Photon propagation and detection in single‐photon emission computed tomography—An analytical approach

Abstract

An analytical theory of photon propagation and detection in single-photon emission computed tomography (SPECT) for collimated detectors is developed from first principles. The total photon detection kernel is expressed as a sum of terms due to the primary and the Compton scattered photons. The primary as well as contributions due to every order of Compton scattering are calculated separately. The model accounts for the three-dimensional depth dependence of the collimator holes as well as for nonhomogeneous attenuation. No specific assumptions about the boundary or the homogeneity of the attenuating medium are made. The energy response of the detector is also modeled by the theory. Analytical expressions are obtained for various contributions to the photon detection kernel, and the multidimensional integrals involved are calculated using standard numerical integration methods. Theoretically calculated projections and scatter fractions for the primary and the first through second scattering orders are compared with our own experimental results for a small cylindrical primary radiation source immersed at various positions in a uniform cylindrical phantom. Also, theoretically calculated scatter fractions for a small spherical (pointlike) source in a uniform elliptic phantom are compared with experimental and Monte Carlo simulation results taken from the recent literature. The results from the analytical method are essentially exact and are free from the inaccuracies inherent in the numerical simulation methods used to deal with the photon propagation and detection problem in SPECT so far. The method developed here is unique in the sense that it provides accurate theoretical predictions of results averaged over an infinite number of simulations or experiments. We believe that our theory enhances an intuitive understanding of the complex image formation process in SPECT and is an important step toward solving the inverse problem, that of reconstructing the primary radiation source distribution from the measured gamma camera projections.

Accuracy and Precision of Radioactivity Quantification in Nuclear Medicine Images

“…Both analytical and Monte Carlo–based scatter modeling have been used. Analytical scatter models can be broken down into exact 11,12 and approximate methods. 13-15 The exact methods tend to be computationally intensive, especially for modeling several scatter orders.…”

Quantitative determination of Gd‐DTPA concentration in T1‐weighted MR renography studies

“…Background signal of approximately 50% was subtracted from the raw kidney counts, consistent with other reports (16). To correct for tissue attenuation, the count rate for each kidney was multiplied by a factor exp͑␣d͒, where ␣ ϭ 0.124 cm -1 is the effective attenuation coefficient, which also accounts for the scattering effects (17,18), and d is the kidney depth. Kidney depths were determined from each subject's axial T 1 -weighted MR images as the distance from the posterior surface of the body to the kidney center at the kidney's mid-height.…”

Approximate 3D iterative reconstruction for SPECT

“…The system matrix was calculated from analytical expressions of photon transport through attenuating media. 15 For both mathematical and experimental phantom studies the matrix incorporated the effects of detector response, attenuation, and finite energy resolution for primary and first order scatter assuming an emission energy of 140 keV ͑Tc-99m͒. In order to evaluate the DEW ML-EM method in the experimental phantom study, an additional matrix was calculated that considered only primary photons.…”

“…Since the focus of this paper is on the efficient use of the system matrix once it has been calculated and because the approximate 3D methods may be used independent of the means of calculating the matrix, details of the system matrix computation are not addressed here but have been published previously. 15 To compute the system matrix for the brain phantom at one projection angle considering primary and first order scatter and 0.712-cm pixel size required approximately 1.5 h on a modern, general purpose workstation ͑Sun Microsystems, Inc., Ultra 2, model 1200͒.…”

“…͑Thus the methods can be integrated into various approaches for system matrix calculation including Monte Carlo, slab-derived, 13,14 and numerical integration of analytical photon transport equations. 15 ͒ These methods greatly reduce the computational demand of the reconstruction by including only a subset of the total system matrix and retaining that subset in program memory during the reconstruction. Unlike previous work using this approach which excluded from the reconstruction only the zero value elements of system matrix, 10,12 this study examines the effect on the recon-structed image, and on computational demand, of excluding nonzero elements.…”