2.1.

Experimental Scheme

The experiments use a heterodyne a/LCI interferometer (Fig. 1 ), described previously.¹⁰ Briefly, depth resolution is achieved by using the optical output of a Ti:sapphire laser ( ${\lambda }_{\mathrm{o}}=870\mathrm{nm}$ , coherence $\mathrm{length}=30\mathrm{\mu }\mathrm{m}$ ) in a modified Mach-Zehnder interferometer geometry. This system acquires a single depth scan at one scattering angle in approximately seven seconds, a fraction of the time needed with previous Michelson-based systems. The scheme achieves fast acquisition rates through the use of acousto-optic modulators to offset the signal and reference fields in frequency, resulting in a heterodyne beat signal of $10\mathrm{MHz}$ when the two fields are mixed at beamsplitter BS2, a significantly higher heterodyne frequency than that of previous a/LCI systems.

Fig. 1

Schematic of new a/LCI system. L1 through L5 are lenses, BS1 and BS2 are beamsplitters, D1 and D2 are balanced detectors, and RR is the retroreflector. The shaded area shows scattered light.

The heterodyne a/LCI system employs several imaging lenses to detect the angular distribution of scattered light. The imaging system is aligned using a standard scattering sample where misalignment is evident by uneven or lack of scattered light detection. Lenses L3 and L4 ( $f_3=10\mathrm{cm}$ , $f_4=3.5\mathrm{cm}$ ) comprise a $4f$ imaging system that collects the scattered light and images the phase and amplitude of the scattered field onto the plane of lens L2. The unequal focal lengths of L3 and L4 cause a magnification of the scattered field by 10/3.5, which also has the effect of scaling down the angular distribution by 0.35. The numerical aperture of this $4f$ system sets the range of detectable angles to $0.65\mathrm{rad}$ . The angular range and sampling of the angular distribution govern the resolution and range of the detected spatial correlations. The impact of these parameters on probing long range correlations in the cell array samples is discussed later.

Both the reference field and the scattered field pass through imaging lenses before they are combined at BS2. Lens L2 $(f_2=10\mathrm{cm})$ is mechanically translated to serially scan the detected scattering angle. Lens L1 $(f_1=10\mathrm{cm})$ is included to alter the wavefront of the reference field to compensate for the effects of L2. The role of lens L5 $(f_5=10\mathrm{cm})$ is to alter the wavefront of the input beam to the sample, such that L4 recollimates it rather than focusing it. The result is a collimated beam reaching the sample with a $0.35\mathrm{mm}\text{diameter}$ and a corresponding diffraction angle of $1.8\mathrm{mrad}$ . The narrow angular distribution of the input beam is essential, since using light traveling at many angles will smear the angular scattering patterns. In addition to this role, the lateral displacement of L5 relative to L4 allows the full angular aperture of the lenses to be used to detect angular scattering distributions. The input beam strikes the sample obliquely, avoiding obscuring the scattered light by specular reflection. Note, however, that the exact backscattering direction is antiparallel to the input beam, regardless of the orientation of the sample surface.

2.2.

Image Analysis

To assess the accuracy of the a/LCI nuclear morphology measurements, ImageJ software [National Institutes of Health (NIH), Bethesda, MD] was used to execute quantitative image analysis (QIA) of stained cell samples. Cell nuclei were stained with Hoechst 33342 (Invitrogen, Carlsbad, CA) and visualized using fluorescence microscopy (Axiovert 200, Zeiss, Oberkochen, Germany). The major and minor axis of each cell nuclei was measured manually and the geometric mean determined to assess the nuclear diameter. QIA of a large number of cell nuclei was used to determine the average nuclear size for the cell samples.

To assess the accuracy of the long range correlation measurements, QIA was executed to obtain autocorrelation data of fluorescence micrographs using ImageJ software and Mathematica (Wolfram Research, Champaign, IL). The average nearest-neighbor distance was found by computing the autocorrelation function $\Gamma \left(r\right)$ for a horizontal image segment:

2.3.

Cell Culture

NIH/3T3 fibroblasts, which have a nuclear diameter on the order of $10\mathrm{\mu }\mathrm{m}$ and a spindly irregular cell structure roughly three times the nuclear diameter, were grown in DMEM with 10% calf serum (Invitrogen), at $37°\mathrm{C}$ in 5% ${\mathrm{CO}}_2$ . Culture media was supplemented with 100-units/mL penicillin and $100\text{-}\mathrm{\mu }\mathrm{g}∕\mathrm{mL}$ streptomycin. Cells near confluence were detached from the tissue culture flask using 0.25% trypsin-EDTA (Invitrogen) and seeded onto micropatterned or unpatterned control substrates at a density of $120,000\mathrm{cells}∕{\mathrm{cm}}^2$ . The culture medium was changed 3-h postseeding to remove floating, dead cells. The a/LCI data, phase contrast, and fluorescence microscopy images were acquired $24\mathrm{h}$ after initial seeding.

2.4.

Cell Arrays

Cell arrays were fabricated using the procedures outlined by Hyun ¹³ and Ma ¹⁴ A comb polymer was micropatterned using an oxidized poly(dimethyl siloxane) (PDMS) stamp presenting micrometer-sized negative features. The stamp was then spin-coated with a 100-mg/mL solution of the comb polymer in a 20:80 (v/v) ${\mathrm{H}}_2\mathrm{O}$ /methanol mixture, and brought into conformal contact with the clean, unmodified glass surface. The comb polymer pattern was cured in an oven under vacuum at $60°\mathrm{C}$ overnight. The resulting surface was then incubated with a $50\text{-}\mathrm{\mu }\mathrm{g}∕\mathrm{mL}$ fibronectin solution in phosphate buffered saline (PBS) for $1\mathrm{h}$ at room temperature and then rinsed with deionized water.

2.5.

Cell Array Design for Angle-Resolved Low Coherence Interferometry

In previous cell array studies, NIH/3T3 cells were cultured on a square micropatterned array with $10\text{-}\mathrm{\mu }\mathrm{m}\text{-}\mathrm{diameter}$ adhesion sites and $40\text{-}\mathrm{\mu }\mathrm{m}$ spacing between cells.^{13, 14} This optimal arrangement, resulting in isolated cells on individual adhesion sites, has a center-to-center distance between sites of $50\mathrm{\mu }\mathrm{m}$ , and thus is termed a $50\text{-}\mathrm{\mu }\mathrm{m}$ period array. The $50\text{-}\mathrm{\mu }\mathrm{m}$ period array, although optimal for isolating individual 3T3 cells, is not well suited for examining long range correlations using a/LCI. In past a/LCI experiments, 30 to 40 points were used to sample the angular distribution over a $0.3\text{-}$ to 0.4-rad range.^{2, 10, 11} This range was chosen to examine nuclear features in the $5\text{-}\mathrm{to}$ $15\text{-}\mathrm{\mu }\mathrm{m}$ range and thus was only able to detect spatial correlations as large as $40\mathrm{\mu }\mathrm{m}$ . Thus, the data acquisition and cell array parameters needed to be altered to observe long range correlations with a/LCI.

Three steps were taken to enable a/LCI detection of long range correlations: 1. the sampling of the a/LCI probe was increased to 60 to 70 points; 2. the angular range was decreased to $0.32\mathrm{rad}$ , and 3. the array period was reduced to 20 and $30\mathrm{\mu }\mathrm{m}$ . By decreasing the angular step size, correlations over greater length scales are detected at the expense of increasing the data collection time. To compensate for this increase, the range of the angular measurements was also reduced. The net result was sensitivity to spatial correlations up to $70\mathrm{\mu }\mathrm{m}$ . To effectively examine the long range correlations, it was still necessary to reduce the array period. For the experiments presented here, array periods of 20 and $30\mathrm{\mu }\mathrm{m}$ were used.

Figures 2a and 2b show $20\text{-}$ and $30\text{-}\mathrm{\mu }\mathrm{m}$ period arrays, respectively, with cell nuclei visualized using fluorescence microscopy. The array sites were imaged using phase microscopy and superimposed on the fluorescence microscopy image of the nuclei. The arrays imaged in Figs. 2a and 2b show that although good registration was achieved at a number of sites, the array spacing was small enough to permit cells to bridge multiple sites. The effects of this registration are discussed later.

Fig. 2

Images of cell nuclei cultured on (a) $20\text{-}\mathrm{\mu }\mathrm{m}$ period microarray substrate and (b) $30\text{-}\mathrm{\mu }\mathrm{m}$ period microarray substrate. Cell nuclei are stained with Hoechst 33342 and imaged with fluorescence microscopy, and cell array sites are imaged using phase contrast microscopy and superimposed. Scale bars are $10\mathrm{\mu }\mathrm{m}$ .

2.6.

Theory

The light scattered by cells in the array can be analyzed using an elastic scattering formalism. In a previous theoretical examination¹⁵ of the scattering of low coherence light by an ensemble of scatterers, Eq. (19) showed that the heterodyne signal at frequency $\delta$ due to wave vector component $k$ for interferometrically detected scattered light can be written as:

The light scattered from a particular depth can be selectively detected by specifying the path length $\mathrm{\Delta }l$ of the reference field and integrating the selected signal across wavevector magnitudes. For a Gaussian distribution of frequencies centered at wavevector $k_o$ , the demodulated mean square signal is given by:

In a previous study,² it was shown that Fourier transformation of a/LCI measurements gives the correlation function of the density of the sample in the Born approximation. Similarly, Eq. 4 can be Fourier transformed to yield information about the spatial correlation of the cell samples:

3.1.

Angle-Resolved Low Coherence Interferometry Data

Raw a/LCI data for light scattered by cells grown on a $20\text{-}\mathrm{\mu }\mathrm{m}$ period array are shown in Fig. 3 , consisting of 57 points over an angular range of $0.32\mathrm{rad}$ . The striking feature in this data is the sharp oscillations in the angular distribution. The spatial correlations can be examined by taking the power spectrum for this data (Fig. 4 ). As shown previously,² the power spectrum of the angular distribution of the scattered field gives the two point spatial correlation of the sample in the Born approximation. The correlation data in Fig. 4 show correlations persisting over a $50\text{-}\mathrm{\mu }\mathrm{m}$ range.

Fig. 3

Raw a/LCI data for light scattered by cells cultured on an array with $20\text{-}\mathrm{\mu }\mathrm{m}$ period. The data are the mean square heterodyne signal, given as a function of angle relative to the backscattering direction. Note the high frequency oscillations, characteristic of long-range order.

Fig. 4

Power spectrum of data shown in Fig. 3. The vertical line indicates the cutoff distance for the low-pass filter used in the a/LCI nuclear size analysis.

To obtain nuclear size information, the data are low-pass filtered to suppress oscillations corresponding to correlations occurring over length scales greater than $20\mathrm{\mu }\mathrm{m}$ . This cutoff was chosen to encompass the nucleus sizes observed in the micrographs. The data are then processed by removing the slowly varying background using a second-order polynomial, as described previously for a/LCI analysis.^{2, 6, 10} The processed data are compared to similarly processed predictions of the Mie theory with the nuclear size determined by minimizing the chi-squared value between the data and Mie theory scattering distributions for ensembles of particles.¹⁵ For the data shown in Fig. 3, the best fit was determined to be a Gaussian distribution of sizes with a $12.5\pm 0.2\text{-}\mathrm{\mu }\mathrm{m}$ mean value, a 2.5% standard deviation (SD), and a refractive index of 1.043 relative to the cytoplasm. A comparison between the processed data and best-fit Mie theory is shown in Fig. 5 . For experiments with $20\text{-}$ and $30\text{-}\mathrm{\mu }\mathrm{m}$ arrays, as well as cells grown on a substrate with no micropatterning as a control, a total of 27 samples were probed with a/LCI. The average nuclear diameter for these 27 samples was found to be $12.8\pm 0.6\mathrm{\mu }\mathrm{m}$ (Table 1 ) with an average SD of the size distribution of 2.5% of the mean diameter and an average refractive index of $1.043\pm 0.008$ . The uncertainty in the mean diameter measurement is given by the 95% confidence interval for the average, defined by the standard deviation observed across the measurements, divided by the square root of the sample size multiplied by the appropriate t-distribution value (2.06). The a/LCI nuclear size data reported here differs slightly from previously presented results of this study.¹⁶ The difference is due to the use of an improved analysis method, which allows for larger nucleus sizes (up to $20\mathrm{\mu }\mathrm{m}$ ), broader size distributions (up to 15% of the mean value), and a corrected center wavelength.

Fig. 5

Oscillatory component of scattering distribution (solid line), obtained via a/LCI analysis, shown compared to the best-fit oscillatory component of Mie theory scattering distribution (dashed line). The nuclear size obtained here is $12.4\mathrm{\mu }\mathrm{m}$ .

Table 1

Nuclear diameter statistics from a/LCI and QIA data.

To examine the long range correlations, the detected signal is divided by the form factor, consisting of the obtained Mie solution and background distribution (not shown). The resulting distribution is then Fourier transformed to give the long range correlations [Fig. 6a ], as shown in Eq. 7. Similarly processed long range correlations are shown for the $30\text{-}\mathrm{\mu }\mathrm{m}$ period array [Fig. 6b], and for the control sample consisting of cells cultured on an unpatterned substrate [Fig. 6c]. The data for the $20\text{-}\mathrm{\mu }\mathrm{m}$ period array, shown in Fig. 6a, shows a sharp peak at a correlation distance of $24.5\pm 2.7\mathrm{\mu }\mathrm{m}$ with a full width at half maximum (FWHM) of $4.3\mathrm{\mu }\mathrm{m}$ . The data for the $30\text{-}\mathrm{\mu }\mathrm{m}$ period array shows a sharp peak centered at a correlation distance of $31.0\pm 2.8\mathrm{\mu }\mathrm{m}$ [Fig. 6b], however, this peak is atop a broad pedestal with a FWHM of $27.6\mathrm{\mu }\mathrm{m}$ . The data for the unpatterned control sample [Fig. 6c] show several peaks with comparable amplitudes and thus do not exhibit as clear of a correlation peak as seen with the patterned arrays. The distribution for the unpatterned arrays shown in Fig. 6c has a mean value of $30.4\pm 2.8\mathrm{\mu }\mathrm{m}$ with a FWHM of $22.2\mathrm{\mu }\mathrm{m}$ . The average correlation distance seen for the $20\text{-}\mathrm{\mu }\mathrm{m}$ arrays was $24.5\pm 3.0\mathrm{\mu }\mathrm{m}$ (ten samples), while that seen for the $30\text{-}\mathrm{\mu }\mathrm{m}$ array was $31.9\pm 4.3\mathrm{\mu }\mathrm{m}$ (six samples). These results are summarized in Table 2 . As a final note, the long range correlation data presented here also differ from the initial presentation of this study,¹⁶ as a new, more rigorous analysis method has been used based on the theory presented earlier.

Fig. 6

Long range correlations for the in-vitro samples. (a) $20\text{-}\mathrm{\mu }\mathrm{m}$ period array with a narrow correlation peak at $24.1\mathrm{\mu }\mathrm{m}$ , (b) $30\text{-}\mathrm{\mu }\mathrm{m}$ period array with a narrow correlation peak at $31.0\mathrm{\mu }\mathrm{m}$ , and (c) control sample without micropatterned array showing a broad distribution with a peak at $30.4\mathrm{\mu }\mathrm{m}$ .

Table 2

Nuclear spacing statistics from a/LCI and QIA data for both 20- and 30-μm period arrays. Correlation statistics for QIA are also presented from cells cultured without microarray substrates.

3.2.

Image Analysis

The diameter for 172 nuclei was determined using QIA, and the resulting distribution was found to form a Gaussian distribution of sizes with a mean of $13.1\mathrm{\mu }\mathrm{m}$ and a SD of $1.9\mathrm{\mu }\mathrm{m}$ , resulting in a 95% confidence interval of 12.8 to $13.4\mathrm{\mu }\mathrm{m}$ . These results are compared to the a/LCI results in the discussion in Sec. 4. In addition to using QIA to analyze the nuclear diameter, the long range correlations were examined by calculating the autocorrelation function for fluorescence microscopy images of cells in the $20\mathrm{\mu }\mathrm{m}$ , $30\mathrm{\mu }\mathrm{m}$ , and unpatterned arrays. The average spacing for the $20\text{-}\mathrm{\mu }\mathrm{m}$ period arrays was found to be $24.0\pm 1.1\mathrm{\mu }\mathrm{m}$ , which included data for 38 cell pairs. The average spacing for the $30\text{-}\mathrm{\mu }\mathrm{m}$ period arrays was found to be $30.1\pm 1.4\mathrm{\mu }\mathrm{m}$ , which included data for 61 cell pairs (Table 2). Representative data are presented for the $20\text{-}$ and $30\text{-}\mathrm{\mu }\mathrm{m}$ arrays in Figs. 7a and 7b with the nearest-neighbor spacing determined to be $23.7$ and $29.3\mathrm{\mu }\mathrm{m}$ , respectively. The correlations are seen to extend for multiple neighbor spacings, with the second neighbor spacing being measured at $48.4$ and $58.5\mathrm{\mu }\mathrm{m}$ , respectively, or roughly double the first neighbor spacing for the $20\text{-}$ and $30\text{-}\mathrm{\mu }\mathrm{m}$ arrays. For cells cultured without an underlying microarray pattern, the average nearest-neighbor distance determined by QIA was found to be $35.0\pm 3.5\mathrm{\mu }\mathrm{m}$ . Comparison to the a/LCI long range correlation data is presented next.

Fig. 7

1-D spatial autocorrelation function of nuclear fluorescence microscopy images (Fig. 2) on (a) $20\text{-}\mathrm{\mu }\mathrm{m}$ period substrate and (b) $30\text{-}\mathrm{\mu }\mathrm{m}$ period substrate. The nonzero correlation peaks are 23.7 and $48.4\mathrm{\mu }\mathrm{m}$ for the $20\text{-}\mathrm{\mu }\mathrm{m}$ period array and 29.3 and $58.5\mathrm{\mu }\mathrm{m}$ for the $30\text{-}\mathrm{\mu }\mathrm{m}$ period array. The first peak is the nearest-neighbor spacing and the second peak is the second neighbor spacing. Note the good agreement to the correlation distances in Fig. 6 as determined by a/LCI.