Letter pubs.acs.org/Langmuir
Statistical Analysis of Photonic Crystal Spectra for the Independent Determination of the Size and Refractive Index of Cells Justin William Stewart and Anna Pyayt* IBIS Lab, Department of Chemical and Biomedical Engineering, University of South Florida, 4202 E. Fowler Avenue, Tampa, Florida 33620, United States ABSTRACT: Photonic crystal flow cytometry is a very attractive platform due to its great sensitivity in combination with a very compact design. Previous studies have demonstrated the possibility to use spectral processing for the measurement of a wide range of parameters, from simple object counting to independent analysis of the buffer solution and immersed microscale objects. Here we propose to go to the next level and simultaneously determine the shape and the refractive index of the cells.
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INTRODUCTION Over the last 50 years, interest in new devices created with photonic crystals (PhCs) has grown greatly due in part to their capability of manipulating light on extremely small scales.1 Here, we conduct a theoretical study using a one-dimensional photonic crystal containing a series of periodic pores. Traditionally, PhCs have found many applications depending on the overall structure of the crystal. One-dimensional (1-D) PhCs are mostly being used in the field of thin film optics to create coatings for lenses and mirrors which are highly wavelength-selective, as well as for the production of colorshifting inks that appear as different hues depending on the angle at which they are viewed.2 Two-dimensional (2-D) PhCs are most commonly used for the manufacturing of highly lossless photonic crystal fibers (PCFs). These are capable of confining light within a core surrounded by periodic structures that exhibit a photonic band gap for the operating wavelength as well as control chromatic dispersion and filter light of undesired wavelengths.3 Meanwhile, 3-D PhCs are not as widely available, though they have already demonstrated potential in the field of optically integrated circuits.1 Additionally, PhCs are being applied to novel devices for refractive index-based sensing4 as an alternative to other refractive indexbased sensing methods.5−7 Yet, PhCs remain an area of great focus with a lot of additional research as they allow direct manipulation and control of light on the micrometer scale and can be easily integrated into lab-on-a-chip devices, which makes the miniaturization of many apparatuses and sensing equipment possible. One potential application of PhCs has been for the development of miniaturized flow cytometry systems for the high-throughput analysis of cells.8 In photonic crystal-based microflow cytometry, individual cells, which have been spatially focused using a sheath fluid, flow over the surface of a 1-D PhC. © 2015 American Chemical Society
As cells pass over the surface, interactions between light resonating within the PhC and the cell near the PhC surface result in changes in the transmission spectrum. By studying the transmission signals, simulations have demonstrated that information such as the number of cells, the shape of cells, and the cell size can be extracted for cells with a known refractive index.8 However, in practice the refractive index of each cell is typically unknown and likely to vary from cell to cell. Taking this into consideration, further statistical analysis of transmission signals, using the theorem of central moments, must be performed to determine information such as the size and refractive index of the cell. The theorem of central moments is widely used in analyzing probability distribution functions as well as for characterizing flow patterns in tracer studies.9 However, the same equations show promise for resolving cell characteristics in PhC microflow cytometry. Here, we further build upon the model of a 1-D PhC microflow cytometer. First we analyze the wavelength sensitivity by probing the surface of the PhC with a cantilever-like probe. This allows for the selection of an optimal wavelength to perform microflow cytometry on blood cells of varying refractive index and diameter. After this, transmission signals recorded for each blood cell are analyzed using the theorem of central moments to determine the cell diameter. Upon obtaining the cell size through these methods it is then possible to determine the refractive index of each cell. Received: April 1, 2015 Revised: May 28, 2015 Published: June 14, 2015 7173
DOI: 10.1021/acs.langmuir.5b00960 Langmuir 2015, 31, 7173−7177
Letter
Langmuir
Figure 1. Three-dimensional model of the photonic crystal microflow cytometer. (a) Isometric view with blood cells. (b) Top-down projection of etched channels. (c) Isometric detailed view of PhC pores. (d) Cross-sectional view of PhC pores with blood cells at the surface.
Figure 2. (a) Heat map showing regions where the PhC is sensitive to light scattering over the spectrum range of 515−532 nm. (b) Change in transmission due to a blood cell (diameter, 7 μm; refractive index, 1.387) shown for three wavelengths exhibiting notable sensitive regions on the PhC surface (521, 524, and 528.5 nm).
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SIMULATION SETUP AND MODEL PARAMETERS The photonic crystal microflow cytometer was simulated using OptiFDTD. Changes in the PhCs spectral characteristics were monitored while blood cells passed over the PhC surface. Figure 1 shows a 3-D model of the proposed device with a stream of red blood cells as they flow through the microfluidic channels of the microscale flow cytometer. The PhC is composed of a 0.6-μm-thick, 1 -μm-wide strip of silicon nitride (refractive index 2.03) deposited on top of thermally grown silicon dioxide (refractive index 1.54), which acts as a lower cladding.10,11 The nitride structure includes eight periodic slits 0.5 μm wide, 1.5 μm from center to center and etched completely through the nitride strip into the silicon oxide layer, for a total depth of approximately 2.7 μm. Cells are guided toward the photonic crystal through microfluidic channels shown in Figure 1 as a 5-μm-thick layer of SU-8 (refractive index 1.70) on both the left and right sides above the silicon nitride structure.12 In FDTD simulations the device was completely submerged and filled with plasma (refractive index 1.3515). Transmission spectra were calculated over the range of 515 to 532 nm and normalized with respect to the input light. Initially the sensitivity of the PhC in different spectral regions was tested to determine the wavelengths of light which would be most appropriate for the optimum detection of cells. To
study the PhC sensitivity, a silicon nitride cantilever tip was simulated to probe different regions of the PhC while transmission spectra were calculated. The probe tip was permitted to touch the photonic crystal surface while the resulting change in the transmission spectrum was recorded for the current tip position and repeated over the entire length of the PhC. Once the most sensitive wavelength for scanning was determined, the flow of blood cells was simulated and the change in transmission was monitored at this wavelength. Blood cells varied in size from 6 to 8 μm in diameter.13 The refractive index of cells was also varied from 1.387 to 1.401 according to the range from the literature.14 The flow of cells was modeled on the basis of the incremental displacement (δy) of the cells through an observation plane. The shapes of the blood cells were approximated as biconcave discs and were assumed to be perfectly symmetric around their center. By this assumption, the simulation of cell displacement was required only from the center (δy = 0 μm) to the edge of the cell (δy = R, μm), and the results were reflected across the origin to obtain the full transmission signal for complete displacement through the observation plane. In practice, if the cells are not symmetric because of, for example, sickle cell disease, then this 7174
DOI: 10.1021/acs.langmuir.5b00960 Langmuir 2015, 31, 7173−7177
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Figure 3. Characteristic signals of red blood cells (diameter 7 μm) with varying refractive indices. (a) Change in transmission at a specific wavelenghth from PhC spectra shown for the three cells. (b) Transmission signal after transformation into transmission density functions. λ = 528.5 nm.
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ANALYSIS OF TRANSMISSION SIGNALS USING CENTRAL MOMENTS As previously stated, research on PhC transmission signals have demonstrated a capability to determine the characteristics of cells as they flow past the surface of the crystal.8 However, these findings were performed for cells of a fixed single value of the refractive index. However, as with most biological substances, the refractive index can vary over a certain range and is not consistent from object to object. For blood cells, this range is approximately 1.387 to 1.401.14 Despite the added variability in refractive index, characterizing cells with photonic crystals can still be performed using additional analysis and statistical methods. To observe the influence of the cell refractive index on the resulting transmission signals (T), blood cells with average refractive indices of 1.387, 1.394, and 1.401 were modeled for the range of diameters of cells (6−8 μm). The change in transmission (T) of the PhC for 7 μm blood cells over the range of possible refractive indices is shown in Figure 3a. It may be noted that for variations in refractive index the only significant difference between each transmission signal is with the amplitude of the signal. As the cell refractive index increases, more light is scattered from the PhC, yet the overall shape of the signal appears to be completely unaffected. Each of the recorded transmission signals (T) was then transformed into transmission density functions (E) by using the following expression9
asymmetry will be detected in the measurements of transmission signals.
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PHOTONIC CRYSTAL WAVELENGTH SENSITIVITY
Results of the PhC sensitivity study are shown in Figure 2a. The heat map displays the amplitude of the scattering losses for a range of wavelengths (515−532 nm) as well as the corresponding location on the PhC surface. Here, a sensitivity value of 1 corresponds to the wavelength with the maximum light scattered from the crystal, and 0 indicates that transmission was unaffected. From Figure 2a it may be clearly noticed that distinct regions with higher sensitivity to the presence of an object form along on the photonic crystal surface for specific wavelengths of light. There are three wavelengths corresponding to the highsensitivity patterns: 520, 524, and 528.5 nm. For light at 520 nm, the PhC is in its least-sensitive regime, which is clearly reflected in low changes in transmission while detecting a red blood cell (diameter, 7 μm; refractive index, 1.387) presented in green in Figure 2b. Such wavelengths with low sensitivity might not be appropriate for analyzing cell properties; however, they might still be useful for performing fluorescence studies with dyes that emit light near those wavelengths. However, at a wavelength of 524 nm, the photonic crystal becomes moderately sensitive to objects near the surface, yet these sensitive regions are located along the left and right sides of the PhC and not in the center where flowing cells would be spatially focused by the microfluidic system. As a result, cells are being scanned not in the center but along the lobes and edges and thus shape is not reproduced (blue curve in Figure 2b). The most sensitive wavelength of light, in this study, was found to be approximately 528.5 nm. In Figure 2a at this wavelength the region of the PhC most susceptible to influence by objects at the surface is roughly located around the center of the crystal. As a result, the signal corresponding to the flow of the red blood cell has the highest amplitude in Figure 2b and also spatial information is preserved and the concave disc shape can be observed. As this wavelength provided the most well-defined and strongest signal for the cell detection, 528.5 nm was used for the remainder of this study.
E (y ) =
T (y ) −∞
∫∞ T(y) dy
(1)
where E and T are both functions of cell displacement in the microfluidic channel (y). Results of this transformation have been plotted for the same 7-μm-diameter blood cells, presented in Figure 3b. It can be clearly seen that upon converting the data into transmission density functions the effects of the blood cell refractive index appear to vanish, as each density function (E) displays approximately the same distribution with only minor differences. Because of this, statistical analysis of transmission density functions using the theorem of central moments could be applied, providing a reliable technique for measuring cell characteristics when variations in refractive index arise in photonic crystal microflow cytometry. 7175
DOI: 10.1021/acs.langmuir.5b00960 Langmuir 2015, 31, 7173−7177
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Figure 4. Method for determing blood cell diameter. (a) Relationships between cell diameter and area under transmission curves are no longer effective once the refractive index can change. (b) By applying the theorem of central moments, a strong correlation exists between blood cell diameter and the resulting standard deviation of their transmission signals (μm). λ = 528.5 nm.
due to only the nature of the simulations in this study and would not be 0 μm in practice. In application, multiple transmission signals of several cells would be continuously collected and recorded with respect to time. Time can then be converted into displacement values through multiplication by the velocity of cells passing across the PhC surface. Each individual signal would then be numerically processed by computer software using these equations. For experimental data the average displacement must be calculated for each signal studied. Once the average displacement is known, the second central moment σ2, known simply as the variance, could be calculated using eq 3. In Figure 4b, the standard deviation (square root of variance) of the transmission signals is plotted with their corresponding blood cell diameters as well as their refractive index. It can be noticed that a significantly stronger correlation (R2 = 0.997) between the cell diameter and signal standard deviation appears when compared to the previous results in Figure 4a, and this linear relationship agrees for the range of blood cell refractive indices studied. Using this relationship as well as the previously mentioned equations, transmission signals collected from a PhC device can be used to determine the diameter of a blood cell, regardless of the cell’s refractive index. After the size of the cell has been determined with the abovementioned techniques, it is then possible to calculate the refractive index of the cell by analyzing the original transmission signal (T), such as those shown in Figure 3a. In Figure 5, the blood cell refractive index is shown with respect to resulting areas under transmission curves for all cell diameters studied. By using this plot the refractive index can be obtained by integrating under transmission curves, and because the diameter is already known the line that must be followed can also be determined.
Prior studies of some cell parameters with photonic crystal flow cytometry showed that there is a strong correlation between the areas under transmission signals and the diameters of the objects passing over the PhC surface.8 Using this information, it is possible to obtain the size of objects by integrating their transmission signals. Nonetheless, for blood cells and other biological entities, where the refractive index is unknown and not consistent for each cell being studied, integration of the transmission signals is no longer sufficient for determining the size of the cells. In Figure 4a, the area under the transmission curves for blood cells of various refractive indices has been plotted against their corresponding blood cell diameters. Data shown in blue correspond to blood cells with a refractive index of 1.387, points in green are cells with a refractive index of 1.394, and points in red are cells with a refractive index of 1.401. Relationships between the transmission signal areas and blood cell diameters do exist, as indicated by the dashed lines; however, these occur only among cells with the same refractive index. In addition, these individual relationships for cells with a constant refractive index exhibit only moderately strong linear correlations, as the data appears to oscillate above and below the lines of best fit. To resolve the issues that arise when the cell refractive index can vary, we will apply the equations for the first and second central moments, given in eqs 2 and 3, on the previously described transmission density functions (E).9 ∞
y̅ =
∫−∞ yT(y) dy ∞
∫−∞ T(y) dy
∞
=
∫−∞ yE(y) dy
(2)
−∞
2
σ =
∫∞ (y − y ̅ )2 T(y) dy −∞
∫∞ T(y) dy
∞
=
∫−∞ (y − y ̅ )2 E(y) dy
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CONCLUSIONS By using the theorem of central moments to analyze transmission signals from photonic crystals, we have been able to independently determine properties of blood cells, such as diameter and their refractive index. The setup was modeled using finite difference time domain (FDTD) simulations for a 1-D photonic crystal, where changes in transmission were recorded as cells flowed past the surface of the crystal. Previous
(3)
Here, y ̅ is the first central moment, otherwise known as the average displacement (μm). For symmetric distributions such those found in our blood cell model, the average displacement simply corresponds to the displacement in the center of the signal. The value of y ̅ holds no major significance at this point but must be determined for later calculations. In Figure 3b, it may be clearly seen that the value of y ̅ is 0 μm; however, this is 7176
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(11) Gosh, G. Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals. Opt. Commun. 1999, 163, 95−102. (12) Parida, O. P.; Bhat, N. Characterization of Optical Properties of SU-8 and Fabrication of Optical Components; ICOP; CSIO: Chandigarh, India, 2009. (13) Gregory, T. The Bigger the C-Value, the Larger the Cell: Genome Size and Red Blood Cell Size in Vertebrates. Blood Cells, Mol., Dis. 2001, 27, 830−843. (14) Rappaz, B.; Barbul, A.; Charriere, F.; Kuhn, J.; Marquet, P.; Korenstein, R.; Depeursinge, C.; Magistretti, P. Erythrocytes volume and refractive index measurement with a digital holographic microscope. Proc. SPIE 2007, 6445, 644509. (15) Cheemalapati, S.; Ladanov, M.; Winskas, J.; Pyayt, A. Optimization of dry etching parameters for fabrication of polysilicon waveguides with smooth sidewall using capacitively coupled plasma reactor. Appl. Opt. 2014, 53, 5745−5749. Figure 5. Blood cell refractive index as a function of the area under transmission change curves for known cell diameters. λ = 528.5 nm.
PhC techniques for studying the characteristics of objects are valid whereas the objects of interest have a consistent refractive index. Yet for cells, the previously mentioned methods are not entirely sufficient unless the refractive index is known for each cell being studied. By analyzing transmission density functions from a statistical approach, it should be very simple to program such equations into computer software, allowing for more sophisticated and precise analysis of cells, biological objects, and many other complex microparticles whereas wellestablished fabrication methods can be used for future device fabrication.15
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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