1. INTRODUCTION

Digital holographic microscopy (DHM) is a non-invasive, high-resolution, whole-field imaging technique for microscopic specimens, particularly translucent samples [1–6]. The phase imaging capability of DHM provides intrinsic contrast for transparent biological samples and also permits quantitative analyses of their 3D structures and refractive index (RI). DHM usually uses a monochromatic plane wave for illumination; consequently, both lateral resolution and axial sectioning are worse in comparison to a conventional microscope employing Koehler illumination. DHM also suffers from speckle noise due to the employed coherent illumination. Off-axis illumination [7–10], structured illumination [6,11], and speckle illumination [12–16] were introduced to improve the lateral and axial sectioning capability and to suppress speckle noise in DHM or, more generally, phase imaging. Low-coherence and incoherent illuminations were also used to reduce speckle noise of DHM [17,18]. Another variant, dark-field opposed-view digital holographic microscopy (OV-DHM), collects scattered light concurrently from both opposite views and, therefore, improves the contrast of internal structures, as well as the signal-to-noise ratio [19,20]. Recently, OV-DHM was shown not only to reduce out-of-focus background and speckle noise, but also to provide autofocusing and field-of-view (FOV) extension [21]. Similar to other non-conventional illumination-based autofocusing methods [19,20], the image plane of a sample is identified in OV-DHM by finding the minimal variation between the two object waves propagating in opposite directions. Consequently, the sample can be refocused by propagating the object waves to their image planes in the computer. The FOV can be extended by combining two object waves propagating at an angle ($\sim 10\mathrm{mrad}$) with respect to each other.

DHM, which is based on interferometry and thus has optical path length measurement accuracy on the nanometer scale, is often employed to investigate the thickness or the RI of biological samples. Of note, the RI of cells is of great significance because it provides fundamental information about the composition and organizational structure of cells [22]. For a cell with refractive index $n_{\text{cell}}$, suspended in a medium with refractive index $n_{\text{medium}}$, the phase shift of light of wavelength $\lambda$ passing through the cell with respect to the medium is

In this paper, we determine the RI of suspended HeLa cells by using OV-DHM. The analysis is based on a description of the cells by a spheroidal model with constant cross-sectional aspect ratio (axial/lateral), which we derived from laser scanning confocal microscopy measurements. The proposed strategy is an easy-to-use approach that—unlike other RI determination methods—avoids the difficulty of imaging the same cell in confocal and DHM modes by employing the spheroidal model. Furthermore, we demonstrate that OV-DHM has the ability to efficiently suppress out-of-focus background. This background suppression as well as the autofocusing ability of OV-DHM contribute to minimizing errors of the RI determination due to defocusing and speckle noise.

2. OV-DHM TECHNIQUE

A schematic diagram of our home-built OV-DHM setup is shown in Fig. 1. It is based on a common-path Sagnac interferometer, comprised of a polarization-maintaining beam splitter (PBS) and two mirrors, ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$. A 561-nm continuous-wave (CW) laser is coupled into a $1\times 2$ fiber splitter, the two outputs of which are used as the reference wave and the light irradiating the object from opposite sides. The laser output from the object-wave end of the fiber splitter is split by the PBS into two components with orthogonal (horizontal and vertical) polarizations, respectively. The horizontally (vertically) polarized component passes through the Sagnac configuration in a clockwise (counterclockwise) fashion. Two telescope systems with $20\times$ magnification, ${\mathrm{MO}}_1-{\mathrm{L}}_1$ and ${\mathrm{MO}}_2-{\mathrm{L}}_2$, are inserted between mirrors ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ to image the sample in opposite views. The distance between the objectives ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$ is about twice their working distance. The sample is placed close to the middle plane between ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$ (Plan $25\times /0.4$, Nanjing Yingxing Optical Instrument Company, Nanjing, China). The two object waves $O_1$ and $O_2$ arising from light passing through the sample in opposite directions are further magnified by the two telescopes by a factor of 1.5 and superimposed with a common reference wave $R$ via a non-polarizing BS. The reference wave $R$ is linearly polarized at an angle of 45° with respect to the polarizations of $O_1$ (horizontal) and $O_2$ (vertical). Two holograms, $I_1={|O_1+R|}^2$ and $I_2={|O_2+R|}^2$, are measured sequentially, by rotating polarizer $P$ to the horizontal and vertical directions, respectively, with a complementary metal–oxide–semiconductor (CMOS) camera ($1920\times 1200\mathrm{pixels}$, $5.86\mathrm{\mu m}/\text{pixel}$, 54 fps, DMK 23UX174, Imaging Source, Bremen, Germany). The CMOS camera is aligned so as to collect in-focus images of the middle plane between ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$ for both clockwise and anti-clockwise directions. A small angle of the reference wave $R$ of $\sim 10\pm 0.1\mathrm{mrad}$ with respect to the object waves allows separation of the real image, twin image, and $dc$ terms of the generated carrier frequency hologram in the Fourier domain, while maintaining the highest spectral components of the object waves with diffraction-limited resolution [27]. An exemplary hologram of OV-DHM and its spectrum are shown in Figs. 2(a) and 2(b), respectively. We mention in passing that this OV-DHM configuration can be further upgraded with two CCD cameras to record both holograms $I_1$ and $I_2$ simultaneously.

 

Fig. 1. Experimental OV-DHM setup. BS, beamsplitter; CCD, charge-coupled device; ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, tube lenses; ${\mathrm{MO}}_1$ and ${\mathrm{MO}}_2$, microscope objectives; PBS, polarization-maintaining beamsplitter; ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$, mirrors; $O_1$ and $O_2$, object waves linearly polarized along the horizontal (0°) and vertical (90°) directions, respectively; P, polarizer; R, reference wave linearly polarized at 45°.

Download Full Size | PDF

 

Fig. 2. Exemplary hologram and spectrum of OV-DHM. (a) Overview and close-up of OV-DHM hologram; (b) spectrum of the hologram in (a). ${\kappa }_x$ and ${\kappa }_y$ are the carrier-frequencies of the off-axis hologram in the $x$ and $y$ directions, respectively. The dashed rectangle in (b) indicates the spectral region selected for hologram reconstruction.

Download Full Size | PDF

Due to the symmetry of the OV-DHM configuration, a sample at a defocus distance $\mathrm{\Delta }d$ in hologram $I_1$ (clockwise) will have an opposite defocus distance $-\mathrm{\Delta }d$ in hologram $I_2$ (counterclockwise). Without loss of generality, two object waves, $O_{r1}$ and $O_{r2}$, can be reconstructed and refocused to a plane at an arbitrary distance $\mathrm{\Delta }d(-\mathrm{\Delta }d)$ from the hologram $I_1(I_2)$ by using the angular spectrum method [28]:

A key issue in refocusing an image from the out-of-focus hologram plane is to find the distance $\mathrm{\Delta }d$ between the hologram plane and the image plane, since a slight out-of-focus displacement will reduce the accuracy of the phase measurement. Hitherto, there have been many reports on image plane detection, based on amplitude analysis [29], spectral norms [30], non-conventional illumination [6,7,31], and other approaches [32,33]. In OV-DHM, focus determination follows the following strategy: Because holograms $I_1$ and $I_2$ have opposite defocus, the difference between the intensity of the two object waves $I_{\text{Diff}}(x,y)={|O_{r1}(\mathrm{\Delta }d)|}^2-{|O_{r2}(-\mathrm{\Delta }d)|}^2$ is minimal for the correct distance $\mathrm{\Delta }d=M^2\mathrm{\Delta }z$ employed in the reconstruction, with $\mathrm{\Delta }z$ being the defocus in the sample domain and $M$ the effective magnification factor of the OV-DHM system. Accordingly, the focus criterion can be defined as

For a specific refocused plane, only in-focus cells will have the same images in $O_{r1}$ and $O_{r2}$, whereas out-of-focus components are different [see Fig. 2(d)]. Thus, out-of-focus components as well as speckle noise can be suppressed by averaging $O_{r1}$ and $O_{r2}$. Of note, out-of-focus background suppression can be further enhanced if the two object waves are not coaxial but propagate at a slight angle ($\sim 10\mathrm{mrad}$). In this case, only in-focus cells appear at the same locations and are enhanced upon averaging, whereas out-of-focus cells are shifted laterally and are averaged out.

3. OUT-OF-FOCUS BACKGROUND SUPPRESSION OF OV-DHM

The auto-focusing and out-of-focus background suppression of OV-DHM were demonstrated with phase imaging of human cervical cancer (HeLa) cells. HeLa cells (LGC Standards GmbH, Wesel, Germany) were maintained at 37°C and 5% ${\mathrm{CO}}_2$ in Dulbecco’s modified Eagle’s medium (DMEM), containing 10% fetal bovine serum (FBS) and antibiotics ($60\mathrm{\mu g}/\mathrm{mL}$ penicillin and $100\mathrm{ng}/\mathrm{mL}$ streptomycin, both from Invitrogen, Carlsbad, CA). After 24 h incubation, the cells were detached by using 500 μL trypsin and mixed with an agarose-phosphate buffered saline solution (3%, weight/weight) at 37°C. After cooling to room temperature (24°C), a transparent gel formed with cells entrapped. Figures 3(a) and 3(b) show amplitude and phase images of $O_{r1}$ (left) and $O_{r2}$ (right) after refocusing to the focal plane, with $\mathrm{\Delta }{z}_{1/2}=-210/+210\mathrm{\mu m}$, obtained by the focus criterion in Fig. 3(e) for selected region 1, indicated by the white rectangle in Fig. 3(b). For comparison, Fig. 3(c) shows phase images of $O_{r1}$ (left) and $O_{r2}$ (right) refocused to the focal plane with $\mathrm{\Delta }{z}_{1/2}=-45/+45\mathrm{\mu m}$, obtained by the focus criterion in Fig. 3(f) for region 2 [green rectangle in Fig. 3(c)].

 

Fig. 3. OV-DHM imaging on HeLa cells suspended in an agarose-water gel (3%, weight/weight). Reconstructed amplitude (a) and phase (b) images of the opposite-view object waves $O_{r1}$ (left) and $O_{r2}$ (right) reconstructed with $\mathrm{\Delta }{z}_1=-210\mathrm{\mu m}$ ($\mathrm{\Delta }{z}_2=210\mathrm{\mu m}$). (c) Phase images of $O_{r1}$ (left) and $O_{r2}$ (right) reconstructed with $\mathrm{\Delta }{z}_1=-45\mathrm{\mu m}$ ($\mathrm{\Delta }{z}_2=45\mathrm{\mu m}$). Scale bar in (c), 40 μm. (d) Comparison of out-of-focus background in $O_{r1}$, $O_{r2}$, and $(O_{r1}+O_{r2})/2$ for the same region [indicated by the red rectangle in (b)]. Standard deviations of the three images, $O_{r1}$, $O_{r2}$, and $(O_{r1}+O_{r2})/2$ in (d) are 0.49, 0.4, and 0.21, respectively. (e) and (f) Focus criterion curves of selected (e) region 1 [white rectangle in (b)] and (f) region 2 [green rectangle in (c)]. In the figure, the abbreviations “Amp.” and “Pha.” refer to amplitude and phase; “Obj.1,” “Obj.2,” and “Aver.” indicate the two opposite-view object waves along the clockwise and anti-clockwise directions and the average of the two (in phase).

Download Full Size | PDF

Figures 3(a)–3(c) show clearly that only in-focus cells have identical images in $O_{r1}$ and $O_{r2}$. Out-of-focus cells are different in $O_{r1}$ and $O_{r2}$ images and thus can be suppressed by averaging $O_{r1}$ and $O_{r2}$. To quantify the ability of OV-DHM to suppress out-of-focus background, an area [red rectangle in Fig. 3(b)] was selected in $O_{r1}$ and $O_{r2}$, and $(O_{r1}+O_{r2})/2$ and compared in Fig. 3(d). The visibility of the out-of-focus background in the images of $O_{r1}$, $O_{r2}$, and $(O_{r1}+O_{r2})/2$ was quantified by the standard deviation, calculated from the three rectangular regions ($50\mathrm{\mu m}\times 80\mathrm{\mu m}$) in Fig. 3(d), yielding 0.49, 0.40, and 0.21, respectively. Accordingly, OV-DHM has the capability of reducing out-of-focus background by averaging the two object waves.

4. REFRACTIVE INDEX MEASUREMENT ON CELLS BY OV-DHM

Live HeLa cells adhere to the bottom of cell containers and thus have a flat, carpet-like appearance [34]. If they die or are detached from surfaces by using trypsin, they round up and become ellipsoids [35], shaped by the tension of cell membrane and gravity. A triaxial ellipsoid is described by the equation $x^2/a^2+y^2/b^2+z^2/c^2=1$, with $a$, $b$, and $c$ being the lengths of the principal half axes along $x$, $y$, and $z$, respectively. In the agarose preparation, the cells are compressed along the principal axis $z$, chosen to be perpendicular to the glass slide supporting the cells, whereas the two other extensions are nearly equal, giving the cells an almost spheroidal shape [35]. For the OV-DHM experiment, the sample is introduced into the setup such that the $z$ axis is along the axial direction of the microscope. Depending on the tension of the cell membrane and the density of the cells with respect to the surrounding medium, the cellular extensions may vary [36,37], but the ratio $S=c/r_0$, with $c$ and $r_0=(a+b)/2$ being the axial and lateral extensions, is essentially fixed for a given cell type.

The 3D ellipsoid model of suspended HeLa cells was studied by using our home-built laser-scanning confocal fluorescence microscope [38,39]. In the experiment, HeLa cells were seeded on eight-well Lab-Tek II chambered cover glass (Thermo Fischer Scientific, Waltham, MA) and cultured in DMEM for 24 h. Afterwards, the cell membranes were stained with CellMask Green (Invitrogen, Darmstadt, Germany) in DMEM for 5 min. The cells were thoroughly washed twice with phosphate-buffered saline, and then embedded into agarose-water gel with the protocol described above. Then, the cells were imaged under the confocal microscope using a bandpass filter (600/37 nm) for fluorescence detection. A series of 3D images was obtained by scanning different regions ($90\mathrm{\mu m}\times 90\mathrm{\mu m}\times 30\mathrm{\mu m}$) of the sample, each with $128\times 128\times 128$ voxels. Exemplarily, Figs. 4(a) and 4(b) show orthogonal views and a 3D view of cells from a selected region. The suspended cells are approximately round in $x–y$ view, while they are elliptical in the $x–z$ and $y–z$ views. For quantitative analysis, the cross-sectional intensities along two lines through the center of a cell along $y$ and $z$ [red and blue lines in Fig. 4(a)] are shown in Fig. 4(c). From these data, the overall axial (lateral) extension of the cell is $12.3\pm 0.1$ ($21.8\pm 0.1$) μm. The axial-to-lateral aspect ratio $S$ of the cell is thus $0.56\pm 0.005$. This procedure was performed for 50 cells; the statistics of axial/lateral ($c/r_0$) and lateral/lateral ($a/b$) ratios are shown in Fig. 4(d), yielding averages of $0.56\pm 0.04$ and $1.05\pm 0.03$, respectively. Here $r_0=(a+b)/2$ is the average lateral extension, and the error bars indicate standard deviations of all measurements. The data show that a spheroidal model (i.e., an ellipsoid with $a=b$) is a good approximation for the suspended HeLa cells, with an almost constant aspect ratio between the axial and lateral extensions. In general, the spheroidal model can be further simplified to a 3D spherical model ($a=b=c$) if cells or tissues are prepared in a rotatory cell culture system [35]. More generally, 3D bioprinting [37] allows one to produce a designed shape for live cells or tissues and thus give more freedom to extend the model.

 

Fig. 4. 3D elliptical model of suspended HeLa cells. (a) Perpendicular views of the HeLa cells. (b) 3D view of cells in volume rendering based on a maximum intensity projection. The inset in (b) shows schematically the 3D elliptical model of a suspended cell. (c) Intensities along the blue and red lines in panel (a). (d) Ratios $c/r_0$ and $a/b$ of 50 suspended cells. The range 25–75% of all data points is included in the boxes; the lines in the boxes show the median. The means are indicated by small squares; vertical lines represent the maximum spread of the data.

Download Full Size | PDF

With the spheroidal model of the suspended HeLa cells, the cell thickness $d$ along the axial direction can be computed from the lateral extension $r_0$ and the aspect ratio $S$. By using Eq. (1) together with the thickness of the cell along the axial direction for a point $(x,y)$ in the lateral plane, $d(x,y)=2z(x,y)$, we obtain

The same HeLa cell sample used for fluorescence imaging was also employed for the RI measurement. Figure 5(a) shows the wrapped phase image of cells reconstructed from the object wave $O_{r1}$ in OV-DHM, after refocusing by a distance $\mathrm{\Delta }{z}_1=-45\mathrm{\mu m}$. The correct refocusing distance, determined by finding the minimal variation between $O_{r1}$ and $O_{r2}$ for the selected cell, helps to minimize the phase error due to image defocus. Even a slight defocus will introduce a moderate error on the phase measurement [40]. The phase distributions of $O_{r1},O_{r2}$, and $(O_{r1}+O_{r2})/2$ along the red dashed line across the cell in Fig. 5(a) were extracted and are shown in Fig. 5(b). It is apparent that the green curve is less noisy due to the averaging operation on $O_{r1}$ and $O_{r2}$. Fitting this curve with Eq. (4) and setting $y=0$ yields $r_0=10.00\pm 0.05\mathrm{\mu m}$ and $\mathrm{\Delta }n=0.0462\pm 0.0005$. The RI of the agarose-water gel (3%, weight/weight) was $1.3481\pm 0.0004$, measured by using a refractometer (PAL-RI, ATAGO, Tokyo, Japan) at 24°C. Therefore, the RI of the cell was $1.394\pm 0.003$ at 24°C. Ten cells were analyzed by the same procedure, yielding $n_{\text{cell}}=1.39\pm 0.01$, with the error denoting the standard deviation within the ensemble. The obtained $n_{\text{cell}}$ agrees with the result $n_{\text{cell}}=1.399\pm 0.004$ measured by dual-wavelength DHM [26], implying that the OV-DHM approach provides a precise RI determination for the HeLa cells.

 

Fig. 5. Refractive index measurement on suspended HeLa cells. (a) Wrapped phase image of HeLa cells reconstructed from $O_{r1}$ in OV-DHM. (b) Phase distributions of $O_{r1}$ (black), $O_{r2}$ (red), and ($O_{r1}+O_{r2})/2$ (green) along the line across the cell center, indicated by the red dashed line in (a). The violet solid curve is the fit of the green curve using Eq. (4).

Download Full Size | PDF

5. CONCLUSIONS

This paper presents a unique technique for RI determination of cells by using OV-DHM. Simultaneous determination of both $d$ and $\mathrm{\Delta }\phi$ is circumvented by using a spheroidal model for detached cells, which was obtained by confocal microscopy. The proposed strategy is not entirely stand-alone because it needs a confocal microscope to quantify the parameters of the spheroidal model. However, the difficulty of imaging the same cell in confocal and DHM modes is avoided by using the spheroidal model. OV-DHM was employed to determine the phase $\mathrm{\Delta }\phi$ of a light passing through a cell. The image plane within the sample was identified by finding the minimal variation between the two object waves; consequently, refocusing was performed by propagating the waves to the common image plane. Out-of-focus background can be suppressed by averaging the two object waves $O_{r1}$ and $O_{r2}$ because in-focus contributions are enhanced by the averaging operation, whereas out-of-focus contributions have different defocus distances and thus are at least partially cancelled, as are noise contributions (e.g., coherent noise). The autofocusing and out-of-focus light-suppressing ability of OV-DHM contributes to improving the accuracy of RI determination.

3D profiles of suspended HeLa cells in agarose gel were measured by laser-scanning confocal microscopy, revealing a spheroidal cell model with a constant axial-to-lateral aspect ratio. Based on this model, the RI can be determined by using a stand-alone phase-imaging approach (including, but not limited to, OV-DHM), since the thickness of the cell can be calculated from the phase image for all lateral positions. The RI depends on the composition of the cell, and the surrounding medium and will be maintained under constant conditions, no matter whether the cell is detached or not.

Our RI measurement strategy based on an ellipsoid model is not limited to HeLa cells studied here but also suitable for many other cell types. Different model shapes, e.g., spheres, can also be imposed by using the newly developed 3D bioprinting techniques [37].