Mathematical Sciences: Positron Emission Tomography: Modelling, Analysis and Algorithms

数学科学：正电子发射断层扫描：建模、分析和算法

• 批准号: 9623077
• 负责人: Bernard Mair
• 金额: $ 12万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623077&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Positron; Emission; Tomography

9623077 Mair The proposers investigate a refinement of the Shepp-Vardi probabilistic model for positron emission tomography (PET), and the corresponding expectation-maximization maximum-likelihood (EM-ML) reconstruction algorithm. This model replaces the usual finite linear system with an integral equation in which the unknown is a Borel measure. The research includes an analysis of the convergence properties of this algorithm, numerical methods of implementation by spline and wavelet bases, and an analysis of the smoothing effects of these bases. The convergence results provide mathematical reasons in terms of the regularity properties of Borel measures, for the seeming divergence of the numerical implementations of the finite dimensional EM-ML algorithm. Also, since the EM-ML algorithm has been recently extended to linear integral equations in which the kernel and the unknown function are nonnegative, these results provide important insight into the open problem of convergence of this algorithm. In addition, the research introduces, analyzes, and develops novel algorithms for dealing with data errors due to accidental coincidences and attenuation. The proposers also investigate the mathematical properties of the probability functions which link the emissions to the detector geometry in both PET and single photon emission computed tomography (SPECT). Preliminary work for SPECT links these functions to the classical Poisson kernel. This study is important for developing accurate, efficient, fast, alternatives to the EM-ML reconstruction algorithm. %%% The mathematical realization of many problems in science, engineering, and medicine, give rise to inverse problems which are ill-posed in the sense of Hadamard, and in which positivity plays an important role. This proposal deals mainly with the particular example of such a problem occuring in the nuclear imaging procedure of PET. In this procedure, a patient is given a radiopharmaceutical which is absorbed disprop ortionately by various regions of the organ of interest and emits positrons according to the amount absorbed. These emissions are collected by PET scanners which surround the region of interest and then used in reconstruction algorithms which generate images containing important information about the metabolism of an organ or region of interest. This information is extremely useful for blood flow and metabolic activity studies. For instance, it is a valuable tool in the diagnosis of tumors, in determining their rates of growth; in psychological studies for mapping activation areas of the brain to cognitive tasks, in determining the effects of various drugs on the brain, and in quantitative measures of the health of the human heart. The proposers develop a refined mathematical model for the PET process and introduce novel mathematical and numerical methods for the reconstruction of PET images. These reconstruction algorithms are sufficiently flexible to deal with significant errors in the PET data, caused by anatomical obstructions which prevent some of the emitted photons from being registered in the appropriate detector. An important component of this work is the development and evaluation of efficient, reliable, accurate algorithms for reconstructing PET images. Although primarily directed to PET studies, the results obtained in this reserach are also applicable to other medical and engineering problems, such as those which occur in liver biopsies, nondestructive evaluation of manufactured items, and the restoration of blurred images from outer space. ***

9623077 Mair提案人研究了正电子发射断层扫描（PET）的Shepp-Vardi概率模型的改进，以及相应的期望最大化最大似然（EM-ML）重建算法。该模型用一个积分方程代替了通常的有限线性系统，其中未知量是一个Borel测度。研究内容包括分析该算法的收敛性、样条基和小波基的数值实现方法以及这些基的平滑效果。 收敛结果提供了数学原因的博雷尔措施的正则性方面，似乎发散的数值实现的有限维EM-ML算法。此外，由于EM-ML算法最近已扩展到线性积分方程，其中的内核和未知函数是非负的，这些结果提供了重要的洞察到该算法的收敛性的公开问题。此外，研究介绍，分析和开发新的算法，用于处理由于偶然的巧合和衰减的数据错误。提议者还研究了概率函数的数学性质，该概率函数将PET和单光子发射计算机断层扫描（SPECT）中的发射与探测器几何结构联系起来。SPECT的初步工作将这些功能与经典的泊松核联系起来。该研究对于开发准确、高效、快速的EM-ML重建算法的替代方案具有重要意义。 在科学、工程和医学中，许多问题的数学实现都产生了逆问题，这些逆问题在阿达玛意义上是不适定的，其中正性起着重要的作用。 该建议主要涉及在PET的核成像过程中发生的这样的问题的具体实例。在该过程中，给患者施用放射性药物，该放射性药物被感兴趣的器官的各个区域不适当地吸收，并根据吸收的量发射正电子。这些发射由围绕感兴趣区域的PET扫描仪收集，然后用于重建算法，所述重建算法生成包含关于感兴趣器官或区域的代谢的重要信息的图像。这些信息对于血流和代谢活动研究非常有用。例如，它是一个有价值的工具，在肿瘤的诊断，在确定其增长速度;在心理学研究中映射大脑的激活区域的认知任务，在确定各种药物对大脑的影响，并在定量测量人类心脏的健康。 提出者为PET过程开发了一个精确的数学模型，并为PET图像的重建引入了新的数学和数值方法。这些重建算法足够灵活，可以处理PET数据中由解剖学障碍物引起的显著误差，这些解剖学障碍物阻止一些发射的光子在适当的探测器中被配准。这项工作的一个重要组成部分是开发和评估有效，可靠，准确的算法重建PET图像。 虽然主要是针对PET研究，在这项研究中获得的结果也适用于其他医疗和工程问题，如那些发生在肝活检，非破坏性评估的制造项目，并恢复模糊的图像从外层空间。 ***