Open Access
Add to BibSonomy
Abstract
This article presents a simple and high-speed approach for tracking colloidal spheres in three dimensions. The method uses the curvature of the wavefront as determined by the transport of intensity equation (TIE) technique. Due to the fact that the TIE is applicable under partially coherent light, our technique is fully compatible with standard bright field microscopes, requiring no demanding environmental stability requirements or restrictions on the noise produced by related laser speckles. The method was validated experimentally to determine the sedimentation and diffusion coefficients of two different sizes of microspheres, 20 and 3 microns. The 3D position of the microspheres was calculated with an accuracy greater than 350 nm. Moreover, we examined the calculated 3D positions to determine the parameters of the microsphere interaction with its surrounding media, such as the sedimentation and diffusion coefficients. The results show that the measured sedimentation and diffusion of the microspheres have a good agreement with predicted values of about 2% and 10%, respectively, demonstrating the robustness of our proposed method.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Three-dimensional (3D) particle tracking is one of the most promising methods for investigating the dynamics of components in a biophysics model system. This approach has a wide variety of applications in natural science, including colloidal science, fluid mechanics, and cell biology. Colloidal particle 3D trajectories can be used to interpret a variety of physical phenomena and properties, including colloidal particle interactions [1–4], the 3D motion of bacteria [5–8] and fluorescent particles [9], transport in living cells [10,11], and the viscoelastic properties of a fluid [9,12]. Additionally, in certain instances, complete 3D information in microparticle tracking is required for accurate interpretation of measured data, for example, in the study of anisotropic and chiral microparticles, as well as the surface effects of boundaries [9,13].
To properly understand the three-dimensional motion of microparticles, a fast and accurate microscopy technique is required. Quantitative microscope imaging in three dimensions has advanced significantly via the use of interferometry and digital holography methods [14–17]. Numerous 3D particle tracking systems based on interferometric techniques have been suggested [4,18–20]], which often involve complex optical setups and data processing. Additionally, several techniques based on Fresnel particle tracking [21] and stereoscopic imaging [22] are discussed. Additionally, more complex tools like optical tweezers and scanning confocal microscopy have been employed [6,23]. Also, several fluorescence microscopic-based techniques have been developed that may be used to estimate the radius of fluorescent particles in defocus images [9,24,25]. The majority of these methods need coherent light and sophisticated optical setups and are only applicable to fluorescent particles, restricting their use in standard microscopy equipment.
In this research, we introduce a novel TIE-based technique for the three-dimensional tracking of spherical colloidal particles. The method uses the TIE equation to extract the phase distribution corresponding to the spherical particle. We use a simple and cost-effective optical system to simultaneously image the particle in two distinct imaging planes to reconstruct the particle’s phase distribution using the TIE equation. We found that the phase distribution changes linearly along the depth axis in the favorable region, making it simple to determine the particle’s location in a three-dimensional chamber. Furthermore, we evaluate the TIE technique by performing standard experiments on the sedimentation and diffusion of colloidal particles to demonstrate the method’s capability and robustness.
2. Theory and model
2.1 Transport of intensity equation (TIE)
The TIE relates the intensity distribution, $I$, to the phase difference (PD), $\phi$, by [26]
Here, by considering periodic boundary conditions, we used fast Fourier transform method to extract phase from two image intensities [27–30]
2.2 Paraxial approximation of the spherical wave
We aim to track the 3D random movement of a spherical microparticle using the TIE technique in conjunction with a standard bright-field microscope. To accomplish this, we investigate the use of phase information to estimate the particle’s depth location. The phase distribution of the passing wave through the microparticle is calculated using the TIE equation introduced in 2.1, which necessitates at least two simultaneous images of the microparticle in two different imaging planes. Since the TIE equation is a slow varying approximation to the Helmholtz equation, one of exact solution is the paraxial approximation of a spherical wave (Gaussian beam) [32] as given by
where $C$ is the curvature of wavefront, $R$ indicates the radius of the spherical wave curvature and $W_0$ denotes the radius of the beam waist, demonstrating the spherical wave’s curvature and the spherical wave radius have a linear and inverse relationship to the depth (z-axis), respectively (Fig. 1). This concept may also be used to reconstructing the wavefront of light which interacts with an object in front of the detection lens. For instance, if a transparent microsphere is located in front of the objective lens, it induces a quadratic distortion to the incident light wavefront related to the microsphere’s position and its geometry and refractive index. Therefore, considering the TIE concept and Eq. (4) can obtain a relationship between the microsphere’s position and its associated wavefront. This relationship will be used to calibrate imaging systems for different microspheres using a predetermined database to identify the depth position of a moving particle.
Fig. 1. The wavefront variation of a converging beam along the optical axis. The schematic figure depicts that the wavefront radius curvature is grown over the propagation when approaching the beam waist.
3. Experiment
3.1 Optical setup
To use the TIE approach, we developed a compact setup (dashed box in Fig. 2) which enables us to record two intensity images from two distinct depths simultaneously. The compact system consists of a 50:50 beam splitter (BS) and two identical CMOS cameras (Proline, 1/3 570$\times$720) offset by a displacement $\delta z$ controlled by a micropositioner with a 10 $\mu$m resolution. The arrangement is connected to an inverted bright-field microscope’s output channel (Olympus, IX-71). The illumination part of the microscope consists of a semi-coherent light source, a bandpass filter (BPF, $\lambda =550\pm 30$ nm), and a condenser lens (0.55 NA). The diffracted light from the sample is then collected by an objective lens (Olympus, UPlanFLN 20$\times$ and 60$\times$). Afterward, a tube lens generates the far-field images on the CMOS sensors, which include two depth-dependent information that can be used to reconstruct the phase by the TIE.
Fig. 2. The schematic configuration of the experimental setup. The optical setup consists of two main parts; the microscope (solid-box) and a homemade accessory (dashed-box); The microscope is consist of a light source (LS), an optical filter (F), condenser lens (C), sample holder (S), microscope objective lens (MO), mirror (M), and tube lens(TL). The homemade accessory is built up by beam splitter (BS) and two cameras (Cam A) and (Cam B) to record two intensity distributions simultaneously. The cameras are parked in different positions along the optical axis by separation distance $\Delta z$ to record the different depth images.
3.2 Calibration
To calibrate the TIE setup, we conducted two sets of experiments using microspheres with diameters of 3 and 20 $\mu$m. The microspheres are separated into two distinct water-filled chambers so that they cannot be displaced randomly. To acquire the depth image, the particles are scanned stepwise along the optical axis (z-axis) using a motorized positioner with a step size of 280$\pm$20 nm. The positioner enables progressive switching of the particle’s focussed image between cameras A and B. The 20$\times$ objective lens (Olympus, UPlanFLN 20/0.50 /0.17) is used to record a pair of images of 20 $\mu$m microsphere by cameras A and B, which are positioned $8.35$ mm apart from each other (Fig. 3(a)). Then, for each pair of acquired images, the phase distribution for the particular position is obtained using Eq. (3) to determine the wavefront curvature. The second example is performed to examine the system’s capability to track the smaller particles within the 3D chamber using the 3 $\mu$m microsphere. For this purpose, a 60$\times$ objective lens (Olympus, UPlanFLN-OI 60 /0.65-1.25 /0.17) is used as a high resolution and magnification system to obtain the microsphere image. The depth of field of the objective lens and size of the microsphere is smaller than the previous example, requiring a smaller scanning depth at the sample place. Accordingly, it results in closer effective imaging planes, which have measured empirically about $6.68$ mm. As shown in Fig. 3(b), we have found that the wavefront curvature follows a linear trend when the microsphere is within the particular range of $\delta z$. The extracted relationships between the curvature and the microsphere depth position ($z$) show that the predictable position along the z-axis is in order of the microsphere size (Fig. 4), which can be used as a database to track a freely moving microsphere in a three-dimensional chamber.
Fig. 3. The wavefront radius curvature extraction via the TIE method. (a) first and second rows: recorded intensity distributions of the particle with cameras A and B at different depths. Third row: the extracted phase distribution via the TIE method for the recorded images. (b) the calculated wavefronts radius curvature along with the depth position of the particle. The solid red line indicates the linear behavior of the wavefront curvature.
Fig. 4. The calibration of the depth position versus the wavefront curvature for (a) 20 $\mu$m and (b) 3 $\mu$m particles. The solid red line indicates the linear behavior of the depth position versus wavefronts curvature.
The resolution is determined by comparing the collected experimental data to the corresponding regression model. The regression model accurately reflects the ideal linear trend of the z position; however, the recorded data deviating from the model limits the measurements. Therefore, the depth measurement limitation could be defined as the maximum deviation as the peak-to-valley (PV) of the recorded data around the regression model, which provides the highest resolution achievable for the microsphere depth prediction. To calculate the average resolution, we have measured the three different PVs from different recorded datasets. First, the regression model of each dataset is calculated individually, and then the associated PV is extracted from its deviation of the regression model resulting a 350 nm resolution on average.
4. Results and discussion
Since our method uses phase-based position prediction, it allows us to effectively calculate the position of random moving particles with high precision even when the microsphere drifts due to changes in environmental parameters such as temperature. In the first experiment, we have imaged a randomly moving 20-micron particle, which was sedimented 18 microns for 95 seconds. The imaging process begins when the microsphere is detected by camera A, parked close to the microscope, and ends when the second camera (Cam B) detects the high contrast image farther away from the microscope. The pair captured images are then used to predict the 3D location of the microsphere within the relatively large chamber (Fig. 5(a)). To do this, first, the lateral position (x-y plane) of the microsphere is obtained from the intensity center of the images. Then, the axial position (z-axis) is determined based on the TIE and calibration database presented earlier in Sec. 3.2 The calculated 3D position of the microsphere can help determine the parameters of the microsphere motion in interaction with its surrounding media, such as sedimentation and diffusion.
Fig. 5. 3D tracking of freely moving 20 $\mu$m particle. (a) the trajectory of the 20 $\mu$m swimmer particle. The colors represent the video frames’ time sequence. (b) the z position of the particle over time, demonstrating the sedimentation phenomena. The sedimentation coefficient is obtained 0.166 $\mu$m/s by fitting a line to the particle z position over time. (c) the mean square displacement in the three directions x, y, and z is denoted by red, green, and blue, respectively. The dashed lines show the mean square displacement, the solid lines show the fitted linear curves, and the colored areas represent the error bar of the experiments.
Fig. 6. The movement analysis of the 3D tracking of a 3 $\mu$m particle. (a) the z position of the particle over time. The red line shows a fitted line with a slope of 0.093 $\mu$m/s representing the sedimentation coefficient. (b) the mean square displacement in the three directions x, y, and z corresponds to red, green, and blue. The dashed line shows the mean square displacement, the solid lines show the fitted linear curves, and the colored areas represent the error bar of the experiments.
The location of the microsphere along the z-axis indicates the sedimentation phenomenon. The changing position of the microsphere along the z-axis illustrates a linear trend that defines the sedimentation coefficient $SC=0.166\pm$0.002 $\mu$m/s. The measured value is strongly matched by the estimated theoretical value of 0.167 $\mu$m/s (Fig. 5(b)) [33], confirming the capability of our proposed method.
The data can be used to characterize the random particle’s diffusion activity in three dimensions. The diffusion coefficients ($D$) are investigated by calculating the mean square displacement (MSD) for all three axes, x, y, and z. The slope values of the MSD graphs are extracted as the $D$ for each axis independently, which are equal to $D_x = 0.024\pm$0.004 $\mu$m$^2$/s, $D_y = 0.021\pm$0.005 $\mu$m$^2$/s, and $D_z = 0.020\pm$0.010 $\mu$m$^2$/s (Fig. 5(c)). The calculated coefficients along the three axes are approximately similar and are matched with the theoretical prediction (0.021 $\mu$m$^2$/s for 20 $\mu$m microsphere) by less than 5% error. As a result, it verifies the potential of the TIE method to determine the precise diffusion coefficient in three dimensions.
In the second experiment, to demonstrate our proposed method’s capability to track a smaller particle, we imaged a free-moving silica microsphere with a diameter of 3 $\mu$m in the 3D chamber. Since the 3$\mu$m microsphere is smaller than the previous example, the expected sedimentation takes longer due to the particle’s faster random interaction with its surrounding medium. To record the particle’s movement, we have used a high NA objective lens (NA=0.7) to increase the resolution and magnification of the recorded images, resulting in a precise phase reconstruction and 3D position of the microsphere. The z coordinate of the reconstructed 3D position -data is used to extract the sedimentation speed of the microsphere, $SC = 0.093\pm$0.002 $\mu$m/s, which verifies the theoretical calculated result, $SC = 0.091 \mu$m/s (Fig. 6(a)). As expected, the 3 $\mu$m particle sedimentation takes approximately 1.8 times longer than the procedure for the 20 $\mu$m particle in an identical situation. Furthermore, the diffusion of the 3 $\mu$m particle in three dimensions is measured $Dx = 0.139\pm$0.035 $\mu$m$^2$/s, $Dy = 0.152\pm$0.036 $\mu$m $^2$/s, and $Dz = 0.122\pm$0.023 $\mu$m$^2$/s, which is quite consistent with the theoretical prediction, 0.140 $\mu$m$^2$/s (Fig. 6(b)). The result demonstrates that the 3 $\mu$m particle interaction is approximately 5.7 times stronger than the interaction of the 20 $\mu$m particle with its surrounding medium.
5. Conclusion
We have developed a low-cost phase imaging system and method based on the TIE technique and simultaneous imaging in three-dimensional particle tracking. The system is built entirely from commercially available optical components. The whole optical system assembled comfortably inside a small footprint of $120\times 90\times 40$ mm$^3$ and can be added to any traditional optical microscope. The system’s size and portability are critical for making this technology more accessible to scientists and expanding the possibilities for sophisticated biological applications. Compared to interferometry and holography techniques, the proposed TIE approach does not require a coherent light source, which eases experimental and computational concerns. The experimental results demonstrate that simultaneous imaging of randomly moving microspheres in conjunction with the TIE method enabled a precise estimation of the microsphere position within more than twice the depth of focus of the imaging system. The 3D tracking results allow us to obtain the microsphere motion parameters such as sedimentation and diffusion in a fairly large chamber. Our empirical findings are consistent with the theoretical methods that have been reported before [33], indicating the high accuracy of our proposed technique in estimating the 3D location of particles with an accuracy of approximately 350 nm. This proof-of-concept application shows how the benefits of TIE phase reconstruction may be easily used to advance 3D particle tracking with a diverse set of biological end-users. The flexibility to apply this method on conventional microscopes makes it appropriate for a variety of applications. For instance, it can be used in different disciplines such as evaluation of the interactions between colloidal particles, investigating the fundamental questions in statistical physics, probing the viscoelastic properties of soft media, and measuring the dynamical properties of single polymers (such as lipid droplets). We anticipate that our proposed method is also valid as long as the object’s overall form retains a spherical-like shape. However, if the object’s asymmetry is broken down into another form, such as ellipsoid-like, our suggested analysis approach should be evaluated with more parameters in addition to the wavefront curvature, such as center intensity, intensity distribution, or it may be gained by a computational technique such as machine learning.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. 40(24), 4152–4159 (2001). [CrossRef]
2. R. Yasuda, H. Noji, M. Yoshida, K. Kinosita, and H. Itoh, “Resolution of distinct rotational substeps by submillisecond kinetic analysis of f 1-atpase,” Nature 410(6831), 898–904 (2001). [CrossRef]
3. M. Duocastella, C. Theriault, and C. B. Arnold, “Three-dimensional particle tracking via tunable color-encoded multiplexing,” Opt. Lett. 41(5), 863–866 (2016). [CrossRef]
4. M. A. Charsooghi and A.-R. Moradi, “Surface proximity effect in sedimentation investigated by digital holographic microscopy,” Appl. Opt. 57(7), B179–B183 (2018). [CrossRef]
5. H. C. Berg, “How to track bacteria,” Rev. Sci. Instrum. 42(6), 868–871 (1971). [CrossRef]
6. G. Soni, B. J. Ali, Y. Hatwalne, and G. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84(4), 2634–2637 (2003). [CrossRef]
7. R. Thar, N. Blackburn, and M. Kühl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66(5), 2238–2242 (2000). [CrossRef]
8. K. Bente, S. Mohammadinejad, M. A. Charsooghi, F. Bachmann, A. Codutti, C. T. Lefèvre, S. Klumpp, and D. Faivre, “High-speed motility originates from cooperatively pushing and pulling flagella bundles in bilophotrichous bacteria,” eLife 9, e47551 (2020). [CrossRef]
9. M. Speidel, A. Jonáš, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003). [CrossRef]
10. M. De Brabander, G. Geuens, R. Nuydens, M. Moeremans, and J. De Mey, “Probing microtubule-dependent intracellular motility with nanometre particle video ultramicroscopy (nanovid ultramicroscopy),” Cytobios 43(174S), 273–283 (1985).
11. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. 26(1), 373–399 (1997). [CrossRef]
12. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18(13), 13563–13573 (2010). [CrossRef]
13. F. Chasles, B. Dubertret, and A. Boccara, “Full-field optical sectioning and three-dimensional localization of fluorescent particles using focal plane modulation,” Opt. Lett. 31(9), 1274–1276 (2006). [CrossRef]
14. E. A. Akhlaghi, A. Darudi, and M. T. Tavassoly, “Reconstructing the phase distribution of two interfering wavefronts by analysis of their nonlocalized fringes with an iterative method,” Opt. Express 19(17), 15976–15981 (2011). [CrossRef]
15. A. Moradi, E. A. Akhlaghi, F. Hajizedeh, and S. N. S. Reihani, “Digital holography based submicron thermometry,” Opt. Express 24(25), 28678–28685 (2016). [CrossRef]
16. V. Abbasian, Y. Ganjkhani, E. A. Akhlaghi, A. Anand, B. Javidi, and A.-R. Moradi, “Super-resolved microsphere-assisted mirau digital holography by oblique illumination,” J. Opt. 20(6), 065301 (2018). [CrossRef]
17. V. Abbasian, E. A. Akhlaghi, M. A. Charsooghi, M. Bazzar, and A.-R. Moradi, “Digital holographic microscopy for 3d surface characterization of polymeric nanocomposites,” Ultramicroscopy 185, 72–80 (2018). [CrossRef]
18. S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. Van Blaaderen, P. Van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15(26), 18275–18282 (2007). [CrossRef]
19. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45(16), 3893–3901 (2006). [CrossRef]
20. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15(4), 1505–1512 (2007). [CrossRef]
21. Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Fresnel particle tracing in three dimensions using diffraction phase microscopy,” Opt. Lett. 32(7), 811–813 (2007). [CrossRef]
22. R. Racca and J. Dewey, “A method for automatic particle tracking in a three-dimensional flow field,” Exp. Fluids 6(1), 25–32 (1988). [CrossRef]
23. S. N. S. Reihani, M. A. Charsooghi, H. R. Khalesifard, and R. Golestanian, “Efficient in-depth trapping with an oil-immersion objective lens,” Opt. Lett. 31(6), 766–768 (2006). [CrossRef]
24. M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micron-scale using a single camera,” Exp. Fluids 38(4), 461–465 (2005). [CrossRef]
25. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7(7), 2043–2045 (2007). [CrossRef]
26. M. R. Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]
27. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001). [CrossRef]
28. N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics, (Optical Society of America, 2010), p. JMA7.
29. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]
30. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997). [CrossRef]
31. M. Krenkel, M. Bartels, and T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21(2), 2220–2235 (2013). [CrossRef]
32. B. E. Saleh and M. C. Teich, and O. FUNDAMENTALS, Photonics (Wiley, 2019).
33. M. A. Charsooghi, E. A. Akhlaghi, S. Tavaddod, and H. Khalesifard, “A matlab program to calculate translational and rotational diffusion coefficients of a single particle,” Comput. Phys. Commun. 182(2), 400–408 (2011). [CrossRef]
