A MATHEMATICAL MODEL OF SINGLE-PHOTON EMISSION COMPUTED TOMOGRAPHY (RADON TRANSFORM, COMPTON SCATTER, ATTENUATION, NUCLEAR MEDICINE)

Single-photon emission computed tomography (SPECT) is a nuclear-medicine imaging technique that has been shown to provide clinically useful images of radionuclide distributions within the body. The problem of quantitative determination of tomographic activity images from a projection data set leads to a mathematical inverse problem which is formulated as an integral equation. The solution of this problem then depends on an accurate mathematical model as well as a reliable and efficient inversion algorithm. The effects of attenuation and Compton scatter within the body have been incorporated into the model in the hopes of providing a more physically realistic mathematical model.;With the use of the single-scatter approximation and an energy-windowed detector, the effects of Compton scatter are incorporated into the model. The data is then taken to be the sum of primary photons and single-scattered photons. The scattered photons are modeled by a scatter operator acting on the original activity distribution within the object where the operator consists of convolution with a given analytic kernel followed by a boundary cut-off operation. A solution is given by first applying the inverse attenuated Radon transform to the data set. This leads to a Fredholm integral equation to which a Neumann series solution is constructed. Again simulations are performed to validate the accuracy of the assumptions within the model as well as to numerically demonstrate the reconstruction procedure.;The attenuated Radon transform is the mathematical basis of SPECT. In this work, the case of constant attenuation is reviewed and a new proof of the Tretiak-Metz algorithm is presented. A space-domain version of the inverse attenuated Radon transform is derived. A special case of this transform that is applicable when the object is rotationally symmetric, the attenuated Abel transform is derived, and its inverse is found. A numerical algorithm for the implementation of the inverse attenuated Radon transform with constant attenuation is described and computer simulations are performed to demonstrate the results of the inversion procedure.