Solution of Sparse High-Dimensional Linear Inverse problems with Application to Analysis of Dynamic Contrast Enhanced Imaging Data

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The project is motivated by analysis of Dynamic Contrast Enhanced (DCE) imaging data. DCE imaging provides a noninvasive measure of tumor angiogenesis and has great potential for cancer detection and characterization. It offers an extremely useful tool for the evaluation and optimization of new therapeutic strategies as well as for long-term evaluation of therapeutic impacts of anti-angiogenic treatments. The current project will be greatly beneficial for a) reducing health care costs by replacing expensive and invasive tests by analysis of medical images, and by shortening hospital stays for stroke patients due to better monitoring of drug effectiveness;b) the medical practice, since development of novel path-breaking methodologies for analysis of DCE imaging data will potentially improve clinical outcomes by providing non-invasive tools for cancer detection and characterization and for longitudinal evaluation of therapeutic impact of anti-angiogenic treatments; c) medical research, since the software for analysis of DCE imaging data will be freely available to everyone who carries out examination of such data and, thus, will contribute to design of new methodologies; d) other types of medical imaging techniques, since methodologies resulted from the proposal will contribute to understanding of reconstruction of sparse high-dimensional functions in the presence of noise; ande) various fields of science such as geophysics and astronomy, which rely on solution of noisy inverse problems. The current project presents an integral effort of merging applications and theory. Mathematically, the problem reduces to solution of a noisy version of a matrix-variate Laplace convolution equation based on discrete measurements. We shall use very modern techniques designed for recovery of sparse representations that led to many successful developments in image and signal analysis and other applications. However, since majority of those methods rely on very stringent assumptions, only few of them have been adopted for solution of inverse ill-posed problems. In addition, recently, acquisition of new types of data brought to light new types of the linear inverse problems where the function of interest is itself vector or matrix-valued. So far, the new challenge has been addressed by separate recovery of each component of the solution. However, recent developments in the area of sparse matrix estimation allow for much more coherent solution of the problem. Furthermore, in many applications, the operator itself is unknown and is estimated from data. Although often the uncertainty in the operator is ignored, we are planning to account for operator uncertainty. The goal of the present proposal is to extend techniques based on penalized optimization or exponential weights to solution of sparse ill-posed high-dimensional linear inverse problems where the operator may be not be known exactly and the function of interest is matrix-variate. We are planning to put a solid theoretical foundation under the proposed new methodologies, develop computational algorithms for their implementation, and apply the newly constructed algorithms to solution of Laplace convolution equation and, subsequently, to analysis of DCE imaging data obtained in ongoing REMISCAN studies.