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The gamma camera and data acquisition electronics used on our laboratory are

designed to the FastSPECT II specifications [34]. The gamma camera is connected to

multi-channel pulse-height analyzers to read and digitize the output from the 9 PMTs

depicted in figure 2.5. The pulse-height analyzers are connected to a large memory buffer

via an ethernet cable, where data is stored in list-mode format. The electronics are then

controlled using LabVIEW [55]. The pulse height analyzers, measure the height of the

pulses generated from each PMT. For a single event, 9 numbers representing the pulse

height of each PMT are collected in a temporary memory buffer. During acquisition a list

of all events including their acquisition time and pulse height are accumulated over time

then stored for post processing. Data collected in this manner is known as a “list-mode”

acquisition [9, 78]. Using list-mode data, the projection images are calculated from the

involves two tasks; position estimation and the rejection of scatter events in the

acquisition process.

Ideally, all the energy of a gamma ray photon would be converted directly to

visible light, where all the visible photons are emitted along the same trajectory and

location of the incident gamma photons. As illustrated in figure A.1 a gamma ray

interaction with the NaI crystal results in a spread of visible photons. The energy of an

incident gamma ray is therefore spread over a range of pulse heights corresponding to the

range in energies of the secondary visible photons. Plotting the histogram of pulse heights

collected from a NaI detector, a peak corresponds to the energy of the gamma emitter.

The “photopeak” obtained from emissions of 99m_{Tc correspond to an energy of 140 keV.}

Ideally, a 100 keV gamma ray photon will produce an average of visible

photons in a NaI crystal [86]. The geometry of the crystal and the conversion from visible

5000_±p5000

Figure A.1: Frequency spectrum of 99m_{Tc with a photopeak corresponding to 140 keV} (adapted from 14-4 in [23]). 140 keV photopeak (99m_Tc) Count s FWHM Digital units Compton Valley Compton Plateau

photons to electrons in the PMT can effect how broad the photopeak is for a given NaI

detector. The Full Width at Half Max (FWHM) of the photo peak is a measure of the

efficiency of the detector for a given gamma emitter [86].

The path of the incident gamma ray may also be redirected through non-ideal

interactions with the NaI crystal prior to conversion to visible light. These interactions are

known as scatter events and make up the low energy events in the energy histogram.

Scattering of gamma ray photons within the range of energies used clinically are

attributed to Compton scattering and the photoelectric effect [23, 86], and these

interactions are illustrated in figure A.2. In Compton scattering, a gamma ray photon

interacts with a free electron in the NaI crystal. In these collisions, some energy from the

incident gamma ray is transferred to the electron. To conserve momentum, the gamma

photon’s energy is reduced, and the trajectory is redirected. Subsequent scintillation

events within the NaI crystal will result in pulse heights corresponding to an energy lower

than the initial energy of the incident photon. In the photoelectric effect, all of the gamma

ray’s energy is transferred to a free electron within the NaI crystal. The free electron is

deflected and repeatedly interacts with the NaI crystal. The deflection interactions with

the crystal generate multiple low energy visible photons until the electron’s kinetic

energy is absorbed.

Additional features in the energy histogram can be attributed to background

events. Interactions between the gamma photons and lead shielding/collimators produce

x-rays that can interact with the NaI crystal [23, 86]. Gamma photons that pass through

the NaI crystal can interact with other components of the detector, such as the light guide.

back towards the NaI crystal, generating visible photons that are reflected off the back of

the detector face. These events are known as “backscatter” events [86]. Both lead shield

interactions and backscatter events form characteristic photopeaks in the Compton

plateau. Additionally, scattering events may occur within the subject/object being imaged,

which will increase the frequency of scattering events in the detector’s energy spectrum.

The list-mode pulse heights from each PMT therefore contain events due to

scattering. In order to calculate a projection image from the list-mode data the event’s

position must be determined and events falling within the Compton plateau must be

rejected, as the location of these events will not be consistent with the collimator

projection model. However, rejecting events using the energy spectra, known as energy

windowing [22], is complicated by the fact that the pulse heights from each PMT are not

only sensitive to the energy of events but also position. Rather, as part of the camera’s

calibration, the response of each PMT is measured by physically placing a well-

✓

Compton Scattering Photoelectric Effect

e-

e -

e -

e-

e -

Figure A.2: Scattering in the NaI crystal: (a) Compton scattering, (b) photoelectric effect (figures adapted from [85]).

collimated point source on a 78 x 78 grid of points spaced 0.15 mm apart on the face of

the detector. For each grid point, data is acquired until 5000 events are counted. The

mean response for all 9 PMTs at each grid point, known as the Mean Detector Response

Function (MDRF), is used to generate a look-up-table (LUT) from which events obtained

during an acquisition can be compared, as pictured in figure A.3 for one of our

laboratory’s cameras.

The _{location on the 78 x 78 grid for a given event is estimated using a}

maximum-likelihood position estimator and treating the MDRF as the mean estimate for

all locations [22, 9, 43]. Finally, a “likelihood window” [22, 43] procedure is used

to accept or reject the event. Using the MDRF, the likelihood of the event occurring at the

(x, y)

(x, y)

Figure A.3: In situ mean response of each of the 9 PMTs to a well collimated point source positioned on a 78 x 78 grid in front of the camera (based on figure 12.2 in [21]).

estimated location with an energy of 140 keV is calculated. If the likelihood is above a

fixed threshold the event is accepted. In this way, list-mode-data is sorted to a grid

location or pixel. The entire projection is then the 2D grid of pixels where the value of a

given pixel is the number of events assigned to its location. Using a high likelihood

threshold, likelihood windowing demonstrates a similar probability of accepting 140 keV

events to energy windowing, considered the gold standard, and Bayesian windowing

Appendix B

Maximum Likelihood Expectation Maximization