2.1.

Effects of Boundary Roughness on PAT Speckles: Simulation Studies

To analyze the effect of boundary roughness on PAT speckles, we simulate PAT of absorbing objects with boundaries having various degrees of roughness, which is quantified by the root-mean-squared (RMS) value ($\delta$) and the correlation length ($\xi$) of the boundary height.¹⁸ Our numerical phantoms are composed of a large number of absorbers, which are statistically independently and homogeneously distributed inside and outside the absorbing objects. The absorbers inside the absorbing objects have five times the average cross sections of those in the background. The boundary-profile functions of the absorbing objects are assumed to follow a stationary Gaussian stochastic process. The mean and the standard deviation of the stochastic process represent the mean boundary location and the standard deviation of the surface height (the RMS value $\delta$), respectively. The correlation function of the stochastic process is $G(r-{\stackrel{\rightharpoonup }{r}}^{\prime })=\exp (-|\stackrel{\rightharpoonup }{r}-{\stackrel{\rightharpoonup }{r}}^{\prime }{|}^2/{\xi }^2)$, where $\stackrel{\rightharpoonup }{r}$ and ${\stackrel{\rightharpoonup }{r}}^{\prime }$ represent two position vectors on the mean boundary plane. Numerically, a boundary profile function is generated by convolving a zero-mean Gaussian distribution of random numbers for the surface height with the Gaussian correlation function.¹⁸

The PAT simulation parameters are set to the same values as those of a custom-designed 512-element ring-array PAT system.¹⁹ Each ultrasonic transducer element is cylindrically focused in the elevational direction; therefore, an in-plane two-dimensional (2D) image can be reconstructed. The simulated mechanical-electrical impulse response (EIR) has a center frequency of 5 MHz (100% bandwidth). Using the Field II program,^17,20 we calculate the spatiotemporal response of every ultrasonic transducer element due to all the absorbers in the imaging region. The PAT image is then reconstructed from these spatiotemporal responses.²¹ The in-plane spatial resolution, i.e., the full-width-at-half-maximum (FWHM) of the point spread function (PSF), is $\sim 180\mu \mathrm{m}$.

2.2.

PAT Speckle Statistics: Experimental Validations

We used the same ring-array based PAT system¹⁹ to experimentally study the statistics of PAT speckles. The illumination source was a tunable pulsed laser system based on an optical parametric oscillator (OPO; Vibrant HE 315I, Opotek, Inc.). The laser pulse had a repetition rate of 10 Hz and a pulse width of 5 ns. After collimation, the laser beam was homogenized through a ground glass before it reached the top surface of the tissue phantom. The absorbing objects were made of gelatin (10% gelatin by weight) mixed with graphite particles in various concentrations.

3.1.

Effects of Boundary Roughness on PAT Speckles

The simulation results are tabulated by the RMS height $\delta$ and the correlation length $\xi$ in Fig. 1. As the RMS height $\delta$ increases, the boundary height fluctuates more; as the correlation length $\xi$ decreases, the position of each point on the boundary becomes less correlated with that of its neighbors; if $\delta =0$ or $\xi \to \infty$, the boundary becomes perfectly smooth [Fig. 1(a)]. Each cell of the table shows the true absorber distribution at the top and the corresponding reconstructed PAT image at the bottom. Segments of the boundaries of the absorbing objects within the dashed frames are extracted and shown as solid curves with a horizontal magnification of two times. The reconstructed PAT images show observable boundary buildups as well as interior speckles. In Fig. 1(a) ($\xi \to \infty$), and Fig. 1(b), 1(e), and 1(h) ($\xi =360\mu \mathrm{m}$, which equals twice the in-plane resolution), the reconstructed and true boundaries agree well (correlation coefficient $\chi =0.92-0.95$) while the visibility of interior speckles ($V$) decreases slightly with increasing $\delta$. In Fig. 1(c), 1(f), and 1(i) ($\xi =180\mu \mathrm{m}$, which equals the in-plane resolution), some of the features of the true boundaries cannot be recovered by the reconstructed boundaries ($\chi =0.73-0.77$). In Fig. 1(d), 1(g), and 1(j) ($\xi =30\mu \mathrm{m}$, which equals $1/6$ of the in-plane resolution), the height fluctuations of the reconstructed and true boundaries become less correlated ($\chi =0.26-0.27$). Moreover, the amplitude of the reconstructed boundary vanishes with increasing $\delta$, increasing the visibility of interior speckles. In all the cases, however, the average powers of the interior speckles without normalization to the boundary signals are equal. Therefore, the variations in speckle visibilities are due to the changes in the amplitudes of the reconstructed boundaries.

Fig. 1

Simulated PAT of absorbing targets with boundaries having various degrees of roughness at a spatial resolution of 180 μm. The boundary roughness is quantified by the RMS value ($\delta$) and the correlation length ($\xi$) of the boundary height. (a) PAT of absorbing targets with smooth boundaries. (b) to (j) PAT of absorbing targets with rough boundaries. In each cell, the simulated absorber distribution is plotted in the top row, and the corresponding PAT image is shown in the bottom row. Segments of the boundaries from both plots are plotted as surface height versus lateral position (solid curves). The correlation coefficient ($\chi$) between the true and the reconstructed boundaries as well as the speckle visibility ($V$) is computed. The color bar defined in (a) is employed for all sub-figures.

In Fig. 2(a) and 2(b), the correlation coefficient ($\chi$) between the reconstructed and the true boundary profiles is quantified as functions of the correlation length $\xi$and the RMS height $\delta$. We found that $\chi$ depends solely on $\xi$ because the correlation coefficient between $\chi$ and $\delta$ is 0.08. If $\xi >180\mu \mathrm{m}$ (the in-plane resolution), all the boundary features can be resolved, and thus the reconstructed boundaries agree with the true boundaries. Conversely, if $\xi <180\mu \mathrm{m}$ (the in-plane resolution), features are too fine to be resolved. Therefore, even when the boundary fluctuations are small ($\delta =30\mu \mathrm{m}$), the reconstructed boundaries do not agree with the true boundaries. In Fig. 2(c) and 2(d), the speckle visibility ($V$) is plotted as functions of the correlation length $\xi$ and the RMS height $\delta$. Because the interior speckle amplitudes follow Gaussian distribution, we define boundary features as the reconstructed image regions that have magnitudes more than three time of the standard deviation of the interior speckle amplitudes. We found that $V$ depends on both $\xi$ and $\delta$. In Fig. 2(c), $V$ decreases as $\xi$ increases when $\xi <180\mu \mathrm{m}$ (the in-plane resolution). If the correlation lengths are smaller than the in-plane resolution, shorter correlation lengths usually introduce more randomized phase-delay variations, especially when the RMS heights are larger than the in-plane resolution so that effective roughness presents. An extreme example is that if $\xi \ll 180\mu \mathrm{m}$and $\delta \gg 180\mu \mathrm{m}$ the reconstructed boundaries become fully developed speckles and the speckle visibility $V\to 1$. As $\xi$ increases further, the speckle visibilities $V$ at various $\delta$ gradually converge to the value of $V$ for the smooth boundary, because $\xi \to \infty$ indicates smooth boundaries regardless of $\delta$. In Fig. 2(d), if $\xi <180\mu \mathrm{m}$, $V$ increases as $\delta$ increases, which introduces more randomized phase-delay variations. If $\xi \ge 180\mu \mathrm{m}$, however, some reconstructed boundary segments [e.g., the paired arrows in Fig. 1(f)] have higher signal strengths than the reconstructed smooth flat boundaries. The geometric shapes of these features match the ring-shaped ultrasonic detection aperture, and therefore the phase variations of the PA partial waves from the surface absorbers are smaller than those from the smooth flat boundaries. As a result, the average boundary strengths of these boundaries are slightly higher than those of smooth boundaries, and thus the speckle visibilities are slightly lower.

Fig. 2

Effects of the RMS height $\delta$ and the correlation length $\xi$ on PA image boundaries. Effects of (a) the correlation length $\xi$ and (b) the RMS height $\delta$ on the correlation coefficient $\chi$ between the real and the reconstructed boundaries. Effects of (c) the correlation $\xi$ length and (d) the RMS height $\delta$ on the speckle visibility $V$.

The analysis above indicates that PAT speckles may appear in rare cases. For example, we may observe speckles when imaging melanoma, if the melanoma has rough boundaries whose correlation length is much smaller than and the RMS value is much greater than the imaging resolution. In contrast, when imaging blood vessels with most of the PAT systems (center frequencies ranging from 1 to 50 MHz), we observe only the boundary signals. Here, the RBCs are treated as individual absorbers. The RBC density in the blood of an average adult is $\sim 5\times {10}^6/{\mathrm{mm}}^3$. The blood vessel walls are considered smooth on the scale of the acoustic wavelengths. According to our analysis, the interior speckles are suppressed by the prominent boundary signals.

3.2.

PAT Speckle Statistics

Photographs of the phantoms with high, medium, and low graphite particle concentrations and with smooth boundaries are shown in Fig. 3(a)–3(c). The corresponding PAT images are shown in Fig. 3(e)–3(g). As predicted in our previous study, we observed strong boundary buildups, which suppress the internal speckle patterns [Fig. 3(i)]. Also, the boundary features are most prominent at the highest particle concentration [Fig. 3(e)], and interior speckles become noticeable at the medium particle concentration [Fig. 3(f)] and become apparent at the lowest particle concentration [Fig. 3(g)]. The phantom with low particle concentration was imaged 10 times and 100 times, and the averaged reconstructed interior textures are shown in Fig. 3(j) and 3(k). Because both the phantom and the imaging system are stationary in each experiment, such averaging does not diminish the speckles but does reduce random noises. The similarity between the two interior images, with a correlation coefficient of 0.996, confirms that the interior texture is due to speckles rather than random noises. For comparison, a phantom with medium particle concentration and rough boundaries [Fig. 3(d)] was studied, where both the correlation length $\xi$ and the RMS height $\delta$ of the boundary profile are $\sim 60\mu \mathrm{m}$. In the PAT images of the phantoms, the rough boundaries [Fig. 3(h)] produced 2.8 times weaker boundary amplitudes than the smooth boundaries at the same particle concentration [Fig. 3(f)].

Fig. 3

Phantom experiments. Photographs of gelatin phantoms with smooth boundaries and with (a) high, (b) medium, and (c) low graphite particle concentrations. (d) Photograph of a gelatin phantom with rough boundaries and with medium graphite particle concentration. (e) to (h) Corresponding PAT images obtained with a ring-array PAT system. (i) to (k) Interior PAT speckle patterns.

We quantified the first order statistic of the PA speckles by plotting the histogram of the interior speckle amplitude (without the envelope detection) [Fig. 4(a)]. Since the coherent interference of ultrasonic waves can be described as a random walk process, the amplitude of the PA speckles follows a Gaussian distribution. The mean of the speckle amplitude is zero, because the PSF does not contain DC. The standard deviation of the Gaussian distribution ($\sigma$) which represents the square root of the average speckle power, is proportional to the product of the average particle absorption cross section $\overline{C}$ and the square root of the particle concentration ($\sqrt{\rho }$). In Fig. 4(b), we show that $\sigma$ is proportional to $\sqrt{\rho }$, while the boundary magnitude is proportional to $\rho$. As a consequence, the speckle visibility ($V$) is inversely proportional to the square root of the particle density $\sqrt{\rho }$ (15) for smooth boundary targets [Fig. 4(c)]. The second-order statistic of the speckles is shown in Fig. 4(d). In the classic speckle theory, the autocorrelation of the fully developed speckle pattern carries only the information of the system PSF rather than that of the target texture. The autocorrelation of the system PSF is shown in Fig. 4(e), which agrees with the autocorrelation of the speckle patterns [Fig. 4(f)].

Fig. 4

Experimental PAT speckle statistics. (a) First-order speckle statistics. (b) Dependence of the square root of the average powers of both the boundaries (red solid line) and the interiors (blue solid line) in the PAT images on the absorber density. (c) Dependence of the PA speckle visibility on the absorber density. (d) Second-order speckle statistics. (e) Auto-correlation of the system PSF. (f) One-dimensional radial plots of the auto-correlation of the speckles and the auto-correlation of the system PSF.