Stopping power and range estimations in proton therapy based on prompt gamma timing: motion models and automated parameter optimization.

OBJECTIVE: Particle therapy treatments are currently limited by uncertainties of the delivered dose. Verification techniques like Prompt-Gamma-Timing-based Stopping Power Estimation (PGT-SPE) may allow for reduction of safety margins in treatment planning. Approach: From Prompt-Gamma-Timing measurements, we reconstruct the spatiotemporal distribution of prompt gamma emissions, which is linked to the average motion of the primary particles. The stopping power is determined by fitting a model of the average particle motion. Here, we compare a previously published implementation of the particle motion model with an alternative formulation and present two formulations to automatically select the hyperparameters of our procedure. The performance was assessed using Monte-Carlo simulations of proton beams (60 MeV to 219 MeV) impinging on a homogeneous PMMA phantom. Main results: The range was successfully determined within a standard deviation of 3 mm for proton beam energies from 70 MeV to 219 MeV. Stopping power estimates showed errors below 5 % for beam energies above 160 MeV. At lower energies, the estimation performance degraded to unsatisfactory levels due to the short range of the protons. The new motion model improved the estimation performance by up to 5 % for beam energies from 100 MeV to 150 MeV with mean errors ranging from 6 % to 18 %. The automated hyperparameter optimization matched the average error of previously reported manual selections, while significantly reducing the outliers. Significance: The data-driven hyperparameter optimization allowed for a reproducible and fast evaluation of our method. The updated motion model and evaluation at new beam energies bring us closer to applying PGT-SPE in more complex scenarios. Direct comparison of stopping power estimates between treatment planning and measurements during irradiation would offer a more direct verification than other secondary-particle-based techniques.

Scattering and Perturbation Theory for Discrete-Time Dynamics.

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasienergy which is defined only modulo 2π. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative quantum field theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyze the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.

Solutions of a Two-Particle Interacting Quantum Walk.

We study the solutions of an interacting Fermionic cellular automaton which is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particle solutions of the automaton recently in the literature, which exploits the symmetries of the evolution operator. In the two-particle sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one-dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum and of the coupling constant. Even in the special case of vanishing coupling, the walk manifests bound states, for finitely many isolated values of the total momentum. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.