*in vivo*.

## 1.

## Introduction

Blood flow speed is an important functional parameter of the biological tissue.^{1} Accurately measuring blood flow speed is critical for diagnosing many diseases, such as burns,^{2} stroke,^{3} and atherosclerosis.^{4} Presently, photoacoustic tomography,^{5}^{–}^{7} which combines optical excitation and acoustic detection, has attracted great attention for measuring blood flow speeds due to its excellent capability to detect blood using hemoglobin as the endogenous contrast.^{8}^{–}^{21} So far, many photoacoustic methods have been reported for blood flow measurement, based on the Doppler effect,^{9}^{,}^{19}^{,}^{20} thermal tagging,^{10}^{,}^{12} and signal-fluctuation correlation.^{13}^{,}^{17}^{,}^{18} Although the Doppler- and correlation-based methods have been successfully demonstrated *in vivo*, both methods depend on tracking varying photoacoustic signal signatures, which requires the absorbers to be distributed with sufficient spatial heterogeneity. By contrast, the thermal-tagging-based methods can be applied to both homogenous and inhomogeneous flowing media. Sheinfeld and Eyal^{10} pioneered the photoacoustic thermal-tagging flowmetry by investigating the dependence of the photoacoustic signals on temperatures and flow speeds, using two lasers as the thermal-tagging and photoacoustic excitations, respectively. Later, this method was improved by Wang et al.^{12} by employing a focused high-intensity-focused ultrasound (HIFU) transducer as the thermal-tagging source on acoustic-resolution photoacoustic microscopy (AR-PAM). The blood flow at depths of several millimeters can be measured, taking advantage of the deep-penetration of the HIFU heating. However, this method was not readily applicable for *in vivo* blood flow measurement, since the HIFU heating and photoacoustic detection are placed at the opposite sides of the sample. Furthermore, only the cooling process can be monitored because of the acoustic interference between the HIFU transducer and the photoacoustic detection. Zhang et al.^{22} further expanded the thermal-tagging method using optical-resolution photoacoustic microscopy (OR-PAM) and two independent light sources for thermal-tagging and photoacoustic excitation, which have limited its applications in traditional OR-PAM systems where only a single light source is typically available.

To address the above issues, we present a thermal-tagging-based photoacoustic flowmetry using OR-PAM with only a single light source. Compared with AR-PAM, OR-PAM can provide higher spatial resolutions, and thus a better thermal-tagging efficiency, at the cost of penetration depth.^{16} In addition, because OR-PAM has higher light-usage efficiency and uses less laser energy, a high pulse repetition rate can be explored for measuring fast blood flow speeds. In this work, instead of diffused optical heating^{10} or focused ultrasound heating,^{12} we use a focused laser beam with a short pulse width and a high repetition rate for both thermal tagging and photoacoustic excitation. Such a simplified configuration enables reflection-mode thermal tagging and photoacoustic imaging using the same light source, which is highly desired for *in vivo* blood flow measurements. In addition, we have developed the mathematical model to describe the thermal-tagging process and a practical fitting method to quantify the flow speeds based on the measured temperature-dependent photoacoustic signals. In the following sections, we will present the mathematical model together with proof-of-concept experiments and further validate it via *in vivo* blood flow measurements.

## 2.

## Theory and Method

The principle of photoacoustic thermal flowmetry originates from the well-established dependence of photoacoustic signal amplitude on the temperature of the flowing medium, which can be modulated by external heating and affected by the flow speed.^{23}^{,}^{24}

## 2.1.

### Local Temperature Change by Laser Heating

To quantify the flow speed using the temperature-dependent photoacoustic signals in OR-PAM, the relationship between the local medium temperature and its flow speed must first be modeled. As the train of focused laser pulses heats the medium, the local temperature within the heated volume increases from its baseline temperature before the heating. Note that the thermal diffusion in the heated volume may not be negligible in our model, meaning that the heated volume may be larger than the optical focus, depending on the total heating time. The temperature $T$ can be modeled as^{10}

## Eq. (1)

$$\frac{\partial T(\overrightarrow{r},t)}{\partial t}=\nabla [\alpha (\overrightarrow{r})\nabla T(\overrightarrow{r},t)]-v(\overrightarrow{r})\nabla T(\overrightarrow{r},t)+s(\overrightarrow{r}),$$Equation (1) is further simplified using a lumped model.^{12} Both sides of Eq. (1) are weight-averaged within the heated volume by introducing a weighted spatial integration as

## Eq. (3)

$$\frac{\partial}{\partial t}[{\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})T(\overrightarrow{r},t)\mathrm{d}V]={\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})\{\nabla [\alpha (\overrightarrow{r})\nabla T(\overrightarrow{r},t)]-v(\overrightarrow{r})\nabla T(\overrightarrow{r},t)+s(\overrightarrow{r})\}\mathrm{d}V,$$## Eq. (4)

$$\overline{T(t)}={\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})T(\overrightarrow{r},t)\mathrm{d}V,$$## Eq. (5)

$$\frac{\partial \overline{T(t)}}{\partial t}={\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})\{\nabla [\alpha (\overrightarrow{r})\nabla \frac{T(\overrightarrow{r},t)}{\overline{T(t)}}]-v(\overrightarrow{r})\nabla \frac{T(\overrightarrow{r},t)}{\overline{T(t)}}\}\overline{T(t)}\mathrm{d}V+{\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})s(\overrightarrow{r})\mathrm{d}V,$$^{12}Equation (5) can then be rewritten as

## Eq. (6)

$$\frac{\partial \overline{T(t)}}{\partial t}=-({C}_{\alpha}+{C}_{v})\overline{T(t)}+{C}_{s},$$## Eq. (7)

$${C}_{\alpha}={\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})\nabla [\alpha (\overrightarrow{r})\nabla \frac{T(\overrightarrow{r},t)}{\overline{T(t)}}]\mathrm{d}V,$$## Eq. (8)

$${C}_{v}=-{\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})v(\overrightarrow{r})\nabla \frac{T(\overrightarrow{r},t)}{\overline{T(t)}}\mathrm{d}V,$$## Eq. (9)

$${C}_{s}={\oint}_{\mathrm{\Omega}}{\omega}_{1}(\overrightarrow{r})s(\overrightarrow{r})\mathrm{d}V.$$After solving Eq. (6), which is a first-order differential equation, the average temperature within the heated volume can be obtained as

## Eq. (10)

$$\overline{T(t)}={e}^{-({C}_{\alpha}+{C}_{v})t}[\overline{T({t}_{0})}-\frac{{C}_{s}}{{C}_{\alpha}+{C}_{v}}]+\frac{{C}_{s}}{{C}_{\alpha}+{C}_{v}},$$## 2.2.

### Photoacoustic Sensing of the Local Temperature

The next step is to establish how the local temperature can be sensed by photoacoustic signals. In our method, the laser pulse width is short enough to satisfy both the thermal and stress confinements, thus the initial photoacoustic pressure rise is given by^{25}

## Eq. (13)

$$P(\overrightarrow{r},t)=\mathrm{\Gamma}(\overrightarrow{r},t)\eta (\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r}),$$Substituting Eq. (14) into Eq. (13), we obtain

## Eq. (15)

$$P(\overrightarrow{r},t)=a{\eta}_{\mathrm{th}}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r})+bT(\overrightarrow{r},t){\eta}_{th}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r}).$$Following the same approach of obtaining the spatial-weighted average temperature within the heated volume, the spatial-weighted average pressure amplitude is obtained by introducing the second weight spatial integration as

## Eq. (16)

$$\overline{P(t)}={\oint}_{\mathrm{\Omega}}{\omega}_{2}(\overrightarrow{r})P(\overrightarrow{r},t)\mathrm{d}V=a{\oint}_{\mathrm{\Omega}}{\omega}_{2}(\overrightarrow{r}){\eta}_{th}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r})\mathrm{d}V+b\overline{T(t)}{\oint}_{\mathrm{\Omega}}{\omega}_{2}(\overrightarrow{r}){\eta}_{th}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r})\frac{T(\overrightarrow{r},t)}{\overline{T(t)}}\mathrm{d}V,$$## Eq. (17)

$${C}_{1}=a{\oint}_{\mathrm{\Omega}}{\omega}_{2}(\overrightarrow{r}){\eta}_{\mathrm{th}}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r})\mathrm{d}V,$$## Eq. (18)

$${C}_{2}=b{\oint}_{\mathrm{\Omega}}{\omega}_{2}(\overrightarrow{r}){\eta}_{\mathrm{th}}(\overrightarrow{r}){\mu}_{a}(\overrightarrow{r})F(\overrightarrow{r})\frac{T(\overrightarrow{r},t)}{\overline{T(t)}}\mathrm{d}V,$$## Eq. (20)

$$\overline{P(t)}={e}^{-Ct}[{C}_{2}\overline{T({t}_{0})}-\frac{{C}_{s}{C}_{2}}{C}]+{C}_{1}+\frac{{C}_{2}{C}_{s}}{C},$$At the end of this section, the maximum measurable flow speed of the proposed method is estimated here. Assuming that at least two measurements (or laser pulses) are required for accurate fitting of the thermal constant $C$, then the total flow distance $d$ during two laser pulses is calculated by

where $f$ is the laser repetition rate. Theoretically, $d$ should be shorter than the maximum transverse length of the heated volume. For the first-order approximation, the maximum transverse length $L$ is estimated as the optical focal zone where NA denotes the numerical aperture of the focusing lens. Hence, we haveAs a result, the maximum measurable flow speed can be improved by increasing the laser repetition rate or enlarging the heating zone. Note that the number of measurements needed majorly depends on the signal-to-noise ratio of the imaging system and the measured flow speeds.

## 2.3.

### Optical-Resolution Photoacoustic Thermal Flowmetry

The schematic of a reflection-mode OR-PAM system is shown in Fig. 1(a). The pulse-laser beam from an Nd:YAD laser (IS8II-E, Edgewave, Würselen, Germany), with a wavelength of 532 nm and a pulse width of 5 ns, was expanded by two convex lenses and then focused by a focusing lens with a focal length of 50 mm (AC127-050-A, Thorlabs, Newton, New Jersey). Defibrinated bovine blood (Quad Five, Ryegate, Montana) was flowed in a transparent plastic tube (inner diameter: 0.3 mm, Dow Corning, Cat. 508-001, Midland, Michigan) driven by a syringe pump (NE-1000, New Era, Farmingdale, New York). The blood flow speeds were precisely controlled by adjusting the syringe pump’s translation speeds. The flowing blood was then thermally tagged and photoacoustically excited by a train of laser pulses. The laser pulse energy was 37 nJ, and the light was focused about $50\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ below the sample surface. The optical fluence at the sample surface was $\sim 37\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mJ}/{\mathrm{cm}}^{2}$, which is slightly higher than the ANSI limit ($20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mJ}/{\mathrm{cm}}^{2}$) but below the damage threshold.^{26} A photodiode (DET025A, Thorlabs, Newton, New Jersey) sampled a small portion of the laser pulse to measure the energy of each laser pulse. A function generator (DG1022, Beaverton, Oregon) was used to generate 50 kHz trigger signals to synchronize the laser firing and data acquisition (DAQ). A focused ring ultrasonic transducer (central frequency: 30 MHz, bandwidth: 30 MHz) detected the photoacoustic signal amplitudes of the heated region. The photoacoustic signals were amplified by 51 dB and sampled at 500 MHz. The lateral resolution of the system is $\sim 5\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, and the axial resolution is $\sim 42\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$.

Peak-to-peak amplitudes of the photoacoustic signals were extracted as a function of heating time (or number of laser pulses). We used 500 laser pulses at 10 kHz for each heating cycle that had a duration of 50 ms. Each heating cycle was 100 ms apart from the next to ensure the heated blood had enough time to flow out of the heating zone before the start of the next heating cycle, as shown in Fig. 1(b). For each heating cycle, the average temperature within the heated volume was elevated from its baseline, resulting in an increase in the average photoacoustic signal amplitude within the same volume. We repeated the heating for 40 cycles to improve the measurement accuracy.

## 3.

## Results and Discussion

Calibration experiments were first performed with different flow speeds ranging from 0 to $54\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ in $6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ increments. We monitored the laser pulse energy during each heating cycle using the photodiode readings, as shown in Fig. 2(a). A pulse energy fluctuation of 2.2% was observed, showing the relatively high stability of heating power. The photoacoustic signal amplitude upon each laser pulse was acquired during each heating cycle. At each flow speed, 40 heating cycles were performed and the results were averaged. Figure 2(b) shows the averaged time courses of the change in photoacoustic signal amplitudes at six representative flow speeds ranging from 6 to $36\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$. The baseline photoacoustic signals obtained with the initial laser pulses were subtracted from all the signals. The results have shown that the measured photoacoustic signal amplitudes initially increased with the heating time before reaching equilibrium. A higher flow speed resulted in a shorter temperature rising time and a lower equilibrium temperature, which was consistent with Eq. (20). The equilibrium temperature was reached when the initial volume of the low-temperature blood had flowed out of the heating zone. Once the equilibrium was reached, the blood volume within the heating zone became thermally “stationary,” even though blood was still flowing through. Data fitting was then performed on the measured photoacoustic signal amplitudes using Eq. (21). The fitting curves are shown as the black solid lines in Fig. 2(b), which correspond to the six different flow speeds. The thermal constants in Eq. (11) at all flow speeds can thus be obtained from the fittings.

As expected in Eq. (11), the thermal constant should be linearly dependent on the flow speed. This linear relationship can be readily validated by fitting the thermal constant as a function of the flow speed. To determine the slope ${k}_{2}$ and the intercept ${k}_{1}\alpha $, a total of 10 thermal constants were obtained at 10 different flow speeds, as shown in Fig. 3(a). The fitting has a coefficient of determination (${R}^{2}$ value) of 0.99, showing a strong linear dependence of the thermal constant on flow speed, which is consistent with Eq. (11). ${k}_{2}$ and ${k}_{1}\alpha $ are fitted as $35.44\pm 2.34$ and $511.74\pm 78.05$, respectively. These two calibration values will be used for quantifying unknown flow speeds if we can assume the thermal diffusivity $\alpha $ (material-dependent) of the biological tissue is consistent. After the system was calibrated, we further measured 10 more different flow speeds based on the calibration values. Figure 3(b) shows the measured flow speeds agreed well with the true flow speeds ranging from 3 to $57\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$. The averaged measurement error was 6.2%. Here, it is necessary to point out that due to the relatively long heating time of 50 ms, the thermal diffusion in the flowing medium cannot be neglected. Therefore, the heating zone was effectively much larger than the optical focal zone. Increasing the laser repetition rate will effectively shorten the effective heating zone.

Validation experiments for maximum measurable flow speeds were also performed at the flow speeds from 60 to $102\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$, as shown in Fig. 3(c). The results show an increased measurement error of more than 20%. The maximum measurable speed of the system can be improved by increasing the laser repetition rate. As a simple demonstration, we used another Nd:YAG laser (VPFL-G-20, V-gen, Tel Aviv, Israel) with a laser repetition rate of 50 kHz. Figures 3(d) and 3(e) show the results with the flow speeds ranged from 11.8 to $236\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$. The averaged measurement error is 6.3%. When the flow speed is beyond the maximum measurable speed, the increased measurement error is shown in Fig. 3(f). The effective optical NAs for the EdgeWave laser beam (10 kHz) and the V-gen laser beam (50 kHz) are 0.1 and 0.13, respectively. The maximum measurable flow speeds are $46.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ for the 10 kHz laser and $178.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ for the 50 kHz laser, according to Eq. (24). Additionally, because the fitting process is more robust when the local temperature reaches the equilibrium, a higher laser repetition rate also limits the ability to measure slow flows when the total number of pulses per heating cycle remains the same.

To further test the proposed flowmetry, *in vivo* measurements were performed by measuring the blood flow speeds of the microvasculature in a mouse ear. The *in vivo* study was conducted on a female Swiss Webster mouse (10 weeks old and 32.1 g in weight), with the protocol approved by the Institutional Animal Care and Use Committee of Duke University. The region of interest for the blood flow measurement was first imaged by the OR-PAM system, as shown in Fig. 4(a). The *in vivo* blood flow speed measurements were then performed with a 10 kHz laser pulse repetition rate. Like the phantom experiments, each heating cycle had 500 laser pulses and was repeated 40 times to improve the measurement accuracy. The laser pulse energy was 34 nJ. The three measurement positions on the mouse ear were indicated by the red solid circles in Fig. 4(a). The changes in photoacoustic signal amplitudes versus heating time were shown in Fig. 4(b), and the major changes at the beginning of the heating from 0 to 15 ms were shown in Fig. 4(c). The three thermal constants were fitted as 602.6, 545.7, and $546\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{s}}^{-1}$, corresponding to respective blood flow speeds of 2.56, 0.96, and $0.99\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$, which generally agree with our previous study.^{9} Note that point 1 and point 2 are at different branch levels of an artery, with point 1 on the mother branch and point 2 on the daughter branch. Hence, the flow speed at point 1 should be faster than point 2.

## 4.

## Conclusion

In conclusion, we have developed a new photoacoustic thermal flowmetry with a single light source, which can be readily applied on traditional reflection-mode OR-PAM systems. The same light source is used for simultaneous thermal tagging and photoacoustic imaging. We have developed a mathematic model to establish the relation between the flow speed and the thermal constant extracted from the detected photoacoustic signal amplitudes. We have performed calibration experiments to determine the system-dependent and medium-dependent constant factors. We have verified the new method on flowing blood phantoms at speeds up to $236\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ and validated its effectiveness for *in vivo* measurements by measuring the blood flow speeds of the microvasculature in a mouse ear.

Compared with the previous photoacoustic flowmetry,^{9}^{,}^{10}^{,}^{12}^{,}^{13}^{,}^{17}^{–}^{21} the new method can be applied to both homogenous and inhomogeneous media and can measure relatively high flow speeds with high spatial resolutions. However, due to the limited penetration of OR-PAM, the new method is only applicable for quasiballistic regimes ($\sim 1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ deep in the skin). To increase the penetration depth, we can apply near-infrared wavelength light^{27}^{,}^{28} or wavefront-shaping technologies that can help focus the light into deep scattering tissues.^{29}^{,}^{30} In addition, due to the symmetry of the heating (imaging) zone, the proposed method is not sensitive to the lateral flow direction. The lateral flow direction can be determined by scanning the flowing media with two opposite scanning directions, as we have demonstrated before.^{9} Moreover, a high laser repetition rate on the level of hundreds of kilohertz is required for the high-speed flow measurements, such as blood flow in major arteries, to ensure a high measurement accuracy; for the slow-speed measurements, such as in microvasculature, a laser repetition rate of tens of kilohertz is needed. In addition, the focal spot size of the light beam is important. A relatively large spot size is preferred for measuring the high-speed blood flows. Finally, the imaging time of the proposed method is a concern for 2-D flow mapping, especially for slow flows. We can potentially shorten the imaging time by increasing the laser repetition rate or decreasing the laser focal spot size, which can decrease the dwell time at each scanning position.

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## Biography

**Wei Liu** is a postdoctoral associate in biomedical engineering at Duke University. He received his BE degree in optics in 2010 and PhD in physics in 2015 from Harbin Institute of Technology, China. Before joining the photoacoustic imaging group at Duke, he stayed at Nanyang Technological University, Singapore, as a research fellow. His current research interest focuses on photoacoustic imaging, biophotonics and biomedical instrumentation.

**Bangxin Lan** is a master’s student in the Department of Biomedical Engineering at Duke University. His major research focus is in high-speed photoacoustic imaging.

**Leo Hu** is an undergraduate student at Duke University, currently pursuing both a BSE in biomedical engineering and a BSE in electrical and computer engineering. Currently, he is researching high-resolution photoacoustic imaging.

**Ruimin Chen** received his BS degree from University of Electronics Science and Technology of China, Chengdu, China, and his PhD from University of Southern California (USC), Los Angeles, California, in 2006 and 2014, respectively, both in biomedical engineering. Currently, he is a research associate in the NIH Resource Center for Medical Ultrasonic Transducer Technology of USC. His research interests include high-frequency ultrasonic transducers and arrays, piezoelectric material characterization, and photoacoustic imaging.

**Qifa Zhou** received his PhD from Xi’an Jiaotong University, Xi’an, China in 1993. Currently, he is a professor of ophthalmology, biomedical engineering at the University of Southern California. His research focuses on the development of piezoelectric high-frequency ultrasonic transducers for biomedical ultrasound and photoacoustic imaging, including intravascular imaging, cancer imaging, and ophthalmic imaging. He is also actively exploring ultrasonic mediated therapeutic technology including trans-sclera drug delivery, and ultrasound for retinal and brain stimulation.

**Junjie Yao** is an assistant professor of biomedical engineering at Duke University, and a faculty member of Duke Center for *In-Vivo* Microscopy and Fitzpatrick Institute for Photonics. He received his BE and ME degrees from Tsinghua University, and his PhD in biomedical engineering at Washington University, St. Louis. His research interest is in photoacoustic tomography technologies in life sciences, especially in functional brain imaging and early cancer detection.