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In this paper we propose a solution to the need for a fast particle transportalgorithm in Online Adaptive Proton Therapy capable of cheaply, but accuratelycomputing the changes in patient dose metrics as a result of changes in thesystem parameters. We obtain the proton phase-space density through the productof the numerical solution to the one-dimensional Fokker-Planck equation and theanalytical solution to the Fermi-Eyges equation. Moreover, a correspondingadjoint system was derived and solved for the adjoint flux. The protonphase-space density together with the adjoint flux and the metric (chosen asthe energy deposited by the beam in a variable region of interest) allowedassessing the accuracy of our algorithm to different perturbation ranges in thesystem parameters and regions of interest. The algorithm achieved negligibleerrors ($1.1 \times 10^{-6}$ $\%$ to $3.6 \times 10^{-3}$ $\%$) for smallperturbation ranges ($-40 \text{ HU to } 40 \text{ HU}$) and small to moderateerrors ($3 \text{ $\%$ to } 17 \text{ $\%$}$) -- in line with the well-knownlimitation of adjoint approaches -- for large perturbation ranges ($-400 \text{HU to } 400 \text{ HU}$) in the case of most clinical interest where the regionof interest surrounds the Bragg peak. Given these results coupled with thecapability of further improving the timing performance it can be concluded thatour algorithm presents a viable solution for the specific purpose of OnlineAdaptive Proton Therapy.

A deterministic adjoint-based semi-analytical algorithm for fast response change computations in proton therapy