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

9623077研究了正电子发射断层扫描(正电子发射断层扫描)的谢普-瓦迪概率模型的改进，以及相应的期望最大化最大似然(EM-ML)重建算法。该模型用一个未知量为Borel测度的积分方程代替了通常的有限线性系统。研究内容包括分析该算法的收敛特性，用样条基和小波基实现的数值方法，以及对这些基的平滑效果的分析。根据Borel测度的正则性，收敛结果为有限维EM-ML算法的数值实现看似发散提供了数学原因。此外，由于EM-ML算法最近被推广到核和未知函数为非负的线性积分方程，这些结果为该算法的收敛这一公开问题提供了重要的见解。此外，研究还介绍、分析和开发了处理意外重合和衰减引起的数据错误的新算法。作者还研究了在PET和单光子发射计算机断层扫描(SPECT)中将发射与探测器几何形状联系起来的概率函数的数学性质。SPECT的初步工作将这些函数与经典的泊松核联系起来。这项研究对于开发准确、高效、快速的EM-ML重建算法的替代算法具有重要意义。科学、工程和医学中许多问题的数学实现产生了在Hadamard意义下不适定的反问题，而正性在其中起着重要的作用。这一建议主要涉及在正电子发射计算机断层扫描的核成像过程中出现这种问题的具体例子。在这一过程中，患者被给予放射性药物，该药物被感兴趣器官的不同区域相反地吸收，并根据吸收的量发射正电子。这些辐射由围绕感兴趣区域的PET扫描仪收集，然后用于重建算法，该算法生成包含有关感兴趣器官或区域新陈代谢的重要信息的图像。这些信息对血液流动和代谢活动的研究非常有用。例如，它是一种有价值的工具，用于诊断肿瘤，确定肿瘤的生长速度；心理学研究，将大脑的激活区域映射到认知任务；确定各种药物对大脑的影响；以及对人类心脏健康的定量测量。作者为PET过程建立了一个精确的数学模型，并引入了重建PET图像的新的数学和数值方法。这些重建算法足够灵活，可以处理PET数据中的重大错误，这些错误是由于解剖障碍导致的，这些障碍阻止了一些发射的光子被记录在适当的探测器中。这项工作的一个重要组成部分是开发和评估高效、可靠、准确的PET图像重建算法。虽然这项研究主要是针对PET研究，但本研究的结果也适用于其他医学和工程问题，如肝脏活检、制成品的无损评估以及从外层空间恢复模糊图像。***

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. ***