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The image processing task of the recovery of an image from a noisy or compromised image is an illposed inverse problem. To solve this problem, it is necessary to incorporate prior information about the smoothness, or the structure, of the solution, by incorporating regularization. Here, we consider linear blur operators with an efficiently-found singular value decomposition. Then, regularization is obtained by employing a truncated singular value expansion for image recovery. In this study, we focus on images for which the image blur operator is separable and can be represented by a Kronecker product such that the associated singular value decomposition is expressible in terms of the singular value decompositions of the separable components. The truncation index k can then be identified without forming the full Kronecker product of the two terms. This report investigates the problem of learning an optimal k using two methods. For one method to learn k we assume the knowledge of the true images, yielding a supervised learning algorithm based on the average relative error. The second method uses the method of generalized cross validation and does not require knowledge of the true images. The approach is implemented and demonstrated to be successful for Gaussian, Poisson and salt and pepper noise types across noise levels with signal to noise ratios as low as 10. This research contributes to the field by offering insights into the use of the supervised and unsupervised estimators for the truncation index, and demonstrates that the unsupervised algorithm is not only robust and computationally efficient, but is also comparable to the supervised method. 

Prostate cancer is a serious public health concern in the United States. The primary obstacle to effective long-term management for prostate cancer patients is the eventual development of treatment resistance. Due to the uniquely chaotic nature of the neoplastic genome, it is difficult to determine the evolution of tumor composition over the course of treatment. Hence, a drug is often applied continuously past the point of effectiveness, thereby losing any potential treatment combination with that drug permanently to resistance. If a clinician is aware of the timing of resistance to a particular drug, then they may have a crucial opportunity to adjust the treatment to retain the drug’s usefulness in a potential treatment combination or strategy. In this study, we investigate new methods of predicting treatment failure due to treatment resistance using a novel mechanistic model built on an evolutionary interpretation of Droop cell quota theory. We analyze our proposed methods using patient PSA and androgen data from a clinical trial of intermittent treatment with androgen deprivation therapy. Our results produce two indicators of treatment failure. The first indicator, proposed from the evolutionary nature of the cancer population, is calculated using our mathematical model with a predictive accuracy of 87.3% (sensitivity: 96.1%, specificity: 65%). The second indicator, conjectured from the implication of the first indicator, is calculated directly from serum androgen and PSA data with a predictive accuracy of 88.7% (sensitivity: 90.2%, specificity: 85%). Our results demonstrate the potential and feasibility of using an evolutionary tumor dynamics model in combination with the appropriate data to aid in the adaptive management of prostate cancer. 

Over time, tumor treatment resistance inadvertently develops when androgen deprivation therapy (ADT) is applied to metastasized prostate cancer (PCa). To combat tumor resistance, while reducing the harsh side effects of hormone therapy, the clinician may opt to cyclically alternates the patient’s treatment on and off. This method, known as intermittent ADT, is an alternative to continuous ADT that improves the patient’s quality of life while testosterone levels recover between cycles. In this paper, we explore the response of intermittent ADT to metastasized prostate cancer by employing a previously clinical data validated mathematical model to new clinical data from patients undergoing Abiraterone therapy. This cell quota model, a system of ordinary differential equations constructed using Droop’s nutrient limiting theory, assumes the tumor comprises of castration-sensitive (CS) and castration-resistant (CR) cancer sub-populations. The two sub-populations rely on varying levels of intracellular androgen for growth, death and transformation. Due to the complexity of the model, we carry out sensitivity analyses to study the effect of certain parameters on their outputs, and to increase the identifiability of each patient’s unique parameter set. The model’s forecasting results show consistent accuracy for patients with sufficient data, which means the model could give useful information in practice, especially to decide whether an additional round of treatment would be effective.