The present proposal is motivated by analysis of Dynamic Contrast Enhanced Computed Tomography (DCE-CT)data. DCE-CT provides a non-invasive measure of tumor angiogenesis and has great potential for cancer detection and characterization. It offers an in vivo tool for the evaluation and optimization of new therapeutic strategies as well as for longitudinal evaluation of therapeutic impacts of anti-angiogenic treatments. The difficulty of the problem stems from the fact that DCE-CT is usually contaminated by a high-level of noise and does not allows to directly measure the function of interest. Mathematically, the problem reduces to solution of a noisy version of Laplace convolution equation based on discrete measurements, an important problem which also arises in mathematical physics, population dynamics, theory of superfluidity and fluorescence spectroscopy. However, exact solution of the Laplace convolution equation requires evaluation of the inverse Laplace transform which is usually found using Laplace Transforms tables or partial fraction decomposition. None of these methodologies can be used in stochastic setting. In addition, Fourier transform based techniques used for solution of a well explored Fourier deconvolution problem are not applicable here since the function of interest is defined on an infinite interval while observations are available only on on a finite part of its domain and it may not be absolutely integrable on its domain. In spite of its practical importance, the Laplace deconvolution problem was completely overlooked by statistics community. Only few applied mathematicians took an effort to solve the problem but they either completely ignored measurement errors or treated themas fixed non-random values. For this reason, estimation of a function given noisy observations on its Laplace convolution on an a finite interval requires development of a completely novel statistical theory. The objective of the present proposal is to fill in this gap and to develop a path-breaking transformative statistical methodology for solution of various aspects of Laplace deconvolution problem: formulation of fundamental theoretical results, algorithmic developments and, finally, application of the newly derived techniques to analysis of DCE-CT data.
The current proposal presents an integral effort of merging applications and theory. Results of this effort will be greatly beneficial for a) the medical practice since development of novel path-breaking methodologies for analysis of DCE-CT data will potentially improve clinical outcomes by providing non-invasive tool for cancer detection and characterization as well as for longitudinal evaluation of therapeutic impacts of anti-angiogenic treatments. First, DCE-CT can be used used for assessment of intra-tumor physiological heterogeneity, thus offering an in vivo tool for the evaluation and optimization of new therapeutic strategies. Second, DCE-CT provides a non-invasive tool for cancer detection and characterization as well as for longitudinal evaluationof therapeutic impact of anti-angiogenic treatments, and therefore, can act as a tool for improvement of those treatments. b) the medical research since algorithmic developments and the software for interpretation of DCE-CT data will contribute to design of new methodologies for non-invasive longitudinal evaluation of tumor angiogenesis, cancer detection and characterization. Software will be freely available to anyone who carries out examination of such data and can be used in cancer and medical imaging research. c) various fields of science since data in the form of noisy measurements of the Laplace convolution of a function of interest with a known or estimated kernel appear in many areas of natural science. Analysis of decay curves in fluorescence spectroscopy is one but not the only example. However, due to the theoretical and methodological challenges associated with the solution of Laplace deconvolution problem, these data are usually analyzed in an "ad-hoc" manner, or the formulation is abandoned overall in favor of a much less precise but easier treatable set-up. Novel path-breaking methodologies which will be constructed as a result of this proposal will benefit all those applications. d) training and development of the future work force and promoting interdisciplinary research by carrying out various educational activities, attracting and training Ph.D., M.S. and undergraduate students, teaching a Special Topics graduate course, organizing interdisciplinary seminars and promoting interdisciplinary research and diversity.