Energy sampling requirements for SPECT angular response functions

Objectives Photon-tracking Monte Carlo simulations of single photon emission computed tomography (SPECT) are inherently inefficient. One proposal to speed such simulations is to use angular response functions (ARFs) to model the collimator/detector. Unfortunately, generation of ARFs is also extremely compute intensive: a table is needed for each incident energy, for each collimator/detector combination, and for each energy window of interest. The method is only useful when the same collimator/detector/energy window combination will be used for a large number of simulations. If the number of tables needed, or the number of photons needed to generate a table, can be reduced, the method will be feasible for a wider range of applications. In this work we examine the former as a function of energy: how many tables are needed?

Methods We simulated Ga-67 imaged with a MEPG collimator and 1cm NaI(Tl) detector with 9% FWHM energy resolution at 140 keV. A 3.7cm thick glass backscatter compartment was included behind the detector. Ga-67 has 10 emission energies spanning the range 91-888 keV; we used 3 energy windows: 84-103, 166-203 and 270-330 keV, the acquisition windows for clinical Ga-67 studies. Using GATE’s ARF calculation package, we simulated 4 billion decays to calculate tables for each energy window at each Ga-67 emission energy and for several intermediate energies. We compared normalized profiles through the incident and azimuthal angle indexes of the tables to determine how quickly the shape of ARFs vary with energy. We summed the elements in tables to determine how their amplitudes vary with energy. For each energy window we estimated tables (1) by linearly interpolating tables from two bracketing incident energies, or (2) by adjusting the amplitude of a table at a nearby incident energy to the amplitude of the table at the desired incident energy (saving time as amplitudes are calculated precisely with far fewer decays). Estimated tables were compared to simulated tables at the same incident energies.

Results At high (> 600 keV) and low (< 150 keV) energies the normalized profiles vary slowly with energy. Around 300 keV the normalized profiles vary quite quickly. The amplitudes of the tables vary linearly with energy at high energies, but are non-linear at lower energies, particularly for energies near an ARF’s energy window. At high energies ARFs are well approximated using linear interpolation with bracketing energies over 50 keV in either direction; at low energies ARFs are well approximated by adjusting the amplitude of an ARF from an energy up to 20 keV different. At energies around 300 keV the normalized profiles vary particularly quickly; we have not yet tested energy intervals less than 20 keV, but found both approximation methods insufficient at that interval.

Conclusions We expect that we can model the high energy range using sampling at 100 keV intervals for energies >500 keV. Sampling at 20 keV intervals should be sufficient for for photon energies <200 keV. We need to further investigate the range between 200 and 500 keV using smaller intervals to determine the sampling requirements. These conclusions only apply to simulations using MEGP collimators. However conclusions about the energy sampling necessary for Ga-67 should be generalizable to many other isotopes.