Prostate cancer is the most common malignancy among men. To confirm an initial detection by a prostate-specific antigen test, magnetic resonance imaging (MRI) is used, but MRI is costly. Ultrasound is a costeffective imaging modality and shows promising results in diagnosing prostate cancer, especially the dynamic contrast-enhanced ultrasound. Dynamic contrast-enhanced ultrasound (DCEUS) is an imaging modality that allows the imaging of the injected microbubbles by exploiting their non-linear acoustic scatter. Because of their size, comparable to red blood cells, ultrasound contrast agents can flow through the vascular tree down to the microvessels, enabling the visualization and, possibly, quantification of the angiogenic processes associated with cancer growth. Although several techniques are applied to DCEUS to reduce noise, speckle noise still exists.

Speckle noise occurs due to the coherent imaging of many microbubbles in one resolution cell. In existing methods, a low-rank matrix decomposition is applied to the DCEUS acquisitions using singular value decomposition, and the despeckling is done by keeping the highest few singular vectors and values. The DCEUS acquisitions come in a tensor format, rich with higher-order structure. The application of the matrixbased denoising technique does not utilize the original tensor structure. This dissertation focuses on despeckling through higher-order tensor decomposition methods. We tackle the following research question: "How can low-rank tensor decomposition methods be leveraged to effectively denoise DCEUS acquisitions of the prostate for improved prostate cancer detection?" We apply tensor decomposition methods that utilize orthogonal factors. In the spatial domain, the orthogonality allows the separation of the malignant and benign regions and the separation of the tissue and the vasculature. In the time domain, the orthogonality allows for capturing the components that correspond with the bubble movement and rejects the components related to the noise.

Despeckling of DCEUS through low-rank tensor decomposition has not been conducted before, and we propose tensor estimation algorithms for this application by utilizing established tensor decomposition frameworks. We start our research by modeling speckle noise as white Gaussian noise (WGN) with sparse outliers. We assess the performance of convex tensor estimation algorithms through simulation. We propose a novel weighting scheme for the soft-thresholding of the singular values. Instead of iterative thresholding, we can truncate the tensor and deviispeckle the DCEUS acquisitions. We propose a rank estimation method for DCEUS acquisitions. Instead of modeling speckle noise as WGN with sparse outliers, we minimize its negative log-likelihood and propose a gradient-based denoising algorithm.

Next, we investigate the classification performance of prostate cancer by comparing the proposed algorithms with the literature. We use the area under the receiver-operator characteristic curve (ROC-AUC) metric to assess the classification performance. For the voxel-based cancer diagnosis of 94 prostate cancer patients, truncated multilinear singular value decomposition has a better performance for the majority of the prostate cancer markers when the ROC-AUC metric is used. A rank estimation technique incorporating WGN with sparse outliers, followed by truncated multilinear singular value decomposition (tr-MLSVD), is the best-performing denoising method for DCEUS. In the context of the main research question, the cancer diagnosis performance of DCEUS acquisitions improves the majority of the time when a tensor-based denoising technique is used. On average, the tensor-based denoising techniques yield approximately a 1.6% relative improvement in the ROC-AUC metric compared to the literature. This translates to billions of additional correct voxel-level malignancy discriminations in our clinical study, which may significantly impact downstream classification and localization performance.

We conclude with a theoretical study on the lower bound of the tensor decomposition method that performs the best for despeckling DCEUS. We calculate a lower bound for estimating the components of MLSVD when the ranks are known. In general, the CCRB that lower bounds the variance of the unbiased estimates of the components of MLSVD does not exist due to the non-uniqueness of the decomposition. However, when the mode-n singular values are unique, the CCRB exists. Additionally, when the multilinear ranks are high and modal singular values are well-separated, it is a tight bound. Such cases do not typically occur with real data such as DCEUS, highlighting the limited modeling capability of the CCRB.