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Point spread function modeling for photoacoustic tomography – I: three-dimensional detection geometries

Chenxi Zhang, Zhijian Tan, and Chao Tian

Open Access Open Access

Abstract

Photoacoustic computed tomography (PACT) has been under intensive investigation as a promising noninvasive biomedical imaging modality. Various acoustic detector arrays have been developed to achieve enhanced imaging performance. In this paper, we study the effect of the detection geometry on image quality through point spread function (PSF) modeling based on back-projection image reconstruction. Three commonly-used three-dimensional detection geometries, namely, spherical, cylindrical, and planar detector arrays, are investigated. The effect of detector bandwidth and aperture on PSF in these detection geometries is also studied. This work provides a performance evaluation tool for acoustic detector arrays used in PACT and can be helpful in the design and selection of detector arrays in practical imaging applications.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Photoacoustic computed tomography (PACT) has gained much attention as a promising biomedical imaging modality and has experienced tremendous development in the past two decades [1–4]. In PACT, the energy of laser pulses is absorbed by chromophores in biological tissues, inducing a rapid thermal expansion and exciting ultrasound waves subsequently. The outgoing ultrasound signals are recorded by an ultrasound detector for final image formation of energy deposition distribution in tissues. For a typical PACT system, ultrasound detector is an essential component, and an exact image reconstruction requires enclosed signal detection geometries, such as unbounded planar, unbounded cylindrical, or closed spherical surfaces, which consist of an adequate density of point detectors [5,6]. In this way, the entire ultrasound signals generated from the acoustic sources can be captured to ensure complete measured data for image reconstruction.

However, due to practical constraints, non-ideal detection geometries rather than ideal detection geometries are used in practice. Common non-ideal detection geometries include linear [7], curved [8], circular [9], planar [10], cylindrical [11], and partially-spherical [12] detector arrays. Such non-ideal detection geometries record only part of photoacoustic signals and introduce image distortions and artifacts, degrading spatial resolution in practical experiments [13–15]. To understand the imaging characteristics of the non-ideal detection geometries, it is necessary to model the point spread function (PSF) of the geometries and understand the PSF evolution behavior in space. Accurate PSF modeling facilitates quantitative image analysis and the development of resolution enhancement approaches. For example, it has been demonstrated that deconvolving photoacoustic images with modeled PSF can recover more details and improve image quality significantly [16]. As such, PSF modeling is critical to PACT and other imaging modalities.

Current PSF modeling approaches in PACT can be roughly classified into two main categories, namely, experimental methods and analytical methods. The experimental methods give an estimation of PSF by measuring the photoacoustic signals of small-scale beads [16–18]. By contrast, the analytical methods give exact or approximate analytical expressions by incorporating different factors in an imaging system [19,20]. It is always based on ideal enclosed detection geometries. However, there are few reported studies on the PSF of non-ideal detection geometries, which are more commonly used in practical experiments. Recently, we studied the characteristics of the PSFs of three typical two-dimensional (2D) detection geometries and the results are reported in a companion paper [21].

In this work, we analyze the degradation mechanism of PSF in PACT and study the impact of the shape of detection geometries on PSF. In particular, the characteristics of the PSFs of three typical three-dimensional (3D) detection geometries, including spherical, cylindrical, and planar, are numerically investigated. We also discuss the impact of detector bandwidth and detector aperture size on PSF. Moreover, PSFs of each geometry with typical detector bandwidths and detector aperture sizes are also presented.

2. Methods

2.1 PSF in an ideal PACT system

An ideal PACT system assumes an infinite number of detectors having a point-like aperture and unlimited bandwidth, forming a closed detection geometry that encloses the photoacoustic source. In this case, the acoustic inverse problem can be analytically solved and exact image reconstruction can be achieved [22]. The PSF in an ideal PACT system can be mathematically characterized by a Dirac delta function as [23]

$$\textrm{PSF}({{\mathbf r^{\prime}}_\textrm{s}}) = \delta ({{\mathbf r^{\prime}}_\textrm{s}} - {{\mathbf r}_\textrm{s}})$$
where rs is the position of the point source and ${{\mathbf r^{\prime}}_\textrm{s}}$ is the position of the point of interest (POI).

2.2 PSF in a non-ideal PACT detection geometry

However, in the case of non-ideal PACT detection geometries that fail to fully enclose the photoacoustic source, there is no exact analytical solution to the acoustic inverse problem. To establish the PSF model, we perform numerical simulations by using a small unit-intensity spherical absorber (diameter: 200 µm) as the photoacoustic source. The initial photoacoustic pressure of such a source excited by a sufficiently short heating pulse can be expressed by [24]

$${p_0}({{\mathbf r^{\prime}}_\textrm{s}}) = U(a - |{{{{\mathbf r^{\prime}}}_\textrm{s}} - {{\mathbf r}_\textrm{s}}} |),$$
where U(·) is the Heaviside function and a denotes the radius of the spherical absorber (a = 100 µm). The photoacoustic signal detected at position rd and time t can be theoretically calculated by
$${p_{\textrm{ideal}}}({{\mathbf r}_\textrm{d}},t) = \frac{{R - {v_0}t}}{{2R}}U(a - |{R - {v_0}t} |),$$
where R = |rs - rd| is the distance from the source position rs to the detection position rd and v0 is the speed of sound in the acoustic medium. The spherical absorber generates an N-shaped acoustic wave, whose normalized amplitude spectrum can be expressed as [25]
$$P(\omega ) = \frac{{({{\omega a} / {{v_0}}})\cos ({{\omega a} / {{v_0}}}) - \sin ({{\omega a} / {{v_0}}})}}{{{{({{\omega a} / {{v_0}}})}^2}}},$$
where ω = 2πf is the angular frequency and f is the ultrasound frequency. The center frequency of the N-shaped wave can be estimated as [26]
$${f_\textrm{c}} = \frac{{{v_0}}}{{3a}}.$$

Assuming that the speed of sound v0 is 1500 m/s, acoustic detectors with a center frequency of 5 MHz are used to record the outgoing photoacoustic signals. After obtaining the acoustic signals, the initial pressure distribution of the spherical absorber can be reconstructed by the back-projection (BP) algorithm proposed by Xu and Wang [23]:

$${p_0}({{\mathbf r}_\textrm{s}}) = \int_\Omega {b({{\mathbf r}_\textrm{d}},t)\frac{{d\Omega }}{\Omega }} ,$$
where the BP term b(rd, t) is determined by the detected photoacoustic signals p(rd, t) and takes the form of
$$b({{\mathbf r}_\textrm{d}},t) = 2\left[ {p({{\mathbf r}_\textrm{d}},t) - t\frac{{\partial p({{\mathbf r}_\textrm{d}},t)}}{{\partial t}}} \right]\delta \left( {t - \frac{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}}{{{v_0}}}} \right),$$
where Ω denotes the entire solid angle covered by the detection surface with respect to the source, and dΩ is the solid angle subtended by the detection element dσ at rd with respect to the source. It can be expressed as
$$d\Omega = \frac{{d\sigma }}{{{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}^2}}}\left( {{{\mathbf n}_\textrm{d}} \cdot \frac{{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}}}{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}}} \right),$$
where nd is the unit normal vector of the detection surface pointing to the source.

For convenience, we define the angle Ω as the ‘imaging angle’. It is equal to 4π steradian for the closed spherical and infinite cylindrical geometries and 2π steradian for the infinite planar geometry. These detection geometries are ideal and can yield exact image reconstruction. However, in practical PACT imaging scenarios, non-ideal versions of these detection geometries are commonly used due to restrictions of experimental conditions, leading to limited imaging angles as shown in Fig. 1. In these cases, numerical calculation rather than analytical deviation in ideal cases [Eq. (1)] is required to obtain the PSFs.

 figure: Fig. 1.

Fig. 1. Schematic showing the definition of the imaging angle in a spherical detection geometry (a) and a planar detection geometry (b). ROI: region of interest; POI: point of interest; Ω: imaging angle.

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In a non-ideal PACT detection geometry, the PSF can be written as

$$\textrm{PSF}({{\mathbf r^{\prime}}_\textrm{s}}) = \delta ({{\mathbf r^{\prime}}_\textrm{s}} - {{\mathbf r}_\textrm{s}}) \ast {p_0}({{\mathbf r}_\textrm{s}}),$$
where * represents the convolution operator. According to the theoretical formula, the PSF in PACT is determined by both the BP signal b(rd, t) and the imaging angle Ω, among which the imaging angle plays a more important role. Given that the BP signal mainly depends on the bandwidth and aperture size of the detectors while the imaging angle is determined by the acoustic detection geometry and its relative position with respect to the POI, we will separately study the PSF of each 3D detection geometry and investigate the impact of detector bandwidth and aperture on PSF.

3. Results

3.1 PSF in an ideal closed spherical detection geometry

As mentioned above, a closed spherical geometry is regarded as an ideal detection geometry in PACT and the PSF is a Dirac delta function. To study the characteristics of the PSF in a closed spherical geometry, we established a 100-mm-diameter spherical detector array to image the preset 200-µm-diameter spherical absorber [see Fig. 2(a)]. There are 215 (32768) point-like acoustic detectors distributed over the detection surface using the Golden section spiral method in the k-Wave toolbox [27]. The detectors are assumed to be bandwidth-unlimited and can respond to the whole frequency bands of the signals. A Cartesian coordinate system is established with the center of the spherical detector array as the origin. Figures 2(b) and (c) show the spatial responses of the detection geometry at the origin and the position (0, 0, 10) 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 results indicate that the spherical absorber can be perfectly recovered regardless of its position, demonstrating a distortion-free PSF.

 figure: Fig. 2.

Fig. 2. Typical PSF in a closed spherical detection geometry. (a) Schematic showing a closed spherical detection geometry. (b) and (c) PSFs at the origin and the position (0, 0, 10) mm. The PSFs are shown in 3D at half maximum intensity and 2D in the x-y plane, x-z plane, and y-z plane, respectively.

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3.2 PSF in a spherical detection geometry

3.2.1 Detection-geometry-impacted PSF

Constrained by the reality, non-ideal versions of a closed spherical detection geometry are commonly used in practical PACT imaging scenarios. To investigate the impact of detection geometry on PSF, we performed numerical simulations of PSFs in non-ideal spherical detection geometries.

Considering the diameter of the preset spherical absorber (diameter: 200 µm), detectors with a center frequency of 5 MHz are used. Idealized conditions are assumed except for the shape of detection geometry. Figure 3 shows the photoacoustic signal recorded by such a detector and the distance between the POI and the detector is 25 mm. The recorded photoacoustic signal approximates the N-shaped wave [Fig. 3(a)] in theory and its frequency components are well captured [Fig. 3(b)]. Therefore, the corresponding BP signal is nearly identical with the theoretical one. It should be noted that, to ensure that the high-frequency contents of the photoacoustic signal can be captured, a large cutoff frequency is used, leading to negativity and oscillation signals in the BP signal [Fig. 3(c)]. This might introduce severe negativity artifacts in simulated PSF models.

 figure: Fig. 3.

Fig. 3. Photoacoustic signal recorded by an ideal detector. (a) Photoacoustic wave. (b) Normalized Fourier spectrum of corresponding PA signal. (c) Calculated BP signal.

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To model the PSF in spherical detection geometry, a hemispherical detector array, which has a diameter of 100 mm and 4096 point-like detectors uniformly distributed over its surface using the Golden section spiral method, is used (see Fig. 4). A Cartesian coordinate system is established with the center of the hemispherical detector array as the origin. The spatial responses of the hemispherical geometry at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm are shown in Figs. 5(a) – (c), respectively. It can be seen that the POI having an imaging angle equal to 2π steradian [Fig. 5(b)] is almost exactly reconstructed, which is similar to the results in the ideal spherical geometry (Fig. 2). However, the POIs having an imaging angle greater than or less than 2π steradian [Figs. 5(a) and (c)] are deformed with obvious artifacts around the POIs. It is worth noting that the weak-intensity artifacts around the POI are caused by limited number of detectors, while the opposite values of artifacts in z direction [Figs. 5(a) and (c)] is caused by their complementary imaging angles.

 figure: Fig. 4.

Fig. 4. Schematic showing a hemispherical detection geometry. Point-like detectors are distributed over a hemisphere with a diameter of 100 mm.

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 figure: Fig. 5.

Fig. 5. Typical PSF in a spherical detection geometry. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. (d) and (e) The effect of the solid angle V of the spherical detection geometry on PSF under different imaging angles Ω (i.e., at different positions on the z axis). FWHMT: tangential FWHM; FWHMR: radial FWHM.

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Figures 5(d) and (e) study the impact of the solid angle V of the spherical detection geometry on the tangential and radial full width at half maximum (FWHM) of the PSF at different spatial locations (i.e., different imaging angles Ω). The results show that when the imaging angle Ω is larger than 2π steradian, the PSF will not degrade in the tangential direction (x or y direction) and only slightly expands in the radial direction (z direction). In contrast, when the imaging angle Ω is smaller than 2π steradian, the FWHM of the PSF is inversely proportional to the imaging angle in the tangential direction and approaches the theoretical value in the radial direction as the imaging angle increases.

3.2.2 Bandwidth-impacted PSF

Apart from the shape of detection geometry, a non-ideal detection system influences the imaging performance of a PACT system through several other detector parameters. Among them, finite detector bandwidth is one of the main determinants. Xu and Wang studied the impact of detector bandwidth on spatial resolution and gave the analytical expression of bandwidth-impacted PSF [19]. However, the work was performed under the assumption of ideal closed detection geometry. In this section, we extend it to non-ideal detection geometries and establish the PSF models in practical detection systems.

When detector bandwidth is considered, the detected photoacoustic signal, pbw(rd, t), can be expressed as the convolution of the ideal signal, pideal(rd, t), and the impulse response of the detector, b(t), i.e.

$${p_{\textrm{bw}}}({{\mathbf r}_\textrm{d}},t) = b(t) \ast {p_{\textrm{ideal}}}({{\mathbf r}_\textrm{d}},t).$$

By replacing p(rd, t) with pbw(rd, t) in Eq. (7), a bandwidth-degraded BP signal can be written as

$${b_{\textrm{bw}}}({{\mathbf r}_\textrm{d}},t) = 2\left[ {{p_{\textrm{bw}}}({{\mathbf r}_\textrm{d}},t) - t\frac{{\partial {p_{\textrm{bw}}}({{\mathbf r}_\textrm{d}},t)}}{{\partial t}}} \right]\delta \left( {t - \frac{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}}{{{v_0}}}} \right).$$

Then, the bandwidth-impacted PSF can be expressed as

$$\textrm{PS}{\textrm{F}_{\textrm{bw}}}({{\mathbf r^{\prime}}_\textrm{s}}) = \delta ({{\mathbf r^{\prime}}_\textrm{s}} - {{\mathbf r}_\textrm{s}})\ast \int_\Omega {{b_{\textrm{bw}}}({{\mathbf r}_\textrm{d}},t)\frac{{d\Omega }}{\Omega }} ,$$
which indicates that detector bandwidth affects PSF through contaminating the BP term.

Figure 6 is a group of numerical simulations showing how finite detector bandwidth degrades a BP signal. The unit-intensity spherical absorber (diameter: 200 µm) is imaged by a detector with unlimited and limited bandwidth, respectively. The distance between the POI and the detector is 25 mm. The bandwidth-limited detector used in this example has a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. The approximately N-shaped signals recorded by the two detectors in the time domain and the frequency domain are shown in Figs. 6(a) and (b), respectively. The results reveal that limited bandwidth distorts and broadens photoacoustic signals in the time domain and loses both low-frequency and high-frequency components of the signals. Consequently, negative values are introduced to the corresponding BP signal, which has a compressed main lobe and oscillates at the edge [red arrow in Fig. 6(c)]. The BP signal integration of all detectors over a detection geometry will eventually result in a degraded PSF with decreased FWHM.

 figure: Fig. 6.

Fig. 6. The effect of bandwidth on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a detector with a bandwidth of 80% (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a). B: bandwidth. Red arrows denote the oscillation signals.

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The effect of limited detector bandwidth on PSF in a spherical detection geometry is demonstrated in Fig. 7. The detectors in this case have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. Other simulation settings are kept the same as those in Section 3.2.1. Figures 7(a) – (c) show the spatial responses of the hemispherical array at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. The modeled PSFs in this case have a similar shape to those in the bandwidth-unlimited hemispherical array [Figs. 5(a) – (c)]. However, the images become blurred rather than having a uniform intensity and sharp edges due to the presence of negativity artifacts and oscillation artifacts at the edge.

 figure: Fig. 7.

Fig. 7. Typical PSF in a hemispherical detection geometry with limited detector bandwidth. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. The detectors have a fractional bandwidth of 80%. (d) and (e) The effect of detector bandwidth on PSF under different imaging angles. B: bandwidth of the detectors.

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To further investigate the impact of bandwidth on PSF 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 z axis (i.e. different imaging angles). Figures 7(d) and (e) show the quantitative results of the FWHMs of the PSFs in the tangential and radial directions, respectively. The results indicate that the FWHM of the PSF decreases in general compared with that in an ideal case regardless of the value of the imaging angle. The tangential FWHM (x or y direction) maintains the inverse relationship with the imaging angle when the imaging angle is less than 2π steradian, and it is less affected by the imaging angle when the imaging angle is greater than 2π steradian [see Fig. 7(d)]. In contrast, the radial FWHM (z direction) generally shows a positive correlation with the imaging angle and becomes closer to the ideal values when the imaging angle becomes greater [see Fig. 7(e)]. Besides, detectors with a greater fractional bandwidth yield PSFs with narrower FWHMs. This is because the smaller the detector bandwidth is, the more significant its impacts on the photoacoustic signal will be (i.e., signal broadening and the introduction of extra negative and oscillation signals).

3.2.3 Aperture-impacted PSF

Thus far, the study has been based on the assumption that acoustic detectors have an infinitely small aperture. However, realistic acoustic detectors always have a finite aperture size. When the detector aperture size is considered, the real photoacoustic signal, papert(rd, t) can be characterized by a surface integral of the ideal photoacoustic signal, pideal(rd, t), over the detector aperture [28,29], i.e.

$${p_{\textrm{apert}}}({{\mathbf r}_\textrm{d}},t) = \int\!\!\!\int {w({{\mathbf r}_i}){p_{\textrm{ideal}}}({{\mathbf r}_i},t)d\sigma } ,$$
where dσ denotes a surface element at point ri and w(ri) denotes a weighting factor representing the contribution of each element to the total signal of the detector. Replacing p(rd, t) with papert(rd, t) in Eq. (7), we obtain the aperture-degraded BP signal:
$${b_{\textrm{apert}}}({{\mathbf r}_\textrm{d}},t) = 2\left[ {{p_{\textrm{apert}}}({{\mathbf r}_\textrm{d}},t) - t\frac{{\partial {p_{\textrm{apert}}}({{\mathbf r}_\textrm{d}},t)}}{{\partial t}}} \right]\delta \left( {t - \frac{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}}{{{v_0}}}} \right).$$

The aperture-impacted PSF can be further given by

$$\textrm{PS}{\textrm{F}_{\textrm{apert}}}({{\mathbf r^{\prime}}_\textrm{s}}) = \delta ({{\mathbf r^{\prime}}_\textrm{s}} - {{\mathbf r}_\textrm{s}}) \ast \int_\Omega {{b_{\textrm{apert}}}({{\mathbf r}_\textrm{d}},t)\frac{{d\Omega }}{\Omega }}. $$

Similar to detector bandwidth, detector aperture also affects the PSF through degrading the BP term.

Figure 8 is an example showing how the detector aperture size affects the photoacoustic signals. A POI is responded to by two detectors, one with a point-like aperture and the other with a finite aperture size. The finite detector aperture has a small flat rectangular shape with a width of 0.5 mm and a height of 10 mm. As Fig. 8(a) shows, by incorporating the detector aperture, the theoretical N-shaped photoacoustic signal is severely distorted and loses its main frequency components [Fig. 8(b)]. Signal aberrance is also apparent in the corresponding BP signal. The resulting asymmetry contaminates the final PSF and introduces artifacts, especially in limited view angle imaging scenarios. In addition, extra negative signals are also introduced to the photoacoustic signal as indicated by the green arrow, leading to artifacts in BP signals [see Fig. 8(c)] and reconstructed images. The deviation of the position of the artifacts from the position of the POI is determined by both the detector aperture size and the distance between the POI and the detector.

 figure: Fig. 8.

Fig. 8. The effect of aperture size on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a detector with a rectangular shape (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a). D: detector aperture size, represented by width times height. Green arrows denote the extra introduced signals.

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To study how detector aperture size affects the PSF in a spherical detection geometry, a hemispherical detector array, which comprises 1024 detectors spaced over a hemisphere using the Golden section spiral method [27], is exploited for simulation. The hemisphere has a diameter of 100 mm and the detectors have a disk-like aperture with a diameter of 2 mm, as shown in Fig. 9. Figures 10(a) – (c) illustrate the spatial responses of the detection geometry at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. It can be seen that the PSF at the origin, which corresponds to an imaging angle of 2π steradian, exhibits little distortion. However, in contrast, the PSFs at other locations are significantly blurred and possess a similar shape as the detector aperture.

 figure: Fig. 9.

Fig. 9. Schematic showing a hemispherical detection geometry with finite detector aperture. The detectors have a disk-like aperture with a diameter (d) of 2 mm.

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 figure: Fig. 10.

Fig. 10. Typical PSF in a hemispherical detection geometry with finite detector aperture. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. The diameter of detector aperture is 2 mm. (d) and (e) The effect of aperture size on PSF at different positions. Z: z coordinate of the POI; d: diameter of detector aperture. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.

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Figures 10(d) and (e) show quantitative evaluation results of the dependence of the FWHM of the PSF on its spatial position. The simulation is performed by varying the z position of the POI and using detectors with variable aperture sizes (diameter d = 0.1, 1, and 2 mm). It is obvious that the FWHM of the PSFs symmetrical about the origin are roughly identical, which is because they have complementary imaging angles. In addition, when the aperture size of the detector is smaller than the size of the POI (200 µm in this study), the PSF is mainly determined by the shape of the detection geometry. However, when the aperture size of the detector is larger than the size of the POI, the PSF expands in the tangential direction (x or y direction) and contracts in the radial direction (z direction). As Fig. 10(d) shows, when the POI is away from the origin, the tangential FWHM of the PSF can be estimated by the equation RT = (rs/rd)d, where rd is the radius of the spherical detection geometry, rs is the distance between the POI and the origin, and d is the diameter of the detector aperture.

3.2.4 Combined PSF

Finally, we attempt to analyze the combined impacts of the shape of detection geometry, detector bandwidth, and detector aperture on PSF. Following the derivations in the above sections, we can express the degraded photoacoustic signal as

$${p_{\textrm{ba}}}({{\mathbf r}_\textrm{d}},t) = b(t) \ast \int\!\!\!\int {w({{\mathbf r}_i}){p_{\textrm{ideal}}}({{\mathbf r}_i},t)d\sigma } .$$

By replacing p(rd, t) with pba(rd, t), Eq. (7) becomes the BP signal affected by both detector bandwidth and aperture as

$${b_{\textrm{ba}}}({{\mathbf r}_\textrm{d}},t) = 2\left[ {{p_{\textrm{ba}}}({{\mathbf r}_\textrm{d}},t) - t\frac{{\partial {p_{\textrm{ba}}}({{\mathbf r}_\textrm{d}},t)}}{{\partial t}}} \right]\delta \left( {t - \frac{{|{{{\mathbf r}_\textrm{s}} - {{\mathbf r}_\textrm{d}}} |}}{{{v_0}}}} \right).$$

The combined PSF in a non-ideal detection geometry can be formulated as

$$\textrm{PS}{\textrm{F}_{\textrm{ba}}}({{\mathbf r^{\prime}}_\textrm{s}}) = \delta ({{\mathbf r^{\prime}}_\textrm{s}} - {{\mathbf r}_\textrm{s}}) \ast \int_\Omega {{b_{\textrm{ba}}}({{\mathbf r}_\textrm{d}},t)\frac{{d\Omega }}{\Omega }} .$$

Figure 11 presents a simulation showing how detector bandwidth and detector aperture size together affect the photoacoustic signals. Broadened by limited bandwidth and distorted by finite aperture, the resultant photoacoustic signal looks quite different from the theoretical N-shaped wave. At the same time, oscillation and redundant negative signals also arise as the red and green arrows indicate. Therefore, signal aberrance occurs in BP signals along with apparent artifacts. Based on Eq. (18), the integration of BP signals of the entire detection surface will produce blurred photoacoustic images with artifacts.

 figure: Fig. 11.

Fig. 11. The combined effect of limited bandwidth and finite aperture on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a rectangular-shaped detector with a bandwidth of 80% (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a).

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To study how detector bandwidth and aperture size affect the PSF in a spherical detection geometry, a hemispherical detector array with a diameter of 100 mm and 1024 detectors is employed. Each detector is set to have a disk-like aperture with a diameter of 2 mm and a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. Figure 12 shows the typical PSFs at different positions. The results reveal that impacted by bandwidth and aperture, the PSF becomes blurred and expanded especially along the tangential direction. The quantitative results of the FWHM in Figs. 12(d) and (e) indicate that the expansion of the PSF in the tangential direction is dominated by the aperture size, while the change in the radial direction is mainly impacted by the bandwidth.

 figure: Fig. 12.

Fig. 12. Typical PSF in a hemispherical detection geometry with limited detector bandwidth (B = 80%) and finite detector aperture (d = 2 mm). (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. (d) and (e) Calculated FWHMs of the PSF at different positions.

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3.3 PSF in a cylindrical detection geometry

3.3.1 Detection-geometry-impacted geometry

A bounded cylindrical detection geometry is another type of geometry commonly used in PACT. It possesses a full view angle in the lateral direction but has limited detection view in the elevational direction. As shown in Fig. 13, a cylindrical detector array with a diameter (D) of 100 mm and a height (H) of 160 mm is used for PSF modeling in this case. The detection geometry consists of 16 circular arrays 10 mm apart from each other. Each circular array has 256 evenly-spaced point-like detectors. A Cartesian coordinate system is established with the center of the detector array as the origin. The spatial responses of the cylindrical geometry at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm are illustrated in Figs. 14(a) – (c), respectively. The PSFs are almost perfectly reconstructed in the xOy plane but show expansion in the elevational direction (z direction) compared with the PSF in the ideal spherical geometry in Fig. 2. The PSFs at the three positions show little visual difference.

 figure: Fig. 13.

Fig. 13. Schematic showing a finite cylindrical detection geometry. Multiple layers of circular arrays constitute a cylindrical array with 100 mm in diameter and 160 mm in height.

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 figure: Fig. 14.

Fig. 14. Typical PSF in a finite cylindrical detection geometry. (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Figure 15 is a group of simulations studying the dependence of PSF on the size of the cylindrical geometry under the same detector density. Figure 15(a) – (f) separately show the quantitative analysis of radial, tangential, and elevational FWHM of the PSF at different spatial positions along the z axis and the x axis. The results reveal that the PSFs are barely affected by the size of the cylindrical array in the radial [Figs. 15(a) and (d)] and tangential [Figs. 15(b) and (e)] directions. However, in the elevational direction, severe PSF expansion occurs when the height of the cylindrical geometry is small [Figs. 15(c) and (f)]. This is reasonable considering that a smaller height yields a more limited imaging angle and thus a worse resolution in the elevational direction. In addition, in this case, the PSFs close to the center of the geometry are more significantly affected. This is because the increase of the angle between the acoustic paths of detectors contributing to PSF in the elevational direction and a larger included angle will weaken the elimination of the positive components by the negative components of BP signals during signal integration.

 figure: Fig. 15.

Fig. 15. The effect of the size of the cylindrical detection geometry on PSF. (a) – (c) The effect of height (H) of the cylindrical geometry on PSF along the z axis. (d) – (f) The effect of height (H) of the cylindrical geometry on PSF along the x axis. (g) – (i) The effect of the ratio of height to diameter (H/D) of the cylindrical geometry on PSF at the position (0, 0, 0). The ratio is changed by varying the diameter (D) of the cylindrical geometry. The height (H) is fixed to 160 mm in this case. FWHMR: radial FWHM; FWHMT: tangential FWHM; FWHME: elevational FWHM; Z: z coordinate of the POI; X: x coordinate of the POI.

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An investigation of the impact of the height to diameter ratio (H/D) on PSF is presented in Figs. 15(g) – (i). The simulation was performed by varying the diameter of the cylindrical geometry while the height was fixed. The results show that the height to diameter ratio has no effect on the PSF in the radial [Fig. 15(g)] and tangential [Fig. 15(h)] directions but exhibits an inversely proportional relationship with the elevational FWHM [Fig. 15(i)]. When the value of the ratio is greater than two, the elevational FWHM of the PSF is much less broadened and the PSF approaches the PSF in the ideal closed spherical geometry (Fig. 2), indicating that the imaging angle is sufficient for reasonable image reconstruction.

3.3.2 Bandwidth-impacted PSF

To evaluate the impact of detector bandwidth on PSF in a cylindrical detection geometry, a cylindrical detector array consisting of bandwidth-limited detectors is employed. The simulation configurations in this case are identical with those in Section 3.3.1, except for detector bandwidth. The detector has a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. The spatial responses of the bandwidth-limited cylindrical detector array at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm are presented in Figs. 16(a) – (c), respectively. It is demonstrated that the limited bandwidth leads to a degraded PSF with negativity artifacts at the edge and thus reduces the FWHM of the PSF.

 figure: Fig. 16.

Fig. 16. Typical PSF in a cylindrical detection geometry with a detector bandwidth of 80%. (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Figure 17 shows the quantitative results of the dependence of the FWHM of the PSF on the spatial position of the POI and the fractional bandwidth of the detector. It can be seen that the FWHM of the PSF becomes smaller in general compared with that in an ideal case regardless of the value of the imaging angle. This is consistent with the findings discovered in a spherical detection geometry in Section 3.2.2. Since the height to diameter ratio (H/D) in this case is 1.6, the cylindrical geometry guarantees a relatively sufficient imaging angle for the POI. Therefore, the impact of bandwidth in the radial [Figs. 17(a) and (d)], tangential [Figs. 17(b) and (e)], and elevational [Figs. 17(c) and (f)] directions is almost independent of the spatial position of the POI. Furthermore, it is again demonstrated that the larger the detector bandwidth is, the narrower the PSF will be.

 figure: Fig. 17.

Fig. 17. The effect of detector bandwidth on PSF at different positions. (a) – (f) The effect of bandwidth on PSF along the z axis and the x axis, respectively.

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3.3.3 Aperture-impacted PSF

To study how detector aperture size affects the PSF in a bounded cylindrical detection geometry, a cylindrical detector array (D = 100 mm, H = 160 mm) consisting of 16 circular detector arrays is used. There are 64 detectors with a flat rectangular-shaped surface (width w = 1 mm, height h = 0.25 mm) evenly distributed over each circle (see Fig. 18). Typical PSFs at different positions are shown in Fig. 19. The results illustrate that the PSF contracts in size and sharp edges are recovered in the radial direction due to full-view signal detection in the x-y plane. In addition, the PSF substantially expands along the elevational direction, which is affected by the height of the aperture. For the POIs away from the z axis, the PSF also extends along the tangential direction and has a rectangular shape, which resembles the shape of the detector aperture.

 figure: Fig. 18.

Fig. 18. Schematic showing a cylindrical detection geometry with finite detector aperture. The detectors have a rectangular-shaped aperture with a width (w) of 1 mm and a height (h) of 0.25 mm.

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 figure: Fig. 19.

Fig. 19. Typical PSF in a cylindrical detection geometry with finite detector aperture (1.0 × 0.25 mm2). (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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To investigate the dependence of the PSF on the size of the detector aperture, the spatial responses of detectors with different aperture sizes are studied. Figure 20 shows how the width of the detector aperture affects the PSF on the z axis and the x axis. The simulation was performed with a variable detector width (w = 0.1, 0.5, and 1.0 mm) but a fixed height (h = 0.1 mm). For the POIs on the z axis, the aperture width does not affect the PSF in all radial [Fig. 20(a)], tangential [Fig. 20(b)], and elevational [Fig. 20(c)] directions. In comparison, as Figs. 20(d) – (f) show, for the POIs on the x axis and away from the origin, the aperture width affects the PSF if it is larger than the diameter of the POI. Similar to the case in the spherical detection geometry (see Section 3.2.3), when the investigated PSF is away from the z axis, its tangential FWHM [Fig. 20(e)] can be similarly estimated by the equation RT = (rs/rd)w, where rd is the radius of the cylindrical geometry and rs is the distance between the POI and the origin.

 figure: Fig. 20.

Fig. 20. The effect of the width of detector aperture on PSF at different positions. (a) – (c) Dependence of the FWHM of the PSF on z position. (d) – (f) Dependence of the FWHM of the PSF on x position. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.

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Figure 21 is a group of simulations showing how the height of the detector aperture affects the PSF. The simulation was performed with a variable detector height (h = 0.1 mm, 0.5 mm, and 1.0 mm) but a fixed width (w = 0.1 mm). The results reveal that the aperture height compresses the radial [Figs. 21(a) and (d)] and tangential [Figs. 21(b) and (e)] FWHMs of the PSFs regardless of the spatial position of the POI. The larger the aperture height is, the narrower the radial and tangential FWHMs will be. In addition, when the aperture height is larger than the size of the POI (200 µm), the elevational FWHM of the PSF is approximately equal to the aperture height, as shown in Figs. 21(c) and (f). Otherwise, the elevational FWHM is only influenced by the cylindrical detection geometry.

 figure: Fig. 21.

Fig. 21. The effect of the height of detector aperture on PSF at different positions. (a) – (c) Dependence of the FWHM of the PSF on z position. (d) – (f) Dependence of the FWHM of the PSF on x position.

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3.3.4 Combined PSF

To study the characteristics of the PSF in a bandwidth-limited and finite-aperture cylindrical detection geometry, a cylindrical detector array (D = 100 mm, H = 160 mm) with limited bandwidth and finite detector aperture is exploited. The detector array consists of 1024 detectors evenly distributed over 16 circles. The detectors have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. They also have a rectangular aperture (width w = 1.0 mm, height h = 0.25 mm). The spatial responses of the cylindrical detector array at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm are shown in Figs. 22(a) – (c), respectively. It is again demonstrated that the final PSF is simultaneously impacted by the geometric shape, detector bandwidth, and detector aperture.

 figure: Fig. 22.

Fig. 22. Typical PSF in a cylindrical detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 0.25 mm2). (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Figure 23 presents the quantitative results of the dependence of the FWHM of the PSF on the position of the POI. Similar to the results in the spherical detection geometry (see Section 3.2.4), the PSF in the radial direction [Figs. 23(a) and (d)] is mainly influenced by the detector bandwidth while the PSF in the elevational direction [Figs. 23(c) and (f)] is mainly impacted by the aperture height. In addition, the PSF in the tangential direction [Figs. 23(b) and (e)] depends on both the detection geometry and the width of the detector aperture.

 figure: Fig. 23.

Fig. 23. Quantitative analysis of the PSF in a cylindrical detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 0.25 mm2). (a) – (c) Calculated FWHMs of the PSF along the z axis. (d) – (f) Calculated FWHMs of the PSF along the x axis.

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3.4 PSF in a planar detection geometry

3.4.1 Detection-geometry-impacted PSF

Compared with a cylindrical detection geometry that has a full view angle in the lateral direction, a finite planar detection geometry possesses limited detection view. To study the characteristics of the PSF in a planar detection geometry, a planar detector array (size: 50 × 50 mm2) with 64 × 64 point-like detectors uniformly spaced on the xOy plane is used (see Fig. 24). A Cartesian coordinate system is established with the center of the detector array as the origin. Figures 25(a) and (b) show the spatial responses of the planar detector array at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. Unlike the isotropic PSF in the ideal spherical geometry (Fig. 2), the PSFs in this case are symmetric axially and laterally. In addition, the planar geometry broadens the PSF substantially in the lateral direction (x or y direction). Compared with the PSF under a smaller imaging angle [Fig. 25(b)], the one under a larger imaging angle [Fig. 25(a)] exhibits less broadening in the lateral direction due to the fact that more useful photoacoustic signals are collected for PSF reconstruction. It is worth mentioning that the artifacts in the cross-sectional images are caused by the finite size of the linear detection geometry and the limited number of the detectors [4,30].

 figure: Fig. 24.

Fig. 24. Schematic showing a finite planar detection geometry. Point-like detectors are uniformly distributed over the xOy plane, constituting a square-shaped detection surface with a length of 50 mm.

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 figure: Fig. 25.

Fig. 25. Typical PSF in a finite planar detection geometry. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. (c) and (d) The effect of the size of the planar geometry on PSF under different imaging angles. FWHML: lateral FWHM; FWHMA: axial FWHM; L: length of the planar geometry.

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Figures 25(c) and (d) give a quantitative study of the PSF at different spatial positions along the z axis by varying the size of the planar detection geometry (length L = 30, 40, and 50 mm). It demonstrates that the FWHM of the PSF is inversely proportional to the imaging angle Ω (i.e., the size of the planar detection geometry) in the lateral direction [Fig. 25(c)] but is not affected in the axial direction [Fig. 25(d)].

3.4.2 Bandwidth-impacted PSF

Figure 26 shows typical PSFs in a bounded planar detector array with a limited bandwidth. The simulation settings are the same as those in Section 3.4.1, except for detector bandwidth. The detectors have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. Figures 26(a) and (b) present the spatial responses of the planar detector array at positions (0, 0, 15) mm and (0, 0, 25) mm. Compared with the PSFs in Figs. 25(a) and (b), the PSFs in this case maintain a similar shape but contain more negativity and ringing artifacts, which results in decreased lateral and axial FWHMs.

 figure: Fig. 26.

Fig. 26. Typical PSF in a planar detection geometry with limited detector bandwidth. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. The detectors have a fractional bandwidth of 80%. (c) and (d) The effect of bandwidth on PSF under different imaging angles.

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To further investigate the impact of bandwidth on PSF under different imaging angles, a group of numerical simulations is conducted as Figs. 26(c) and (d) show. Detectors with a fractional bandwidth of 60%, 80%, and 100% are used to respond to the POIs located at different positions on the z axis (i.e., different imaging angles). The results indicate that the FWHM of the PSF becomes smaller in general compared with that in an ideal case regardless of the value of the imaging angle. The lateral FWHM [Fig. 26(c)] of the PSF exhibits an inverse relationship with the imaging angle. The axial FWHM shows a positive correlation with the imaging angle and becomes closer to ideal values [the black diamonds in Fig. 26(d)] when the imaging angle becomes larger. In addition, detectors with a greater fractional bandwidth yield PSFs with narrower FWHMs.

3.4.3 Aperture-impacted PSF

To study how detector aperture size affects the PSF in a bounded planar detection geometry, a planar detector array (size: 50 × 50 mm2) with 32 × 32 detectors evenly spaced on the xOy plane is used. The detectors are set to have a small flat square-shaped surface (width w = 1 mm), as shown in Fig. 27. Figures 28(a) and (b) present the spatial responses of the planar detector array at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. It is manifest that the PSF is significantly expanded in the lateral direction and narrowed down in the axial direction due to the presence of the detector aperture.

 figure: Fig. 27.

Fig. 27. Schematic showing a planar detection geometry with finite detector aperture. The detectors have a square-shaped aperture with a width (w) of 1 mm.

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 figure: Fig. 28.

Fig. 28. Typical PSF in a planar detection geometry with finite detector aperture. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. The detector aperture is 1.0 mm in width and height. (c) and (d) The effect of aperture size on PSF under different imaging angles. The gray dashed lines indicate the size of detector aperture.

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A further investigation of the dependence of the PSF on detector aperture size is carried out and the quantitative results of the lateral and axial FWHMs are shown in Figs. 28(c) and (d). The simulation was performed with a variable detector aperture size (w = 0.1, 0.3, and 0.5 mm). It reveals that the aperture size affects the PSF mainly in the lateral direction. Specifically, when the size of the detector aperture is smaller than the ideal lateral FWHM of the PSF, it has no impact on the lateral FWHM, which depends on the shape of the detection geometry (that is, it exhibits inverse proportion to the imaging angle Ω). Otherwise, it expands the lateral FWHM, which approximately equals the aperture size.

3.4.4 Combined PSF

To evaluate the impact of detector bandwidth and aperture size on the PSF in a planar detection geometry, a typical planar detector array (size: 50 × 50 mm2) with 32 × 32 detectors evenly spaced on the xOy plane is exploited. The detectors have a small square-shaped surface (width w = 1 mm). They also have a Gaussian-shaped bandpass frequency response with a fractional bandwidth of 80%. Figures 29(a) and (b) show the spatial responses of the planar detector array at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. The results indicate that the PSFs simultaneously exhibit the characteristics (i.e. PSF blurring and ringing) impacted by detector bandwidth and the characteristics (i.e. PSF broadening) impacted by detector aperture. It is worth noting that in the lateral direction, the PSF is dominated by the detector aperture, while in the axial direction, it is closer to the bandwidth-impacted PSF presented in Section 3.4.2. These findings agree with the quantitative results analyzing the dependence of the FWHMs of the PSF on the position of the POI in Figs. 29(c) and (d).

 figure: Fig. 29.

Fig. 29. Typical PSF in a planar detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 1.0 mm2). (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. (c) and (d) Calculated FWHMs of the PSF under different imaging angles.

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4. Conclusion

In this work, we investigated the degradation mechanism of PSF and the impact of the shape of detection geometry, detector bandwidth, and detector aperture on PSF in BP-based PACT. PSF modeling in an ideal closed spherical detection geometry and three commonly-used non-ideal detection geometries, including spherical, cylindrical, and planar configurations, were studied. According to the theoretical PSF formula [see Eq. (9)], the PSF in PACT is determined by both the BP signals and the imaging angle. In other words, the recorded photoacoustic signals and the detection geometry contribute to the degradation of PSF. In addition, the shape of the detection geometry dominates the PSF. It is clear that the non-ideal detection geometry loses useful photoacoustic signals, thus distorting the PSF in the image reconstruction process; the limited detector bandwidth and finite detector aperture affect the signal shape and further contaminate the PSF.

Our results demonstrate that the shape of detection geometry does not affect the PSF in the axial or radial direction, but broadens the PSF in the lateral or tangential direction. Specifically, when the imaging angle of the POI is smaller than 2π steradian, the lateral or tangential FWHM of the PSF is inversely proportional to the imaging angle; otherwise, the lateral or tangential FWHM is almost independent of the imaging angle.

Detector bandwidth blurs and broadens recorded photoacoustic signals in the time domain and loses both low-frequency and high-frequency contents of signals. Consequently, negative values are introduced to corresponding BP signals, leading to shrinkage and oscillation at the edge (see Fig. 6). The BP signal integration over the non-ideal detection surface eventually results in a degraded PSF with negativity and ringing artifacts. Except for the decreased FWHMs in all directions, the bandwidth-impacted PSF has a similar shape to the PSF produced by an ideal detector array.

Detector aperture may significantly distort recorded photoacoustic signals and lose their frequency components. As a result, the BP signals for image reconstruction may greatly differ from the theoretical ones (see Fig. 8). The resultant signal contaminates the PSF and introduces distortion, especially under limited-view imaging scenarios. Specifically, detector aperture will broaden the PSF in the lateral or tangential direction and the elevational direction. The shape of the aperture-impacted PSF resembles the shape of the detector aperture.

In practice, PSF is the result of the joint influence of the shape of detection geometry, detector bandwidth, and detector aperture. Among them, the shape of detection geometry plays the dominant role by determining the main shape of the PSF. Detector bandwidth affects the axial or radial FWHM, and detector aperture changes the shape of the PSF by affecting the lateral or tangential FWHM and the elevational FWHM.

The findings in this study help researchers choose the most suitable type of detection geometry according to practical applications and determine its diameter, bandwidth, and aperture size according to the requirements of spatial resolution. Conversely, the findings can also be helpful in the interpretation of photoacoustic images obtained by a practical PACT imaging system. Furthermore, combining the modeled spatially-variant PSFs with deconvolution methods, the quality of photoacoustic images can be significantly enhanced without precise and complex experiments.

Funding

National Natural Science Foundation of China (12174368, 61705216, 62122072); National Key Research and Development Program of China (2022YFA1404400); Anhui Provincial Department of Science and Technology (18030801138, 202203a07020020); 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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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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Figures (29)
Fig. 1.
Fig. 1. Schematic showing the definition of the imaging angle in a spherical detection geometry (a) and a planar detection geometry (b). ROI: region of interest; POI: point of interest; Ω: imaging angle.

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Fig. 2.
Fig. 2. Typical PSF in a closed spherical detection geometry. (a) Schematic showing a closed spherical detection geometry. (b) and (c) PSFs at the origin and the position (0, 0, 10) mm. The PSFs are shown in 3D at half maximum intensity and 2D in the x-y plane, x-z plane, and y-z plane, respectively.

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Fig. 3.
Fig. 3. Photoacoustic signal recorded by an ideal detector. (a) Photoacoustic wave. (b) Normalized Fourier spectrum of corresponding PA signal. (c) Calculated BP signal.

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Fig. 4.
Fig. 4. Schematic showing a hemispherical detection geometry. Point-like detectors are distributed over a hemisphere with a diameter of 100 mm.

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Fig. 5.
Fig. 5. Typical PSF in a spherical detection geometry. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. (d) and (e) The effect of the solid angle V of the spherical detection geometry on PSF under different imaging angles Ω (i.e., at different positions on the z axis). FWHMT: tangential FWHM; FWHMR: radial FWHM.

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Fig. 6.
Fig. 6. The effect of bandwidth on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a detector with a bandwidth of 80% (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a). B: bandwidth. Red arrows denote the oscillation signals.

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Fig. 7.
Fig. 7. Typical PSF in a hemispherical detection geometry with limited detector bandwidth. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. The detectors have a fractional bandwidth of 80%. (d) and (e) The effect of detector bandwidth on PSF under different imaging angles. B: bandwidth of the detectors.

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Fig. 8.
Fig. 8. The effect of aperture size on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a detector with a rectangular shape (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a). D: detector aperture size, represented by width times height. Green arrows denote the extra introduced signals.

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Fig. 9.
Fig. 9. Schematic showing a hemispherical detection geometry with finite detector aperture. The detectors have a disk-like aperture with a diameter (d) of 2 mm.

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Fig. 10.
Fig. 10. Typical PSF in a hemispherical detection geometry with finite detector aperture. (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. The diameter of detector aperture is 2 mm. (d) and (e) The effect of aperture size on PSF at different positions. Z: z coordinate of the POI; d: diameter of detector aperture. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.

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Fig. 11.
Fig. 11. The combined effect of limited bandwidth and finite aperture on photoacoustic signals. (a) Comparisons of the signals recorded by an ideal detector (black) and a rectangular-shaped detector with a bandwidth of 80% (blue). (b) Comparisons of the Fourier spectra of the two signals in (a). (c) Comparisons of the calculated BP signals from the two signals in (a).

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Fig. 12.
Fig. 12. Typical PSF in a hemispherical detection geometry with limited detector bandwidth (B = 80%) and finite detector aperture (d = 2 mm). (a) – (c) PSFs at positions (0, 0, -10) mm, (0, 0, 0) mm, and (0, 0, 10) mm, respectively. (d) and (e) Calculated FWHMs of the PSF at different positions.

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Fig. 13.
Fig. 13. Schematic showing a finite cylindrical detection geometry. Multiple layers of circular arrays constitute a cylindrical array with 100 mm in diameter and 160 mm in height.

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Fig. 14.
Fig. 14. Typical PSF in a finite cylindrical detection geometry. (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Fig. 15.
Fig. 15. The effect of the size of the cylindrical detection geometry on PSF. (a) – (c) The effect of height (H) of the cylindrical geometry on PSF along the z axis. (d) – (f) The effect of height (H) of the cylindrical geometry on PSF along the x axis. (g) – (i) The effect of the ratio of height to diameter (H/D) of the cylindrical geometry on PSF at the position (0, 0, 0). The ratio is changed by varying the diameter (D) of the cylindrical geometry. The height (H) is fixed to 160 mm in this case. FWHMR: radial FWHM; FWHMT: tangential FWHM; FWHME: elevational FWHM; Z: z coordinate of the POI; X: x coordinate of the POI.

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Fig. 16.
Fig. 16. Typical PSF in a cylindrical detection geometry with a detector bandwidth of 80%. (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Fig. 17.
Fig. 17. The effect of detector bandwidth on PSF at different positions. (a) – (f) The effect of bandwidth on PSF along the z axis and the x axis, respectively.

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Fig. 18.
Fig. 18. Schematic showing a cylindrical detection geometry with finite detector aperture. The detectors have a rectangular-shaped aperture with a width (w) of 1 mm and a height (h) of 0.25 mm.

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Fig. 19.
Fig. 19. Typical PSF in a cylindrical detection geometry with finite detector aperture (1.0 × 0.25 mm2). (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Fig. 20.
Fig. 20. The effect of the width of detector aperture on PSF at different positions. (a) – (c) Dependence of the FWHM of the PSF on z position. (d) – (f) Dependence of the FWHM of the PSF on x position. The gray dashed lines indicate the estimated tangential FWHM of the PSF, RT.

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Fig. 21.
Fig. 21. The effect of the height of detector aperture on PSF at different positions. (a) – (c) Dependence of the FWHM of the PSF on z position. (d) – (f) Dependence of the FWHM of the PSF on x position.

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Fig. 22.
Fig. 22. Typical PSF in a cylindrical detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 0.25 mm2). (a) – (c) PSFs at positions (0, 0, 0) mm, (0, 0, 10) mm, and (10, 0, 0) mm, respectively.

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Fig. 23.
Fig. 23. Quantitative analysis of the PSF in a cylindrical detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 0.25 mm2). (a) – (c) Calculated FWHMs of the PSF along the z axis. (d) – (f) Calculated FWHMs of the PSF along the x axis.

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Fig. 24.
Fig. 24. Schematic showing a finite planar detection geometry. Point-like detectors are uniformly distributed over the xOy plane, constituting a square-shaped detection surface with a length of 50 mm.

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Fig. 25.
Fig. 25. Typical PSF in a finite planar detection geometry. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. (c) and (d) The effect of the size of the planar geometry on PSF under different imaging angles. FWHML: lateral FWHM; FWHMA: axial FWHM; L: length of the planar geometry.

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Fig. 26.
Fig. 26. Typical PSF in a planar detection geometry with limited detector bandwidth. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. The detectors have a fractional bandwidth of 80%. (c) and (d) The effect of bandwidth on PSF under different imaging angles.

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Fig. 27.
Fig. 27. Schematic showing a planar detection geometry with finite detector aperture. The detectors have a square-shaped aperture with a width (w) of 1 mm.

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Fig. 28.
Fig. 28. Typical PSF in a planar detection geometry with finite detector aperture. (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. The detector aperture is 1.0 mm in width and height. (c) and (d) The effect of aperture size on PSF under different imaging angles. The gray dashed lines indicate the size of detector aperture.

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Fig. 29.
Fig. 29. Typical PSF in a planar detection geometry with limited detector bandwidth (80%) and finite detector aperture (1.0 × 1.0 mm2). (a) and (b) PSFs at positions (0, 0, 15) mm and (0, 0, 25) mm, respectively. (c) and (d) Calculated FWHMs of the PSF under different imaging angles.

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Equations (18)

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PSF ( r ′ s ) = δ ( r ′ s − r s )
p 0 ( r ′ s ) = U ( a − | r ′ s − r s | ) ,
p ideal ( r d , t ) = R − v 0 t 2 R U ( a − | R − v 0 t | ) ,
P ( ω ) = ( ω a / v 0 ) cos ⁡ ( ω a / v 0 ) − sin ⁡ ( ω a / v 0 ) ( ω a / v 0 ) 2 ,
f c = v 0 3 a .
p 0 ( r s ) = ∫ Ω b ( r d , t ) d Ω Ω ,
b ( r d , t ) = 2 [ p ( r d , t ) − t ∂ p ( r d , t ) ∂ t ] δ ( t − | r s − r d | v 0 ) ,
d Ω = d σ | r s − r d | 2 ( n d ⋅ r s − r d | r s − r d | ) ,
PSF ( r ′ s ) = δ ( r ′ s − r s ) ∗ p 0 ( r s ) ,
p bw ( r d , t ) = b ( t ) ∗ p ideal ( r d , t ) .
b bw ( r d , t ) = 2 [ p bw ( r d , t ) − t ∂ p bw ( r d , t ) ∂ t ] δ ( t − | r s − r d | v 0 ) .
PS F bw ( r ′ s ) = δ ( r ′ s − r s ) ∗ ∫ Ω b bw ( r d , t ) d Ω Ω ,
p apert ( r d , t ) = ∫ ∫ w ( r i ) p ideal ( r i , t ) d σ ,
b apert ( r d , t ) = 2 [ p apert ( r d , t ) − t ∂ p apert ( r d , t ) ∂ t ] δ ( t − | r s − r d | v 0 ) .
PS F apert ( r ′ s ) = δ ( r ′ s − r s ) ∗ ∫ Ω b apert ( r d , t ) d Ω Ω .
p ba ( r d , t ) = b ( t ) ∗ ∫ ∫ w ( r i ) p ideal ( r i , t ) d σ .
b ba ( r d , t ) = 2 [ p ba ( r d , t ) − t ∂ p ba ( r d , t ) ∂ t ] δ ( t − | r s − r d | v 0 ) .
PS F ba ( r ′ s ) = δ ( r ′ s − r s ) ∗ ∫ Ω b ba ( r d , t ) d Ω Ω .
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