1.

Introduction

Photoacoustic (PA) imaging can provide images proportional to the distribution of the absorbed optical energy density. It has already demonstrated its ability to detect breast cancer, skin related diseases, and brain tumors in small-animal studies.^1–5 Generation of photoacoustic signals can be classified according to two excitation modes: continuous-wave modulation and pulsed mode. The shape of the time-resolved photoacoustic signal depends on the physical characteristics of the irradiated body and the properties of the optical exciting pulse.^6,7

It is well known that the temporal amplitude and profile of photoacoustic signals are spatial mappings of the absorbed optical energy, which is the product of the optical absorption coefficient ${\mu }_a$ and the fluence $F\overrightarrow{(r)}$. The local laser fluence $F\overrightarrow{(r)}$ at the absorber is determined by a combination of optical absorption and optical scattering of background medium. Considerable attention has been devoted to methods for extracting the tissue optical absorption coefficient ${\mu }_a$ from the detected photoacoustic signals. Currently, these methods may be divided into two categories. One is based on the temporal profiles of photoacoustic signal combined with a light transport model.^8–16 The other involves solving the photoacoustic wave equation and frequency-domain coupled with photon diffusion equation with iterative algorithms.^17–20

Spectra of photoacoustic signals are also used to quantify optical absorption coefficients.²¹ This method is calibration-free since it deals only with the relative change in various acoustic frequencies. However, this method uses two optical wavelengths and ignores the wavelength-dependent fluence.

In this paper, we present a model for determining the optical absorption coefficient of a heterogeneous absorber using photoacoustic imaging. In this model, the ratio of the amplitude of the peak signal from the top boundary to that from the bottom boundary (TBR) of the absorbing sample is an exponential function of the absorption coefficient. The theoretical component of this method can demonstrate that the optical absorption coefficient could be recovered using TBR. This method is self-calibrating, since it depends only on the relative change of the peak signals from both the top and bottom boundaries of the absorbing target. In addition, we performed experiments to validate this model based on wavelet transform. Finally, the method is applied to determine the absorption coefficient of an absorber with different concentrations of ink.

2.

Theory and Methods

2.1.

Photoacoustic Signal Generation and Detection

The photoacoustic measurement geometry used in this study is depicted in Fig. 1. Photoacoustic signals are generated in the phantom experiment by irradiating the surface with nanosecond pulses of NIR laser light. The absorption of the optical energy leads to near-impulsive heating of the irradiated volume followed by rapid thermoelastic expansion. The photoacoustic pressure is governed by the following wave equation:

Eq. (1)

$${\nabla }^{2}p(\overrightarrow{r},t)-\frac{1}{c^2}\frac{{\partial }^{2}p(\overrightarrow{r},t)}{\partial t^2}=\frac{-\beta }{C_p}\frac{\partial \mathrm{\Phi }(\overrightarrow{r},t)}{\partial t},$$

where $p$ is the wave pressure, $\overrightarrow{r}=(x,y,z)$ denotes the position, $t$ is the time, $c$ is the speed of the acoustic wave in the medium, $\beta$ is the thermal expansion coefficient, $C_p$ is the constant pressure specific heat capacity, and $\mathrm{\Phi }$ is the absorbed optical energy density. When the laser pulse is sufficiently short, the optical energy density $\mathrm{\Phi }$ can be modeled as a $\delta$ function in time,

Eq. (2)

$$\mathrm{\Phi }(\overrightarrow{r},t)=\mathrm{\Phi }(\overrightarrow{r})\delta (t).$$

The optical energy density $\mathrm{\Phi }(\overrightarrow{r})$ is a product of optical absorption coefficient ${\mu }_a(\overrightarrow{r})$ and optical fluence $F(\overrightarrow{r})$,

Eq. (3)

$$\mathrm{\Phi }(\overrightarrow{r})={\mu }_a(\overrightarrow{r})F(\overrightarrow{r}).$$

The initial acoustic pressure $p(\overrightarrow{r})$ is proportional to the absorbed energy density $\mathrm{\Phi }(\overrightarrow{r})$,

Eq. (4)

$$p(\overrightarrow{r})=K\mathrm{\Phi }(\overrightarrow{r}),$$

where $K$ is a proportionality coefficient related to the ultrasonic parameters, the Grüneisen parameter, and the efficiency of the conversion of heat to pressure. Since the photoacoustic signals usually involve a wide range of frequencies, $p(\overrightarrow{r})$ could be expressed as the sum of signals at different acoustic frequencies $f:p(\overrightarrow{r})=\sum _{}^{}p(\overrightarrow{r},f)$.

Fig. 1

Cross-sectional schematic of photoacoustic imaging with focused transducer. The phantom is made of Agar gel with a cylindrical target (length l and diameter d) with a distance of $z_c$ from the surface of the phantom. The scanning of the exciting laser beam is from left to right with several typical A-lines labeled as lines k, l, m, n, o, and p. Lead-pencil absorbers are placed at C, D, E, F, and G.

The measured photoacoustic signal $S(\overrightarrow{r})$ should be the convolution of $p(\overrightarrow{r})$ with the detector geometry,²² given by $S(\overrightarrow{r})=\sum _{}^{}S(\overrightarrow{r},f)$. The two common computational challenges for photoacoustic imaging are directional sensitivity of the detector and acoustic attenuation during the geometric spreading of the propagating wavefronts.²³ To circumvent the two challenges, we used a focusing transducer featuring extended focal zone, which is much larger than the dimensions of the signal source. We also used the following approximation,²⁴

Eq. (5)

$$p(\overrightarrow{r})=\sum _{}^{}p(\overrightarrow{r},f)=\sum _{}^{}S(\overrightarrow{r},f)\exp ({\alpha }_f{D}_{\mathrm{st}}),$$

for correcting initial acoustic amplitude $p(\overrightarrow{r})$ from measured signal $S(\overrightarrow{r})$, where ${\alpha }_f$ is the attenuation coefficient dependent on the frequency, and $D_{\mathrm{st}}$ is the distance between the photoacoustic signal source and detector. In Eq. (5), the attenuation coefficient ${\alpha }_f$ satisfies a power law dependence on frequency: ${\alpha }_f={\alpha }_{0f}{|f|}^n$, where $n$ is a real positive constant, $a_{0f}$ is the attenuation constant, and $f$ is the acoustic frequency. For many biological tissues, $n\approx 1$, and the average attenuation coefficient is $a_{0f}\approx 0.5\mathrm{dB}·{\mathrm{MHz}}^{-1}·{\mathrm{cm}}^{-1}$.²⁴

3.

Results

Figure 2(a) and 2(b) shows the typical A-line photoacoustic signals and the corresponding wavelet transform. Figure 2(a) shows the bipolar nature of the temporal photoacoustic signals from the top surface of the target. The signals from the bottom surface appear to be monopolar simply because they are weak and are buried in the tails of top surface signals. Result in Fig. 2(b) indicates that the wavelet coefficient below scale 16 is approximately homogeneous due to noise, and the wavelet coefficient of the photoacoustic signal increases from scales 16 to 80. The Fourier frequencies corresponding with the scales from 16 to 80 are from 6.25 MHz to 1 MHz. The wide-bandwidth-focused transducer is sensitive to the frequency in this range. In order to improve the signal-to-noise ratio (SNR), the superposition of wavelet coefficients from scales 16 and 80 is used, as shown in Fig. 2(c). The maximal values in Fig. 2(c) indicate the locations of the boundaries of the target (A and B). The size of the absorber is calculated by multiplying the time interval between A and B with the speed of sound in the medium.

Fig. 2

PA signal and wavelet transform of a typical photoacoustic A-line (line k in Fig. 1): (a) raw data; (b) the corresponding wavelet coefficients; (c) superposition of wavelet coefficients from scales 16 to 80. A and B denote the positions of the top and the bottom of the ink block, respectively.

Since the acoustic attenuation coefficient depends on the frequency, the measured peak photoacoustic signals at both boundaries are chosen from the wavelet transform, as shown in Fig. 3(a). According to Eq. (5), the data in Fig. 3(a) are corrected, and the results are shown in Fig. 3(b). Results in Fig. 3(c) demonstrate that the frequency dependence of the profile and amplitude of the ratio of wavelet coefficient of the top boundary to that of the bottom boundary is different after correction.

Fig. 3

Wavelet coefficients versus acoustic frequencies at top and bottom boundaries of the sample before correction (a), after correction (b), and the ratio of wavelet coefficient of the top boundary to that of the bottom boundary (c).

The peak values of wavelet coefficient of pencil leads (C, D, E, F, and G), as shown in Fig. 1, were used to extract the absorption coefficient based on Eq. (7). Figure 4 shows the plot of the peak values of wavelet coefficients after the correction using Eq. (5) versus the depth of the absorber. For a target with 0.1% ink, the $\mathrm{\Delta }{\mu }_a$ was determined as $2.7\pm 0.2{\mathrm{cm}}^{-1}$ based on Eq. (7). In comparison, using the restored amplitudes at the top and bottom boundaries (A and B as shown in Fig. 1), we obtained an absorption coefficient of $2.9\pm 0.3{\mathrm{cm}}^{-1}$ based on Eq. (11). These results validated our proposed method using TBR of the target.

Fig. 4

The peak values of wavelet coefficient of different pencil leads as a function of depths inside the target, after signal correction using Eq. (5). The fitting curve is used to determine the absorption coefficient of the target using Eq. (7).

With the corrections for the attenuation of the peak amplitudes at the top and bottom boundaries using Eq. (5), we obtained absorption coefficients of samples based on Eq. (11). The absorption coefficients of six samples with different concentrations (0.1 to 2%) of ink are given in Fig. 5. The absorption coefficients obtained using our new method ranged from 2.9 to $5.6{\mathrm{cm}}^{-1}$, linearly related to the concentration of the ink from 0.1 to 1%. However, at the highest concentration (2%), the absorption coefficient is noticeably lower below the linear fit, which may be explained by nonlinear optical absorption coefficient.²⁸

Fig. 5

Absorption coefficients of absorbers with different concentrations of ink. Data represent $\text{mean}\pm \text{standard}\text{deviation}$ ($n=10$).

4.

Discussion

Since biological tissues are strong scattering media, diffusion approximation of local fluence rate should be used for characterizing the optical properties. In the diffusive region, the light fluence along the axial direction is reduced due to absorption and scattering.¹⁵ The fluence can be expressed as $F_0=C_0\exp (-{\mu }_{\mathrm{eff}}z)$, where $C_0$ is a constant, and ${\mu }_{eff}=3{\mu }_a[{\mu }_a+{\mu }_s(1-g)]$. Since our method for determining absorption coefficient of target using the TBR depends on the incident fluence on the surface, it can be used for strong scattering background medium. Moreover, the scattering coefficient and the anisotropy of highly scattering medium can be determined by optical coherence tomography (OCT).²⁹ Thus we could combine the photoacoustic imaging and OCT for simultaneous determination of absorption coefficient of heterogeneous absorbers and the optical scattering properties of the background.

Our method is self-calibrating, since it deals only with the relative change in the signals from the top and bottom boundaries of the absorbing target. In addition, this method uses different wavelengths of laser light to study the wavelength-dependent absorption property of targets.

Tissue absorption coefficient is an important factor in imaging and therapy based on photothermal interactions. Different methods have been used to determine the absorption coefficients of different target tissues. Yet a reliable accurate method is still not available. This study proposes a novel method to noninvasively determine absorption coefficient using photoacoustic method. This method is particularly useful in determining the outcomes of selective photothermal interaction for cancer treatment with absorption enhancement of target tumors using either chemical dye^30–32 or nanoparticles.³³ Using accurate absorption coefficients, simulations of photothermal effects^34,35 can yield more reliable predictions for laser irradiation of biological tissues.

5.

Conclusions

In summary, we demonstrated the feasibility of a new method using TBR of absorbing target to quantify the optical absorption coefficient with a focusing photoacoustic imaging technique based on the wavelet transform. The wavelet transform is used to locate the boundaries of the target and to extract photoacoustic signals at the specific ultrasound frequency. The theoretical analysis reveals that the TBR is an exponential function of the absorption coefficient of a heterogeneous absorber and is independent of absolute optical fluence, the ultrasound parameters, and the Grüneisen parameter. Our method was validated experimentally using an ink-enhanced absorbing target buried in a low-scattering medium. This method could be beneficial to absorption-based imaging and therapy by providing an accurate absorption coefficient.