Dynamic Cell Imaging in PET With Optimal Transport Regularization

Abstract

We propose a novel dynamic image reconstruction method from PET listmode data that could be particularly suited to tracking single or small numbers of cells. In contrast to conventional PET reconstruction our method combines the information from all detected events not only to reconstruct the dynamic evolution of the radionuclide distribution, but also to improve the reconstruction at each single time point by enforcing temporal consistency. This is achieved via optimal transport regularization where in principle, among all possible temporally evolving radionuclide distributions consistent with the PET measurement, the one is chosen with least kinetic motion energy. The reconstruction is found by convex optimization so that there is no dependence on the initialization of the method. We study its behaviour on simulated data of a human PET system and demonstrate its robustness even in settings with very low radioactivity. In contrast to previously reported cell tracking algorithms, our technique is oblivious to the number of tracked cells. Without any additional complexity one or multiple cells can be reconstructed, and the model automatically determines the number of particles. For instance, four radiolabelled cells moving at a velocity of 3.1 mm/s and a PET recorded count rate of 1.1 cps (for each cell) could be simultaneously tracked with a tracking accuracy of 5.3 mm inside a simulated human body.

MATHMOD 2022 Discussion Contribution Volume

“…The details for labelling the resin spheres as well as the algorithm for the triangulation and elimination of random coincidence was described by Cheng et al (2011). It has been shown that the conventional PET and PEPT tracking algorithms need improvement to be optimized for a single particle (Jung et al 2020, Schmitzer et al 2019) and multiple particles (Langford et al 2016, Langford et al 2017 tracking.…”

Towards a mathematical theory of trajectory inference

“…Learning with optimal transport as a regularizer One main source of inspiration for our work is [65], where the authors set up a learning problem for trajectory inference where the regularization term comes from optimal transport, and the data-fitting term is a log-likelihood. This approach was similar to the one followed in [14,12] where the authors set up a learning problem with an optimal transport regularizer.…”

On the extremal points of the ball of the Benamou–Brenier energy

“…Moreover, it motivated recent developments in unbalanced optimal transport theory [20,21,32,33], that is, when the marginals are arbitrary positive measures. Finally, as the Benamou-Brenier energy provides a description of the optimal flow of the transported mass at each time t, which is a valuable information in applications, it was recently employed as a regularizer for variational inverse problems [13,15,28,34,49] (see also a forthcoming paper by Bredies, Carioni, Fanzon and Walter). The goal of this paper is to characterize the extremal points of the unit ball of the Benamou-Brenier energy B at (3), and of a coercive version of it, which is obtained by adding the total variation of ρ to B.…”