To characterize the transport of nanoscopic and microscopic particles (e.g., active movement of vesicles along microtubules in cells or the flow of blood cells in vessels), it is important to understand many biological processes [1,2]. A variety of techniques have been proposed to quantify the dynamics of particle flow, including particle image velocimetry (PIV) [3,4] and optical Doppler tomography (ODT) [5–8]. PIV, which involves continuous imaging of many small tracer particles to analyze the speed and direction of the flow pattern, is limited in temporal resolution. In ODT, the flow velocity at a specified location in the conduit is determined by measuring the interference between the backscattered light from a sample and a reference arm and analyzing the Doppler frequency shift. ODT is a noninvasive, noncontact optical technique; currently the ODT time resolution is mainly limited by the necessity to record hundreds of holograms for each swept wavelength [9]. Another major difficulty for Doppler flow imaging is to separate the Doppler flow signal from background noise; e.g., induced by tissue micromotion and heterogeneity [7]. Fluorescence correlation spectroscopy (FCS) [10–14] has also proven to be a valuable tool for the quantitative assessment of particle flow. In standard FCS experiments, a laser beam is focused into a sample to create a tiny observation volume of ≤1fL, from which the fluorescence emitted by particles moving through the volume is registered as a function of time [15]. Time autocorrelation analysis of the intensity yields quantitative information about diffusion and flow. Two-beam FCS [16] has proven to be a valuable tool to determine flow parameters, including both flow velocity and flow direction using cross-correlation. Compared to single-beam FCS, two-beam FCS can capture greater flow velocities and provides information about the flow direction [16]. However, both FCS and two-beam FCS require either intrinsically fluorescent particles, or particles labeled with fluorescent moieties. The unavoidable photobleaching limits its application and calls for techniques that do not rely on fluorescence.

The refractive index (RI) is an intrinsic optical parameter related to the electrical permittivity of the material. The RIs of biological particles, subcellular organelles, or cells differ from their (usually aqueous based) surrounding biofluid; therefore, the RI can provide the necessary contrast against the background. The RI of subcellular organelles can be measured with high accuracy using quantitative phase microscopy (QPM) [17–22]. Cheong et al. [23] employed label-free holographic (inline holography) flow cytometry to study avidin binding to biotinylated polystyrene spheres. Min et al. [24] combined off-axis digital holographic microscopy (DHM) with flow cytometry to determine the features of different cells. Merola used tomographic flow cytometry to characterize the flow of red blood cells and diatom algae in a microfluidic channel [25]. All these methods exploit phase information to create the necessary contrast. Notably, the measurable flow speed is limited due to the particle trajectory-based analysis.

Here, we present two-beam phase correlation spectroscopy (2B-ΦCS), a label-free approach to quantify particle flow, such as in vivo blood flow or particle flow through a standard microfluidic chip [26–28]. In 2B-ΦCS, phase fluctuation time traces are acquired by quantitative phase imaging, and correlation analysis yields information on the concentration and velocity of flowing particles. 2B-ΦCS overcomes photobleaching and phototoxicity issues inherent in fluorescence-based approaches, and thus can be widely applied in many fields, including biomedicine.

To implement 2B-ΦCS with a microfluidic chip [Fig. 1(a)], we drove a fluid containing particles through a channel with a width of 1 mm and a height h of 100 μm, thus avoiding defocus [Fig. 1(b)]. The sample was imaged (perpendicular to the flow direction) by partially coherent point-diffraction digital holographic microscopy (PC-pDHM) using a 10×/0.45 objective. With its common path configuration, PC-pDHM has an ultrahigh phase stability against external disturbances and a high sensitivity to refractive index changes in the sample. The illumination beam propagated vertically through the channel. The transmission image was magnified by a microscope system and recorded by a camera after interfering with a reference wave. A sequence of holograms was recorded by a camera (DMK33UX174, The Imaging Source, LLC, Charlotte, NC, USA) at a rate up to 162s−1, limiting the time resolution to ca. 6 ms. Phase images are reconstructed using the conventional off-axis hologram reconstruction algorithm [Fig. 1(c)].

Two circular regions of interest (ROIs), shown by the red and green circles in Fig. 1(c), were selected in the phase image series. Two phase-time traces, F1(t) and F2(t), were calculated by integrating the phases of all pixels within the red and green ROIs, respectively, for every time point, t [Fig. 1(d)]. From the two time traces, two autocorrelation functions (ACFs) and a cross-correlation function (CCF) were calculated [Fig. 1(e)] as {Gc(τ)=⟨Fc(t)·Fc(t+τ)⟩⟨Fc(t)⟩2−1,G×(τ)=⟨F1(t)·F2(t+τ)⟩⟨F1(t)⟩⟨F2(t)⟩−1,(1)where Gc(τ) represents the ACFs of the two traces, with time lag τ and trace index c=1,2; and G×(τ) is the CCF of the two traces. Angular brackets indicate averaging over all the pairs of phase values within the recorded data set.

In close analogy to FCS [16,29], the amplitude of Gc(τ) is equal to the reciprocal of the mean number N of the particles in the two observation volumes [Fig. 1(b)]. The width of Gc(τ) represents the average residence time of a particle in the observation volume, whereas the peak of G×(τ) gives the time a particle takes to move from the first to the second region. For diffusional motion superimposed with a constant velocity flow, the ACFs and CCF can be globally fitted with the model functions {Gc(τ)=1N(1+ττd)−1exp{−(ττf)2(1+ττd)−1},G×(τ)=1N(1+ττd)−1exp{−[(d0r0−ττf)2(1+ττd)−1]}.(2)

Here, τf=r0/vf is the correlation time of the particle flow (i.e., the average time a particle takes to transit the circular ROI with radius r0), and d0 is the spatial displacement between the centers of two ROIs. Note that the fitting model is equivalent to the two-beam FCS [16] for flow analysis [29,30], with the particle phase as the observable instead of the fluorescence intensity. The 2D free diffusion of the measured species was captured by the term (1+τ/τd)−1, with the diffusional correlation time τd. By using nonlinear least-squares fits of Eq. (2) to the experimental Gc(τ) and G×(τ) curves, the parameters could be determined. The concentration of the particles was calculated from N using C=N/(πr02h). Furthermore, the average flow velocity of the particles was determined using vf=r0/τf. For typical biofluid applications, such as fluid flow in biological vessels, τd is one or two orders of magnitude greater than τf, so we neglected diffusion in the analyses of the different applications presented below.

In general, 2B-ΦCS can be applied to measure flow velocities spanning a broad range by adjusting the d0 parameter because a larger d0 is obviously more suitable for larger flow velocities. Our present time resolution of 6 ms is sufficient to monitor the fluid flow with speeds in the range of 100–300μms−1 (see Section 2.E), selecting two ROIs with suitable radii and mutual displacement.

To demonstrate the enormous sensitivity enhancement of phase over intensity correlation analysis, we performed time correlation analysis using either the amplitude or the phase as the observable to measure the particle flow. To this end, 5 μm diameter yeast cells (Angel Yeast Co., Ltd., Yichang, China) were diluted in phosphate-buffered saline (PBS) solution to a concentration of ∼0.6gL−1 (Section 3.D). The solution was pumped through the microfluidic channel sandwiched between two cover slips and imaged by PC-pDHM for 200 s at a frame rate of 30s−1. Amplitude and phase images were reconstructed from all 6000 images, shown three times in Figs. 2(a) and 2(b), respectively. Two circular regions, with radii r0=8μm and a mutual displacement d0=24μm along the flow direction were selected for the amplitude and phase image series, and amplitudes and phases within the circular regions were integrated to form two time traces. From the resulting amplitude-time and phase-time traces, ACFs and CCF were calculated with Eq. (1) and fitted with Eq. (2). This process was performed with 10 groups of amplitude and phase image sequences to obtain the mean ACFs and CCF shown in Figs. 2(c) and 2(d), respectively. Obviously, the phase images have much higher contrast than the amplitude images; therefore, they have a much-enhanced SNR. The fit of the ACFs and CCF curves in Fig. 2(d) with the model in Eq. (2) yielded C=(3.2±0.1)×105μL−1 and vf=77.8±0.9μms−1.

We further tested 2B-ΦCS for measuring the flow velocity of polymethyl methacrylate (PMMA, KM-5050, diameter 5 μm, DG KM New Material Co., Ltd., Dongguan, China) beads dissolved in an aqueous (PBS) solution. The solution flowed through a microfluidic channel with an electrical pump at three different speeds. For each of the three measurements, we took 6000 DHM holograms in 200 s. The procedures and ROI setting were the same as in Section 2.B. As expected, the peak of the CCF shifts to shorter times with an increase in the flow speed [Fig. 3(a)]. The global fit of Gc(τ) and Gx(τ) with Eq. (2) further revealed vf=114±1μms−1, 60±3μms−1, and 50±1μms−1 for the three flow speeds. The 2B-ΦCS results agreed within the error found by PIV-based analysis, yielding 113±16μms−1, 66±15μms−1, and 60±12μms−1, but had markedly smaller uncertainties [Fig. 3(b)]. Note that in PIV-based analysis 30 beads were randomly selected for velocity determination.

The large error here was mainly due to the parabolic flow velocity of the selected particles with respect to their distance to the surfaces of the channel (Poiseuille’s law). Thus, 2B-ΦCS had the merit of a simpler analysis and yielded more reliable results since more particles were included. However, the measurement yielded the flow velocity averaged over the velocity distribution of all registered individual particles.

We further explored the capability of 2B-ΦCS to determine the concentration of flowing particles. The PMMA particles were prepared with three concentrations: 1.2×104μL−1, 2.6×104μL−1, and 4.0×104μL−1 that were determined by simply counting the particle number in the field of view and dividing it by the volume. Figure 4(a) shows the two (essentially identical) ACFs and CCF for three different concentrations.

It is obvious that the amplitudes decrease with an increase in the PMMA concentration. The fit of the correlation curves with Eq. (2) gives the concentrations as C=(1.2±0.1)×104μL−1 (blue), (2.6±0.1)×104μL−1 (green), and (3.9±0.1)×104μL−1 (red), which was in excellent agreement with the counting results.

To measure the concentration of red blood cells (RBCs) in vitro, blood was drawn from the rat [Sprague Dawley (SD) rat, Air Force Medical University, Xi’an, China] and diluted with PBS at a volume ratio of 1:100. The solution was measured, as illustrated in Fig. 5(a), using the same procedure as in Section 2.C. Phase images were reconstructed from the recorded holograms; a few selected ones are shown in Fig. 5(b). We selected two circular ROIs with r0=12μm and d0=36μm. ACFs and CCF were calculated and fitted with Eq. (2) to obtain the concentration of RBCs in the solution. The measurement was repeated 35 times; for each measurement, eight sets of regions were selected for 2B-ΦCS analysis to yield 280 independent data points, which is shown as a histogram in Fig. 5(c). A Gaussian fit revealed that an average concentration of RBCs was (4.7±1.9)×106μL−1 (mean±s.d.).

Furthermore, the RBC diameter could be assessed with the phase image correlation approach mentioned in Section 3. A phase image was transformed into a time trace F(xk), and the ACF of F(xk) was calculated with the first equation in Eq. (1). The fit of the ACF with Eq. (3) (Section 3) provided the average particle diameter. The measurement was repeated 30 times, and six phase images were selected from each measurement for the 2B-ΦCS analysis to yield 180 data points which were compiled in a histogram [Fig. 5(d)], revealing an overall average diameter of 4.6±0.4μm (mean±s.d.). Notably, the measured concentration and radius agreed well with the reported concentration [(3.8–5.5)×106μL−1] [30] and diameter range (4.63±0.04μm) of rat RBCs [31].

The in vivo characterization of blood flow is important in both clinical diagnosis and biomedical research. In this section, 2B-ΦCS was used to monitor blood flow in live zebrafish embryos, as depicted in Fig. 6(a). Live embryos (three to four days post fertilization) were anesthetized and imaged at a rate of 100s−1. An exemplary phase image reconstructed from the PC-pDHM hologram series [Fig. 6(a), right hand side] shows RBCs in a vein (posterior cardinal vein, PCV) and an artery (dorsal aorta, DA) in phase contrast. For 2B-ΦCS analysis, we again selected two circular regions, from which the phase values were extracted to form time traces. As an example, ACFs and CCF calculated from the DA data are shown in Fig. 6(b). The ACFs and CCF displayed artifacts within the time range of 0.1–0.5 s; i.e., the curves drop below zero. This was caused by the periodic acceleration of the blood due to the heartbeat. The global fit of the ACFs and CCF curves with the model in Eq. (2) revealed a flow speed in the artery of 179±13μms−1. Figure 6(c) shows the statistics on the flow velocities of RBCs in arteries (red square) and veins (gray square) characterized by 2B-ΦCS on 15 zebrafish embryos, yielding average flow velocities of 290±110μms−1 (mean±s.d.) in the DA and 120±50μms−1 (mean±s.d.) in the PCV. Thus, blood flows roughly twice as fast in the DA as in the PCV. PIV-based analysis on the same phase images gave 270±210μms−1 (mean±s.d.) for DAs and 90±67μms−1 (mean±s.d.) for PCVs, showing agreement with 2B-ΦCS within the error. The considerable spread of the mean over the population likely reflects (size) variations within the zebrafish ensemble [32]. This experiment demonstrates that 2B-ΦCS can be applied to in vivo measurements of blood flow in live zebrafish.

Partially coherent point-diffraction DHM (PC-pDHM) [35] was used to acquire quantitative phase images of all samples. The PC-pDHM used in our experiments is equipped with a 10×/0.45 microscope objective. The spatial resolution of the system was 0.8 μm upon illumination with 532 nm light. The off-axis hologram has 1920×1080pixels and a carrier frequency of 9.6μm−1. Compared to conventional DHM, PC-pDHM has a common path configuration and is, therefore, robust against environmental disturbances. Furthermore, the use of partially coherent illumination endows PC-pDHM with a high SNR. For 2B-ΦCS measurements, the samples, either the microfluidic chip or the live zebrafish embryo, were mounted on top of the objective of the PC-pDHM, and off-axis holograms were recorded sequentially in time. Both amplitude and phase images were reconstructed from the recorded holograms using an angular spectrum-based approach [35]. Notably, the defocus of the sample can be digitally compensated by propagating the reconstructed wave from the hologram plane to the image plane. 2B-ΦCS analyses were performed on the phase image series using Eqs. (1) and (2).

For 2B-ΦCS on flowing particles and blood cells, a transparent microfluidic device was fabricated as follows. Two rectangular strips of double-sided adhesive tape were placed on a glass substrate (25×75mm microscope slide), leaving a 1 mm wide gap. A channel was formed by using a microscope cover glass as a top cover. The resulting channel had a depth of 100 μm (measured by confocal microscopy), which ensured that particles of various sizes could pass the channel and, furthermore, avoided defocus of particles in different layers during PC-pDHM imaging. Fluid inlet and outlet ports were made by drilling two holes (1.5 mm diameter, 30 mm apart) at the two ends of the channel and connecting them to microfluidic tubes with an inner diameter of 0.5 mm.

The radius of an RBC can be assessed from a single phase image using phase image correlation analysis [36], which also uses phase image sequences of the sample. In this method, a time trace F(xk) is generated by rearranging the pixelated phase values in the phase image along a raster scanning pathway. From this sequence, the ACF of F(xk) was calculated with the first equation in Eq. (1) and fitted with Gc(ξ)=1Nexp(−ξ2·σ2rp2).(3)

Here, rp is the average particle radius, N is the average particle number in the observation volume πrp2h, and σ is the pixel size of the CCD camera. ξ is an independent variable that represents the pixel displacement. This method yields the particle radius rp from a single-phase image; hence, it is fast.

Natural highly active dry yeast (diameter 5 μm, Angel Yeast Co., Ltd.) and PMMA microparticles (PMMA, KM-5050, diameter 5 μm, DG KM New Material Co., Ltd.) were used without further purification. A total of 30 mg of yeast was dissolved in 50 mL deionized water to generate 0.6 g/L concentration for the velocity measurements shown in Fig. 2. Similarly, PMMA microparticles were dissolved in deionized water to generate three different concentrations of 2, 3, and 4gL−1 for the concentration measurements shown in Figs. 3 and 4. Both yeast and PMMA solutions were vortexed before the 2B-ΦCS experiments to ensure monodispersity.

Blood was freshly drawn from the rat and stored in an anticoagulation tube. Before the 2B-ΦCS measurement, we diluted the rat blood a hundred fold with a PBS solution, filled it into a syringe, and pumped it through the microfluidic chamber at a constant rate for data acquisition.

Live wild-type zebrafish embryos (strain AB, Shanghai FishBio Co., Ltd.) were cultured at 28°C (14 h in light and 10 h in the dark each day) for three or four days post fertilization, and the nutrient solution was exchanged regularly. Before the 2B-ΦCS measurement, the embryos were anesthetized by mounting in 0.006% tricaine solution to suppress spontaneous body movement during the measurement. Then, the animals were placed on a glass slide and transferred to the PC-pDHM microscope to collect image sequences of the zebrafish vessels.

In this work, we introduced 2B-ΦCS and demonstrated that it is a powerful tool to study the concentration, size and velocity of flowing particles. In 2B-ΦCS, time-dependent RI variations are observed by quantitative phase imaging approaches, including, but not limited to, interferometric approaches, such as the DHM used in this study. 2B-ΦCS uses the same pair correlation approach as FCS and allows measurement of particle dynamics over a wide particle size range, even with diameters smaller than the microscope’s resolution limit.

Unlike FCS, 2B-ΦCS does not suffer from photobleaching and phototoxicity introduced by fluorescence labeling since it uses the RI difference of target particles against the surrounding medium, which is an intrinsic property providing an alternative contrast mechanism. In contrast to conventional single-focus FCS, 2B-ΦCS is based on the analysis of an image series of a rectangular sample region. Therefore, it can assess the dynamics at different image locations, which is beneficial when studying heterogeneous flow.

2B-ΦCS can be implemented by using a microfluidic device, which provides a convenient environment to study flowing particles. The width and height of the microfluidic channel should not be too small; otherwise, the flow velocity tends to have a parabolic distribution across the channel according to Poiseuille’s law, and the velocity measured by 2B-ΦCS will be an average over the velocity profile within the observation volume.

There are related approaches that use phase fluctuations to extract dynamics information from the sample. Ma et al., for example, proposed the phase correlation imaging (PCI) method [33] to quantify the diffusion of intercellular biomolecules. In PCI, a correlation map is calculated from the acquired images along a time sequence and analyzed in the spatial frequency domain to determine the size and diffusion coefficient of the particles. 2B-ΦCS uses the concept of pair correlation along the flow direction. Hence, it is more straightforward and more precise for flow velocity determination and can capture a wider velocity range by choosing a suitable displacement for the two circular areas in the phase images. Importantly, 2B-ΦCS can be used with any phase imaging technique; hence, it is compatible with existing holographic flow cytometry approaches (inline holography, off-axis holography, and tomography) to obtain quantitative phase images [23–25]. The combination with 2B-ΦCS analysis yields a much faster analysis than the trajectory-based one and enables measurements on much denser samples, although our work demonstrates 2B-ΦCS analysis with transmission DHM, reflection DHM (or quantitative phase microscopy in general), and even OCT is also possible.

2B-ΦCS has been demonstrated for both in vitro inspection of RBCs flowing in a microfluidic device and in vessels of live zebrafish embryos, providing valuable information on the concentration and flow velocity of RBCs. Notably, we observed a blood flow twice as fast in an artery as in the vein of a live zebrafish embryo [34]. We anticipate that our technique will find future application in biological imaging and biomedicine in general, and perhaps also in industrial applications.