Zooming out: from a voxel-wise dynamic PET modeling problem to a full FOV of implicit inverse problems

This thesis investigates mathematical and computational methods for modeling tracer dynamics in PET imaging, with a focus on advancing quantitative analysis beyond conventional metrics. The work originates from a clinical observation in FDG-PET scans of lung cancer patients, where voxel-wise relative Patlak analysis revealed unexpected tracer release in certain tumor regions—an effect likely associated with inflammatory infiltrates. This finding highlighted the limitations of standard semi-quantitative indices like SUV and motivated a deeper exploration of kinetic modeling. We first examine compartmental models, particularly the 2-compartments and Sokoloff models, to estimate biologically meaningful kinetic parameters through systems of ODEs. These models provide a more robust framework for interpreting tracer kinetics, especially in oncology, where irreversible accumulation patterns are of interest. Building on this foundation, we address the inverse problem of parameter estimation from noisy temporal data. We reinterpret it as an implicit inverse problem, where parameters are not directly observable but constrained by the dynamics of the system. To solve this, we propose a homotopy-based optimization strategy that gradually transitions from a highly regularized to a minimally regularized formulation. This path-following approach is coupled with gradient descent, adjoint-state gradient computation, and Newton–Raphson integration for the forward model. The thesis concludes with a broader framing of these challenges within the emerging field of ODE learning, where data-driven techniques aim to recover unknown dynamics. By bridging clinical imaging and abstract inverse problems, this work contributes novel tools for quantitative modeling in both biomedical applications and general dynamical systems analysis.