In its simplest form, a detector acquires data from a thin axial or transverse cross section containing radioactivity, by linear scanning from multiple directions around the cross section, as shown in Figure 13.1. In practice, data are simultaneously acquired from multiple axial cross sections by using a scintillation camera (detector head) in SPECT or multiple axial blocks of detectors in PET, mainly to reduce the data acquisition time. In an advanced form, data are simultaneously acquired also from multiple directions around the cross section by using multiple detector heads in SPECT or the coincidence detection mechanism in PET, as explained later. In these cases, the principle of transverse tomography is the same and consists of two steps: acquisition of linear projection data for each thin cross section from multiple directions, and reconstruction of the cross sections from these data.

Pixel Width, Matrix, and Number of Projections. Let us consider a simple case where a single detector scans a cross section linearly from many directions. Let us also assume that as the detector moves from one position to the next along a line, it receives counts from N columns
that are perpendicular to the direction of a scan and chosen to be equally spaced. This is shown in Figure 13.2 for the two positions of the detector in two different linear scans (projections). Two columns perpendicular to each other have only a small area in common. This area is called a pixel, and the entire cross section can be imagined to consist of these small pixels distributed in a square matrix of size N × N. When scans are performed from different directions, different combinations of pixels, but always perpendicular to the direction of scan, contribute to the counts of the detector at different positions along a scan line. These are called projections of a plane on a line, and the number M of these projections is a critical parameter in the reconstruction of the cross section. For accurate and artifact-free reconstruction, M should be nearly equal to N. From the projection data, the radioactivity of each pixel is computed using the methods described in the next section.

Data acquisition with a single detector is very time-consuming. A variety of methods has been developed to reduce the acquisition time or to increase the overall sensitivity. For example, to avoid the linear motion, a linear array of N detectors can be used simultaneously to gather data from a projection. This increases sensitivity N-fold compared with a single detector. Further gains can be achieved by using multiple linear arrays, stacked to make a 2D array of detectors, gathering data simultaneously from many cross sections. This is the reason why a scintillation camera is routinely used for SPECT data acquisition in clinical nuclear medicine. Use of multiple-head scintillation cameras (e.g., 2 or 3) allows for further gains in sensitivity due to simultaneous acquisition of data from multiple directions. In cardiac applications, other novel dedicated methods of data collection have been quite successful, resulting in further gains in sensitivity, as described in Chapter 14.

ECT imaging results in the generation of multiple transverse (axial) cross sections, thus generating volumetric or 3D images. As discussed in Chapter 11, the individual image elements in a 3D image are referred to as voxels. However, for a given cross section, we can continue to refer to them as pixels. The number of pixels in a cross section is determined by the number of columns, N, in a scan direction and is simply N × N. In the case of Figure 13.2, this number is 9 × 9 = 81. For a given cross section, the number of pixels and the size of the pixel Δx are inversely related. The smaller the pixel size is, the larger the number of pixels or the matrix size. In routine clinical nuclear medicine, the matrix size is typically 128 × 128, resulting in 16,384 pixels, but this can vary. What determines the pixel size is discussed next.

Pixel Width, Resolution, and Sensitivity. Pixel width, Δx (and therefore the matrix size N × N and number of linear projections M) is an important consideration in data acquisition because it pertains to the amount of time required to complete a scan. Pixel width depends on the resolution obtainable in the tomogram, which in turn depends on the resolution of the collimator used in SPECT or the size of the detector used in PET. A rule of thumb is that the pixel width Δx should be less than the resolution of the collimator used in the SPECT or the size of the detector used in the PET, and is typically set to be around half of it. To improve the resolution of an imaging device, one can reduce the size of the hole in the collimator of the detector (SPECT) or use smaller size detectors (PET). In the case of SPECT, this results in two things: (1) it reduces the sensitivity of the detector as more events are stopped by the new collimator. As a result, to achieve the same statistical reliability, one has to spend more time at each location. In addition, (2) the reduction in Δx as a result of higher resolution increases the number of pixels N along the scan line. which necessitates more linear projections M from different angles for optimum sampling. This again requires more time. Thus, we face a similar (actually worse) dilemma in SPECT as we face in planar imaging; that is, improvement in resolution costs dearly in terms of sensitivity and a compromise has to be made between the two. In the case of PET, the above two concerns are less relevant, as coincidence detection (Chapter 15) naturally addresses them. However, using smaller-sized detectors results in greater manufacturing costs and greater complexity of the associated electronics.

In CT, because of the large number of photons available from an x-ray tube (hence, relatively smaller statistical errors compared with SPECT or PET), the typical resolution is about 0.5 mm, and the number of projection directions is 1,000 or more. In nuclear medicine, where the available number of photons in a typical cross section is relatively small, typical resolution for SPECT is in the range of 1 cm and the number of linear scans (projections) is 60 or more for cardiac applications, where 180-degree coverage is used (to keep the detectors closer to the heart) and ˜120 for other applications involving 360-degree coverage. Newer cardiac-specific SPECT systems have a resolution of about 6 mm. For PET, a typical resolution is about 5 to 6 mm.

Reconstruction of a transverse cross section from multiple linear projections through it is a general inverse problem that is encountered in diverse fields. The first analytic solution of this problem was provided by Radon way back in 1917. However, because of the complexity involved in the computations, practical realization of the solution was achieved quite later with the advent of and easy access to large digital computers. There exist two general classes of image reconstruction algorithms, analytic and iterative. Of the analytic methods, the most successful method is known as filtered back-projection (FBP), which is a modified, more accurate form of back-projection. This has been widely used in CT, SPECT, and PET. At the same time, iterative methods have achieved increasing use in both SPECT and PET imaging. We provide an overview of these methods.

Back-Projection (a Simple Explanation). Let us consider a simple cross section containing two radionuclide sources as depicted in Figure 13.3A. When data are acquired from this cross section with an appropriate detector and by scanning from multiple directions (line projections), the responses of the detector are shown as scan 1, scan 2, scan 3, scan 4, and so on. Note that in this particular example, the detector receives counts only at one or two discrete locations in each scan, and these locations are determined by perpendicular lines from each source to the detector. In the back-projection method of reconstruction, as the name implies, data received at one location of the detector is back-projected along the column perpendicular at that location, and so on. Because no prior knowledge exists about which or how many pixels in that column contain radioactivity (which is actually what we aim to estimate), one assumes that counts were equally contributed from all pixels in that column. This is depicted in Figure 13.3B. Counts received by the detector in scan 1 at the two locations are distributed equally in the 12 pixels of columns 3 and 7, respectively. When a similar procedure is repeated for other scans (2, 3, 4, etc.),
the reconstruction of the cross section is complete. As can be seen, the two original radioactive sources have been reproduced at the correct locations (two highest-intensity pixels in the reconstructed image), although not as point sources but as star patterns. As a result, a number of pixels (shown as shaded) that contained no radioactivity in the original cross section are reconstructed inaccurately as containing some radioactivity. If projection angles (linear scans) are acquired increasingly, the artifacts will look less like a star and more like a uniform blur near a given focal uptake. In any case, back-projection results in inaccuracies.