The goal of the ODT [1] is the recovering of the sample scattering potential defined as V ( r ) = k 0 2 ( n 2 ( r ) − n m 2 ) from a series of intensity measurements. Here, k ₀ is the wavenumber in vacuum, r is a position vector, and n(r) and n _m are the RI of the sample and surrounding media, correspondingly. The scattering potential is, in general, a complex valued function V ( r ) = P ( r ) + i A ( r ) with real and imaginary parts, P ( r ) = k 0 2 ( n re 2 ( r ) − n im 2 ( r ) − n m 2 ) and A ( r ) = 2 k 0 2 n re ( r ) n im ( r ) , respectively, that are related to the real (n _re) and imaginary (n _im) parts of the sample RI (n = n _re + i n _im) Here, n _m is assumed to be real.

Let us consider that the sample satisfies the first-order Born approximation, which is suitable for weakly absorbing and low scattering samples. Under this approximation, the 3D intensity distribution I(r) measured in a bright-field microscope (e.g., a stack of through-focus intensity images) can be written as the convolution of the point spread function (PSF), h (r), of the microscope and the sample’s scattering potential, as demonstrated in Ref [10–14]. By splitting the PSF into its real and imaginary contributions, one obtains the functions h _P (r) and h _A (r) representing the microscope response to a point scatter, δ(r), and to a point absorber, iδ (r), correspondingly. Then, I(r) can be understood as the linear superposition of the real and imaginary contributions of the scattering potential, respectively, convolved with h _P (r) and h _A (r), as it follows [11]. I ( r ) = B + P ( r ) ⊗ h p ( r ) + A ( r ) ⊗ h A ( r ) with B being the background intensity (un-scattered light). p(r) and therefore n _re(r) can be easily recovered from this equation for the case of a non-absorbing sample. For weakly absorbing sample, it can be supposed that the real and imaginary parts of the RI are proportional n _im = εn _re, where ε takes a small positive value (in the range 10^-5–10^-3) [13, 14]. Then introducing the effective point spread function h E F F ( r ) = h P ( r ) + 2 ε h A ( r ) , the expression is linearized with respect to P(r). I ( r ) = B + P ( r ) ⊗ h E F F ( r )

The deconvolution process is usually carried out in the Fourier space [13, 14]. The analytical expressions for the phase and absorption optical transfer functions (which are the 3D Fourier transform of h P ( r ) and h A ( r ) , respectively) can be found in Ref [17]. However, better results are obtained using numerically calculated transfer functions, which take into account the experimental sample illumination [18]. The real part of the RI recovered from P(r) is considered for the cell dynamic analysis in Result and Discussions.

One of the advantages of PC-ODT is that it can be easily implemented in conventional transmission wide-field microscope (e.g., Nikon Eclipse Ti-U inverted) equipped with a quasi-monochromatic LED illumination source filtered with a band-pass filter (depending on the desired working wavelength). In this work, the sample was illuminated with quasi-monochromatic partially coherent light (λ = 560 nm, band-pass filter with FWHM = 30 nm), and a dry condenser lens (Thorlabs-CSC200 NA_c = 0.78) and imaged with an oil-immersed objective (Nikon Plan-Apo, 60x, n _imm = 1.518, NA_o = 1.45) with a tube lens (f_TL = 200 mm). The lateral and axial resolutions of the setup defined as the Nyquist distance [19] are 125 nm and 270 nm, correspondingly.

The 3D through-focus intensity stack required for PC-ODT can be obtained in two different ways: mechanically or optically. The mechanical refocusing consists of changing the distance between the stage of the sample and the objective lenses (for instance, with a motorized piezo stage). However, it is undesirable for high-resolution microscopy (immersed objectives). Conversely, the optical refocusing considered here allows for axial scanning of the sample without moving it physically. It is achieved by incorporation of an optical refocusing module (ORM) after the intermediate image space, as it is shown in Figure 1. An electrically tunable lens (ETL) with a variable focal length is a crucial element of this module that enables a fast measuring of a through-focus intensity stack I (r).

The high-speed ETL (Optotune EL-10-30-C-VIS-LD-MV device in our case) is located in the Fourier conjugated plane with respect to the central slice, z _defocus = 0 μm of the intermediate 3D image. The camera sensor records a 2D image (intensity distribution) corresponding to the transverse xy-object slice located in the microscope’s focal plane (defocus distance z _defocus = 0 μm). The position of the focused plane is optically shifted according to electric current applied to the ETL, therefore yielding the axial z-scanning of the sample almost in real time. Finally, I (r) is acquired by an sCMOS camera (Hamamatsu, Orca Flash 4.0, 16-bit gray-level, pixel size of p = 6.5 μm, in our case) and stored in a computer for its processing.

The shift of the imaged plane ( z defocus ) depends on the magnification M of the considered microscope, as well as the relay lenses projecting the image onto the camera detector. Specifically, the shift distance is given by [20] z defocus = − n m f R L 1 2 P ETL / M 2 , where f_RL1 is a focal distance of the relay lens RL1, and P_ETL is the power of the ETL lens from the interval of [ P ETL min ,P ETL max ] = [ − 1.5 , + 3.5 ] dpt . The total magnification of the system is M s = M × f RL 2 / f RL 1 .

The culture and preparation of COS-7 cells (hail from the African green monkey kidney) considered in the next section were conducted as follows. First, the cells were cultured in Dulbecco’s modified Eagle medium (DMEM) supplemented with 10% fetal bovine serum (FBS) and incubated at 37°C and 5% CO₂ for 4 days. 12 h before fixation, the cells were trypsinized and transferred to the coverslip. Afterward, the cells are immersed in phosphate-buffered saline (PBS) and sealed between coverslips, to prevent evaporation. The RI of the PBS is similar to water.

The fast monitoring of the sample provided by PC-ODT enables the study of cells with fleeting dynamics (wobbling, flickering, etc.), which exhibit noticeable changes in the time scale below 0.5 s. For evaluating the capability of PC-ODT for the study of intracellular motion, we analyze the behavior of living fibroblast-like COS-7 cell for 9 s. The prepared sample has been considered to observe the process of cell attachment to the substrate. The intensity stack comprises 50 2D images acquired with 2-ms camera exposure time (0.1 s for one stack). During 9 s 25 of such stacks have been measured. Then, the 3D RI distribution in the volume of 500 × 500 × 50 pixels, corresponding to 60 × 60 × 14 μm³, has been obtained by applying the PC-ODT deconvolution procedure explained in Partially Coherent-Optical Diffraction Tomography Principle. The temporal evolution of two RI tomograms at z = 0 μm (the cell is attached to the substrate) and z = 4 μm is presented in Supplementary Video S1. Alternatively, the temporal evolution of 3D RI distribution can be observed in Supplementary Video S2, where the dynamic cell is shown as a volumetric reconstruction with a color-coded depth and brightness-coded RI applied for fast and easy cell 3D visualization. We have developed this volumetric reconstruction by using a customized ImageJ macro based on “Z-stack Depth Color Code” and “Clear Volume” plugins [21].

In Figure 2, we show the time evolution of RI at three characteristic planes, at z = 0 μm, z = 1 μm, and z = 4 μm. The slices at z = 0 μm and z = 1 μm (Figure 2A) and (Figure 2B) show intensive activity of lamellipodia, while the indicated filopodium and rear protrusion are immobile. Lamellipodia are a cytoskeletal protein actin projection on the leading edge of the cell. Together with filopodia, they organize process of cell migration. However, while lamellipodia enable an explorative, rather fast random, walk (up to 1.2 μm/s according to Ref [22]), the filopodia promote a more slow ballistic migration guided by external inputs [23]. The sub-second-scale rearrangement of barbed actin filaments of lamellipodia is observed in Figures 2A,B (see also Supplementary Videos S1 and S2). The recent comparative fluorescence and RI microscopic study [24] allows easy identification of the cell organelles in the RI images. Thus (see insets of Figure 2A), the mitochondria network and lipid droplets can be found. At the slice at z = 4 μm (Figure 2C) several organelles including nucleoli immersed in the nucleus are observed. We underline that fibroblast cells often have several nucleoli in the nucleus as in the cases studied in this work. The RI distributions in the plane z = 4 μm demonstrate that the cell exhibits intensive intracellular trafficking, nucleus plication, and counterclockwise rotation.

This cell activity is also reflected in Figure 2C, where 3D rendering of the entire cell at different moments of time is shown. The visualization principle is the same as in the Supplementary Video S2.

Let us consider now more rapid changes of this cell. In Figure 3, four consecutive frames (each pair separated by only 0.36 s) have been analyzed for the planes z = 0 μm and z = 4 μm. Along with RI distributions, we also include speed map plots (see enlarged images, green arrows) of the structures inside the nucleus for studying the rotation motion of its internal organelles. The speed maps have been obtained by using PIVlab plug-in (a digital particle image velocimetry tool) available for MATLAB [25] that performs a multipass window deformation ensemble correlation widely applied in particle velocimetry applications [26]. Thus, by considering the correlation between consecutive image pairs, one can obtain the velocity of each region of the cell. Note that every speed arrow corresponds to a region of 7 × 7 pixels. Thanks to the streamlines of this speed vector field, one is able to characterize the motion of organelles inside the entire cell. A higher activity is observed inside the nucleus, in the form of a relatively fast rotation. It is observed that the speed field magnitude is variable within the range 0.18–1.4 μm/s, with certain tendency to vorticity particularly noticeable at z = 4 μm. The nucleus is rotated around an axis perpendicular to the substrate with a speed about 3 degrees/s.

Note that the fibroblast cells often display a high nuclear rotational activity [27]. The nuclear rotation and oscillation are associated with cell movement, mitosis, and necrosis or are induced by external factors, such as mechanical shear stress [28], and have been reported in several publications, but on the larger temporal sampling (minutes). The entire nucleus rotation of the order 10-20 degrees/min (0.2-0.3 degrees/s) has been recently observed using C-ODT [24]. Here, significantly more rapid nucleus rotation (3 degrees/s) is observed (see also Supplementary Videos S1 and S2). This example illustrates the capability of PC-ODT for fast and high-spatial resolution studies of living cells, in particular their behavior under environmental changes, external force applications [22], differentiation, mitosis, etc.

Apart from short-term dynamics, long-term processes such as cell mitosis (division) or necrosis (death) are also of great interest in biomedicine. Thus, it is important to prove the applicability of PC-ODT for this type of studies, where the observation can take minutes or even hours. Here, we consider the process of induced necrosis of two COS-7 cells under nutrient deprivation. We scan a field of view (60 × 60 × 16 μm³). 3D intensity stack has been measured within 2 hrs in 5-min regular intervals, and the RI has been recovered. The temporal evolution of 3D RI distribution can be observed in Supplementary Video S3, as a volumetric reconstruction with a color-coded depth and brightness-coded RI applied for fast and easy cell 3D visualization (as in Supplementary Video S2). In Figure 4A, two different z-slices for z = 0 μm and z = 4.5 μm of the sample RI are presented, at four different times (t = 0 min, 15 min, 60 min, and 120 min). At z = 0 μm, the cells are attached to the substrate. At z = 4.5 μm, the cell nucleus with several nucleoli is observed. Also in Figures 2A,B, the mitochondrial network and lipid droplets can be identified. Volumetric cell reconstruction with a color-coded depth and brightness-coded RI for different moments of time is shown in Figure 4B.

The different phases of the cell cycle have been observed during this long-term study. First, the cell swells progressively (from t = 0 min up to t = 60 min, see Figures 4A,B) until a maximum size is reached. Later, the multiple disruption of the outer cell membrane starts, resulting in the efflux of intracellular contents and a dramatic loss of cell volume (compare RI slices in Figure 4A). This behavior can be observed in Supplementary Video S3.

For quantitative characterization of the observed necrosis process, we analyze the average dry mass concentration (DMC), which can be easily obtained from RI contrast. Indeed, it is widely accepted [29–32] that for biological samples immersed in an aqueous medium, there exists a linear relationship linking the DMC of the sample and the real part of its RI as DMC(r)=[n(r)-n _m ]/a, where a represents the so-called specific refractive index increment, n is the sample’s RI, and n _m is the water RI. Although the cell chemical composition is variable, an average value around a = 0.0018 dL/g is usually used within the light visible range when nucleated cells are considered [30, 32].

To obtain the average DMC, the reconstructed RI stack must be split into cell and background (surrounding medium). The applied segmentation algorithm takes into account the gradient and the absolute value of the sample RI. The 3D image regions that simultaneously satisfy both | ∇ n ( r ) | >0 . 002 and n ( r ) > 1 . 333 conditions are considered as the cell sample. Only the RI values belonging to those regions are used to obtain the DMC. The temporal evolution of the averaged (mean) DMC is shown in Figure 5. During the first hour of experiment, the mean DMC significantly decreases (from 2.422 g/dL at t = 5 min to 1.685 g/dL at t = 60 min). In this period of time, the volume of the cell grows remarkably, but the overall dry mass does not change significantly. After 1 hr of experiment, the DMC reaches a plateau around 1.65 g/dL that can be interpreted in terms of two processes with opposing effects: the outer membrane suffers multiple micro-ruptures and the cell tends to shrink, which should again increase DMC, but at the same time, all the cell contents leak into the extracellular space.

The cell death phenomenon is a topic of great interest in biology and medicine [33–36], and PC-ODT seems to be a suitable tool for this study. Note that data for long-term cell temporal evolution observation can be acquired automatically by programmable activation of both ETL and camera sensor.