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Abstract
Point spread function (PSF) modeling is important for the characterization of the imaging performance of a photoacoustic computed tomography (PACT) system. This work aims to study the degradation mechanism of PSF in PACT and investigate the impact of the shape of detection geometry on PSF. PSF modeling of three typical two-dimensional detection geometries, including circular, curved, and linear detector arrays, is presented. Based on the non-ideal detection geometries, the effect of detector bandwidth and detector aperture on PSF is also investigated. Moreover, PSFs of each geometry with typical detector bandwidths and typical detector aperture sizes are presented. Experiments are conducted to validate the results. The proposed PSF modeling approach and corresponding results can help predict and interpret the quality of photoacoustic images produced by a practical PACT system. It is beneficial for the design of detector arrays for enhanced imaging performance.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
By energy conversion from light to sound, photoacoustic imaging is capable of realizing multi-contrast imaging of biological tissues with high spatial resolution and deep penetration depth [1–4]. Photoacoustic computed tomography (PACT) is one of the most important implementations of photoacoustic imaging. It can achieve real-time imaging with a spatial resolution of tens to hundreds of micrometers by utilizing multi-element ultrasound detector arrays [5–7]. The most commonly used detector arrays in PACT include linear, curved, and circular configurations for two-dimensional (2D) imaging [8–10], and planar, cylindrical, and spherical configurations for three-dimensional (3D) imaging [11–14]. To evaluate the imaging performance of different detection geometries, point spread function (PSF) is an effective measure. Researchers have entered into in-depth research on PSF investigation. In 2003, Xu and Wang proposed analytical expressions for PSF related to detector bandwidth and detector aperture in ideal spherical, cylindrical, and planar detection geometries [15]. Haltmeier and Zangerl extended the analytical study for approximate point detectors and approximate line detectors in 2010 [16]. Tian and coworkers further studied the impacting factors of PSF from the perspective of whole PACT imaging systems [12]. In addition, a few studies reported experimental PSFs by measuring the photoacoustic signals of small-scale beads [17,18]. However, no work studies the characteristics of PSF in non-ideal detection geometries in PACT through numerical modeling. Recently, we investigated the characteristics of the PSF in 3D detection geometries, the results of which are reported in a companion paper [19]. It is found that the detection geometry has a direct effect on PSF, especially in the lateral or tangential direction. In this paper, we study the characteristics of PSF in 2D detection geometries and discuss the impact of geometric shape, detector bandwidth, and detector aperture on PSF. Experimental results are presented to support the findings in this study.
2. Methods
Similar to the approach used in the study of 3D detection geometries [19], a small unit-intensity spherical absorber (diameter: 200 µm) was used as a photoacoustic source to model the PSF in 2D detection geometries. A pulsed laser with an infinitely short duration is used to irradiate the source, and detectors configured in different detection geometries are exploited to collect generated photoacoustic signals. Considering the diameter of the preset photoacoustic source, the center frequency of detectors is set to be 5 MHz [20]. Then, the initial acoustic pressure of the source is recovered using the back-projection (BP) algorithm proposed by Xu and Wang [21]. It can be written as
According to the aforementioned equations, the PSF in PACT is determined by both the BP signal and the imaging angle Ω, among which the angle Ω plays a more important role. The BP signal, which is determined by recorded photoacoustic signals, mainly depends on the detector bandwidth and detector aperture. In comparison, the solid angle Ω, which reduces to a plane angle in 2D PACT imaging, depends on the acoustic detection geometry and its relative position with respect to the POI. For convenience, we define the plane angle Ω as the ‘imaging angle’. As Fig. 1 illustrates, a circular detection geometry guarantees a full imaging angle of 2π radian regardless of the position of the POI while the imaging angle in a linear detection geometry is inherently smaller than π radian. In contrast, PACT imaging using curved detector arrays is slightly complicated. Specifically, the imaging angle in a curved detection geometry can either be less than π radian (i.e., outside the detection region [22]), equal to or greater than π radian (i.e., inside the detection region).
Fig. 1. Schematic showing the definition of the imaging angle in a circular detection geometry (a), a curved detection geometry (b), and a linear detection geometry (c). ROI: region of interest; POI: point of interest; Ω: imaging angle.
3. Results
3.1 PSF in a circular detection geometry
3.1.1 Detection-geometry-impacted PSF
To study the characteristics of the PSF in a circular detection geometry, a circular detector array with a geometric radius of 25 mm and 1024 evenly-distributed point detectors is considered, as shown in Fig. 2. A Cartesian coordinate system is established with the center of the detector array as the origin. Figures 3(a) and (b) present the spatial responses of the circular array at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. The 3D reconstructed result at half maximum intensity and corresponding 2D bipolar images in the x-y plane, x-z plane, and y-z plane are displayed. The PSFs are both isotropic in the x-y imaging plane with negligible distortion. By comparison, in the elevational direction, the PSF is significantly elongated because one row of detectors is used for signal reception in the elevational direction. In particular, when the POI is at the origin, the PSF is elevationally isotropic due to the symmetry of the imaging scenario; otherwise, the PSF bends toward the nearest detector.
Fig. 2. Schematic showing a circular detection geometry. Point-like detectors are evenly spaced over a circle with a diameter of 50 mm.
Fig. 3. Typical PSF in a circular detection geometry. (a) and (b) PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. (c) – (e) The effect of the diameter of the circular detection geometry on PSF at different positions on the x axis. FWHMR: radial FWHM; FWHMT: tangential FWHM; FWHME: elevational FWHM; X: x coordinate of the POI; R: radius of the circular geometry; Distance: the closest distance between the POI and the detection geometry (Distance = R – X).
Figures 3(c) – (e) give a further verification of the conclusions by changing the diameter of the circular geometry. The simulation is performed by varying the x position of the POI and the radii of the circular detector arrays are set to 20 mm, 25 mm, and 30 mm. The results reveal that the radial [Fig. 3(c)] and tangential [Fig. 3(d)] full width at half maximum (FWHM) of the PSF are independent of the diameter of the circular array and the position of the POI, which is because the POI is fully enclosed by the detection geometry in the x-y imaging plane. In contrast, the elevational FWHM of the PSF [Fig. 3(e)] depends on the distance between the POI and the detection geometry and shows no dependence on the diameter of the geometry.
3.1.2 Bandwidth-impacted PSF
To evaluate the impact of detector bandwidth on PSF in a circular detection geometry, a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80% was imposed on the detectors in the numerical simulation in Section 3.1.1. Figures 4(a) and (b) show the modeled PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. Compared with the PSFs in Figs. 3(a) and (b), the PSFs in this case have a similar shape but are affected by negative values at the edge, resulting a decreased FWHM in the radial, tangential, and elevational directions.
Fig. 4. Typical PSF in a circular detection geometry with limited detector bandwidth. (a) and (b) PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. The detectors have a fractional bandwidth of 80%. (c) – (e) The effect of detector bandwidth on PSF at different positions on the x axis. B: bandwidth of the detectors.
Figures 4(c) – (e) further investigate the dependence of PSF on the spatial locations of the POI and the fractional bandwidth. Detectors with 60%, 80%, 100%, and unlimited bandwidth are used to image the POIs on the x axis, respectively. Compared with the results of bandwidth-unlimited detectors, an overall reduction of FWHMs is observed for the PSF in all three directions. In addition, detectors with a smaller fractional bandwidth have PSFs with larger FWHMs, which is because in this case, the recorded photoacoustic signals suffer more from the problem of signal broadening and distortion. The finding agrees with that discovered in 3D detection geometries [19].
3.1.3 Aperture-impacted PSF
To study how detector aperture size affects the PSF in a circular detection geometry, a circular detector array (radius R = 25 mm) consisting of 256 evenly distributed finite-aperture detectors is used. The detector aperture is rectangular-shaped and has a width of 0.5 mm and a height of 10 mm, as shown in Fig. 5. The spatial responses of the circular detector array at positions (0, 0, 0) mm and (10, 0, 0) mm are illustrated in Figs. 6(a) and (b), respectively. Compared with the detection-geometry-impacted PSFs in Figs. 3(a) and (b), the aperture-impacted PSFs attenuate in intensity especially near the x-y imaging plane, leading to a reduction of the FWHM. Besides, even with an imaging angle of 2π radian, the PSF away from the center [Fig. 6(b)] expands along the tangential direction due to the asymmetric detection circle with respect to the POI, which is different from the PSF at the center [Fig. 6(a)]. Note that artifacts with weak intensity occur around the POI due to the detector aperture, which is depicted in the companion paper [19].
Fig. 5. Schematic showing a circular detection geometry with finite detector aperture. The detectors have a rectangular-shaped aperture with a width (w) of 0.5 mm and a height (h) of 10 mm.
Fig. 6. Typical PSF in a circular detection geometry with finite detector aperture. (a) and (b) PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. The detector aperture is 0.5 mm in width and 10 mm in height. (c) – (e) The effect of the width of detector aperture on PSF at different positions on the x axis. (f) – (h) The effect of the height of detector aperture on PSF at different positions on the x axis. d: detector aperture size, represented by width times height. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.
A further study of the dependence of the PSF on detector aperture size is carried out and the quantitative results of the FWHMs are presented in Figs. 6(c) – (h). Figures 6(c) – (e) are a group of simulations showing how the width of the detector aperture affects the PSF on the x axis. The simulation was performed with a variable aperture width (w = 0.1, 0.3, and 0.5 mm) but a fixed height (h = 10 mm). It can be seen that when the POIs move away from the origin, the aperture width starts affecting the PSF once it is greater than the diameter of the POI, especially in the tangential direction [Fig. 6(d)]. Specifically, the tangential FWHM can be theoretically estimated by the equation RT = (rs/rd)w, where rd is the radius of the circular geometry and rs is the distance between the POI and the origin. This is consistent with the conclusions drawn in spherical and cylindrical detection geometries [19]. Figures 6(f) – (h) show how the height of the detector aperture affects the PSF. The simulation was performed with a variable aperture height (h = 7.5, 10, and 12.5 mm) but a fixed width (w = 0.5 mm). The results reveal that the PSFs are almost independent of the aperture height in this case.
3.1.4 Combined PSF
To analyze the combined impacts of the shape of detection geometry, detector bandwidth, and detector aperture on PSF in a circular detection geometry, 256 detectors evenly spaced on a circle with a radius of 25 mm are used for PSF modeling. The detectors have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. They also have a rectangular aperture (width w = 0.5 mm, height h = 10 mm). Figures 7(a) and (b) show the spatial responses of the circular detector array at positions (0, 0, 0) and (10, 0, 0) mm, respectively. The results indicate that impacted by bandwidth and aperture, the PSFs get blurred and expanded especially along the tangential direction when the POI is away from the origin. The quantitative results of the dependence of the FWHM of the PSF on the position of the POI are presented in Figs. 7(c) – (e). It is manifest that the PSF in the radial direction mainly depends on the detector bandwidth, the PSF in the tangential direction is mainly determined by the detector aperture, while in the elevational direction, the PSF is dominated by the distance between the POI and the circular detection geometry.
Fig. 7. Typical PSF in a circular detection geometry with limited detector bandwidth (80%) and finite detector aperture (0.5 × 10 mm2). (a) and (b) PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. (c) – (e) Calculated FWHMs of the PSF at different positions on the x axis.
3.1.5 Experimental validation
To validate the conclusions drawn by the proposed PSF modeling method, a set of experiments were performed. A single black microsphere (Cospheric, LLC, Santa Barbara, California) with a diameter of 100 µm embedded in a gel phantom is used as the photoacoustic source. The laser source for signal excitation is provided by a tunable laser system and the wavelength is fixed at 690 nm. A circular detector array with a radius of 25 mm is used for signal reception. The detector array consists of 256 elements with a small flat rectangular aperture. The aperture width and height of each element are 0.51 mm and 8.0 mm, respectively. The center frequency is 7.5 MHz and the bandwidth is 73%. The images are reconstructed by the BP algorithm. Figures 8(a) and (b) show the reconstructed results of the microsphere at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. The experimental results show a similar shape as the simulation results in Figs. 7(a) and (b). The quantitative results of the dependence of the FWHM of the PSF on the position of the microsphere are presented in Figs. 8(c) – (e). It is seen that the experimental FWHMs are consistent with the simulation results, demonstrating the validation of the findings in this study.
Fig. 8. Experimental results of a circular detector array. (a) and (b) PSFs at positions (0, 0, 0) mm and (10, 0, 0) mm, respectively. (c) – (e) Calculated FWHMs of the PSF at different positions on the x axis.
3.2 PSF in a curved detection geometry
3.2.1 Detection-geometry-impacted PSF
Curved detection geometry is a non-ideal case of the aforementioned circular detection geometry, the imaging angle of which is always less than 2π radian. To investigate the characteristics of the PSF in a curved detection geometry, a semicircular detector array is used, as shown in Fig. 9. The semicircular detector array has a geometric radius of 25 mm and 1024 evenly-distributed point-like detectors. A Cartesian coordinate system is established with the center of the circle as the origin. The spatial responses of the detector array at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm are separately shown in Figs. 10(a) – (c). When the imaging angle is greater than or equal to π radian [Figs. 10(a) and (b)], the PSF is isotropic in the x-y imaging plane with negligible deformation. By comparison, when the imaging angle is smaller than π radian [Fig. 10(c)], the PSF becomes distorted in the x-y imaging plane, exhibiting expansion in the tangential direction. However, the PSF becomes elongated and bends to the direction of the curved array in all three cases, which is similar to the results in circular detection geometry (see Section 3.1.1). Furthermore, the closer the POI is to the curved array, the larger the bending curvature of the PSF becomes, leading to a smaller FWHM.
Fig. 9. Schematic showing a semicircular detection geometry. Point-like detectors are evenly distributed over a half circle with a diameter of 50 mm.
Fig. 10. Typical PSF in a curved detection geometry. (a) – (c) PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. (d) – (f) The effect of the central angle V of the curved detection geometry on PSF at different positions on the x axis (i.e. under different imaging angles).
To generalize the conclusion drawn from the semicircular detection geometry, another three groups of simulations using curved detection geometries with central angles of 2π/3, 5π/6, and 7π/6 radian were conducted. The quantitative results of the FWHM of the PSF are shown in Figs. 10(d) – (f). It reveals that the central angle of the curved geometry does not affect the PSF in the radial direction regardless of the position of the POI [see Fig. 10(d)]. This is different from the result in the tangential direction, which is closely related to the imaging angle Ω. Specifically, as Fig. 10(e) illustrates, when the imaging angle is smaller than π radian, the tangential FWHM of the PSF is inversely proportional to the imaging angle; however, when the imaging angle is greater than π radian, it is independent of the imaging angle. In the elevational direction, as shown in Fig. 10(f), the distance between the POI and the curved geometry is the only impacting factor when the POI is inside the detection geometry. Otherwise, the imaging angle also affects the elevational FWHM of the PSF and a larger imaging angle leads to a slower degradation of the FWHM when the POI moves away from the detection geometry along the x axis.
3.2.2 Bandwidth-impacted PSF
To study how detector bandwidth affects the PSF in a curved detection geometry, a semicircular detector array consisting of bandwidth-limited detectors is exploited. The detectors in this case have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. Other simulation settings are identical with those in Section 3.2.1. Figures 11(a) – (c) present the spatial responses of the detector array at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. The results reveal that the PSFs have a similar shape as the corresponding ones in the bandwidth-unlimited curved detector array [Figs. 10(a) – (c)], but are affected by negative values at the edge.
Fig. 11. Typical PSF in a semicircular detection geometry with limited detector bandwidth. (a) – (c) PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. The detectors have a fractional bandwidth of 80%. (d) – (f) The effect of detector bandwidth on PSF at different positions on the x axis.
To investigate the impact of bandwidth on PSF at different positions (i.e. under different imaging angles), a group of numerical simulations is conducted. Detectors with a fractional bandwidth of 60%, 80%, and 100% are used to respond to the POIs located at different positions on the x axis. The quantitative results of the radial, tangential, and elevational FWHMs of the PSF are shown in Figs. 11(d) – (f). It is again demonstrated that the FWHM of the bandwidth-impacted PSF becomes smaller in general compared with that in an ideal case regardless of the position of the POI. When the POI is located inside the detection region (i.e. the imaging angle is greater than π radian), the imaging situation is similar to that using a circular detector array, leading to a similar evolution behavior of the PSF in space as that in Figs. 4(c) – (e). In contrast, when the POI is located outside the detection region (i.e. the imaging angle is smaller than π radian), the radial FWHM exhibits positive correlation with the imaging angle and becomes closer to ideal values [the black diamonds in Fig. 11(d)] when the imaging angle becomes larger. The tangential and elevational FWHMs maintain the same relationship with the position of the POI as those in an ideal case [the black diamonds in Figs. 11(e) and (f)]. Moreover, the larger the detector bandwidth is, the narrower the PSF will be.
3.2.3 Aperture-impacted PSF
To study the influence of detector aperture on the PSF in a curved detection geometry, detectors having a rectangular-shaped aperture with a width of 0.5 mm and a height of 10 mm are used as Fig. 12 shows. There are 128 such detectors uniformly distributed on a semicircle with a radius of 25 mm. Figures 13(a) – (c) present the modeled PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. It can be seen that the PSF at the origin [Fig. 13(b)], which corresponds to an imaging angle of π radian, exhibits less deformation in the x-y imaging plane. By comparison, the PSFs away from the origin [Figs. 13(a) and (c)] are significantly expanded in the tangential direction. In addition, when the imaging angle is smaller than π radian, the aperture-impacted PSF is further elongated in the elevational direction [Fig. 13(c)] compared with that in an ideal case [Fig. 10(c)].
Fig. 12. Schematic showing a semicircular detection geometry with finite detector aperture. The detectors have a rectangular-shaped aperture with a width (w) of 0.5 mm and a height (h) of 10 mm.
Fig. 13. Typical PSF in a semicircular detection geometry with finite detector aperture. (a) – (c) PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. The detector aperture is 0.5 mm in width and 10 mm in height. (d) – (f) The effect of the width of detector aperture on PSF at different positions on the x axis. (g) – (i) The effect of the height of detector aperture on PSF at different positions on the x axis. d: detector aperture size, represented by width times height. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.
Figures 13(d) – (i) give a further investigation of the dependence of the PSF on the size of the detector aperture. Figures 13(d) – (f) show how the width of the detector aperture affects the PSF on the x axis. The simulation was performed with three semicircular detector arrays with varying aperture widths (w = 0.1, 0.3, and 0.5 mm) but a fixed aperture height (h = 10 mm). The results reveal that the radial and elevational FWHMs of the PSF [Figs. 13(d) and (f)] are almost independent of the detector aperture width. However, as shown in Fig. 13(e), when the POIs are away from the origin, the PSF expands in the tangential direction and the tangential FWHM can be estimated by the equation RT = (rs/rd)w, where rd is the radius of the curved detection geometry and rs is the distance between the POI and the origin. Figures 13(g) – (i) are the results showing the impact of the height of the detector aperture on the PSF on the x axis. The simulation was performed with three semicircular detector arrays with varying aperture heights (h = 7.5, 10, and 12.5 mm) but a fixed aperture width (w = 0.5 mm). As the results indicate, the radial, tangential, and elevational FWHMs of the PSF of the three detector arrays show little difference from each other, respectively.
3.2.4 Combined PSF
To study the impact of detector bandwidth and aperture size on the PSF in a curved detection geometry, a semicircular detector array (radius R = 25 mm) consisting of 128 evenly-spaced detectors is used for PSF modeling. The detectors have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. They also have a rectangular-shaped aperture (width w = 0.5 mm, height h = 10 mm). The modeled PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm are shown in Figs. 14(a) – (c). It is obvious that the final PSF combines the detection-geometry-impacted PSF, the bandwidth-impacted PSF, and the aperture-impacted PSF together, which agrees with the findings discovered in a circular detector array in Section 3.1.4. It is worth mentioning that the PSF in the radial direction is dominated by the detector bandwidth while the PSF in the tangential direction mainly depends on the detector aperture. Moreover, the PSF in the elevational direction is determined by both the detection geometry and the detector aperture. These conclusions can be further demonstrated by the quantitative results in Figs. 14(d) – (f).
Fig. 14. Typical PSF in a semicircular detection geometry with limited detector bandwidth (80%) and finite detector aperture (0.5 × 10 mm2). (a) – (c) PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. (d) – (f) Calculated FWHMs of the PSF at different positions on the x axis.
3.2.5 Experimental validation
To evaluate the proposed method in a curved detection geometry, a set of experiments using a semicircular detector array are conducted. Except for the element number of the array (128 in this case), other experimental parameters are the same as those in Section 3.1.5. A Cartesian coordinate system similar to that in Fig. 12 is established. The reconstructed results at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm are shown in Figs. 15(a) – (c), respectively. Regardless of the noises in the experiment, the obtained results are basically consistent with the simulation results in Figs. 14(a) – (c). Figures 15(d) – (e) show the dependence of radial, tangential, and elevational FWHMs of the images on the x position of the microsphere. It is demonstrated again that the experimental PSFs exhibit similar evolutionary trends in space as the simulated PSFs.
Fig. 15. Experimental results of a semicircular detector array. (a) – (c) PSFs at positions (-10, 0, 0) mm, (0, 0, 0) mm, and (10, 0, 0) mm, respectively. (d) – (f) Calculated FWHMs of the PSF at different positions on the x axis.
3.3 PSF in a linear detection geometry
3.3.1 Detection-geometry-impacted PSF
Compared with the aforementioned curved detection geometry, linear detection geometry provides a simpler imaging scenario with an imaging angle that is always smaller than π radian. To study the characteristics of the PSF in a linear detection geometry, a linear detector array (length L = 50 mm) with 1024 point-like detectors evenly spaced along the y axis is used, as Fig. 16 shows. The spatial responses of the linear array at positions (15, 0, 0) mm and (25, 0, 0) mm are presented in Figs. 17(a) and (b), respectively. The PSFs are symmetric both axially (x direction) and laterally (y direction) and the linear detection geometry distorts the PSF in the lateral direction. Similar to the above two detection geometries, the PSF in the elevational direction (z direction) is significantly elongated and bends toward the linear array. In addition, compared with the PSF under a larger imaging angle [Fig. 17(a)], the PSF under a smaller imaging angle [Fig. 17(b)] exhibits more broadening in the lateral direction and has a smaller bending curvature in the elevational direction.
Fig. 16. Schematic showing a linear detection geometry. Point-like detectors are evenly distributed along the y axis, constituting a detection line with a length of 50 mm.
Fig. 17. Typical PSF in a linear detection geometry. (a) and (b) PSFs at positions (15, 0, 0) mm and (25, 0, 0) mm, respectively. (c) – (e) The effect of the size of linear detection geometry under different imaging angles (i.e. at different positions on the x axis). FWHMA: axial FWHM; FWHML: lateral FWHM; L: length of the linear detection geometry.
A further study of the dependence of the PSF on the linear detection geometry is carried out by changing the length of the linear array and the imaging angle (i.e. the position of the POI). The quantitative results of the FWHM of the PSF are shown in Figs. 17(c) – (e). It can be seen that in a linear geometry, the axial FWHM of the PSF is hardly impacted by the length of the geometry or the imaging angle [see Fig. 17(c)]. However, the lateral FWHM of the PSF exhibits an inverse relationship with the imaging angle Ω [see Fig. 17(d)]. Moreover, it is confirmed that the elevational FWHM of the PSF is only impacted by the distance between the POI and the linear geometry and shows no dependence on the length of the detection geometry.
3.3.2 Bandwidth-impacted PSF
Figure 18 presents typical PSFs in a linear detection geometry with limited detector bandwidth. The same simulation configuration parameters as those in Section 3.3.1 are set in this case, except for detector bandwidth. The detectors have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. The spatial responses of the linear detector array at positions (15, 0, 0) mm and (25, 0, 0) mm are shown in Figs. 18(a) and (b). It is again observed that limited bandwidth introduces negativity and oscillation artifacts at the edge, yielding a degraded PSF. However, the bandwidth-impacted PSFs have a similar shape to those in the bandwidth-unlimited linear detector array [Figs. 17(a) and (b)].
Fig. 18. Typical PSF in a linear detection geometry with limited detector bandwidth. (a) and (b) PSFs at positions (15, 0, 0) mm and (25, 0, 0) mm, respectively. The detectors have a fractional bandwidth of 80%. (c) – (e) The effect of detector bandwidth on PSF under different imaging angles.
To study how detector bandwidth affects the PSF under different imaging angles (i.e. at different positions), a group of numerical simulations is conducted by varying the detector bandwidth (B = 60%, 80%, and 100%) and the x position of the POI. Similar to the results in the circular and curved detection geometries, an overall reduction of the FWHM of the PSF is apparent in all directions compared with the ones in an ideal case [black diamonds in Figs. 18(c) – (e)]. The axial FWHM shows a positive correlation with the imaging angle while the lateral FWHM exhibits an inverse relationship with the imaging angle. Besides, the axial, lateral, and elevational FWHMs all become closer to the ideal values when the imaging angle becomes larger. This is consistent with the findings discovered in a planar detection geometry [19]. In addition, detectors with a greater fractional bandwidth yield a smaller FWHM of the PSF.
3.3.3 Aperture-impacted PSF
To study the impact of detector aperture size on PSF in a linear detection geometry, a linear detector array (length L = 50 mm), which comprises 128 evenly-distributed detectors with finite aperture size, is used for PSF modeling. As shown in Fig. 19, the detectors have a small flat rectangular-shaped aperture and have a width of 0.25 mm and a height of 10 mm. Figures 20(a) and (b) show the modeled PSFs at positions (15, 0, 0) mm and (25, 0, 0) mm, respectively. Compared with the PSFs in Figs. 17(a) and (b), the PSFs here still exhibit axial symmetry but broaden in the lateral direction. By comparison, the PSF remains a similar size in the elevational direction.
Fig. 19. Schematic showing a linear detection geometry with finite detector aperture. The detectors have a rectangular-shaped aperture with a width (w) of 0.25 mm and a height (h) of 10 mm.
Fig. 20. Typical PSF in a linear detection geometry with finite detector aperture. (a) and (b) PSFs at positions (15, 0, 0) mm and (25, 0, 0) mm, respectively. The detector aperture is 0.25 mm in width and 10 mm in height. (c) – (e) The effect of the width of detector aperture on PSF at different positions on the x axis. (f) – (h) The effect of the height of detector aperture on PSF at different positions on the x axis. d: detector aperture size, represented by width times height. The gray dashed lines indicate the width of detector aperture.
To demonstrate the dependence of PSF on the size of the detector aperture and the spatial position of the POI (i.e. the imaging angle Ω), a group of numerical simulations is performed. The quantitative results showing how the width and height of the detector aperture affect the PSF on the x axis are presented in Figs. 20(c) – (h), respectively. The results indicate that the aperture size affects the PSF primarily in the lateral direction [Figs. 20(d) and (g)], second in the axial direction [Figs. 20(c) and (f)], but has no impact on the PSF in the elevational direction [Figs. 20(e) and (h)]. For the lateral FWHM of the PSF, the imaging angle Ω and the width of the rectangular detector w are two major impacting factors. Specifically, when the width of the detector aperture is smaller than the ideal lateral FWHM of the PSF, it has no impact on the lateral FWHM, which exhibits the same evolution behavior in space as the ideal ones (that is, inversely proportional to the imaging angle Ω). When the width of the detector aperture is greater than the ideal lateral FWHM of the PSF, the degraded lateral FWHM is limited by the width of the detector aperture. Moreover, the smaller the rectangular aperture height is, the more severe the PSF expands in the lateral direction, especially when the imaging angle is small [Fig. 20(g)].
3.3.4 Combined PSF
To investigate the characteristics of the PSF in a bandwidth-limited and finite-aperture curved detection geometry, a linear detector array with a length of 50 mm and 128 evenly-spaced detectors is employed. The detectors are set to have a rectangular-shaped aperture with a width of 0.25 mm and a height of 10 mm and have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. The typical PSFs at different positions are presented in Fig. 21. It is again demonstrated that the combined PSF is simultaneously influenced by the geometric shape of the detector array, detector bandwidth, and detector aperture. Furthermore, the quantitative results of the FWHM in Figs. 21(c) – (e) reveal that in the axial direction, the PSF is mainly affected by the detector bandwidth, while in the lateral direction, the PSF is dominated by the detector aperture. In contrast, the PSF in the elevational direction is determined by the detection geometry.
Fig. 21. Typical PSF in a linear detection geometry with limited bandwidth (80%) and finite detector aperture (0.25 × 10 mm2). (a) and (b) PSFs at positions (15, 0, 0) mm and (25, 0, 0) mm, respectively. (c) – (e) Calculated FWHMs of the PSF at different positions.
3.3.5 Experimental validation
Figure 22 shows the experimental measurement results of a microsphere by a linear detector array (L7-4, Philips, Netherlands). The linear array has 128 elements with a center frequency of 5 MHz and a bandwidth of 80%. Each element has a width of 0.25 mm and a height of 7.5 mm. The spacing between two adjacent elements is 0.298 mm. Other experimental settings are kept the same as those in Section 3.1.5. A Cartesian coordinate system similar to that in Fig. 19 is established. Figures 22(a) and (b) present the experimental results of the microsphere at positions (15.4, 0, 0) mm and (26.9, 0, 0) mm. The reconstructed images show a similar shape as the simulation results in Figs. 21(a) and (b). Figures 22(c) – (e) show a further analysis of the dependence of FWHMs on the x position of the microphere. The FWHMs of the experimental PSFs show similar evolutionary trends in space as the FWHMs of the simulated PSFs, especially in the axial and elevational directions. In addition, the PSF exhibits more broadening in the lateral direction in the experiment than in the simulation, which may be because the height of the detector element is smaller in the experiment [7.5 mm in the experiment and 10 mm in the simulation, Fig. 20(g)].
Fig. 22. Experimental results of a linear detector array. (a) and (b) PSFs at positions (15.4, 0, 0) mm and (26.9, 0, 0) mm, respectively. (c) – (e) Calculated FWHMs of the PSF at different positions on the x axis.
4. Conclusion
In this work, we numerically studied the impact of acoustic detection geometries on PSF in PACT imaging based on the BP image reconstruction algorithm. Three typical 2D detection geometries, namely, circular, curved, and linear detector arrays, were investigated. The analysis of the characteristics of the PSFs in these detection geometries with limited detector bandwidth and finite detector aperture was also presented. Results indicate that the geometric shape is the fundamental determinant of the characteristics of the PSF, which is closely related to the imaging angle enclosed by the detection geometry. Limited imaging angle loses useful photoacoustic signals, thus distorting the PSF. Limited detector bandwidth introduces negativity and oscillation signals into BP signals and compresses the size of the PSF. In addition, finite detector aperture broadens photoacoustic signals in the time domain and distorts corresponding BP signals. As a result, the aperture-impacted PSF expands on the basis of the geometry-impacted PSF, especially in the tangential or lateral direction. The experimental results validate the reliability and applicability of these findings. This work is expected to help predict and interpret PACT image quality and promote high-performance imaging for practical biomedical applications.
Funding
National Key Research and Development Program of China (2022YFA1404400); Anhui Provincial Department of Science and Technology (18030801138, 202203a07020020); National Natural Science Foundation of China (12174368, 61705216, 62122072); University of Science and Technology of China (YD2090002015); Institute of Artificial Intelligence at Hefei Comprehensive National Science Center (23YGXT005).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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