We consider detection of high-energy photons in positron emission tomography (PET) using thick scintillation crystals.
Abstract
We consider detection of high-energy photons in positron emission tomography (PET) using thick scintillation crystals. Parallax effect and multiple Compton interactions in this type of crystals significantly reduce the accuracy of conventional detection methods. In order to estimate the scintillation point coordinates based on photomultiplier responses, we use asymptotically optimal nonlinear techniques, implemented by feed-forward neural networks, radial basis functions (RBF) networks, and neuro-fuzzy systems. Incorporation of information about angles of incidence of photons, significantly improves accuracy of estimation. The proposed estimators are fast enough to perform detection, using conventional computers. Monte-Carlo simulation results show that our approach outperforms the conventional Anger algorithm.
Introduction
Detection of high-energy photons emitted as the result of positron decay is one of the most important low-level stages in PET imaging. A typical detector used in PET is based on the so-called Anger scintillation camera. Incident high-energy gamma quanta, generated due to positron decay, produce scintillation effect in the crystal. As the result, a shower of low energy photons in the visible and UV spectra is emitted. These photons are collected by an array of photo-multipliers (PMTs), optically coupled to the scintillation crystal, and invoke electric impulses in them. The PMT responses are utilized in estimation of the scintillation point coordinates.
We consider a non-collimated Anger camera, based on thick crystals with high photon penetration depth such as NaI(Tl). Application of such thinck crystals in PET scanners is desirable, due to their low cost and very high light output; they were previously used primarily in gamma ray astronomy.
The majority of existing detection algorithms are based on centroid arithmetic, usually combined with correction maps. Their application appears, however to be problematic in the case of thick crystals due to significant parallax observed at large radiation incidence angles.
Our work presents a solution for these problems, incorporating side information on the photon incidence angle into the process of position estimation. We use localized, asymptotically optimal, nonlinear estimators, implemented by feed-forward neural networks, radial basis functions (RBF) networks, and neuro-fuzzy systems. As a byproduct, we get accurate position estimation over the entire area of detector including the edges. This is difficult to obtain with centroid arithmetics algorithms.
The solution
Scintillation detector can be considered to be a complicated non-linear stochastic system that maps the photon line of flight (LOF) into a vector of PMT responses. The stochastic aspects of this mapping are related to the random nature of position of the fist interaction within the crystal, possible multiple Compton interactions and the number of visible/UV photons in each interaction, registered by PMTs. Statistical effects of these factors depends heavily on the incidence angle, which can be estimated with reasonable accuracy using approximate LOF coordinates, measured by a pair of opposite detectors (obtained, for example, from the Anger algorithm).
Given the incidence angle, LOF is defined by planar coordinates y on the surface of the crystal. For every incidence angle, we implement an optimal nonlinear estimator of y of the form 
, where 
is a family of functions, parameterized by the vector of parameters W.
A reasonable criterion for estimator optimality is the expectation of some error function, for example, the expected squared error
We are interested in forms of , that possess the property of a universal approximator: when the number of parameters W is large enough, any bounded function f(x) can be approximated with given accuracy over a bounded domain by an appropriate choice of W.
Given the PMT responses to a set of known LOFs
(referred to as a training set), we find such W, that minimizes the mean-squared error (MSE) on the training set, i.e:
This process is referred to as training. When the training set is sufficiently large, the MSE approximates the expected squared error with any desired accuracy. Under such conditions, a universal approximator with sufficient parameters approaches the optimal non-linear estimation. In this work we used three type of universal approximators implemented as artificial neural networks.
Our scheme is based on a combination of coarse and fine estimators (Figure 1). The core of the scintillation coordinates estimation algorithm is a set of fine estimators, implemented as neural networks. Fine estimators are trained on scintillation events in different (possibly overlapping) regions and at different incidence angles. Coarse estimators, based, for example, on the Anger algorithm determine the rough position and incidence angle of the photon. According to this information, the appropriate fine estimator is selected. Such a combination of estimators allows reduction in the size of each network and accelerates the training.
The sets of neural networks are trained independently on simulated (or on measured) PMT responses resulting from scintillation events in appropriate regions and angles.
Figure 1 – Block diagram of a practical ANN-based scintillation coordinates estimation algorithm: estimation of scintillation coordinates in detector 1 using side information from detector 2
Simulation results
In order to test the proposed approach and compare it with other algorithms, we performed a Monte Carlo simulation of ray tracing and gamma quanta interaction in a scintillation detector. The simulation was performed using a slightly modified version of TRIUMF detector modeling platform introduced by Tsang et al. The main goal of the simulation was to show the advantages of localized angle-dependent estimators. The region of interest and the incidence angle involved in the estimator selection were assumed to be known. A model of a NaI(Tl) scintillation crystal of size 210x210x45 mm, separated with a 20 mm glass light guide was simulated (Figure 2).
Figure 2 – Detached NaI scintillation detector used in the simulation
Three tests were performed in order to analyze the effectiveness of different scintillation coordinates estimation algorithms. The tests were performed in small regions of the detector at different incidence angles. The conventional Anger algorithm with non-linear pulse amplification, commonly used in PET, was compared to localized estimators implemented using localized linear regression (LLR), multilayer perceptron (MLP), radial-basis function network (RBF) and NEFPROX fuzzy system.
Figure 3 – RMS error histogram for incidence angle of 0 degrees
Figure 4 – RMS error histogram for incidence angle of 10 degrees
Simulation results showed that the use of localized estimators outperformed the existing approach by up to 30% for normal photons and by more than 40% for photons impinging at the detector surface at 10 degrees. For larger incidence angles the proposed estimation methods showed even more impressing results. The proposed estimators had almost zero bias and smaller error standard deviation and FWHM compared to the Anger algorithm.
Figure 5 – RMS error of the compared algorithms
Conclusions
The proposed method of scintillation coordinates estimation in PET, based on artificial neural networks, incorporates information about the incidence angle in the estimation algorithm. This approach is capable of estimating directly the photon line of flight, given PMT responses from a pair of detectors. The proposed algorithm allows compensation for the parallax effect, it reduces the resolution degradation due to multiple Compton scattering and increased effective detection area. Our approach outperforms conventional scintillation coordinates estimation algorithms in simulation studies.
Our approach can be implemented in software or hardware using conventional computers.
Acknowledgment
We are especially grateful to our project supervisors Dr. Michael Zibulevsky and Prof. Yehoshua Zeevi for their very kind help and guidance throughout this work, without which it could not be possible; to the VISL engineer Johanan Erez for his infinite patience and to GE Medical Systems staff for their valuable notes and comments. We would like to thank Dr. Christian Morel from Institut de Physique des Hautes Energies Lausanne, Switzerland for kindly supplying us with his previous works on the subject and the TRIUMF Group, Canada for making us available an outstanding simulation platform. Our special thanks to Gunnar Raetsch from The German National Research Center for Information Technology for his excellent MATLAB RBF framework and to Dr. Detlef Nauck and Prof. Rudolf Kruse from University of Magdenburg, Germany for the NEFPROX code. We would also like to express our gratitude to the Ollendorf Minerva Center Fund for supporting this project and to all the people not mentioned here, who helped us throughout the work.