Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data

Abstract

In medical SPECT imaging, we seek to simultaneously obtain the internal radioactive sources and the attenuation map using not only ballistic measurements but also first order scattering measurements. The problem is modeled using the radiative transfer equation by means of an explicit nonlinear operator that gives the ballistic and scattering measurements as a function of the radioactive source and attenuation distributions. First, by differentiating this nonlinear operator we obtain a linearized inverse problem. Then, under regularity hypothesis for the source distribution and attenuation map and considering small attenuations, we rigorously prove that the linear operator is invertible and we compute its inverse explicitly. This allows to prove local uniqueness for the nonlinear inverse problem. Finally, using the previous inversion result for the linear operator, we propose a new type of iterative algorithm for simultaneous source and attenuation recovery for SPECT based on Neumann series and a Newton-Raphson algorithm.

Algorithms for joint activity–attenuation estimation from positron emission tomography scatter

“…It should be noted that due to computational complexity, 2-D considerations are not uncommon in recent studies regarding image reconstruction from scattered photons [20, 26, 43]. Furthermore, a more sophisticated imaging model that offers more realistic system matrix components A λ and would be subject to the same measurement equations and lead to the same derivation of MLGA.…”

PET/MRI attenuation estimation in the lung: A review of past, present, and potential techniques

“…Motivated by the fact that attenuation in PET is mainly due to Compton scattering, and that scatter events represent up to 40% of the total recorded coincidences, several groups have considered the possibility of using scattered emission data as an additional source of information to overcome reconstruction ambiguities in MLAA, especially without ToF data. These attempts share similarities with SPECT studies, and extend the earlier suggestion of using individual photon energy information . The idea of deriving quantitative information from scatter essentially relies on the possibility of estimating scattering angles through photon energy measurements, although this is limited by the energy resolution of the PET detectors.…”

(An overview of) Synergistic reconstruction for multimodality/multichannel imaging methods

“…An example is the reconstruction of both attenuation and tracer concentration from detection of both unscattered and scattered photons, the latter having reduced energy. This has been demonstrated in both SPECT [29][30][31] and PET [32][33][34]. -MCMI: Multi-Time Imaging.…”