2.1.

Imaging System

The PAM system used in this study is shown in Fig. 1(a). A tunable pulsed optical parametric oscillator (OPO) laser (Vibrant 355 II HE, Opotek, Carlsbad, California) with a repetition rate of 10 Hz and a pulse width of 5 ns was used to illuminate the imaging targets through a custom-made light delivery system. A spherically focused ultrasonic transducer (V315, Olympus, Tokyo, Japan) was used to sense the excited photoacoustic pressures from the targets. The electrical signals from the transducer were then amplified and acquired by a data acquisition card at a speed of $50\mathrm{MS}/\mathrm{s}$ for digitization. Further details of the system can be found in our previous publications,¹⁴ and thus are not repeated here. Figure 1(b) illustrates the basic principles of virtual-detector PAM. In short, the focal point of the ultrasonic transducer was treated as a virtual detector, with its divergence angle determined primarily by the NA of the transducer. Through mechanical scanning of this virtual detector, it resembles the detection of a linear ultrasonic array and thus can recover the information of the out-of-focus regions through reconstruction. In this study, the ultrasonic transducer has a center frequency of 10 MHz, a diameter of 19 mm, and a focal length of $\sim 25\mathrm{mm}$. As a result, the full width half maximum (FWHM) of the focal point of the transducer is $\sim 250\mu \text{m}$, with a focal zone of $\sim 2.5\mathrm{mm}$, and a divergence angle of $\sim 20\mathrm{deg}$ [Fig. 1(b)].

Fig. 1

(a) Overall architecture of the photoacoustic microscopy (PAM) system and (b) illustration of the concept of virtual-detector PAM. PD, photodiode; FWHM, full width at half maximum.

2.2.

Reconstruction Model

In general, during photoacoustic wave generation, the acoustic pressure $p(\mathbf{r},t)$ at position $\mathbf{r}$ and time $t$ satisfies the wave equation:

On the other hand, CS is known to recover sparse or compressible signals with under-sampled measurements. To apply CS-based reconstruction, it is required that the signals shall possess sparsity in certain domains. Fortunately, most medical images are sparse in certain domains by finding an appropriate sparse transform $\psi :\mathbf{x}=\psi \theta$, where $\theta$ is the original image and $\mathbf{x}$ is the transformed one.

In PAT, if the measurement data from the ultrasonic transducers is $\mathbf{y}$, the measurement matrix corresponding to the imaging system is $\mathbf{K}$, then we have $\mathbf{y}=\mathbf{K}\theta ={\mathbf{K}\mathrm{\Psi }}^{-1}\mathbf{x}$. When CS is incorporated into the photoacoustic imaging process, the reconstruction with sparse transform $\psi$ can be implemented by solving the following constrained optimization problem:

It is noteworthy that, the incoherence between the measurement matrix and the sparse transform is another important condition to guarantee the successful application of CS. In previous works, this condition has been proven to be satisfied in photoacoustic reconstruction.^16,17

To recover the photoacoustic images of the physically out-of-focus regions in PAM, a CS-based reconstruction model according to Eq. (3) was developed in our work

Equation (4) can also be written as

In Eq. (4), $\mathrm{\Psi }$ is the sparse transform—in this study, a four-level Daubechies wavelet transform is used. In addition to the item of $l_1$ norm in a sparse domain, the total variation penalty of the signals is also incorporated into the objective function $F$ to improve the reconstruction accuracy. Further, $\alpha$ and $\beta$ (0.06 and 0.1, respectively, in this study) are the regularization parameters determining the trade-off between the data consistency and the sparsity, which should be determined appropriately. Overweighed values could result in distortion of the reconstruction. On the contrary, if their proportions to the objective function were too small, their support would become ineffective. In our experiments, they were empirically determined by trying different combinations and choosing the optimal ones. Finally, ${\mathbf{K}}_{\text{angle}}$ is the measurement matrix incorporating the effect of the divergence angle, which is defined as

3.1.

Phantom Results

To validate the effectiveness of CS-PAM, phantom experiments were performed. To fabricate the phantom, three human hairs with diameters of $\sim 50\mu \text{m}$ were fixed in parallel on the surface of a hollow plastic holder, with a $\sim 4.2\mathrm{mm}$ separation between hair-1 and hair-2, and a $\sim 0.9\mathrm{mm}$ separation between hair-2 and hair-3. In a series of experiments, the height of the ultrasonic transducer was varied (while the optical illumination on the hairs was unchanged), so that the phantom was imaged under both in-focus and out-of-focus circumstances. For the out-of-focus cases, the phantom was placed at 2.5, 5.0, and 7.5 mm away from the focal point, respectively. To complete a B-scan for each case, the imaging head was mechanically scanned along $x$, with a step size of 50 μm and a total scanning range of 6.4 mm.

Figure 2(a) shows the directly acquired PAM images under both in-focus and out-of-focus conditions. As expected, the photoacoustic images get blurred more and more when the hairs are away from the ultrasonic focal point. However, upon performing image reconstruction with either back-projection or compressed-sensing strategy, using the scanning focal point as virtual detectors, the spatial resolution and contrast of the photoacoustic images of the out-of-focus hairs are clearly improved [Figs. 2(b) and 2(c)]. Further, compared with that using BP-PAM, the separation between hair-2 and hair-3 is resolved even better using CS-PAM. Finally, compared with the results from BP-PAM, the reconstructed signal (optical absorption) amplitudes and distribution using CS-PAM also agrees better with that of the corresponding in-focus hair images (the assumed gold standard). To further demonstrate this improvement, the maximum amplitude projections (MAP, along $z$-axis) of the B-scan images are plotted in Fig. 3 for comparison. These plots confirm that, overall, CS-PAM has provided better contrast-to-noise ratio (CNR) and spatial resolution over BP-PAM. In one particular case [Fig. 3(d)], in which the hairs are seriously out of focus, CS-PAM has offered $\sim 4$-fold improvement in CNR and $\sim 10\%$ improvement in spatial resolution, compared with that of BP-PAM.

Fig. 2

Photoacoustic images of the hair phantom. (a) Directly acquired photoacoustic B-scan images of in-focus and out-of-focus hairs; (b) and (c) B-scan images reconstructed with BP-PAM and CS-PAM, respectively.

Fig. 3

Plots of the maximum amplitude projections (MAP) of the hair phantom’s B-scans. The three peaks from left to right represent the MAP plots of hairs 1, 2, and 3, respectively. (a) Plot (from directly acquired PAM image) with the phantom at the focal point; (b)–(d) Plots with the phantom at $\sim 2.5$, 5, and 7.5 mm away from the focal point, respectively. The arrows indicate the FWHMs of the peaks; CNR, contrast-to-noise ratio.

3.2.

In Vivo Results

To validate the performance of CS-PAM in vivo, the back of an anesthetized nude mouse weighing $\sim 25\mathrm{g}$ was imaged. The transducer height was varied in experiments, so that two photoacoustic images were acquired, first when the skin surface of the mouse back was in focus, and then when it was $\sim 4\text{-}\mathrm{mm}$ out of focus. The laser (at 570 nm) fluence used in this study was $\sim 2\mathrm{mJ}/{\mathrm{cm}}^2/\text{pulse}$ on the skin surface, well below the $20\text{-}\mathrm{mJ}/{\mathrm{cm}}^2$ American National Standards Institute laser safety limit. All animal experiments described here were carried out in compliance with the approved protocols of Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences.

The image acquired when the mouse back was in focus is assumed as the gold standard [Fig. 4(a)], while the directly acquired PAM image when the mouse back was out of focus is shown as a control [Fig. 4(b)]. Figures 4(c) and 4(d) show the corresponding image reconstructed with BP-PAM and CS-PAM, respectively. Note that the images in Figs. 4(a)–4(d) are shown as the MAP—the maximum photoacoustic amplitudes projected along the depth direction toward the skin surface. To further compare the reconstructed images, representative B-scan images corresponding to the cross sections as indicated by dash lines 1 and 2 in Fig. 4(a) are also shown [Figs. 4(a1)–(d1) and (a2)–(d2)]. In all in vivo experiments, the comparison between BP-PAM and CS-PAM reconstructions was performed using identical sparse sampling data, acquired with a scanning step size of 50 μm. Through in vivo imaging, we can see that: (1) similar to that in the phantom experiments, the directly acquired PAM images of the out-of-focus targets (mouse-back vessels) were both blurred and of degraded CNR [Fig. 4(b)]; (2) in contrast to that in the phantom experiments, BP-PAM did not provide obvious improvement in image quality for the out-of-focus targets in vivo; and (3) CS-PAM essentially maintained its great performance in improving the image quality in the in vivo case. Presumably, the deteriorated performance of BP-PAM in vivo was due to the small divergence angle of the virtual detector, together with the relatively low CNR (compared with that in the phantom experiments). In CS-PAM, however, the use of the CS-based reconstruction model has partially compensated the effect of the small divergence angle.¹³

Fig. 4

In vivo PAM of the back of a mouse. (a) Directly acquired PAM image with the skin surface in-focus; (b) directly acquired PAM image with the skin surface $\sim 4\mathrm{mm}$ out-of-focus; (c) and (d) Recovered images when the skin surface is $\sim 4\mathrm{mm}$ out-of-focus, using BP-PAM and CS-PAM, respectively. (a1)–(d1) and (a2)–(d2) B-scans corresponding to the cross sections along line 1 and line 2, respectively, in (a).