2.1.

Microscopy Measurement and Three-Dimensional Reconstruction

We have investigated two prostate cell types that are denoted in this report as PC3 for the cancer cell line and PCS for the normal one. A PC3 human prostate cancer cell line of high metastatic potential (CRL-1435, ATCC) was maintained in RPMI-1640 (Gibco BRL, Life Technologies) supplemented with 10% fetal calf serum. The culture media were supplemented with penicillin $100\mathrm{U}/\mathrm{ml}$, streptomycin $100\mathit{\mu }g/\mathrm{ml}$, and glutamine $0.1\mathrm{mg}/\mathrm{ml}$. The normal human prostate epithelial cells (PCS440010, ATCC) were maintained in the prostate epithelial cell basal medium (PCS440030, ATCC) supplemented with the prostate epithelial cell growth kit (PCS440040, ATCC). The adherent cells in their logarithmic phases of growth were detached from culture plates with trypsin–EDTA solution, resuspended in culture medium, and kept on ice before the confocal and p-DIFC measurements. Viability of the suspended cells was checked by a trypan blue exclusion test before measurement and percentages of viable cells were found to be $\sim 95\%$. The concentrations of the cell suspension samples were adjusted to a value of about $1\times {10}^6\text{cells}/\mathrm{mL}$ for p-DIFC measurement.

For confocal imaging, the cells were first double-stained for nucleus and mitochondria with fluorescent dyes (Syto-61 and Mitotracker-Orange, Life Technologies) with protocol detailed in Zhang et al.²⁰ A laser scanning confocal microscope (LSM 510, Zeiss) was used to acquire image stacks with a $63\times$ water-immersion objective of 1.2 in NA and a $4\times$ digital zoom option provided by the Zeiss image acquisition software to reduce pixel size to $0.07\mu \mathrm{m}$. The number of two-dimensional (2-D) fluorescence image slices in each stack ranges from about 40 to about 70 with 0.5 μm in the translation step in air along the direction perpendicular to the slices. The acquired confocal image slices in a stack were segmented using an in-house developed software.^10,14,20 The segmented slices were then used to add image slices through interpolation to make final slice separation, after correction due to light refraction, approximately the same as the pixel size of the image slices to obtain cubic voxels for 3-D reconstruction. The details of the image segmentation and reconstruction have been provided elsewhere.²⁰ A total of 29 voxel-based parameters were calculated to quantify the morphology of the different organelles of cytoplasm, nucleus, and mitochondria in reconstructed cells. We used the SPSS software (Version 19, IBM) to perform the two-sample $t$-tests of the morphology parameters and assess the statistical significance of the differences in the 3-D parameters between the two cell types. The definitions and a table of 29 morphologic parameters are provided online.²¹

2.2.

Diffraction Imaging Flow Cytometric Measurement

Design details of the p-DIFC system have been published elsewhere for cell positioning through hydrodynamic focusing in a square flow channel and the imaging of scattered light.^13–15,19 Briefly, a continuous-wave solid state laser (MGL-III-532-100, CNI) was used to produce an incident beam of 532 nm in wavelength and up to 180 mW in power. A spherical lens of 75 mm in focal length was used to focus the incident beam onto the core fluid carrying the cells with a spot diameter of about $30\mu \mathrm{m}$. The profile of a linearly polarized incident beam propagating along the $z$-axis is close to a Gaussian distribution, and the power was measured as $P_0$ before the focusing lens and adjusted with neutral density filters. The loss of the optical power by the index-mismatch interfaces from the focusing lens to the imaged cell was estimated to be about 17%. Figure 1 presents the schematic of the p-DIFC system.

Fig. 1

Schematic of an experimental p-DIFC system for acquisition of s- and p-polarized diffraction images. BE: beam expander; WP: half-wave plate; M: mirror; ND: neutral density filters; FL: focusing lens; FC: flow chamber; CL: condenser lens; OB: objective; WF: 532 nm wavelength filter; PBS: polarizing beam splitter; TL: tube lenses; BS: beam splitter; PMT: photomultiplier; CCD: camera. The $x$-axis and $z$-axis are labeled by red lines.

The polarization direction was set to one of the three directions of horizontal (hor), vertical (ver), or 45 deg from horizontal with a half-wave plate. The light scatter from flowing cells was collected by an infinity-corrected $50\times$ objective of 0.55 in NA (378-805-3, Mitutoyo) within an angular cone, which was centered at the scattering polar angle ${\theta }_{\mathrm{s}}=90\mathrm{deg}$ along the $x$-axis and of a cone angle ${\theta }_{\mathrm{sm}}$ in water. A polarizing beam splitter was employed to divide the scattered light into the s- and p-polarized beams for acquisition of two cross-polarized diffraction images of $640\times 480\text{pixels}$ and a 12-bit pixel depth by two CCD cameras (LM075, Lumenera). Camera-triggering signals were produced with a photomultiplier, and the exposure time was set to 0.3 ms to reduce image blurring for the imaged cells flowing at a speed of about $6\mathrm{mm}/\mathrm{s}$.

To vary the angular cone of the imaged light and increased image contrast, the imaging unit consisting of the objective, optics, and cameras was translated toward the flow chamber by a distance of $\mathrm{\Delta }x=150\mu \mathrm{m}$ from the focusing position conjugate to the imaged cell or core fluid. It has been shown that at these nonconjugate positions, the acquired images present patterns of diffraction in high-fidelity because of the unique correspondence between the angular positions of the coherent light scatter and the imager pixel positions.^14,16 Furthermore, the maximum cone angle ${\theta }_{\mathrm{sm}}$ of the scattered light passing through the exit pupil of the objective decreases linearly as $\mathrm{\Delta }x$ increases, which allows the variation of the angular cone viewed on the acquired images. At $\mathrm{\Delta }x=150\mu \mathrm{m}$, ${\theta }_{\mathrm{sm}}=23.3\mathrm{deg}$ for the objective used in the imaging unit.¹⁶ The throughput of the p-DIFC measurement was maintained at about $1\text{cell}/\mathrm{s}$ and mainly limited by the frame rate of cameras triggered externally. Before extraction of p-DIFC image parameters, the acquired image pairs were first filtered with an in-house developed preprocessing software. The overexposed and underexposed image pairs were removed, which were defined, respectively, as those with one image of saturated pixels more than 1% of the total pixels or both images of average pixel intensities $<2\%$ of the saturated pixel values ($=4095$ for 12-bit images). Additionally, image pairs with strip patterns of high symmetry or large speckles were also removed, since these have been shown to associate with spherical particles or aggregated small particles or cellular debris instead of intact cells.¹⁸

The remaining diffraction image pairs were converted linearly from the 12-bit images of the acquired data into normalized 8-bit images in which the minimum and maximum pixel intensities in the 12-bit image were set to 0 and 255. The bit reduction was designed to speed up the subsequent parameter extraction by the gray-level-co-occurrence-matrix (GLCM) algorithm without significant loss of dynamic range.^17,22 Using the GLCM algorithm, a total of 38 image parameters have been extracted as $g_{\mathrm{im}}$ with $m=\mathrm{1,2},\dots$, 38 to characterize the texture and pixel intensity of the normalized image pair for the $i$’th imaged cell.¹⁹ The list of the 38 diffraction image parameters and their definitions are provided online.²¹

2.3.

Diffraction Image Analysis and Cell Classification

With either 3-D parameters from the confocal image data or parameters of the cross-polarized diffraction image data, the $i$’th imaged cell can be represented by a parameter vector given by ${\mathbf{c}}_i=\sum _{m=1}^{N_m}{\mathrm{g}}_{im}{\mathbf{u}}_m$, where ${\mathbf{u}}_m$ are unit vectors defining a parameter space ${\mathrm{\Sigma }}_p$ of $N_m$-dimension. The set of $N_m$ parameters, $\{g_{im}\}$, consists of either all or a portion of the 3-D parameters with $1\le N_m\le 29$ or of the GLCM parameters with $1\le N_m\le 38$. To classify the imaged cells as represented by their parameter vectors, we chose a statistical learning algorithm of SVM for its well-recognized balance between training complexity and test performance in comparison to other machine-learning algorithms such as the neural network method.^23,24 Instead of direct classification by ${\mathbf{c}}_i$ of the training data in ${\mathrm{\Sigma }}_p$, the SVM approach maps the input vectors into a high-dimensional feature space $E$ by a kernel function $K({\mathbf{c}}_i,{\mathbf{c}}_j)$ with a training data set consisting of $N_{\text{tra}}$ cells.

Specifically, SVM constructs a matrix $Q$ of rank $N_{\text{tra}}$ and defines its elements $Q_{ij}=t_i{t}_{j}K({\mathbf{c}}_i,{\mathbf{c}}_j)$ with the type identifier of $t_i$ and $t_j$ ($=1$ or $-1$ for two types of cells) and the index $i$ or $j$ ranging from 1 to $N_{\text{tra}}$. The mapping from ${\mathrm{\Sigma }}_p$ to $E$ allows classification in $E$ and solves for $\mathbf{\alpha }$ as a quadratic optimization problem by minimizing $(1/2){\mathbf{\alpha }}^{\mathrm{T}}·Q\mathbf{\alpha }-{\mathbf{e}}^{\mathrm{T}}·\mathbf{\alpha }$ under the constraints of positive definite $\mathbf{\alpha }$ and ${\mathbf{t}}^{\mathrm{T}}·\mathbf{\alpha }=\mathrm{\Delta }$.²⁵ By defining ${\mathbf{\alpha }}^{\mathrm{T}}=({\mathrm{\alpha }}_1,\dots ,{\mathrm{\alpha }}_{N_{\mathrm{tra}}})$ and ${\mathbf{e}}^{\mathrm{T}}=(1,\dots ,1)$ in $E$, one can obtain $\mathbf{\alpha }$ together with a bias parameter $b$ from a training data set in terms of the vectors ${\mathbf{c}}_i$ with $N_m$ parameters as components and a kernel function. These define an SVM model with a decision function $F$ as the classifier

By assessing the performance, we define the following numbers to measure the outcomes of the classification according to the values of $F$: TP as the number of correctly identified image pairs acquired from the given PC3 cells with $F>0$, TN as the number of correctly identified image pairs from the PCS cells with $F<0$, FP as the number of image pairs of PCS cells that are incorrectly identified as PC3 cells with $F>0$, and FN as the number of image pairs of PC3 cells that are incorrectly identified as PCS cells with $F<0$. SVM models were evaluated by their classification accuracy $A$ on a given data set from the above values as

3.1.

Confocal Measurement and Quantitative Characterization of Three-Dimensional Morphology

We have performed confocal imaging and 3-D reconstruction of the detached PC3 and PCS cells after double-staining of the nucleus and mitochondria. Following reconstruction, a total of 29 parameters were obtained.²⁰ Selected confocal image slices and perspective 3-D views of two PC3 and two PCS cells are presented in Fig. 2. Table 1 lists the mean values and standard deviations of 17 key parameters together with the $p$-values to test the statistical significance of the difference between the two prostate cell types. From these data, one can clearly see that the PC3 cells’ cell and nuclear volumes are larger on average than those of the PCS cells. Similar morphologic differences of statistical significance can also be observed in the cell shapes as indicated by the distribution of the membrane voxels’ distances $R_{\mathrm{c}}$ to the centroid.

Table 1

Morphologic parameters of the two prostate cell types.

Fig. 2

Perspective views of 3-D structures (not to scale) and selected slices from confocal image stacks acquired from two PC3 and two PCS cells with nucleus and cytoplasm stained and imaged in red (top rows) and mitochondria and cytoplasm stained in green (bottom rows) channels. The cell type and values of cell volume in $\mu {\mathrm{m}}^3$, nucleus-to-cell volume ratio, and mitochondria-to-cell volume ratio are: (a) PC3, 2140, 28.1%, 4.06%; (b) PC3, 2073, 32.4%, 11.3%; (c) PCS, 1185, 46.2%, 22.8%, and (d) PCS, 1451, 40.9%, 6.89%. $\mathrm{Bar}=10\mu \mathrm{m}$.

To examine the difference in morphology closely, we provide in Fig. 3 the scatter plots of the 3-D parameters selected from Table 1 with $p$-values $<0.05$ for the imaged cells. While most of the cells in each type overlap each other in these scatter plots, the PC3 cells as a group appear to have significantly smaller spreads in their values of cellular and nuclear parameters than those of the PCS cells, which can also be noted from the standard deviations of most of the other parameters in Table 1. Taken together, the quantitative characterization provides insight into the morphologic differences between the two cell types and demonstrates that the 3-D parameters alone are not sufficient for accurate classification, which is confirmed by the SVM classification results to be presented later.

Fig. 3

Scatter plots of 40 PC3 cells and 38 PCS cells with different combinations of 3-D parameters. See Table 1 for definitions of the symbols used as axis labels.

3.2.

Measurement and Analysis of Diffraction Images

Cross-polarized diffraction image pairs have been acquired from cell suspension samples of about 2 mL in volume with the p-DIFC system shown in Fig. 1 in three measurements carried out in different weeks to confirm the repeatability of acquired data and subsequent classification. In each measurement, a small portion of the PC3 or PCS cell suspension sample was loaded into the core fluid syringe followed by alignment of the imaging unit to the same off-focus position of $\mathrm{\Delta }x$ and adjustment of the incident laser beam power $P_0$. About 1000 to 2000 cells were imaged from each cell sample for one of the three incident beam polarizations at the directions of ver, hor, and 45 deg.

Figure 4 shows examples of the normalized 8-bit image pairs acquired from single PC3 and PCS cells for three incident beam polarizations. It is clear from these normalized 8-bit images and the range of pixel values in the raw 12-bit images that both PC3 and PCS cells present stronger s-polarized light scatter for an incident beam that is also s-polarized (ver). Similarly, stronger p-polarized scatter can be observed for images acquired with an incident beam of p-polarization (hor). For 45 deg polarization of the incident beam, however, both PC3 and PCS cells yield much stronger scattered light of s-polarization, which can be understood by the fact that molecular dipoles induced by the incident laser beam within the illuminated cell are of higher efficiency to emit s-polarized than p-polarized light as scatter along the side directions. The dependence of scattered light intensity on polarization is shown in Fig. 5.

Fig. 4

Examples of normalized 8-bit cross-polarized image pairs of two PC3 and two PCS cells acquired in measurement #1 for each incident beam polarization with white for pixel intensity 255 and black for 0. Each image is labeled with the cell type, polarization of the incident beam, polarization of the scattered light, and maximum, average, and minimum pixel intensities of the acquired 12-bit images.

Fig. 5

Scatter plots of $N_{\text{tot}}$ imaged cells with the average pixel intensity of the acquired s-polarized (s-$I_{\mathrm{av}}$) versus that of the p-polarized diffraction image (p-$I_{\mathrm{av}}$) acquired in measurement #1 with different incident beam polarizations: (a) vertical or s-polarized with $N_{\text{tot}}=716$ for PC3 cells and $N_{\text{tot}}=668$ for PCS cells; (b) horizontal or p-polarized with $N_{\text{tot}}=681$ for PC3 cells and $N_{\text{tot}}=623$ for PCS cells; (c) 45 deg with $N_{\text{tot}}=770$ for PC3 cells and $N_{\text{tot}}=378$ for PCS cells. The values of incident beam power $P_0$ are labeled.

3.3.

Classification of Two Prostate Cell Types and Comparison

With the 3-D morphology or p-DIFC image parameters, we performed cell classification study by the SVM algorithm to obtain the best SVM model with the highest value of accuracy $A$. For SVM classification with the 3-D parameters, the data were divided into a training data set of 30 cells/type and a test data set with the rest of cells. The data for GLCM parameters extracted from diffraction images were similarly divided, and Table 2 provides the number of cells in the training and test data sets acquired in three p-DIFC measurements.

Table 2

Experimental parameters and classification results with diffraction images.

The search for the best SVM model to classify cells started by evaluation of the individual performance of 29 3-D parameters or 38 p-DIFC image parameters with different kernel functions based on the averaged values of $A$ as $A_{\mathrm{av}}$ using a scheme of five-fold cross-validation with the training data set. The scheme divides the data into five equal parts with one part being used as a test data assembly and the remaining four parts as a training data assembly. The procedure was iterated five times with $A$ calculated each time to obtain $A_{\mathrm{av}}$ followed by ranking of the single parameters in the order of decreasing $A_{\mathrm{av}}$, which depends on the kernel functions used in SVM calculations. Different SVM models were then formed by a parameter vector ${\mathbf{c}}_i$ for cell $i$ in the training data, with $N_m$ selected parameters in the same sequence of ranking as components, and the corresponding kernel function. Each SVM model was trained in the feature space $E$ with the training data and then applied to the test data to obtain $A_{\mathrm{av}}$ for evaluation.

For SVM classification with the 3-D parameters, the single parameters with highest $A_{av}=71.7\%$ for the training data set are the cell’s equivalent spherical radius (${\mathrm{ER}}_{\mathrm{c}}$) using the kernel functions of the polynomial or RBF. The corresponding parameter for a sigmoid kernel function is the nucleus to cell volume ratio (${\mathrm{Vr}}_{\mathrm{nc}}$) with $A_{\mathrm{av}}=68.3\%$ and cell volume ($V_{\mathrm{c}}$) for linear with $A_{\mathrm{av}}=71.7\%$. SVM models obtained by including additional 3-D parameters ($N_m>1$) according to their ranks have been found to produce slightly larger or smaller values of $A_{\mathrm{av}}$ in comparison to the single parameter model with the top-ranked one. The performance results of SVM models with $N_m$ up to 10 are presented in Fig. 6 with different kernel functions for both training and test data sets. It is obvious that 3-D morphology parameters extracted from confocal image stacks of cells with stained nucleus and mitochondria do not yield accurate markers for classification of the two prostate cell types, which is consistent with the data shown in Fig. 3.

Fig. 6

Averaged accuracy $A_{\mathrm{av}}$ versus the maximum number of 3-D parameters $N_m$ used for SVM classification with four different kernel functions for: (a) training data set and (b) test data set. The lines are for visual guide.

To investigate classification with the p-DIFC image parameters, an SVM model was first optimized with the training data set. Table 2 includes the values of $A_{\mathrm{av}}$, $N_m$, and kernel functions of the best SVM models established for the diffraction image pair data acquired with three different incident beam polarizations in three measurements. One can clearly see that the p-DIFC parameters provide a much improved performance in comparison to the 3-D parameter for classifying the two prostate cell types. However, $A_{\mathrm{av}}$ decreases significantly if we combine all data from the three measurements together as shown by the bottom section of Table 2. Similar decreases were observed by applying the best SVM model trained by the data of one measurement to the data of different measurements (not shown).

To demonstrate the effectiveness of the SVM algorithm with the p-DIFC image parameters, scatter plots of the training results are presented in Fig. 7 for three cases of cell classification with the best SVM model in each case on data acquired in the same measurement. The data show clearly that the SVM algorithm provides a powerful tool to improve cell classification with extracted image parameters by mapping them from the parameter space ${\mathrm{\Sigma }}_p$ into the feature space $E$ using a kernel function. In the case of Fig. 5(a), the top two ranked single GLCM parameters of dissimilarity and sum average²² extracted from p-polarized images yield, respectively, classification accuracies $A_{\mathrm{av}}$ of 91.0% and 87.7% for the training data. These values of $A_{\mathrm{av}}$ with single parameter values are significantly smaller than the accuracy of 99.1% that can be achieved with the best SVM model of $N_m=10$ parameters and the linear kernel function. A similar improvement in classification can be observed in the other two cases: the values of $A_{\mathrm{av}}$ were found to increase, respectively, from 77.9% for s-IDM and 74.4% for s-DIS alone to 93.5% with $N_m=9$ and a linear kernel function in the case of Fig. 5(b) and from 61.3% for s-DIS and 61.2% for s-DEN alone to 91.3% with $M=10$ and a polynomial kernel function in the case of Fig. 5(c). The GLCM parameters extracted from the normalized diffraction image pairs are available online for readers to investigate other classification methods.²⁶

Fig. 7

Scatter plots of training data with values of decision function $F$ versus the top two ranked p-DIFC image parameters used by the best SVM model established for: (a) data acquired in measurement #1 with vertical incident beam polarization and $N_{\text{tra}}=500$ for each of the two cell types, p-DIS: dissimilarity of p-polarized images, p-SAV: sum average of p-polarized images; (b) data acquired in measurement #3 with vertical incident beam polarization and with $N_{\text{tra}}=800$, s-IDM: inverse difference moment of s-polarized images; (c) same as (b) except with horizontal polarization, s-DEN: difference entropy of s-polarized images. The cells with $F>0$ (above the line of $F=0$) are classified by the SVM model as PC3 cells and those with $F<0$ (below the line) as PCS cells. The values of TP, FN, TN, FP, $N_m$, kernel function, and $A_{\mathrm{av}}$ of the best SVM model are labeled.