1. Introduction

The capability to image subtle mechanical motion at cellular and sub-cellular scale can be used to study how extracellular particles interact with cultured cells and more generally how cells interact with their environment. Knowledge gained from cellular and sub-cellular scale dynamic imaging studies will advance our understanding in the basic science of cell biology, and will assist the development of drugs and drug delivery systems. Hence there is a need for a functional microscopic imaging technology that performs quantitative motion imaging with cellular/sub-cellular resolution.

Phase resolved optical imaging techniques have nanoscale (nanometer) sensitivity to displacement (the change of optical path length) and provide a label-free approach to image the structure and dynamics of cells [1–6]. Particularly, Doppler techniques that measure the change of phase over time allows the tracking of fast dynamics. One of the technologies developed to image fast dynamic events is Doppler optical coherence tomography (D-OCT) [7–9], a functional extension of optical coherence tomography (OCT) [10]. D-OCT performs phase resolved cross-sectional imaging, quantifies the change of phase over time introduced by the moving object, and extracts dynamic information about the target of interest. D-OCT has found a wide range of applications in blood flow measurement, vessel visualization, optical elastography, etc [11–13]. For example, D-OCT has been used to measure human retinal blood flow for clinical management of ophthalmology diseases [14,15]. Although researchers have demonstrated high-speed D-OCT imaging with fast OCT engine and GPU accelerated signal processing [16,17], OCT and D-OCT imaging in the en face plane remains slow. This is because conventional OCT system (and D-OCT) acquires Ascans (depth profiles) sequentially at different lateral positions, prioritizes B-mode scanning and obtains an en face image in $(x,y)$ plane after the scanning of entire 3D volume ($x,y,z)$ is accomplished [10]. The time delay between the acquisition of different Ascans prevents simultaneous measurement of Doppler signals at different locations within the en face plane [18]. An alternative OCT imaging approach is based on a full field interferometer that uses a spatially incoherent light for area illumination and performs parallel signal detection in the en face plane [19,20]. However, full field OCT systems usually rely on mechanical scanning of the optical path length for depth resolved measurement and signal demodulation, resulting in a limited temporal resolution and phase stability for effective Doppler measurement.

To address the unmet need for quantitative motion imaging at cellular and sub-cellular scale, we investigate a full field Doppler phase microscopy (FF-DPM) technology based on optical computation. FF-DPM performs parallel Doppler imaging for all the pixels within the en face plane. FF-DPM is a functional extension of optically computed phase microscopy (OCPM), a novel imaging technology recently developed by our team [21]. OCPM processes signal photons from the sample using a low coherence interferometer, an optical computation module, and accompanying post processing algorithms to achieve depth resolved imaging of amplitude and phase. As demonstrated in our previous studies, OCPM is an ideal imaging platform for label-free cell imaging. Notably, with depth segmentation achieved by optical computation, FF-DPM performs parallel Doppler measurement within the en face plane. This is desirable to image cultured cells that form a thin film attached to the substrate of a petri dish. By imaging dynamic events ($m(\boldsymbol {r},t)$) with high spatial ($\boldsymbol {r}$ representing 3D spatial coordinate) and temporal ($t$) precision, FF-DPM has the potential to become a powerful tool to investigate the dynamic activities for a population of cells as well as individual cells. Although Doppler measurement is known to be sensitive to axial motion along $z$ direction rather than lateral motion in $(x,y)$ plane, the nanometer level displacement sensitivity allows FF-DPM to detect subtle 3D motion that has a non-zero axial component. Our results clearly show unique motion characteristics for particles adhered to a cell and free particles through en face Doppler imaging.

The key innovation of this study is the use of an optical computation strategy to perform depth resolved phase quantification and Doppler measurement. We envision the FF-DPM to have a significant impact, as it provides a unique capability to quantitatively measure subtle motion for models based on cultured cells. In this manuscript, we first describe the principle of FF-DPM. To validate motion tracking capability of FF-DPM, we present results obtained from samples that were mechanically actuated by a piezo transducer (PZT). Afterwards, we demonstrate FF-DPM imaging of magnetic particles suspended in cell culture medium and cultured with cells. We then summarize the manuscript, and discuss the results.

2. Principle

The configuration of the FF-DPM system is depicted in Fig. 1. Briefly, it uses a low coherent light source (LED center at 730nm with 20nm bandwidth) to illuminate a Michelson interferometer (box on the left side of Fig. 1) that has a reference and a sample arm. The objective lenses used in the reference arm and sample arm are identical (Nikon Apochromatic Water Dipping Objectives, 40X, NA=0.8). The Michelson interferometer’s output is interfaced with an optical computation module (the box on the right side of Fig. 1). The optical computation module consists of a diffraction grating (GH50-12V, Visible Reflective Holographic Grating from Thorlabs, 1200/mm, $50mm \times 50mm \times 9.5mm$), a lens and a spatial light modulator (SLM, LC-R 720 Reflective by Holoeyes). SLM has found a wide range of applications in optical imaging [22], and plays a critical role in optical computation in our FF-DPM system. After SLM modulation, the light goes through the grating again and is detected by a CMOS camera (Basler acA2000). A frame grabber (NI 1433, National Instruments) acquires data from the camera and streams the data to the computer. The data acquisition is synchronized by a trigger signal from the SLM.

 

Fig. 1. FF-DPM system configuration. Ref lens: reference lens; OBJ lens: objective lens; BS: beamsplitter; PBS: polarization beam splitter; SLM: spatial light modulator.

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To discuss the principle of FF-DPM, we represent the field reflectivity (complex) of the sample as Eq. (1) ($t$: time; $k=2\pi /\lambda$: wavenumber; $(x,y)$: transverse coordinate; $z$: axial coordinate along the direction of light propagation; $A(x,y,z)$: complex sample reflectivity; $\delta d(x,y,t)$: axial displacement). For the 3D coordinate system, $z$ is the primary direction of light propagation, $x$ is along the direction of the grating’s periodic groove structure and $y$ is orthogonal to $x$. The objective of FF-DPM is to measure the displacement $\delta d(x,y,t)$ and the velocity $v(x,y,t)=\frac {\partial \delta d(x,y,t)}{\partial t}$ at a selected depth $z$.

Assuming the reference mirror is located at $z=0$ and has a reflectivity $A_{ref}$, we denote the output (field) from the interferometer (superposition of reference light and sample light shown as Eq. (1)) as $\eta \sqrt {M_0(k)}(i_{sample}+A_{ref})$ where $M_0(k)$ represents the source spectrum (intensity) and $\eta$ is proportional to the amplitude of optical field incident into the interferometer. The output of the interferometer is dispersed by a grating and is focused by an achromatic doublet lens (Lens1) to the SLM plane. The grating has groove structure along $x$ direction and disperses the interferometric light along the same direction (Fig. 2(a)). On the other hand, signals from the same $y$ coordinate at the sample space ($(x_1,y_1)$ and $(x_2,y_1)$ in Fig. 2(a)) are mapped to a corresponding row of pixels with the same $y$ coordinate at the SLM plane, as the grating does not alter light propagation along $y$ direction. Hence the light intensity at the SLM is proportional to $M_0(k)|(i_{sample}+A_{ref})|^2$ and can be expressed as Eq. (2). To perform optical computation, the SLM imposes a sinusoidal pattern (Eq. (3) and Fig. 2(b)) to modulate the light incident within the SLM ($i_{SLM}$). Modulated by the SLM, the optical signal goes through the grating again, detected by the camera, and can be expressed as Eq. (4). The integral with regard to $k$ represents light detection without discriminating wavelength (or wavenumber $k$). For cultured cells with thickness significantly smaller than the axial resolution of the imaging system, the depth profile of the sample can be modeled as an impulse function located at depth $z_{cell}$: $A(x,y,z)=\delta (z-z_{cell})a(x,y)$. With $z_{cell}=z_{SLM}/2$, Eq. (4) becomes Eq. (5) when $\beta$ is a constant. As illustrated in Eq. (2)–Eq. (5), SLM modulation along $k$ direction effectively selects signals from depth $z_{SLM}/2$, by optically calculating the inner product between two sinusoidal functions ($e^{kz_{SLM}}$ and the interference spectrum $i_{SLM}$). This strategy has been used in previous studies on scan-less volumetric imaging [23]. On the other hand, SLM modulation along $y$ direction introduces a phase term $\alpha _yy$ that enables the extraction of complex field reflectivity from the real measurement and enables quantitative measurement of the phase [21].

 

Fig. 2. (a) illustration of the mapping between spatial coordinates in the sample plane and spectra seen by the SLM; (b) illustration of the modulation pattern projected by the SLM.

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To obtain Doppler signal, we apply Hilbert transform ($\mathcal {H}$) to Eq. (5), by performing Fourier transform, eliminating negative frequency components, and performing inverse Fourier transform [4]. The result, denoted as $\mathcal {H}(i_{camera})$, is complex with a real part equal to $i_{camera}$. Afterwards, we demodulate $\mathcal {H}(i_{camera})$ along y direction by applying frequency shifting in spatial frequency domain, which leads to Eq. (6) [21]. The Doppler phase is obtained by computing the phase difference between $i_{complex}(x,y,t+\delta t)$ and $i_{complex}(x,y,t)$ using Eq. (7) [16]. Assuming a constant velocity within the time interval $\delta t$, we calculate the axial velocity using Eq. (8) where $\lambda _0$ represents the central wavelength of the light source and $n$ represents the refractive index. Equation (7) and Eq. (8) clearly demonstrate the capability of FF-DPM to perform parallel measurement of Doppler signal in $(x,y)$ plane, which is critical for quantitative observation of motion at cellular and sub-cellular scales. Notably, when the phrases Doppler signal or Doppler image are used in subsequent discussions, we refer to $\Delta \phi _{Doppler}(x,y,t)$ or $v(x,y,t)$ depending on how we specify it.

3. Results

The axial and lateral resolution of FF-DPM were the same as OCPM, and were experimentally characterized via established protocols [21]. The axial resolution was measured by assessing the width of the axial point spread function and the value was measured to be 8.4 $\mu m$. We imaged a 1951 USAF resolution target to evaluate the lateral resolution and the value was measured to be 1 $\mu m$. The range of trackable motion depended on the data acquisition rate. The camera used in the FF-DPM has a 340 Hz maximum frame rate, and the SLM refreshes at a rate of 60Hz. Currently, the frame rate of Doppler imaging is limited by the exposure time of camera. To collect enough signal photons from the weakly scattering sample (cultured cells and particles introduced into the cell culture medium), we used an exposure time that is approximately 100ms, and operated the system at a 7.7Hz frame rate. During the exposure time, the camera collected photons for each frame of image. A small camera exposure time would lead to a low signal to noise ratio and poor image quality. $\delta t$ in Eq. (7) and Eq. (8) is 0.13s. To avoid phase wrapping artifact, $|\Delta \phi _{Doppler}(x,y,t)|$ in Eq. (7) and Eq. (8) must be smaller than $\pi$ [24,25]. This implies a maximum measurable speed of 1.1$\mu m/s$. On the other hand, the phase sensitivity of the OCPM platform was measured to be 0.064rad ($\delta \phi =0.064rad$). Under the same frame rate, the lower limit of detectable speed is 0.016$\mu m/s$ ($v_{min}=\frac {\lambda _0}{4\pi n\delta t}\frac {\delta \phi }{\sqrt {2}}$).

We first validated FF-DPM’s capability in quantitative motion tracking. We used a glass slide as the sample. The glass slide was attached to a piezo transducer (PZT, TA0505D024W Thorlabs). The PZT has a dimension of 5.0mm*5.0mm*2.4mm and consists of stacked piezoelectric ceramic layers chips that are sandwiched between inter-digitated electrodes. The PZT controller (KPZ101) actuated the PZT with a step-wise voltage to control the axial motion of the sample: $V(t)=\beta \lfloor \frac {t}{T_0} \rfloor$ with $T_0=1s$. We took 2D measurements ($i_{camera}(x,y,t)$ in Eq. (4)) using the camera, and obtained a stack of en face Doppler images ($\Delta \phi _{Doppler}(x,y,t)$ in Eq. (7)). We obtained noise suppressed Doppler signal $\Delta \overline {\phi }_{Doppler}(t)$ by averaging data within an area (14$\mu m$ by 14$\mu m$) at the center of the image, calculated the instantaneous velocity ($\overline {v}(t)=\dfrac {\lambda \Delta \overline {\phi }_{Doppler}(t)}{4\pi n \delta t}$), and obtained the displacement that is the cumulative sum of the velocity ($\delta \overline {d}(t)=\int _{t_0}^t \overline {v}(\tau ) \,d\tau$). In Fig. 3, we show results obtained when the PZT was actuated by different voltages. Figure 3(a) and (b) shows Doppler measurements when the sample moved away ($\beta <0$) and towards ($\beta >0$) the equal optical path length plane, respectively. The blue, green and red curves correspond to $\beta =\mp 3V$, $\beta =\mp 2V$, and $\beta =\mp 1V$, respectively. The black dotted lines represent linear fitting of the experimental data, the left-y axis represents the accumulated phase ($\int _{t_0}^t\Delta \overline {\phi }_{Doppler}(\tau ) \,d\tau$) and the right-y axis represents the displacement ($\delta \overline {d}(t)$). The step-wise motion observed in Fig. 3 is consistent with the step-wise driving voltage applied to the PZT. We performed linear fitting on $\delta \overline {d}(t)$ to estimate the average velocity ($\delta \overline {d}(t)=\hat {v}t$) (see the insets in Fig. 3(a) and (b)). For different voltages ($V(t)=\beta \lfloor \frac {t}{T_0} \rfloor$) applied to the PZT ($\beta$=−3V, −2V and −1V), we obtained $\hat {v}$=−0.122$\mu m/s$, −0.084$\mu m/s$ and −0.042$\mu m/s$, corresponding to a ratio of 2.89 : 2.00 : 1. For $\beta$=3V, 2V and 1V, we obtained $\hat {v}$=0.109$\mu m/s$, 0.072$\mu m/s$ and 0.037$\mu m/s$, corresponding to a ratio of 2.94 : 1.94 : 1. On the other hand, we assume the PZT to have a constant responsivity $\epsilon$ that is the axial displacement induced by a unit voltage change. We can calculate the average velocity of the PZT driven by voltage $V(t)=\beta \lfloor \frac {t}{T_0} \rfloor$ to be $v=\beta \epsilon T_0$. With $\beta =\mp 3V, \mp 2V, \mp 1V$, the axial velocities of the PZT are expected to have a ratio of 3:2:1. Experiments results obtained from Doppler imaging is consistent with theoretical estimation.

 

Fig. 3. Doppler tracking of accumulated phase (left) and displacement (right) of the sample(glass slide) (a) when the PZT was driven by negative voltages (red: $\beta =-1V$; green: $\beta =-2V$; blue: $\beta =-3V$); (b) when the PZT was driven by positive voltages (red: $\beta =1V$; green: $\beta =2V$; blue: $\beta =3V$);

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To validate the robustness of Doppler tracking for different amplitude signal, we used FF-DPM to track the motion of an object that had heterogeneous reflectivity in the en face plane. We place a Variable Line Grating Target (Thorlabs) on the PZT and drove the PZT using voltage $V(t)=\beta \lfloor \frac {t}{T_0} \rfloor$ with $T_0=1s$ and $\beta = -2V$. The target thus moved away from the equal optical path length plane over time. When the sample was actuated by the PZT, we took a series of 2D measurements ($i_{camera}(x,y,t)$ as in Eq. (4)) and obtained en face Doppler images ($\Delta \phi _{Doppler}(x,y,t)$ as in Eq. (7)). Results are summarized in Fig. 4. Figure 4(a) shows a 2D measurement taken by the camera. Areas with large signal magnitude correspond to high reflectivity coating of the resolution target, while areas with low signal magnitude correspond to glass substrate. To compare motion tracking results at different regions, we selected a ROI from a highly reflective region and another ROI from a low reflectivity region (marked as red boxes in Fig. 4(a)). We averaged the Doppler signal within each ROI, and converted the phase shift value to instantaneous velocity. In Fig. 4(b), velocities obtained from Doppler analysis are consistent for different ROIs, validating robustness of Doppler tracking for different amplitude signal. Moreover, the measured instantaneous velocities have has discrete peaks, because the PZT was driven by a step-wise voltage from the controller and took discrete steps. We further demonstrated the capability for simultaneous structural and Doppler imaging. We obtained $I_{complex}$ as expressed in Eq. (6) and normalized the value such that the magnitude of data ranged from 0 to 1. We fused structural image ($|I_{complex}|$) and Doppler image ($\Delta \phi$) together, using a RGB color map. We assigned the following values to the R, G and B channels of a pixel ([R, G, B]): $[|I_{complex}|-\kappa \Delta \phi,|I_{complex}|,|I_{complex}|]$ for $\Delta \phi <0$ and $[|I_{complex}|,|I_{complex}|+\kappa \Delta \phi,|I_{complex}|]$ for $\Delta \phi >=0$, with $\kappa =1$. This effectively overlay the structural image with a red shade for a negative velocity and with a green shade for a positive velocity. Figure 4(c) is the overlay of structural image and Doppler image obtained at time point indicated by the black arrow(t=0.4s) in Fig. 4(b). Due to the small instantaneous speed, the contribution of the Doppler channel is minimal. Figure 4(d) is the overlay of structural image and Doppler image obtained at time point indicated by the red arrow(t=1.0s) in Fig. 4(b). Due to the large instantaneous speed, the contribution of the Doppler channel is clearly observable. Motion tracking results (overlaid structural-Doppler image) over an extended period of time can also be found in Visualization 1. We also present results of phase tracking at discrete time points along $x$ dimension in Fig. 4(e). The left-y axis and right-y axis correspond to the accumulated phase ($\int _{t_0}^t\Delta \phi (x,y_0,\tau ) \,d\tau$) and the accumulated displacement, respectively. As illustrated in Fig. 4(e), the sample moved away from the objective over time. Moreover, the glass substrate corresponds to a larger phase retardation compared to region with coating due to a longer optical path length and the actual height of the chrome coating is approximately 100$nm$, which is consistent with the result we computed from the phase difference.

 

Fig. 4. FF-DPM imaging of variable line resolution target. (a) a measurement taken by the camera(image before computation); (b) velocity tracking over time for different ROIs; (c) overlaid structural-Doppler image when the speed was small; (d) overlaid structural-Doppler image when the speed was large; (e) displacement over time along $x$ axis (blue dotted line is where the measurement is).

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After validating the capability of FF-DPM in tracking the global motion of an object (Fig. 3 and Fig. 4), we used FF-DPM to image moving particles in different environments. In the first experiment, we suspended magnetic particles in cell culture medium. Magnetic particles (Sera-Mag Carboxylate-Modified[E3] Magnetic Particles, density $1.7g/cm^3$,Diameter 1 $\mu m$) were purchased from Fisher Sci, and used directly without further purification. We controlled the motion of the particles by moving a permanent magnet in the horizontal plane ($(x,y)$ plane) at a 1 Hz frequency. We took a stack of FF-DPM images ($\Delta \phi _{Doppler}(x,y,t)$), and overlaid the Doppler measurement with amplitude image using the same approach described before. We represented a positive Doppler phase shift using the color of green and negative Doppler phase shift using the color of red. One frame of the overlaid image is shown in Fig. 5(a). Sequentially acquired Doppler images can be seen in Visualization 2.

To illustrate how the Doppler measurements quantify the motion field with spatial and temporal resolution, we selected an ROI (red box in Fig. 5(a)) and show the measurement results in Fig. 5(b) and (c) (overlaid structural-Doppler image obtained at $T_0=9.62s$ and $T_0+\delta t$). Figure 5(d) and (e) show the amplitude images corresponding to measurements taken at $T_0$ and $T_0+\delta t$). According to results in Fig. 5(b) and (c), the magnetic particles and their motion patterns are clearly visible with Doppler contrast. In comparison, amplitude images in Fig. 5(d) and (e) show vaguely visible magnetic particles. The number of particles that could be observed within the field of view depended on the particle concentration. In Visualization 2, the motion of multiple magnetic particles could be observed. The motion of these magnetic particles followed a similar temporal pattern, because they were all excited by the same magnetic field. Residual fringe artifacts exist in Fig. 5, due to the challenge to fully subtract the DC signal. However, this does not affect Doppler imaging significantly, because Doppler signal is obtained by calculating the phase difference between frames, while the residual fringe artifacts are static. In Fig. 5(f), we compare Doppler signal obtained from a spatial location without magnetic particles ($(x_y,y_y)$ as indicated by the yellow arrow in Fig. 5(a)) and Doppler signal at a spatial location with magnetic particles ($(x_b,y_b)$ as indicated by the black arrow). $\phi _{Doppler}(x_b,y_b,t)$ (black curve in Fig. 5(f)) shows a much larger amplitude compared to $\phi _{Doppler}(x_y,y_y,t)$ (red curve in Fig. 5(f)), because of the motion of the particle. As shown in Fig. 5(f), it took approximately 1s for the particle to accomplish a cycle of motion, which was consistent with the period of the changing magnetic field. Figure 5(g) and (h) illustrate how the Doppler signal from the moving magnetic particles takes a pinwheel shape. Each magnetic particle ($Fe_3O_4$) could be considered as a small magnet that aligned itself along the magnetic line of force of the moving permanent magnet (Fig. 5(g)). When the permanent magnet took a counterclockwise rotation in the en face plane (from Fig. 5(g) right to left), the northern hemisphere of the magnetic particle rolled along the arrow in the top figure of Fig. 5(h), resulting in positive axial velocity (green) at one half (left) of the hemisphere and negative velocity (red) at the other half (right) of the hemisphere. On the other hand, the southern hemisphere of the magnetic particle rolled along the arrow in the bottom figure of Fig. 5(h), resulting in negative axial velocity (red) at one half (left) of the hemisphere and positive velocity (green) at the other half (right) of the hemisphere. The northern and southern hemispheres of the magnetic particle take different rotation direction, because the axis of the particle changes over time as illustrated Fig. 5(g). Results in Fig. 5 suggest that FF-DPM can be used to quantitatively study the complicated motion of multiple particles.

 

Fig. 5. FF-DPM images of magnetic particles actuated by external magnetic field: (a) a frame of overlaid structural-Doppler image where red and green colors indicates different motion direction; (b) overlaid structural-Doppler image within the ROI at $T=T_0$; (c) overlaid structural-Doppler image within the ROI at $T=T_0+\delta t$; (d) amplitude image within the ROI at $T=T_0$; (e) amplitude image within the ROI at $T=T_0+\delta t$; (f) $\phi _{Doppler}(x_b,y_b,t)$ (black line - correspond to the black arrow in fig5 (a)): Doppler signal at a spatial location with magnetic particle; $\phi _{Doppler}(x_y,y_y,t)$ (red line - correspond to the yellow arrow in fig5 (a)): Doppler signal at a spatial location without magnetic particle; (g) magnetic particles align themselves with the magnetic line of force; (h) the motion of northern hemisphere of the magnetic particles and the motion of southern hemisphere of the magnetic particles.

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We further demonstrated the capability of FF-DPM tracking the dynamic interaction between particles and cells. We used C2C12 cell line that is a subclone of myoblasts. The cells were cultured in Dulbecco’s Modified Eagle Medium (DMEM) supplemented with 10% fetal bovine serum, 1% penicillin / streptomycin at 37$^{\circ }$ in a humidified 5% $CO_2$ incubator. The concentration of the magnetic particles was diluted to approximately $0.001mg/ml$, such that the cell viability was not affected. We cultured cells with the magnetic particles for 3 hours. Under a microscope, it could be observed that some particles remained in the cell culture medium and other particles attached to the cells.

To excite mechanical motion, we translated the permanent magnetic field in $(x,y)$ plane at a 1Hz rate. We took a stack (200 frames, $N_{frame}=200$) of measurements ($i_{camera}$) over time, and obtained Doppler images $\Delta \phi _{Doppler}(x,y,t_i)$ with $i=1, 2,\ldots, N_{frame}$. We overlaid the morphological image and Doppler image using an RGB color map and show one of the frame in Fig. 6(a). Locations with a large magnitude of axial velocity appear to be red (negative velocity) or green (positive velocity)in Fig. 6(a). Clearly, Doppler signal arose from the motion of the magnetic particles. Real time Doppler tracking can be seen in Visualization 3 that shows a series of images (overlaid structural-Doppler image) obtained over time. To quantitatively assess the magnitude of motion during the observation period, we calculated the temporal variation of the Doppler phase: $\sigma _{\phi }^2(x,y)=\frac {1}{N_{frame}}\Sigma _{i=1}^{N_{frame}} (\Delta \phi _{Doppler}(x,y,t_i)-\Delta \overline {\phi }_{Doppler}(x,y))^2$, where $\Delta \overline {\phi }_{Doppler}(x,y)=\frac {1}{N_{frame}}\Sigma _{i=1}^{N_{frame}}\Delta \phi _{Doppler}(x,y,t_i)$. We present the result in Fig. 6(b). As illustrated in Fig. 6(b), "free" particles in the medium had larger magnitude of axial motion and larger lateral area with prominent motion signal. On the other hand, particles attached to the cells might be partially or fully internalized by the cell and the movement of these particles seemed to be restricted (small variance) axially and transversely. To highlight the difference, we enclose a particle attached to the cell a by green box and enclose a particle out of cell by a red box in Fig. 6(a) and (b). By simultaneously imaging the motion of different particles, FF-DPM revealed spatial and temporal characteristics of particle motion under different conditions. Our results suggest FF-DPM can be used to reveal the dynamic interaction between particles and cells at single-particle, single-cell level.

 

Fig. 6. FF-DPM imaging of magnetic particles cultured with C2C12 cells. (a) overlaid structural-Doppler image with red and green colors indicating negative and positive axial velocity; (c) temporal variation of Doppler signal highlights the magnetic particles.

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4. Discussion and conclusions

In this manuscript, we present a full field Doppler phase microscopy (FF-DPM) technology that addresses the unmet need for quantitative motion imaging at different spatial locations in the en face plane with cellular and sub-cellular resolution. The key innovation is to utilize an optical computation strategy to achieve depth resolved imaging and phase quantification. The system had a 8.4 $\mu m$ axial resolution and 1 $\mu m$ transverse resolution. The maximum and minimum measurable speeds were 1.1$\mu m/s$ and 0.016$\mu m/s$, respectively. We validated the motion tracking capability of FF-DPM by imaging samples actuated by a piezo transducer (PZT). In addition, we used FF-DPM to image the motion of magnetic particles. Our results clearly show unique motion characteristics for particles adhered to a cell and free particles through en face Doppler imaging.

Although FF-DPM and OCPM are based on the same system configuration, the specific difference between OCPM and FF-DPM is the physical parameter used to create image contrast. OCPM image performs quantitative phase measurement to map the nanometer scale topology of the sample. FF-DPM uses a Doppler algorithm the calculate the phase shift between adjacent frames. The smallest velocity that can be tracked by FF-DPM depends on the random variation of the measured phase ($\sigma _{\phi }^2$). To reliably measure subtle motion, it is desirable to measure the phase with higher confidence. This can be achieved by improving the signal to noise ratio in amplitude measurement [26], or by utilizing a common path configuration to improve the phase stability [27]. As shown in Fig. 3(a) and (b), the speeds tracked by FF-DPM were slightly different (0.055$\mu m/s$ versus 0.048$\mu m/s$, 0.109$\mu m/s$ versus 0.093$\mu m/s$, 0.158$\mu m/s$ versus 0.141$\mu m/s$,), when negative or positive voltages were applied to the PZT. This is because the PZT has hysteresis, and its deformation characteristics are different when the voltage increases or decreases [28]. In fact, the manufacture provided a responsivity ($\epsilon =0.037\mu m/V$) with a 15% uncertainty. We can also estimate the responsivity to be $\hat {\epsilon }=0.053\mu m/V$). The difference can be attributed to difference in mechanical loading, temperature and operating voltage range. Although Doppler imaging is limited to the measurement of axial motion, it is sensitive to nanometer scale displacement, and the motion sensitivity is much higher than techniques for transverse motion tracking, Hence FF-DPM is useful to image 3D motion that has a non-zero axial components. Similar to traditional holographic particle tracking technology [29–31], we recover the structure of the object using the phase of optical signal. Nevertheless, our system does not need the use of a laser that has both spatial and temporal coherence. Instead, we use a low coherent light source (LED) for illumination. The phase is extracted from a temporally coherence gated signal (in axial dimension), while the one-to-one mapping between the transverse coordinate in the sample plane and detector plane eliminates the need for spatial coherence. Moreover, the contrast of Doppler phase microscopy derives from the phase difference (in time), at a specific depth selected by the SLM modulation pattern.

As a proof-of-concept, we quantitative image the cell and nanoparticles by our system. Compared with other system, the most significant advantage of our system is that it can image and compute the phase of the en face plane simultaneously. In addition, we can track en face plane at different depth by altering the modulation pattern at the SLM. Improving and enhancing the robustness of our system when affected by noise is necessary to image the smaller motion or motion from smaller particles, such as lipid particles used to deliver therapeutics in mRNA Covid vaccines.