As described in 5, non-neoplastic HMT-3522 S1 cells were cultured in 3D in the presence of Matrigel™ for up to 12 days to induce acinar morphogenesis. Malignant HMT-3522 T4-2 cells were cultured under similar conditions for a maximum of 11 days to avoid the overgrowth of tumor nodules. DNA was stained with DAPI to visualize the limits of the nuclear volume and NuMA proteins were labeled with Texas red. Three-dimensional images were acquired using a Zeiss 410 confocal laser-scanning microscope with planapochromatic 63×, 1.4 numerical aperture lens. The resulting voxel dimensions of the 3D images were 0.08 × 0.08 μm in the plane of the slide and 0.5 μm along the optical direction.

We used three image datasets to test our phenotype clustering approach. The first dataset contains 2673 non-neoplastic S1 cells taken from 77 confocal images. Images 1–25, 26–45, 46–61, and 62–77 are S1 cells cultured for 12 days, 10 days, 5 days, and 3 days respectively. The second dataset contains 3535 malignant T4-2 cells taken from 44 images. Images 1–14, 15–26, 27–36, and 37–44 are T4-2 cells cultured in 5 days, 10 days, 11 days, and 4 days respectively. The third dependent dataset contains both malignant T4-2 and non-neoplastic S1 cells taken from the direct combination of all the 121 images. The time points were selected to span the growth progression of the non-neoplastic cultured cells. Optical sections from 3D images of individual nuclei, showing representative NuMA staining for each of the phenotypes, are displayed in the Methods section.

Using an automated image analysis method developed earlier 5, we extracted the local bright staining features of NuMA protein and quantified their radial distribution in each nucleus in all the 121 S1 and T4 images. In this way, we obtained 2673 and 3535 LBF distributions for S1 and T4 cells respectively. Each distribution is represented by the normalized density of bright NuMA protein feature as a function of the normalized distance from the perimeter of the nucleus to its center (see Methods for further details).

Using traditional approaches of fuzzy C-means clustering, Gaussian mixture model clustering (with a spherical kernel), K-means, hierarchical clustering (with a complete link scheme), and spectral clustering 67891011121314, we divided the dataset into a number of clusters according to the similarities of their LBF distributions. Figure 1 shows the results for each of these traditional approaches when the dataset of 2673 non-neoplastic S1 cells is divided into 8 clusters. The final result, as we show below, is not dependent on the number of clusters. Each cluster is represented by the centroid (curve) and standard deviation (small vertical bar) of the LBF distributions in the cluster. Clearly, the different methods cluster the data in different ways.

Table 1 shows the consistencies between these clustering results evaluated by pair-wise F-measure (see Methods). The results show that quantitatively the consistencies between the clusters produces from each approach are unsatisfactory. For instance, the F-measures between the hierarchical clustering and the Gaussian mixture model, fuzzy C-means, K-means, and spectral clustering are 0.5205, 0.5270, 0.4543, and 0.5365 respectively (the fourth row in Table 1). The F-measures between the spectral clustering and the Gaussian mixture model, fuzzy C-menas, hierarchical clustering, and K-means are 0.6282, 0.6177, 0.5365, and 0.6253 respectively (the sixth row in Table 1).

As shown in Table 1, different clustering methods may generate different results for the same dataset and the agreement between them can be low. This is because each clustering method assumes certain data distributions and cluster characteristics. For instance, the Gaussian mixture model assumes clusters satisfy the Gaussian distribution. K-means works well for clusters of convex shapes. Thus, some algorithms might perform well for specific datasets and not for others. In general, no single clustering method can successfully handle different types of cluster structure. In addition, even different initializations and parameter settings of the same method, for instance, K-means and Gaussian mixture model, may generate different clustering results. As a result, selecting an optimal clustering method is non-trivial or even impossible in many cases. A reasonable way to get a reliable partition of a dataset is to derive a consensus from multiple clustering results, the assumption being that the judgment made by a committee is more robust and unbiased than those made by individuals. This idea, called ensemble clustering, has been investigated in some literatures and several major benefits have been identified 15161718192021. First, ensemble-clustering can improve the robustness of clustering. The clusters generated tend to be less sensitive to noise, outliers, initialization, or sampling variations compared to individual clustering methods. Second, ensemble clustering does not need a priori information about the number of clusters, but can effectively determine the most probable number of clusters. Third, ensemble clustering can detect outliers. This ability is closely associated with the ability of determining the number of clusters.

Several different ensemble-clustering methods have become available. In 15, a voting algorithm based on hierarchical clustering of the co-association matrix (which represents how often each pair of data appears in the same cluster) is used to derive the consensus clusters. In 16, Strehl and Ghosh developed an evidence accumulation and a hypergraph representation ensemble clustering method. In 17, Topchy et al proposed a mutual-information-based method. In 20, Fischer and Buhmann developed a bootstrap algorithm by first relabeling the data in each clustering result to find the correspondence and then using a voting scheme to find consensus.

In this work, we used a probabilistic ensemble approach based on Bayesian latent variable induction 212223 (see Methods). Assuming that the clustering results generated by individual methods, i.e., Gaussian mixture model, fuzzy C-means, K-Means, hierarchical clustering, and spectral clustering, are independent of each other, the Bayesian latent variable induction method is able to obtain the statistically optimal combination of individual clustering results as shown by Chickering and Heckerman in 21. A similar probabilistic ensemble approach has also been adopted by Topchy in 18 where accurate consensus was obtained from unreliable individual clustering results.

Using the probabilistic ensemble clustering approach (see Methods for detail), we derived the statistically optimal consensus from different data partition results generated by the five traditional clustering methods mentioned above. Figure 2 shows the result of combining the clusters generated by the five traditional approaches as shown in Figure 1 using the probabilistic ensemble approach. The number of clusters, 16, is automatically determined as a result of finding the consensus.

Table 2 further shows the comparison of our method with traditional methods in terms of the number of clusters predefined in individual clustering methods (the second row) and those automatically determined by the probabilistic ensemble clustering approach (the third row) for the dataset containing both S1 and T4-2 cells. Clearly, the number of clusters automatically determined by the probabilistic ensemble approach does not vary significantly with the number of clusters predefined for individual clustering methods. When the number of clusters predefined changes from 8 to 26, the number of clusters identified by the probabilistic ensemble clustering approach is much more stable, ranging from 19 to 25.

With clusters reliably determined, we then calculated the number of LBF distributions falling into each cluster for each of the 8 populations of cells, i.e., non-neoplastic S1 cells cultured for 3 days, 5 days, 10 days, and 12 days, as well as malignant T4-2 cells cultured for 4 days, 5 days, 10 days, and 11 days. By doing so, we obtained a cluster histogram for each of the 8 populations of cells. Figure 3a shows the 20 clusters automatically determined by combining the clustering results of Gaussian mixture model, fuzzy C-means, hierarchical clustering, K-means, and spectral clustering using the probabilistic ensemble clustering for the dataset containing 2673 non-neoplastic S1 cells and 3535 malignant T4-2 cells. The number of the clusters predefined for these baseline methods is 14 (as shown in Table 2). In fact, the cluster histograms and the phenotype trees built in later step are insensitive to the number of clusters predefined for traditional clustering methods as will be shown in the Methods section. The 20 clusters in Figure 3a are ordered from the left to the right and the top to the bottom according to their peak locations. The first 8 clusters are approximately flat. In the 9^th to the 20^th clusters the peak location shifts from the left to the right. Figure 3b shows the cluster histograms for the 8 populations of cells. For S1 cells, the cluster histograms (the top row in Figure 3b) are remarkably different between the early stage (e.g. S1 Day 3) and the completion of acinar morphogenesis (e.g., S1 Day 12). The peak of the histogram gradually shifts from the left to the right as the number of days in culture increases, indicating a gradual modification during the 12-day in vitro morphogenesis process. This is consistent with the fact that NuMA staining is diffusely distributed within the nuclei of proliferating cells, but aggregates into foci of increasing size as cells arrest proliferation and complete acinar morphogenesis. Therefore, the cluster histograms statistically reflect the phenotype of non-neoplastic S1 cells. Moreover, the peak of the histogram profile does not change significantly for malignant T4-2 cells cultured for different numbers of days (bottom row in Figure 3b). This is also consistent with the fact that NuMA staining is diffusely distributed within T4-2 nuclei despite the number of days in culture. Interestingly, the cluster histograms of malignant T4-2 cells differ significantly from those of non-neoplastic S1 cells. The consistency of cluster histograms and cell types indicates that it is meaningful to develop a method to predict cell phenotypes and their sub-categories based on cluster histograms.

Using the approach introduced in the Methods section, we have constructed phenotype trees to show how the phenotypes, defined by the behavior of the cells in 3D culture, can be hierarchically grouped and the statistical significance of each grouping calculated. Figure 4a shows the phenotype tree built for non-neoplastic S1 cells. At the first level in this figure, the four phenotypes of S1 cells were divided into two groups. Of the multiple ways to create two groups from four phenotypes, our method found that having S1 cells at day 12 and day 10 in one group and S1 cells at day 3 and day 5 in the other resulted in the highest confidence value, of 0.9286 (Figure 4a). In the second level of the tree, our method divided S1 cells into three phenotype groups. The results showed that having S1 cells at day 12 and day 10 as one group, S1 cells at day 5 as the second group, and S1 cells at day 3 as the third provided the highest confidence value of 0.8511. This was lower than the confidence of dividing S1 cells into two groups. Finally, the method divided S1 cells into four groups which resulted in a confidence value of 0.6822 (Figure 4a). This phenotype tree indicates we can distinguish S1 cells at day 3 and 5 from those cultured at day 10 and 12 days with high confidence.

Using the same approach, we constructed the phenotype trees for malignant T4-2 cells and for the combination of S1 and T4-2 cells, as shown in Figure 4b and Figure 4c respectively. Figure 4b shows that we can distinguish T4-2 cells cultured at day 4, day 5, day 10 from those cultured at day 11 in relatively high confidence (0.8591; the first level of Figure 4b). However, if we want to distinguish T4-2 cells cultured for different numbers of days, the confidence drops to 0.5748. Figure 4c shows that we can distinguish S1 and T4-2 cells with very high confidence (0.9419; see the first level of Figure 4c). However, the confidence drops as level increases. The certainty in distinguishing all the 8 phenotypes drops to 0.5508 at the highest level of the tree. In general, the phenotype trees provide us a way to evaluate how the phenotypes, defined by the behavior of the cells in 3D culture, can be hierarchically grouped and the statistical significance between each grouping calculated.

Using Zeiss 410 confocal laser-scanning microscope with planapochromatic 63×, 1.4 numerical aperture lens, we acquired hundreds of 3D images of non-neoplastic S1 and malignant T4-2 cells cultured for up to 12 days. Figure 7 shows optical sections from the middle of 3D images of individual nuclei, showing representative NuMA staining for each of the phenotypes described in this work.

In an earlier study, an image analysis method was developed to extract the local bright staining features of NuMA protein and quantify their radial distribution in each individual nucleus (5, also see Figure 8). The technique first used a model-based method to automatically segment individual nuclei in the DAPI-stained channel of the confocal images. It then divided the brightness at each point within a nucleus by the local average brightness in a region surrounding that point in the NuMA-stained channel, thus isolating the local brightness features (LBF) of each nucleus. Then, the radial distribution of these bright features was computed using a distance transform. The transform calculates the shortest distance of each point within a nucleus to the nuclear boundary and in doing so, divides each nucleus into a set of concentric terraces of equal thickness. In each terrace, the density of local bright features was calculated as the number of bright pixels divided by the total number of pixels. To account for variations in the number of terraces per nucleus due to variations in nucleus size and shape, the density per terrace was normalized so that the average density of bright features was 1 for each nucleus, and the distances from nuclear perimeter were also normalized to the range of [0, 1.0]. Through the above process, a radial distribution of LBF was derived for each nucleus, represented by the normalized density of bright features as a function of the normalized distance from the perimeter of the nucleus to its center.

Our phenotype clustering algorithm is based on the radial distribution of LBFs. To group the LBF distribution of thousands of nuclei into clusters of similar patterns, we first tested traditional clustering approaches, including the most widely used K-means, fuzzy C-means clustering, Gaussian mixture model (with a spherical kernel), hierarchical clustering (with the complete link scheme), and the spectral clustering methods 67891011121314.

Since different clustering methods generate different clusters, we computed the pair-wise F-measure score to evaluate the consistencies between different clustering results. The F-measure is defined as follows. For any two data partition U and V, denote the ith cluster in partition U as u_i, and the jth cluster in partition V as v_j. The proportion of data in u_i that is also in v_j is R = |u_i ⋂ v_j|/|u_i|, and the portion of data in v_j that is also in u_i is P = |u_i ⋂ v_j|/|v_j|. Define F(i, j) = 2PR/(P+R). The score to measure the consistency of the partition V with partition U is F₀ = [Σ|u_i|max_jF(i, j)]/[Σ|u_i|], where |u_i| is the number of data point in u_i. To make it symmetrical, the final F-measure is defined as F = (F₀+F₀')/2, where F₀' denotes the transpose of F₀.

The probabilistic ensemble clustering approach we used to derive the consensus clusters from multiple clustering results is based on general Bayesian latent variable induction 212223. Let us suppose we have M different clustering approaches, generating M data partition C_i (i = 0,..., M) of the same dataset D containing N data points. Our purpose is to infer the optimal consensus data partition L from the multiple partitions C_i. We notice that one simple yet reasonable assumption is that we can treat all the M clustering results C₁,..., C_M as independent samples drawn from the same underlying distribution L. In another words, we can assume that the distributions of C₁,..., C_M are conditionally independent of each other given the latent variable L. This assumption allows us consider the following Bayesian latent variable induction model.

Let us suppose the ith clustering approach divides the dataset into r_i clusters, then each C_i has r_i states (categorical labels), i.e., 1,..., r_i. Initially the consensus L may divide the dataset into k clusters (the final value k* is automatically determined; see below), then L has k states, i.e., 1,..., k. Since each LBF distribution vector in the dataset is assigned a cluster label by C_i, it takes a specific state value on C_i. Denote s = (C₁ = c₁, C₂ = c₂,...., C_M = c_M), where c_i (i ∈ [0, M]) takes one state in 1,..., r_i.

Upon initialization of the latent variable L, we randomly assign each of the N data points one of the k states. Given a data s which is assigned state label c_i by the ith clustering method C_i, we derive its probability of taking state label l (where l ∈ [1, k]) in consensus L, i.e., P(L = l|s). Based on the conditional independence assumption, we have

P ( L ( j ) = l | s ( j ) ) ∝ P ( s ( j ) | L ( j ) = l ) = ∏ i = 1 M P ( C i ( j ) = c i | L ( j ) = l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6A52@

where j denotes the jth data in the dataset D, P(C_i = c_i|L = l) (i ∈ [0, M]) can be easily obtained by counting and normalizing the occurrence frequency of data that are assigned the state label c_i by the clustering method C_i, given the data is assigned the state label l in L. Once P(L = l|s) is available, we use it to resample and update the state label of each data in L. The above process repeats until all the data do not change states. This will lead to the estimation of an optimal consensus function L for a specified number of clusters, k.

We observe that when the data samples (LBFs) are independent of each other, the likelihood of the latent variable L which has k states can be estimated as

P ( k | D ) = ∏ j = 1 N P ( k | s ( j ) ) = ∏ j = 1 N ∑ l = 1 k P ( k , L ( j ) = l | s ( j ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6663@

It is apparent that we can maximize the likelihood in Eq. (2) to find the best k over a specified range. In practice, we can often avoid iteration in Eq. (2) by directly assigning a big k. After convergence in solving Eq. (1), there are k* (k ≥ k*) states in L that have non-zero number of data points. This k* value is the statistically optimal k value automatically determined.

Once we obtained reliable clusters of LBF distributions of individual nuclei, we analyzed how the cells belonging to different phenotypes, defined by the behavior of the cells, (i.e., S1 and T4-2 cells cultured in different days) were distributed across the various LBF clusters. For this purpose, we counted the number of nuclei whose LBF distribution fell into each cluster for each phenotype, i.e., S1 cells cultured for 3, 5, 10, and 12 days, and T4-2 cells cultured for 4, 5, 11, and 12 days. By doing so, we obtained the cluster histogram of each phenotype, represented by the percentile of nuclei as a function of clusters. The cluster histograms do not only directly link to predefined phenotypes (as shown in Figure 3) but also provided more detail information compared to cell malignancy and days in culture.

Taking the non-neoplastic S1 cells cultured for different days as an example, our method in constructing the tree is as follows. For all the N images of S1 cells, we assume images of the same day are of the same phenotype and morphogenesis progresses montotonically, as defined by biologists. This allowed us to group the images sequentially, leading to ∑i=1P−1CP−1i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdoeadnaaDaaaleaacqWGqbaucqGHsislcqaIXaqmaeaacqWGPbqAaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemiuaaLaeyOeI0IaeGymaedaniabggHiLdaaaa@3A97@ possible ways of grouping the different phenotypes, where C denotes the combination operation and P is the number of defined cell phenotypes. For instance, if P = 4, then the total number of possible ways of grouping phenotypes is 7 (i.e., ∑i=13C3i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdoeadnaaDaaaleaacqaIZaWmaeaacqWGPbqAaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaeG4mamdaniabggHiLdaaaa@3673@). Among these 7 cases, 3 cases (i.e., C31 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqdaqhaaWcbaGaeG4mamdabaGaeGymaedaaaaa@2FCC@) correspond to grouping the four macroscopically defined phenotypes into 2 groups, 3 cases (i.e., C32 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqdaqhaaWcbaGaeG4mamdabaGaeGOmaidaaaaa@2FCE@) correspond to grouping them into 3 groups, and 1 case (i.e., C33 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqdaqhaaWcbaGaeG4mamdabaGaeG4mamdaaaaa@2FD0@) corresponds to grouping them into 4 groups. These 7 cases are shown in Figure 9a. Different colors in each row represent different groups. The first three bins correspond to dividing the S1 cells cultured for 3 days, 5 days, 10 days and 12 days into 2 groups, the next three bins correspond to dividing the cells into 3 groups, and the 7^th bin corresponds to dividing the cells into 4 groups.

Our next step is to determine the likelihood of these potential groupings. Assume we want to divide the predefined phenotypes into p groups (where p = 2,3,4 in the above example). We then grouped the cluster histogram of the 77 S1 cell images into the same number of clusters. To improve reliability we again used multiple clustering algorithms, including K-means, fuzzy C-means clustering, hierarchical clustering, Gaussian Mixture model, and spectral clustering, as used in generating the LBF clusters (see Figure 9b). We then paired each clustering result with the phenotype grouping under consideration, and calculated the degree of agreement between them using the F-measure. We then selected the maximum F-score as the confidence of the corresponding cell phenotype grouping (see Figure 9c). By repeating the process for each potential phenotype grouping, we finally obtained the value of the confidence as the function of the different cases of phenotype grouping.

To further test the sensitivity of this method to the number of clusters predefined when generating the clusters of LBF distributions using the five traditional clustering approaches, we repeated the process for different numbers of clusters predefined for the traditional methods and obtained a set of confidence values for each phenotype grouping case as indicated by the colored dots in each bin of Figure 9d. The result exhibits a central tendency, indicating that the method is insensitive to the number of clusters predefined in clustering the LBF distributions. We then took the median of the confidence values obtained under different number of clusters on each bin as the overall confidence value of the corresponding phenotype grouping.

Given p, the number of groups that the predefined phenotype should be grouped into, we selected from all the phenotype grouping cases that have the same number of groups the one that has the maximum confidence value, as the most likely phenotype grouping case under the given p. For instance, if we want to group the predefined phenotypes into 2 groups, i.e., p = 2, there are three phenotype grouping cases, corresponding to the first three bins in Figure 9d and the first three rows in Figure 9a. The second case has the maximum confidence value (indicated by the left-most dashed ellipse in Figure 9d, which corresponds to the second row of Figure 9a) and is thus taken as the right way of grouping the predefined phenotypes into 2 groups. This means that S1 cells cultured for 10 and 12 days (i.e., images 1–45) belong to one group, and those cultured for 3 and 5 days belong to another (i.e., images 46–77). Using this approach, we determined the most likely phenotype grouping for p = 3 and p = 4, which correspond to the 6^th and 7^th bin in Figure 9d and the 6^th and 7^th row in Figure 9a respectively. These three phenotype groupings constitute the first to the third level of the phenotype tree as shown in Figure 4a.