Developing a phenomenological model of the proton trajectory within a heterogeneous medium required for proton imaging

To develop an accurate phenomenological model of the cubic spline path estimate of the proton path, accounting for the initial proton energy and water equivalent thickness (WET) traversed. Monte Carlo (MC) simulations were used to calculate the path of protons crossing various WET (10-30 cm) of different material (LN300, water and CB2-50% CaCO3) for a range of initial energies (180-330 MeV). For each MC trajectory, cubic spline trajectories (CST) were constructed based on the entrance and exit information of the protons and compared with the MC using the root mean square (RMS) metric. The CST path is dependent on the direction vector magnitudes (\P-0,P-1\). First, \P-0,P-1\ is set to the proton path length (with factor Lambda(Norm)(0,1) = 1.0). Then, two optimal factor Lambda(0,1) are introduced in \P-0,P-1\. The factors are varied to minimize the RMS difference with MC paths for every configuration. A set of Lambda(opt)(0,1) factors, function of WET/water equivalent path length (WEPL), that minimizes the RMS are presented. MTF analysis is then performed on proton radiographs of a line-pair phantom reconstructed using the CST trajectories. Lambda(opt)(0,1) was fitted to the WET/WEPL ratio using a quadratic function (Y = A + BX2 where A = 1.01,0.99, B = 0.43,-0.46 respectively for Lambda(opt)(0), Lambda(opt)(1)). The RMS deviation calculated along the path, between the CST and the MC, increases with the WET. The increase is larger when using Lambda(Norm)(0,1) than Lambda(opt)(0,1) (difference of 5.0% with WET/WEPL = 0.66). For 230/330 MeV protons, the MTF10% was found to increase by 40/16% respectively for a thin phantom (15 cm) when using the Lambda(opt)(0,1) model compared to the Lambda(Norm)(0,1) model. Calculation times for Lambda(opt)(0,1) are scaled down compared to MLP and RMS deviation are similar within standard deviation. Based on the results of this study, using CST with the Lambda(opt)(0,1) factors reduces the RMS deviation and increases the spatial resolution when reconstructing proton trajectories.