  Positron emission tomography (PET) is an important modality for diagnosing
diseases such as cancer and Alzheimer's disease, capable of revealing the
uptake of radiolabeled molecules that target specific pathological markers of
the diseases. Recently, positronium lifetime imaging (PLI) that adds to
traditional PET the ability to explore properties of the tissue
microenvironment beyond tracer uptake has been demonstrated with time-of-flight
(TOF) PET and the use of non-pure positron emitters. However, achieving
accurate reconstruction of lifetime images from data acquired by systems having
a finite TOF resolution still presents a challenge. This paper focuses on the
two-dimensional PLI, introducing a maximum likelihood estimation (MLE) method
that employs an exponentially modified Gaussian (EMG) probability distribution
that describes the positronium lifetime data produced by TOF PET. We evaluate
the performance of our EMG-based MLE method against \st{traditional} approaches
using exponential likelihood functions and penalized surrogate methods. Results
from computer-simulated data reveal that the proposed EMG-MLE method can yield
quantitatively accurate lifetime images. We also demonstrate that the proposed
MLE formulation can be extended to handle PLI data containing multiple positron
populations.
