Molière maximum likelihood proton path estimation approximated by cubic Bézier curve for scatter corrected proton CT reconstruction.

A maximum likelihood approach to the problem of calculating the proton paths inside the scanned object in proton computed tomography is presented. Moliere theory is used for first time to derive a physical model that describes proton multiple Coulomb scattering, avoiding the need for the Gaussian approximation currently used. To enable this, the proposed method approximates proton paths with cubic Bezier curves and subsequently maximizes the path likelihood through parametric optimization, based on the Moliere model. Results from the Highland formula-based Gaussian approximation are also presented for comparison. The simplex method is utilized for optimisation. The scattering properties of the material(s) of the scanned object are taken into account by appropriately calculating the scattering parameters from the stopping power map that is calculated/updated at every iteration of the algebraic reconstruction process. Proton track length constraint imposed by the proton energy loss is also accounted for. The method is also applied in the case that no exit angle data are measured. Geant4 Monte Carlo simulations were performed for model validation. Our results show that use of Moliere probability density function for modelling the multiple Coulomb scattering presents a modest 2% accuracy improvement over the Gaussian approximation and most-likely-path method. Simulations of voxelized phantom showed no essential benefit from the inclusion of the material information into the optimization, while path optimization with energy constraint slightly increased path resolution in a bone/water interface phantom. Method error was found to depend on energy, proton track-length within the medium, and proportion of data filtering.