- Methodology article
- Open Access
Using iterative cluster merging with improved gap statistics to perform online phenotype discovery in the context of high-throughput RNAi screens
- Zheng Yin†^{1, 3},
- Xiaobo Zhou†^{1},
- Chris Bakal†^{2},
- Fuhai Li^{1},
- Youxian Sun^{3},
- Norbert Perrimon^{2} and
- Stephen TC Wong^{1}Email author
https://doi.org/10.1186/1471-2105-9-264
© Yin et al; licensee BioMed Central Ltd. 2008
- Received: 30 November 2007
- Accepted: 05 June 2008
- Published: 05 June 2008
Abstract
Background
The recent emergence of high-throughput automated image acquisition technologies has forever changed how cell biologists collect and analyze data. Historically, the interpretation of cellular phenotypes in different experimental conditions has been dependent upon the expert opinions of well-trained biologists. Such qualitative analysis is particularly effective in detecting subtle, but important, deviations in phenotypes. However, while the rapid and continuing development of automated microscope-based technologies now facilitates the acquisition of trillions of cells in thousands of diverse experimental conditions, such as in the context of RNA interference (RNAi) or small-molecule screens, the massive size of these datasets precludes human analysis. Thus, the development of automated methods which aim to identify novel and biological relevant phenotypes online is one of the major challenges in high-throughput image-based screening. Ideally, phenotype discovery methods should be designed to utilize prior/existing information and tackle three challenging tasks, i.e. restoring pre-defined biological meaningful phenotypes, differentiating novel phenotypes from known ones and clarifying novel phenotypes from each other. Arbitrarily extracted information causes biased analysis, while combining the complete existing datasets with each new image is intractable in high-throughput screens.
Results
Here we present the design and implementation of a novel and robust online phenotype discovery method with broad applicability that can be used in diverse experimental contexts, especially high-throughput RNAi screens. This method features phenotype modelling and iterative cluster merging using improved gap statistics. A Gaussian Mixture Model (GMM) is employed to estimate the distribution of each existing phenotype, and then used as reference distribution in gap statistics. This method is broadly applicable to a number of different types of image-based datasets derived from a wide spectrum of experimental conditions and is suitable to adaptively process new images which are continuously added to existing datasets. Validations were carried out on different dataset, including published RNAi screening using Drosophila embryos [Additional files 1, 2], dataset for cell cycle phase identification using HeLa cells [Additional files 1, 3, 4] and synthetic dataset using polygons, our methods tackled three aforementioned tasks effectively with an accuracy range of 85%–90%. When our method is implemented in the context of a Drosophila genome-scale RNAi image-based screening of cultured cells aimed to identifying the contribution of individual genes towards the regulation of cell-shape, it efficiently discovers meaningful new phenotypes and provides novel biological insight. We also propose a two-step procedure to modify the novelty detection method based on one-class SVM, so that it can be used to online phenotype discovery. In different conditions, we compared the SVM based method with our method using various datasets and our methods consistently outperformed SVM based method in at least two of three tasks by 2% to 5%. These results demonstrate that our methods can be used to better identify novel phenotypes in image-based datasets from a wide range of conditions and organisms.
Conclusion
We demonstrate that our method can detect various novel phenotypes effectively in complex datasets. Experiment results also validate that our method performs consistently under different order of image input, variation of starting conditions including the number and composition of existing phenotypes, and dataset from different screens. In our findings, the proposed method is suitable for online phenotype discovery in diverse high-throughput image-based genetic and chemical screens.
Keywords
- Gaussian Mixture Model
- Reference Dataset
- Reference Distribution
- Gradient Vector Field
- Iterative Cluster
Background
Metazoan cells have the ability to adopt an extraordinarily diverse spectrum of cell shapes. For example, the cuboidal, polarized morphology of epithelial cells differs markedly from that of neuronal cells, which extend long, thin, and highly-branched projections. The shape of an individual cell is the result of a complex interplay between the activity of thousands of genes and the cell's environment. Understanding this interplay is a fundamental challenge in developmental and cell biology. Currently, there are two key aspects to deciphering cellular morphogenesis on genome-scale. The first is determining the individual functional contributions of every gene towards the regulation of cell shape, and the second is to describe how complex relationships between cell shape genes affect morphology. With the advent of high-throughput RNA interference (RNAi) screening technologies, particularly in model systems such as Drosophila melanogaster [1], it is now possible to systematically query the involvement of genes in the regulation of different cellular processes and functions. Typically, RNAi-based genetic screens involve the acquisition of relatively low-content, single-dimensional data which is easily analyzed using conventional and unbiased means and thus feasible to perform on genome, or multi-genome scales [1, 2]. In order to facilitate similar analysis of image-based screens, we and other researchers have recently developed novel image segmentation algorithms to rapidly quantitate hundreds of different parameters at a single-cell level in an automated fashion [3–6], and we have demonstrated that such image segmentation algorithms can be used in the context of genetic screens [7]. Notably however, this and other similar screens [8] have been 50–100 fold smaller in scale than typical low-dimensional screens and are not yet genome-scale. The reduced scale of these screens is due, largely in part, to the fact that the expert opinion of cell biologists is still an essential and rate-limiting aspect in the analysis of many image-based datasets. Although human intervention is not required in screens where the potential phenotypic outcomes are few or binary in number (e.g. an image-based screen where a particular marker is determined to be nuclear or non-nuclear), such intervention is currently necessary in order to identify novel/subtle phenotypes in image-based datasets of genetic or chemical perturbations where the dynamic range of cellular phenotypes cannot be predicted before the data is collected. For example, in genome-scale screens for regulators of cell shape, it is impossible to predict a priori the diversity of morphologies that will ultimately be present in the dataset. The failure to accurately measure this phenotypic variation will lead to concomitant classification errors, especially false negatives, and misleading results. Current methodologies usually employ a two-step procedure to maximize the amount of variation that is captured in a particular image-based analysis. First, 100–600 phenotypic features are measured on a single-cell level (automatically but somehow exhaustively), and second, supervised techniques assisted by biologists are used to both reduce dimensionality of feature space and carrying out classification on the images. The biologist has to at least perform preliminary qualitative visual scoring of a small part of the dataset in order to gain a crude assessment of the phenotypic variance that is present in this subset. Unfortunately, it is impossible to perform such analysis in the course of screens where millions of images are acquired, thus the ability of these screens to identify new phenotypes is greatly limited. The issues of defining meaningful phenotypes and describing them using informative feature subsets are closely related. Automated feature space reduction schemes have been implemented in the context of high content screen, including feature extraction methods examined in [9], factor analysis in [10] and SVM-RFE method in [11]. These methods allow more effective modelling of existing phenotypes, and also prompt the necessity of updating informative feature sets so that they can not only model the existing, but also discover the novel.
Cluster analysis is widely used to reveal the structure of unlabeled datasets. Specifically, there are a number of methods that have been developed in order to estimate cluster numbers from a dataset such as using a series of internal indices [12], jump methods [13], and weighted gap statistics [14]. Moreover, supervised approaches to cluster validation such as using re-sampling strategy [15], prediction strength [16], methods based on mixture models and inference of Bayesian factors[17, 18], or strategies which are application-specific [19] have also been previously implemented. Nevertheless, most existing methods are subject to certain hypothesis on a fixed dataset, and cannot be directly used for online phenotype discovery where new images continuously extend the dataset and millions of cells are involved. Improper assumptions on data structure may cause incorrect division or merging of biologically meaningful phenotypes. To avoid this problem, such methods combine each new image with the whole existing dataset (regardless of the large difference in cell numbers) and frequently re-run from the very beginning.
Methods for online phenotype discovery should be sensitive and flexible to various phenotypes and avoid frequent re-modelling involving complete existing datasets. As a kernel machine based novelty detection method, one-class SVM is used for "off-line" phenotype discovery [20]. However, two major points limit its application to high-throughput image-based screens, especially for screens of cell shape regulators. First, in one-class SVM all the test samples are classified into two classes, "novel" and "known", however many high-throughput RNAi datasets may potentially contain multiple diverse and unique novel phenotypes which should not necessarily be grouped together. Subsequent cluster analysis would be needed to identify and model different novel phenotypes following the use of one-class SVM. Second, each time a novel phenotype is discovered using one-class SVM, the support vectors need to be modified so that the newly discovered phenotype are included as "known" in the following loops, otherwise it will continuously be identified as novel in future. As mentioned earlier, in a typical RNAi screen on 1,000–10,000s genes with dozens of images for each RNAi and 100s of cells in each image, such updating would involve millions of cells and is intractable.
Results
Synthetic dataset
Overcoming large sample size difference between two clusters
Difference between sample numbers of distinct clusters could bias cluster number estimation. We propose to tackle this problem by using GMMs as reference distributions for existing phenotypes in gap statistics and validate our method using simulation.
Each simulated dataset consists of observations from two populations ℙ_{1} and ℙ_{2}, each population are sampled from a two dimensional Gaussian distribution with means (0,0) for ℙ_{1} and (0,3) for ℙ_{2}, and an identity covariance for both groups. Gap statistics [12] are used to estimate number of clusters from the experiment dataset ℙ_{1} ⋃ ℙ_{2}. This method uniformly samples different reference datasets from the support of ℙ_{1} ⋃ ℙ_{2}, and here the number of reference datasets is set to 20. Then the experiment dataset ℙ_{1} ⋃ ℙ_{2} and the 20 reference datasets are clustered into candidate cluster numbers k = 1... K, and we set K = 10. For each clustering result, we measure the compactness of obtained clusters using "within cluster dispersion". For each cluster number k, such dispersion are measured separately on experiment dataset and each reference dataset, and gap statistic for k, denoted as gap(k), is defined as the average value of difference between the dispersion on experiment dataset and that on each reference dataset, meanwhile we obtained standard deviation of such difference across 20 reference dataset, and denoted as s_{ k }.
When samples from ℙ_{1} and ℙ_{2} have identical number of 100, gap statistics can correctly judge sample number as 2, then we set sample number from ℙ_{1} as 200, 300, 500, 700, 900 and 1000, and sample number from ℙ_{2} are fixed as 100. Figure 2, left and Figure 2, middle show that when number difference is over six-fold, gap statistics does not work correctly.
A ten-fold difference of cell numbers between existing phenotypes and new images is not the worst situation we would face in online phenotype discovery. Based on the results given in Figure 2, middle, samples from two obviously different populations would be merged together. We propose to solve this bias through fitting a GMM model for existing phenotypes, as well as using GMM as reference distribution for each existing cluster in gap statistics.
Next, we consider ℙ_{1} as existing cluster with known distribution model. If we use Gaussian model for ℙ_{1} as reference dataset for gap statistics, it gives the correct result in 87.4% occasions across 500 experiments even with ten-fold difference in sample number. Figure 2, right shows one gap curve with ten-fold difference, where Gap(1)-(Gap(2)-s_{2}) = -0.0006, meanwhile Gap(2)>Gap(3)-s_{3}, thus, the estimated cluster number is 2 rather than 3 (although Gap(3)>Gap(2)), because data point for k = 2 is higher than the bottom of error bar for k = 3.
Cross validation results on overcoming the bias of cluster size difference. By using distribution models as reference distribution gap statistics can give correct result even under 10-fold difference.
Difference between sample number of ℙ_{1} and ℙ_{2} | Average cluster number estimation accuracy % (Uniform reference distribution) | Average cluster number estimation accuracy % (GMM as reference distribution for ℙ_{1}, uniform reference for ℙ_{2}) |
---|---|---|
Equal | 100 | 100 |
2-fold | 88.5 | 98.1 |
3-fold | 81.8 | 93.3 |
5-fold | 69.2 | 91.0 |
7-fold | <20 | 89.5 |
9-fold | <20 | 88.9 |
10-fold | <15 | 87.4 |
Simulating typical cells using seven types of polygons
2000 polygons from each of seven types were generated and used as training dataset, i.e. the set of existing phenotypes, and another 2000 polygons were generated for each type to as testing dataset. In each experiment, we started from a certain set of existing phenotypes and built GMM from training samples, meanwhile, we iteratively chose two of seven polygon types and selected 100 polygons apiece from the testing dataset to form a synthetic test image, altogether 70 images can be formed for one experiment. Using the model estimated from the training set, we can identify existing and novel type of polygons from these synthetic images, and observe the performance of our method under different number of novel phenotypes and order of image input.
Performance under different sets of existing phenotypes
Figure 4 shows our method as having consistent accuracy around 85% for different polygons under different conditions, and the best performance is seen when the number of existing phenotypes are 3 and 4. When the number of existing phenotype is 6, more false negatives appears as novel samples are merged into existing phenotypes, and thus prompts the importance of cluster validation and more refined multiple hypothesis tests.
Box and whisker plots for performance under different conditions
Performance comparison with SVM based methods
One-class SVM [20] tackles the novelty detection problem of differentiating novel phenotypes from known ones by estimating a distribution from the core structure of existing dataset and model such distribution using a series of support vectors. It then labels each testing samples as "known" or "novel" using the model built upon support vectors. Compared with SVM used in classification, a parameter ν ∈ (0, 1) (denoted as 'Nu' in figures) is involved in one-class SVM. This parameter is used to define the core structure of the existing dataset, and it has two roles, i.e. the asymptotic upper bound of training data which are labelled as outliers and the lower bound of the fraction support vectors in training samples. However, one-class SVM itself cannot be used to handle problems such as the restoration of multiple existing phenotypes. We modify one-class SVM to fit it into the scenario of online phenotype discovery. Each new image is combined with the support vectors trained from the existing samples and novelty detection is carried out using one class SVM with Gaussian kernels of width 0.5 and various parameters ν to define the scale of support vectors and outliers. After novelty detection, each test sample labelled as "known" is subject to multiple linear SVM classifiers (trained from one pair of existing phenotypes) and assigned into one of multiple existing phenotypes according to majority vote among classifiers. We detailed one-class SVM and our modification in [Additional file 1].
Cell culture, image segmentation, morphological feature extraction and selection
Cell culture and image acquisition
As a next step, we implemented our methods in the context of a high-throughput image-based screen. In particular, we focused on a novel dataset of images acquired in the course of a genome-scale RNAi screen for regulators of Drosophila Kc167 cell shape that have hemocyte-like properties (Bakal et al, unpublished). By using dsRNA to target and inhibit the activity of specific genes/proteins, the role of individual genes in regulating morphology can be systematically determined. Briefly, Kc167 cells are bathed in the presence of individual dsRNAs targeting all known Drosophila protein kinases and phosphatases in 384-well plates (detailed protocols are available at [22]). Following a 5-day incubation period, the cells are fixed and stained with reagents in order to visualize the nuclear DNA (blue channel in all images), polymerized F-actin (green), and α-tubulin (red). For each well, sixteen images from each of the three channels (blue, green and red) were acquired in an automated fashion using an Evotec spinning-disk confocal with a 60× water objective. Auto-focusing is performed in a two-step fashion by first focusing on the bottom well at each individual site, and then moving the objective by the same Z-distance (in this case 3 μ m above the bottom of the well) at each site. The images were captured at a binning of 2 and have a resolution of 661*481 pixels.
Image segmentation
To analyze the morphology of single cells, it is necessary to first delineate the boundaries of individual cells. Direct segmentation of the cell bodies in the F-actin and α-tubulin channels is difficult due to the complex morphology of cellular boundaries. Segmentation of nuclei in the DNA channel is relatively easier, and its segmentation results provide the rough position information of the cell bodies. Herein, we utilize a two-step segmentation procedure [5, 23] including nuclei segmentation on DNA channel, and cell body segmentation of images derived by combining images from the DNA, F-actin and α-tubulin channels.
In nuclei segmentation, the nuclei are first separated from the background by using a background correction based adaptive thresholding method [24]. However, the clustered nuclei cannot be separated by the adaptive thresholding method. To separate the clustered nuclei, the centres of the nuclei are first detected using a gradient vector field (GVF) based detection method [24]. Specifically, we filter the nuclei image using a Gaussian filter, which suppresses the noise and generate local maxima inside cells, and these local maxima correspond to the nuclei centres. However, there are still some local maxima due to noise. To further eliminate the noisy local maxima, we detect the true cell centres using GVF method. It is a well-known fact that in an electric field, the electric field lines point to the positive electrodes, and the free negative electrons move along the electric field lines and stop at these electrodes. In GVF, the gradient-vector lines also point to the local maxima. Analogous to the electron moving inside the electron field, we put one particle on each detected cell pixel and pushed it along the gradient vector lines. Consequently, these particles stop at these local maxima. Since no or very few particles stop at non-maxima and noisy local maxima, the true cell centres can be identified by choosing the points that have many particles [24]. After the centres of nuclei are detected, the nuclei are segmented using the marker-controlled watershed algorithm.
To use both F-actin and α-tubulin channels information, we combine the two channels' signal as I = (I_{F-actin} + I_{α-tubulin})/2, where I, I_{F-actin} and I_{α-tubulin}denote the combined image, F-actin channel image and I_{α-tubulin}channel image respectively. We then segment the cell bodies using the combined image. First the cell bodies are separated from the background using the aforementioned adaptive thresholding algorithm. The nuclei segmentation results facilitate the segmentation of cell bodies by providing the rough position information of cell bodies. Herein, we employ the marker-controlled watershed and the nuclei segmentation results to segment the individual cell bodies. To reduce the over-segmentation of cell bodies a feedback system proposed in [5] is employed. Three scoring models, which measure the morphological appearance, gradient and edge intensity of cell pairs respectively, are built to identify the over-segmented cell bodies, and guide the merging procedure [5]. Detailed shape and boundary information of nuclei and cell bodies is obtained after the two-step segmentation procedure.
Morphological feature extraction and feature selection
Cellular phenotype identification depends on choosing a rich set of descriptive features, which is one of the most critical steps for pattern recognition problems. To capture the geometric and appearance properties, 211 morphology features belonging to five categories are extracted following [23]. The selected features include a total of 85 wavelet features (70 of them from Garbor wavelet transformation [25] and 15 features from 3-level CDF97 wavelet transformation [26]), 10 geometric region features describing the shape and texture characteristics of cells [23], 48 Zernike moments features with selected order of 12 [27], 14 Haralick texture features [28] and a total of 54 phenotype shape descriptor features (36 features of ratio length of the central axis projection and 18 features of area distribution over equal sectors) [23]. A feature selection procedure is necessary to de-noise the dataset and describe it in the most informative way. As the datasets and phenotype models are being updated adaptively, an unsupervised feature selection without relying on phenotype labels is used to supply a stable feature subset. It is based on iterative feature elimination using k nearest neighbour features following [29]. In this study, an informative subset of fifteen features is selected to quantify the segmented cells.
Online phenotypes discovery in the context of RNAi high-throughput screenings
Fitting GMM model for existing phenotypes
Case 1: merging cells in existing phenotypes
Cross validation results on merging cells into existing phenotypes
Phenotype | Average merging accuracy and standard deviation % | Typical Mistakes |
---|---|---|
Normal | 98.6 (1.7) | left alone |
LPA | 93.8 (2.9) | Merged into Normal, left alone |
CCA | 92.4 (2.4) | left alone |
Our method can identify and merge cells into original phenotypes well. In the third column of Table 2, we list the typical mistakes made during the merging loops. Some cells with normal and LPA phenotypes are not merged correctly, and such mistakes suggest the existence of previously undefined phenotypes in such images.
Case 2: discovering new phenotypes: cross validation based on known phenotypes
Cross validation results on discovering new phenotypes
Phenotype to be identified | # of cells to be identified | Average accuracy with standard deviation % | Typical Mistakes |
---|---|---|---|
Normal | 1000 | 95.2 (3.5) | Left alone |
LPA | 1000 | 90.3 (3.1) | Merged into Normal, left alone |
CCA | 600 | 89.6 (2.7) | Merged into Rho1 |
Rho1 | 600 | 87.4 (4.2) | Left alone |
Case 2 shows our method's ability of identifying novel phenotypes. We hypothesize that the relatively low accuracy for CCA and Rho phenotypes can be attributed to the small number of samples and incomplete understanding of which phenotypes is the biological representative for the entire treatment class. High classification accuracies for normal cells in both case 1 and case 2 provide strong validation of the ability of our methods to identify wild-type cells. While the overlap of normal and LPA serves as a starting point for novel phenotypes discovery.
Case 3: identifying multiple novel phenotypes from online image input and performance comparison with SVM based methods
In this case, we still used the test dataset in case 2, which included a total of 3,200 cells from four phenotypes. In each group of experiments we started from the models (available from previous step) of two existing phenotypes, and all 3,200 test cells were divided into 32 images with cells from two phenotypes in one image, and all images were input with 50 different orders. Altogether, three groups of experiments were carried out, and in each experiment, normal phenotype were paired with one of the other three phenotypes, to serve as sets of ''existing phenotypes''. Both our method and SVM based methods were used in each experiment, and we can thus validate our method's ability to deal with multiple novel phenotypes well as the performance under different order of image input.
Having multiple phenotypes in a single image is a challenge in the analysis of image based high-throughput screens. Our method successfully tackle such cases and identify multiple phenotypes from online image input, and can therefore provide a better perspective for further quantification of the whole image, en route to the identification for the role of each gene.
Identifying "rl/tear-drop" phenotype in the context of Drosophila genome-scale RNAi screen
We also tested our method using two published dataset from Drosophila RNAi screen [7] and HeLa cell cycle phase detection [ref. s3 in Additional file 1], compared its performance with SVM based methods and validated our method's ability of handling dataset from various organisms. These experiments are described in [Additional file 1], the results on Drosophila dataset from [7] are reported in [Additional file 2] and results on HeLa dataset are reported in [Additional file 3, 4].
All other functions are developed in Matlab 7.0 and ran in PC with Intel^{®} Core™ 2 T7200 2.00 GHz CPU and 2.00 GB of RAM. Starting from four existing phenotypes, the average running time for our method based on improved gap statistics is 1.8 seconds on a group of 100 segmented cells, 10.2 seconds on a group of 600 cells and 19.4 seconds on a group of 1000 cells. Considering the fact that cell number in each image is seldom over 300 in the reported high content screen, our method is suitable for online application.
Discussion
Online identification and validation of novel morphological phenotypes are major challenges in specific high-throughput image-based screens. Manual phenotype labelling of high-throughput image-based data is a laborious and inordinately time-consuming process, while available automatic identification methods usually classify cells into a limited set of predefined phenotypes which may be determined through biased means and will not be updated according to the online image input. As millions of images are now generated during the course of a comprehensive genome-scale screen, new methods are needed to effectively identify novel phenotypes in such massive databases. Here we report the development of an online phenotype discovery method which models existing phenotypes, compares cells in new images with existing phenotype models through cluster analysis, assigns some new cells to existing phenotypes, and finally identifies and validates novel phenotypes online.
GMM is used for modelling existing phenotypes and gap statistics, with GMM as reference distribution for existing phenotype, plays a key role in cluster analysis and merging. We built GMM for existing phenotypes, sampled datasets from the model and used them as reference distribution in gap statistics method, following this pipeline we can cover the complete properties of phenotypes more efficiently. Furthermore, gap statistics are dealing with only one existing phenotype plus a part of the new image in each merging loop, and the content of new image is iteratively updated with the merging procedure. We present Additional file 5 to validate the idea of modeling existing phenotype using GMM, the detailed information of GMM estimated from four existing phenotype in our real dataset are reported along with histograms for some typical feature.
For analysis of high content screen data, many researchers choose to summarize the information of single cells or objects to supply a normalized signature for higher level concepts (e.g. treatment conditions, genes, complexes, etc). Thus it is critical to identify different phenotypes related to a same treatment condition. Our method can be used in identifying multiple phenotypes in single well and supply detailed insight into related questions. The performance of our methods relies on the quality of image processing, feature selection, phenotype modelling, and cluster analysis methods. Using iterative cluster merging, our future goals are to build more reliable phenotype models and to construct complete pipelines of cluster analysis with detailed validation procedures to obtain more reliable definition of clusters.
Conclusion
We propose an online phenotype discovery method for high-throughput RNAi screen, which can be used in the course of many image-based screens. This method is based on adaptive phenotype modelling and iterative cluster merging using improved gap statistics. Given datasets for existing phenotypes, the method can build a model of each existing phenotype, identify novel phenotypes in images obtained from ongoing screening and assign newly obtained cell images into different phenotypes. Compared with traditional novelty detection techniques, our approach avoids frequent re-modelling involving the huge existing dataset and can handle multiple existing phenotypes in a flexible manner. Implementation of our methods in the analysis of images acquired during a genetic screen for regulators of Drosophila cell morphology demonstrates the power of these computational tools in efficiently discovering meaningful new phenotypes.
Methods
Online cluster discovery: problem formulation
and the total number of existing cells is $u={\displaystyle \sum _{m=1}^{{K}_{0}}{u}_{m}}$. Similarly, when a new image $\mathbb{E}$ is obtained, the i-th cell in this image is also described using p features, and denoted by e_{ i }≜ [e_{i 1}, e_{i 2}... e_{ ip }], and $\mathbb{E}={\left\{{e}_{i}\right\}}_{i=1}^{v}$, where ν is the number of cells in $\mathbb{E}$. New images are continuously obtained, and each new image $\mathbb{E}$ contains tens of cells while there are thousands cells for each $\mathbb{S}$_{ m }, thus v <<u_{ m }<u.
Given a new image $\mathbb{E}$, we need to adaptively determine number of new phenotypes K_{ new }, based on K_{0}, $\mathbb{S}$ and $\mathbb{E}$. Cells in $\mathbb{E}$ while belonging to some existing phenotype $\mathbb{S}$_{ m }should be identified, and used to update model for $\mathbb{S}$_{ m }. It is unfeasible to involve every single cell in $\mathbb{S}$ into cluster discovery, because the large scale of $\mathbb{S}$ could bias cluster analysis towards existing phenotypes and add computation burden. On the other hand, "new cluster" identified only according to $\mathbb{E}$ is vulnerable to outliers. Thus an efficient method to utilize $\mathbb{S}$ is necessary.
Outline of the proposed approach
- (1)
Phenotype modelling. A GMM is fit to each existing phenotypes using Expectation-Maximization (EM) algorithm following [21].
- (2)Sampling existing phenotype and combining existing information with the new image. We sample from the GMM of one existing clusters, say $\mathbb{S}$_{ m }, m ∈ {1, 2... K_{0}}, get the sample set ${\mathbb{S}}_{m}^{\text{'}}$, and put ${\mathbb{S}}_{m}^{\text{'}}$ together with new image $\mathbb{E}$, we denote this combined set as $\mathbb{F}$, thus$\mathbb{F}={\mathbb{S}}_{m}^{\text{'}}\cup \mathbb{E}$(2)
- (3)
Estimating the cluster number in $\mathbb{F}$. An improved gap statistics method is used in which we take reference dataset from the range of feature values of ${\mathbb{S}}_{m}^{\text{'}}$ and $\mathbb{E}$ separately, and use GMM as the reference distribution for reference samples obtained from the support of ${\mathbb{S}}_{m}^{\text{'}}$.
- (4)
Defining clusters on $\mathbb{F}$. Based on the estimated cluster number from step 3, a partition of $\mathbb{F}$ is obtained using Partitioning Around Medoids (PAM) [31] method.
- (5)
Merging samples from $\mathbb{E}$ to existing phenotypes.
- (a)
If some samples from $\mathbb{E}$ are assigned to a same cluster as at least 95% samples from ${\mathbb{S}}_{m}^{\text{'}}$, they are considered as a candidate for merging.
- (b)
Validate merging operation using a statistical test with Bonferroni correction. For each merging candidate, calculate its p value under the GMM for $\mathbb{S}$_{ m }, reject the merging operation and keep this sample in $\mathbb{E}$ if p value is smaller than 1/K_{0}, or else merge this candidate into $\mathbb{S}$_{ m }and delete it from $\mathbb{E}$.
- (6)
Returning to step 2 to sample another existing phenotype and start new merging loop.
- (7)
Updating phenotype models. After each existing phenotype $\mathbb{S}$_{ m }, m ∈ {1, 2... K_{0}}merges their counterparts in $\mathbb{E}$, define clusters left in $\mathbb{E}$ as new phenotypes and estimate GMM for them.
Through modelling and re-sampling, $\mathbb{S}$ becomes more flexible and re-useable, allowing us to cover complete properties of phenotypes. Following data modelling and sampling, the information from existing phenotypes is combined with new image one by one in the loops from step 3 to 5. Thus in each single loop, the task of estimating cluster number is simplified to identifying difference between new image $\mathbb{E}$ and only one existing phenotype. After each $\mathbb{S}$_{ m }merges its counterpart in $\mathbb{E}$, clusters left in $\mathbb{E}$ are identified as new phenotypes.
Cluster modelling and sampling
where N denotes Gaussian distribution. We denote the number of Gaussian terms for phenotype $\mathbb{S}$_{ m }as Q_{ m }and define parameters for $\mathbb{S}$_{ m }as ${\pi}_{m}={\left\{{\pi}_{m,t}\right\}}_{t=1}^{{Q}_{m}},{\mu}_{m}={\left\{{\mu}_{m,t}\right\}}_{t=1}^{{Q}_{m}},{\Sigma}_{m}={\left\{{\Sigma}_{m,t}\right\}}_{t=1}^{{Q}_{m}}$. Initially, the covariance matrix Σ_{m, t}is set to be diagonal. We use Expectation-maximization (EM) algorithm to estimate {π_{ m }, μ_{ m }, Σ_{ m }} from $\mathbb{S}$_{ m }. In the initialization of EM algorithm, Q_{ m }is set to four, and $\mathbb{S}$_{ m }is first partitioned into Q_{ m }clusters using fuzzy C-means clustering method, and then initial parameters are estimated using the standard vector quantization method. For each class, Q_{ m }is reduced to the minimum possible using minimum description length (MDL) technique, following [21]. We obtain random samples from the GMM to form set ${\mathbb{S}}_{m}^{\text{'}}$ having i.i.d $\mathbb{S}$_{ m }, ${\mathbb{S}}_{m}^{\text{'}}$, and ${\mathbb{S}}_{m}^{\text{'}}$ is combined with new image $\mathbb{E}$ to form $\mathbb{F}$. When estimating cluster number in $\mathbb{F}$ using gap statistics, GMM is used as reference distribution for ${\mathbb{S}}_{m}^{\text{'}}$.
Estimating cluster numbers using improved gap statistics
To estimate the number of clusters from an unlabeled dataset, many existing methods focus on the within cluster dispersion W_{ k }, resulting from clustering datasets (e.g. $\mathbb{F}$) into k clusters, C_{1}, C_{2},... C_{ k }with C_{ r }denoting the indices of samples in clusters r and f_{i, j}denotes the value of j-th feature measured from i-th data point. Based on ${D}_{r}={\displaystyle \sum _{i,{i}^{\text{'}}\in {C}_{r}}{\displaystyle \sum _{j=1}^{p}{({f}_{i,j}-{f}_{{i}^{\text{'}},j})}^{2}}}$, we have ${W}_{k}={\displaystyle \sum _{r=1}^{k}\frac{1}{2{n}_{r}}{D}_{r}}$. W_{ k }tends to decrease monotonically as the number of clusters k increases, but from some k on, such decrease flattens markedly. Statistical folklore has it that error measure based on W_{ k }should have an "elbow" at the desirable cluster number, thus different criterions based on W_{ k }are defined.
Gap statistics [12] method utilizes the output of any clustering algorithm under different k, compares the change of W_{ k }to the dispersion expected under a reference null distribution, the gap between the logarithms of these two dispersions are employed to detect cluster number. Gap statistics can detect homogenous non-clustered data against the alternative of clustered data [14]. This ability is critical when all cells in new image $\mathbb{E}$ belong to the same phenotype.
The reference distribution is a null model of data structure. In [12], reference datasets are sampled uniformly either from the range of observed values for each feature, or the range of a box aligned with the principle components of data. However, it is encouraged to estimate reference distribution from existing samples rather than simply using uniform distribution because the bounding box of the whole dataset always includes some "blank" area. The existing cluster definition can help us focus on where the data really lies, and avoid generating a reference dataset violating the properties of original one. Here, we sampled ${\mathbb{S}}_{m}^{\text{'}}$ from GMM of each existing phenotype and it makes sense to use this GMM as the reference distribution for ${\mathbb{S}}_{m}^{\text{'}}$.
We discuss more detail of this strategy in the [Additional file 1], and supply a figure as [Additional file 6] to illustrate the motivation and innovation of our strategy.
Gap statistics method repeatedly carries out clustering using a set of candidate cluster numbers, and pick up the number supplying best within cluster dispersion as estimated cluster number. GMM is an accurate model for ${\mathbb{S}}_{m}^{\text{'}}$, and using GMM as reference distribution can avoid the risk of split biological meaningful clusters and retain biological properties of existing phenotypes. The selection of K and B controls the number of clustering operation to be carried out and greatly influences the complexity of the whole methods. An effective way of utilizing existing phenotypes can greatly reduce K and make gap statistics method more suitable for online phenotype discovery in high-throughput image-based screens.
Cluster definition and merging
We use Partitioning Around Medoids (PAM; also known as K-medoids) [31] to do clustering on combined set $\mathbb{F}$. PAM provides better flexibility and robustness of choosing suitable dissimilarity measurements for different applications [32] and more efficient compared to Fuzzy clustering methods, especially in our cases where clustering are carried out frequently.
after all merging loops, clusters left in $\mathbb{E}$ are defined as new phenotypes.
- (1)
Merging operations only happens when some cells in $\mathbb{E}$ are assigned into $\mathbb{F}$_{1}, together with more than 95% of samples ${\mathbb{S}}_{m}^{\text{'}}$. And such cells are considered candidates for merging operation.
- (2)
For each merging candidate in $\mathbb{E}$, we carry out statistical test with Bonferroni correction. We calculate the p value for each candidate with respect to the GMM for ${\mathbb{S}}_{m}^{\text{'}}$, i.e. possibility of obtaining a value at least as extreme as (if not more) this candidate under the GMM. The corrected p value for each candidate is defined as its p value with respect to the GMM of ${\mathbb{S}}_{m}^{\text{'}}$ divided by number of existing phenotypes K_{0}. If the corrected p value is lower than 0.05/K_{0}, the merging operation is rejected and we keep that candidate in $\mathbb{E}$.
In the merging loops, we focused on identifying samples of existing phenotypes from $\mathbb{E}$. While some cells of $\mathbb{E}$ are merged into existing phenotypes, novel clusters gradually stand out.
Notes
Declarations
Acknowledgements
The authors would like to thank research members of The Methodist Hospital Research Institute, Radiology Department, and we thank colleagues in the Department of Genetics at Harvard Medical School, for technical support and comments. CB is a Fellow of the Leukemia and Lymphoma Society. NP is an Investigator of the Howard Hughes Medical Institute. The research is funded by The Methodist Hospital Research Institute (Stephen T.C. Wong).
Authors’ Affiliations
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