The Monte Carlo method is employed in this study to simulate the proton irradiation of a water-gel phantom. Positron-emitting radionuclides such as 11C, 15O, and 13N are scored using the Particle and Heavy Ion Transport Code System Monte Carlo code package. Previously, it was reported that as a result of 16O(p,2p2n)13N nuclear reaction, whose threshold energy is relatively low (5.660 MeV), a 13N peak is formed near the actual Bragg peak. Considering the generated 13N peak, we obtain offset distance values between the 13N peak and the actual Bragg peak for various incident proton energies ranging from 45 to 250 MeV, with an energy interval of 5 MeV. The offset distances fluctuate between 1.0 and 2.0 mm. For example, the offset distances between the 13N peak and the Bragg peak are 2.0, 2.0, and 1.0 mm for incident proton energies of 80, 160, and 240 MeV, respectively. These slight fluctuations for different incident proton energies are due to the relatively stable energy-dependent cross-section data for the 16O(p,2p2n)13N nuclear reaction. Hence, we develop an open-source computer program that performs linear and non-linear interpolations of offset distance data against the incident proton energy, which further reduces the energy interval from 5 to 0.1 MeV. In addition, we perform spectral analysis to reconstruct the 13N Bragg peak, and the results are consistent with those predicted from Monte Carlo computations. Hence, the results are used to generate three-dimensional scatter plots of the 13N radionuclide distribution in the modeled phantom. The obtained results and the developed methodologies will facilitate future investigations into proton range monitoring for therapeutic applications.