## Abstract

The theory of photoacoustic tomography (PAT) imaging using a 4f acoustic lens imaging system has been investigated, and the theoretical results show that a 4f acoustic lens has the ability of imaging and guarantees axial and lateral unit magnification of image. A system, a 4f acoustic lens imaging system combining with time-resolved technique, is developed to acquire PAT images. The 4f acoustic lens is able to image initial photoacoustic (PA) pressure distribution, which exactly resembles the absorption distribution, onto an imaging plane. Combining with time-resolved technique, the linear transducer array is adopted to acquire the PA pressure distribution to reconstruct the PAT images. Experimental results demonstrate that the system is able to obtain PAT images and the images contrast sharply with their backgrounds.

© 2007 Optical Society of America

## 1. Introduction

Recently, there has been a wide interest in photoacoustic tomography (PAT) imaging, which is a noninvasive imaging modality for visualizing both the structural and functional information of biological tissues. It combines both high optical contrast and high acoustic penetration depths. In PAT, a short laser pulse is used to heat biological tissues, and a photoacoustic (PA) effect results in the emission of acoustic signals that can be measured by use of a wide-band ultrasonic transducer. The PA pressure is linearly directly proportional to local optical absorption in the biological tissue, and the PA signals carry information about the geometrical structure and optical properties of biological tissues. Thereby, the PA signals are associated with their physiological and pathological status. The ultrasonic waves are free from the strong scattering and decaying of light in biological tissue, and outside tissue, they can be recorded by an acoustic transducer to map optical absorption distribution. The PAT image can display the structure and function of the biological tissue. In experimental, the imaging modality has been applied to imaging breast cancer, skin cancer, brain tumor, blood concentrations and vasculature structure [1–8].

Many researches have been done to get PAT images [9–14]. However, the original absorption distribution can only be obtained by image reconstruction, so it is hard to get realtime PA images. To get real-time PA images, an acoustic lens, which is able to image the initial PA pressure distribution onto an image space in real time without the necessity of computational reconstructions, has been developed and applied to PAT imaging [15–17]. However, to display the three-dimensional (3D) real object, it requires the images have the same magnification on axial and lateral. In optical imaging, the 4f imaging system guarantees axial and lateral unit magnification of the images. So, the 4f acoustic lens can guarantee that.

In this paper, the theory of PAT imaging using a 4f acoustic lens imaging system has been investigated, and the theoretical results show that a 4f acoustic lens has the ability of imaging and guarantees axial and lateral unit magnification of an image. A 4f acoustic lens imaging system combined with time-resolved technique, which is developed to acquire PAT images.

## 2. Theoretical model

#### 2.1. Principle of 4f acoustic lens imaging

The induced PA waves are linearly directly proportional to the local optical absorption and they take on the characteristics of wave, such as interference and diffraction. According to the imaging theory of Fourier, if an acoustic lens allows spatial Fourier transform, then it is capable of imaging. However, to display the three-dimensional real object, the same magnification on axial and lateral is needed. To make sure that, the 4f acoustic lens imaging system is designed, as Fig. 1 shows, which similar to the 4f imaging system in the optical imaging.

In this system, the two confocal lenses L_{1} and L_{2} is combined, the front focal plane of the lens L_{1} is the object plane O, the image is inputted from this plane. The back focal plane of the lens L_{2} is the imaging plane I, the image is outputted from this plane. The confocal plane is named the transform plane T. The system is named for short the OTI imaging system. When the object is illuminated by laser, the ultrasonic signals induced from the object plane can be considered as coherent signals, the whole OTI imaging system is coherent imaging system. According to Abbe theory of image formation, the imaging process of the system will be discussed as follows.

The imaging process of coherent ultrasonic system has two steps as follow: The first step is the Fraunhofer diffraction from the O plane to the T plane, its action is dividing frequency. The second step is also the Fraunhofer diffraction, its action is sum frequency, and that is, the synthesized spectrum is to output images. If the ultrasonic is able to free pass the transform plane, the image can be emersion completely.

The amplitude of acoustic pressure on the object plane is supposed as *P _{O}* (

*x*,

*y*, on the transform plane

*t*(

*u*,

*v*) =1, so

*P*

_{2}(

*u*,

*v*) =

*P*

_{1}(

*u*,

*v*) , that is

*P*(

_{T}*u*,

*v*) ; on the imaging plane, the amplitude of acoustic pressure is

*P*(

_{I}*x*′,

*y*′) . As Fig. 2 shows, the transforms of acoustic pressure-amplitude are Fourier transforms in the process of two Fraunhofer diffractions.

That is

Omitting all coefficients, so

Substitution of Eq. (3) into Eq. (4) gives

$$\phantom{\rule{22.5em}{0ex}}=\int \underset{-\infty}{\overset{+\infty}{\int}}{P}_{O}\left(x,y\right)\left\{\underset{-\infty}{\overset{+\infty}{\int}}\mathrm{exp}\left[\frac{-\mathrm{ik}\left(x+\mathrm{x\prime}\right)}{f}u\right]\mathrm{du}\right\}\times \left\{\underset{-\infty}{\overset{+\infty}{\int}}\mathrm{exp}\left[\frac{-\mathrm{ik}\left(y+\mathrm{y\prime}\right)}{f}v\right]\mathrm{dv}\right\}\mathrm{dxdy}\propto \int \underset{-\infty}{\overset{+\infty}{\int}}{P}_{O}\left(x,y\right)\delta \left(x+\mathrm{x\prime}\right)\delta \left(y+\mathrm{y\prime}\right)\mathrm{dxdy}$$

$$\phantom{\rule{7.5em}{0ex}}\phantom{\rule{0.2em}{0ex}}={P}_{O}{\left(-\mathrm{x\prime},-\mathrm{y\prime}\right)}_{o}\phantom{\rule{17.5em}{0ex}}$$

So the outputted image is same to the inputted image; in the formula, the minus in the front of *x*′,*y*′ means the image is inverted.

#### 2.2. The lateral magnification and the axial magnification of the imaging system

According to the relation between the lateral magnification and the axial magnification of ideal optical imaging system, in the ideal acoustic imaging system, the lateral magnification and the axial magnification also have the relation as follows:

In the formula, *α* is the lateral magnification of the imaging system, *β* is the axial
magnification, *f* is the object focal length, *f*′ is the image focal length, *n*′ is the refractivity
of medium in object space, and *n* is the refractivity of medium in image space. If the medium in object space is same to the image space, the formula (6) can be simplified as

The equation shows that the image of a cube isn’t a cube, except the cube is on the position of *β* = ±1 .In the 4f imaging system, *α* = 1 because of *β* = -1, the image of a cube is also a cube. Therefore, the 4f acoustic imaging system guarantees axial and lateral unit magnification of the images, a displacement of the object plane by a distance Δ*Z* results in a displacement of the corresponding focused image plane by a distance Δ*Z* , just as Fig. 3 shows. In addition, the acoustic lens has the property of long focal depth, which means that the distance Δ*Z* of the image plane of the lateral magnification *α* = 1 will be long, about 1–2cm. And it can be described by [17]

Where *f* is the focal length of an acoustic lens, *a* is the radius of the acoustic lens, and *λ* is the wavelength of sound.

## 3. Imaging system and method

Figure 4 shows the experimental setup. A Q-switched Nd: YAG laser is adopted to provide 1064 -nm laser pulses with a FWHM of 8 ns. The laser beam is expanded to heat the imaging object and produce the PA signal. The distribution of optical absorption generates proportionate PA waves. The acoustic pressure distribution is imaged onto the image planes by a 4f acoustic lens in the scattering media. Into the image plane, a linear detector detects the acoustic pressure distribution. The linear detector consists of 64 PZT probes with a center frequency of 1 MHz frequency. The radius of every probe is about 0.25mm, and the central distance between two neighboring probes is 1.5 mm. And the linear detector is fixed on a scanning stage. The scanning stage, which is controlled by a computer, drivers the linear detector vertically to detect the planar PA signals. The signals are chosen and amplified in turn by the electronic switch of 64 lines and input a Boxcar (made in American Stanford Research System Inc. Company). The Boxcar triggered by the laser Q switch integrates the signals during a fixed delay time, when the peak value of the signal is transformed to the corresponding voltage of dc. These voltage signals are sent to a computer through an acquisition card (ADC, Model: Advantech PCL-818HG) and transformed into corresponding 256 gray levels to display the image. An oscillograph (Model: TDS1002) monitors the PA signals and the sampling gate of the Boxcar.

Similar to an optical imaging system, the PA signals from an object plane need the same delay time to reach the image plane. So the Boxcar is used to hold on a fixed delay time to acquire the PA signal, which guarantees the acquired signals are from the same image plane. So we can reconstruct the different absorption distribution planes by detecting the acoustic pressure on the corresponding focused image planes. However, because of long focal depth of an acoustic lens, a determined object plane can be precisely imaged onto the imaging plane within a range distance inside focal depth by an acoustic lens. The PA signals from different planes require different time to reach a same detecting plane. Therefore, on this same detecting plane, we can use the time-resolved technique to distinguish photoacoustic signals from different planes. So, onto one detecting plane, we can get corresponding photoacoustic tomography images of different object planes without moving the detector.

## 4. Experimental results and discussion

#### 4.1. Two-dimensional (2D) PAT imaging

To demonstrate the feasibility of IOT imaging system, a series of experiments were conducted with turbid phantoms prepared from an insoluble in water milk. It is demonstrated that the 4f acoustic imaging system has the ability of 2D PAT imaging. A sample such as Fig. 5(a), which three black adhesive tape points was adhered to a piece of polymethylmethacrylate, was in 3% milk. The sample was heated by laser, and PA signals from the sample are imaged on the imaging plane by the acoustic lens. The imaging plane is scanned by a detector to signals to reconstruct the PAT image. The PAT image is smaller than the sample because of the limited probes of the detector. The original image is magnified six times as Fig. 5(b). The acoustic image is in a perfect accordance with the sample, it demonstrates the 2D imaging ability of this system.

As Fig. 6(a) and Fig. 7(a) show the samples made of black adhesive tape are the patterns ‘V’ and ‘C’, and they are glued on the polymethylmethacrylate. The PAT image is obtained as Fig. 6(b) and Fig. 7(b) by scanning on the image plane. The PA images are in a perfect accordance with the sample; they are clear and contrast sharply with their backgrounds. All the sizes of the images are approximately equal to the original objects’, and it shows the lateral magnifications are unit for the 4f acoustic lens. The PA image is inverted compared to that of object for the acoustic lens forms the inverted real image. Otherwise, the pictures were altered little by the finite aperture and lens aberrations. So, in other to obtain the clear image, the acoustic lens should be revised and a 2D ultrasonic detector array with multiple small elements should be developed.

#### 4.2. The resolution of the imaging system

Similar to optical imaging system, the impulse response of the imaging system shows the characteristic of imaging. In experiment, the pulse laser heated a black adhesive tape point and generated the PA waves. The black adhesive tape point’s diameter is about 0.8 mm, and it acted as a point acoustic source. Figure 8(a) is the black adhesive tape point, and Fig. 8(b) is the impulse response curve of experimental result. According to Rayleigh criterion for resolution, the lateral resolution equals to the full width at half maximum of the impulse response curve, and it can be worked out about 4mm.

In experiment, the lateral resolution can also be measured by adjusting the distance between two points. Experimental result shows that the imaging system can resolve two points when the distance is 3mm between them, and the PA image is as Fig. 9. And the PA image is only one big point when the distance between two points is less than 3mm. There are some error between the above two data but they are approximately equal.

#### 4.3. The imaging property of the acoustic lens on axis

In the imaging modality, the three-dimensional object is sliced, and every slice is a 2D x-y plane. Along the z-axis, the three-dimensional object is segmented some x-y planes. The imaging property of the x-y plane has been shown, and the image is equal to the object. Here, the imaging property of the z-axis will be discussed. The sample, which two different patterns are adhered to the front as well as at the back of a piece of polymethylmethacrylate, is adopted in the experiment, just as Fig. 10. The polymethylmethacrylate is about 15mm thick, so the distance between the two object planes is about 15mm. Figure 11 shows the PA signals of two different imaging planes. According to the axial unit magnification of the 4f acoustic lens, the distance between the two object planes must be equal to the distance between two image planes. The distance of image planes can be expressed as

where *v* is the acoustic velocity, and Δ*t* is the time difference of two PA signals reached the detector.

Figure 11 shows the time difference is Δ*t* = 8.21×10^{-5}
*s*-7.65×10^{-5}
*s*, = 5.6×10^{-6}
*s*, = 5.6μ*s*, and*v* = 1.49*mm*/μ*s*, so *D* = *v*∙Δ*t* = 1.49×5.6*mm* = 8.344(*mm*). The distance between the two object planes is about 15mm in polymethylmethacrylate, so it must be transformed to the distance in solution. The transforming relation is *d* = *s*/*n*, where s is the distance between the two object planes, *n* is a relative refraction index, and $n=\frac{{v}_{\mathrm{polymethylmethacrylate}}}{{v}_{\mathrm{water}}}=\frac{\frac{2.7\mathrm{mm}}{\mathrm{\mu s}}}{\frac{1.49\mathrm{mm}}{\mathrm{\mu s}}}=1.8.$ So the distance is $d=\frac{s}{n}=\frac{15\mathrm{mm}}{1.8}=8.333\mathrm{mm}$. This just proves the distance between the two object planes
is equal to the distance between two image planes, and the acoustic lens has the property of axial unit magnification.

The imaging modality is to slice the three-dimensional object, then how close two slices can be distinguished? It is the depth resolution (DR), and the DR can be expressed as^{17}

where *τ* is the temporal impulse width of PA signals and *v* is the velocity of PA signals in the biological tissue. In general, *v* is a constant, thus DR is determined by *τ*, where *τ* relates to the response characteristic of the detector.

## 5. Conclusion

The PAT imaging theory of a 4f acoustic lens imaging system has been present, and the theoretical results show that a 4f acoustic lens is able to image the PA pressure distribution onto an image space and guarantees axial and lateral unit magnification of image. Experimental results demonstrate that the system is able to obtain PAT images of samples inside optically turbid medium, and the reconstructed images are agree well with the sample. Furthermore, in 3D PAT imaging, the IOT imaging system may provide the PA images with the same magnification on axial and lateral, and the 3D structure of the object can be imaging really.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (grant No.60377009), National 863 Program Project of China (grant No. 2006AA02Z4B4) and the Natural Science Foundation of Guangdong Province, China (Grant No.05005926).

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