Moehrs Sascha, Defrise Michel, Belcari Nicola, Guerra Alberto Del, Bartoli Antonietta, Fabbri Serena, Zanetti Gianluigi
Department of Physics, University of Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy.
Phys Med Biol. 2008 Dec 7;53(23):6925-45. doi: 10.1088/0031-9155/53/23/018. Epub 2008 Nov 12.
Iterative image reconstruction algorithms for positron emission tomography (PET) require a sophisticated system matrix (model) of the scanner. Our aim is to set up such a model offline for the YAP-(S)PET II small animal imaging tomograph in order to use it subsequently with standard ML-EM (maximum-likelihood expectation maximization) and OSEM (ordered subset expectation maximization) for fully three-dimensional image reconstruction. In general, the system model can be obtained analytically, via measurements or via Monte Carlo simulations. In this paper, we present the multi-ray method, which can be considered as a hybrid method to set up the system model offline. It incorporates accurate analytical (geometric) considerations as well as crystal depth and crystal scatter effects. At the same time, it has the potential to model seamlessly other physical aspects such as the positron range. The proposed method is based on multiple rays which are traced from/to the detector crystals through the image volume. Such a ray-tracing approach itself is not new; however, we derive a novel mathematical formulation of the approach and investigate the positioning of the integration (ray-end) points. First, we study single system matrix entries and show that the positioning and weighting of the ray-end points according to Gaussian integration give better results compared to equally spaced integration points (trapezoidal integration), especially if only a small number of integration points (rays) are used. Additionally, we show that, for a given variance of the single matrix entries, the number of rays (events) required to calculate the whole matrix is a factor of 20 larger when using a pure Monte-Carlo-based method. Finally, we analyse the quality of the model by reconstructing phantom data from the YAP-(S)PET II scanner.
用于正电子发射断层扫描（PET）的迭代图像重建算法需要扫描仪的复杂系统矩阵（模型）。我们的目标是为YAP-(S)PET II小动物成像断层扫描仪离线建立这样一个模型，以便随后将其与标准的ML-EM（最大似然期望最大化）和OSEM（有序子集期望最大化）一起用于全三维图像重建。一般来说，系统模型可以通过解析、测量或蒙特卡罗模拟获得。在本文中，我们提出了多射线方法，这可以被视为一种离线建立系统模型的混合方法。它纳入了精确的解析（几何）考虑以及晶体深度和晶体散射效应。同时，它有潜力无缝模拟其他物理方面，如正电子射程。所提出的方法基于多条从图像体积到探测器晶体或从探测器晶体到图像体积追踪的射线。这种射线追踪方法本身并不新颖；然而，我们推导了该方法的一种新颖数学公式，并研究了积分（射线端点）点的定位。首先，我们研究单个系统矩阵元素，结果表明，与等距积分点（梯形积分）相比，根据高斯积分对射线端点进行定位和加权能得到更好的结果，特别是在仅使用少量积分点（射线）的情况下。此外，我们表明，对于单个矩阵元素的给定方差，使用基于纯蒙特卡罗的方法计算整个矩阵所需的射线（事件）数量要大20倍。最后，我们通过重建来自YAP-(S)PET II扫描仪的体模数据来分析模型的质量。
