Energy Scale-Factor Estimation for Use in Photomultiplier Tube Energy Calibration Using C-SPECT

Abstract

An array of photomultiplier tubes (PMTs) provides energy readout for gamma cameras, leading to event selection and positioning. However, operational and environmental changes, such as temperature, can cause PMTs to "drift" away from their nominal energy readouts and, therefore, require a correction procedure to return to their reference energies. We present two methods for determining the energy-scale change of each PMT using data collected on C-SPECT, a dedicated cardiac single photon emission computational tomography (SPECT) scanner. A scan of a vertical line source of ^99mTc provides the data from which we produce an energy histogram for each of the 130 PMTs. Each energy histogram is composed of events passing an energy-fraction selection to give events closest to the PMT center. We consider energy fractions ranging from 0.25% to 5.00%. For our analysis, we use bootstrapping to create data realizations as well as emulating energy-scale changes (simultaneously and independently for all PMTs) in the data. Using the average relative error as a measurement of the accuracy and the standard deviation from taking bootstrapped replicates of our data as a measurement of our precision, we determine the energy-scaling to within -0.05% ± 0.03% (mean and standard deviation, respectively).

Keywords: C-SPECT; Energy Calibration; Gamma Camera; Photomultiplier Tubes (PMT); SPECT.

Fig. 1:

Photographs of the C-SPECT system. Top: Photograph of the fully operational C-SPECT system. Bottom: Top-down photograph of its detector system during assembly.

Fig. 2:

A photo (left) and diagram (right) of the layout of the 13 PMTs which provide the energy readout for an individual detector module. For size reference, the PMTs are 51 mm in diameter and the module is 90×168 mm. The 4 different colors highlight the different PMT groups (Central, Outer, Edge, and Corner) that represent the different positions of the PMT relative to the crystal (dark-gray rectangle). The Bulk PMT category is made up of all the non-Outer PMTs, which are contained within the dashed line in the diagram. The Overlap PMT category (not indicated) is the Edge and Corner PMTs combine, which overlap the detector module edge on either side of the central column of PMTs.

Fig. 3:

Energy-fraction histograms for an example Central (left) and Edge (right) PMT (see Fig. 2). The vertical lines indicate the effective-energy fraction for a given energy-fraction selection percentage. The insert details the same histogram but zoomed in to energy fraction values from 0.55 to 0.75. The engineering prefix of M (×10⁶) is used to denote the number of events in each histogram bin.

Fig. 4:

Example energy histograms for a Central PMT (left), Edge PMT (middle), and Corner PMT (right). Each energy histogram plot uses a bin-width of 5 a.u. and shows the composition of 12 different energy-fraction selection percentages (from 0.25% to 3.00%). The engineering prefix k (×10³) is utilized to express the number of events in each histogram bin.

Fig. 5:

The effective-energy fraction versus the emulated-energy scaling (k) for an energy-fraction selection of 1.00% (left) and the effective-energy fraction versus the energy-selection percentage with no energy scaling, i.e., k = 1 (right). The results are shown for the 4 different PMT groups for a single detector module using the average over 25 replicates. The uncertainty (standard deviation) is much smaller than the marker.

Fig. 6:

Fitted photopeak energy using the photopeak fitting method versus energy-fraction selection percentage for an example Central PMT (blue) and an Edge PMT (red). The open circles markers are the mean fitted photopeak energy of the data using 100 replicates whereas the error bar is the standard deviation of the same. Note that for most data points the error bar is smaller than the marker.

Fig. 7:

The MLE k relative error for each PMT (indexed from 1 to 182) for an energy-fraction selection percentage of 1.25%. The black points with errors bars are the mean and standard deviation across the 100 bootstrapped replicates. To better distinguish the individual results a red line at 0% error has been added. The color bands demarcate the 4 PMT groups. The three regions with comparably low uncertainty are the Bulk PMTs (Central, Corner, and Edge PMTs). The two other regions (in green) are, together, the Outer PMTs.

Fig. 8:

The MLE k relative error versus the energy-fraction selection percentage for Bulk PMTs (left) and for Outer PMTs (right). The data points (blue and red markers) represent the average across the 100 replicates and all PMTs in the Bulk and Outer groups, respectively. The error bar is the standard deviation of the same.

Fig. 9:

The average MLE k relative error versus the emulated energy-scale factor (k) is shown for an energy-fraction selection of 1.25% for the Bulk and Outer PMT groups (shown using blue and red markers, respectively). The error bars represent the standard deviation of the 25 replicates and the 11 or 2 PMTs of each PMT group category. The behavior at k = 1 follows from results shown in Section III.B.2.

Fig. 10:

The average MLE k relative error for Bulk and Outer PMTs versus the energy-fraction selection percentage. The error bar represents the standard deviation across the 25 replicates, the 50 non-unity energy scaling, and the number of PMTs in the Bulk and Outer PMT groups while the value of the marker is the mean of the same.

Fig. 11:

The MLE k relative error is shown for an energy-fraction selection of 1.00% for all 182 PMTs with 25 replicates. The top plot is a scatter plot versus the emulated (true) energy-scale factor for Bulk and Outer PMTs using blue and red markers, respectively. The bottom plot shows the same data at the same selection percentage, but the 25 realizations have been averaged for each unique PMT and plotted against PMT index like Fig. 7. The error bar represents the standard deviation of the 25 realizations. The blue line (yellow band) represents the mean (95% confidence across) the Bulk PMTs. Likewise, the red line (green band) represents the same but for the Outer PMTs.

Fig. 12:

The MLE k relative error versus energy-fraction selection percentage for Bulk, Outer, and all PMTs (shown in blue, red, and magenta, respectively). The central values of the data points are the mean over the 25 realizations and all associated PMTs for the given selection percentage. The error bars are the standard deviations of the same.