A PET scanner counts coincident events between pairs of detectors. The straight line connecting the centers of two detectors is called a line of response (LOR). Unscattered photon pairs recorded for a specific LOR arise from annihilation events located within a thin volume centered around the LOR. This volume typically has the shape of an elongated parallelipiped and is referred to as a tube of response.

To each pair of detectors da,db is associated an LOR Ld,db and a sensitivity function y/d,db (r = (x,y, z)) such that the number of coincident events detected is a Poisson variable with a mean value

where t is the acquisition time and f(r) denotes the tracer concentration. We assume that the tracer concentration is stationary and that f(r) = 0 when V(x2 + y2) > Rf, where RF denotes the radius of the field-of-view (FOV). The reconstruction problem consists of recovering f(r) from the acquired data pd,db, {da, db} = 1 • • • ,Nlor, where NLOR, the number of detector pairs in coincidence, can exceed 109 with modern scanners.

The model defined by Eq. (1) is linear and hence implies that nonlinear effects due to random coincidences and dead time be pre-corrected. In the absence of photon scattering in the tissues, the sensitivity function vanishes outside the tube of response centered on the LOR. In such a case, the accuracy of the spatial localization of the annihilation events is determined by the size of the tube of response, which in turn depends on the geometrical size of the detectors and on other factors such as the photon scattering in the detectors, or the variable depth of interaction of the gamma rays within the crystal (parallax error, figure 2.26).

We have so far considered a scanner comprising multiple small detectors. Scanners based on large-area, position-sensitive detectors such as Anger cameras can be described similarly if viewed as consisting of a large number of very small virtual detectors.

Analytic reconstruction algorithms assume that the data have been pre-corrected for various effects such as randoms, scatter and attenuation. In addition, these algorithms model each tube of response as a mathematical line joining the center of the front face of the two crystals(1). This means that the sensitivity function ¥dd (r) is zero except when r e Ld,db. With this approximation, the data are modeled as line integrals of the tracer distribution:

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