CHAPTER 1: GENERAL INTRODUCTION
1.4 TOMOGRAPHIC RECONSTRUCTIONS IN SPECT
Many algorithms have been developed for tomographic reconstruction from projection data. Up until the past decade, the most popular algorithm was filtered back- projection, which was adapted from X-ray Computed Tomography [32]. Although
computationally efficient, filtered back-projection did not allow the reconstruction of quantitative images in SPECT. This was due primarily to the inability to incorporate the physics of gamma ray propagation into the filtered back-projection algorithm, and to a lesser extent, the inaccurate modeling of Poisson noise in gamma ray detection [33]. Today, filtered back-projection has been replaced extensively with iterative reconstruction. Although much more computationally demanding, iterative reconstruction can incorporate gamma ray attenuation, Compton scatter, and distance- dependent resolution effects, as well as modeling noise properly [34].
We provide a brief introduction to iterative reconstruction. The reader is referred to [35] for a detailed review. First, we consider the process of measuring a distribution of radioactivity using a gamma camera. Let the vector F, with size M×1 and elements
f1…fM, be a discretized representation of the distribution of radioactive disintegrations in
a region that we are interested in measuring. The elements of F represent the individual
volume elements, or voxels, of the object. Let the vector G, with size N×1 and elements
g1…gN, represent the pixel-by-pixel measurements of the radioactivity as recorded by a
gamma camera as it rotates around the source of radioactivity – the elements of a sinogram. Thus G would be a measure of the projection of F. This can be written
mathematically as:
G
=
AF
(1.3) Where A is an N×M matrix. Each element in A, corresponds to the probability that the gamma ray emitted in a particular voxel is detected by a particular pixel at a particular gamma camera position (during its rotation around the radioactive source). For example, elements in A represent the probability that voxel fi emits gamma rays that are recordedby the gamma camera in pixel gj. Equation (1.3) is commonly referred to as the forward
projection, and A the transition matrix. The attractiveness of iterative reconstruction lies in the flexibility of the transition matrix. The transition matrix can include extremely detailed models of the physics of both photon propagation in non-uniform media and detection with a gamma camera. Various groups have developed methods to include attenuation effects using radioisotope transmission images [36, 37] or X-ray CT [38, 39, 40] or even MRI [41]. Incorporation of Compton scatter, although more computationally demanding, has received considerable attention [42, 43], as has distance-dependent resolution effects [44].
One of the most popular iterative reconstruction algorithms is Maximum- Likelihood Expectation Maximization (MLEM), introduced to the nuclear medicine society by Shepp and Vardi [45], and Lange and Carson [46]. We provide a brief introduction to the algorithm, but the reader is referred to Lange and Carson [46] for a detailed derivation. The objective of tomographic reconstruction is to solve Equation (1.3) by finding the best estimate of F: the mean number of radioactive disintegrations in
the measurement region that can produce the measurements G with the highest
likelihood assuming Poisson statistics. The MLEM formula is designed to find this best estimate of F. It can be written as [46]:
∑
∑
∑
= = = +=
N n M nm m k m nm n N n nm k m k ma
f
a
g
a
f
f
1 1 1 1 (1.4)Where = new estimate 1 + k m f = old estimate k m f
= measured number of counts
n
g
n = bin number m = pixel number k = iteration number
anm = probability photon emitted from mth pixel will be detected in nth bin
This equation seems complex at first presentation, but is simply a set of successive projection and back-projection operations. It can be expressed more simply in words [47]:
Image (k+1) = Image (k) × Normalized Backprojection of Measured Projections (1.5)
Projection of image (k)
The MLEM algorithm requires a starting estimate of the image to be reconstructed, referred to as F0 in Equation (1.4) or Image0 in Equation (1.5). This is often a uniform disk of the value 1. As the MLEM iterations proceed, the image gradually converges to the tomographic representation of the original distribution of radioactivity that was measured. My research utilized a modified version of MLEM known as Ordered Subsets Expectation Maximization (OSEM) [48]. Studies have shown that OSEM converges to a useful image faster than MLEM, and so it is sometimes termed an accelerated version of MLEM. Similar equations and process previously described are used by both algorithms, but are implemented differently. For the case of
MLEM a projection/back-projection calculation must be done for every projection angle before an updated estimate can be generated. In OSEM, only a portion of the projection angles are projected and back-projected for each iteration. Successive iterations utilize different subsets, gradually stepping through all projection angles.
Attenuation, scatter, and distant-dependent resolution are important forms of image degradation. For the purpose of this thesis we will discuss methods to compensate for all these physical effects, but will concentrate more on attenuation correction since this thesis’ primary focus is on methods of attenuation correction.