GND-PCA Method for Identification of Gene Functions Involved in Asymmetric Division of C. elegans

Abstract

Due to the rapid development of imaging technology, a large number of biological images have been obtained with three-dimensional (3D) spatial information, time information, and spectral information. Compared with the case of two-dimensional images, the framework for analyzing multidimensional bioimages has not been completely established yet. WDDD is an open biological image database. It dynamically records 3D developmental images of 186 samples of nematodes C. elegans. In this study, based on WDDD, we constructed a framework to analyze the multidimensional dataset, which includes image segmentation, image registration, size registration by the length of main axes, time registration by extracting key time points, and finally, using generalized N-dimensional principal component analysis (GND-PCA) to analyze the phenotypes of bioimages. As a data-driven technique, GND-PCA can automatically extract the important factors involved in the development of P1 and AB in C. elegans. A 3D bioimage can be regarded as a third-order tensor. Therefore, GND-PCA was applied to the set of third-order tensors, and a set of third-order tensor bases was iteratively learned to linearly approximate the set. For each tensor base, a corresponding characteristic image is built to reveal its geometric meaning. The results show that different bases can be used to express different vital factors in development, such as the asymmetric division within the two-cell stage of C. elegans. Based on selected bases, statistical models were built by 50 wild-type (WT) embryos in WDDD, and were applied to RNA interference (RNAi) embryos. The results of statistical testing demonstrated the effectiveness of this method.

1. Introduction

Gene function analysis plays a more and more important role in modern society. Many studies on large-scale, data-based gene function analysis have been carried out and achieved fruitful results [1,2,3,4]. Among them, research based on differential interference contrast (DIC) imaging and RNA interference (RNAi) techniques have become prevalent and have gained extensive applications [5,6]. In [7], Narayanaswamy R. et al. developed spotted cell microarrays for measuring cellular phenotypes on a large scale to identify genes involved in the response of yeast to mating pheromones. In [8], Hamahashi et al. presented a system that automatically measures the cell division pattern of Caenorhabditis elegans (C. elegans). They also statistically analyzed two inferences on the spatial arrangement of cells and the time span between the end of the four-cell stage and the beginning of the eight-cell stage of C. elegans for both par-1 mutation embryos and wild-type (WT) embryos. In [9], Yang et al. Applied PCA-based gene function analysis method to identify the concrete functions of involved genes in the asymmetric division of C. elegans. However, although many works effectively identified gene functions based on image data and statistical testing, they usually applied manually selected features. Furthermore, to our knowledge, it is still required to build a complete framework that includes necessary and important processing techniques when processing a multidimensional dataset.

In this study, we propose a data-driven scheme to automatically identify important factors hidden in multidimensional datasets. It includes necessary and important preprocessing steps, such as image segmentation, image registration, size registration, and time registration. After that, generalized N-dimensional principal component analysis (GND-PCA) [10] is applied to extract features of the processed image set. GND-PCA can be viewed as an extension of PCA in multidimensional cases. It can linearly approximate a multidimensional tensor by a set of tensor bases, and each tensor base reflects a certain feature of the image set. At last, these bases are selected as features to build statistical models to screen RNAi embryos in WDDD.

The advantages of this approach are as follows: first, the identification of important factors is data-driven and completed automatically, without the requirement of experience; second, the scheme includes necessary and important preprocessing steps in handling a multidimensional biological dataset; third, the identification of important factors are objective, and we can sort them by energy just like PCA; and finally, GND-PCA can retain spatial structure in a dataset and overcome the problem of small sample size.

To verify the performance of the proposed method, a task on identifying gene functions for C. elegans is selected. The nematode C. elegans is widely applied to the analysis of gene functions. One of the reasons is that it is one of few animals in which embryonic genes have been surveyed by genome-wide RNAi screening [11]. Another reason is that its cell lineage and cell fate map have also been revealed [12].

In this paper, we focus on the genes involved in controlling the cleavage from P0 (the fertilized egg) to its two daughters, P1 and AB, and using “asymmetric division” indicates this specific process. Usually, for the first two blastomeres of a WT embryo, the anterior blastomere (AB) is larger than the posterior blastomere (P1), and the position of AB indicates the anterior of the ultimately developed mature body. Furthermore, the early asymmetric division provides the location for chemical signals and genetic materials, which ensures the subsequent division and development are carried out under precise control. It is precisely because of the existence of asymmetric division that the development of C. elegans has cellular diversity [13,14]. The asymmetric division also exists in the embryonic development of Drosophila melanogaster and mouse [15,16]. Some genes have been identified to be involved in the process [17,18]. The currently identified genes related to the process are collected in the category of “P1/AB Asynchrony of Divisions” in Phenobank [19].

Because the proposed method requires a large number of WT embryos to carry out the analysis, we use Worm Developmental Dynamics Database (WDDD) as the database. WDDD includes the records of 136 RNAi and 50 WT embryos developing from 1 to 16 cells [20], which makes it a desired platform for our study. One of the merits of WDDD is that there are as many as 50 sets of quantitative data from WT embryos. Therefore, building models based on these data is reliable. Using these models as benchmarks, the abnormality of RNAi embryos recorded in WDDD can be examined.

The content and structure of this paper are as follows: in Section 2, we briefly introduce the basic theory of GND-PCA; Section 3 is the introduction of preprocessing steps and using GND-PCA to extract important factors, including the analysis of their geometric meanings; the statistic testings are presented in Section 4; and Section 5 summarizes the method.

2. Basic Theory of GND-PCA

2.1. Background Knowledge

PCA has been widely used in many research fields, such as data mining and pattern recognition. Essentially, it is a transformation from the original data space to its subspace where the data can be represented compactly and efficiently. When the dimension of the training vectors is larger than the number of training samples, the generalization performance of PCA will be injured. When dealing with 2D data, the original method of PCA transforms the data into a 1D vector, which makes it hard to handle. Under the influence of [21], Hong H. et al. proposed a method called G2D-PCA [22]. G2D-PCA calculates the bases on both row- and column-mode subspaces, and it can handle 2D data both accurately and efficiently.

With the development of imaging techniques, a large number of three-dimensional biological images are generated for scientific research. However, efficiently handling these datasets in lower spaces and extracting effective features are still difficult problems. A 3D image can be viewed as a third-order tensor; thus, tensor-based theories and techniques are beneficial to deal with the problem. There are many research achievements in this field. In [23], Vasilescu M. et al. proposed interesting TensorFaces to dig out distinctive features in a face image dataset. Soltani S. et al. applied dictionary learning based on tensor analysis to address tomographic reconstruction [24]. For a new work on the analysis of 3D medical images using tensors, see [25]. In [10], authors proposed GND-PCA to handle multidimensional data in a data-driven way. The basic idea of GND-PCA is to find the best approximation of a set of tensors on sub-mode spaces. Compared with the previous methods, GND-PCA focuses not only on the representation of a dataset but also on the analysis of the meaning of bases (or components) and the location of a sample on the set of bases. In the following part of this section, we will briefly review the algorithm of GND-PCA.

2.2. Definitions and Preliminaries

To illustrate the theory, we first briefly introduce the related terms and symbols. In this paper, scalars are denoted by italic letters, i.e., ($a,b,\dots$) or ($A,B,\dots$). Bold lower case letters, i.e., ($a,b,\dots$), are used to represent vectors, and matrices are denoted by bold upper case letters, i.e., ($A,B,\dots$). Higher-order tensors (no less than third-order tensors) are denoted by calligraphic upper-case letters, i.e., ($\mathcal{A},\mathcal{B},\dots$). An Nth-order tensor $\mathcal{A}$ is defined as a multi-array with N indices, where $\mathcal{A}\in {ℝ}^{I_1\times I_2\times \dots \times I_N}$ and $ℝ$ is the field of real numbers. Elements of the tensor $\mathcal{A}$ are denoted as $a_{i_1\dots i_n\dots i_N}$, where $1\le i_n\le I_n$. The order of a tensor is the number of dimensions, also known as ways or modes. For a tensor $\mathcal{A}$, $\left|\right|\mathcal{A}\left|\right|$ is the Frobenius norm of $\mathcal{A}$, which is defined by $\Vert \mathcal{A}\Vert =\sqrt{\langle \mathcal{A}\cdot \mathcal{A}\rangle }$, where $\langle •\rangle$ is the inner product. For two tensors $\mathcal{A},\mathcal{B}\in {ℝ}^{I_1\times I_2\times \dots \times I_N}$, their inner product is defined as $\langle \mathcal{A}\cdot \mathcal{B}\rangle ={\displaystyle {\sum }_{i_1}{\displaystyle {\sum }_{i_2}\dots }{\displaystyle {\sum }_{i_N}{a}_{i_1{i}_2\dots i_N}{b}_{i_1{i}_2\dots i_N}}}$.

An Nth-order tensor $\mathcal{A}$ can be viewed as a set of “mode-n vectors” by varying the nth indices and fixing the other indices. When arranging all the mode-n vectors sequentially along the nth dimension, $\mathcal{A}$ can be unfolded. The unfolded (N − 1)th-order tensor is also called the “mode-n matrix” $A_{(n)}$, $A_{(n)}\in {ℝ}^{I_n\cdot (I_1\times \dots \times I_{n-1}\times I_{n+1}\times \dots \times I_N)}$. In Figure 1, an example of the unfolding process for a third-order tensor is illustrated. For a tensor $\mathcal{A}\in {ℝ}^{I_1\times I_2\times \dots \times I_N}$ and a matrix $U\in {ℝ}^{J_n\times I_N}$, their product is called “mode-n product”, denoted by $\mathcal{A}\times {}_{n}U$; here $\mathcal{A}\times {}_{n}U\in R^{I_1\times I_2\times \dots \times I_{n-1}\times J_n\times I_{n+1}\times \dots \times I_N}$, and $n=1,\dots ,N$. The ($i_1\times i_2\times \dots \times i_{n-1}\times j_n\times i_{n+1}\times \dots \times i_N$) entry of $\mathcal{A}\times {}_{n}U$ is ${\displaystyle {\sum }_{i_n}{a}_{i_1\dots i_{n-1}{i}_n{i}_{n+1}\dots i_N}}{u}_{j_n{i}_n}$.

2.3. GND-PCA

Given a series of the Nth-order tensors with zero-mean ${\mathcal{A}}_i\in {ℝ}^{I_1\times I_2\times \dots \times I_N}$, $i=1,\dots ,M$, and ${\displaystyle {\sum }_{i=1}^M{\mathcal{A}}_i}=0$, we hope to find another series of lower rank—$(J_1,J_2,\dots ,J_N)$ tensors $\stackrel{\wedge }{{\mathcal{A}}_i}$ (the order of $\stackrel{\wedge }{{\mathcal{A}}_i}$ is the same as ${\mathcal{A}}_i$, which means $\stackrel{\wedge }{{\mathcal{A}}_i}\in {ℝ}^{I_1\times I_2\times \dots \times I_N}$) which can most accurately approximate the original tensors, where $J_1<I_1$, …, $J_N<I_N$. The new series of tensors $\stackrel{\wedge }{{\mathcal{A}}_i}$ can be decomposed by N matrices $U^{(n)}\in {ℝ}^{I_n\times J_n}$ and $n=1,\dots ,N$, with orthogonal columns according to the Tucker model [26] (see Equation (1)), where ${\mathcal{B}}_i\in {ℝ}^{J_1\times J_2\times \dots \times J_N}$ is a core tensor.

$$\stackrel{\wedge }{{\mathcal{A}}_i}={\mathcal{B}}_i\times {}_{1}U^{(1)}\times {}_{2}U^{(2)}\times \dots \times {}_{n}U^{(n)}\times \dots \times {}_{N}U^{(N)}$$

(1)

The orthogonal matrices $U^{(n)}$ can be determined by minimizing the cost function shown by Equation (2).

$$S={\displaystyle \sum _{i=1}^M{\Vert {\mathcal{A}}_i-\stackrel{\wedge }{{\mathcal{A}}_i}\Vert }^2}={\displaystyle \sum _{i=1}^M{\Vert {\mathcal{A}}_i-{\mathcal{B}}_i\times {}_{1}U^{(1)}\times {}_{2}U^{(2)}\times \dots \times {}_{N}U^{(N)}\Vert }^2}$$

(2)

In Equation (2), once the N matrices $U^{(n)}$ are known, we have Theorem 1 [10].

Theorem 1.

Given fixed N matrices $U^{(n)}$, the tensor ${\mathcal{B}}_i$ that minimizes the cost function, Equation (2), is given by ${\mathcal{B}}_i={\mathcal{A}}_i\times {\left({}_{1}U^{(1)}\right)}^T\times {\left({}_{2}U^{(2)}\right)}^T\times \dots \times {\left({}_{N}U^{(N)}\right)}^T$, and the minimization of the cost function is equal to the maximization of the cost function $S^{\prime }={\displaystyle {\sum }_{i=1}^M{\mathcal{A}}_i\times {\left({}_{1}U^{(1)}\right)}^T\times {\left({}_{2}U^{(2)}\right)}^T\times \dots \times {\left({}_{N}U^{(N)}\right)}^T}$.

To obtain N matrices, an iterative procedure is designed by alternately fixing $N-1$ matrices to confirm the remaining matrix. Theorem 2 is the mathematical statement of the solving method.

Theorem 2.

Given fixed $N-1$ matrices $U^{(1)}$,…, $U^{(n-1)}$, $U^{(n+1)}$,…, and $U^{(N)}$, maximizing the cost function $S^{\prime }$ is given by the matrix $U^{(n)}$, whose columns are the first $J_n$ leading eigenvectors of the matrix ${\displaystyle {\sum }_{i=1}^M{C}_{i(n)}\cdot C_{i(n)}^T}$; here $C_{i(n)}$ is the mode-n matrix of ${\mathcal{C}}_i={\mathcal{A}}_i\times {({}_{1}U^{(1)})}^T\dots \times {({}_{n-1}U^{(n-1)})}^T\times {({}_{n+1}U^{(n+1)})}^T\times \dots \times {({}_{N}U^{(N)})}^T$.

According to Theorem 1 and Theorem 2, GND-PCA is designed as an iteration algorithm [10] to obtain the N optimal matrices U⁽ⁿ⁾. Because a 3D medical image is treated as a third-order tensor in this study, we use a special case N = 3 to illustrate the solving procedure.

A third-order tensor $y_i\in {ℝ}^{I_1\times I_2\times I_3}$, $i=1,\dots ,K$ can be unfolded into three matrices along each mode-space and represented by $y_{i(1)}\in {ℝ}^{I_1\cdot (I_2\times I_3)}$, $y_{i(2)}\in {ℝ}^{I_2\cdot (I_3\times I_1)}$, and $y_{i(3)}\in {ℝ}^{I_3\cdot (I_1\times I_2)}$. The three unfolding procedures, also called the mode matricization of a third-order tensor, are visualized in Figure 1. For each unfolded matrix, the eigenvector matrix $U^{(n)}=\left[\begin{array}{cccc}{u}_1^{(n)}& u_2^{(n)}& \dots & u_{J_n}^{(n)}\end{array}\right]$, $n=1,2,3$, and $J_n\le I_n$ associated with the first $J_n$ largest eigenvalues can be calculated from the covariance matrix $y_{i(n)}{y}_{i(n)}^T$. For each $U^{(n)}$, the order of the eigenvectors, $u_1^{(n)},u_2^{(n)},\dots ,u_{J_n}^{(n)}$, satisfies the corresponding eigenvalues ${\lambda }_1^{(n)}\ge {\lambda }_2^{(n)}\ge \dots \ge {\lambda }_{J_n}^{(n)}$. The initialization of the optimal set of bases $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, $U_{opt}^{(3)}$ can utilize the above $U^{(n)}$, $n=1,2,3$. Then, an iteration algorithm, which calculates one of $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$, respectively, when setting the other two as fixed, is used to determine the optimal set of matrices by approximating each ${\mathcal{A}}_i$ using $\stackrel{\wedge }{{\mathcal{A}}_i}$ according to Equation (2). The approximation is illustrated in Figure 2. In this way, in the k-th iteration, $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$ will be calculated one by one. When these three pairs of $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$ computed at the k-th iteration and at (k + 1)-th iteration satisfy the given convergence conditions, the procedure is finished. Algorithm 1 summarizes the procedure of GND-PCA.

Algorithm 1: The training algorithm of GND-PCA

IN: a series of Nth-order tensors, ${\mathcal{A}}_i\in {ℝ}^{I_1\times I_2\times \cdots \times I_N}$, $i=1,\dots ,M$. OUT: N Matrices $U^{(n)}\in {ℝ}^{I_n\times J_n}$($J_n<I_n$, $n=1,\dots ,N$) with orthogonal column vectors.   1. Initial values: k = 0 and $U_0^{(n)}$ whose columns are determined as the first $J_n$ leading eigenvectors of the matrices ${\displaystyle {\sum }_{i=1}^M{A}_{i(n)}\cdot A_{i(n)}^T}$.   2. Iterate for k until convergence ($\Vert •\Vert$ is the Frobenius norm) Maximize $S^{\prime }={{\displaystyle {\sum }_{i=1}^M\Vert {\mathcal{C}}_i\times {({}_{1}U^{(1)})}^T\Vert }}^2$, ${\mathcal{C}}_i={\mathcal{A}}_i\times {({}_{2}U^{(2)})}^T\times \dots \times {({}_{N}U^{(N)})}^T$ Solution: $U^{(1)}$ whose columns are the first $J_1$ leading eigenvectors of ${\displaystyle {\sum }_{i=1}^M{C}_{i(1)}\cdot C_{i(1)}^T}$ Set $U_{k+1}^{(1)}=U^{(1)}$. Maximize $S^{\prime }={{\displaystyle {\sum }_{i=1}^M\Vert {\mathcal{C}}_i\times {({}_{2}U^{(2)})}^T\Vert }}^2$, ${\mathcal{C}}_i={\mathcal{A}}_i\times {({}_{1}U^{(1)})}^T\times {({}_{3}U^{(3)})}^T\times \dots \times {({}_{N}U^{(N)})}^T$ Solution: $U^{(2)}$ whose columns are the first $J_2$ leading eigenvectors of ${\displaystyle {\sum }_{i=1}^M{C}_{i(2)}\cdot C_{i(2)}^T}$ Set $U_{k+1}^{(2)}=U^{(2)}$. … Maximize $S^{\prime }={{\displaystyle {\sum }_{i=1}^M\Vert {\mathcal{C}}_i\times {({}_{N}U^{(N)})}^T\Vert }}^2$, ${\mathcal{C}}_i={\mathcal{A}}_i\times {({}_{1}U^{(1)})}^T\times \dots \times {({}_{N-1}U^{(N-1)})}^T$ Solution: $U^{(N)}$ whose columns are the first $J_N$ leading eigenvectors of ${\displaystyle {\sum }_{i=1}^M{C}_{i(N)}\cdot C_{i(N)}^T}$ Set $U_{k+1}^{(N)}=U^{(N)}$. k = k + 1   3. Set $U_{opt}^{(1)}=U_k^{(1)}$, $U_{opt}^{(2)}=U_k^{(2)}$, …, $U_{opt}^{(N)}=U_k^{(N)}$.

3. GND-PCA Procedure

3.1. Data Preprocessing

Before using GND-PCA to compress data, the 3D images will be processed to obtain better performance. The preprocessing steps include the following:

(1)

Embryonic image segmentation based on shape index;

(2)

Embryonic image registration based on AP-axes;

(3)

Embryonic image resampling;

(4)

Time registration.

3.1.1. Image Segmentation Based on Shape Index

Before registering images, an image segmentation method based on shape index is applied to detect embryonic boundaries [27]. Due to the limitations of micro-imaging technology, DIC images are often clear in the middle layers and blurry in the outer layers along the imaging direction. Moreover, the intensity in the background of images varies greatly and diversely. These two factors greatly influence the performance of classic image segmentation methods. Threshold algorithms are classical in image segmentation, and many adaptive or neighborhood-based techniques have been presented to cope with the problem of nonuniform illumination [28,29,30]. Image segmentation methods based on deformable templates [31,32] and local feature extraction [33,34] are also widely applied in biomedical images. In recent years, using deep learning in feature detection and image processing has become a hot topic [35]. Jelli E. et al. applied deep learning to segment cells in 3D images [36]. Greenwald N. F. et al. constructed TissueNet using deep learning techniques to segment tissue types in [37]. Generally, the traditional intensity-based image segmentation methods are greatly affected when the background light intensity varies diversely. On the other hand, image segmentation methods based on feature extraction and deformable templates are ineffective in the case of blurry imaging. Segmentation results from deep learning methods are usually accurate; however, they require labeled samples and intensive computation to reach that.

Shape index (SI) is a descriptor to quantify the pure shape of a 2D surface [38]. The computation of SI is shown in Equation (3); here, κ₁ and κ₂ are two principal curvatures of the surface. The range of SI is [−1, 1], and the different value represents the different curved surface. The basic idea of using SI in image segmentation is to treat local intensity changes as a surface shape, and then use shape index to quantify these surface shapes. Because the scale of intensity change in the background is often greater than the scale of the local intensity change derived from the details of objects in images, and image blurring is essentially a smoothing to the surface of a local intensity change, these two factors have little impact on the calculation of shape index. For cellular images, there are a large number of cytoplasmic microparticles within boundaries, and the shape of their local intensity is spherical. Therefore, using SI to detect is both accurate and robust. The segmentation results on DIC images show that, compared with traditional segmentation methods, the segmentation method based on shape index can achieve more robust results.

$$SI=\frac{2}{\pi }\arctan \frac{{\kappa }_2+{\kappa }_1}{{\kappa }_2-{\kappa }_1}, {\kappa }_1\ge {\kappa }_2.$$

(3)

3.1.2. Image Registration Based on AP-Axes

AP-axes are important in embryonic development because they reflect the direction of the head and tail. After the step of image segmentation, embryonic boundaries are segmented. Then, the ellipsoid-fitting method [39] is used to compute the AP-axes of embryos. The traditional method of computing AP-axes is PCA. Because of the blur existing in DIC imaging, the segmentation on a 3D DIC image is often incomplete along the imaging direction. Therefore, the segmentation result often includes only inner layers. As a result, when using PCA, it makes the computed AP-axes deflect away from the imaging direction. Compared with PCA, the ellipsoid-fitting method can decrease the influence of incomplete boundaries. After computing the AP-axes, the steps of registration are as follows:

(1)

Rotate the AP-axes to the x-axis;

(2)

Rotate the middle point of the two nuclear centers of P1 and AB to the z-axis around the x-axis;

(3)

Set the middle point of the two nuclear centers of P1 and AB to [300, 300, 300] at the registered image.

Figure 3 shows the original images and the registered results of 50 WT embryos. For each 3D image at a certain time point, we only show the slice that includes the center.

3.1.3. Resampling Nuclear Images

Embryos may have different sizes, just like eggs. Some may be larger than others. To make the comparison more identical, we also compute the length of the AP-axes and zoom the corresponding nuclei. After that, we check the spatial span of nuclei along the x, y, and z axes and find that in the y and z axes, the coordinates of nuclei range between [176, 420] and [177, 417], respectively. Therefore, we cut off the redundant parts of images in y and z coordinates (from 1 to 100 and 501 to 600), and the rest of the images are 600 × 400 × 400. In practical computation, images with the size of 600 × 400 × 400 still occupy high memory, and the computation is slow. Therefore, we resample the images to 300 × 200 × 200 to make the computation more efficient.

3.1.4. Time Registration

In WDDD, different embryonic samples exist at different times during the two-cell stage, which makes them difficult to compare. Therefore, we also register the time points for each embryo sample. Sphericity can be used to judge whether a nucleus stays in stable development or it begins to divide. Based on the same idea, we choose three time points, $T_1$, $T_2$, and $T_3$, as the key time nodes for each embryo to express the starting, middle, and ending points.

(1)

$T_1$ is the starting time when the sphericity of both P1 and AB are larger than 0.6;

(2)

$T_2$ is the middle time of $T_1$ and $T_3$;

(3)

$T_3$ is the starting time when the sphericity of either P1 or AB is less than 0.6.

The computation of sphericity is based on [40]. Equation (4) is the formula where $V_{ol}$ is the volume of a nucleus and $S_A$ is the surface area of a nucleus. When the body of an object is a standard sphere, the computed sphericity is 1, which is also the maximum value of sphericity. As the object extends in one direction, it becomes an ellipsoid, and its value of sphericity decreases. Thus, for a long and thin object, the value of its sphericity is close to 0. As we know, when a nucleus divides into two, it will first extend in one direction and then split into two, and at last, the two split parts will become spherical again. Therefore, sphericity is an ideal indicator in the process.

$$S_{phericity}={\pi }^{\frac{1}{3}}\frac{{(6V_{ol})}^{\frac{2}{3}}}{S_A}$$

(4)

3.2. The Procedure of Analyzing Gene Functions Based on GND-PCA

Similar to PCA, before using GND-PCA, we need to decentralize these images. We suppose the original images are $I_1\times I_2\times I_3$, and the compressed volumes are $J_1\times J_2\times J_3$ in discussion. The tensor decomposition model is Tucker decomposition [26] (see Figure 2). Before describing the specific steps of GND-PCA, we explain several key terms.

• Eigenvector Matrix: $U^{(1)}$, $U^{(2)}$, and $U^{(3)}$ are called eigenvector matrices (also called projective matrices) in Tucker decomposition. Their sizes are $I_1\times J_1$, $I_2\times J_2$, and $I_3\times J_3$, respectively. The calculated eigenvector matrices by GND-PCA are marked as $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$;
• Base: The tensors built by the Kronecker product of three columns of $U^{(1)}$, $U^{(2)}$, and $U^{(3)}$ are called bases or base images: $Bas{e}^{i,j,k}=u_i^{(1)}\circ u_j^{(2)}\circ u_k^{(3)}$;
• Core tensor: When using GND-PCA to project images, the built $J_1\times J_2\times J_3$ volumes are also called core tensors. It refers to ${\mathcal{B}}_i$ in Figure 2;
• Coefficient: An entry in a core tensor is called a coefficient, and is marked by $b_{i,j,k}$, $i=1,\dots ,J_1$, $j=1,\dots ,J_2$, and $k=1,\dots ,J_3$. Sometimes we need to refer to a position in the core tensor to distinguish the value (coefficient) and the location in the core tensor; we call the location the element of a core tensor.

3.2.1. Determining the Size of Core Tensors

In PCA, we usually do not use all the eigenvalues and eigenvectors, because most of the energy or information of data is embodied by several leading eigenvalues and the corresponding eigenvectors. Here, leading eigenvalues means these eigenvalues have larger values than other abandoned eigenvalues. Similarly, we usually use the core tensors of GND-PCA and the corresponding base images to represent data. It is easy to see, and a larger size of core tensors retains more information and can rebuild original images more precisely. At the same time, however, it generates more trivial features.

To select an appropriate size of core tensors, we choose a set of sizes in advance based on experience, which includes 6 × 4 × 4, 9 × 6 × 6, 12 × 8 × 8, 15 × 10 × 10, 18 × 12 × 12, 21 × 14 × 14, 24 × 16 × 16, 27 × 18 × 18, 30 × 20 × 20, 33 × 22 × 22, 36 × 24 × 24, 39 × 26 × 26, 42 × 28 × 28, and 45 × 30 × 30. Then, for each size of core tensors, we use the following three steps to determine the right size:

(1)

project all decentralized WT nuclear images $Nu{c}_i$, $i=1,\dots ,N$, to core tensors using GND-PCA, and save three projective matrices $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$;

(2)

use the core tensors and $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$ to rebuild WT nuclear images $Nuc{\mathrm{Re}}_i$, $i=1,\dots ,N$;

(3)

compare the similarity between $Nu{c}_i$ and $Nuc{\mathrm{Re}}_i$, $i=1,\dots ,N$.

In this study, we use normalized correlation (NC) to evaluate the similarity. The NC of two volumes, $I_A(x,y,z)$ and $I_B(x,y,z)$, is defined as

$$NC=\frac{{\displaystyle {\sum }_{x,y,z}{I}_A(x,y,z)\cdot I_B(x,y,z)}}{\sqrt{{\displaystyle {\sum }_{x,y,z}{I}_A^2(x,y,z)}}\cdot \sqrt{{\displaystyle {\sum }_{x,y,z}{I}_B^2(x,y,z)}}}=\frac{\langle I_A,I_B\rangle }{\Vert I_A\Vert \cdot \Vert I_B\Vert }$$

(5)

In Equation (5), $I_A(x,y,z)$ and $I_B(x,y,z)$ are zero-mean volumes. NC has a value between 0 and 1. The more similar the two volumes are, the larger the NC value is. Figure 4 shows the computed NC and the central slice of rebuilt nuclear images. Setting 0.85 as the threshold, based on the computational results, we choose 15 × 10 × 10 as the size of core tensors.

3.2.2. Using GND-PCA Extracting Features

Once the size of core tensors is determined, we can use all decentralized WT nuclear images to obtain projective matrices $U_{opt}^{(1)}$, $U_{opt}^{(2)}$, and $U_{opt}^{(3)}$, based on Algorithm 1. After that, each 3D image will be projected onto a core tensor with a size of 15 × 10 × 10. For a computed core tensor ${\mathcal{B}}^m$, $m=1,\dots ,M$, M is the number of nuclear images, and each entry $b_{i,j,k}$ at $(i,j,k)$, $i=1,\dots ,15$, $j=1,\dots ,10$, and $k=1,\dots ,10$, represents a specific feature. The value of the mth nuclear image on the $(i,j,k)$ feature is noted as $b_{i,j,k}^m$. Sometimes we also call $b_{i,j,k}^m$ the projected value on the base $(i,j,k)$ of the mth nuclear image. Thus, after computation, for each element in 15 × 10 × 10 core tensors, the M nuclear images project to M values.

3.2.3. Sorting and Selecting by ‘Energy’

As we have seen, after determining the size of core tensors, the number of computed features is 15 × 10 × 10 = 1.5 × 10³. It is still too large for analysis one by one. In PCA, we usually choose a small number of eigenvalues based on the accumulated energy. The energy corresponding to an eigenvalue actually reflects the variance in projected values on eigenvectors. In GND-PCA, we use the standard deviation of projected values on certain bases to represent their energy and sort bases by energy. In analyzing bases, too many numbers will greatly increase the computational burden. By setting the threshold of accumulated energy as 0.85, we selected the top bases. The total number is 240.

Table 1. The leading 20 bases and their std values (Ord is the order).

Ord	(x,y,z)	std	Ord	(x,y,z)	std	Ord	(x,y,z)	std	Ord	(x,y,z)	std
1	(1,1,1)	52.91	6	(4,1,1)	24.27	11	(1,3,1)	19.03	16	(3,1,2)	16.46
2	(3,1,1)	49.77	7	(1,1,2)	21.97	12	(1,1,3)	18.56	17	(6,1,1)	16.39
3	(2,1,1)	45.79	8	(1,2,1)	21.61	13	(7,1,1)	18.23	18	(3,2,2)	15.94
4	(2,2,1)	32.84	9	(5,1,1)	20.51	14	(3,2,1)	17.11	19	(3,3,1)	15.24
5	(2,1,2)	31.98	10	(1,2,2)	20.29	15	(3,1,3)	16.99	20	(8,1,1)	13.34

shows the leading 20 bases and their std values.

3.2.4. Building Base Images

In a core tensor, each coefficient corresponds to a base image. Theoretically, base images are the GND-PCA counterparts of eigenvectors in PCA. We can observe the geometric meaning of a base by shifting the mean image along the direction of the base. The construction of the base image ${\mathcal{G}}^{i,j,k}$ corresponding to the coefficient $b_{i,j,k}$ is the Kronecker product of the three columns $u_i^{(1)}$, $u_j^{(2)}$, and $u_k^{(3)}$, shown in Equation (6).

$${\mathcal{G}}^{i,j,k}=u_i^{(1)}\otimes u_j^{(2)}\otimes u_k^{(3)}$$

(6)

3.2.5. Geometric Meaning of Selected Bases

To observe the geometric meaning of a certain base corresponding to the coefficient $b_{i,j,k}$, the key is to avoid the influence of other bases. Therefore, a common way is to rebuild an image which is the linear combination of a mean image $Nu{c}_{mean}$ and a base image ${\mathcal{G}}^{i,j,k}$. Equation (7) shows the way to rebuild images, and the rebuilt images are called characteristic images. The range of k is between [−2 and 2].

$$Nu{c}_{mean}\pm k\cdot std(b_{i,j,k}^n)\cdot {\mathcal{G}}^{i,j,k}, n=1,\dots ,N.$$

(7)

In Equation (7), adjusting k will cause a change in characteristic images, and this process reveals the geometric meaning of the base $b_{i,j,k}$.

3.3. Geometric Meaning of Bases

Figure 5 shows rebuilt characteristic images for the leading 10 bases when setting k as −2, 0, and 2, respectively. After checking the geometric meaning of each selected base, we find the first, second, and third bases are highly related to the size variation of P1 and AB. Other bases are more related to local morphological variation.

To verify the geometric meaning of the three bases, we first check whether the first, second, and third bases are really related to the size variance in P1 and AB. That can be performed by computing the volume of P1 and AB from the rebuilt characteristic images. Figure 6 shows the computational results. Observing the results, we can find the first base and second base are mainly related to the size change in AB and P1, respectively; the third base is mainly involved in the reverse change in them in size.

4. Statistical Test

4.1. Preparing RNAi Embryos

Using 10 embryos from 5 genes, dcn-1, lect-754, mcm-5, par-2, and par-3, similar to [9], we analyze their functions according to the category classification in [19]. These genes are involved in the asymmetric division of P1 and AB. Before using GND-PCA, the RNAi nuclear images also need to be processed, just as the WT nuclear images.

(1)

Use image segmentation to obtain embryonic boundaries;

(2)

Compute AX-axes according to the segmented boundaries;

(3)

Build interpolated nuclear images according to the original coordinates of nuclear image voxels;

(4)

Register the interpolated nuclear images based on the computed AP-axes;

(5)

Standardize the length of AP-axes and zoom the nuclear coordinates;

(6)

Resample nuclear images to 300 × 200 × 200;

(7)

Compute three key time points based on nuclear sphericity;

(8)

Decentralize these nuclear images based on computed WT mean image Nuc_mean.

After preprocessing steps, we removed the two RNAi embryos related to gene mcm-5. The reason is that their nuclear sphericities are always less than 0.6, which makes it impossible to compute the three key time points.

4.2. Statistical Testing of RNAi Embryos Based on Selected Bases

It is known that the selected genes are involved in the asymmetric division of P1 and AB, and silencing them may cause an abnormality in size for P1 or AB. Because we verified that the first, second, and third bases are related to the size variation of P1 and AB, we can use the WT projected coefficients on the three bases to build models and then use these models to test RNAi embryos. In addition, for each embryo, there are three projected coefficients on certain bases from 3 key time points $T_1$, $T_2$, and $T_3$. The comparison should be based on the three coefficients simultaneously. Thus, we build a vector (denoted as a projective vector) for each embryo, which includes three entries to store the three coefficients, and the testing is based on these vectors.

The steps of the testing are as follows (for base ($i,j,k$)):

(1)

Compute the mean value of 50 WT projective vectors ${}^{i,j,k}v_{mean}$, $i=1,\dots ,15$, $j=1,\dots ,10$, and $k=1,\dots ,10$;

(2)

Obtain the 50 Euclidean distance between each WT projective vector ${}^{i,j,k}v_w$ to ${}^{i,j,k}v_{mean}$, $w=1,\dots ,50$, and build a normal model if they obey a Gaussian distribution (using ks-test to check);

(3)

For each projective vector ${}^{i,j,k}v_r$ of the 8 RNAi embryos, $r=1,\dots ,8$, compute the Euclidean distance between ${}^{i,j,k}v_r$ and ${}^{i,j,k}v_{mean}$ and test its abnormality based on the built model.

Using 50 WT projective vectors on ${\mathcal{G}}^{1,1,1}$, ${\mathcal{G}}^{3,1,1}$, and ${\mathcal{G}}^{2,1,1}$, the results of the testing on the 8 RNAi embryos can be seen in

Table 2. The parameters and testing results on ${\mathcal{G}}^{1,1,1}$, ${\mathcal{G}}^{3,1,1}$, and ${\mathcal{G}}^{2,1,1}$.

Base	${\left({}^{\mathit{i},\mathit{j},\mathit{k}}\mathbf{v}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}\right)}^{\mathit{T}}$			Parameters		The CDF of RNAi Embryos 1
				μ	σ	let-754		par-3		par-2		dcn-1
[1,1,1]	58.04	−33.13	15.62	54.68	29.52	0.609	0.208	0.255	0.965	1.000	1.000	0.982	1.000
[3,1,1]	29.79	15.55	−21.05	71.51	40.52	0.568	0.777	0.886	0.962	0.490	0.842	0.774	0.970
[2,1,1]	48.18	−23.10	−27.00	56.00	28.73	0.497	0.999	0.781	0.866	0.995	0.998	0.907	0.829

¹ The significance level of each Gaussian distribution is 0.05 in Table 2.

. All the fitted models passed the ks-test.

4.3. Comparative Analysis of the Test Results

As shown in Table 2, using the first three rebuilt bases can identify that let-754, par-2, and dcn-1 play important roles in the asymmetric division of P1 and AB. Because the purpose and the used dataset are consistent with [9], the comparative analysis of the two results is worthwhile.

For the convenience of explanation, we first briefly state the method and main conclusions in [9]. First, five characteristics related to size and shape (volume, surface, diameter, diameter min/max, and compactness) were selected. Second, their mean values and standard deviations (10 features altogether) were computed by all time points within the existence of both P1 and AB. Third, based on the difference between P1 and AB on the 10 features, PCA was applied to extract principal components. Fourth, Gaussian models were built based on 50 WT samples on the selected components. Finally, RNAi samples from dcn-1, lect-754, mcm-5, par-2, and par-3 were tested. The results are as follows:

(1)

The first component (its variance proportion is 40.0%) is mainly related to the size difference between P1 and AB. RNAi samples on lect-754, par-2, par-3, and dcn-1 are tested as abnormal;

(2)

The second component (its variance proportion is 20.4%) is mainly involved in the variation of the size difference. RNAi samples on dcn-1, mcm-5, and lect-754 tested abnormal;

(3)

The third component (its variance proportion is 11.6%) is mainly associated with the shape difference. RNAi samples on dcn-1, par-2, and par-3 tested abnormal.

By comparing the results from [9] and the results using GND-PCA, we can draw the following conclusions:

(1)

GND-PCA correctly recognizes that the size difference between P1 and AB is an important phenomenon in the development of C. elegans. The three leading bases are all related to it. However, the selected features in [9] are based on expertise, while the computed bases here are purely learned from data;

(2)

In [9], lect-754, par-2, par-3, and dcn-1 have been tested to be involved in the size difference between P1 and AB by the first component. However, using the three leading bases, we can precisely point out that lect-754 is probably involved in the reverse change in size between P1 and AB; dcn-1 is mainly related to the development of AB, and par-2 may be associated with the two aspects;

(3)

In [9], mcm-5 has been tested to be related to the variation of the size difference between P1 and AB. In this study, it is clear that mcm-5 is vital in the beginning development of both P1 and AB, because after knocking the gene out, both of them could not change their shapes from cylinder to sphere.

5. Conclusions

In this study, we introduce a complete scheme based on GND-PCA to handle a multidimensional bioimage dataset and apply it to the identification of gene functions in the development of C. elegans. The constructed characteristic images and the results of statistical tests demonstrate the effectiveness of this method. By applying and verifying this method, we can draw the following conclusions:

(1)

A multidimensional image analysis system based on the GND-PCA method can effectively reduce data size and automatically extract valuable features. This process requires minimal expert knowledge, but can yield valuable results. For other image datasets with different characteristics, other more effective methods of image segmentation and registration can be chosen, which makes it feasible to use the GND-PCA method for analysis. The advantage makes this method highly practical;

(2)

The statistical tests in this paper are only conducted on the first three bases. By analyzing the characteristic images corresponding to the first 10 bases, it can be seen that different bases often correspond to different shape features. Therefore, using GND-PCA enables this method to automatically extract the most informative features according to the characteristics of the datasets themselves. These extracted features are powerful weapons for analyzing these datasets;

(3)

The illustration of this study is aimed at a specific issue. However, for a dataset with a labeled subset of samples, the dataset can be used to automatically extract effective features, whereas the labeled samples can be used to automatically screen the extracted features. After that, the selected features can be applied to unlabeled samples. In this way, the applicability of the method will be further enhanced, and this is also a valuable research topic in the future;

(4)

It should be pointed out that to effectively apply the method, it is not only necessary to have a certain understanding of the relevant datasets, but also to understand the relevant theories of GND-PCA. These two requirements have certain limitations on the application of this method.