Noninvasive quantification of abnormal particles hidden in a granular mixture from deep tissue is still a challenge of medical examination. In this study, we have theoretically deduced the power spectrum of the photoacoustic signals from the random mixture of particles with non-uniform sizes. It is revealed that there is an approximate linear relationship between the content of abnormal particles and the spectral slope. This finding provides a parameter of equivalent diameter for the abnormal particle detection. The experimental studies sensitively differentiate and quantify a trace of big micro-particles mixed in small micro-particles. Since the abnormal particles are associated with many important physiological and pathological processes, this study might provide a noninvasive way to assess the related diseases, such as microthrombosis, through monitoring the abnormal particles.
© 2015 Optical Society of America
Quantitative detection of abnormal particles in a random granular mixture has important biomedical significances. For example, microthrombosis or the aggregation of red blood cells  mixed in the erythrocyte is one major determinant of hemodynamics, including blood viscosity, blood microcirculation, and so on [2,3 ]. The hyperaggregation could affect the normal physiological function and be related with the diabetics [4,5 ], thrombosis  and rheumatoid arthritis , etc. Therefore, quantitative assessment of these abnormal particles from the normal erythrocytes would be very meaningful to the evaluation of some pathophysiological states.
Potential candidates for the detection of abnormal particles include the erythrocyte sedimentation rate (ESR), syllectometry, and ultrasonic measurement. ESR is based on the different sedimentation rate of cell clumping and single cell in the stationary vertical tube . And it is an invasive and slow method to characterize red blood cell aggregation in vitro. Syllectometry detects the abnormal particles through measuring the scattered light induced by different particles , whereas, syllectometry method is only available for in vitro measurements, because of the limitation of strong optical scattering in tissues. Ultrasonic tissue characterization could provide a noninvasive tool to monitor the abnormal particles in vivo . This method distinguishes the abnormal particles by examining the difference in the backscattered ultrasound signals of aggregated cells and normal cells. However, the ultrasound method usually has a low contrast and sensitivity in comparison with the optical method. Generally, the current methods still have their limitations in sensitive and noninvasive examination of the abnormal particles in the particle mixtures.
Photoacoustic (PA) imaging (PAI) is a hybrid non-invasive imaging modality, which combines the merits of both the high resolution of ultrasonography in deep tissue and the functional contrast of optical imaging [10–16 ]. Moreover, the nonionizing wave used in PAI is much safer than the ionizing radiation (e.g., x-ray) for biological tissue. Especially, hemoglobin in the tissue can be a strong absorber for several absorption bands of light . These merits make PAI become a burgeoning modality in medical imaging. The axial resolution of PAI usually depends on broadband of PA signal detection and its lateral resolution depends on the central frequency of the transducer. Therefore, the conventional PAI is often ineffective in detecting the micron-size particles in deep tissues .
In order to break this limitation, inspired by the spectrum analysis of ultrasonic backscatter signals in tissue characterization [18–20 ], recent studies have shown that the spectral characteristics of PA signals are closely related to the microstructure in tissues [17,21–25 ]. PA spectrum analysis shows its potential in characterizing some kinds of microstructures, such as the erythrocyte aggregation [23,24 ]. Further theoretical analysis and experimental studies revealed that the spectral slope of PA signals can quantitatively measure the characteristic sizes of micro-particles . Moreover, this measurement is device-independent and available even when the particle size is smaller than the wavelength corresponding to the central frequency of the transducer . Saha provides a computer model to study PAs from mixtures of different populations of absorbers . Using this model, Saha revealed abnormal particles and polydisperse particles could affect the spectral characteristics of PA signals [26,27 ]. Hysi et al. found that the PA spectra depends on the red blood cell aggregation in their experiments . These pioneering studies suggest that the spectral parameter could have the potential for monitoring the abnormal particles.
However, there is still a lack of an explicit analytic theory and a quantitative parameter to describe the spectral properties of PA signals from particles with the non-uniform characteristic size.,They are incompetent in quantifying non-uniform particle mixtures.
The purpose of this study is to quantify the spectral parameters of the non-uniform particle mixture. Based on this finding, we could quantitatively examine the abnormal particles hidden in particle mixtures.
Let’s consider the scenario given in Fig. 1 . After the pulse laser illumination, the generated PA pressure p(r, t) owing to absorption of electromagnetic energy obeys the equation,Eq. (1), the PA spectrum S(f) measured at r ' can be given as 
For a random suspension of the granular mixture, A(r) can be given as the sum of individual granular absorber Ai,Eq. (3) into Eq. (2), we have,
Expanding ϕ(f, ai) into Taylor series and ignoring the higher order, it has ϕ(f, ai) ≈ϕ(f, a 0) + ϕa'(f, a 0) (ai – a 0), a 0 can be an arbitrary value close to the particle diameter. Then Eq. (4) can be rewritten as,
Equation (6) involves two random parameters. One is related to the granular size ai and another depends on the position. The two random variables mi and Ri are independent variables and they are uncorrelated. The expectations in Eq. (6) can be calculated separately, that is,
When the particles of abnormal size are mixed into the granular suspension, the PA spectral characteristics will be changed. Comparing Eq. (7) with Eq. (4), it can be found that the spectrum of a granular mixture can be approximate to the spectrum of the random particles of the uniform size â. The equivalent size of the granular mixture depends on the size and amount of abnormal particles hidden in the suspension.
The power spectrum S(f) is measurable and the function |φ(f, â)|2 can be theoretically predicted by . For a sphere with a diameter of â, it has A(r – r i) = 1 for |r – r i| ≤ â and A(r – r i) = 0 for |r – r i| > â. |φ(f, â)|2 can been given as 24]. Since the equivalent size relies on the dimension and amount of abnormal particles in the mixed suspension, we can use â as an indicator to quantify the abnormal particles. This is our basic idea of the abnormal particle detection.
3. Experiment and results
Phantom experiments have been performed to verify the theory before the proposed method can be used in practical biomedical application. Figure 2 shows the experimental setup and a phantom picture. A Q-switched Nd:YAG laser is employed as the energy source which has the wavelength of 532 nm, a pulse reputation rate of 10 Hz and a pulse energy of 80 mJ. After the beam expansion and diffusion, the laser pulses are vertically irradiated on the phantoms, which are immerged in water. The generated PA signals are detected by a focused ultrasound transducer (V310, Panametrics) with a focal length of about 25 mm, a radius of 3 mm, a center frequency of 4.39 MHz and a bandwidth of 4.4 MHz at –6 dB. The directivity of a transducer could decrease the spectral slope slightly . However, when the transducer radius ρ is much smaller than its focal length F, e.g., ρ/F < 1/6 (ρ/F = 3/25 in the experiment), the influence of the transducer aperture can be ignored. A computer-controlled stepper motor drives the transducer in a circular horizontal orbit around the phantom and stops with the step of 3° to detect PA signals. To reduce the noise, the signals are detected repeatedly for 30 times and averaged at one position. The signals are amplified (SA-230F5, NF) and recorded at the rate of 60MHz (PCI-5105, NI) for the following analysis. All recorded signals are used in the following spectral analysis.
Without loss of generality, binary mixtures of spherical particles are considered in the experiments. One kind of black dyed polymer microsphere with a small diameter of a s = 49.1 ± 3.6 μm (BK50T, Thermo) plays the role of normal particles. Another kind of big microsphere with a diameter of a b = 199 ± 10 μm (BK200T, Thermo) is used to simulate abnormal particles. The mass of a big particle is almost 66 times as that of a small particle. Five phantoms made of agar are prepared for the experiment. They have the shape of a cylinder with a radius of about 2 cm and a thickness of 1cm. The two kinds of particles are evenly embedded into the phantoms. Each phantom contains the granular mixture with the same total mass M of 10 mg, but different mass ratios, i.e., m b/M = 0.0, m b/M = 0.25, m b/M = 0.50, m b /M = 0.75 and m b /M = 1.00, respectively. m b is the total mass of big particles. With the density of 1.05 g/cm2 for polymer particle, the total volume of particles in each sample is about 9.52 mm3, which occupies 0.08% of the volume of the sample. The number density of particles in each sample are 11.6 × 103 particles/cm3, 8.74 × 103 particles/cm3, 5.89 × 103 particles/cm3, 3.03 × 103 particles/cm3, and 0.18 × 103 particles/cm3 for m b/M = 0.0, 0.25, 0.5, 0.75 and 1.00 respectively.
Figure 3 gives the experimental results. Figure 3(a) shows two typical waveforms of the PA signals detected from the granular mixtures with the mass ratio of m b/M = 0.75 and m b/M = 0.25. Because of the random distribution of particles, the signals are noise-like. According to the signal waveforms, it is almost impossible to quantify the abnormal particles in the mixture, even hard to distinguish their existence. Applying Fourier transform to the signals, their spectra S(f) are obtained, as shown in Fig. 3(b), where the five solid lines correspond to the five granular mixtures with different mass ratios, respectively. The spectra from the five granular mixtures are associated with the abnormal particles mixed into phantoms.
However, the real PA spectra S(f) could also be modified by the response S 0(f) of the measurement system. Calculation of S 0(f) is usually not easy. In order to remove the effect of the system response S 0(f), the calibration spectrum C(f) = S 0(f)∙|φ c(f, a c)|2 is measured from the micro-particles with an uniform diameter of 49 ± 3.6 μm (BK50T, Thermo), as shown with the dashed line in Fig. 3(b). Since the spectra of the granular mixture and the calibration spectrum are measured by the same system, they involve the same term of S 0(f). Dividing the power spectrum S(f) by the calibration spectrum C(f), the calibrated spectrum Sc(f) = S(f)/C(f) = |φ(f, â)|2/|φ c(f, a c)|2 is only related to microstructure characteristics of the granular mixture, but not the measurement system. Figure 3(c) gives the calibrated spectra with the solid lines. Although the phantom of m b/M = 0.0 and the phantom for calibration contain the same kind of particles, the particles are randomly distributed in them. Their spectra could have some slight differences. Therefore, the calibrated spectrum of phantom m b/M = 0 is not exactly zero. When the abnormal (big) particles are mixed into the normal (small) particle suspensions, the PA spectra of the mixtures will be changed. The more abnormal particles exist, the narrower the spectrum will be. The mixtures with a large mass ratio, e.g., m b/M = 0.75 and 1.00 have a narrower spectral bandwidth than those mixtures with a small mass ratio, e.g., m b/M = 0 and 0.25. The spectral characteristics clearly imply the existence of abnormal particles.
Quantitative comparison of the theoretical prediction and experimental measurements can quantify the amount of abnormal particles in the granular mixtures. Spectral slope is employed as an indicator to judge whether the predicted spectra match the measurements best. Performing a linear regression [dashed line in Fig. 3(c)] over the usable bandwidth 0.5~6.5 MHz, we can determine the spectral slope corresponding to each phantom. A relative low frequency and a wide bandwidth are beneficial for detecting signals from deep tissue and reducing the influence of spectral fluctuation on the slope estimation. Additionally, in the measured PA signal, its low frequency part below 0.5 MHz is strongly affected by the background signal produced by the laser illumination on the container . Moreover, the particle signal in the high frequency part is weak. Therefore, in order to get a good signal-to-noise ratio, we chose the spectrum in the range from 0.5 ~6.5 MHz, which is also close to the band of the transducer. The slopes corresponding to the five granular mixtures are −0.01 dB/MHz for m b/M = 0.0, −0.20 dB/MHz for m b/M = 0.25, –0.59 dB/MHz for m b/M = 0.50, −0.74 dB/MHz for m b/M = 0.75, and −1.08 dB/MHz for m b/M = 1.00. Integrating Eq. (9), the calibrated spectrum Sc(f) = |φ(f, â)|2/|φ c(f, a c)|2 and its spectral slope can be theoretically predicted. Figure 3(d) gives the predicted spectral slope as a function of the equivalent diameter â with a red line. Then, the equivalent diameters of the granular mixtures are estimated according to the measured spectral slopes and the predicted slope-diameter curve, as shown in Fig. 3(d) [blue lines].
The experiments are repeated five times for each phantom. The mean values and standard deviations of the equivalent diameter extracted in the five trials are given as a function of the mass ratio in Fig. 4(a) . According to our theory â = Σi(mi/M)ai and summing the same kind of particles, the equivalent characteristic diameter of binary mixtures can be written as â = (m s/M)a s + (m b/M)ab, where m s and m b are the total mass of small particles and big particles, respectively. It is found that the equivalent diameter â is an approximately linear function of the mass ratio m b/M [solid line in Fig. 4(a)]. The experimental results [empty dots in Fig. 4(a)] agree with the theoretical prediction well. The average relative error between the theoretical prediction and experimental measurement is only about 8.2%. Moreover, the standard deviations of five measurements are small, which indicates that the method has a good stability and repeatability for the detection of abnormal particles. Finally, we also give the equivalent diameter as a function of the number ratio of abnormal particles in the granular mixture. As shown in Fig. 4(b), the equivalent diameter increases sharply as the number ratio rises from 0 to only 0.5%. That means even though only 5 abnormal particles are mixed in 1000 normal particles, they can be sensitively and quantitatively examined by the proposed method.
In summary, we have theoretically quantified the spectrum of the PA signals from the random mixture of particles with non-uniform size. It is revealed that there is an approximate linear relationship between the content of abnormal particles and the spectral slope. Based on this finding, we successfully differentiate and quantify a trace of big micro-particles mixed in small micro-particles. Since the abnormal particles could be related to many important physiological and pathological processes, this study may provide a noninvasive and sensitive way to examine the relevant diseases, such as microthrombosis.
This work was supported by the National Basic Research Program of China under Grant No. 012CB921504, the NSF of China under Grant Nos. 11422439, 11274167, 11274171, and SRFDP Grant No. 20120091110001.
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