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We have proposed and demonstrated a very sensitive volume measurement scheme for a live cell with a quantitative phase microscopy (QPM) utilizing auto-focusing and numerical edge detection schemes. An auto-focusing technique with two different focus measures is applied to find the focus dependent errors in our live cell volume measurement system. The volume of a polystyrene bead sample with 3 μm diameter has been measured for the validity test of our proposed method. We have shown that a small displacement of an object from its focusing position can cause a large volume error. A numerical edge detection technique is also used to accurately resolve the boundary between a cell and its suspension medium. We have applied this method to effectively suppress errors by the surrounding medium of a single red blood cell (RBC).
©2009 Optical Society of America
1. Introduction
Volume changes in a live bio cell are known to be originated from the concentration variations of soluble cell constituents such as ions, metabolites, osmotic pressure, and its health condition. These volume variations are essential characteristics of a cell for many pathological conditions and these can also represent the overall surrounding environments of a cell. An accurate and rapid measurement of a cell volume or its morphological variation is very important in biomedical science for monitoring or diagnosing cells, cellular growths, and their functions [1, 2]. Several kinds of cell volume measurement methods have been proposed such as electrophysiological method, light scattering method, and optical 3D imaging method. Electrophysiological method includes the patch clamped capacitance measurement technique [3] and scanning ion conductance microscopy technique [4]. These methods are technically complicated to implement due to the interventional measurement technique and cannot measure repeatedly for a sample since these require attachments of electrodes to a sample. Light scattering methods such as coulter counting and flow cytometry techniques are normally used for the mean volume measurement of cell populations [5, 6]. Especially, mean corpuscular volume (MCV) measurement with light scattering methods have been used as a versatile tool to measure the average red blood cell volume, which is a standard measure for diagnosing anemia. These techniques cannot measure an individual cell volume or its variation. Optical imaging techniques to measure the volume of an individual cell have been developed utilizing optical sectioning microscopy [3, 7], fluorescence microscopy [8, 9], video enhanced contrast optical microscopy [10], and spatial filtered microscopy [11]. These optical techniques can provide the absolute volumetric value of a single cell with the cost of sophisticated optical instruments as well as laborious experimental and computational procedures. Recently, several attempts have been reported on cell volume measurements using various QPM systems such as Fourier phase microscopy [12, 13], Hilbert phase microscopy [14], Diffraction phase microscopy [15, 16], Digital holographic microscopy [17, 18], and Heterodyne Mach-Zehnder phase microscopy [19]. In these techniques cell volume can be obtained fast and easily just by converting measured phase data into thickness information with a nanometer level sensitivity. However, relatively little work has been done on the errors related to these volume measurement techniques. When a phase image of a cell is out of focus, the spatial resolution of an imaging system will be degraded and the boundary of a cell can be smeared into or out of a boundary. The phase value itself can be altered from its original value in an off-focused phase image. These may work as error sources for the accuracy of volume measurement using QPM.
In this paper, we present the off-focusing effects in cell volume measurement using QPM. By introducing an edge detection and auto-focusing techniques in phase image acquisition methods for a live cell, we have analyzed accuracy in cell volume measurement techniques with QPM. The basic experimental configuration for our proposed method consists of Mach-Zenhder interferometer which employs an inverted optical microscopy system with a monochromatic source and a charge coupled detector (CCD) for detecting a spatial interferogram. QPM can retrieve a quantitative phase image with a subnanometer path length sensitivity from a single interferogram measurement [14, 15]. This makes it possible to measure the volume variation of a live cell within a millisecond time scale. Since the volume of a cell is obtained by converting phase data into thickness by applying certain refractive index data for a cell, the volume measurement speed using QPM can be as fast as the image acquisition rate and it depends only on the frame rate of a CCD. We have applied several auto-focusing techniques [20, 21] to the cell volume measurement and compared the errors in measured cell volumes for a live cell. We have also shown that an edge detection technique which can accurately resolve the boundary between a cell and its surroundings can also effectively enhance the system accuracy in cell volume measurement using QPM.
2. Experimental Setup
Our cell volume measurement system is based on the Hilbert phase microscopy introduced by Ikeda et al. [14]. It is one of the QPM systems with a monochromatic light source and a CCD as a detection system. We measure the cross-correlation interferogram of a cell sample in the spatial domain, which contains the spatial phase information of a sample. The spatial dependent interferogram can be generally expressed as
where IR and IS are the irradiance in the reference arm and the sample arm, respectively, qx, and qy are the constant spatial frequency offsets of the interferogram with respect to x and y axes, and ϕ(x, y) is the phase of a sample. We define qmax as the maximum spatial frequency offset with respect to either x or y directions. The maximum tilted angle, θmax between the reference and the sample beams can be determined by the wavelength of the laser and the maximum spatial frequency qmax with
When the pixel to pixel space of a CCD is △ x, the maximum spatial frequency we can measure with this CCD becomes 1/(2 △ x) by the Nyquist-Shannon sampling theorem. Since the pixel to pixel space △ x of our CCD is 7.4 μm, we have qmax = (2×7.4 μm)-1. The maximum angle between the reference and the sample waves, θmax is calculated by Eq. (2) to be 36 milliradian. We have chosen the tilted angle between the reference and the sample beams to be 12 milliradian along with both x and y axes. This makes the constant spatial frequency offset of the system along the x and y axes become q ; qx = qy = (6×7.4 μm)-1. This means that one period of a sinusoidal interference data is sampled with 6 pixels in our detecting CCD for both x and y axes. After using the high frequency filtering and the Hilbert transformation of a measured interferogram, we can obtain the quantitative phase ϕ(x, y) of a sample by just taking a single shot interferogram measurement [14].
Schematic diagram of the experimental setup is shown in Fig. 1. This setup is designed for the quantitative phase measurement of a transparent sample. The basic configuration is the Mach-Zehnder interferometer with an optical source and a CCD for detecting an interferogram [14]. A diode pumped solid state (DPSS) laser is used as an optical source with a center wavelength of 532 nm. The laser light is split into two beams by a beam splitter (BS) and passes to the reference and the sample arms of the Mach-Zehnder interferometer. We have put a pinhole in the reference arm to eliminate phase error in measured interferogram associated with the wavefront noise of the reference beam. An inverted microscopy system is built in the sample arm of the interferometer with a 100 × objective lens that allows 0.3μm transverse resolution. The output fields from these two arms are combined with another BS and then the combined beam is detected with the CCD through a tube lens of 200 mm focal length. Two linear polarizers (LP1 & LP2) are used to adjust input light intensity to the interferometer and a neutral density filter (NDF) in the reference arm of the interferometer is used to maximize the visibility of an interferogram recorded by the CCD. The tilted reference beam with respect to the sample beam forms an interferogram on the CCD with a linearly modulated fringe patterns tilted with respect to x axis by an angle of 45 degrees. We have used a CCD from Imperx (IPX-VGA210-L) which has an image acquisition rate of 207 frames per second at a full resolution of 640×480 pixels. The pixel to pixel distance of the CCD is 7.4 μm. The optical magnification between an object and its detected image by the CCD is calibrated to be about 110 for our system.
3. Off-focusing effects in QPM
3-1. Polystyrene beads as a test sample
We have investigated the effect of defocusing on the accuracy of cell volume measurement technique utilizing QPM. Polystyrene beads (Duke Scientific Co.) whose average size is known was used as a test sample. The average diameter of the beads is 3.0 μm and the standard deviation of the diameter size is 0.027 μm. The refractive index of the polystyrene bead is 1.59. The polystyrene beads were sandwiched with a cover glass and a slide glass and are floating on glycerol whose refractive index is 1.47.
3-2. Auto-focusing methods
The focusing position of an intensity image for a reflective or a scattering object is identified in general by spatial frequency analysis with various auto-focusing techniques [20]. Those methods are based on the fact that high spatial frequency components are enhanced in the intensity image of reflective or scattering objects when they are focused. Unlike the intensity images of these reflective or scattering objects, it has been reported that the phase image of a transparent phase object does not show the highest spatial frequency components with its fine fringe structures at a focused position [21]. Since we have observed similar trends in the phase image of a phase object, we have also used the intensity images of a phase object to find its focus position. Figure 2 is obtained with a 100 × objective lens and the depth of field in our system is 0.4μm. Since the maximum thickness of our sample is about 3 μm, the depth of field is much smaller than the sample by an order of magnitude and this makes the sample to become out of focus easily with small shifts in the longitudinal direction. Figure 2(a) shows measured interferograms of a polystyrene bead for 5 different focusing conditions; we have moved the polystyrene bead sample using a translation stage from -2 μm to + 2 μm with a step size of 1μm. From these 5 interfegrams shown in Fig. 2(a), we have retrieved 5 intensity and 5 phase images. In order to quantify this focusing effect on our cell volume measurement system, we have adapted two auto-focusing schemes based on the intensity images of a phase object to obtain the optimal focus position of the object sample in our proposed QPM system. The first auto-focusing scheme uses the squared gradient algorithm (SGA) and the second one uses the Laplacian filtering algorithm (LFA) [21]. These auto-focusing schemes utilize only the intensity images. When the reconstructed amplitude of an image measured by our QPM setup is f(x,y), two-dimensional (2D) spatial integral of the gradient square of the amplitude (|∇f(x, y)|2) within a given window of an image works as the focus measure of the SGA. The mathematical expression for the focus measure of SGA is [20]
In the LFA, the Laplacian of the amplitude (∇2 f(x, y)) is calculated for each data position within a window of interest and the 2D spatial integration of this is used as the focus measure. This second focus measure of LFA can be written as [20]
Fig. 2. (a) Measured interferogram images of a polystyrene bead for 5 different object positions from -2 μm to +2 μm with the step size of 1 μm. (b) The reconstructed amplitude images from the interferograms displayed in Fig. 2(a). (c) Calculated focus measures of SGA and LFA for the five reconstructed amplitude images obtained in Fig. 2(a). Each graph is normalized by its peak value.
Figure 2(b) shows the reconstructed intensity images of the polystyrene bead, which are calculated from the interferograms displayed in Fig. 2(a). The reconstructed intensity image is obtained by extracting absolute value from a complex two dimensional image measured by the QPM setup illustrated in Fig. 1. Figure 2(c) shows calculated focus measures of SGA and LFA for the five reconstructed intensity images of Fig. 2(b). Each graph is normalized by its peak value. Both SGA and LFA parameters show minimum value at the focus plane. Unlike the cases for the images of reflective or scattering objects it shows that these focus measures have minimum values at the focus position. It is due to the fact that a phase object shows the smoothest variation of its intensity at the focus position [21]. This shows that the optimal focal position can be experimentally obtained by adjusting the position of a sample with a translation stage while monitoring either SGA or LFA of an amplitude image.
3-3.Thickness and volume calculation
Fig. 3. (a) Calculate phase images of the polystyrene bead obtained from the interferograms displayed in Fig. 2(a). (b) The thickness line profiles of the sample indicated by each arrow in Fig. 3(a). (c) The thickness line profile of the sample at the focus position (z = 0) and the simulated thickness profile for a perfect sphere.
Figure 3(a) shows the calculated phase images of the polystyrene bead corresponding to the five different interferograms shown in Fig. 2(a) with five different object positions. Quantitative phase information can be easily converted into a 2D thickness profile by using
where Δn is the refractive index difference between a sample and its surrounding medium (glycerol in our case), ϕ(x,y) and t(x,y) are the phase and the physical thickness of the sample, respectively. Figure 3(b) shows the thickness profiles of the sample across the center of each bead indicated by each arrow in Fig. 3(a). The thickness profile of the sample at the focus position (z = 0) and the theoretically simulated thickness of the circular shape polystyrene bead sample are shown in Fig. 3(c). Simulated thickness line profile is calculated assuming a perfect sphere of 3 μm diameter without considering any diffraction effect. The measured thickness profile is in good agreement with the simulated result except near the edge of the profile. This is due to the finite spatial resolution of the imaging system used in our setup. The measured polystyrene bead volume at the focus position is 14.74 μm3, while the expected actual volume is 14.14 μm3. The experimental results of our proposed method agree well with the actual volume within 4 % error. The measured volume is slightly larger than the actual volume due to the diffraction effect of the objective lens and the different refractive index between the polystyrene bead and the glycerol. The experimental results are quantitatively measured in a focus position of the sample to remove the spurious phase distortions due to defocusing effects.
The volume changes of the polystyrene bead are calculated for the four different object positions and are plotted in Fig. 4. As is expected from the thickness plots shown in Fig. 3(b), the measured volume has almost linear relation with the object position. Error in the measured volume of a bead can be different as much as 30% even when the object is only 2 μm off from its optimum focus position. This indicates that there can be large errors in volume measurements utilizing a QPM setup unless phase images are measured at the focus position of an object with a very accurate autofocus scheme.
3-4.Accuracy in volume measurement
Figure 5 shows the measurement errors of optical path length in a series of phase images without a sample. Images are taken at the rate of 100 frames per second with the CCD in our setup. The optical path length fluctuations are averaged for 25 samples over an area of 5×5 pixels of the interferogram at each frame. Since the standard deviation of the measured optical path lengths for 25 CCD pixels is 0.61 nm, the standard deviation corresponding to a single CCD pixel becomes 3.05 nm (=√25 ×0.61 nm). As the pixel to pixel distance is 7.4 μm for the CCD we have used, and the optical magnification of our imaging setup is 110, the unit area corresponding to each pixel of our CCD becomes (7.4 μm/110)2 = 4.53 × 10-3 μm2. The volume of a cell is calculated by the sum of volume elements within the cell boundary, where each volume element corresponding to a single pixel in a CCD is the measured path length for that pixel times its unit area, 4.53 × 10-3 μm2. Therefore, accuracy in the optical path length measurement of each CCD pixel determines the accuracy of our volume measurement method. The standard deviation of a single volume element becomes the standard deviation of the corresponding optical path length times the unit area divided by Δn, which is (3.05 × 10-3 μm) × (4.53 × 10-3 μm2)/0.06 = 2.3 × 10-4 μm3. The volume of a cell is calculated by adding a certain number of volume elements within a cell boundary. When the cell volume is the sum of N volume elements, the standard deviation of the cell volume becomes √N time the standard deviation of a single volume element: √N × 2.3 × 10-4 μm3. This means that the accuracy of a measured cell volume becomes worse for a cell with a larger area.
4. The measurement of a live cell volume
4-1.RBC as a sample
In order to demonstrate the volume measurement of a biological sample, we have measured the volume of a single live RBC. Fresh blood was collected from a mouse. 10 μl of the blood was diluted by 50-fold in a low K+ solution (mM: KCl 2, NaCl 145, HEPE-Na 10, MgCl2 0.15 and 1 mg/ml BSA). After washed twice, it was adjusted to an appropriate concentration for a volume measurement. We have taken the quantitative phase image of a live RBC in a focus position by using our auto-focusing technique explained in the previous section. The quantitative phase image of a single RBC and its two dimensional thickness profile is shown in Fig. 6(a) and 6(b). Figure 6(c) and 6(d) show the thickness line profiles along the vertical and the horizontal directions indicated by two arrows in Fig. 6(a). Thickness is calculated with the refractive index of the RBC to be 1.399, where the error of measurement in refractive index was estimated to be ± 0.006 [22, 23], while the refractive index of the surrounding medium is 1.34. Unlike the smooth background line profiles of Fig. 3 for a polystyrene bead sample which is floating on glycerol, the background line profiles of the RBC in the surrounding area of the sample in Fig. 6 have small but fluctuating non-zero values. This is due to the spatial inhomogeneities in the surrounding medium. Since the volume of an RBC is calculated by integrating two dimensional thickness data within a cell boundary, these non-zero values in the background may cause errors in measuring RBC volumes. We have applied an edge detection technique to eliminate this background fluctuation effect on cell volume calculation.
Fig. 6. (a) and (b) Quantitative phase images of a single RBC. (c) and (d) Thickness line profiles along the vertical and the horizontal arrows indicated in Fig. 6(a).
4-2. Edge detection scheme for accurate cell volume measurement
RBC samples are floating on blood plasma. Because of the spatial inhomogeneities in the surrounding medium, the surrounding area around an RBC has non-zero contribution in volume measurement. In order to exclude this fluctuating background effect in the volume calculation of an RBC, we need to calculate the volume integral only within the cell boundary of an RBC. For this purpose, we have adapted a standard edge detection scheme developed by Canny [23]. First, the phase image ϕ(x,y) shown in Fig. 6(a) is convolved with a 5 × 5 Gaussian filter whose full width at half maximum is about 1.5. Small-amplitude high-frequency noises in an image can be effectively suppressed by this Gaussian filter. Then, the filtered phase image function ϕ1(x,y) is differentiated with respect to x and y axes to calculate the an edge-strength function g(x,y) defined as
By selecting a proper thresh value for this calculated function g(x,y), we can find the boundary of the RBC in its phase image. We have also adapted the non-maximum suppression process introduced by Canny [23].
Fig. 7. (a) Calculated boundary line of an RBC calculated from the phase image of Fig. 6(a). (b) The dilated binary boundary image modified from Fig. 7(a). (c) Center-filled binary mask modified from Fig. 7(b). (d) Background suppressed phase image by multiplying the binary mask of Fig. 7(c) to the original phase image in Fig. 6(a).
Figure 7(a) shows the calculated binary boundary image corresponding to the phase image of an RBC shown in Fig. 6(a). Red pixels in Fig. 7(a) represent the calculate boundary of the RBC. Since the phase image of an RBC is a slowly varying function, the phase values on these detected boundaries are not completely zero. Therefore, we have made the thin boundary lines in Fig. 7(a) to be thick by convolving the original binary boundary image of Fig. 7(a) with the same 5 × 5 Gaussian filter used above. Then, we have generated another binary boundary image by selecting a proper threshold value for this convoluted edge image. Figure 7(b) shows this newly generated boundary image whose shape is same as that of Fig. 7(a) except that its boundary line width is broadened roughly by the width of the Gaussian function used in this procedure. We have made this thick boundary line whose outmost line becomes zero. As the inner part of this donut-shape boundary is the area we need to use to calculate the volume of the RBC, we have assigned the inner area of this donut-shape boundary to be one to make a circular shape binary image whose value is one for the area of RBC and zero for the background area of blood plasma. Fib. 7(c) shows this new circular binary image which will work as a binary filter to eliminate the background effect in our RBC volume measurement. This clearly shows the area of the RBC sample and that of the surrounding blood plasma. The area of this circular mask involves with 7927 CCD pixels. Since the unit area corresponding to each CCD pixel is 4.53 × 10-3 μm2, the area of the binary mask shown in Fig. 7(c), which is an approximated area of the RBC sample, is calculated to be about (4.53 × 10-3 μm2) × 7927 = 35.90 μm2. Finally, we have multiplied this circular binary mask to the original phase image of an RBC shown in Fig. 6(a). Figure 7(d) illustrates this new background-free phase image of the RBC, where the fluctuating phase of surrounding background is forced to be zero. All of these numerical procedures were performed by a simple MatLab program.
The volume of a single RBC is measured to investigate the dynamic changes of a cell on a millisecond time scale. Figure 8(a) shows temporal variations in the measured volumes of a single RBC at a measurement rate of 100 images per second. It shows the difference in RBC volume measurement between cases with and without the background noise suppression scheme utilizing numerical image processing techniques based on the Canny’s boundary detection technique [24]. The upper and the lower graphs are without and with the background noise suppression technique, respectively. The upper graph shows that the calculated volume of the RBC shown in Fig. 6(a) was 57.54 μm3 without the background noise suppression scheme, while the standard deviation of it was 0.74 μm3. As shown in the lower graph, when the background fluctuation noises were excluded in volume calculation with our proposed technique, the calculated volume of the RBC became 54.02 μm3 with a standard deviation of 0.52 μm3. From the results of section 3-4, the expected standard deviation of a cell volume involved with N = 7927 CCD pixels becomes √7927 × (2.3 × 10-4 μm3) = 2.05 × 10-2 μm3. This is much smaller that the measured value of 0.52 μm3. This relatively large standard deviation in the measured RBC volume is originally caused by the membrane fluctuation of RBC [25]. With our background noise suppression technique, the measured volume of the RBC is corrected by 6.1 % from 57.54 μm3 to 54.02 μm3, and the standard deviation of repeated volume measurements is reduced by 30 % from 0.74 μm3 to 0.52 μm3. By using a conventional simple edge detection technique, we have demonstrated noticeable improvement in the volume of a live RBC. We have measured the average volume of 40 different RBCs with our proposed method. Their mean volume is 49.53 μm3. We have also measured the average volume of RBCs in a bold sample of 11×106 cells/μl concentration by using an automatic hematology analyzer (Celltac Alpha MEK-6318, Nihon Kohden, Tokyo, Japan). Its mean corpuscular volume is 47.6μm3. The experimental results of our proposed method agree well with the results obtained by the impedance volume analyzer within 4% error.
Fig. 8. Measured volumes of a single RBC with and without our background suppression method at a measurement speed of 100 frames per second.
3. Conclusion
We have demonstrated two major improvements in measuring the volume of a live cell utilizing QPM. The proposed ideas are the use of (1) the auto-focusing technique to obtain the optimal focusing position of a sample and (2) the numerical analysis method to detect the cell boundary in cell volume calculation. Details of this technique have been demonstrated by measuring the volume of a polystyrene bead and a live RBC. We have shown that the optimal focus position of the sample can be obtained easily at the minimum focus measure value calculated from the reconstructed amplitude image of a phase object. Possible errors in volume measurement caused by the defocusing of a sample are analyzed in detail. We have also demonstrated the differences in the measured volume of an RBC by comparing results obtained with and without our edge detection technique. Since our proposed volumetric analysis method can measure a cell volume with high sensitivity, we anticipate that the improvements of our proposed technique in volume measurement can make the cell measurement method with QPM be a standard tool for a live cell volume measurement especially for RBC samples.
Acknowledgment
This work was supported by Creative Research Initiatives (3D Nano Optical Imaging Systems Research Group) of MEST/KOSEF.
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