Direct MAP estimation of attenuation sinogram using TOF PET emission data

Objectives It has been shown recently (Defrise et. al.) that the attenuation sinogram can theoretically be derived from ideal TOF PET Emission Data using a 2-step method. Here we propose and evaluate a novel one-step Maximum a Posteriori (MAP) approach to directly estimate the attenuation sinogram from TOF PET Emission data with high signal-to-noise ratio (SNR).

Methods By incorporating two differential matrixes (with respect to s and Φ) into the discretized TOF data consistency condition, we obtain a linear function B=AX, where X is the unknown attenuation sinogram, B is a function of the measured TOF data, and A can be computed from the measured data and the differential matrices. Based on this linear equation, we developed a MAP estimator, which includes a least-square data-fitting term to account for the noise in B and a quadratic penalty term to impose the smoothness of X, in order to directly estimate the unknown attenuation X up to a constant. The optimization problem is solved using conjugate gradient and Armijo-backtracking line search. The true values in a small region of the sinogram are used to estimate the constant. We performed realistic simulations while modeling Poisson noise of an anthropomorphic phantom to validate our approach and compare its performance to the 2-step method.

Results In the noise-free case, the attenuation sinogram estimated by the MAP method yielded significantly (p<0.01) smaller bias as compared to the 2-step method. In the presence of Poisson noise, the MAP estimation yielded accurate estimates of the attenuation sinogram while the 2-step approach departed significantly from the true attenuation distributions, p<0.01. Bias and standard deviations in four ROIs are listed in table 1.

Conclusions In both noise-free and noisy cases, the MAP method achieved superior estimation accuracy and precision of the attenuation sonogram compared to the 2-step approach, making a promising practical approach for clinical PET data with state-of-art TOF.

Research Support NIH R21CA149587 and NIH R01EB013293