Index Terms—: Successive Subspace Learning, Clinical Decision-Making System, MRI, Amyotrophic Lateral Sclerosis

A. MRI Data Acquisition and Preprocessing

We collected T2-weighted MRI of the head and neck from a total of 20 controls and 26 patients via a fast spin-echo sequence (Philips Ingenia, Best, Netherlands) with the following parameters: TR/TE = 1,107/80 ms and interpolated voxel size = 0.78×0.78×5 mm³. The statistics of the collected data are shown inTable I. There were no significant differences in age, gender, or weight between the patient and healthy control groups. Patients were assessed at a relatively early stage of disease, as evidenced by well-preserved ALS functional rating scale-revised (ALSFRS-R) and Medical Research Council (MRC) composite muscle scores, and median disease duration of just over a year from symptom onset (median delay from symptom onset to diagnosis in ALS is approximately a year). We note that in one patient, weight was not available (due to advanced disability). Additional acquisition details are described in [10].

A head and neck image atlas was constructed with group-wise diffeomorphic registration [28], as shown inFig. 2. Then, registration between each subject and the atlas was carried out to generate the 3D deformation field, each with the size of 704×704×50×3. Based on the manually identified brain and tongue masks as shown inFig. 2, we cropped the region of interest (ROI) with the size of 330×220×30, which was slightly larger than the exact ROI.

B. Our VoxelHop Framework

The input to our VoxelHop is the multi-channel 3D deformation field$x\in {ℝ}^{S_0\times S_0\times K_0\times C_0}$, whereS₀ andK₀ denote the horizontal and vertical dimensions, respectively, andC₀ denotes the number of channels of the input data (we setC₀=3 for the deformation field). In this work, we use 3D deformation fields as our input, since volume differences between the atlas and individual subjects, as embedded in the deformation fields, play a crucial role in the classification task. It is worth noting that the atlas serves as a reference volume for which the deformation field obtained can be an objective and salient descriptor on the volume difference. Additionally, the vertical dimensions are the same for different channels, but this can be easily generalized to different vertical dimensions for each channel.

The inputx is fed intoI cascaded multi-channel VoxelHop units (M-VoxelHop) andI − 1 max-pooling operations to extract the features (i.e., attributes in [24], [29]) at different spatial scales in the unsupervised Module 1. Then, the features extracted at thei-th M-VoxelHop,i ∈ 1,2, ⋯ ,I, are aggregated, followed by carrying out the supervised label-assisted regression (LAG) module for further dimensional reduction to generate aM′-dimensional feature vector in Module 2. In Module 3, theM′-dimensional feature vectors of all M-VoxelHop units are concatenated to form aM′ ×I-dimensional feature vector for the final classification task. A block diagram of our framework is illustrated inFig. 1.

Our modules are trained based on the previous one rather than independent of each other. For example, the supervised dimension reduction in Module 2 is trained on the feature extracted by Module 1. Then, the classifier in module 3 is trained with the feature extracted in Module 2. We note that both Modules 2 and 3 are supervised by the ground-truth label, while the PCA-like Module 1 is used to efficiently extract the general and discriminative information from the information theory perspective. We detail each step below.

A. Implementation Details

All the experiments were implemented using Python on a server with a Xeon E5 v4 CPU/Nvidia Tesla V100 GPU with 128GB memory. We also used the widely adopted deep learning library, Pytorch, to implement the 3D CNN approaches. For a fair comparison, we downsampled the deformation fields to the size of 110×110×30×3, which was consistent with the input of the 3D CNN approaches.

B. Framework Details

The architecture of our five-stage multi-channel VoxelHop for the input$x\in {ℝ}^{110\times 110\times 30\times 3}$ is detailed inTable II. With five blocks of VoxelHop and maxpooling, the horizontal dimension was reduced to 3 × 3. By configuring the pooling in the aggregation at the fifth stage, i.e.,$P_5=\frac{1}{2}{S}_5$,$Q_5=\frac{1}{2}{K}_5$, the output had the size of 1×1×(1×F₅)×3. As a result, we were able to configure, at most, VoxelHop with six stages for the input$x\in {ℝ}^{110\times 110\times 30\times 3}$, and dropped the aggregation at the sixth stage to maintain the horizontal size of 1 × 1.

To investigate the proper choice ofF_i at each stage, we examined the relationship between the number of Saab AC filters and the energy preservation ratio, as shown inFig. 6. We can see that the leading AC filters account for a large amount of energy, while the energy drops, as the index gets larger. In addition, we plot five energy thresholds, where the orange, green, blue, red, and purple dots represent the cumulative energy ratio of 95%, 96%, 97%, 98%, and 99%, respectively. This indicates that different energy ratios can be selected to balance the classification performance and complexity. In this work, we chose the number of Saab AC filters in the unsupervised dimension reduction procedure in a way to preserve the total energy ratio to 98%.

We compared our VoxelHop with a series of 3D CNN approaches. The VGG [6] and ResNet [7] are the popular networks in 2D computer vision, which have been adopted for single-channel 3D medical data [32] as a strong backbone for many applications [4]. In order to adapt them for multi-channel 3D deformation fields, we followed [31] to configure independent convolutional layers for each channel. We then concatenated the extracted features in the first fully connected layer. The detailed structure of 3D VGG [32], [31] is shown inFig. 5 andTable III. We used the threshold of 0.5 for the final prediction of the 3D CNN approaches.

To aggregate features spatially in Module 2, we applied the pooling of (4 × 4 × 4F_i)-to-(1 × 1 ×F_i) or (2 × 2 × 2F_i)-to-(1 × 1 ×F_i) at different VoxelHop units to reduce the spatial dimension of feature vectors. Specifically, we have

Moreover, we empirically setα=10 andL=3 in the LAG unit of Module 2. The performance was stable for a relatively large range ofα ∈ [5, 20] andL ∈ [2, 5].

C. Experimental Results

For quantitative analysis, we carried out leave-one-out cross-validation, where we used the same hyperparameters for all folds. We simply used the standard Randomized Search Cross-Validation¹ for hyperparameter selection in our cross-validation. Briefly, the accuracy was calculated by running each learning method 46 times, each time removing one of the 46 training subjects, and testing on the training subject that was removed. The final results were computed by averaging all of the 46 folds. We tested both our framework and the comparison methods five times, and the standard deviation was reported.

Without backpropagation, a fold training for multi-channel VoxelHop was completed within 20 mins with a single CPU, while the training of 3D CNNs took about two to three hours to achieve the convergence with a V100 GPU, and we set the batch size to 2. The PCA-like processing in our VoxelHop can be costly w.r.t. computation. However, our framework only required one time forward pass, and did not need iterative training. In contrast, each iteration of CNNs was fast, while the training usually took 200 epochs (i.e., 9,000 times forward pass and 4,500 times backpropagation for the setting of batch size=2).

The accuracy and the area under the curve (AUC) of our proposed M-VoxelHop are given inTable IV. We used the prefic “M−” to denote the multi-channel version of VoxelHop or 3D CNNs. The receiver operating characteristic (ROC) curves of VoxelHop and 3D VGG and 3D ResNet are given inFig. 7. The proposed five-stage M-VoxelHop with 98% energy ratio achieved superior accuracy and AUC over the compared 3D CNNs in our ALS classification task.

We also analyzed the performance of different input sizes, as shown inTable IV. By applying the maxpooling operation to input data with the size of 110×110×30×3 at each stage, we were able to set up one to six stages.Fig. 8 shows an ablation study of configuring different stages. We can see that cascading multiple SSL operations effectively improve classification accuracy. Since the spatial size of the extracted features was small at the late stages, e.g., 3×3×(3×F₅)×3 at the fifth stage, the additional sixth stage did not substantially contribute to the performance. The result of the AUC score is also given inTable IV. Of note, the use of five or six stages for$x\in {ℝ}^{110\times 110\times 30\times 3}$ yielded similar results, while the sixth stage added an additional cost.

We also used the downsampling ofx for a fair comparison even though our VoxelHop was flexible for a larger input size, and without the downsampling, there could be more information contained in the input sample$x\in {ℝ}^{330\times 220\times 30\times 3}$. With the six-stage VoxelHop framework, we achieved a state-of-the-art AUC score of 0.9427 and a classification accuracy of 93.48%. We note that we also definedF_i by keeping 98% energy. To demonstrate the effectiveness of the cropping operation, we inputted the original sample$x\in {ℝ}^{704\times 704\times 50\times 3}$ to the five-, six-, and seven-stage VoxelHop. Of note, we only considered the bulbar region, including the brain and tongue. By doing so, we were able to reduce the signal-to-noise ratio of the input sample resulting from other regions.

InTable V, we compared the classification performance with different inputs, including the original 3D MRI volumes. We found that deformation fields (DF) with SyN outperformed other input data, partly because DFs as in VBM provide volume expansions and compressions at each voxel in an objective manner. In other words, the size or volume difference of the tongue and brain relative to an atlas was the most important cue for our classification task as shown inTable V. In addition, the combination of 3D MRI volumes with DFs was likely to introduce redundant information, thus leading to worse performance. In addition to the registration with SyN [28], we also evaluated our framework with the registration using diffeomorphic demons (DD) [37]. The performance of our VoxelHop was on par with that of 3D ResNet for both registration methods; therefore, our proposed framework was robust against different registration approaches. With the segmentation masks, we were able to remove the unnecessary parts, and alleviated the difficulty of information extraction. For both our VoxelHop and 3D ResNet, there was a 1–2% improvement w.r.t. accuracy of using the segmentation masks. Of note, accurate segmentation of the brain and tongue region was not necessary; instead, we only cropped the ROI to include the brain and tongue region.

InTable VI, we compared our VoxelHop with the classic subspace learning, e.g., Eigenface [35] and PCANet [36] which use only one or multiple stages of PCA for classification. In addition, we were able to use the one-stage PCA coefficients as the feature, and kept the rest the same like our VoxelHop. However, it was extremely difficult to apply the one-stage PCA directly. This is because the input dimension is 110×110×30 = 363,000, which makes the covariance matrix dimension to be (363,000) × (363,000); thus it can be impractical for the implementation. An alternative is to replace the Saab transform at each stage with the standard PCA, while keeping the rest the same. VoxelHop with the Saab transform outperformed its multi-stage PCA version, since the latter could have the sign confusion problem.

The number of AC filters was defined by the energy ratio at each stage, thus affecting the performance and complexity.Table VII shows the classification performance w.r.t. the energy ratio set in our VoxelHop units. The use of the threshold of 99% achieved the best performance, while it increased the number of AC filters, as shown inFig. 6. Therefore, the threshold of 98% was a good trade-off for our ALS disease classification.

The cross-entropy-guided feature selection was developed to simplify the LAG module. We defined the selected numberN_i based on the proportion.Fig. 9 shows the AUC score of keeping different proportion features. We can see that the top 30% features contributed to the performance, and the AUC score was stable with the top 50% features. We used the AUC metric, since the accuracy was not sensitive despite a relatively small number of datasets used in this work. The accuracy was saturated for 40% feature selection. Therefore, we simply dropped the last 40% features in all of our experiments, which largely reduced the to be processed features in the subsequent LAG unit and maintained the performance.

In order to demonstrate the robustness of our VoxelHop with the fewer training samples, we further randomly removed 5, 10, 15, and 20 training samples in each leave-one-out evaluation fold.Fig. 10 shows the AUC of our VoxelHop and 3D ResNet of using fewer training data. We note that we removed the control and patient subjects iteratively to keep the datasets balanced between two categories as demonstrated inFig. 10, where the performance drop of 3D ResNet was more pronounced than our VoxelHop framework, when we removed more training data.

A. Summary of Results

In this work, we presented a lightweight and transparent SSL framework, and applied it to a small number of 3D medical imaging data for classifying between ALS patients and healthy controls. To the best of our knowledge, this is the first attempt at analyzing both the brain and tongue to differentiate controls from patients using T2-weighted structural MRI [38]. ALS is a relentlessly progressive neurodegenerative disease [39], and MRI has been widely used to study ALS [40] to date. In full-blown ALS, the diagnosis is generally clear-cut, and actually can usually be made clinically. However, this is challenging in early diagnosis or disease progression monitoring, due to the lack of a fully validated biomarker, and so clinical trials are reliant on tools such as the ALSFRS-R questionnaire, which have limitations. When combined with considerable heterogeneity in terms of disease progression rate, this necessitates that clinical trials are currently prolonged and expensive. Thus, developing an objective biomarker and decision-making system to rapidly make go/no-go decisions is crucial. Prior research showed that ALS patients exhibited atrophy of gray matter in frontotemporal regions [14], and atrophy of internal muscles of the tongue [15]. In this work, therefore, we used brain and tongue regions simultaneously for our analysis through deformation fields obtained via registration between a head and neck atlas and all the subjects. Our framework achieved an accuracy of 93.48% and an AUC score of 0.9394, which was better than the state-of-the-art 3D CNN classification approaches, including 3D VGG, ResNet, and DenseNet.

B. Comparison of VoxelHop and 3D CNNs

In this subsection, we provide a thorough comparison between VoxelHop and 3D CNNs. CNNs are, in general, well-suited to analyzing 3D input data [4], by extending their 2D counterparts, while many of CNNs were first applied to 2D input data. For example, the performance of a 3D version of VGG [6] and ResNet [7] has been demonstrated in many applications [32]. Targeting multi-channel 3D input data, parallel convolutional layers were applied to each channel independently [30], [31]. The three-channel version of 3D VGG is detailed inTable III, and the corresponding architecture is provided inFig. 5. Both VoxelHop and 3D CNNs constructed the successively growing neighborhoods, and used spatial pooling to reduce the redundancy of neighborhood overlapping.

Although VoxelHop and 3D CNNs have a similar high-level concept, they are different in their model construction, training procedures, and training complexities. We list the differences between VoxelHop (SSL) and CNNs inTable X and elaborate on the details below.

TABLE VIII.

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