A Collection Input Based Support Tensor Machine for Lesion Malignancy Classification in Digital Breast Tomosynthesis

Abstract

Digital breast tomosynthesis (DBT) with improved lesion conspicuity and characterization has been adopted in screening practice. DBT-based diagnosis strongly depends on physicians’ experience, so an automatic lesion malignancy classification model using DBT could improve the consistency of diagnosis among different physicians. Tensor-based approaches that use the original imaging data as input have shown promising results for many classification tasks. However, DBT data are pseudo-3D volumetric imaging as the slice spacing of DBT is much coarser than that of the in-plane resolution. Thus, directly constructing DBT as the third-order tensor in a conventional tensor-based classifier with introducing additional information to the original DBT data along the slice-spacing dimension will lead to inconsistency across all three dimensions. To avoid such inconsistency, we introduce a collection input based support tensor machine (CISTM)-based classifier that uses the tensor collection as input for classifying lesion malignancy in DBT. In CISTM, instead of introducing the third dimension directly into the geometry construction, the third-dimension structural relationship is related by weight parameters in the decision function, which is dynamically and automatically constructed during the classifier training process and is more consistent with the pseudo-3D nature of DBT. We tested our method on a DBT dataset of 926 images among which 262 were malignant and 664 were benign. We compared our method with the latest tensor-based method, KSTM (kernelled support tensor machine), which does not consider the unique non-uniform resolution property of DBT. Experimental results illustrate that the CISTM-based classifier is effective for classifying breast lesion malignancy in DBT and that it outperforms the KSTM-based classifier.

1. INTRODUCTION

As a pseudo-3D mammography technology, digital breast tomosynthesis (DBT) enables cross-sectional imaging and achieves better characterization and localization of lesions than 2D mammography (Moy, 2019). DBT is associated with higher specificity and a greater proportion of breast cancers detected than 2D mammography alone (Conant et al., 2019). Therefore, DBT has been increasingly adopted in screening practice (Bernardi et al., 2018; Alsheik et al., 2019).

Imaging-based diagnosis is mainly based on physicians’ judgement and, thus, strongly depends on their training and experience. Machine learning-based approaches could improve the consistency of diagnosis and lesion malignancy classification among physicians using DBT. Although computer-aided diagnosis (CADx) has been extensively investigated for mammograms (Chan et al., 2005; Chan, 2015; Samala et al., 2016a; Chan et al., 2018; Kim et al., 2017), little development has been devoted to DBT-based CADx.

The current medical imaging-based classification approaches can be divided into three categories: (1) handcrafted feature-based strategies (Lambin et al., 2017; Cirujeda et al., 2016; Li et al., 2016; Valdora et al., 2018); (2) feature learning-based strategies (Yan et al., 2018; Antropova et al., 2017; Sun et al., 2016; Kooi et al., 2017); and (3) voxel/pixel-based strategies (Huang et al., 2017; Khamis et al., 2017; Zuluaga et al., 2015).

As a handcrafted feature-based method, radiomics has been applied to breast lesion malignancy classification using DBT (Veena and Padma, 2019; Zhou et al., 2019). Veena et al. (Veena and Padma, 2019) developed a radiomics model that uses a traditional probabilistic neural network (PNN) classifier after manually extracting quantitative features. Zhou et al. (Zhou et al., 2019) proposed a multi-objective–based feature selection algorithm that selects an optimal feature subset after feature extraction, then uses support vector machine (SVM) to classify breast lesion malignancy in DBT. However, the generic nature of the features selected may not fully reflect the unique characteristics of particular cancers (Chung et al., 2015), and the robustness of the classification is often hindered by the feature selection methods. Moreover, these radiomics methods may not fully utilize the DBT information. For the feature learning-based models, deep learning using the intrinsic features learned automatically has shown potential for classification. However, deep learning requires large-scale annotated datasets, which is often a challenge for medical imaging, including DBT (Samala et al., 2018). Among the voxel/pixel-based strategies, tensor-based methods that construct the model using an intrinsic original structure from 3D data as input have been applied successfully to predict distant failure after lung stereotactic body radiation therapy (Li et al., 2018).

To make effective use of all information in DBT, we have adopted a tensor-based method to construct a model that predicts lesion malignancy. However, the third dimension of DBT is much coarser than the in-plane resolution, which makes DBT fundamentally different from other traditional 3D imaging, such as positron emission tomography (PET) and computed tomography (CT), in which the natural 3D geometry structure makes the tumor suitable as a third-order tensor. Therefore, special processing is needed to handle the third dimension of DBT in tensor-based approaches. Motivated by the kernelled support tensor machine (KSTM) (Li et al., 2018) and multilinear rank support tensor machine (MRSTM) (Zhou et al., 2018), we developed a collection input based support tensor machine (CISTM) for DBT.

In the CISTM-based classifier, the data relationship of the third dimension is treated as a collection, i.e. there is no fixed geometric relationship between slices in the collection, which is weaker than the 3D relationship and consistent with the pseudo-3D nature of DBT. Specifically, each slice of DBT in CISTM is considered as one tensor. All slices of DBT are used as a collection of multiple tensors for the input of CISTM and the relationship between different tensors (i.e., slices) are automatically considered during the model training. In CISTM, so the decision function can be considered a multivariate function in the tensor space where the corresponding weight parameters defining the separating hyperplane can intrinsically reflect the relationship of these tensors in the collection. Since the resolution in the third dimension fundamentally differs from that in-plane for DBT, directly constructing DBT as the third-order tensor may be inconsistent with the intrinsic geometry of the tumor, and directly constructing DBT as the second-order tensor by splicing planes along the direction of rows or columns may lose the third dimension’s structural information. Thus, in this work, CISTM is applied to DBT in the following way: (1) each slice is directly treated as a second-order tensor to preserve the natural geometric structure of the plane; and (2) instead of introducing the third dimension directly into the geometry construction, the third dimension structural relationship is related by the weight parameters in the decision function and determined gradually during alternating training. In this way, the relationship between these tensors in each input collection is not determined artificially. Instead, it is determined by the dataset in the training process. In contrast, traditional tensor-based classifiers require that each input is a tensor and that the relationship of the third dimension in DBT is artificially fixed first, which may result in adding inconsistent information to the original data. Therefore, considering the DBT’s pseudo-third dimension, we applied the CISTM-based classifier to classify lesion malignancy in DBT.

2. Materials and methods

This section first describes our dataset and preprocessing, then briefly reviews the latest tensor-based method KSTM (S. Li et al., 2018) for comparison with our method in DBT, and lastly introduces the proposed CISTM by formulating the decision function and solving it with an alternating algorithm.

2.1. Dataset and preprocessing

In this study, the DBT cases were acquired in craniocaudal (CC) and mediolateral oblique (MLO) views with a Hologic DBT system with an angular range of 15°. In our dataset, each breast has both CC and MLO views. There were 926 available samples among which 262 were malignant and 664 were benign. The malignancy status was validated through biopsy. The DBT data are composed of 20-86 slices per image with 2457×1890 pixels per slice. Tumors within each DBT were segmented by a semi-automatic approach. Eight radiologists, each with more than three years of experience in breast cancer diagnosis, chose one slice of each DBT and then contoured tumors on the chosen slices. These contours were manually checked and modified, if needed, by two radiologists with more than five years of experience each in breast cancer diagnosis. These contours were then fine-tuned by the Chan-Vese model (Wang et al., 2010) for more consistent segmentation. The tumor in all other slices was then segmented by a contour-guided segmentation method (Chen et al., 2018a) that incorporated similarity information on tumor shape and position among adjacent slices under the guidance of the initial tumor contour from the chosen slice for each image.

A series of data preprocessing processes was conducted as follows: all slices of the views (CC or MLO) were interpolated to the same resolution (0.1 mm) using a linear interpolation method, as the in-plane resolutions differed across images (ranging from 0.0859 mm to 0.1099 mm). The final DBT data of size 767×695×20 was extracted from segmented tumors. The size of the in-plane was determined by the largest tumor, and zero peripheral filling was applied to regions outside the tumor. The size of the third dimension was determined by the tumor with the minimal slices. For each DBT, the final data was extracted from the central slice of the tumor plus ten slices above and nine slices below. The DBT processing pipeline for our dataset is illustrated in Figure 1.

Figure 1:

Pipeline of DBT processing in our dataset.

2.2. Kenneled Support Tensor Machine (KSTM)

Consider a training set of n data points ${\left\{X_n,y_n\right\}}_{n=1}^N$, where each tensor $X_n\in R^{J_1\times \cdots \times J_D}$ belongs to a class labeled by $y_n\in \left\{-1,1\right\}$. Let $Ø:R^{J_1\times \cdots \times J_D}\to R^{I_1\times \cdots \times I_D}$ be a transformation from tensor space $R^{J_1\times \cdots \times J_D}$ to a feature tensor space $R^{I_1\times \cdots \times I_D}$, satisfying $Ø\left(X_n\right)={\left({Ø}_{i_1\times i_2\times \cdots \times i_D}\left(X_n\right)\right)}_{I_1\times I_2\times \cdots \times I_D}$. The KSTM aims to find a decision function $f: R^{J_1\times \cdots \times J_D}\to \{-1,1\}$, which is formulated as

$$f\left(X\right)=\mathrm{sign}\left(Ø\left(X\right){\prod }_{d=1}^D{\times }_d{w}^{\left(d\right)}+b\right),$$ (1)

where $X\in R^{J_1\times \cdots \times J_D}$ is a tensor of order $D,{\prod }_{d=1}^D{\times }_d$ is the mode product between a tensor and D vectors, sign is the sign function, b is the bias to be determined, and the weight parameters $w^{\left(d\right)}\in { R}^{I_d}$, d=1,2,…,D are estimated by solving the following minimization problem:

$$\begin{array}{c}{\min }_{w^{\left(d\right)},b,\xi }\frac{1}{2}{\prod }_{d=1}^D{\Vert w^{\left(d\right)}\Vert }^2+C{\sum }_{n=1}^N{\xi }_n,\hfill \\ \mathrm{s.t.}{y}_n\left(Ø\left(X_n\right)\prod _{d=1}^D{\times }_d{w}^{\left(d\right)}+b\right)\ge 1-{\xi }_n,\hfill \\ \hfill {\xi }_n\ge 0,d=1,2,\dots D,n=1,2,\dots N,\hfill \end{array}$$ (2)

where ${ \xi }_n$ is the misclassification error of the n^th training sample $X_n$, and C≥0 is a trade-off parameter.

Since the optimization problem (2) is not convex, KSTM utilizes an alternating strategy to train the weight vectors for every mode of the tensor for the classifier. In each alternation step, KSTM optimizes only one weight by fixing, in order, D-1 weights in (2) inherited from the previous step. This is equivalent to alternately finding the following decision function:

$$f\left(X\right)=\mathrm{sign}\left({\left(w^{\left(l\right)}\right)}^{\text{T}}{Ø}_l\left(X{\prod }_{1\le d\le D}^{d\ne 1}{\times }_d{P}_d\left(w^{\left(d\right)}\right)\right)+b\right),$$ (3)

where $w^{\left(d\right)},d=1,2,\dots ,D$ and $d\ne l$ are fixed, ${Ø}_l:R^{J_l}\to R^{I_l}$ is a projective mapping from $R^{J_l}$ to ${\mathrm{ }R}^{I_l}$, and $P_d:R^{I_d}\to R^{J_d}$ is a projective mapping from ${\mathrm{ }R}^{I_d}$ to ${\mathrm{ }R}^{J_d}$. The optimization problem in each alternation step is reformulated as an SVM problem with respect to $w^{\left(l\right)}$ as a weight vector.

2.3. Collection Input based Support Tensor Machine (CISTM)

Because the resolution of the third dimension of DBT is very different from the resolution in-plane, instead of directly constructing DBT as a tensor, as in KSTM, we constructed DBT as a tensor collection in the proposed CISTM scheme to minimize the inconsistency introduced when adding artificial information to the original data.

Consider a training set of n data points ${\left\{X_n,y_n\right\}}_{n=1}^N$, in which each tensor collection ${\chi }_n\in R^{J_1\times \cdots \times J_D}\times R^{J_1\times \cdots \times J_D}\times \cdots \times R^{J_1\times \cdots \times J_D}$ consists of M tensors of order D belonging to a class labeled by ${ y}_n\in \{-1,1\}$. As a simple example, let M=2, D=3. Thus, ${\chi }_n$ is just a collection of two third-order tensors. When M=1, CISTM is simplified to KSTM.

CISTM aims to find a decision function $f:R^{J_1\times \cdots \times J_D}\times R^{J_1\times \cdots \times J_D}\times \cdots \times R^{J_1\times \cdots \times J_D}\to \left\{-1,1\right\}$, which is formulated as

$$f\left(\mathit{\chi }\right)=\mathrm{sign}\left({\sum }_{\mathit{m}=\mathbf{1}}^{\mathit{M}}\left(Ø\left(X_{\mathit{m}}\right){\prod }_{d=1}^D{\times }_d{w}^{\left(d\mathit{m}\right)}\right)+b\right),$$ (4)

where $\chi ={\left\{X_m\right\}}_{m=1}^M$ is a collection of M tensors, $X_m\in R^{J_1\times \cdots \times J_D}$ is a tensor of order D, ${\times }_d$ is the mode product between a tensor and a vector, b is the bias to be determined, and the weight parameters $w^{\left(dm\right)}\in R^{I_d},d=1,2,\dots ,D,$ and m= 1,2,…,M are estimated by solving the following minimization problem:

$$\begin{array}{l}\underset{w^{\left(d\mathit{m}\right)},b,\xi }{\min }\frac{1}{2}{\sum }_{\mathit{m}=\mathbf{1}}^{\mathit{M}}{\prod }_{d=1}^D{\Vert w^{\left(d\mathit{m}\right)}\Vert }^2+C{\sum }_{n=1}^N{\xi }_n,\\ \mathrm{s.t.}{y}_n\left({\sum }_{\mathit{m}=\mathbf{1}}^{\mathit{M}}\left(Ø\left(X_{n\mathit{m}}\right){\prod }_{d=1}^D{\times }_d{w}^{\left(d\mathit{m}\right)}\right)+b\right)\ge 1-{\xi }_n,\\ {\xi }_n\ge 0,d=1,2,\dots D,n=1,2,\dots ,N,\mathit{m}=1,2,\dots \mathit{M},\end{array}$$ (5)

where ${\chi }_n={\left\{X_{\mathit{nm}}\right\}}_{m=1}^M$ is the n^th training sample, which is a collection of $X_{nm}\in R^{J_1\times \cdots \times J_D}$; ξ_n is the misclassification error of the n^th training sample; and C≥0 is a trade-off parameter.

Motivated by the iterative procedures of KSTM (Li et al., 2018) and MRSTM (Zhou et al., 2018), we propose an alternating strategy to find a solution for the nonconvex optimization problem defined in equation (5). We first divide the weight parameters into D weight sets ${\left\{w^{\left(1m\right)}\right\}}_{m=1}^M,{\left\{w^{\left(2m\right)}\right\}}_{m=1}^M,\cdots ,{\left\{w^{\left(Dm\right)}\right\}}_{m=1}^M$. In each alternation step, CISTM optimizes only one weight set by fixing, in order, D-1 weight sets in (5) inherited from the previous step. This is equivalent to alternately finding the following decision function:

$$f\left(\mathit{\chi }\right)=\mathrm{sign}\left({\sum }_{\mathit{m}=\mathbf{1}}^{\mathit{M}}\left({\left(w^{\left(l\mathit{m}\right)}\right)}^{\text{T}}{Ø}_l\left(X_{\mathit{m}}{\prod }_{1\le d\le D}^{d\ne l}{\times }_d{P}_d\left(w^{\left(d\mathit{m}\right)}\right)\right)\right)+b\right),$$ (6)

where $w^{\left(dm\right)},d=1,2,\dots ,D$ and $d\ne l$ are fixed, ${Ø}_l:R^{J_l}\to R^{I_l}$ is a projective mapping from $R^{J_l}$ to ${\mathrm{ }R}^{I_l}$, and $P_d:R^{I_d}\to R^{J_d}$ is a projective mapping from ${\mathrm{ }R}^{I_d}$ to ${\mathrm{ }R}^{J_d}$. Note that the bold fonts in equations (4)–(6) highlight the differences between KSTM and CISTM.

Let $w_{\left(l\right)}$ be a vector composed of the vectors in the l^th weight set ${\left\{w^{\left(lm\right)}\right\}}_{m=1}^M$ defined in equation (7) and $x_{\left(l\right)}$ be a corresponding vector composed of ${\left\{{Ø}_l\left(X_m{\prod }_{1\le d\le D}^{d\ne 1}{\times }_d{P}_d\left(w^{\left(dm\right)}\right)\right)\right\}}_{m=1}^M$ defined in equation (8).

$$w_{\left(l\right)}={\left({\left(w^{\left(l1\right)}\right)}^{\text{T}}{\left(w^{\left(l2\right)}\right)}^{\text{T}}\dots {\left(w^{\left(lM\right)}\right)}^{\text{T}}\right)}^{\text{T}}.$$ (7)

$$x_{\left(l\right)}={\left({\left({Ø}_l\left(X_1{\prod }_{1\le d\le D}^{d\ne l}{\times }_d{P}_d\left(w^{\left(d1\right)}\right)\right)\right)}^{\text{T}}{\left({Ø}_l\left(X_2{\prod }_{1\le d\le D}^{d\ne l}{\times }_d{P}_d\left(w^{\left(d2\right)}\right)\right)\right)}^{\text{T}}\cdots {\left({Ø}_l\left(X_M{\prod }_{1\le d\le D}^{d\ne l}{\times }_d{P}_d\left(w^{\left(dM\right)}\right)\right)\right)}^{\text{T}}\right)}^{\text{T}}.$$ (8)

Then, the weight parameters defined in equation (6) can be estimated by solving the following minimization problem:

$$\begin{array}{l}\underset{w^{\left(l\right)},b,\xi }{\min }\frac{1}{2}{\Vert w_{\left(l\right)}\Vert }^2+C{\sum }_{n=1}^N{\xi }_n,\\ \mathrm{s.t.}{y}_n\left({\left(w_{\left(l\right)}\right)}^{\text{T}}{x}_{\left(l\right)n}+b^{\left(l\right)}\right)\ge 1-{\xi }_n,\\ {\xi }_n\ge 0,l=1,2,\dots D,n=1,2,\dots ,N,\end{array}$$ (9)

where $x_{\left(l\right)n}$ is a vector obtained from the n^th training sample ${\chi }_n={\left\{X_{\mathit{nm}}\right\}}_{m=1}^M$ by using equation (8).

Finally, the optimization problem in each alternating step is reformulated as an SVM problem with respect to $w_{\left(l\right) }$ as a weight vector. The detailed iterative procedures to solve (9) are summarized in Table 1. Once the iterative CISTM training model defined in (9) is solved, the class label of a test sample $\chi$ can be predicted using the following classification:

$$f\left(\chi \right)=sign\left({\sum }_{n=1}^N\left({\alpha }_n^{\left(1\right)}{y}_n\text{K}\left(x_{\left(l\right)n},x_{\left(l\right)}\right)\right)+b^{\left(l\right)}\right),$$ (10)

where $x_{\left(l\right)}$ is a vector obtained from the test sample $\chi$ by using (8), the kernel function K is optional, and ${\alpha }_n^{\left(l\right)}$ and $b^{\left(l\right)}$ are similar to the SVM. In our experiments, we used the radial basis function (RBF) kernel function with a gamma parameter γ, i.e. $\text{K}\left(\text{x},{\text{x}}^{\prime }\right)=\exp \left(-\gamma {\Vert \text{x}-{\text{x}}^{\prime }\Vert }^2\right)$.

Table 1:

Detailed procedure of the CISTM iterative algorithm.

Input	The training set ${\left\{{\chi }_n,y_n\right\}}_{n=1}^N$, a trade-off parameter C, and a threshold parameter ε.
Output	$w_{\left(d\right)}\left(d=1,2,\dots ,D,d\ne l\right),{\alpha }_n^{\left(l\right)}\left(n=1,2,\dots N\right),$ and b(l).
Step 1	Initialize $w_{\left(d\right)}\left(d=1,2,\dots ,D\right)$ as random vectors, and the initial $P_d$ in (3), (6), and (8) are defined as $P_d\left(w^{\left(dm\right)}\right)=w^{\left(dm\right)}$.
Step 2	For l=1,2,…,D, let $w_{\left(d\right)}\left(d=1,2,\dots ,D,d\ne 1\right)$ be known, then obtain $x_{\left(l\right)n}\left(n=1,2,\dots ,N\right)$ by using (8).
Step 3	Solve the dual problem of (9) by an SVM solver to obtain ${\alpha }_n^{\left(l\right)}\left(n=1,2,\dots ,N\right)$.
Step 4	Set $P_l\left(w_{\left(l\right)}\right)={\sum }_{n=1}^N{\alpha }_n^{\left(l\right)}{y}_n{x}_{\left(l\right)n},$ where $P_l\left(w_{\left(l\right)}\right)={\left({\left(P_l\left(w^{\left(l1\right)}\right)\right)}^{\text{T}}{\left(P_l\left(w^{\left(l2\right)}\right)\right)}^{\text{T}}\dots {\left(P_l\left(w^{\left(lmM\right)}\right)\right)}^{\text{T}}\right)}^{\text{T}}.$
Step 5	If $\left|{\sum }_{d=1}^D\left(\langle P_d\left(w_{\left(d\right)}^{\left(t\right)}\right),P_{\left(d\right)}\left(w_{\left(d\right)}^{\left(t-1\right)}\right)\rangle {\Vert P_d\left(w_d^{\left(t\right)}\right)\Vert }^{-2}-1\right)\right|<\epsilon ,$ where t represents the tth iteration. Then set $w_{\left(d\right)}=w_{\left(d\right)}^{\left(t\right)}\left(d=1,2,\dots ,D,d\ne l\right)$ and output $w_{\left(d\right)}=\left(d=1,2,\dots ,D,d\ne l\right)$, then go to Step 6. Else let l=l+1 (if l<D) and l=1 (if l=D), then go to Step 2.
Step 6	Obtain $x_{\left(l\right)n}\left(n=1,2,\dots ,N\right)$ by using $w_{\left(d\right)}=\left(d=1,2,\dots ,D,d\ne l\right)$ and (8), then obtain the final ${\alpha }_n^{\left(l\right)}\left(n=1,2,\dots ,N\right)$ and $b^{\left(l\right)}$ by solving the optimization problem (9) using an SVM solver. Finally, output ${\alpha }_n^{\left(l\right)}\left(n=1,2,\dots ,N\right)$ and ${ b}^{\left(l\right)}$, then stop.

3. Experiments and analysis

3.1. Setup

First, 284 DBT images (about 30%), which comprised 78 benign and 206 malignant images, were selected randomly as an independent testing dataset, and the remaining images (about 70%) were used as the training dataset. In our experiments, images from the same breast were assigned to the same subset (training set or test set) to keep the two subsets independent. A three-fold cross validation method was performed in the training dataset to choose the optimal parameters. The trade-off parameter C in equation (9) and the gamma parameter γ used in the RBF kernel function was chosen from {2⁻⁴, 2⁻³, …, 2⁴}. The threshold parameter ε in Table 1 was empirically set to 10⁻².

To demonstrate CISTM’s performance, we compared it with the latest tensor-based method KSTM (Li et al., 2018). According to the different requirements of the input type in each classifier, after the same data preparation, each data is constructed as a tensor in KSTM-based classifier, whereas in CISTM-based classifier each data is constructed as a collection of 20 tensors (i.e., with each slice being used as one tensor) as shown in Figure 2(d). Since the input data requirement of KSTM is a tensor, three types of tensors were constructed based on different slices’ relationships, as shown in Figure 2 (a)–(c).

Figure 2:

Input data for different classifiers: (a) 2D in KSTM1; (b) 2D in KSTM2; (c) 3D in KSTM3; (d) Pseudo-3D in CISTM.

The 2D data shown in Figures 2(a) and 2(b) and the 3D data shown in Figure 2(c) were obtained by directly splicing all DBT slices along the horizontal, vertical, and third directions, respectively. However, the pseudo-3D data shown in Figure 2(d) was not fixed with any explicit assumption of structural relationship among these slices, because this structural relationship was determined during the process of classifier training. The KSTM-based classifiers using tensors defined in Figures 2(a), 2(b), and 2(c) are called KSTM1, KSTM2, and KSTM3, respectively.

For quantitative assessment, we used the following four criteria: (1) the area under the receiver operating characteristic (ROC) curve (AUC); (2) classification accuracy (ACC), calculated as (TP + TN) / (TP + TN + FP + FN); (3) sensitivity (SEN), calculated as TP / (TP + FN); and (4) specificity (SPE), calculated as TN / (TN + FP). In the above calculation formulas, TP indicates true positive cases, TN indicates true negative cases, FP indicates false positive cases, and FN indicates false negative cases. Each STM classifier will output malignancy probability ranging from 0 to 1. The output probability of each sample from the classifier was used for ROC analysis. All methods were repeated five times. The mean and standard deviation for each evaluation criterion were calculated from the results of five rounds of the experiment, where the sensitivity and specificity were calculated by applying a threshold of 0.5 on the output malignancy probability.

3.2. Results and analysis

We summarized the performance of CISTM and KSTM in classifying breast lesion malignancy using DBT by listing the mean and standard deviation values of AUC, accuracy, sensitivity, and specificity obtained for each method in Table 2. The KSTM3-based classifier performed the worst among the three KSTM-based classifiers, while the KSTM1- and KSTM2-based classifiers obtained approximately the same level of performance in the accuracy. The CISTM-based breast lesion malignancy classifier achieved the highest AUC (0.802), accuracy (0.750) and sensitivity (0.785) scores among the methods investigated in this study.

Table 2:

Performance of CISTM and KSTM for breast lesion malignancy classification using DBT. The best results are in bold.

Methods	Input Data		Independent testing
	Dimension	Size	AUC	ACC	SEN	SPE
KSTM1	2 D	767×13900	0.790±0.020	0.734±0.016	0.726±0.041	0.737±0.020
KSTM2	2 D	15340×695	0.801±0.017	0.738±0.012	0.772±0.037	0.726±0.019
KSTM3	3 D	767×695×20	0.707±0.039	0.660±0.034	0.705±0.097	0.643±0.033
CISTM	Pseudo-3D	(767×695) ×20	0.802±0.014	0.750±0.015	0.785±0.040	0.737±0.026

Figure 3 shows bar plots of the four evaluation criteria values obtained by the four methods in each round of the experiment. Results obtained by the KSTM1- and the KSTM2-based classifiers differed substantially except in the third round of the experiment. The KSTM3-based classifier obtained the lowest AUC score in each experiment. The CISTM-based classifier obtained the most stable values of AUC across different rounds of the experiment. The results in Table 2 and Figure 3 demonstrate that the CISTM-based method obtained better performance than the KSTM-based methods for classifying lesion malignancy using DBT.

Figure 3:

Bar plots of four evaluation criteria obtained by four methods in each round of the experiment.

The confusion matrix showing TP, FP, TN, and FN for the four methods in one random independent testing experiment is presented in Table 3. The CISTM-based classifier obtained higher TP and TN, and lower FP and FN than the KSTM-based classifiers.

Table 3:

Confusion matrix results for four classifiers.

Methods	Tumor	Predicted Benign	Predicted Malignant
KSTM1	Actual Benign	149	57
	Actual Malignant	23	55
KSTM2	Actual Benign	151	55
	Actual Malignant	20	58
KSTM3	Actual Benign	138	68
	Actual Malignant	32	46
CISTM	Actual Benign	158	48
	Actual Malignant	19	59

The ROC curves for the four breast lesion malignancy classification algorithms in one random independent testing experiment are shown in Figure 4. The proposed CISTM-based classifier outperformed the KSTM-based classifiers for lesion classification in DBT.

Figure 4:

ROC curves for four algorithms using DBT.

We have analyzed the applicability of the CISTM and KSTM to different diseases on the test dataset in Table 4. As shown in the table, the proposed method has different classification accuracies for different disease conditions. The KSTM3-based classifier obtained the lowest accuracy scores among the different disease conditions, while the KSTM1- and KSTM2- based classifiers obtained approximately the same level of performance in fibroadenomas and breast cystic diseases. The CISTM-based classifier achieved the highest accuracy scores among the different disease conditions.

Table 4:

Accuracies of the CISTM and KSTM to different diseases.

Diseases (Number)	Invasive Ductal Carcinomas (46)	Fibroadenomas (48)	Breast Cystic Diseases (58)	Others
				Malignant	Benign
KSTM1	0.739	0.771	0.570	0.679	0.779
KSTM2	0.783	0.792	0.570	0.679	0.798
KSTM3	0.674	0.583	0.483	0.536	0.635
CISTM	0.783	0.833	0.603	0.714	0.827

We also investigated the statistical significance of the difference between the proposed method and the other methods in Table 5 using the paired t-test. Based on the p-values, the proposed method is significantly better than the other methods at a significance level of 0.05.

Table 5:

P-values in t-test between CISTM and the other three methods.

Methods	P-values
KSTM1	0.0443
KSTM2	<0.0001
KSTM3	<0.0001

4. Discussion and Conclusion

In this study, we proposed a CISTM-based classifier that takes a tensor collection as input to classify breast lesion malignancy using DBT. This CISTM-based classifier considers the DBT data’s pseudo-third dimension’s relationship during the classifier training process. To demonstrate CISTM’s performance, we compared it with a recent tensor-based method, KSTM (Li et al., 2018). Experimental results on DBT indicate that the CISTM-based classifier can achieve higher AUC, accuracy and sensitivity than the KSTM-based classifier. The CISTM method’s superior performance can be attributed to the pseudo-third dimension of DBT. In traditional tensor-based methods, the tensor must be constructed as input data at the beginning. Regardless of whether DBT is constructed as 2D or 3D data, the relationship of the third pseudo-dimension is artificially added, which may introduce inconsistent information to the original data, as the resolution of the third dimension of DBT is much coarser than the in-plane resolution. In the proposed CISTM, however, the pseudo-third dimension of DBT is not directly assumed in the tensor construction. Instead, CISTM dynamically and automatically considers the DBT data’s pseudo-third dimensional relationship during the training process.

In our experiments, the final ROIs of size 767×695×20 was extracted from 20 consecutive slices centered around the tumor at each CC and MLO view. The size of the in-plane was determined by the largest tumor. The size of the third dimension was chosen based on the following observations/reasons. While more slices may contain additional information of a tumor, including more peripheral slices (along z-axis direction) does not necessarily improve model performance as objects in the out-of-focus plans can be blurred due to the nature of DBT. Indeed, ROIs composed of different consecutive slices have been adopted in several previous studies (Kim et al., 2017; Samala et al., 2016b; Samala et al., 2018) on DBT analysis. For example, Samala et al. (Samala et al., 2016b; Samala et al., 2018) used the ROIs extracted 5 consecutive slices center around the tumor while Kim et al. (Kim et al., 2017) selected 51 slices of DBT in their experiments. In our dataset, the third dimension of original DBT ranged from 20 to 86. In all experiments performed in this work, we used 20 slices of DBT centered around the tumor. We also initially investigated the influence of slice numbers by comparing the performance CISTM using 20 or 86 slices DBT. Results with the same experimental setup and parameters in one random independent experiment in summarized in Table 6 below. CISTM using 20 slices and 86 slices obtained approximately the same level of performance in the AUC, accuracy and specificity. We have also investigated the statistical significance of the difference between them using the paired t-test. Based on the p-values (0.8704), they are not significantly different at a significance level of 0.05. In a future study, a systematic study on the influence of number of slices as input is worthy of further investigation.

Table 6:

Performance of CISTM at different number of slices.

Input Data Size	Independent testing
	AUC	ACC	SEN	SPE
(767×695) ×20	0.805	0.764	0.756	0.767
(767×695) ×86	0.794	0.757	0.743	0.762

In our experiment, two views (CC and MLO) DBT of the same breast were assigned to the same subset (training set or test set). Meanwhile, within each subset, the two views DBT were considered as independent samples as the appearance of tumors at CC and MLO views are very different. While this approach has been adopted in previous publications (Samala et al., 2016b; Kim et al., 2017) on DBT analysis, some correlation may exist between CC and MLO views from the same breast. In tensor-based approaches, we may concatenate DBT data from CC and MLO views to form one tensor. The performance of CISTM may be further improved with concatenating CC and MLO views although the computational burden increases as the size of tensor size enlarged with this structure. A future study is warranted to explore the potential benefit of the concatenating tensor.

In the dataset used in this study, there are mainly three disease conditions: invasive ductal carcinomas, fibroadenomas and breast cystic diseases. We have analyzed the applicability of the proposed method to different diseases on the test dataset in Table 4. As shown in the table, the proposed method has different classification accuracies for different disease conditions. Among these three major conditions, the CISTM method performs worst for cystic condition. The current CISTM method only predicts two categories (benign or malignant) without considering subcategory of disease conditions. In a future study, we will develop a model to predict diseases conditions with the hope to improve tumor classification accuracy for subcategory such as cystic condition.

This work mainly focused on the tensor-based methods modified from the KSTM using a tensor collection as the input. Therefore, handcrafted feature extraction and feature selection are not needed in the proposed method. However, handcrafted features can incorporate domain knowledge into the feature extraction process and may provide complementary information to tensor-based methods. Thus, CISTM’s performance may be improved by combining it with handcrafted feature-based methods. This could be achieved by fusing the outputs from CISTM and handcrafted feature-based methods through evidential reasoning (Chen et al., 2018b; Chen et al., 2019) which warrants a future study.

In summary, we have developed a CISTM method for lesion classification in DBT, and we have evaluated whether the tensor collection representation is more suitable for DBT than the traditional tensor representation in tensor-based methods. Our experimental results have illustrated that the CISTM-based classifier is feasible and effective for classifying breast lesion malignancy in DBT, as it outperformed a conventional STM that does not consider DBT’s unique non-uniform resolution property.