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Our research problems are motivated by multimodality breast cancer imaging data produced by Boppart Lab [84] at University of Illinois at Urbana-Champaign. We applied the proposed method to multiphoton imaging data for breast cancer diagnosis. To better visualize the biological tissue at cellular and molecular levels, [84]’s multiphoton microscope generates multimodal images using two-photon auto-fluorescence (2PAF), three-photon auto-fluorescence (3PAF), second-harmonic generation (SHG) and third-harmonic generation (THG). Two-photon-fluorescence microscopy is commonly used to visualize tissue morphology and physiology at a cellular level, and three- photon-fluorescence with longer wavelength can reach deeper levels of the tissue and thus provide higher resolution [11, 28]. Figure 4.5 illustrates the four modalities of 2PAF and 3PAF, SHG and THG for normal rat’s breast tissue and cancerous rat’s breast tissue. The new technique is able to identify cancer cell clusters in a specimen which are not easily identified by histology imaging.

Figure 4.5 provides the multiple contrast mechanisms produced by four-modality microscope imaging, which highlight the structural components of tissues. In contrast to the normal rat’s tissue, multiple modalities clearly indicate a large number of biological tumor-associated micro-vesicles (circled in Figure 4.5) which appear spatially aligned in a tubular formation on the cancer rat’s tissue, particularly in the 3PAF image. In addition, Figure 4.5 also shows that the microvesicles are visible in the THG and 3PAF images, but are not obvious in the SHG and 2PAF images through visualization, indicating that there is a critical need to integrate all modalities for more efficient detection of TMVs using novel statistics and machine learning tools.

Furthermore, although different individuals have imaging at very different locations, different modalities from the same individual are observed from the same tissue and thus share some com- mon structures, which could be informative for capturing the spatial locations and formations of TMVs. Therefore, it is crucial to utilize homogeneous information from multimodality imaging within individuals.

boundary area, therefore we study a segmented imaging of 150 × 150 pixels more closely at the boundary area for both the normal rat and the cancerous rat (see Figure 4.6). In addition, due to the limited sample size at the current experimental stage, we generate more sample images using a resampling technique. Specifically, every original image is segmented into nine subregions with no overlapping, and each subregion has a size of 50 × 50 pixels. Additional sample images are generated by randomly sampling from the original subregions with replacement as well as adding certain noise to the subregions samples. The noise added to each pixel is generated from N (0, σ2), where σ is set as ˆσm/5, and ˆσm (m = 1, . . . , 4) is the sample standard deviation for the mth- modality imaging. Consequently, we generate a training data set and a testing data set with a total of 40 subjects for each data set, where each subject has four-modality images taken in the same subregion.

We compare the proposed IMTL method to the four models described in Section 4.3. The tuning parameters associated with the latent rank for the TR, the MPCA, the HOCPD and the IMTL are selected in order to minimize the prediction error rates in the validation set, which is generated following the same resampling procedure as described above.

Table 4.4 provides the prediction results on the testing set, which illustrates that the proposed method outperforms the other methods significantly in terms of achieving the highest overall pre- diction accuracy rate, sensitivity and specificity. In addition, both the proposed IMTL and the HOCPD utilizing all four-modality imaging outperform all other methods, showing the prediction power improved by integrating multimodality information. Figure 4.6 displays several common tubular spatial structures of TMVs shared by the tumor imaging on modalities 1, 2 and 4. The proposed IMTL method applying an individualized layer to different imaging modalities from the same subject performs the best for capturing important heterogeneous TMVs’ patterns and thus enhances the prediction power for cancer detection. The VPL method and the tensor regression model perform inadequately with prediction accuracy below 55%. This is because the locations of the TMV’s vary heterogeneously for different subjects and the signals of the TMV’s are weak compared to the modality background (e.g., third modality). Therefore the VPL and the TR are

not powerful at capturing the TMV’s effects in predicting disease outcomes.

4.7

Discussion

In this article, we propose an individualized multilayer tensor learning model incorporating imag- ing covariates to predict targeted responses. In the proposed two-stage model, we first extract important features from tensor covariates incorporating different layers to achieve dimension re- duction through tensor decomposition techniques, and then fit a prediction model with the extracted features. We illustrate the proposed method through numerical studies and data application on both single-modality and multimodality imaging data.

A major contribution of the proposed method is that we achieve tensor decomposition through utilizing an additional layer of individual structure in addition to population-shared modality- specific structure following the CANDECOMP/PARAFAC decomposition. Our method is mo- tivated by a multiphoton multimodality imaging study for breast cancer diagnosis, where tumor locations of imaging can vary for different individuals, yet the multimodality images from the same individual share important spatial information. Most existing methods assuming fixed signal locations are either infeasible or inefficient in our setting. In contrast, the proposed individualized layer is capable of capturing within-subject spatial features through integrating different modali- ties’ imaging information for the same individual. Both simulation studies and real data analyses demonstrate that the proposed method can achieve higher diagnostic accuracy compared to other competing methods.

In the proposed method, we only consider a linear transformation for dimension reduction on the tensor data, e.g., the CP decomposition. Due to the complex nature of imaging data, it will be our next step to employ nonlinear transformation techniques such as manifold dimension reduction. Moreover, it is worth future research to develop supervised feature extraction through constructing a constraint tensor decomposition conditional on outcome responses.