1. Introduction

The measurement of cell motility is important for the study of living cells sampled from biological tissues. For example, the motility of cancer cells correlates with their metastatic potentials [1]. As for nanometer-scale measurements, the voltage-induced cell membrane motion was examined by applying atomic force microscopy (AFM), and its relationship to the charge density of the cell surface was demonstrated [2]. In recent years, the biophysical properties of cancer cells have been investigated by AFMs in nanometer scale [3].

Considering the intactness of the cell samples, it is desirable to use a non-invasive label-free technique to image the dynamic morphology of the cells. In this regard we expect that an optical technique that uses interferometry would be ideal. There are two optical schemes available to measure the cell morphology. One is quantitative phase microscopy (QPM), which allows the measurement of the phase of optical fields in the sub-wavelength resolution. So far, Hilbert Phase Microscopy (HPM) [4–6], Quantitative Phase Microscopy by white-light interferometry [7], and Digital Holographic Microscopy (DHM) [8,9] have been used for the imaging of living cells. The other is Optical Coherence Tomography (OCT), which allows the acquisition of weak light reflected from spatially localized regions [10–12].

The key technique of QPM is to compensate the phase drifts and phase noise associated with the measurement system by monitoring the phase-referenced signals [4,13]. HPM and the DHM have the nanometer-scale resolution of the optical path [6,8]. However, they do not allow the discrimination of multiple reflection surfaces [4,9,14]. OCT, on the other hand, allows the discrimination of reflection signals attributed to multiple surfaces in a sample. However, many cases of conventional OCT have omitted the phase information of reflected light for obtaining the distribution of the amplitude of reflectance [10,12].

Increasing numbers of researchers in recent years have integrated the advantages of the quantitative-phase imaging technique with OCT in order to measure the dynamic morphology of a particular surface with sub-cellular resolution. Yang et al. developed a phase-referenced interferometer with a low-coherence light source in order to observe the phase fluctuation associated with cell surface motion [13]. Moreover, they employed the phase-referenced low-coherence technique to measure nerve displacement during action potential [15,16]. Concerning the OCT approach, Jeong et al. developed Fourier Domain Digital Holographic OCT (FD-DHOCT) to quantify the degree of cell motility by an intensity-based statistical technique [17,18], and Ellerbee et al. were able to capture an image of the phase fluctuation of chick embryos by applying Multi-Dimensional Spectral Domain Phase Microscopy (MD-SDPM) [19]. To distinguish a particular cell surface, some researchers used nanoparticles to enhance the reflection signals of cells by OCT [20–22]. These studies showed nanometer-scale motions of cells. However, in a label-free state the reflected light from the cell surface was not sufficiently distinguished from that of other boundaries, including the glass surface or the membrane of intracellular organelles, and when the researchers needed to take the signal from the membranes only, it was necessary to use contrast agents.

This paper shows the three-dimensional distribution of quantitatively assessed phase fluctuation caused by the motion of multiple surfaces in cultured cells. We used high-numerical-aperture (NA) illumination with a low-coherence light source while controlling the phase-referenced optical path difference at the nanometer scale [23]. The longitudinal spatial resolution (FWHM (Full Width at Half Maximum) of the interference fringe) of the imaging system was 0.93 μm, while the transversal resolution was 0.56 μm (diffraction limit). The results quantitatively revealed the depth-resolved fluctuations of the intracellular surfaces, allowing us to measure the plasma membrane, the reflecting surfaces in cytoplasmic region and the surface of the substrate independently.

2. Methods

Figure 1(a) depicts the experimental setup of Low-Coherent QPM based on a Linnik configuration. Light emitted from a halogen lamp (Nikon, LV-UEPI 50W) passed through a Linnik interferometer equipped with two identical water-immersion objective lenses (Nikon, CFI Fluor 60 × W, NA = 1.0). The reflected wavefronts from the sample and the reference mirror were focused onto the 12-bit CCD camera (Hamamatsu, C9300-201) in order to obtain interference images.

 

Fig. 1 Schematic illustration of the experimental Setup. (a): Whole setup, (b): Interference fringe observed on the CCD camera while moving the PZT2, (c): Detail of the sample arm. IR-LD: Infrared laser diode, PZT: Piezoelectric transducer, PD: Photo detector, ND: Neutral density filter, M: Mirror, L: Lenses, BS: Beam splitter, DM: Dichroic mirror (cutoff: 900nm).

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The cells under testing were cultured on a glass slide to which both an AR coating for the bandwidth of 550-950 nm and a reflection enhancement coating for 1.3 μm were applied. The AR coating was necessary because the light reflected from the cell membrane and intracellular membranes was so weak that the reflection from the glass surface would easily overwhelm the cell reflection even if coherence gating was used. The reflection enhancement coating caused the glass surface to work as a reference plane for the feedback control of optical path difference (OPD).

The emission light from the halogen lamp was spectrally filtered by a glass long-pass filter having a cutoff wavelength of 695 nm (Shott, RG695), in order to match the AR coatings of the optical components. Figure 1(b) shows the one-point interferogram measured on a single pixel of the CCD camera while linearly translating the mirror by PZT2. The fringe spacing Λ was 327 nm per fringe, and the FWHM of the envelope of the interference fringe was estimated to be 0.93 μm. As Dubois et al. reported, in the case of focusing illumination, the fringe spacing Λ became β times more than the estimated in the case of plane waves: $\Lambda =\beta \cdot \lambda /\left(2n_{medium}\right)$ where β related to the numerical aperture NA through the equation $\beta =2/\left(1+\cos \left({\sin }^{-1}NA/n\right)\right)$ [12,24]. We measured the factor β to be 1.13 using a narrow band-pass optical filter to transmit the wavelength of 780 ± 5 nm (FWHM) in front of the lamp. With the measured value of β, we estimated the effective numerical aperture of the entire system to be 0.85 and the original center wavelength λ_c of the incident imaging light to be 773 nm. The resultant diffraction limited lateral resolution d was 0.61 × λ /NA = 0.56 μm.

The feedback control system of the OPD consists of an infrared laser diode (IR-LD) (PD-LD Inc., PL13H0101), an infrared photodetector (PD) (New Focus, MODEL 2053), a piezoelectric transducer (PZT1) (NEC TOKIN, AE0505D08F) to translate the reference mirror longitudinally, and a long-working-distance piezoelectric transducer (PZT2) (Physik Instrumente, P-611.ZS) to translate the sample dish. The optical path of the IR light is adjusted so that it is reflected by the glass surface in the vicinity of the cell (see Fig. 1(c)). The details of this feedback system have been described in our previous publications [7,23].

The PZT1 adjusts the OPD fast (~500 Hz) with a small stroke (<330 nm) at high resolution (<1 nm), while the PZT2 under the sample dish covers a large stroke (100 μm maximum) at coarse resolution (20-nm step). The feedback circuit continuously controls the PZT1 in the reference arm to adjust the OPD by nanometer resolution. Only when the PZT1 exceeds its assigned maximum stroke (330 nm) does the PZT2 complementally move the sample dish to cancel the PZT1 motion. Therefore, by consecutively commanding the OPD scanning at sub-wavelength increments/decrements, the focal plane and equal-path-length plane are scanned simultaneously.

The minimum number of raw images needed to make a phase image is 3 [25]. However, in order to compensate for the non-uniformity of the interference amplitude due to the envelope of the coherence function, a more sophisticated phase-shifting algorithm is preferable. Because Hibino et al have demonstrated that their seven points phase shifting algorithm estimates the phase ϕ more accurately than the traditional four step algorithm, under the presence of the amplitude modulation of the interferogram [26], we adopted their algorithm. Following Hibino’s algorithm, we take seven raw images $I_{-3\pi /2},I_{-\pi },I_{-\pi /2}$ $,I_0,I_{\pi /2}$ $,I_{\pi },I_{3\pi /2}$ to construct a single-phase image. Here I_δφ represents the raw intensity image on CCD with phase shift δφ. Notice that 2π of the phase shift corresponds to the displacement of the PZT2 of Λ (327 nm). Using the optical electric field of the reference and the sample as E _r and E _s, the raw intensity on CCD can be expressed as

Because $C\left(x,y\right)=\eta \left(2\gamma \left(z\right)\left|E_r\right|\left|E_s\right|\right)\left(\cos \phi +i\sin \phi \right),$ R and ϕ are thus

As shown in Fig. 2 , we increased the OPD from $-3\pi /2$ to$3\pi /2$, and then decreased it from $3\pi /2$ to $-3\pi /2$ at intervals of$\pi /2$, while capturing the images. In this cycle, two electric field images were obtained by scanning in the forward and reverse directions, and we averaged them in the complex plane to make one image per measurement cycle. By repeating this phase-shifting cycle a series of electric field images C _n, reflectance images R _n and reflection phase images ϕ _n were obtained at time intervals Δt.

 

Fig. 2 Timing chart of the phase shifting and CCD exposure.

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3. Results

3.1 Phase image and mean squared displacement

The sample is a cell line derived from human breast cancer cells labeled MCF-7. The cells were subcultured on the glass slides two days prior to measurement. The phase shifting was performed at intervals of 104 milliseconds so that phase images were obtained in the intervals of Δt = 1.25 sec. Figure 3(a) shows transmission-phase and reflection-phase images of the living cell sample under testing. The transmission phase image is obtained by using the specular reflection from the bottom glass underneath the cells. The reflection-phase image shown in Fig. 3(b) was taken at the height 2.4 μm above the cell bottom. We captured the reflection-phase image of this cell at this height for 125 seconds. The phase of the reflected light from the cell fluctuated over time, as shown in Fig. 3(c), with the trend in fluctuation being dependent on the position. The fluctuation of the phase of the light reflected from the cell membrane (point A) shows the vertical motion of the membrane. In the case of the fluctuation of the reflection signal from the first surface of the sample, the conversion factor of the phase change Δϕ to the vertical motion Δh is $\Delta h={\lambda }_c\cdot \Delta \phi /\left(4\pi \cdot n_m\right)=52nm/rad\times \Delta \phi$, where n _m is the refractive index of the medium (n _m = 1.335).

 

Fig. 3 Phase images of the living cell under test, including (a) transmission phase image; (b) reflection phase image at the height 2.4 μm above the glass surface; and (c) typical temporal fluctuation of the phase of the reflection signal. The blue and black lines show the temporal fluctuation at the points A and B shown in Fig. 3(b), respectively. The green line shows the temporal fluctuation measured at the top of a 10 μm polystyrene bead immersed in pure water.

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Figure 3(c) also indicates the phase fluctuation of the reflected light from the top of a 10 μm polystyrene bead immersed in pure water (n = 1.333) as a means to compare the mechanical stability of the system. The stability of the phase of the light reflected from the polystyrene bead was 0.023 radian (the standard deviation), which corresponds to 1.2 nm.

In order to quantify the phase fluctuation, we used the mean squared displacement (MSD) $\Delta {\phi }^2\left(\tau \right):$

The MSD, which is the mean squared change of the displacement at times t and t-τ, characterizes the random migration of cells [27] or particle motion in the cells [28]. Recently, the researchers using the quantitative phase imaging technique employed MSD to characterize the cell membrane displacement caused by cell growth of epithelial cells or beating of cardiomyocyte [29,30]. Here we also employ the MSD to characterize the z-directional motion of the membrane in order to quantify the activity in the cells. To calculate the MSDs accurately, it was necessary to compensate for the overestimation of MSD attributable to the random phase fluctuation due to the noise on the CCD. All the images of phase fluctuation in this section were compensated for random noise, and the compensation procedure is given in Section 4.

3.2 Full-field image of phase fluctuation

The MSDs were calculated at every position in the field of view. Figures 4(b) -4(d) show the distribution of the MSD with the τ of 1.25, 3.75 and 6.25 seconds. Figure 4(a) shows the distribution of the reflectance R(x,y) in Eq. (3). The pixels where the fringe contrasts were insufficient were masked out and shown by white color in Figs. 4(b)-4(d). The MSD increases by making the τ larger. The intracellular cytoplasmic region, such as the point B in Fig. 4(b), fluctuates more dynamically than the membrane region, such as at point A. The source of the reflection from the intracellular region such as the point B can be intracellular organelle surface reflection or the inhomogeneity of the concentration of chemicals in cytosol which behaves as a superficial surface. Though we do not have a conclusion about the source of the reflection in cytoplasm, the motion of the objects or superficial surfaces in cytoplasm should reflect the liquidity of the intracellular region. There is a space within the cell where the reflectance was smaller than the surrounding area. Based on the three-dimensional morphology, which will be shown later, we believe this area is a nucleus and the high reflectance granular objects in this space are nucleoli. The reflectance from inside the nucleus was so weak that sufficient interference signals for the phase measurement were not obtained. To see the reflection from inside the nucleus, the employment of a light source with higher power would be useful.

 

Fig. 4 Phase fluctuation of the living MCF7 cell sample: The unit of the x axis and y axis is the micrometer. (a): Reflectance image R(x,y). The unit of color is arbitrary. (b)-(c): Distribution of the MSD: The pseudo color shows the MSD $\Delta {\phi }^2\left(x,y,\tau \right)$ and the unit of the color bar is the radian². The τ is 1.25, 3.75 and 6.25 seconds in the Figs. 4(b), 4(c) and 4(d), respectively. The unit of the x axis and y axis is μm.

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In order to demonstrate the validity of our method, we fixed the sample with 2% paraformaldehyde. After the treatment, the cells died and the membrane and intracellular structures were fixed chemically. Figure 5 shows the phase fluctuation observed in the same cells after fixation. Figure 5(a) shows the reflectance R(x,y), and Figs. 5(b)-5(d) show the distribution of the MSD with the τ of 1.25, 3.75 and 6.25 seconds, respectively. The MSD dramatically decreased, and the intracellular fluctuation, which was distinctive in the living cells, was suppressed to the same level as that of the membrane region. Figure 6 summarizes the behavior of the MSD as a function of τ, where the MSD at the positions A and B in Fig. 4(b) and C and D in Fig. 5(b) are plotted as a function of τ. The right axis of Fig. 6 also indicates the MSD measured by the corresponding vertical motion of the first surface in culture medium. Because the conversion factor of the phase change Δϕ to the vertical motion Δh is $\Delta h=52nm/rad\times \Delta \phi$, the conversion factor between the MSD in phase to the MSD in vertical motion is $\Delta h^2\left(\tau \right)=2704n{m}^2/ra{d}^2\times \Delta {\phi }^2\left(\tau \right)$. The slope of the MSD at the point A, as measured by the nanometer scale, is 31 nm²/sec., which is in agreement with the AFM studies on cell membrane [31].

 

Fig. 5 Phase fluctuation of the MCF7 cell sample after paraformaldehyde treatment. The unit of the x axis and y axis is the micrometer. (a): Reflectance image R(x,y): The unit of the color is arbitrary. (b)-(c): Distribution of the MSD: The pseudo color shows the MSD $\Delta {\phi }^2\left(x,y,\tau \right)$ and the unit of the color bar is the radian.² The τ is 1.25, 3.75 and 6.25 seconds in the Figs. 5(b), 5(c) and 5(d), respectively. The unit of the x axis and y axis is μm.

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Fig. 6 MSD$\Delta {\phi }^2\left(\tau \right)$at the positions A and B in Fig. 4 and C and D in Fig. 5.

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3.3 Three-dimensional tomography of the phase fluctuation

To analyze the fluctuation of the cell membrane and intracellular structure in detail, we captured the phase-fluctuation image at different heights. Figure 7 indicates a chart of phase-shift timing. The phase image at one sectioning height was taken for 10 times (12.5 sec.). Subsequently, the sectioning height was shifted by 327 nm, corresponding to 2π phase shift of the illumination light, and another 10 phase images were taken at this new sectioning height. By repeating this procedure, the stack of en-face phase images at every sectioning height is obtained.

 

Fig. 7 Timing chart of phase shift for tomographic imaging of phase fluctuation.

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By calculating the MSD at each sectioning height, we obtained the tomographic image of MSD as a function of four variables $\Delta {\phi }^2\left(x,y,z,\tau \right)$. Figure 8 shows horizontal, cross-sectional images of the tomographic phase-fluctuation data $\Delta {\phi }^2\left(x,y,z,\tau \right)$ where τ = 3.75 and z = 2.4, 3.9, 5.9 and 7.4 μm. Figure 9 shows the vertical cross-sectional images of the tomographic phase fluctuation, as reconstructed from the overall data $\Delta {\phi }^2\left(x,y,z,\tau \right)$. The sectioning line for Fig. 9 is shown in Fig. 8(a). The cross section of the reflection amplitude is shown in Figs. 9(a) and 9(c), which correspond to the so-called B-scan image of the line-field OCT. In Fig. 9(a) can be seen a spherical membrane inside the cell, which we believe is the nuclear membrane.

 

Fig. 8 Tomographic image of phase fluctuation: The time difference τ for calculating MSD is 3.75 sec. (a): $\Delta {\phi }^2\left(x,y,z=2.4\mu m,\tau =3.75\right)$, (b): $\Delta {\phi }^2\left(x,y,z=3.9\mu m,\tau =3.75\right)$, (c): $\Delta {\phi }^2\left(x,y,z=5.5\mu m,\tau =3.75\right)$, and (d): $\Delta {\phi }^2\left(x,y,z=7.4\mu m,\tau =3.75\right)$. The unit of the x, y and z axes is μm.

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Fig. 9 Vertical cross section of the tomographic phase-fluctuation image. Figs. (a) and (b) are images of the living cell, and the figs. (c) and (d) are images of the paraformaldehyde-treated cell. Figs. (a) and (c) are the cross sections of the reflectance, which is calculated from the amplitude component of the interference image. Figs (b) and (d) are the cross-sectional images of the mean squared displacement of phase fluctuation $\Delta {\phi }^2$ where τ = 3.75 and y = 20μm. The unit of the color bar for the figs. (b) and (d) is the radian.² The unit of the x axis and z axis is μm.

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4. Numerical processing

4.1 Overview

The mean squared displacement (MSD) of the phase $\Delta {\phi }^2\left(\tau \right)$, which is the mean squared change of the phase at times t and t-τ, deviates by the fringe contrast of the interference image. When the fringe contrast is low, the measured phase has a significant degree of error due to the random intensity noise of the CCD [32], so that the MSD is overestimated. The fringe contrast observed on the cell membrane was typically ± 100 counts over 3000 counts of the DC, and the error attributed to the random intensity noise of the CCD was not negligible. To compensate the overestimation of the MSD, we have formulated the phase errors attributed to the CCD noise and shown the manner in which the overestimation of MSD was compensated.

4.2 Estimation of random noise

The signal intensity measured with the CCD camera, at each exposure, has random noise attributable to the readout noise of the analog-to-digital converter as well as the electron counting noise (quantum noise). Figure 10(a) shows the temporal standard deviation ${\sigma }_{CCD}$ of the signal from the pixels of the CCD camera (C9300-201, Hamamatsu, 12-bit, gamma = 1.0) as a function of the temporal average of the count $N_{CCD}$. We uniformly illuminated the CCD with the halogen lamp and took 1,200 images in order to derive the average count $N_{CCD}$ and the temporal standard deviation ${\sigma }_{CCD}$. The measurement was repeated with different (OD = 0 and OD = 0.4) light intensities and exposure times (20msec., 40msec. and 80msec.). The noise characteristic of the CCD camera was consistent with the following equation:

 

Fig. 10 Noise characteristic of the CCD camera C9300-201. (a) Standard deviation σ_CCD of the measured CCD intensity (analog-to-digital converted value) as a function of the average intensity N _CCD. (b) Example of the distribution of the intensity: Black line: measured histogram of the count on the CCD pixels. The mean value is 3261 and the standard deviation is 26.28. Green line: Gaussian curve with the mean value of 3261 and the standard deviation of 26.28. The inset shows a zoom-in of the region contained within the gray box.

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Notice that the electron counting noise ${\sigma }_{electron}$ is given by the count of the electron $N_{electron}$: ${\sigma }_{electron}=\sqrt{N_{electron}}$. Because the well depth of the CCD camera C9300-201 is 20,000 electrons (as indicated by the manufacturer’s operating manual), the electron counting noise should be $\sqrt{20000}=$141 electrons at the maximum, which is 0.7% of the mean intensity. This is consistent with our estimation (${\sigma }_{CCD}=29$ at $N_{CCD}=4096$).

The histogram of the counts on the CCD pixels well fits to the normal distribution with the mean value of N _CCD and the variance of σ _CCD ², as shown in Fig. 10(b). We recorded the raw intensity on the CCD; I(x,y,n), for n = 1 to 1,200 frames while uniformly illuminating it, then cropped a 100 × 100 area in which the uniformity of the illumination was the best. From the 100 × 100 × 1200 samples, we made the histogram shown in Fig. 10(b). The mean value of the spatial standard deviation at every frame was 26.22, and the mean value of the temporal standard deviation at every pixel was 26.17. This result implies that the noise characteristic of the CCD is ergodic.

This uncertainty regarding the intensity of each image affects the complex interference image C written in the Eq. (2). According to the nature of normal distribution, the variance of the summation of normal distributions is the summation of the variances of the normal distributions. Therefore, the standard deviation of the real part and the imaginary part of the interference signal C in the Eq. (2) are estimated to be ${\sigma }_{\mathrm{Re}}=0.433{\sigma }_{CCD}$ and ${\sigma }_{\mathrm{Im}}=0.405{\sigma }_{CCD}$, respectively. To make the following calculation easier, we approximated the probability density function of the interference signal to be cylindrically symmetric as,

This uncertainty of the interference signal is schematically shown in Fig. 11(a) and the uncertainty of the phase (in radian) is written as,

 

Fig. 11 Interference signal shown in the complex plane: (a) Schematic illustration, (b) The experimentally measured probability density function of the interference signal C on a bead surface measured for 125 seconds.

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Therefore, the uncertainty of the phase ${\sigma }_{\phi }$ will make the MSD overestimated as,

To show the validity of this estimation of the MSD, we measured the surface of 10 μm polystyrene beads immersed in pure water as a test target. We recorded 100 phase images with time intervals of 1.25 second. Figure 11(b) shows an example of the temporal fluctuation of the interference signal C on a position (x ₀,y ₀) on the bead, where the real part and imaginary part of $C\left(x_0,y_0,t\right)$ are plotted. The blurring of the interference signal due to the random intensity noise ${\sigma }_{CCD}$ can be seen.

Figure 12 shows the relationship between the estimated overestimation of the MSD and the measured MSD. We have sampled a 30 x 30 pixel area (corresponding to 3.5 μm by 3.5 μm) on the surface of the 10 μm bead. Due to the coherence gating by the white light, the fringe contrast was not uniform on the bead surface, and such a situation was suitable for demonstrating the effect of low fringe contrast. Figure 12 plots the measured MSD as a function of estimated MSD. The result implies that the random intensity noise of the CCD is the dominant factor in the overestimation of the superficial MSD.

 

Fig. 12 Raw mean squared displacement observed on a bead surface as a function of the estimated offset of the MSD. The green lines show $\Delta {\phi }^2=2{\sigma }_{\phi }{}^2$. (a) τ = 1.25, (b) τ = 12.5.

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4.3 Compensating the overestimation of MSD

Because we estimated the overestimation of the MSD quantitatively as Eq. (8), we can compensate the overestimation by subtracting it from the superficial MSD:

Figure 13 shows the MSD on the 10 μm bead surface (3.5 μm by 3.5 μm area). On the pixels where the fringe contrast is low (see Fig. 13(a)), the MSD will be overestimated as shown in Fig. 13(b); however, the overestimation is compensated as shown in Fig. 13(c) after the offset subtraction. We demonstrate how the overestimation of the MSD is successfully compensated also in the cell measurement by comparing the images before and after compensation. For instance, we show the compensation of MSD in the case of the fixed cell shown in Fig. 5. Figure 14(a) shows the raw MSD before compensation. τ = 3.75 was chosen for the demonstration. Figure 14(b) shows the estimated overestimation of the MSD. Then, Fig. 14(c) shows the MSD after compensation. Figure 14(c) is the repetition of Fig. 5(c), but the scale of the color bar is changed.

 

Fig. 13 Compensation of the MSD on the bead surface: (a) Interference fringe, (b) MSD before compensation, (c) MSD after compensation. The unit of the color bar is the radian².

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Fig. 14 The MSD of the fixed cell shown in the Fig. 5 before and after the compensation. τ = 3.75 was chosen for the demonstration; (a) the raw MSD before the compensation, (b) the estimated overestimation of MSD, (c) the MSD after the compensation performed by subtracting the image (b) from the image (a). The unit of the x and y axes is μm. The unit of the color bar which is common in these three images is the radian².

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As shown in the Eq. (8), the random intensity noise offsets the MSD independent of τ. Figure 15 shows the MSD $\Delta {\phi }^2\left(\tau \right)$at the positions A and B in Fig. 4 and C and D in Fig. 5 before the compensation. In the living cell sample, the estimated overestimations of the MSD at the points A and B were 0.0187 and 0.0511 radians², respectively, while in the fixed cell sample, the estimated overestimations of the MSD at the points C and D were 0.0521 and 0.0824 radians², respectively. Because the overestimations of MSD are inversely proportional to the intensity of the reflection signal, the overestimation of MSD tend to be larger in the intracellular region than on the cell membrane. The compensation of the MSD is important especially when we analyze the quiet cells or dead cells in order not to pick up artifact signals.

 

Fig. 15 The raw MSD $\Delta {\phi }^2\left(\tau \right)$ before compensation as a function of τ. (a): The raw MSD of the living MCF7 cell at the positions A and B in Fig. 4. (b): The raw MSD of the fixed MCF7 cell at the positions C and D in Fig. 5. In these figures, the estimated offset of $\Delta {\phi }^2\left(\tau \right)$ is also shown by dashed lines.

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5. Conclusion

Our Low-Coherent QPM visualizes the distribution of the fluctuations of the intracellular surfaces and cell membrane. The image acquisition rate is 1.25 sec. per image, and the lateral resolution is 0.56 μm. The mechanical stability of the system is 0.023 radian (the standard deviation), corresponding to 1.2 nm. Because the uncertainty of the phase attributed to the random intensity noise of the CCD is typically 0.1 radian (the standard deviation) on the cell surface, the dominant source of the uncertainty of the reflection phase is the random intensity noise of the CCD. We estimated the phase noise attributed to the random variance of the intensity observed on the CCD at every exposure. As shown in Fig. 12, the estimated phase noise is consistent with the results of our experimentation.

By subtracting the estimated overestimation of the MSD attributed to the random phase noise, we successfully compensated the systematic error in the MSD. However, in order to reduce the random phase noise it was necessary to enhance the fringe contrast and the increase of the total electron count on the CCD. Therefore, to enhance the fringe contrast, the reduction of the stray lights from optical component and the employment of the more efficient anti-reflection coating on the glass slide would be useful. Moreover, to sufficiently increase the total electron count on the CCD, the use of a light source with higher power and the optimization of the compatibility of the light source and the sensitivity curve of the camera would be useful.

Because the MSD relates to the diffusion constant, which is one of the important characteristics in cell biology [28], the potential application of our setup includes the three-dimensional trace of the intracellular particle’s motion to reveal the diffusion constant of the cytoplasm. Our future works also include the assessment of the cell condition in terms of the diffusion constant after drug treatment. To the best of our knowledge, our system visualized the depth-resolved intracellular motility for the first time in sub-micrometer resolution without the use of contrast agents.