## 1.

## Introduction

Photoacoustic (PA) imaging is a novel hybrid imaging technology that is able to image in great depth the optical energy deposition with high spatial resolution in biological tissue.^{1, 2, 3} Since optical energy deposition reflects the physiologically specific optical absorption contrast, PA imaging has been widely used to study various functional parameters of living biological tissue, especially hemoglobin oxygen saturation^{4, 5, 6, 7}
$\left(\mathrm{s}{\mathrm{O}}_{2}\right)$
.

To measure
$\mathrm{s}{\mathrm{O}}_{2}$
, PA imaging depends on the difference in the molar extinction coefficient spectra of the two forms of hemoglobin, oxygenated hemoglobin
$\left(\mathrm{Hb}{\mathrm{O}}_{2}\right)$
and deoxygenated hemoglobin (HbR). When a blood vessel is irradiated by laser pulses with identical pulse energy at different optical wavelengths, PA signals are generated with different amplitudes due to the optical-wavelength-dependent molar extinction coefficients. Therefore, PA imaging performs spectral measurement and treats HbR and
$\mathrm{Hb}{\mathrm{O}}_{2}$
as the dominant optical absorbers at each wavelength
$\left({\lambda}_{i}\right)$
to calculate their relative concentrations ([HbR] and
$\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]$
, respectively) based on a least squares fit.^{5} The
$\mathrm{s}{\mathrm{O}}_{2}$
is further calculated by
$\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]\u2215(\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]+\left[\mathrm{HbR}\right])$
.

The prerequisite for the aforementioned spectral measurement and inverse calculation (least-squares fit) is that the detected PA signals from blood are linearly related with the optical absorption coefficients of hemoglobin at
${\lambda}_{i}$
, which denotes as the absorption-dependent PA detection. However, when the center frequency of the ultrasonic detector is low with respect to the corresponding optical absorption coefficient,^{8} the detected PA signal will not linearly reflect the optical absorption change in blood. In an extreme case, the detected PA signal may rely on only the incident optical energy regardless of the optical absorption coefficient of blood. This phenomenon is referred to as the saturation effect.^{9} When saturation effect occurs, the inverse calculation of the [Hb] and
$\left[\mathrm{Hb}{\mathrm{O}}_{2}\right]$
becomes inaccurate.

In the *in vivo* PA imaging of
$\mathrm{s}{\mathrm{O}}_{2}$
, the required maximum imaging depth for different applications varies from a few hundred micrometers to several millimeters. In general, increased maximum imaging depth requires the ultrasonic detector to have a lower center frequency because of the wavelength-dependent ultrasonic attenuation in tissue.^{10} Due to the saturation effect, however, a threshold of the ultrasonic center frequency could exist. Using an ultrasonic detector with a center frequency below such a threshold will lead to saturation in the PA signal detection and present an incorrect inverse calculation of
$\mathrm{s}{\mathrm{O}}_{2}$
with an error greater than a preset value.

Here, we report both the theoretical and experimental studies of the saturation effect in functional PA imaging. In our experiments, the change of the optical absorption coefficient was achieved by varying the concentration of the optical absorbing material rather than the optical wavelength. The saturation effect discussed here is limited to the nonlinear dependency of the measured PA amplitude with the optical energy deposition under the adiabatic heating condition induced by short laser pulses. Other nonlinear effects caused by, for example, heat diffusion^{11} are excluded.

## 2.

## Methods and Materials

We used a semi-infinite homogeneous optical absorbing object, as shown in Fig. 1
, to model the optical energy deposition as well as the PA wave generation and detection. This model is valid as an approximation for blood vessels when the optical penetration depth is less than one-tenth of the vessel diameter. Within the spectral range of
$520\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, the optical absorption coefficient of the whole blood (considering the
$\mathrm{s}{\mathrm{O}}_{2}$
to be 95%) is^{12}
$20\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}296\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$
, corresponding to a penetration depth of
$34\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}491\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. In our model, the laser illumination is along the
$z$
axis and the ultrasonic detector is placed directly above the illuminated area.

The depth-resolved optical fluence distribution $F\left(z\right)$ inside the object is described as

where ${F}_{0}$ (in joules per squared centimeter) is the optical fluence on the surface of the object, and ${\mu}_{a}$ (in inverse centimeters) is the optical absorption coefficient of the medium. Hence, the induced depth-resolved PA waves can be described as## Eq. 2

$$\mathrm{PA}(z,{\mu}_{a})={F}_{0}{k}_{1}{\mu}_{a}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-{\mu}_{a}z),$$^{13}

As the peak amplitude of the detected PA signals is usually taken as a measure of the optical energy deposition in a vessel, here we estimate the peak amplitude of the detected PA signal for Eq. 2. If an ultrasonic detector has an infinitely high axial resolution, the detected peak PA amplitude will be
$\alpha {F}_{0}{k}_{1}{\mu}_{a}$
, where
$\alpha $
is the coefficient of the ultrasonic detection system that converts pressure to voltage. For an ultrasonic detection system that has only finite axial resolution, the detected PA signal is an integration of the PA waves within a specific axial pixel. For a given wideband ultrasonic detector at a cut-off wavelength of
${\lambda}_{c}$
, the axial resolution in PA imaging can be approximated^{14} as
${k}_{2}{\lambda}_{c}$
, where
${k}_{2}$
is a dimensionless coefficient between 0.8 to 0.92. Thus, the detected PA peak amplitude can be approximated as

## Eq. 3

$${A}_{\mathrm{PA}}\left({\mu}_{a}\right)=\alpha {\int}_{0}^{{k}_{2}{\lambda}_{c}}{F}_{0}{k}_{1}{\mu}_{a}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-{\mu}_{a}z)\mathrm{d}z=\alpha {F}_{0}{k}_{1}[1-\mathrm{exp}(-{\mu}_{a}{k}_{2}{\lambda}_{c})].$$The experiments were carried out by measuring PA peak-to-peak amplitudes detected from an ink (Fiesta Red, Private Reserve Ink, The Ink Flow, San Antonio, Texas
) filled plastic tube (Tygon^{®} S-54-HL Microbore Tubing 56464, United States Plastic Corp., Lima, Ohio
). The measured optical absorption spectrum of the ink is shown in Fig. 2
. The optical wavelength used for the PA measurement was
$523\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
. The corresponding optical absorption coefficient was
$1028\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$
. The tube has an inner diameter of
$0.5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
and an outer diameter of
$1.5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
.

A schematic diagram of the experimental setup is shown in Fig. 3 . Laser light at $523\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ was generated by a Nd:YLF laser (INNOSLAB; IS8II-E; EdgeWave, Germany ). The pulse width was $6\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$ , the pulse repetition was set to $100\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ , and the laser beam diameter was $10\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The light coming out of the laser first passed through a beam sampler, then a right-angle prism (PS908; Thorlabs, Newton, New Jersey ), and finally a neutral density filter (NE10A; optical density: 1.0; Thorlabs, Newton, New Jersey ) before it illuminated the tube. The light was attenuated because of better pulse energy stability when the laser system is working at a higher driving current.

The PA waves were detected by a high-frequency, unfocused ultrasonic transducer and amplified by a wideband, lownoise preamplifier (ZFL500LN+, Mini-Circuits, Brooklyn, New York ) for $28\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ . The amplified PA signals were then digitized and stored by an oscilloscope (TDS5034B, Tektronix, Beaverton, Oregon ). A fast photodiode (DET10A, Thorlabs, Newton, New Jersey ) detected the sampled laser pulses to generate triggers for data acquisition. Each PA signal was averaged 600 times to minimize the influence from the pulse energy instability.

Three ultrasonic detectors were used in the experiments. Their respective center frequencies and bandwidths
$(-6\phantom{\rule{0.3em}{0ex}}\mathrm{dB})$
were 10 and
$9\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$
(V312, Olympus, Waltham, Massachusetts
), 20 and
$10\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$
(V317, Olympus, Waltham, Massachusetts
), and 40 and
$15\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$
[custom-built by the National Institutes of Health (NIH) Resource Center for Medical Ultrasonic Transducer Technology,^{15, 16} The University of Southern California, Los Angeles]. Their corresponding cut-off wavelengths were 105, 60, and
$32\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. Each transducer was positioned independently to reach its maximum PA amplitude.

## 3.

## Results and Discussions

Figure 4 gives the theoretical calculations of $\mathrm{PA}(z,{\mu}_{a})$ and ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ . Figure 4 shows the normalized depth-resolved PA waves along the $z$ axis in the semi-infinite object calculated from Eq. 2. The ${\mu}_{a}$ was varied from $70\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}630\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$ with a step size of $140\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$ . For simplicity, all the $\mathrm{PA}(z,{\mu}_{a})$ were normalized by ${F}_{0}{k}_{1}{\mu}_{a\phantom{\rule{0.2em}{0ex}}\mathrm{max}}$ , where ${\mu}_{a\phantom{\rule{0.2em}{0ex}}\mathrm{max}}$ is the maximum value of ${\mu}_{a}$ used in simulation study, which was $720\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{-1}$ . As we can see in Fig. 4, higher ${\mu}_{a}$ induces larger PA pressure peak amplitude under the same optical illumination, but the depth-resolved signal decays faster than those with lower ${\mu}_{a}$ values.

When an ultrasonic detector measures the induced PA pressures for anatomical or functional imaging purposes, a time-resolved PA signal will be acquired. In most situations, the peak-to-peak amplitude of the detected PA signal is used to measure functional parameters such as
$\mathrm{s}{\mathrm{O}}_{2}$
. Ideally, the detected PA signal is anticipated to be identical with the induced PA pressures inside the sample; however, because ultrasonic detection is band-limited and thus has only limited axial resolution, the detected PA signal must be integrated within an ultrasonically resolvable depth. Such resolvable depth can be considered as the axial resolution and is related with the ultrasonic center frequency and bandwidth.^{10, 14} In our work, the axial resolution is approximated by
${k}_{2}{\lambda}_{c}$
.

Figure 4 shows the calculated ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ for the three ultrasonic detectors used in the experiments. In the simulation, ${k}_{2}$ was set to 0.92 and the ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ for each ultrasonic center frequency was normalized independently. As we can observe in Fig. 4, when ${\mu}_{a}$ is low, ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ increases proportional to ${\mu}_{a}$ for all three center frequencies; however, when ${\mu}_{a}$ is higher than a certain value, the correlation between ${\mu}_{a}$ and ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ loses its linearity and eventually reaches a plateau, which demonstrates the saturation effect. Moreover, the nonlinearity becomes more severe when the ultrasonic center frequency lowers. In Fig. 4, the PA signal detected by a $10\text{-}\mathrm{MHz}$ ultrasonic detector was saturated at a ${\mu}_{a}$ value lower than those detected by 20- and $40\text{-}\mathrm{MHz}$ ultrasonic detectors. If we consider the ${\mu}_{a}$ value corresponding with the $1\u2215e$ decay of ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ as the “turn point” of the saturation, the linear range for a given ${\lambda}_{c}$ can be found from the simulation results. As seen in Fig. 4, the horizontal line corresponding with the value of $1-{e}^{-1}$ , where the left side of the cross point with each ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ representing the preferred linear measuring range. It is important to understand that such a transition at the $1\u2215e$ “turn point” between linearity and saturation is not abrupt.

The saturation effect limits the selection of the ultrasonic detector’s center frequency and bandwidth in functional PA imaging since the inverse calculation of $\mathrm{s}{\mathrm{O}}_{2}$ requires the ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)$ to be linearly associated with optical absorption. Due to the dramatic variation in the molar excitation coefficients of Hb and $\mathrm{Hb}{\mathrm{O}}_{2}$ within the spectral range of $520\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}600\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ , the PA measurement may be saturated within a certain spectral range while remaining linear within others. The saturation effect was further observed by measuring PA signals generated from the ink phantoms. The same normalization scheme as that used for Fig. 4 was used in the experiments.

Figure 5 shows the change of the detected PA amplitude using the $20\text{-}\mathrm{MHz}$ ultrasonic detector when the ink concentration increases. The temporal profiles of several detected PA waves are also shown in the Fig. 5. Similar temporal shapes can be observed for PA waves within both the linear and saturated ranges. As shown in Fig. 5, the experimental data acquired from three ultrasonic detectors demonstrate the same trend as that in the simulation results, where the saturation starts much earlier for the $10\text{-}\mathrm{MHz}$ transducer than transducers with higher center frequencies.

We further fit the experimental data to verify ${k}_{2}$ . The target fitting function was simplified from Eq. 3 to be $a\text{-}b$ $\left({e}^{-c{\mu}_{a}}\right)$ , where $a$ , $b$ , and $c$ are unknown constants. After fitting was conducted for all three ultrasonic transducers, we acquired $a=0.98\pm 0.016$ , $b=0.82\pm 0.090$ , and $c=0.0076\pm 0.0018$ . Then ${k}_{2}$ was subsequently calculated to be $1.15\pm 0.10$ . Compared with the values reported in Ref. 14, our value is only slightly higher, possibly due to experimental noises and different optical-ultrasonic geometry. Both the simulation and experimental results indicate that the ultrasonic detector used for functional PA imaging should satisfy ${k}_{2}{\lambda}_{c}{\mu}_{a}\u2a7d1$ to avoid saturation. In this situation, Eq. 3 becomes ${A}_{\mathrm{PA}}\left({\mu}_{a}\right)=\alpha {F}_{0}{k}_{1}{k}_{2}{\lambda}_{c}{\mu}_{a}$ by ignoring high-order terms in the Taylor expansion.

## 4.

## Summary

We have used a semi-infinite homogeneous optical absorbing medium to model laser-induced, depth-resolved PA pressure and the resolution-limited detected PA amplitude. The saturation effect was simulated by varying the optical absorption coefficient of the medium. The saturation effect was further observed in ink phantoms using three ultrasonic detectors with center frequencies of 10, 20, and $40\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ . The experimental results agreed with the theoretical studies and the fitted constant agrees with the reported theoretical estimation.

## Acknowledgments

This work was supported in part by start-up funding from the University of Wisconsin-Milwaukee, a grant from The Lynde and Harry Bradley Foundation, and financial support from the Chinese Ministry of Education to J. Wang.

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