continuous rotational sampling strategy results in a massive concentration of element positions and sampling redundancy near the center of the aperture^20,29. This non-uniform geometric weighting may explain why, despite its clear superiority in sidelobe suppression (PSL −14.011 dB to −22.351 dB), the F-AA was outperformed in key fidelity metrics like SSIM and PSNR in the X–Y and X–Z planes by D-AA, R-SA, and F-SA^30,31.

While true full-aperture sampling provides unmatched sidelobe suppression, its highly redundant central sampling (Fig. 5f) comes with diminishing returns^32,33. The F-SA delivers a more balanced performance, achieving near full-aperture fidelity in key beam characteristics (FWHM, Peak Displacement) and quantitative metrics with a fraction of the data^34,35.

The F-SA configuration (Fig. 5e) emerged as the most effective and balanced compromise. Its quasi-uniform spatial distribution (CV 0.2567, Table 1) translates directly to a visually compact and symmetric focal region (Fig. 2f)³⁶. Critically, it was the only sparse array to closely match the F-AA’s fundamental beam characteristics, with a comparable axial FWHM (9.161 mm) and peak displacement (0.860) (Table 2)³⁷. This structural integrity was complemented by strong fidelity metrics, including the highest SSIM (0.711) of any sparse array in the X–Y plane and a competitive Y–Z PSL (-12.538 dB)^34,37. The F-SA successfully balances the competing goals of main lobe fidelity, sidelobe suppression, and sampling efficiency^34,37,38.

The R-SA (Fig. 5d) demonstrated the high risk and variability of stochastic, asymmetric sampling. Its highly irregular spatial distribution (CV 0.4199, Table 1) led to erratic performance metrics^39,40. Morphologically, its focal region was visibly fragmented (Fig. 2e) and, contrary to what one might expect from its high SSIM, was quantitatively broader in the Y–Z plane (FWHM 2.795 × 3.805 mm) than the F-SA (2.469 × 2.675 mm) (Table 2)^39,41. This paradoxical result, achieving the highest Y–Z SSIM (0.738) and PSNR (25.790) despite a broader focus and the worst Y–Z PSL (−5.702 dB), highlights the unreliable and unpredictable nature of a single stochastic geometry^20,40. It is critical to note that the R-SA is one particular case; given the numerous possible combinations, it cannot be considered representative of all random sparse configurations. A full optimization or statistical analysis of random arrays is beyond the scope of this study, but future work will systematically investigate multiple randomized distributions to correlate spatial uniformity metrics (e.g., CV) with beam-quality parameters^39,40.

The simulated acoustic medium incorporated spatially distributed scatterers (0.05 mm radius) with random perturbations of ± 5% in sound speed and density (baseline values: 1540 m/s and 1000 kg/m³), as described in the Methods section. This configuration mimics weak acoustic impedance mismatches typical of biological tissues and generates Rayleigh-distributed backscattered signals resembling speckle patterns in diagnostic ultrasound^42,43. Because these scatterers produced low-amplitude signals, post-processing gain compensation was required, which elevated the background noise level, most notably in configurations with low spatial coherence^12,13. As shown in Fig. 2, this effect was most pronounced for the R-SA (Fig. 2e), emphasizing that spatially regular or quasi-uniform array geometries are essential to minimize post-processing artifacts and maintain PAM reconstruction fidelity.

The D-AA provides a clear example of this geometry-morphology link. Its highly periodic geometry (lowest CV, 0.1245, Table 1) directly manifests as the "weak ring-shaped sidelobe pattern" observed visually in its reconstruction (Fig. 2d) and its competitive Y–Z PSL (−12.858 dB)²⁴. However, highly periodic array geometries can reinforce coherent interference patterns that amplify sidelobe or grating-lobe artifacts when the element spacing approaches or exceeds half the wavelength, as described by the array factor relationship⁴⁴. The consistently poor metrics of the C-LA (Fig. 5b) and 1D-LA (Fig. 5a) (e.g., 1D-LA FWHM of 10.322 mm and PSNR of 9.563 in X–Z) quantitatively confirm that simple 2D or orthogonal scanning, which results in obvious visual artifacts (Fig. 2b, c), is insufficient for high-fidelity 3D PAM⁴⁵.

Limiting the channels to 64 reflects practical hardware constraints for hybrid therapeutic-monitoring systems, which, unlike high-end diagnostic platforms, often prioritize cost and compactness over massive channel counts⁴⁶. From an engineering perspective, synthesizing a 2D aperture through mechanical rotation of a 1D probe provides a practical and cost-effective alternative to 2D matrix arrays^36,47. Mechanical scanning systems are widely used and have demonstrated reliability in clinical 3D ultrasound (e.g., obstetric imaging), offering substantial economic advantages by requiring only standard 1D probes and limited channel electronics⁴⁸. While inherently slower than the electronic steering of matrix arrays and potentially subject to long-term wear or calibration drift, these effects are minimized in this study by the limited rotation range and short acquisition time⁴⁹. The proposed configuration thus reduces transducer complexity and hardware cost while maintaining compatibility with advanced beamforming methods³⁶.

Sequential acquisition introduces potential motion sensitivity. One full 3D dataset (64 rotational steps) corresponds to a total signal acquisition duration of approximately 5–6 ms, with a complete mechanical sweep occurring in less than 1 s. This acquisition window is substantially shorter than typical physiological motion cycles (> 2 s for respiration or peristalsis), indicating that motion-induced incoherence is expected to be minimal under controlled conditions^50–52. Furthermore, motion-tracking and phase-compensation methods established in aberration correction and super-resolution ultrasound studies could be adapted to further mitigate any residual misalignment^53–56.

The proposed method is designed specifically for pre-treatment focal region verification rather than continuous real-time monitoring. Prior to therapeutic HIFU delivery, a single volumetric PAM frame is acquired using low-energy acoustic pulses to visualize and confirm the focal region at the target location. This single-frame acquisition paradigm differs fundamentally from diagnostic ultrasound imaging, which requires continuous frame rates (> 10 Hz) for dynamic visualization. In our application, the mechanical rotation period (< 1 s) is clinically acceptable because only one verification scan is needed before treatment, making sparse, hardware-efficient configurations more practical than real-time systems requiring dense 2D matrix arrays and high-speed electronics.

Beyond demonstrating the feasibility of synthetic aperture reconstruction using rotationally sampled 1D arrays, the proposed framework offers substantial utility as a rapid prototyping platform for sparse array design^57–59. Because both element selection and rotation strategy can be virtually controlled, diverse sparse-aperture configurations can be evaluated without hardware modification^58,60. This flexibility enables efficient exploration of alternative sampling geometries, allowing users to assess beam quality, grating-lobe suppression, and focal precision under design constraints such as element pitch, array aperture, and distribution patter^58–61. Moreover, the framework facilitates adaptive optimization, in which the active aperture can be dynamically adjusted to avoid acoustically obstructive regions (e.g., bone interfaces) or to meet application-specific performance criteria^59,61. This capability highlights the framework’s broader role in accelerating the development of next-generation ultrasound array architectures^58,59.

The selection of spatial discretization in k-space pseudospectral simulations is a critical determinant of both numerical accuracy and hardware feasibility. Our findings highlight that voxel size selection is inherently application-specific; while 5 PPW (0.2 mm) may suffice for general beam morphology, high-fidelity 3D PAM reconstruction, which relies on subtle backscattered signals from weak heterogeneities, demands higher discretization (10 PPW, 0.1 mm) to suppress staircasing and dispersion artifacts below physical signal levels. However, this refinement carries a substantial hardware penalty, increasing GPU RAM requirements nearly six-fold. Researchers are encouraged to conduct explicit convergence testing rather than relying on generic rules, as the “practically converged” voxel size is dictated by the specific interaction between imaging bandwidth, tissue heterogeneity, and available computational resources.

The present study was conducted using realistic nonlinear tissue parameters (B/A = 6) with low-energy tone-burst excitation maintained below cavitation and thermal thresholds. This design reflects the intended application of pre-treatment focal region verification, where non-invasive imaging must precede therapeutic HIFU delivery without inducing biological effects. Under these low-energy conditions, the second harmonic and higher harmonics remain negligible in amplitude. PAM reconstruction relies solely on the fundamental frequency component (1.5 MHz) backscattered from weak acoustic impedance heterogeneities, as described in the Methods section.

The array’s geometric distribution is a key predictor of performance^20,32. Advanced sparse geometries like the Fibonacci spiral offer a scalable and efficient path toward 3D PAM^20,21. While true full-aperture sampling provides unmatched sidelobe suppression, its highly redundant central sampling (Fig. 5f) comes with diminishing returns^20,32. The F-SA delivers a more balanced performance, achieving near full-aperture fidelity in key beam characteristics (FWHM, Peak Displacement) and quantitative metrics with a fraction of the data^20,21,34.

Methods

Numerical modeling of the HIFU and linear probe system

Numerical simulations were conducted to model the acoustic field generated by a high-intensity focused ultrasound (HIFU) transducer and the corresponding signals received by a one-dimensional (1D) linear array. Synthetic sparse aperture configurations were created by mechanically rotating the linear array around a fixed axis. The received radiofrequency (RF) signals from each configuration were recorded for subsequent processing. All simulations were carried out in three dimensions using a MATLAB-based acoustic simulator (k-Wave, version 1.4, released in November 2022).

Figure 4 illustrates the coaxial configuration employed in the simulation, which consists of a spherically concave HIFU transducer with a central aperture and a diagnostic linear ultrasound probe. The probe was positioned at the center of the transducer’s opening and aligned along the shared acoustic axis. This arrangement enabled the probe to receive backscattered signals primarily from the X–Y plane at Z = 0, corresponding to the focal zone.

Fig. 4.

Numerical modeling of a coaxially aligned HIFU transducer and 1D linear array probe for time-domain scattering simulations. (a) Schematic of the system configuration, showing a High-Intensity Focused Ultrasound (HIFU) transducer with a concave aperture and a centrally positioned 1D linear probe. Key geometric parameters include aperture size (AS), radius of curvature (RF), probe length (PL), and focal length (FL). (b) Voxel-based representation of the simulation domain, constructed in a homogeneous water medium, with embedded HIFU transducer, 1D linear probe, and a 2D passive acoustic mapping (PAM) plane positioned at the focal region (Z = 0). (c) 3D PAM imaging volume illustrating multiple rotated array configurations (angle θ) around the central axis, simulating probe rotation for synthetic aperture acquisition. The HIFU transducer surface is defined as the acoustic source to model wave propagation and scattering.

The simulation domain was implemented as a 3D voxel grid, embedding both the transducer and probe in a homogeneous tissue-like medium. Acoustic properties such as sound speed, density, and attenuation were defined to replicate soft tissue characteristics, enabling time-domain modeling of nonlinear acoustic wave propagation, including harmonic generation and the acquisition of scattered signals by the probe elements.

HIFU transducer

The HIFU transducer was modeled as a spherically curved, bowl-shaped source with an aperture size (AS) of 64.0 mm and a central opening (CO) of 42.0 mm (Table 4). The transducer had a rim-to-focus distance (RF) of 31.0 mm and a focal length (FL) of 45.0 mm, defined as the distance from the transducer surface to the geometric focus. A 1D linear ultrasound probe with a probe length (PL) of 25.6 mm was concentrically positioned within the central opening and aligned along the shared acoustic axis (Table 1). The transducer operated at a center frequency of 1.5 MHz, corresponding to an acoustic wavelength of approximately 1 mm in the soft tissue model. A single tone-burst (5-cycle) pulse was transmitted from the transducer surface, serving as the acoustic excitation source.

Table 4.

Geometric dimensions used for modeling the HIFU transducer and 1D linear probe in the simulation domain.

Transducer	Parameter	Value	Unit
HIFU	Number of channels	1	ea
	Aperture size, AS	64.0	mm
	Central opening, CO	42.0	mm
	Probe length, PL	25.6	mm
	Focal length, FL	45.0	mm
	Rim-to-focus, RF	31.6	mm
1D linear probe	Number of channels	128.0	ea
	Element width	0.2	mm
	Element height	3.5	mm
	Pitch	0.2	mm
	Total length	25.6	mm

OD and ID refer to the outer and inner diameters of the concave aperture. PL, FL, and RF represent the probe length, focal length, and radius of curvature, respectively.

Propagation medium and scatterer

The propagation medium was modeled as homogeneous soft tissue, with baseline acoustic properties of 1540 m/s for sound speed and 1000 kg/m³ for density (Table 5). To simulate realistic acoustic scattering conditions representative of biological tissue microstructure, spatially distributed point scatterers were randomly distributed throughout the simulation domain. Each individual scatterer was defined as a single voxel (0.1 mm × 0.1 mm × 0.1 mm) and introduced independent, uncorrelated perturbations of ± 5% to both sound speed and density to the background medium values. The spatial distribution of scatterers was implemented using a uniform random process with no spatial correlation between neighboring scatterer positions, ensuring isotropic scattering characteristics. The ± 5% perturbation amplitude was selected to produce weak acoustic impedance mismatches to mimic the heterogeneity typically found in biological tissues that generate Rayleigh-distributed backscattered signals characteristic of fully developed speckle patterns in ultrasound imaging.

Table 5.

Simulation parameters used for modeling the acoustic field and signal propagation.

Parameter		Value	Unit
x, y, z		66.0	mm
dx, dy, dz		0.1	mm
Dt		25.0	ns
Frequency		1.5	MHz
Pulse length		5.0	cycles
Simulation time		90.0	μs
Medium	Sound speed	1540 ± 5%	m/s
	Density	1000 ± 5%	kg/m3
	Alpha coeff	0.75	dB/MHz/cm
	Alpha power	1.50	N/A
	B/A	6.00	N/A

Spatial and temporal resolutions, transducer frequency, and medium properties (including attenuation and nonlinearity parameters) are specified to replicate realistic soft tissue conditions.

Such perturbations produce weak acoustic impedance mismatches that generate low-level backscattered signals, enhancing the realism of the received RF data without substantially distorting the primary HIFU beam. This approach follows speckle-generating models commonly adopted in ultrasound simulations⁶².

To quantify the practical trade-off between spatial resolution and memory demand, a grid-refinement study was conducted using isotropic voxel sizes of 0.2, 0.1, 0.05, and 0.025 mm. For the 1.5 MHz excitation, these correspond to approximately 5, 10, 20, and 40 points per wavelength (PPW). The GPU memory required to store the full fields increased from approximately 8.5 GB at 0.2 mm to 47 GB at 0.1 mm. While the 0.2 mm grid (5 PPW) satisfied minimal accuracy for linear propagation, the 0.1 mm grid (10 PPW) was required to achieve numerical convergence with < 1% deviation in focal amplitude and position.

Backscattering characteristics and applicability to passive acoustic mapping

The feasibility of backscatter-based passive acoustic mapping (PAM) was assessed under simulation conditions representative of soft tissue. The acoustic medium was defined with a sound speed of 1540 m/s and density of 1000 kg/m³, each varied by ± 5% to introduce heterogeneity. The excitation was a 1.5 MHz focused ultrasound pulse (λ ≈ 1.03 mm). The scatterer corresponded to a single voxel of 0.1 mm × 0.1 mm × 0.1 mm, giving an equivalent spherical radius of 0.05 mm and a size parameter of ka ≈ 0.38, which lies in the Rayleigh-to-Mie transition regime.

1

Although subwavelength in scale, this scatterer produces measurable backscattering due to moderate acoustic impedance contrast and coherent interference of scattered waves. The computed backscattered field retained sufficient amplitude and phase coherence to be captured by the receiving array and reconstructed using delay-and-sum beamforming. The resulting intensity distributions confirmed that localized pressure variations near the focal region could be resolved, supporting the theoretical feasibility of PAM reconstruction using backscattered signals under these acoustic conditions.

Sensor array

A virtual 1D linear array consisting of 128 elements was modeled as the receiving ultrasound probe. Each element was 0.2 mm wide and 3.5 mm high, with a uniform pitch of 0.2 mm, resulting in a total aperture length of 25.6 mm. The probe was positioned coaxially within the 42 mm central opening (CO) of the HIFU transducer, with its center aligned at the geometric focal point. Each element was assumed to be a broadband pressure sensor with a center frequency of approximately 3 MHz, consistent with typical diagnostic ultrasound systems.

To evaluate different spatial sampling strategies, the linear array was mechanically rotated around the acoustic axis intersecting the HIFU focus. Six synthetic aperture configurations were tested, each comprising a total of 64 channels. As illustrated in Fig. 5, these configurations differ in how channels were selected from the probe:

1. 1D Linear Array (1D-LA, No Rotation): The array remained in a single fixed orientation (0°), simulating conventional 2D PAM. 64 uniformly spaced elements were selected from the center of the array, resulting in an effective pitch of 0.4 mm.

2. Cross Linear Array (C-LA, Orthogonal Rotations): Two perpendicular scans were performed at 0° and 90°, simulating a cross-shaped array. 32 uniformly spaced elements were selected at each angle, yielding a total of 64 channels with an effective pitch of 0.8 mm in both axes.

3. Discrete Annular Array (D-AA, Discrete Angular Steps): The probe was rotated at multiple discrete angles to emulate a sparse annular configuration. Elements were distributed along concentric circular paths, with each ring positioned at a different radius from the origin (0, 0). The number of selected elements varied by ring, increasing with radial distance to maintain quasi-uniform angular spacing. A total of 64 synthetic elements were selected across the entire structure.

4. Random Sparse Array (R-SA, Irregular Rotations): 64 elements were selected based on randomly chosen angular orientations and radial positions to form a disk-like sparse aperture with irregular spatial sampling.

5. Fibonacci Spiral Array (F-SA, Quasi-Uniform Spiral): The elements were arranged along 9 spiral leaves with quasi-uniform radial and angular spacing, forming a deterministic Fibonacci spiral pattern. Each spiral contained 7 elements distributed around the central axis, resulting in a total of 63 elements. One additional element was placed at the center to complete the 64-element configuration.

Spiral Leaf Definition: Each spiral leaf, the radial positions r_k and angular positions θ_k of the points were defined as follows⁶³:

2

where r_s = 0.11 is the initial radial increment, r_f = 0.5 is the final radial extent of the spiral, θ_start = 90° is the initial spiral angle, and Δθ is the angular step derived from the full spiral span divided by n-1. The local Cartesian coordinates for each point on a spiral leaf were then calculated as:

3

where ϕ = 58° was an empirically determined offset angle for the base spiral orientation.

Spiral Leaf Rotation and Assembly: To form the complete synthetic aperture, each spiral leaf was rotated by an angle of 2πm/N_s and shifted outward from the origin by a base circle radius R_c = 0.15, as described by:

4

5

where m = 0, 1, …, N_s − 1 indexes the spiral leaf. This transformation ensures a symmetric circular distribution of all leaves.

f. Full Aperture Array (F-AA, Continuous Rotations): For the full-aperture configuration, the 1D linear array was synthetically rotated around the acoustic axis in 1° increments from 0 to 179°, resulting in a total of 180 distinct angular positions. At each angle, all 128 elements of the linear probe were used, yielding a cumulative total of 128 × 180 = 23,040 synthetic receiving channels. This dense sampling approach provided near-continuous aperture coverage, serving as the reference configuration for evaluating reconstruction accuracy and sidelobe suppression across the sparse array designs. All element coordinates were normalized and then mapped to the corresponding voxel indices on the simulation grid and used to extract RF signals from the simulated acoustic field, and their spatial coordinates were later incorporated into the delay-and-sum beamforming algorithm for passive acoustic mapping reconstruction.

Spatial regularity analysis using delaunay triangulation

To evaluate the spatial regularity of each synthetic array configuration, we applied Delaunay triangulation (DT) to the normalized 2D coordinates (y, z) of all array elements^24,64. The DT generates a set of non-overlapping triangles such that no point lies within the circumcircle of any triangle. For each triangulated structure, we calculated the Euclidean edge lengths:

6

To minimize boundary effects, edge lengths exceeding the 90th percentile were excluded. From the remaining edges, we computed the mean , standard deviation , and coefficient of variation CV:

7

which serves as a unitless indicator of spatial uniformity. A lower implies a more regular array geometry, which is hypothesized to enhance acoustic sampling efficiency. Delaunay analysis was not applied to the 1D-LA and C-LAs. The 1D configuration is colinear, making triangulation undefined in 2D. The C-LA, composed of two orthogonal lines, produces degenerate or highly skewed triangles, which are not representative of spatial uniformity.

Passive beamforming and image reconstruction

To evaluate the performance of the 1D linear array and synthetic 2D configurations, specifically the F-SA, for PAM, we implemented a conventional three-stage post-processing pipeline consisting of raw RF signal acquisition, Time-Gain Compensation (TGC), and band-pass filtering centered at the fundamental frequency of 1.5 MHz using a Gaussian filter.

Raw RF Data (Fig. 6a): The initial RF signals acquired from both array types display strong early echoes corresponding to the incident wavefront directly scattered from the medium. The 1D-LA exhibits a structured pattern with consistent echo curvature, while the F-SA shows more spatially distributed and complex patterns due to its non-uniform synthetic geometry.

Fig. 6.

Post-processing workflow for radio-frequency (RF) channel data acquired from two array configurations: a 1D Linear Array and a Fibonacci Spiral Array. (a) Raw RF channel data obtained from synthetic aperture simulations. (b) Application of time-gain compensation (TGC) to enhance deeper signals attenuated by propagation loss. (c) Removal of the strong early-time incident HIFU pulse to isolate weaker backscattered components. (d) Band-pass filtering centered at 1.5 MHz (filter order = 20) to extract the fundamental frequency band and suppress out-of-band noise. The top panels illustrate representative waveforms after each processing stage, while the lower panels display corresponding channel data from the two array configurations.

Time-Gain Compensation (Fig. 6b): To compensate for depth-dependent signal attenuation, an exponential gain function was applied across all channels. This adjustment significantly enhanced signal visibility in deeper regions. The F-SA retained broad spatial sampling characteristics, whereas the 1D-LA maintained its expected linear alignment with depth.

Incident Wave Removal: The received RF signals were temporally windowed to suppress the strong early-time incident HIFU pulse directly received by the array elements. This procedure minimizes contamination of the backscattered components by the high-amplitude transmitted waveform.

Fundamental Frequency Filtering (Fig. 6c): Bandpass filtering was applied centered at the fundamental frequency (1.5 MHz) with a filter order of 20 to suppress broadband noise and reduce background clutter. The filtering process isolated the primary frequency components associated with the fundamental response of the HIFU field, thereby improving contrast in the reconstructed backscattered intensity maps.

For each configuration, we simulated the received time-domain signals on the 64-element linear array at all specified orientations. The HIFU transducer emitted a 5-cycle tone-burst at 1.5 MHz, and signals were recorded for 90 µs to capture both direct arrivals and backscattered fields.

PAM reconstruction was conducted using a conventional delay-and-sum beamforming approach. A 3D Cartesian grid of voxels was defined around the HIFU focal region. For each voxel, the time-of-flight between the voxel and every receiving element (positioned at its respective rotational orientation) was calculated, assuming a constant speed of sound. The signals were temporally aligned and coherently summed across all channels and orientations, effectively synthesizing a large virtual aperture. A spatial apodization function was applied to the summed signals to isolate the impact of array geometry on sidelobe behavior. The resulting 3D intensity map represents the spatial distribution of acoustic energy and is visualized via intensity projections or 2D slices. The 3D PAM intensity at a voxel is calculated as⁶⁵:

where is the apodization weight for the nth channel, is the time-domain RF signal from the nth element, is the position of the nth element, is the assumed focal zone center, is the speed of sound, and is the sampling frequency.

The imaging output represents the summed intensity of scattered pressure signals with apodization applied.

To systematically analyze the spatial characteristics of the reconstructed passive acoustic fields, we segmented the imaging domain into five regions based on the geometric relationship to the HIFU transducer and the receiving probe (Fig. 7). These predefined regions enabled classification of signal contributions by angular orientation and distance from the focus, facilitating a more detailed comparison of spatial sensitivity across array configurations². The regions were defined as follows:

• R1: The primary beam path between the HIFU ceramic and the focal point, representing the main axis of incident HIFU energy propagation.

• R2: The post-focal region located beyond the focal point along the HIFU beam axis.

• R3: A central volume in the center of the HIFU ceramic, lateral to the HIFU beam paths. This region does not lie along the direct HIFU beam path.

• R4: A vertically extended region above the HIFU aperture, outside the direct beam path.

• R5: Diagonal sectors extending from the focal zone outward toward the lower periphery of the aperture, also located outside the main HIFU beam path.

Fig. 7.

Geometric region definitions used in the distance-based analysis of passive acoustic mapping. Five spatial regions (R1-R5) are defined relative to the location of the linear ultrasound probe and the concave HIFU ceramic to analyze signal origin contributions.

Performance metrics

To evaluate the beamforming quality across different array configurations, a set of quantitative metrics was defined based on the reconstructed 3D passive acoustic maps. These metrics include:

• Spatial localization metrics

• Full width at half maximum (FWHM): The beamwidth was quantified in three orthogonal directions-axial (x), lateral (y), and elevational (z)-by measuring the spatial spread of the main lobe at 50% of the peak intensity calculated as:

  9

  The spatial coordinates and correspond to the points where the intensity ) drops to .

• Peak location (PkL): The coordinates (x, y, z) of the intensity maximum in the reconstructed 3D image were extracted for each configuration. These positions were compared to the known simulated focus location (x_sim, y_sim, z_sim) to evaluate localization accuracy.

• Peak displacement (PkD): The deviation of the measured peak location from the simulated ground truth was calculated as:

  10

  This displacement is reported in millimeters for the axial, lateral, and elevational directions.

• Reconstruction fidelity metrics

• Structural similarity index (SSIM): Measures perceptual similarity in structure, luminance, and contrast. Ranges from 0 to 1, with 1 indicating perfect similarity.

  11

  where, , are mean intensity of simulation image and reconstructed image , , are variances⁶⁶.
• Peak signal-to-noise ratio (PSNR): Indicates the ratio between the maximum possible signal power and the power of the noise. Defined as:

  12
• Mean squared error (MSE): Quantifies average squared differences between the reconstructed and reference images:

  13
• Peak sidelobe level (PSL): The peak sidelobe level was extracted from the radial cross-section of the beam profile and reported in decibels (dB) relative to the main lobe peak intensity. It was calculated as:

  14

  A lower PSL indicates better suppression of off-axis energy and improved beam focus.

These metrics were computed across all aperture configurations. The computed values for each metric are later summarized in Tables 3 and 4 in the Results section to facilitate comparative analysis across array configurations.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

G.K. designed and conducted the simulations and drafted the main manuscript.J.H.H. co-designed the simulation, supervised the project, contributed to manuscript writing, and critically revised it for important intellectual content. Both authors reviewed and approved the final manuscript.

Funding

This study was supported by the Focused Ultrasound Foundation (FUS884).

Code availability

The datasets generated and analyzed during the current study are not publicly available as they were entirely created by the authors and not derived from external sources. However, they are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.