PET Reconstruction With Non-Negativity Constraint in Projection Space: Optimization Through Hypo-Convergence

Area of Science:

• Medical Imaging
• Nuclear Medicine
• Computational Science

Background:

• Standard positron emission tomography (PET) reconstruction uses maximum-likelihood expectation-maximization (MLEM), often with a positivity constraint on the activity image.
• This positivity constraint can introduce bias in low-count/high-background PET, compromising accurate quantification.
• Existing methods to allow negative values often alter data statistics by modifying the log-likelihood function.

Purpose of the Study:

• To develop a PET reconstruction technique that incorporates positivity constraints on projections only, avoiding bias introduced by image-space constraints.
• To achieve maximization of the penalized log-likelihood function without constraints by approximating it with a sequence of objective functions.
• To provide a mathematically rigorous proof of convergence for the proposed method.

Main Methods:

• Approximating the penalized log-likelihood function with a sequence of objective functions enabling unconstrained maximization.
• Utilizing hypo-convergence to ensure convergence of maximizers to the original log-likelihood.
• Implementing and comparing the novel method against the alternative direction method of multipliers (ADMM) using simulated data.

Main Results:

• The proposed method converges to a valid maximizer within the feasibility set.
• Demonstrated faster convergence compared to the ADMM method.
• Reduced image bias, particularly in cold regions of necrotic tumors, while maintaining accurate reconstruction in hot regions.

Conclusions:

• The novel PET reconstruction method effectively applies positivity constraints to projections, mitigating bias in low-count scenarios.
• The technique offers improved convergence rates over existing methods like ADMM.
• This approach enhances quantitative accuracy in PET imaging, especially for complex lesion types.