A Method for Point Spread Function Estimation for Accurate Quantitative Imaging

Abstract

Aim: A method to determine the point spread function (PSF) of an imaging system based on a set of 3-D Gaussian functions is presented for a robust estimation of the recovery corrections for accurate activity quantification in positron emission tomography (PET) and single-photon emission computed tomography systems. Materials and Methods: The spatial resolution of the ALBIRA II PET subsystem was determined using a 370-kBq 22Na point source. The measured data were reconstructed with a maximum-likelihood expectation-maximization algorithm. The PSF was calculated based on the National Electrical Manufacturing Association (NEMA) NU-4 2008 protocol and on alternative methods based on three 3-D fitting functions for the xyz-directions: 1) a 3-D Gaussian function (3-D 1-Gauss) and convolutions of this function with a pixel size (3-D ${\text {Gauss}}_{p}$ ) or source dimension of O 0.25 mm (3-D ${\text {Gauss}}_{s}$ ); 2) the sum of two Gaussian functions (3-D 2-Gauss); and 3) three Gaussian functions (3-D 3-Gauss). Goodness of fit and the method based on an Akaike information criterion were used for choosing the best function. A MATLAB-based mathematical source simulation study was performed to quantify the relevance of PSFs calculated from the different methods. Results: Based on the PSFs calculated from the NEMA protocol, the full-width at half-maximum (FWHM) in xyz-directions were 1.68, 1.51, and 1.50 mm. The corresponding results using 3-D Gauss and 3-D ${\text {Gauss}}_{s}$ functions both were (1.87 ± 0.01), (1.70 ± 0.01), and (1.50 ± 0.01) mm and for 3-D ${\text {Gauss}}_{p}$ were (1.84 ± 0.01), (1.67 ± 0.01), and (1.47 ± 0.01) mm. The FWHMs calculated with 3-D 2-Gauss and 3-D 3-Gauss were (1.78 ± 0.01), (1.74 ± 0.01), and (1.83 ± 0.01) mm and (1.76 ± 0.03), (1.72 ± 0.03), and (1.78 ± 0.03) mm, respectively. All coefficients of variations of the fit parameters were ≤29% and the adjusted $R^{2}$ were ≥0.99. Based on Akaike weights $w_{i}$ , the 3-D 3-Gauss method was best supported by the data ( $w_{i} = 100$ %). The simulation study showed a relative error in quantification of spherical lesions in the range of 15%–45% for lesions of diameters 1–5 mm compared to the PSFs based on the NEMA method. Conclusion: An alternative method to calculate the PSFs of imaging systems to accurately correct for recovery effects is presented. The proposed method includes choosing and fitting of 3-D functions, validation of fitting quality, and choosing the function best supported by the data along with an estimation of the uncertainty.