2.1.

Analysis of Light Scattering with the Mueller Matrix

For scatterers of size parameter $\alpha$ ranging from about 1 to 100, rigorous treatment of light as electromagnetic fields is necessary for accurate analysis of light scattering, where $\alpha =2\pi a∕\lambda$ , $2a$ is the characteristic size of the scatterer, and $\lambda$ is the light wavelength in the host medium. The size parameter of human blood cells ranges often between 10 and 100 in the visible and near-IR regions of the optical spectrum. To fully describe the scattering of polarized light, we adopt a widely used formalism with the Stokes vectors for the incident and scattered light fields and Mueller matrix for the scatterer. Numerous papers and books have been published on this topic, ^{23, 24, 25, 26, 27, 28} and we limit the description here only to those relevant to the presentation and discussion of our results.

The linearity of the Maxwell equations enables the use of the real valued Stokes vectors and Mueller matrix for relating the intensity signals as

Fig. 1

Schematic of the experimental setup: P; polarizer; PE; PE modulator; B, beam expander; L, focusing lens; SH, sample holder; IT, immersion tank; A, analyzer; T, apertured tube; PM, photomultiplier; A/D; analog-to-digital converter. Two lock-in amplifiers were used to measure the $f_0$ and $2f_0$ signals simultaneously. Inset: Schematic of the sample holder made of black plastic body with a flat sapphire window S. A reflecting insert I is used to direct the transmitted beam to the side channels inside the holder body, marked by the dashed line, and can be removed for aligning the incident laser beam through the holder.

For scatterers of complex morphology, calculations of the Mueller matrix elements must be pursued numerically by solving the Maxwell equations or the resultant wave equations. Due to the computational complexity of numerical methods, these calculations are mostly practical for scatterers with $\alpha$ less than 100. In contrast, many experimental determinations of the Mueller matrix elements for biological cells have been conducted with hydrosol samples in which multiple cells are illuminated and contribute to the detected signals. The use of multiple cells as the scatterer is necessary because of the difficulty in angle-resolved measurement of the weak signals from a single cell in a large angular range and holding a single cell over a long time ( $100\mathrm{s}$ or longer). Therefore, the effect of multiple cells at random orientations must be considered when the measured and calculated Mueller matrix elements are compared.

For a single scatterer of rigid body, there are at most seven independent elements in the Mueller matrix, which can be derived from a $2\times 2$ amplitude matrix of four complex elements relating the electric fields of the scattered light to the incident.²⁵ In the case of a scatterer with spherical symmetry, the Mueller matrix contains only three independent elements among the eight non-zero elements,²⁵ i.e., $S_{11}=S_{22}$ , $S_{12}=S_{21}$ , $S_{33}=S_{44}$ , $S_{34}=S_{43}$ , and $S_{11}^2=S_{12}^2+S_{33}^2+S_{34}^2$ . Under the conditions of single scattering and random motion, the Mueller matrix of multiple particles in a hydrosol sample is equal to the addition of the matrices of individual particles due to incoherent superposition of the scattered fields.^{24, 28} For particles of spherical symmetry and identical morphology, the resultant Mueller matrix has the identical form with that of an individual particle with only three independent elements. For biological cells, however, the Mueller matrix elements of multiple cells are expected to be independent of each other in general because of varied morphology within the same type or cell line. In published studies, the elements $S_{12}$ and $S_{34}$ attracted active interests for their possible use in distinguishing cells of different structures.^{6, 7, 21} We chose to measure and calculate $S_{11}$ , $S_{12}$ , and $S_{34}$ for our first results reported here.

2.2.

Experimental Method

We constructed a goniometer system with a photoelastic (PE) modulation scheme for determination of the Mueller matrix elements that has been developed and reported by other researchers.^{21, 29} The experimental setup is shown schematically in Fig. 1. A PE modulator (PEM90, Hinds Instrument) with birefringence modulated²⁹ at a frequency of $f_0=50\mathrm{kHz}$ was used to obtain the sinusoidal time dependence of the phase difference $\varphi$ between the $e$ and $o$ components of a linearly polarized laser beam incident on the sample after the focusing lens. The light scattered by the sample along the polar angle ${\theta }_s$ in the horizontal plane $({\varphi }_s\sim 0\mathrm{deg})$ , relative to the incident beam, is passed through an optional analyzer and collected by the photomultiplier. The polarizer $P$ , PE modulator, and analyzer (A) can be represented with their respective Mueller matrices for axial transmission with the Stokes vectors of the detected light given by

By choosing appropriate orientations, the Mueller matrices of ${{M}}_A$ and ${{M}}_{\mathrm{PEM}}$ can be made in such forms that the optical power signal $I_d({\theta }_s,t)$ arriving at the photomultiplier can be expressed³⁰ as different combinations of the elements of ${S}\left({\theta }_s\right)$ . Further, the modulation of the birefringence in the PE modulator leads to the elements of ${{M}}_{\mathrm{PEM}}\left(t\right)$ as the sinusoidal function of time or $f_{0}t$ , allowing a Fourier expansion of $I_d({\theta }_s,t)$ . With this scheme, the elements of ${S}\left({\theta }_s\right)$ can be extracted from $I_d({\theta }_s,t)$ from the dc, fundamental $\left(f_0\right)$ , and second harmonic $\left(2f_0\right)$ components. For our measurements, we oriented the optical axis of the PE modulator at $45\mathrm{deg}$ to the horizontal axis with three different analyzer configurations described below for determination of the elements $S_{11}$ , $S_{12}$ , and $S_{34}$ .

In the first configuration, the analyzer $A$ is taken out and the scattered optical power signal is given by

Three different continuous-wave lasers were used separately to obtain an incident beam at one of three wavelengths of 442, 633, and $850\mathrm{nm}$ with a power between 1.5 and $3.0\mathrm{mW}$ . The incident beam from a selected laser was expanded to a diameter of about $10\mathrm{mm}$ and then focused by a lens of $700\mathrm{mm}$ in focal length to create a long beam waist at the center of the sample holder with a diameter of about $0.2\mathrm{mm}$ . A sample holder of black Plexiglas was designed to reduce stray light background and is illustrated in the inset of Fig. 1. A flat sapphire window of $12\mathrm{mm}$ diameter and a semicircle glass window of $25.0\mathrm{mm}$ inside diameter were glued to the holder at the entrance and exit, respectively, for reducing distortion of the incident and scattered light beam profiles. To further decrease the distortion of the scattered light beam profile at the glass cylinder, the sample holder was immersed in a larger water tank of $243\mathrm{mm}$ diameter with a flat entrance window. The incident beam profile inside the holder was measured and found to be close to Gaussian using a knife-edge method to measure the beam diameter variation along the beam axis. The blackbody of the holder and an apertured tube of $200\mathrm{mm}$ length placed in front of the photomultiplier proved very effective in minimizing the stray light background in the detected signals.

A photomultiplier (R6356, Hamamatsu) was rotated on a circular track of $475\mathrm{mm}$ radius in the horizontal plane around the central axis of the sample holder to measure the modulated light signal $I_d$ at scattering angle ${\theta }_s$ between 11 and $160\mathrm{deg}$ . The electronic system consists of an analog-to-digital (A/D) converter (KPCI-3103, Keithley) for measurement of the dc components and two dual-phase lock-in amplifiers (SR830, Stanford Research) for the first $\left(f_0\right)$ and second-harmonic $\left(2f_0\right)$ components of the signal from the photomultiplier. The electronic signals from the lock-in amplifier have varying phase angles between the signals from the photomultiplier and the reference from the PE modulator controller. These angles change by $180\mathrm{deg}$ during a goniometer scan and correspond to the sign change of the Mueller matrix elements represented by the amplitude of harmonic terms in Eqs. 4 to 5. The relation between the lock-in phase angle of the electronic signals and the sign of the Mueller matrix elements of the sample was determined from the calibration measurements with the polystyrene microspheres. A computer was used to control a stepping motor for rotating the analyzer-detector assembly at $1\mathrm{deg}$ per step and to acquire signals from the A/D converter and lock-in amplifiers.

2.3.

Sample Preparation

Two types of hydrosol samples were used in our experiment: polystyrene microspheres suspended in deionized water for system validation and B cells suspended in culture medium. The hydrosol sample was placed into the holder with a pipette and all measurements were carried out at a room temperature of about $22°\mathrm{C}$ . The sphere suspensions were prepared by diluting from the solution of 10% concentration in weight (5100B, Duke Scientific) which has a mean diameter of $1\mathrm{\mu }\mathrm{m}$ and a coefficient of variance of 3%. Small samples of the suspensions were drawn to be examined for sphere aggregation under microscope before system calibration. If aggregations were observed, a small drop of dispersing agent (TWEEN 20, USB) was added to the suspension to eliminate sphere aggregation.

Cell suspensions were prepared for goniometer measurements from cultured NALM-6 cells. The NALM-6 cell line was derived from human B cells extracted from the peripheral blood of a patient with acute lymphoblastic leukemia.³¹ These cells were cultured in a standard media containing the following: $75\mathrm{ml}$ of AmnioMax basal (Invitrogen Corp., Carlsbad, California), $12.5\mathrm{ml}$ of AmnioMax supplement (Invitrogen Corp., Carlsbad, California), $87.5\mathrm{ml}$ of minimal essential media (Invitrogen Corp., Carlsbad, California), $20\mathrm{ml}$ of fetal bovine serum (Invitrogen Corp., Carlsbad, California), $2\mathrm{ml}$ of $L$ -glutamine (Invitrogen Corp., Carlsbad, California), and $2\mathrm{ml}$ of penicillin/streptomycin (Invitrogen Corp., Carlsbad, California). Fresh culture media was added to the NALM-6 cells 10 days before they were used in each goniometer measurement to ensure that the majority of the cells were in resting $\left(G_0\right)$ phase due to lack of nutrients, as opposed to the addition of fresh media every 3 to 4 days when the NALM-6 cells were maintained in log phase and not prepared for scattering measurements.

2.4.

Confocal Imaging and FDTD Calculations

To account for the complex morphology of biological cells in the modeling, we developed a method to reconstruct the 3-D structure of cultured B cells from their confocal images.²⁰ Cultured B cells were stained with a fluorescence red dye of SYTO 61 for differential staining of the nucleus and cytoplasm followed by confocal imaging to extract the shapes of the nuclear boundary and cytoplasm membrane. The stacks of the confocal images were processed by an in-house developed software to obtain the 3-D structure of the B cells with two elements: the nucleus and the cytoplasm. Different values of real refractive indices for the nucleus $\left(n_n\right)$ , cytoplasm $\left(n_c\right)$ , and host medium $\left(n_h\right)$ outside the cytoplasm membrane were assigned in the FDTD code for calculations of the angle-resolved Mueller matrix elements of a single B cell.

A parallel FDTD code has been built by our group.¹⁸ Briefly, the FDTD method discretizes the two curl Maxwell equations with a Yee cell scheme in the near-field region around the scatterer where the optical heterogeneity is described in terms of spatial distribution of refractive index within the cell.^{14, 32} Once the electric and magnetic fields are solved in the near-field region in the time domain, the far-field scattered light fields in the frequency space are obtained through the Green formalism. The Mueller matrix elements can then be determined from the relation between the incident and scattered light fields as a function of the scattering angles of ${\theta }_s$ and ${\varphi }_s$ when the cell is oriented along a direction of angle ${\theta }_0$ and ${\varphi }_0$ relative to the incident beam.¹⁷ In our FDTD calculations, the size of the discretizing grid cell was set at $\lambda ∕20$ . For each B cell, the matrix elements were obtained as a function of ${\theta }_s$ by averaging over ${\varphi }_s$ at each of 12 orientations of the cell uniformly distributed in the $4\pi$ range of $({\theta }_0,{\varphi }_0)$ . A second averaging was performed over the 12 orientations for each calculated matrix element to take into account of the random orientations of a suspended cell in a hydrosol sample. We also performed cell-averaging with two selected B cells to examine the effect of cell morphology on the FDTD results. The calculated element of $S_{11}$ presented in the following figures were all normalized to $4\pi$ over the solid angle range of $4\pi$ .

3.1.

System Calibration

The goniometer system was first calibrated with microsphere suspensions. To ensure a single-scattering approximation, the sphere concentration was reduced by a factor of about 50,000 from the purchased aqueous solution of 10% by weight to $4.8\times {10}^2\mathrm{spheres}∕\mathrm{\mu }\mathrm{L}$ . At this concentration level, the $S_{11}$ signal was found to change linearly with the concentration and the dc, fundamental, and second-harmonic components were acquired to determine $S_{11}$ , $S_{12}$ , and $S_{34}$ . Data processing consists of four steps: (1) subtraction of background signals measured with deionized water in the sample holder; (2) removal of the effect of different sample volume “seen” by the detector at different angular positions with a volume scaling factor; (3) determination of the element $S_{11}$ within an adjustable constant from the dc signal and other Mueller matrix elements by combining the measured amplitude and phase angle of the lock-in signals; and (4) normalization of the processed signals by dividing different harmonic signals by the dc signal to determine the matrix elements normalized by $S_{11}$ . For the second step, a volume scaling factor was numerically calculated from the system geometry and the angular range of the apertured detector tube, $5.42\times {10}^{-3}\mathrm{rad}$ or about $0.31\mathrm{deg}$ , as a function of ${\theta }_s$ and is presented in Fig. 2 . The validation measurements were carried out at each of the three wavelengths 442, 633, and $850\mathrm{nm}$ and repeated before each measurement of cell suspension as a part of system alignment and calibration. One set of typical results of Mueller matrix elements $S_{11}$ , $s_{12}$ , and $s_{34}$ are shown in Fig. 3 . The measured element $S_{11}$ is plotted with a scaling parameter, corresponding to the constant $C$ in Eq. 4a, by fitting to the calculated curve based on the Mie theory using refractive indices of water as $n_h$ and polystyrene sphere as $n_{\mathrm{sp}}$ (Ref. 33). The angular distributions of other normalized matrix elements $s_{ij}$ were obtained by the ratios of measured elements to the measured $S_{11}$ without any adjustable parameter. One can see from these results that $S_{11}$ agrees well to the calculated values while other elements agree to lesser degrees. Specifically, the characteristic oscillations in the angle-resolved elements match consistently with the predicted values from a single sphere but the oscillation amplitudes are reduced. We attribute the difference between the amplitudes of oscillation to the angle-averaging effect by the multiple spheres contributing to the detected signals, fluctuation in sphere diameter and possible residue sphere aggregation in the suspension causing breakdown in single-scattering approximation. At large scattering angles the element $S_{11}$ becomes very weak and SNRs are consequently low. This causes large fluctuations in $S_{11}$ at these angles and in normalized values of other elements, as manifested by the measured values of $s_{12}$ and $s_{34}$ exceeding 1 near $140\mathrm{deg}$ .

Fig. 2

Calculated sample volume “seen” by the detector versus the scattering angle ${\theta }_s$ .

Fig. 3

Measured and calculated Mueller matrix elements $S_{11}$ , $s_{12}=S_{12}∕S_{11}$ , and $s_{34}=S_{34}∕S_{11}$ versus the scattering angle ${\theta }_s$ for water suspension of polystyrene microsphere with diameter of $1.00\mathrm{\mu }\mathrm{m}$ at $633\mathrm{nm}$ . The solid lines were obtained from the Mie theory with $n_h=1.332$ and $n_{\mathrm{sp}}=1.582$ for a single sphere and the measured $S_{11}$ was fitted to the calculated results with one adjustable constant. The normalized elements are plotted with no fitting.

3.2.

Experimentally Determined Matrix Elements of B-Cell Suspensions

B-cell hydrosol samples were prepared from the NALM-6 cell culture by a 9:1 dilution with 0.9% saline solution to a concentration of about $2.34\times {10}^2\mathrm{cells}∕\mathrm{\mu }\mathrm{L}$ , which was set by a similar signal linearity check to ensure the single-scattering approximation. With the signal acquisition procedures identical to those described in the previous section, the Mueller matrix elements $S_{11}$ , normalized elements $s_{12}$ and $s_{34}$ were determined as a function of scattering angle ${\theta }_s$ at each of the three wavelengths 442, 633, and $850\mathrm{nm}$ . Determination of these elements for each hydrosol sample required three goniometer scans with a different analyzer configuration and took about $60\min$ to complete. Each sample was replaced after three scans with fresh cells, which were kept in an incubator at $37°\mathrm{C}$ during the day of measurements. All measurements were performed at a room temperature of about $22°\mathrm{C}$ . At each wavelength, the measurements were repeated on three different cell samples to obtain three sets of angle-resolved Mueller matrix elements. The measured results are plotted in Fig. 4 with symbols and error bars representing the mean and standard deviation of the three data sets, respectively. The elements $S_{11}$ contains an adjustable parameter that was determined from the microsphere calibration results performed at each wavelength before the cell measurements.

Fig. 4

Angle-resolved Mueller matrix elements of B-cell suspension samples at wavelengths of $\lambda =442$ , 633, and $850\mathrm{nm}$ . The data points and selected error bars represent the mean and standard deviation of three measurements with different cell samples.

3.3.

Confocal Imaging and FDTD Calculations

To justify the comparison between the experimentally determined Mueller matrix elements and calculated elements of a single cell, we analyzed the variation of the morphology of the B cells in culture using a confocal microscope (LSM 510, Zeiss). The confocal images were acquired of 79 B cells randomly selected in the culture. Analysis of these images showed that the majority of cells (87%) contains single nuclei with shapes closely resembling spheres or ellipsoids while the rest (13%) is in the mitosis phase with two nuclei in each cell. Based on these observations, we selected two typical B cells to investigate the effect of cell morphology with cell 8 as an example of single-nucleus cells and cell 10 of two-nucleus cells. Using previously developed software,²⁰ we reconstructed the 3-D structures and imported them into our parallel FDTD code to calculate the Mueller matrix elements for comparison. The calculated elements for each cell were averaged over the scattering angle ${\varphi }_s$ and 12 orientations, as described before, to simulate the effect of random orientations of suspended cells in the hydrosol sample. The FDTD calculations were executed on parallel computing clusters of either 96 or 32 processing elements with Intel $3.06\mathrm{GHz}$ Xeon CPU. The FDTD calculations of matrix elements took about $100\min$ for each orientation and $20\mathrm{h}$ for the orientation-averaged elements for each of cells 8 or 10 at $\lambda =1\mathrm{\mu }\mathrm{m}$ on the 96-processing-element cluster. For shorter wavelengths of 633 and $850\mathrm{nm}$ , the simulation time becomes significantly larger because it scales as ${\lambda }^{-n}$ with $3<n<4$ for a fixed grid cell size set at $\lambda ∕20$ .

To reduce the simulation time, we initially chose the wavelength of $1\mathrm{\mu }\mathrm{m}$ to investigate the effect of cell morphology on the averaged angular distributions of the Mueller matrix elements of $S_{11}$ , $s_{12}$ , and $s_{34}$ . Two cells, 8 and 10, were selected from the confocal image stacks as the representative cases of B cells with one or two nuclei, respectively, using the same set of refractive indices of $n_h=1.350$ , $n_c=1.368$ , and $n_n=1.400$ . The three matrix elements of 12 different orientations and the averaged ones are presented side by side in Fig. 5 , which show similar angular dependence. Based on this conclusion, we limit our FDTD calculations to obtain the orientation-averaged elements for cells 8 and 10 with different values of refractive indices at $\lambda =633$ and $850\mathrm{nm}$ for comparison with the experimentally determined elements. These results are plotted in Fig. 6 to compare the effects of wavelength and the cell averaging. The cell-averaged elements are defined as the weighed sum of elements of cell 8 (87%) and cell 10 (13%), which demonstrated nearly identical angular dependence with those of cell 8 alone, indicating the weak influence of the nuclear shape at the adopted small index contrast on the considered Mueller matrix elements. Based on this conclusion, we carried out FDTD simulations with only cell 8 with three different values of nuclear index for comparison with the measured elements. The element comparison is limited at the wavelength of $850\mathrm{nm}$ to make the simulation time acceptable and results are presented in Fig. 7 .

Fig. 5

Angle-resolved Mueller matrix elements of normalized $S_{11}$ , $-s_{12}=-S_{12}∕S_{11}$ , and $s_{34}=S_{34}∕S_{11}$ calculated by the FDTD method for two B cells of 3-D structures reconstructed from their confocal images with refractive indices set at $n_h=1.330$ for the host medium, $n_c=1.368$ for the cytoplasm, and $n_n=1.4000$ for the nucleus. The thin colored lines are obtained at each of 12 different orientations relative to the incident light beam of wavelength $\lambda =1.00\mathrm{\mu }\mathrm{m}$ and the thick black lines are the averaged elements over the 12 orientations. The insets are the transparent views of B cell 8 (volume= $566\mathrm{\mu }{\mathrm{m}}^3$ ) and cell 10 $\left(628\mathrm{\mu }{\mathrm{m}}^3\right)$ . (Color online only.)

Fig. 6

Angle-resolved Mueller matrix elements of normalized $S_{11}$ , $-s_{12}$ , and $-s_{34}$ calculated by the FDTD method for B cells 8 and 10 of 3-D structures reconstructed from their confocal images with refractive indices set at $n_h=1.336$ for the host medium; $n_c=1.380$ for the cytoplasm, and $n_n=1.430$ for the nucleus at $\lambda =633\mathrm{nm}$ and $n_h=1.330$ , $n_c=1.368$ , and $n_n=1.400$ , at $\lambda =850\mathrm{nm}$ . All calculated elements are the averaged values over 12 orientations for each cell and the figures in the left panel are the averaged elements of cell 8 (87% weight)and cell 10 (13% weight).

Fig. 7

Angle-resolved Mueller matrix elements of $S_{11}$ , $-s_{12}$ , and $-s_{34}$ at $\lambda =850\mathrm{nm}$ . The mean values of the measured results are represented by solid lines and FDTD simulated results by dashed or dash-dotted lines with the reconstructed B cell 8 structure of $n_h=1.330$ , $n_c=1.368$ , and different nuclear indices $n_n$ .

The FDTD calculations at $\lambda =442\mathrm{nm}$ require much longer simulation times and exceed our allocated computing time. To study fully the effect of wavelength, we also employed a coated sphere model³⁴ in which the volumes of the inner sphere and coating were set to be identical to those of cell 8. The same sets of refractive indices of index $n_n$ for the nucleus (inner sphere) and $n_c$ for the cytoplasm (coating) were adopted for the wavelengths of 850 and $633\mathrm{nm}$ and higher values were used for $442\mathrm{nm}$ . The three independent matrix elements of $S_{11}$ , $s_{12}$ , and $s_{34}$ of the coated spheres were obtained and are presented in Fig. 8 . To account for the varied cell morphologies, these elements were averaged over a Gaussian distribution of the inner sphere radius $r_n$ and outer radius of the coating $r_c$ over $\pm 20\%$ of assigned values, $n_n$ over $\pm 3.6\%$ and $n_c$ over $\pm 0.8\%$ to reduce the characteristic angular oscillation for a single sphere of fixed radius.

Fig. 8

Angle-resolved Mueller matrix elements of normalized $S_{11}$ , $-s_{12}$ , and $-s_{34}$ calculated by a coated sphere model with the mean inner radius $r_n=4.143\mathrm{\mu }\mathrm{m}$ and outer radius $r_c=5.133\mathrm{\mu }\mathrm{m}$ . The mean values of the refractive indices were assigned as $n_h=1.342$ , $n_c=1.400$ , and $n_n=1.460$ at $\lambda =442\mathrm{nm}$ ; $n_h=1.336$ , $n_c=1.380$ , and $n_n=1.430$ at $\lambda =633\mathrm{nm}$ ; and $n_h=1.330$ , $n_c=1.368$ , and $n_n=1.400$ at $\lambda =850\mathrm{nm}$ . The elements are the values averaged over the coated spheres of Gaussian distributions with $\pm 20$ % on $r_n$ and $r_c$ and $\pm 3.6$ % on $n_n$ and $\pm 0.8$ % on $n_c$ .