2.1.

Self-Interference Digital Holographic Microscope

The phase projections are captured by an SIDHM.⁹ In comparison to a common transmission DHM based on Mach–Zehnder interferometer configuration,³² the SIDHM is highly insensitive to vibration due to replacing a reference beam with a shearing module realized by a Michelson interferometer (Fig. 1). In the setup, two sheared object wavefronts are superimposed.

Fig. 1

Self-interference DHM: SMF, single mode fiber; CL, condenser lens; SPL, specimen plane; MO, microscope objective; TL, tube lens, M1 and M2, mirrors, $\alpha$, M2 tilt angle; and CCD, CCD camera.

In the self-interference configuration, used for phase tomography, the sample is illuminated in transmission with coherent light from a frequency doubled Nd:YAG laser ($\lambda =532\mathrm{nm}$) via a single mode optical fiber and the microscope condenser (CL). Optical imaging of the sample is performed by a Zeiss achromatic microscope objective [Zeiss LD Achroplan $20\times 0.4\mathrm{Corr}\mathrm{.}$ numerical aperture (NA) 0.4)] with a long working distance (10 mm) and the nontilted mirror (M1) of the Michelson interferometer. The second mirror (M2) is tilted in such a way that an area without a sample is superimposed with the wavefront that is transmitted through the sample. Thus, a reference wave is generated that is not disturbed by the sample. The digital holograms are captured by a charge coupled device (CCD) sensor (Basler pia-2400-12gm camera, pixel size: 3.45 μm) and then evaluated numerically aiming for accurate determination of an object phase ${\varphi }_{\mathrm{o}}$ at a plane conjugate with an object. The first step of this procedure was to calculate the phase in the hologram plane ${\varphi }_{\mathrm{h}}$. This was performed by spatial phase shifting as described in Refs. 33 and 34. The proper operation of SIDHM for phase retrieval in combination with approaches in Refs. 33 and 34 has already been demonstrated and verified in Ref. 35 and in the supplementary information of Ref. 36 by data from silica and polystyrene microspheres. The imaging system provided the overall Abbe resolution $\lambda /2\text{NA}$ of 0.67 μm and spatial sampling of the object equal to 0.18 μm. The further steps of hologram processing for tomographic phase imaging are described in Sec. 2.3.

2.2.

Rotary Fiber Holder

Tomography requires capturing numerous digital holograms of an object from different directions. To fulfill this requirement, we had proposed at first to implement a hollow optical fiber as a cuvette for a cell or a cell group and subsequently place the fiber in an accurate rotary fiber holder. It should be emphasized that such solution requires only a minor modification of SIDHM prior to 3-D cell measurement and it does not involve additional optical components. The idea behind the mechanical rotation of a fiber capillary is to provide a tool capable of cell cultivation as well as measurement of 3-D biological structures. The main feature of this concept is the fact that cells—once placed in a hollow fiber—grow in a similar way as in a Petri dish. This concept is comparable to using a micropipette in order to rotate the sample,¹⁹ although in our case, there is no mechanical intrusion in the specimen. Moreover, by virtue of the capillary attraction, it is not necessary to use plugs to prevent cell culture medium from flowing out. However, in order to avoid strong diffraction and refraction phenomena due to the cylindrical shape of the capillary and also to allow high accuracy holographic measurements, it is necessary to insert the fiber cuvette into an immersion liquid³² as shown in Fig. 2(a). A standard microscope stage has been replaced with a custom-designed one, which consists of a Petri dish support and an Elliot Martock rotary fiber holder. The holder, which offers an angular resolution of $<0.01\mathrm{deg}$, assured high precision of the specimen position for each acquired projection. A cell, once inserted in the hollow fiber, can be properly placed in the field of view through 3-D manipulation of the fiber. The fiber capillary (inner diameter: 212 μm, outer diameter: 300 μm) has been made of Hereaus fused silica with refractive index of $n=1.46071$ at $\lambda =532\mathrm{nm}$. The inner surface of the fiber was treated with 0.5% polyvinyl alcohol in order to prevent cells from attaching to the fiber walls.³⁷ The fiber was placed inside a modified Petri dish (μ-Dish, ibidi GmbH, Munich, Germany), filled with immersion liquid (refractive index $n=1.4636$ at $\lambda =532\mathrm{nm}$) matched to the capillary material.

Fig. 2

The rotary fiber holder setup: (a) scheme and (b) photo. CL, condenser lens; PD, Petri dish; IL, immersion liquid; FC, fiber capillary; MO, microscope objective; $(x,y,z,\varphi )$, movement directions of the fiber capillary; $\mathrm{\Delta }\varphi$, rotation step for $N$ projections; RH, rotary holder; and CS, cover slip.

The bottom of the used Petri dish (PD) is a 0.17 mm cover slip. Since the microscope objective has been designed to work with such CSs, this ensures proper working conditions for the imaging system. The use of the immersion liquid reduced the cylindrical lens effect caused by the shape of the hollow fiber. The remaining lens effect caused by the difference between cell culture medium and fiber refractive indices was removed numerically.³⁸

2.3.

Tomographic Reconstruction

The flow chart of the applied procedures for phase tomography is presented in Fig. 3. In order to obtain a successful 3-D reconstruction of an object, it is necessary to acquire a set of phase images corresponding to a series of angular projections of an object. In this work, 180 projections were captured with the angle of rotation between sequential measurements equal to 1 deg. For each projection, captured with SIDHM within a specified area of interest [see Fig. 3(a)], the integrated phase ${\varphi }_{\mathrm{h}}$ in the hologram plane has been calculated.^33,34 Then, all phases were propagated using a convolution approach to obtain phase ${\varphi }_{\mathrm{o}}$ in the object plane. This approach takes into consideration the diffraction effects and therefore allows for the application of the filtered backprojection³⁹ method for tomographic reconstruction in case of phase objects.⁴⁰ In order to find the object plane (the plane of best focus) for each projection, the numerical autofocusing algorithm was applied.⁴¹ The complex object wave was numerically propagated to the successive planes and then the real amplitude was reconstructed in order to evaluate the focus. Cells in a 3-D collagen matrix are treated as phase-only objects, thus the criterion for the best focused image was based on the lowest detail in the amplitude distribution. The holograms were assessed automatically using the method based on calculating the variance of optical field modulus, as proposed in Ref. 41. Next, the influence of the fiber capillary on the phase distribution has been corrected by means of subtraction of an estimated aberration profile, which was created using areas of a hologram not containing any cells.^38,42 What is more, radial cell position run-out correction has been performed in the specimen plane. The position of a cell under study has been tracked using a phase value threshold and the calculation of the first moment of the area of a cell in the field of view, as described in Ref. 43. Then, the phase images were cropped in order to keep the center of the investigated cell or cell cluster in the middle and thus minimize reconstruction artifacts.⁴⁴ The resulting phase distributions ${\varphi }_{\mathrm{o}\_\mathrm{corr}}$, having undergone the aberration removal and radial run-out correction [Fig. 3(b)], were additionally filtered with a small median filter ($3\times 3\text{pixels}$) and were then used as an input for phase tomography algorithm. The 180 phase projections (acquired at an angular step $\mathrm{\Delta }\alpha =1\mathrm{deg}$) were then used to create sinograms [Fig. 3(c)], and later reconstructed in the second step by means of the Filtered Backprojection method as reported in Refs. 39, 40, 45, and 46. This approach requires the assumption that the object under study introduces only slow-varying phase changes in the object wave. This condition is expected to be fulfilled by the small differences between the cells and the surrounding collagen matrix refractive indices. Furthermore, the object phase is propagated to the center of the object. Thus, the diffraction effects caused by the object are minimized and therefore assumed to be neglectable.

Fig. 3

The flowchart of the procedures for the reconstruction and visualization of the three-dimensional (3-D) refractive index distributions.

The full 3-D distribution of phase values was achieved by sequentially assembling the cross sections of the reconstructed phase differences $\mathrm{\Delta }\mathrm{\Phi }(x,y,z_0÷Z)$. Then, the results for each reconstructed layer $z_i$ have been scaled to the refractive index difference values $\mathrm{\Delta }n$ (relative values of refractive index) according to

The determination of the absolute cellular refractive index value $n$ requires the knowledge of the exact value of the refractive index of the medium that surrounds the cells, which plays a role of an immersion liquid $n_{\mathrm{ref}}$ and thus serves as a reference medium:

The uncertainties of the refractive index can be quantified using the exact differential method:

The first summand of Eq. (3) is the random error $u_{\text{n}\_\mathrm{r}}$ ($[\partial \mathrm{\Phi }(\lambda /2\pi d)]$) of the measurement. The next three summands consider the systematic error $u_{\text{n}\_\mathrm{s}}$. The systematic error $u_{\text{n}\_\mathrm{s}}$ includes the contributions of the errors that originate from the uncertainties of the devices (spectral width of the laser source, pixel size of the camera, etc.) used in the experimental setup ($\partial \lambda (\mathrm{\Phi }/2\pi d)+\partial d(\mathrm{\Phi }\lambda /2\pi d^2)=0.0037$), which are constant for every measurement and the uncertainty $u_{\text{n}\_\mathrm{ref}}$ of the refractive index $n_{\mathrm{ref}}$ of the collagen that surrounds the cells. Unfortunately, the density of the collagen matrix inside the optical fiber is spatially inhomogeneous and depends on the preparation procedure. Thus, the refractive index uncertainty of the collagen matrix can be estimated to be high ($u_{\text{n}\_\mathrm{ref}}\approx 0.01$). The random error resulting from the phase measurements and the filtering procedure is estimated as $u_{\text{n}\_\mathrm{r}}=0.0005$. In summary, the uncertainty of the absolute refractive index values is high ($u_n=0.0142$) and strongly depends on the uncertainty of the refractive index of the reference medium. For this reason, the results in Figs. 4Fig. 5–6 are presented in relative refractive index values, for which the uncertainty is much lower ($u_n=0.0042$). However, the scaling process could be enhanced if the refractive index markers in the form of microspheres with a known refractive index³⁵ were introduced into the measurement volume. This approach is reported in Sec. 3.2.

Fig. 4

The relative refractive index distribution inside a single U937 cell. Refractive index peak to valley value $\mathrm{\Delta }n=0.030\pm 0.004$ (Video 1, MPEG, 2.2 MB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.046009.1].

Fig. 5

The relative refractive index distribution inside an HT-1080 cell group at several cross sections (of the same cell cluster). Refractive index peak to valley value $\mathrm{\Delta }n=0.030\pm 0.004$ (Video 2, MPEG, 4.6 MB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.046009.2].

Fig. 6

The relative refractive index distribution in an HT-1080 cell with an extension, vertical cross sections through the cell. Refractive index peak to valley value $\mathrm{\Delta }n=0.032\pm 0.004$ (Video 3, MPEG, 2.2 MB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.046009.3].

The processing of all the measurement data necessary to reconstruct the tomographic phase images was performed by custom-built MATLAB scripts. In order to improve the data visualization, surface representations were created with a threshold-based filtering of the refractive index values that correspond to the refractive index levels of background of the observed cells.

3.1.

Biological Samples

The chosen biological samples were human malignant lymphoma cells (U937) and human fibrosarcoma cells (HT-1080). Both cell types were cultivated in Dulbecco’s modified Eagle’s medium (Biochrom AG, Berlin) supplemented with 10% fetal calf serum (Biochrom AG, Berlin) at 37°C and 10% ${\mathrm{CO}}_2$ atmosphere. Some of HT-1080 cells were additionally incubated for one night with uncoated silica (${\mathrm{SiO}}_2$) microspheres (Mikropartikel, GmbH, Berlin, Germany) in order to enable incorporation of the particles into the cells by phagocytosis.³⁵ These particular cells were applied to illustrate the procedure of determination of the absolute refractive index values inside a cell. Then, the cells were detached and suspensions of ${10}^6\text{cells}/\mathrm{ml}$ were prepared. To inert the U937 cells, the suspension was mixed with an agar solution to a final concentration of 0.15% agar. The suspended HT-1080 cells were mixed with collagen to a final concentration of 0.16% collagen. The prepared agar and collagen mixtures were introduced into the hollow fibers by capillary attraction. Such mixture, especially the cultivation in collagen, helped to avoid cell tumbling during the rotation in the measurement process. Owing to the viscosity of the medium and the small moment of inertia of cells, there is no visible movement of cells due to the measurement procedures. However, this matter should be separately investigated in the future. Lymphoma cells were observed directly after the preparation. Fibrosarcoma cells were cultured for additional 24 h to allow polymerization of the collagen fibers and permit cells to migrate inside the matrix. HEPES buffer has been added to the collagen cell mixture to compensate for the missing ${\mathrm{CO}}_2$ atmosphere.³⁵ All tomographic measurements were performed at a room temperature ($T=22°\mathrm{C}$) in order to prevent the fast cell migrations during our proof of principle experiments.^35,36 Although this is not the perfect incubation temperature for cells, the specimens were not influenced significantly by this fact, as the measurement was performed within no more than 2 h after the environment had changed. The cells with a typical appearance were selected for the measurements. There was no need for an additional heating chamber in the setup. However, the device is suitable to be integrated into an incubation system for extended time-lapse investigations in the future.

3.2.

Experimental Results

The cells described in Sec. 3.1 were used as the specimens measured in the TPM. For each measurement object, 180 images were acquired at the rate of 1 Hz. At first, a single U937 cell with spherical appearance, which was randomly chosen from the cell distribution in the fiber capillary, was measured. It is the least complex measurement object of the whole measurement series presented in this paper. The reconstructed relative refractive index distribution in this cell is shown in Fig. 4. The nuclear envelope and density changes in the nucleus (see arrows in Fig. 4) are clearly visible.

Next, the capability to measure the clusters of cells is demonstrated. The refractive index distribution in a cluster of five HT-1080 cells in collagen with several individual cross sections through the cells is presented in Fig. 5. Here, nucleoli are visible inside the cells, which greatly help to distinguish between the cells and to count them (see arrows in Fig. 5 and Video 2). Owing to the subcellular resolution, it is possible to observe cell walls and to assess the interaction between the cells. This result appears to be of particular importance when cell clusters in 3-D cell matrices are treated with drugs.

Employing our method shows up prospects to compare such a cell cluster before and after the treatment with drugs or toxins in order to trace internal changes in cell structure within a cell group. Furthermore, we applied our tomographic system to measure a highly nonsymmetrical object, namely to study HT-1080 cells with the developed extensions. The result is shown in Fig. 6.

The results presented in Figs. 4–6 are given in the values of a relative refractive index $\mathrm{\Delta }n$. However, it is also of interest to rescale the data to absolute refractive index values as these values are directly related to the intracellular solute concentrations and the dry mass.^47,48 As it has already described in Sec. 2.3, it is not suitable to use the refractive index of the surrounding medium as a reference in our study. However, the refractive index values can be obtained if an object with a well-known refractive index value, such as silica microspheres that are internalized by living cells, is present in the measurement volume. Such a case is presented in Fig. 7, where two silica beads are clearly resolved inside an HT-1080 cell [see label “1” in Fig. 7(a)]. As the refractive index of uncoated silica microspheres was measured to be equal to $n_{\mathrm{ref}}=1.435\pm 0.003$,³⁵ it is possible to calibrate the measurement of the cell to obtain the absolute refractive index values with total uncertainty (see Sec. 2.3) equal to 0.0072. Rescaling the former relative refractive index variations to the absolute refractive index values provides the full quantitative measurement data. The refractive index value of a nucleolus of the cell is obtained as $n=1.385\pm 0.0072$ and represents the highest refractive index inside the cell—apart from the silica microspheres (see label “2,” Fig. 7). The detected refractive index changes inside the cell (Fig. 7) were found in agreement with the previously published data.^20,35 However, it should be noted that introducing an object with a step refractive index into a cell is a source of a locally strong phase gradient, while the main assumption that allowed for using the FBP algorithm was that only weak spatial variations of phase should be present. This leads to a conclusion that, besides causing possible changes of the intracellular structure, an inappropriate calibration object may cause additional significant errors in the reconstructed refractive index distribution. Nevertheless, in our case, we were able to avoid the impact of diffraction on the FBP algorithm as a result of the propagation applied as described in Sec. 2.3. In fact, the image artifacts were sufficiently small to successfully retrieve the absolute refractive index distribution of a living cell.

Fig. 7

The 3-D refractive index distribution of an HT-1080 cell with incorporated ${\mathrm{SiO}}_2$ microspheres with refractive index $n=1.435\pm 0.003$ and diameter $ø\approx 3.44\mu \text{m}$; (a) vertical cross section through the absolute refractive index (peak to valley value $\mathrm{\Delta }n=0.080\pm 0.007$) (Video 4, MPEG, 2.2 MB) [URL: http://dx.doi.org/10.1117/1.JBO.19.4.046009.4] and (b) horizontal cross section through the same cell, absolute refractive index values. Here, image scale is adjusted to the refractive index of a nucleolus (refractive index peak to valley value $\mathrm{\Delta }n=0.030\pm 0.007$).