# Performance Evaluation of the Current Birmingham PEPT Cameras
## 2 Simple Characterisation of PEPT Systems  
In PEPT, the location of the radioactively labelled tracer particle is determined from triangulation of a number of detected LORs. Because of fundamental uncertainties in the decay process (the distance travelled by the positron before annihilation, and the slight acollinearity of the resulting photons) as well as limitations of the detector systems (e.g., finite crystal sizes), these LORs will exhibit a distribution of distances from the tracer location, which can be characterised by a standard deviation \(w\). Simplistically, provided the tracer does not move significantly during the measurement, using \(N\) LORs, its position can be determined with an uncertainty of order \(\frac{w}{\sqrt{N}}\). The data also include a background of invalid events, such as random coincidences or events where one or both of the photons have scattered before detection. The Birmingham PEPT algorithm iteratively discards these invalid events, whose LORs do not pass close to the tracer position, so that only a fraction \(f\) of the detected events are useful for locating the tracer. By taking a sufficiently large sample of LORs, a stationary tracer can be located with arbitrary precision given by \(\frac{w}{\sqrt{fN}}\). This relationship can be used to determine an effective value of \(w\) for the positron camera; note that in most cases the distribution of LORS will be anisotropic so that the tracer can be located better in one direction than another, but it is still useful to represent the overall 3D uncertainty by a single value. For a moving tracer there will be an optimum sample size, large enough to give adequate statistics but not so large that the tracer moves significantly during the time interval used for location. One way of approaching this is to set this time interval \(\Delta t\) such that the tracer moves a distance roughly equal to the corresponding stationary location precision:  
\[v\Delta t\approx\frac{w}{\sqrt{fN}}=\frac{w}{\sqrt{fR\Delta t}}, \tag{1}\]  
where \(v\) is the tracer speed and \(R\) is the data rate (events per second). Rearranging this equation gives  
\[\Delta t\approx\left[\frac{w^{2}}{v^{2}fR}\right]^{1/3} \tag{2}\]  
and hence an uncertainty of roughly \(\left[\frac{w^{2}}{fR}\right]^{1/3}\) obtained by using  
\[N=R\Delta t\approx f^{-1/3}\left[\frac{Rw}{v}\right]^{2/3}. \tag{3}\]  
These simple approximations work quite well provided the number of events used per location is reasonably large (>>10). For example, the ADAC Forte camera (described below) is capable of recording up to \(10^{5}\) events per second and has an intrinsic spatial resolution of around 9 mm. Assuming that 50% of the detected events are useful (\(f=0.5\)) one might expect that a tracer moving at 1 m/s could be located every 1 mm along its path with an uncertainty of around 1 mm--this corresponds to a sample size \(N=100\) and reflects what is often achieved in practice. At lower data rates and/or higher tracer speeds, the data become sparser and the algorithm finds it harder to discriminate against invalid events. Data rate \(R\) is therefore crucial for tracking at high speeds. We define the sensitivity \(\Gamma\) of the positron camera as the limit, as source activity \(A\) tends to zero of the fraction of positrons emitted by the source, which result in detected events, so that \(R=\Gamma A\). For a symmetrical system and a source at the centre of its FOV, this sensitivity can be expressed as the product \(\Omega\epsilon^{2}\), where \(\Omega\) is the fraction of the total \(4\pi\) solid angle covered by detectors and \(\epsilon\) is the quantum efficiency of these detectors for detecting 511 keV photons--this expression relies on the fact that for a central source, if one photon is emitted in a direction so as to strike a detector, its partner will automatically strike the opposite detector.