Attenuation Pre-Correction in Iterative PET Image Reconstruction: What is the Statistical Cost?

Attenuation pre-correction (APC) is an effective strategy to reduce the computational cost of iterative PET image reconstruction by decoupling object-dependent attenuation effects from the system sensitivity image. However, APC alters the statistical properties of the projection data, leading to a mismatch between the assumed Poisson likelihood and the true data distribution. In this work, we investigate the statistical cost of APC in terms of image variance increase. By modeling APC reconstruction as a misspecified likelihood problem, we derive the variance lower bound using the Godambe information matrix (GIM), which provides a principled alternative to the conventional Fisher information matrix. To enable efficient computation for large-scale three-dimensional PET systems, we introduce a zero-picosecond time-of-flight (ZPT) assumption and derive a closed-form expression for the variance ratio of APC relative to MLEM. Theoretical predictions are validated against simulations using iterative reconstruction. Simulations demonstrate that attenuation heterogeneity along the contributing lines of response is the dominant factor driving APC-induced variance increase, while the impact of TOF resolution becomes secondary for modern TOF PET systems with timing resolution on the order of 200 ps. Three-dimensional evaluations with ZPT across different scanner geometries show large increases (>100%) in lesion variance on whole-body and total-body scanners, while a more moderate increase is observed on a brain-dedicated scanner. Experimental validation using human PET data further confirms the theoretical findings. This work provides a quantitative framework for evaluating the statistical cost of APC PET reconstruction.