  In this paper we propose a solution to the need for a fast particle transport
algorithm in Online Adaptive Proton Therapy capable of cheaply, but accurately
computing the changes in patient dose metrics as a result of changes in the
system parameters. We obtain the proton phase-space density through the product
of the numerical solution to the one-dimensional Fokker-Planck equation and the
analytical solution to the Fermi-Eyges equation. Moreover, a corresponding
adjoint system was derived and solved for the adjoint flux. The proton
phase-space density together with the adjoint flux and the metric (chosen as
the energy deposited by the beam in a variable region of interest) allowed
assessing the accuracy of our algorithm to different perturbation ranges in the
system parameters and regions of interest. The algorithm achieved negligible
errors ($1.1 \times 10^{-6}$ $\%$ to $3.6 \times 10^{-3}$ $\%$) for small
perturbation ranges ($-40 \text{ HU to } 40 \text{ HU}$) and small to moderate
errors ($3 \text{ $\%$ to } 17 \text{ $\%$}$) -- in line with the well-known
limitation of adjoint approaches -- for large perturbation ranges ($-400 \text{
HU to } 400 \text{ HU}$) in the case of most clinical interest where the region
of interest surrounds the Bragg peak. Given these results coupled with the
capability of further improving the timing performance it can be concluded that
our algorithm presents a viable solution for the specific purpose of Online
Adaptive Proton Therapy.
