Circular-scanning variance delay and sum algorithm for photoacoustic imaging

Abstract

Photoacoustic imaging (PAI) commonly uses the delay-and-sum (DAS) algorithm, which is fast but prone to artifacts. This work introduces variance delay-and-sum (VDAS), a novel algorithm for circular-array PAI that utilizes signal variance as a weighting factor. A high variance suggests a high probability of an optical absorber, significantly improving vasculature imaging. Numerical simulations and phantom/ex-vivo mouse ear experiments demonstrate that VDAS enhances image quality, reduces artifacts, and maintains computational efficiency compared to other DAS-based methods.

1. Introduction

Photoacoustic imaging (PAI), also called optoacoustic imaging, as a non-invasive imaging technique has promising potential in biomedical imaging [1–8]. PAI is a hybrid method combining high resolution of ultrasound and high contrast of optical absorption. PAI has been widely applied in biomedical imaging [9–12], oxygen saturation measurement [13–16], cancer diagnosis [17–19], and treatment guidance of major diseases such as malignant tumors [20–23], cardiovascular diseases [24–27] and brain disease [28,29]. The underlying physics of PAI is called photoacoustic effect, which was discovered by Alexander Graham Bell in 1880 [30]. The effect basically consists of two processes, illuminating the biological sample under test with a short-pulsed laser and induces acoustic waves from tissue based on thermo-elastic expansion properties. The US transducers around tissue can measure the raw PA signal. Post-processing the PA signal by imaging algorithms reconstructs a photoacoustic image which is the distribution of optical absorption capability [31–33]. What is more, optical absorption coefficient is highly related to the interior structure and reveals the shape of object. However, reconstructing the image from PA signal is an arduous inverse problem due to its nonlinear and ill-posed properties [34].

The reconstruction algorithms for PAI can be roughly classified into four categories: analytical methods such as delay and sum (DAS) [35] and universal back-projection [36]; numerical methods, such as time reversal [37]; iterative optimization methods, such as model-based algorithms [38]; and deep learning methods, such as upgUNet-based approaches [39]. The time reversal method based on the principle of coordinate invariance inverts the signal propagation process in the time domain. The model-based algorithms demand an accurate forward matrix to optimize the discrepancy between the measured signals and anticipated signals by the forward model. These two kinds of algorithms need high computational cost and time consumption, which prohibits the applications of PAI in some scenarios. Recently, deep learning (DL) technology has developed very rapidly and some DL enabled applications in PAI have been reported [40–43], but the difficulty in acquiring large amount of experimental data for specific samples or even live animals needs to be well addressed before this algorithm can be widely applied in PAI. Therefore, DAS is one of the most computation- and time-efficient algorithms for photoacoustic image reconstruction, which is highly suitable for real-time PAI applications.

However, due to its simple principle, DAS is relatively more sensitive to measurement noise and modeling errors than the other imaging techniques [44]. Many practical limiting factors in real experimental or clinical environments tend to deteriorate image quality, such as limited bandwidth of ultrasonic transducers, limited view of the data acquisition configuration, heterogeneity in medium, and acoustic losses in medium. Some algorithms based on DAS have been proposed to suppress artifacts and ameliorate image quality. For instance, minimum variance DAS (MV-DAS) method aims to reduce the off-axis signals and enhance image quality [45]. Delay multiple and sum (DMAS) [46] and double stage DMAS (DS-DMAS) [47] techniques can improve image resolution by multiplying obtained signals from any two ultrasonic sensors to interchange information. Multi-delay and sum with enveloping (multi-DASE) [48] remove artifacts and improves image resolution by utilizing acoustic pressure distribution at multiple time points. But these new DAS-based mechanisms still suffer from some deficiencies. For example, the reported MV-DAS and DS-DMAS protocols are applicable only for linear array measurement setup, which inevitably results in losses of information of the sample under test and degradation of the image integrity. Since the DMAS algorithm is naturally suitable to a linear array, it requires significantly more additional computational efforts when applied to circular arrays due to the geometric distribution of the circular sampling manner, rendering it considerably more challenging [49–51]. In addition, the DMAS method for a full circular array has not been demonstrated. Moreover, the MV-DAS and multi-DASE approaches are both computationally complex and time consuming since the former is an optimization algorithm, and the latter needs to run the DAS algorithm many times to recover spatial pressure distributions at multiple time points. Since circular transducer arrays are widely used for PAI and very suitable for testing small animals [1,15], improving the image quality and reconstruction speed for circular-array PAI setup is highly meaningful.

Considering that the traditional DAS-based algorithms only independently utilize the signal due to each measurement channel, i.e., do not consider changes or variance between the signals or images due to different channels, this work proposes a novel reconstruction algorithm called variance delay and sum (VDAS) for circular-array PAI configurations. Compared to the traditional DAS algorithm, VDAS can enhance image quality and suppresses artifacts without adding extra system cost or affecting too much the time efficiency. Although coherence factor DAS (CFDAS) [52], which makes use of coherence factor that is similar to variance, can improve lateral resolution and signal-to-noise ratio (SNR), it only works for linear arrays and no counterpart for circular array configurations has been reported. Effectiveness of the VDAS algorithm is first evaluated by simulations. Then experiments are performed using a blood vessel phantom and an ex-vivo mouse ear. All the secured results indicate that the VDAS method is less vulnerable to artifacts and leads to more distinct microstructures than DAS. The proposed VDAS modality is suitable for circular, cylindrical and spherical scanning geometries and especially suitable for reconstructing strip-shaped structures like blood vasculature. This work may contribute to the building of a high-quality and fast imaging speed compact clinical PAI system in the future.

The paper is organized as follows. In Section 2, theories of the VDAS algorithm are introduced. In Section 3, simulation and experiments are implemented to test the feasibility of VDAS. In Section 4, discussions and conclusions are presented.

2. Rationale

2.1. Delay and sum (DAS) algorithm

Photoacoustic waves are generated upon the absorption of pulsed optical energy by a sample under investigation. The physical process has been well studied and the theoretical background has been well established by many works. The time-resolved acoustic pressure signal can be expressed [1,53]

$$P(\overrightarrow{r},t)=\frac{\mathrm{\Gamma }}{4\pi c^2}\underset{V}{\int \int \int }\frac{A({\overrightarrow{r}}^{\prime })}{||{\overrightarrow{r}}^{\prime }-{\overrightarrow{r}}_2||}\frac{\mathrm{d}\phi {(t)}^{\prime }}{\mathrm{d}{t}^{\prime }}\mathrm{d}{\overrightarrow{r}}^{\prime }$$ (1)

where $p(\overrightarrow{r},t)$ is the acoustic signal located at $\overrightarrow{r}$ and time t, $A(\overrightarrow{r})$ is the heat distribution in the sample to be recovered, ${||\bullet ||}_2$ is Euclidean norm, V is the region of the object, $t^{\prime }=t-{||{\overrightarrow{r}}^{\prime }-\overrightarrow{r}||}_2/C$ represents a time-dependent spherical surface, c is the speed of sound, Γ is the Grüneisen parameter describing the conversion efficiency from heat to acoustic wave, and φ(t) is the power of the light pulse.

As illustrated in [ Fig. 1], it is straightforward that $p(\overrightarrow{r},t)$ is only contributed by the object within the region U, which is defined by a set of ${\overrightarrow{r}}^{\prime }$ satisfying $t-T\leqq {||{\overrightarrow{r}}^{\prime }-\overrightarrow{r}||}_2/C\le t$ , where T is the laser pulse width. So, (1) can be rewritten as the following form performing integration in the region U.

$$P(\overrightarrow{r},t)=\frac{\mathrm{\Gamma }}{4\pi c^2}\underset{U}{\int \int \int }\frac{A({\overrightarrow{r}}^{\prime })}{||{\overrightarrow{r}}^{\prime }-\overrightarrow{r}||}\frac{\mathrm{d}\phi {(t)}^{\prime }}{\mathrm{d}{t}^{\prime }}\mathrm{d}{\overrightarrow{r}}^{\prime }$$ (2)

Fig. 1.

Illustration of the region U (the shadow region) that contributes to $p(\overrightarrow{r},t)$ .

The image due to the transducer at ${\overrightarrow{r}}_m$ can be simply reconstructed through inverse mapping (back project), i.e., the “delay” operation of DAS, which is denoted as A_m. Then, the complete reconstructed image of the sample is acquired by summing up the images due to all the transducers, i.e., the “sum” operation of DAS.

Taking circular transducer array as an example, the corresponding reconstructed image is given by

$$\begin{array}{r}{A}_{\text{DAS}}(\overrightarrow{r})=w(\overrightarrow{r})\sum _{m=1}^M{A}_m(\overrightarrow{r})=w(\overrightarrow{r})\sum _{m=1}^M{p}_m({\tau }_m(\overrightarrow{r}))\\ {\tau }_m(\overrightarrow{r})=\frac{{||\overrightarrow{r}-{\overrightarrow{r}}_m||}_2}{c}\end{array}$$ (3)

where $A_{\text{DAS}}(\overrightarrow{r})$ represents the final reconstructed PA image, $A_m(\overrightarrow{r})$ is the reconstructed image individually due to the PA signal measured by the m th transducer and can be directly obtained using the expression $p_m({\tau }_m(\overrightarrow{r}))$ , M is the total number of transducers, and $w(\overrightarrow{r})$ is a weighting factor that is set to 1 in the standard DAS. Due to its low computational complexity, DAS can offer real-time reconstructions [54–56]. However, DAS is prone to image artifacts due to diversified reasons. The coherence factor DAS (CFDAS) uses signal coherence to improve image quality,

$$W_{cf}(\overrightarrow{r})=\frac{{\left({\sum }_{m=1}^M{p}_m({\tau }_m(\overrightarrow{r}))\right)}^2}{M{\sum }_{m=1}^M{p}_m{({\tau }_m(\overrightarrow{r}))}^2}$$ (4)

Because of the inequality of arithmetic and geometric means, we can have $W_{cf}(\overrightarrow{r})\in [0,1]$ .

2.2. Variance delay and sum (VDAS) algorithm

One of the most extensively explored applications of PAI is revealing tissue vasculatures. One remarkable feature of blood vessels is their cylinder-shaped geometry with length much larger than the width. Detailed procedure of the VDAS method is described as follows. For the sake of simplicity, only 2-D scenario is considered in this work, i.e., 2-D samples and circular transducer array.

We consider the signal generated by a strip-shaped sample with uniform absorbed optical energy distribution shown in Fig. 2 to explain the basic principle of VDAS. PA signals obtained by four transducers (m = 1, 2, 3, 4) in a circular array are depicted in Fig. 2. It is straightforward to see that the signal recorded by transducer 1 is much stronger than those by transducers 2∼4 since the corresponding region U is the largest for transducer 1. Images $A_m({\overrightarrow{r}}_a)$ and $A_m({\overrightarrow{r}}_b)$ are respectively proportional to the signal values $p_m\left(\frac{{||{\overrightarrow{r}}_a-{\overrightarrow{r}}_m||}_2}{c}\right)$ and $p_m\left(\frac{{||{\overrightarrow{r}}_b-{\overrightarrow{r}}_m||}_2}{c}\right)$ , with point ${\overrightarrow{r}}_a$ inside the sample (the red dot) and point ${\overrightarrow{r}}_b$ outside the sample (the green dot). $p_m\left(\frac{{||{\overrightarrow{r}}_a-{\overrightarrow{r}}_m||}_2}{c}\right)$ and $p_m\left(\frac{{||{\overrightarrow{r}}_b-{\overrightarrow{r}}_m||}_2}{c}\right)$ are labeled by red and green plus-sign markers, respectively, on each PA signal curve. Because $p_1\left(\frac{{||{\overrightarrow{r}}_a-{\overrightarrow{r}}_1||}_2}{c}\right)$ is much bigger than other labeled PA signal values in Fig. 2, the variance of the v $\left[p_1\left(\frac{{||{\overrightarrow{r}}_a-{\overrightarrow{r}}_1||}_2}{c}\right),\cdots ,p_4\left(\frac{{||{\overrightarrow{r}}_a-{\overrightarrow{r}}_4||}_2}{c}\right)\right]$ for point ${\overrightarrow{r}}_a$ is much bigger than the variance of the 1 × 4 vector $\left[p_1\left(\frac{{||{\overrightarrow{r}}_b-{\overrightarrow{r}}_1||}_2}{c}\right),\cdots ,p_4\left(\frac{{||{\overrightarrow{r}}_b-{\overrightarrow{r}}_4||}_2}{c}\right)\right]$ for point ${\overrightarrow{r}}_b$ . Thus, the variance of $p_m[{\tau }_m(\overrightarrow{r})]$ or $A_m(\overrightarrow{r})$ can be taken advantage of to enhance the image quality of the traditional DAS. To be more specific, if $p_m[{\tau }_m(\overrightarrow{r})]$ has a large variance at a spatial point $\overrightarrow{r}$ , there is a high possibility that $\overrightarrow{r}$ is inside a vessel or an optical absorber, i.e., the variance can be taken as a reliable indicator of high optical absorption in the sample.

Fig. 2.

Schematic description of the principle of VDAS. The sample is a strip-shaped absorptive object (the orange object). Four transducers deployed on a circular ring are used to explain the basic principle and the associated simulated PA signals are plotted as the blue curves. Plus-sign markers on the PA curves correspond to the two points ${\overrightarrow{r}}_a$ and ${\overrightarrow{r}}_b$ , with the former inside the sample and the latter outside the sample.

Accordingly, the proposed circular-array VDAS algorithm employs the variance of $p_m[{\tau }_m(\overrightarrow{r})]$ as the weighting factor to suppress undesired artifacts and the corresponding formula is given in (5).

$$\begin{array}{rl}{A}_{\text{VDAS}}(\overrightarrow{r})& =w(\overrightarrow{r})A_{\text{DAS}}(\overrightarrow{r})={\left[\frac{V(\overrightarrow{r})}{\underset{\overrightarrow{r}}{max}V(\overrightarrow{r})}\right]}^k{A}_{\text{DAS}}(\overrightarrow{r})\\ V(\overrightarrow{r})& =var[p_1({\tau }_1),p_2({\tau }_2),\cdots ,P_M({\tau }_M)]\\ V(\overrightarrow{r})& =var[A_1(\overrightarrow{r}),A_2(\overrightarrow{r}),\cdots ,A_M(\overrightarrow{r})]\end{array}$$ (5)

where k is a parameter tuning the effect of the weighting factor and is set to be 3 in this work to make $A_{DAS}(\overrightarrow{r})$ and $V(\overrightarrow{r})$ same scale, $V(\overrightarrow{r})$ is called variance matrix and its i th pixel is calculated by the variance of the vector $[A_1({\overrightarrow{r}}_i),\cdots ,A_M({\overrightarrow{r}}_i)]$ . More discussions on the selection of the k value are available in Section 4. Generally, a high resemblance between $w(\overrightarrow{r})$ and the optical absorption in the actual sample can lead to a high-quality image. A schematic illustration of the process of the VDAS technique is given in Fig. 3.

Fig. 3.

Schematic description of the complete procedure of the VDAS algorithm. The sample is a 2-D realistic vessel phantom, shown in the square DOI region. A circular array deployed on the dashed circle and enclosing the phantom is applied. A₁ to A_M are the reconstructed images individually due to each transducer. The small blue squares in A₁ to A_M represent an arbitrary pixel ${\overrightarrow{r}}_0$ in the images. Then a vector $[A_1({\overrightarrow{r}}_0),\cdots ,A_M({\overrightarrow{r}}_0)]$ can be established and the variance of this vector forms the pixel ${\overrightarrow{r}}_0$ in a matrix V. The i th pixel of V is the variance of the vector $[A_1({\overrightarrow{r}}_i),\cdots ,A_M({\overrightarrow{r}}_i)]$ . Summation of the images A₁ to A_M leads to the traditional DAS image A_DAS. Finally multiplying normalized V and A_DAS gives the VDAS image A_VDAS.

3. Experiments and results

We conduct both numerical simulations and experiments to test the performance of the proposed VDAS algorithm. We provide both qualitative and quantitative comparison with the CFDAS and DAS imaging techniques. To quantitatively evaluate the reconstructed images, we obtain some performance indicators including peak signal to noise ratio (PSNR) [57], structural similarity (SSIM) [58], and Pearson correlation (PC) [59]. SDNR and PC are employed for simulations since those exact target images are known, while SNR is used for experiments. The PSNR is given as

$$PSNR=20{\log }_{10}\left(\frac{HW}{{\mathrm{\Sigma }}_{\mathrm{i}=1}^H{\sum }_{j=1}^W{(\hat{I}(i,j)-I(i,j))}^2}\right)$$ (6)

where, $\hat{I}$ is the reconstructed image, I denotes the target image or ground truth image, and H and W are the size of image.

The SSIM is defined as

$$SSIM=\frac{(2{\mu }_{\hat{I}}{\mu }_I+C_1)(2{\sigma }_{\hat{I},I}+C_2)}{(u_{\hat{I}}^2+{\mu }_I^2+C_1)({\sigma }_{\hat{I}}^2+{\sigma }_I^2+C_2)}$$ (7)

where, $\sigma$ represents the expectation, ${\sigma }_{\hat{I},I}$ represents the covariance, $\sigma$ represents the standard deviation. c1 and c2 are two constants to avoid a weak denominator and we set c1 = 10⁻⁴ and c2 = 9 × 10⁻⁴ in our experiments.

The PC is given as

$$PC(\hat{I},I)=\frac{{\sigma }_{\hat{I},I}}{{\sigma }_{\hat{I}}{\sigma }_I}$$ (8)

3.1. Numerical simulation results

To test the performance of the proposed VDAS approach, we first carry out simulations using the k-wave toolbox in MATLAB [60]. The 2-D domain of interest (DOI) contains 512 × 512 pixels, and the pixel size is 0.1 mm. The used sample to be imaged is assumed to have a uniform initial pressure of 1 kPa, which is the acoustic wave generation source. The entire simulation region is filled with a homogeneous medium with 1500 m/s speed of sound and 1000 kg/m³ mass density. In total 360 point-wise acoustic transducers are uniformly distributed on a 70 mm radius circular array centered at the DOI center to record the time-resolved PA signals generated from the sample.

In the first trial, a 22.4 mm × 22.4 mm square model placed in the center of the DOI is used, shown in Fig. 4(a). The images reconstructed by the DAS, CFDAS, and VDAS techniques are given in Figs. 4(b)-(d), respectively. The results of DAS and VDAS successfully reconstruct the square model. However, there are some artifacts in the CFDAS result, which include a blurred boundary of the square and an x-shaped artifact at the center. It is seen that the image via the proposed VDAS method exhibits better quality as compared to the traditional DAS method. The insets in Figs. 4(b) and (d) obviously show that the artifacts in the background of the DAS image are much stronger than those in the VDAS image. For further comparison, the zoomed-in images in the red boxes in Figs. 4(b)-(d) are also given, which show that the VDAS image has a clearer boundary than the other two. This can be better visualized in the 1-D images in Fig. 4(e), where the black dashed line is the square boundary. It is found that the peak value of the 1-D DAS and CFDAS image is much higher than the value at the actual boundary, while the value of the VDAS image boundary is close to the peak value, which complies with the underlying physics.

Fig. 4.

Numerical simulation results of a square model. (a) Ground truth of the square model. (b) Reconstructed image using the DAS algorithm. (c) Reconstructed image using the CFDAS algorithm. (d) Reconstructed image using the VDAS algorithm. (e) 1-D images along the dashed yellow line in (b), (c) and (d). The green dashed line represents the actual boundary of the square model.

In the second trail, we apply a complicated blood vessel model displayed in Fig. 5(a). To make the simulation studies more realistic and verify the robustness of the proposed algorithm, we introduce Gaussian noises into the simulated time-domain PA signals with an SNR of 20 dB. PA images obtained by different methods are shown in Figs. 5(b)-(d). The entire profile of the blood vessel phantom can be successfully recovered. For the DAS result, noticeable artifacts can be seen in the purple and red insets. The CFDAS method can efficiently reduce the artifacts, but the image is quite faint. In contrast, the VDAS result has less artifacts and a clear vessel. The calculated PSNR, SSIM, and PC of the three images are given in Table 1. Compared with DAS, VDAS leads to a 32% increase in SSIM and 7% increase in PC, at the cost of slightly increased computation time. The computational speed of VDAS is similar to the conventional DAS algorithm under the same implementation conditions. Thus, the VDAS algorithm has computational complexity of O(M), with M denoting the number of sensors. For the simulation case in Fig. 5(d), the imaging frame rate is 4.35 Hz, which is determined by the number of transducers, number of pixels in the image, and hardware performance. The current implementation condition includes 3500 samples in the time-domain PA signal, 360 transducers, 512 × 512 pixels (0.1 mm × 0.1 mm per pixel) with in the reconstructed 2-D image, and computation by MATLAB using NVIDIA RTX A6000. This frame rate can be further increased by applying parallel computing techniques, a field programmable gate array (FPGA), and a more efficient customized program. We also notice that the computation time for the VDAS algorithm is only 3% more than that for the DAS algorithm, which is not detrimental to its potential applications.

Fig. 5.

Numerical simulation results of the blood vessel model. (a) Ground truth of the blood vessel model. (b) Reconstructed image using the DAS algorithm. (c) Reconstructed image using the CFDAS algorithm. (d) Reconstructed image using the VDAS algorithm.

Table 1. Parameters for Images Obtained by Simulations for Different Algorithms.

Algorithm	PSNR (dB)	SSIM	PC	Reconstruction time (s)
DAS	21.48	0.295	0.822	5.71
CFDAS	18.28	0.130	0.780	5.95
VDAS	21.86	0.435	0.881	5.92

3.2. Experimental system

To further evaluate the proposed VDAS method, we perform experiments using the experimental system showcased in Fig. 6. We apply a Q-switched Nd:YAG (SLI-10, Continuum Surelite) laser with tunable wavelength pumped (tuning range 680-2500 nm) by an OPO laser with 532 nm wavelength as the excitation source. We choose the wavelength of 704 nm as the excitation laser source, which has the maximum energy and can result in the highest SNR among the wavelength tuning range. The pulse repetition rate and the pulse duration of the laser are 10 Hz and 5 ns, respectively. The laser beam illuminates the sample with energy density of 18 mJ/cm² (less than the ANSI limit [61]). The PA signals are collected by a 2.25-MHz ultrasound transducer (V323-SU, Olympus), which is mounted on a 3D-printed holder shown in Fig. 6(b) to perform circular scanning around the sample and controlled by a step motor. The scanning radius is 70 mm, and 120 different scanning positions are formed by a step of 3° that leads to a full-view 2-D imaging system. The sample is situated on an acrylic holder and is immersed in a water tank, together with the transducer, filled with deionized water as the coupling medium. The detected time-domain PA signals are first amplified by 59 dB using a preamplifier (5072PR, Olympus) and then collected by a data acquisition (DAQ) card (NI PXIe-8840, National Instruments) with a sampling frequency of 31.25 MHz. The PA signal at each scanning position is averaged 40 times to reduce noise. The entire experimental system is controlled by LabView.

Fig. 6.

(a) Photograph of the PAI experimental system. The blue line represents the light path. (b) Photograph of the transducer, mounting holder, phantom and acrylic sample holder.

3.3. Phantom experiment

For experimental verification of the proposed VDAS method, we first apply a blood vessel phantom sample shown in Fig. 7(a) made by 3D printing technique (VeroBlackPlus). The phantom is placed in a DOI of 22.4 mm × 22.4 mm with a pixel size of 0.2 mm × 0.2 mm.

Fig. 7.

Experimental results of the blood vessel phantom. (a) Photograph of the blood vessel phantom. (b)-(d) Reconstructed images using the DAS algorithm, CFDAS algorithm and VDAS algorithm, respectively. (e) 1-D images along the yellow dashed line in (b)-(d). (f) 1-D images along the red dashed line in (b)-(d). (g) The standard deviation of the results in the red boxed region in (b)-(d).

The reconstructed images of DAS, CFDAS and VDAS are shown in Figs. 7(b)-(d). There are some noises and artifacts in the imaging results, especially in the regions near the phantom sample, which is partially because the speed of sound in the phantom material is larger than that of the water. The CFDAS result has the clearest background, but the sample seems quite faint and uneven intensity within the region of the vessel. Compared with the other algorithms, the VDAS result not only has low background noise but also has clearly displayed sample. To further demonstrate the merit of the VDAS algorithm, Figs. 7(e) and (f) respectively plot the 1-D images along the dashed yellow line and dashed red line delineated in [Figs. 7(b)-(d)]. It is seen that the vessel phantom part (represented by the highest peak in Figs. 7(e) and (f) can be reliably recovered by the DAS and VDAS algorithms. But the artifacts of the VDAS result are only about half of those of the DAS result. However, the CFDAS result has the lowest peak value, which cannot be considered as a successful reconstruction. Figure 7(g) provides the STD of the red boxed region in Figs. 7(b)-(d). Although the CFDAS method leads to the lowest STD, the vessel sample is not well recovered. The STD of the VDAS algorithm is less than 1/4 of that for the DAS algorithm, which is a big improvement.

3.4. Ex-vivo mouse ear experiment and hair experiment

Then, an ex-vivo mouse ear, presented in [ Fig. 8(a)], is interrogated to further study the feasibility of the VDAS method. The mouse ear was obtained from ShanghaiTech University and the institutional oversight was waived. The mouse ear is fixed on the Acrylic holder using agar (made of 1.5 g agar and 100 g water). The applied excitation laser in this set of ex-vivo experiment is directly provided by the OPO laser, which is a 532-nm green light [62]. The dimension of the DOI in this case is 25 mm × 25 mm with a pixel size of 0.1 mm × 0.1 mm, which can allow the visualization of fine details of the mouse ear.

Fig. 8.

Experimental results of the ex-vivo mouse ear. (a) Photograph of the mouse ear. (b)-(d) Reconstructed image using the DAS algorithm, CFDAS algorithm, and VDAS algorithm. (e) 1-D images along the dashed red line in the insets of (b)-(d). (f) The standard deviation of the results in the yellow boxed region in (b)-(d).

The reconstruction results for the DAS, CFDAS and VDAS techniques are, respectively, shown in Figs. 8(b)-(d). Most of the mouse ear's major blood vessels can be recovered by the DAS and VDAS algorithms with high fidelity. However, the CFDAS result misses many parts of the vessels. The DAS method still leads to an image with strong background artifacts. In the VDAS results given in Fig. 8(d), the artifacts are apparently suppressed. The CFDAS result has few artifacts and fewer vessel details. A small region in the mouse is used to analyze the imaging results, which is shown in the insets of Figs. 8(a)-(d) and contains a section of Y-shaped blood vessel. It is observed that the Y-shaped blood vessel in the inset of Fig. 8(d) due to VDAS has more explicit boundaries than that in Fig. 8(b) and (c). To further compare the results of these algorithms, we plot 1-D images along the red dashed line marked in the insets of Figs. 8(b)-(d). As shown in Fig. 8(e), the blood vessel is located at 1 mm position. But the DAS image value at 1 mm (circled by the green dashed line) is even lower than that around 2 mm, where no vessel exists. For the VDAS image, the peak value occurs at 1 mm, which indicates the correct reconstruction of the vessel. Figure 8(f) presents the STD of the yellow boxed region in Figs. 8(b)-(d)]. The STD of VDAS is only about 1/4 of that for DAS. From the inset of Fig. 8(d), it is seen that the reconstructed thinnest blood vessel is 0.2 mm thick. Therefore, the obtainable image resolution is about 0.2 mm. We then do one more set of experiment using two crossed hairs shown in Fig. 9(a). The recovered image using the VDAS method in Fig. 9(c) is obviously better than that based on the DAS algorithm in Fig. 9(b). The calculated image resolution for this sample is also 0.2 mm, which is obtained on the white dashed line according to the criterion defined in [63]. It is called the half-maximum resolution method and defined as the distance between the midpoints of two intensity peaks when they are on the threshold of becoming indistinguishable.

Fig. 9.

Experimental results of two crossed hairs. (a) Photo of the sample. (b) Reconstructed image using the DAS algorithm. (c) Reconstructed image using the VDAS algorithm. The image resolution is calculated on the white dashed line.

4. Discussion and conclusions

Here, we provide some pros and cons of the proposed VDAS method. Although only 2-D simulation and experimental results are presented, the VDAS approach can also be applied in 3-D cylindrical and spherical measurement geometries as suggested by the provided rationale.

Owing to its fast calculation speed, VDAS is favorable for fast dynamic PAI applications and may contribute to functional PAI [64]. The system configuration for dynamic or functional PAI applications is the same as that for the DAS or VDAS modality.

VDAS is most effective for circular, spherical or other 2-D/3-D transducer arrays or scanning mechanisms that can perform full-view measurements. This is because limited-view measurements may reduce the variance and affect the superiority of VDAS compared to other methods [65]. In other words, a limited-view measurement means that the variance matrix $V(\overrightarrow{r})$ in (3) becomes a new one $U(\overrightarrow{r})=\mathrm{var}[A_1(\overrightarrow{r}),A_2(\overrightarrow{r}),\cdots ,A_N(\overrightarrow{r})]$ , where [1,2,…,N] is a subset of [1,2,…,M] and some pixels of $U(\overrightarrow{r})$ are smaller than those of $V(\overrightarrow{r})$ . We perform a simulation study to show the effect of limited-view measurements.

We also study imaging results with transducers covering limited view angles of 180° and 90°. As can be seen from Fig. 10, the 180° limited-view result slightly increases the artifacts in the image, while the 90° limited-view image quality is obviously degraded, rendering the small branching vessels indistinguishable.

Fig. 10.

Numerical simulation results of the blood vessel model using the VDAS algorithm based on limited-view detection. (a) Sensors covering 180°. (b) Sensors covering 90°.

The parameter k in (5) is a coefficient used to remove artifacts from the obtained images. We have tested different k ranging from 1 to 5 using the VDAS algorithm and compared the impact of k on the SNR, PSNR, and SSIM of the obtained images. It is seen from Fig. 11 that as k increases, the SNR and PSNR are reduced while the SSIM is enhanced. This is mainly because a smaller k can recover more details of the sample at the cost of increased background noises and artifacts. Since the value of k affects different parameters differently, it is suggested to adjust k according to specific application scenarios. For the current work, the value of k is set to be 3, which can be considered as a tradeoff between the performance evaluated based on SNR and SSIM.

Fig. 11.

Numerical simulation results of the blood vessel model using the VDAS algorithm with different k. (a)-(e) uses k from 1 to 5, respectively. (f) Ground truth.

We further perform simulation validations to explore the imaging of a more challenging 3-D sample with anatomically complex structures. Two different views of the ground truth of the sample are given in [ Figs. 12(a) and (b)], which are also shown below. We simulate a cylindrically focused 360-element ring transducer array to detect the PA signals in xoy planes and scan the array along the z axis with a step of 0.1 mm to guarantee the imaging accuracy [66–69]. The ring transducer array is commonly applied in many PA related previous works for 3-D computed tomography. We obtain an image in each xoy plane based on the VDAS method and finally form a 3-D image as shown in Figs. 12(c) and (d). It is clearly observed that the 3-D image faithfully reconstructs the complicated structures of the sample and prove the 3-D imaging ability of the VDAS algorithm in conjunction with a ring transducer array.

Fig. 12.

Numerical simulation 3-D imaging results of a blood vessel model using the VDAS algorithm. (a) and (b) Ground truth images in two different views. (c) and (d) 3-D images corresponding to (a) and (b).

We also perform a B-scan imaging of the vessel sample based on simulations. We uniformly arrange 321 sensors along a 16-cm line at a distance of 7 cm from the center of the sample to acquire PA signals. We also try arranging 161 sensors along an 8-cm line. The results are shown in Fig. 13. It is seen that the B-scan images’ quality is inferior to the VDAS image, which is likely due to the less PA signals detected by the linear array of the B-scan mode.

Fig. 13.

B-scan images using different number of sensors and algorithms.

For adverse effects caused by tissue heterogeneity in in-vivo translations, including variations in speed of sound, scattering, and attenuation, can distort wave propagation and degrade image quality. To address this, we will develop heterogeneity-aware reconstruction methods that integrate adaptive speed-of-sound mapping, aberration correction, and attenuation compensation [38,44]. These strategies can improve the applicability of the VDAS modality in vivo investigations to ensure accurate and robust photoacoustic imaging under realistic biological conditions. For practical in-vivo imaging of a mouse, the proposed circular-scanning system can be realized using a cylindrically focused full-ring ultrasonic transducer array, in which multiple ultrasound elements are evenly distributed along a circular aperture [70]. In practice, the ring transducer is first aligned so that its cross-sectional plane coincides with the region of interest within the mouse body. A short-pulse laser beam is then homogenized and passed through a conical (or annular) optical element to form a ring-shaped illumination around the mouse body at the imaging plane. The ring-shaped illumination band typically has a diameter matched to the target cross-section and is aligned to slightly exceed the acoustic focal plane to avoid strong surface signals. Using this geometry enables simultaneous multi-angle detection and defines the field of view as the region enclosed by the inner circular area, allowing full coverage of the target tissue without mechanical rotation [70]. To ensure stable signal acquisition, the transducer array can be integrated with a conformable hydrogel interface that provides consistent acoustic coupling and allows the array to adhere directly to the skin, avoiding coupling fluctuations during motion. This wearable configuration minimizes acoustic loss and improves imaging stability under physiological conditions. Nevertheless, residual motion artifacts caused by respiration, cardiac pulsation, or involuntary movements remain a significant challenge for in-vivo imaging. To address this issue, future work will focus on integrating motion-compensation strategies that combine both hardware stability and algorithmic correction. Techniques like multi-scale feature-based registration and deformable motion modeling can effectively align adjacent photoacoustic frames by tracking vessels and compensating for inter-frame motion. Meanwhile, physiological gating has been used in functional photoacoustic imaging to reduce motion artifacts caused by breathing and cardiac cycles [55,62]. This combination of wearable hardware and adaptive motion correction is expected to yield a stable, high-fidelity, and real-time photoacoustic imaging platform suitable for practical biomedical applications.

It is necessary to provide more detailed comparison of the DMAS and VDAS algorithms [49]. proposed the NL-p-DAS method to enhance coherent signals and suppress noise. However, it relies on a linear array channel structure, and when extended to a circular array, the geometric structure and coherence calculation need to be redefined [50]. proposed the BB-DMAS method to reduce computational load and improve signal-to-noise ratio. But after migration to a circular array, each angle requires different delay mapping and complex signal processing [51]. proposed the NL-p-SMS method, which combines spectrum processing with nonlinear weighting, and is effective for linear arrays. However, for circular arrays, its one-dimensional Fourier-based method requires complex spectral expansion and interpolation for the circular array, thereby increasing computational costs. These three methods effectively enhance contrast and suppress noise, but all assume the use of linear arrays; for annular arrays, delay, pairing, and spectrum processing need to be redesigned, which makes the implementation process more complex. While statistical weighting methods like VDAS are more directly applicable. Compared with the DMAS + CF work in [71], VDAS retains the linear superposition framework of DAS, and its computational complexity is much lower than the original DMAS. This is because the DMAS method requires pairwise products of the corrected signals with a computational cost of O(N^2) and usually needs parallel computation or hardware acceleration to achieve real-time processing. Although Jeon et al. achieved clinical real-time imaging through optimization, the method itself still has a higher computational burden than DAS or VDAS. The CA-SAFT + 2nd-D-BP method proposed by [72] achieved significant resolution and SNR improvements in the elevational and lateral directions through synthetic aperture and second-order derivative back-projection. But this method requires additional hardware and a more complex data synthesis process, resulting in higher computational and implementation costs. The advantage of our VDAS lies in its direct adaptation to circular arrays, simplicity of implementation, and only bringing a minimal time overhead as compared to the DAS. The VDAS algorithm can also significantly suppress artifacts without increasing computational resources when revealing vascular-like structures.

The VDAS method holds potential to be applied in dual-modality imaging techniques [73] and thermoacoustic imaging discipline [74–76] that shares the same principle as the PAI technique. It can also contribute to the recently developed DL-based imaging technique [77–81] and contrast-enhanced imaging modality [82–84].

To summarize, this work proposes the VDAS imaging algorithm for PAI applications using circular-array configurations. It is shown that variance of the images due to each transducer individually can be used as a good indicator to estimate if a spatial point has an optical absorbing sample or not. The VDAS technique employs the variance as the weighting factor to improve image quality without degrading the reconstruction speed, which is mostly suitable for revealing samples like blood vessels. The VDAS method is first validated by numerical simulation and then experimentally verified by a blood vessel phantom and an ex-vivo mouse ear. All the obtained results show that VDAS is more advantageous in reducing the artifacts and improving the image quality as compared to DAS. The reconstruction time and computational resource consumption of the VDAS approach is almost the same as that for the DAS algorithm. These merits of VDAS pave the way for enhancing PAI applications in various fields like biological clinical diagnosis, detection, and treatment. The idea of making use of the variance is also possible to be explored by other imaging modalities based on DAS, such as acoustic imaging, thermoacoustic imaging, and microwave imaging. Furthermore, the adoption of multimodal imaging techniques combining photoacoustic and ultrasound technologies offers novel approaches for medical imaging diagnostics [85–87].

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.