Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography

Abstract

Reconstruction procedures that account for attenuation in forming maximum-likelihood estimates of activity distributions in positron-emission tomography are extended to include regularization constraints and accidental coincidences. A mathematical model is used for these effects. The corrections are incorporated into the iterations of an expectation-maximization algorithm for numerically producing the maximum-likelihood estimate of the distribution of radioactivity within a patient. The images reconstructed with this procedure are unbiased and exhibit lower variance than those reconstructed from precorrected data.

On complete-data spaces for PET reconstruction algorithms

“…Although the PET applications may often violate the necessary regularity conditions, our analysis predicts heuristically that the ML-IB algorithm, which has a smaller complete-data space, should converge faster than ML-IA. This is corroborated by the empirical findings in [1]. …”

“…Recently, Politte and Snyder proposed two ML-EM algorithms for PET image reconstruction that directly incorporate the effects of known attenuation and accidental coincidences into the statistical measurement model [1]. The algorithms are based on two different complete-data spaces, one of which is a subset of the other.…”

“…A useful complete-data space supplements the observed measurements in a way that facilitates parameter estimation [3]. Although only one complete-data space has been suggested for PET under the simple measurement model used in the early papers [2,4], there will be more choices as investigators consider more comprehensive measurement models, such as those accounting for photon attenuation [5], accidental coincidences [1], deadtime, and scatter [6,7]. This paper illustrates the importance of parsimony in choosing complete-data spaces, and some of the tradeoffs that result.…”

“…The measurement models used in [1] assumed exact knowledge of the survival probabilities, the probability that both photons of a positron-produced pair escape unattenuated. In practice, one must obtain these factors experimentally, typically by a transmission scan that precedes the radiotracer injection.…”

“…In this paper, we show that EM algorithms based on smaller complete-data spaces will typically converge faster. We discuss two practical applications of these concepts: (i) the ML-IA and ML-IB image reconstruction algorithms of Politte and Snyder [1] which are based on measurement models that account for attenuation and accidental coincidences in positron-emission tomography (PET), and (ii) the problem of simultaneous estimation of emission and transmission parameters. Although the PET applications may often violate the necessary regularity conditions, our analysis predicts heuristically that the ML-IB algorithm, which has a smaller complete-data space, should converge faster than ML-IA.…”