Laplace Deconvolution and Its Application to Analysis of Dynamic Contrast Enhanced Computed Tomography Data

本提案的动机是动态对比增强计算机断层扫描（DCE-CT）数据的分析。DCE-CT提供了一种无创的肿瘤血管生成测量方法，在癌症检测和表征方面具有很大的潜力。它为评估和优化新的治疗策略以及纵向评估抗血管生成治疗的治疗效果提供了一种体内工具。这个问题的困难在于DCE-CT通常受到高水平噪声的污染，并且不允许直接测量感兴趣的函数。在数学上，该问题简化为基于离散测量的拉普拉斯卷积方程的噪声解，这是一个重要的问题，在数学物理、种群动力学、超流体理论和荧光光谱学中也会出现。然而，拉普拉斯卷积方程的精确解需要计算拉普拉斯逆变换，这通常是用拉普拉斯变换表或部分分式分解得到的。这些方法都不能用于随机环境。此外，基于傅里叶变换的技术用于解决已深入研究的傅里叶反褶积问题在这里不适用，因为感兴趣的函数是在无限区间上定义的，而观测仅在其域的有限部分上可用，并且在其域上可能不是绝对可积的。尽管拉普拉斯反卷积问题具有重要的实际意义，但却被统计学界完全忽视了。只有少数应用数学家努力解决这个问题，但他们要么完全忽略测量误差，要么将其视为固定的非随机值。由于这个原因，在有限区间内对一个给定拉普拉斯卷积噪声观测值的函数进行估计，需要发展一种全新的统计理论。本提案的目的是填补这一空白，并开发一种开创性的变革性统计方法来解决拉普拉斯反卷积问题的各个方面：基本理论结果的制定，算法的发展，最后，应用新导出的技术来分析DCE-CT数据。目前的方案是应用和理论相结合的整体努力。这项工作的结果将对医疗实践大有裨益，因为用于分析DCE-CT数据的新颖的开创性方法的发展，将通过为癌症检测和表征以及抗血管生成治疗的治疗效果的纵向评估提供非侵入性工具，潜在地改善临床结果。首先，DCE-CT可用于评估肿瘤内的生理异质性，从而为评估和优化新的治疗策略提供一种体内工具。其次，DCE-CT为癌症检测和表征以及抗血管生成治疗效果的纵向评估提供了一种非侵入性工具，因此可以作为改进这些治疗的工具。b)自算法发展以来的医学研究和用于解释DCE-CT数据的软件将有助于设计用于肿瘤血管生成、癌症检测和表征的无创纵向评估的新方法。软件将免费提供给任何进行此类数据检查的人，并可用于癌症和医学成像研究。c)各种科学领域，因为具有已知或估计核的感兴趣函数的拉普拉斯卷积的噪声测量形式的数据出现在自然科学的许多领域。荧光光谱中衰减曲线的分析是一个但不是唯一的例子。然而，由于与拉普拉斯反卷积问题的解决相关的理论和方法上的挑战，这些数据通常以一种“特设”的方式进行分析，或者摈弃整个公式，以支持一种更不精确但更容易处理的设置。由于这一建议而建立的新颖的开创性方法将使所有这些应用受益。d)通过开展各种教育活动，吸引和培养博士、硕士和本科生，教授专题研究生课程，组织跨学科研讨会，促进跨学科研究和多样性，培养和发展未来的劳动力，促进跨学科研究。

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 fora) 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.