Poly (DL-lactic-glycolic acid) (PLGA, block copolymer, 50:50 ratio of PGA and PLA, 40,000~75,000 Da) and poly (ε-caprolactone) (PCL, 114,000 Da, M_w/M_n = 1.43) were obtained from Sigma Aldrich, St Louis, MO. PLGA and PCL, respectively, were dissolved in chloroform to 8% and 5% mass and spin coated on silicon chips (22 × 22 mm). To provide adhesion of these polymers to the silicon during cell culture, the silicon was pretreated with a Piranha etch (70% H₂SO₄/21% H₂O/9% H₂O₂ at 80°C for 1 h) followed by 1 min in a HF acid bath and a final rinse in DI water (filtered at 0.2 μm).

Established from newborn mouse calvaria, [34] the MC3T3-E1 cell line has been shown capable of differentiating into osteoblast and osteocytes in vitro[35]. MC3T3-E1 cells have been shown to exhibit specific bone related protein expression patterns, under different developmental stages, similar to primary mouse calvaria cells[35,36]. This cell line is thus a suitable in vitro model for investigating cell behaviors, regulations of such behaviors, and underlying mechanisms in different osteoblast maturation stages[37]. Since the original MC3T3-E1 cell line has been found phenotypically heterogeneous with regard to cell differentiation, more homogeneous subclones have been established[38]. In this work, MC3T3-E1 subclone 4 (from ATCC, VA), which shows homogenous capabilities of osteogenesis both in vitro and in vivo, [38] was chosen in order to minimize variations due to phenotypical heterogeneities.

Cell proliferation was assayed by BrdU immunohistochemistry. Briefly, PLGA- and PCL-coated wafers were mounted into Costar^® 6-Well TC-Treated Microplates (Corning, NY). The tissue culture treated polystyrene (TCPS) surfaces of the microplate wells were used as controls. After sterilization (70% ethanol solution, 30 min), MC3T3-E1 cells (passage 6) were seeded onto the coated wafers at 4 × 10⁴ cells/cm². This relatively high seeding density was selected to highlight the effects of contact inhibition of cell growth and other space-sensitive cell-to-cell interactions. After seeding, microplates were shaken for 10 min on a shaker (Instrument model, operation frequency) to obtain uniform seeding. Cells were cultured in DMEM (Cellgro^® Dulbecco's Modification of Eagle's Medium, Mediatech, Inc., VA) with 10% fetal bovine serum (ATCC^® SCRC-1002™, ATCC, VA), L-glutamine and streptomycin at 37°C in a humidified 5% CO₂ atmosphere. At 5 h post seeding, surfaces were washed with Dulbecco's Phosphate-Buffered Saline (DPBS, with Ca⁺⁺ and Mg⁺⁺) to remove non-attached cells, and fresh culture medium was then added. At 18 h post seeding, 2 mM BrdU (5-bromo-2-φ-deoxyuridine, Sigma, MO) in PBS was added to the culture medium to reach a final concentration of 20 μM. After 6 h of BrdU incorporation, cells were fixed with 3.6% paraformaldehyde and BrdU incorporation was assayed by immunohistochemistry (primary antibody: mouse anti-BrdU, BD Biosciences, CA; secondary antibody: goat anti-mouse, Rhodamine conjugated, Rockland Immunochemicals, Inc., PA; counter staining: Hoechst 33342, Molecular Probes, Invitrogen Corporation, CA).

Low calcium concentration suppresses contact inhibition of cell growth by deactivating calcium-dependent cadherins[7,39]. This phenomenon was used in this study to validate the local cell metrics, and at the same time the dependency of contact inhibition on calcium was quantitatively studied. In order to investigate the role of Ca⁺⁺ on cell spreading and proliferation, BrdU incorporation experiments in low Ca⁺⁺ medium were performed on TCPS surfaces. Fifteen minutes before the introduction of BrdU, cells were rinsed twice with DPBS (without Ca⁺⁺ and Mg⁺⁺) and afterword cultured in the low Ca⁺⁺ medium (0.5% FBS in Ca⁺⁺ and Mg⁺⁺ free DPBS)[39]. The rest of the protocol was the same as previously described.

Cell locations and proliferation were quantified using fluorescent microscopy (Olympus BX51 Clinical Microscope). A robotic translation stage was used to image predetermined locations on each culture surface using a MicroFire™ monochromic digital camera (SKU S99826, Optronics, CA). The image locations were fixed on a 16 × 20 grid with horizontal and vertical spacing of 1280 μm and vertical spacing of 960 μm. For each location a 1189 × 892 μm² BrdU staining image and Hoechst counter staining image were acquired at a resolution of 1600 × 1200 pixels². All images and contextual information were organized and stored in an Oracle^® 10 g (Oracle, CA) database for further image processing and data analysis.

The Image Processing Toolbox of Matlab™ R14 (MathWorks, MA) was employed for image processing. Due to the volume of image data dynamic, self-adapting algorithms were developed for automated image processing. Binary images of both surface lateral patterns of cell nuclei counter staining were obtained from raw grayscale microscopic images by a variation-adjusted iterative selection method (VAIS), which was modified from the original iterative selection method [40-45] (details in the BrdU thresholding part below).

Binary images of cell nuclei were segmented by the marker-controlled watershed method[42] to separate images of closely-spaced cell nuclei. This process was critical because the nuclei of a pair of recently-proliferated cells were often too close to be distinguished with thresholding alone. Resultant black-and-white cell nuclei images were used as masks by overlaying them with corresponding BrdU staining images to determine the fluorescence intensity of incorporated BrdU. The histogram of BrdU staining intensity per nucleus [see Additional File 1: Figure S.1] is composed of two major peaks: the low intensity peak (background) represents cells at rest, while the high intensity peak (foreground) indicates proliferating cells. The optimal threshold between these two peaks was determined automatically by VAIS. Briefly, starting at an initial threshold T_i = 0.5, the histogram was divided into resting (background) and proliferating (foreground) parts. Means and standard deviations of the foreground and background, respectively denoted as M_bi, M_fi, σ_bi, and σ_fi, were determined by fitting each peak to a Gaussian curve. A new threshold was calculated as T_i+1 = (σ_fiM_fi + σ_biM_bi)/(σ_fi + σ_bi) and was repeated until convergence on a stable threshold. Compared with more common iterative selection methods, which use a simple mean intensity, the modified VAIS procedure is more robust when background and foreground intensities have different variances. Indeed, the variance of the BrdU signal intensity from non-proliferating cells was significantly greater than that of the proliferating cells [see Additional File 1: Figure S.1]. During image processing, data washing in the form of median filtering was performed to remove noise below a certain threshold. Image processing was supervised in order to assure the performance of self-adaptive algorithms and images of poor quality not permitting quantification were occasionally discarded. Proliferation behaviors were determined for every cell and stored in the database along with the cell location on the surfaces.

Cell density and proliferation were described with summary statistics such as number of resting and proliferated cells computed for each image. This provides a set of global metrics for features in each image. As indicated in Figure 1a, global metrics are most naturally understood in terms of conventional summary-statistics, exploratory data analysis, and well-known methods for estimating confidence and significance levels based on an assumed probability distribution. The ability to detect contact inhibition of cell proliferation, a known phenomenon, was used as an indicator of the effectiveness of the global metrics cell density and proliferation averages.

Source codes that implement the algorithms presented in this section have been made available by the authors. [see Additional File 2] Consider that the collection of all cells (A) is composed of either proliferated (P) or resting cells (R), such that A = P + R. The symbol A represents any cell chosen at random, regardless of proliferative status. The proliferating-resting cell distance, PR, is used here to illustrate the definition and properties of local cell metrics, as indicated in Figure 1b. The definitions below are generalizable to any type of cell-cell distance, or any other spatial or temporal metric of cells. Assume that in the k^th image the number of P-class and R-class cells is n_Pk and n_Rk, the distance PR_ijk between the centroids of the nuclei of the i^th P-cell and the j^th R-cell can be calculated readily from the results of image analysis. In the k^th image, the set of all such distances, PR_k is defined as(1)

And for all images an overall set PR can be defined as(2)

A set of N+1 distance bins is defined as(3)

where d₀, d₁, ..., d_N is a user-defined distance scale over which analysis is to be performed. The centroid of each interval in bin_dist is defined as(4)

and the resultant centroid set for bin_dist is(5)

The bin_dist is used to sort set PR into an N-bin histogram(6)

where is the number of PR distances that fall in the interval [d_i-1, d_i), which is centered at .

The total number of elements in set PR is(7)

After normalizing by n_PR, the frequency function LCM is(8)

and represents the R-type cell environment of the P-type cells observed over distances . Frequency functions, denoted as , , , and , may also calculated for cell-to-cell distances PP, AA, RR, and PA in similar manner.

Normalization is necessary to interpret LCMs in a meaningful manner and to compare the probability of cell responses under different cell environments. One method of normalization is to relate observed occurrences to random occurrences. Given the finite image size and generally non-overlapping nature of cultured cells, the distribution of random cell occurrences is not Gaussian. The random distribution for cell-cell distance, f_std, was calculated as the any cell-any cell distribution (f_AA) of 1× 10¹⁰ randomly-chosen nuclei positions on a simulated image 1600 pixels by 1200 pixels. The normalized LCM is(9)

Other LCMs (f_AA, f_PA, f_RR) are normalized similarly, which allows direct comparisons of different types of cell distances on different surfaces.

In addition to normalizing by the standard distribution, f_std, direct ratios between LCMs are used also in our analysis, in which case f_std cancels, as indicated in the next equation.(10)

The ratio r_PR|PA highlights the specific effects of non-proliferated cells on the central proliferating cell relative to the effects of any given cell. Thus, the probability of cell responses under different cell environments can be compared meaningfully. Furthermore, each set of cell-to-cell distances can be decomposed into subsets, which allows investigation of the contribution of each subset to the overall effect. Therefore, ratios of cell backgrounds may be constructed and used as classifiers for screening and identifying significant cell environment patterns. These ratios also define posterior odds (PO) of observing certain proliferation behaviors. For example, consider , and using the subscript i to signify the evaluation at a certain distance , the ratio is calculated as(11)

where(12)(13)

Applying equations (12) and (13) to equation (11),(14)

Thus, is a posterior odds that quantifies how the probability of cell proliferation is changed by the presence of a second cell located at distance , relative to the average proliferation for all cell-cell distances. Computationally, to promote the efficiency of the codes, we defined A_k = {P_k, R_k}={P_1k, P_2k, ..., P_NPk, R_1k, R_2k, ..., R_NRk}, and removed self-to-self cell pairs (PA_ijk where i = j) and identical cell pairs (PA_ijk where i >j) from PA.

Furthermore, each set of cell-to-cell distances can be decomposed into subsets, which allows isolation of each subset's contribution. For example, consider r_PR|PA defined above. As described graphically in Figure 2, since the denominator PA = PP ∪ PR (the union of distance sets PP and PR) and PP ∩ PR = ∅ (∅ = the empty set), one may remove the PR component from PA, and the denominator becomes PP. The result is that the ratio r_PR|PA is transformed into r_PR|PP. By removing the shared, or overlapping, component PR from the denominator, r_PR|PP has higher "contrast" for observing effects of R-cells on P-cells than r_PR|PA.

Local cell metrics are naturally connected to Bayesian analysis, which is a powerful statistical method used for classification[46,47]. Specifically for the PR distance, the Bayesian approach allows one to quantify the local environment of P cells, as the conditional probability of finding an R cell a certain distance PR from a P cell. Based on the definition of f_PR in equation (8), a naïve Bayes model can be established as follows. Consider a "test" cell chosen at random. It is desired to predict the possibility this cell will be in proliferating status, based upon the local environment of non-proliferating cells, which is given by the following conditional probability function(15)

where represents the probability of finding a non-proliferating cell at a distance of from the central, randomly chosen, test cell. Using Bayes's theorem,(16)

In the above function, the components p() and p(prolif) are constants that can be determined from their frequency in the data. The only non-constant component, the class-conditional probability, is given by(17)

Assuming the occurrence probabilities around the non-proliferating cell distances are conditionally independent (uncorrelated), then where i ≠ j (the naïve Bayes assumption). Hence, equation (17) reduces to(18)

A key development is to notice that , which means that under the naïve Bayes assumption the LCM is in fact a class-conditional probability function. The term p() represents the probability of locating the R cells at distances () from any cell, which is . Hence, the Bayes conditional probability function from equation (16) becomes(19)

The naïve Bayes model allows prediction of the probability of proliferation as a function of the LCMs, and , which are easily computed from a training data set, as is p(prolif). The evaluation of this modeling approach will be the subject of forthcoming work.