Measurement of the 12C(p,n)12N reaction cross section below 150 MeV

Objective. Proton therapy currently faces challenges from clinical complications on organs-at-risk due to range uncertainties. To address this issue, positron emission tomography (PET) of the proton-induced 11C and 15O activity has been used to provide feedback on the proton range. However, this approach is not instantaneous due to the relatively long half-lives of these nuclides. An alternative nuclide, 12N (half-life 11 ms), shows promise for real-time in vivo proton range verification. Development of 12N imaging requires better knowledge of its production reaction cross section. Approach. The 12C(p,n)12N reaction cross section was measured by detecting positron activity of graphite targets irradiated with 66.5, 120, and 150 MeV protons. A pulsed beam delivery with 0.7–2 × 108 protons per pulse was used. The positron activity was measured during the beam-off periods using a dual-head Siemens Biograph mCT PET scanner. The 12N production was determined from activity time histograms. Main results. The cross section was calculated for 11 energies, ranging from 23.5 to 147 MeV, using information on the experimental setup and beam delivery. Through a comprehensive uncertainty propagation analysis, a statistical uncertainty of 2.6%–5.8% and a systematic uncertainty of 3.3%–4.6% were achieved. Additionally, a comparison between measured and simulated scanner sensitivity showed a scaling factor of 1.25 (±3%). Despite this, there was an improvement in the precision of the cross section measurement compared to values reported by the only previous study. Significance. Short-lived 12N imaging is promising for real-time in vivo verification of the proton range to reduce clinical complications in proton therapy. The verification procedure requires experimental knowledge of the 12N production cross section for proton energies of clinical importance, to be incorporated in a Monte Carlo framework for 12N imaging prediction. This study is the first to achieve a precise measurement of the 12C(p,n)12N nuclear cross section for such proton energies.


Introduction
Proton therapy is a modality of radiotherapy that offers a superior dose conformation to the tumor volume, as well as a significant reduction of the dose to the organs-at-risk (OARs).This is made possible by the proton energy loss characteristics, such as the finite range of protons and the high dose deposition just before the end of their trajectory, a region known as the Bragg peak.However, studies have indicated discrepancies between planned and actual treatment due to range uncertainties, setup errors, or anatomical changes (Lomax 2008, 2020, Paganetti 2012).
Several in vivo range verification techniques for particle therapy have been developed and tested in order to obtain feedback which offers the possibility of quantifying and reducing the range uncertainties (Knopf and Lomax 2013, Polf and Parodi 2015, Parodi and Polf 2018).These techniques are based on the detection of secondary radiation produced by the interaction of the proton beam with tissues.Secondary radiation primarily C, 15 O, 13 N, 30 P, 38 K) and one is for 12 N.The 12 N reaction channel is limited by the fact that the cross section data for the 12 C(p,n) 12 N nuclear reaction has only been reported for proton energies below 48 MeV with a systematic uncertainty of 15% (Rimmer and Fisher 1968).Note that based on the results of Dendooven et al (2015), the production of 12 N on 12 C integrated for a proton energy of 55 MeV is at least 10 times larger than that on 16 O.Hence the production on 16 O is not considered.
In this work, we describe the measurement of the 12 C(p,n) 12 N reaction cross section within an energy range from 23.5 to 147 MeV using a method that is based on 12 N PET imaging, aiming for a smaller uncertainty than (Rimmer and Fisher 1968).The purpose of this work is to include the cross section values in the database of the RayStation Monte Carlo engine, an essential component of the framework for real-time in vivo range verification in proton therapy via 12 N imaging.

Proton beam properties
The experiments were performed at the Accélérateur Groningen-ORsay (AGOR) cyclotron of the Particle Therapy Research Center (PARTREC), University Medical Center Groningen.Monoenergetic proton beams with a kinetic energy of 66.5, 120, and 150 MeV were used, the energy distribution of the primary beams is given by a full width at half maximum (FWHM) of 0.2%.These beams were transmitted through a horizontal beamline and directed towards the target configuration, which is described in section 2.2.
To monitor the intensity of the beams, an air-filled ionization chamber called beam intensity monitor (BIM) was positioned after the beamline exit.The BIM units (monitor units, MU) in terms of number of protons were calibrated using a low-intensity beam delivered on a 10 × 10 cm 2 plastic scintillator detector which counted individual protons passing through the BIM.Table 1 provides details on the calibration factors.These factors were found through the average value of 10 measurements integrated over 10 s each, the uncertainty of the factors was calculated from the standard deviation of the dataset.
The beam profiles were measured using a harp-type wire grid measurement system in both the vertical and horizontal direction, thus measuring profiles that are integrated along these two perpendicular directions.The beam profiles were found to have a gaussian shape with a FWHM of 7 mm (66.5 MeV), 5 mm (120 MeV), and 4 mm (150 MeV) at the target position.
The irradiation consisted of beam-on pulses with a width of 5 ms, delivered in periods of 35 ms (for 66.5 MeV) and 100 ms (for 120 and 150 MeV).The pulse structure was generated by a waveform generator that controlled the voltage on the electrostatic deflection plates installed in the injection line of the AGOR cyclotron.
The tungsten plates are essential to increase the probability of 12 N positron annihilation, which is very low for the thin targets used due to the high energy of the 12 N positrons (average energy 7.80 MeV and maximum energy 16.3 MeV).Tungsten was chosen because of its high stopping power and the low production of shortlived positron emitters by a proton beam.By using a tungsten plate before and after the target, the positron annihilation is optimized, and the plate before the target can be used as energy degrader, modifying the proton energy incident on the graphite target without significantly broadening the beam on target.Using the aluminum degrader system that is situated 15-25 cm upstream from the target (see figure 1) would broaden the beam on target substantially more.Due to an insufficient amount of tungsten to degrade the 120 MeV beam to 50 MeV, 16 mm of the aluminum degrader system was added to the 3.5 mm tungsten for this measurement.The target configuration was placed at a distance of 35 mm in the downstream beam direction from the center of the fieldof-view (FoV) of the PET scanner and oriented at an angle of 45°with respect to the beamline.This angle was chosen to minimize the attenuation by the tungsten plates of 511 keV annihilation photons which are emitted in the direction of the PET scanner panels.For one irradiation (energy of 49.5 MeV), 16 mm of aluminum degrader in-between the BIM and the target was used because the thickness of tungsten necessary to degrade the 120 MeV beam to this energy, around 7 mm, would considerably increase the attenuation of the annihilation photons.The specifications of the target configuration for each irradiation are presented in table 2.
Each PET data acquisition lasted 60 s, with irradiation during the first 20 s.It is worth noting that for each irradiation, the graphite and tungsten plates were replaced with 'fresh' plates.Due to the limitation on the number of tungsten plates available, these plates were reused within about 30 min.However, due to the low activation of tungsten, this did not affect the accuracy of the measurements.Reactions in tungsten do not produce very short-lived positron emitters and the long-lived activation per mm thickness was measured for the 66.5 MeV beam to be 35 times lower than in the graphite targets.Additionally to the graphite irradiations, polymethyl methacrylate (PMMA, (C 5 O 2 H 8 ) n ) irradiations were carried out to validate the method for measuring the 12 N cross section.This validation was based on well-known cross section measurements for the reaction channels of the typically-used positron emitters, named in this work long-lived positron emitters, including, 12 C(p,pn) 11   Table 2. Irradiation parameters and target configurations (tungsten (W)+graphite+tungsten (W)) used at different energies.The mean energies in the targets were calculated using an analytical energy loss model in one-dimension.The number next to the plus-minus signs indicates the energy loss along the graphite target.targets were 25 × 25 mm 2 plates with thicknesses of 2 mm (for 66.5 and 120 MeV), and 5 mm (for 150 MeV), having a density of ρ = 1.19 ± 0.01 g cm −3 .As these long-lived positrons have a short range, tungsten plates were not necessary in these measurements.The irradiation time was 60 s and the PET data acquisition lasted 660 s, using the time structure presented in figure 2.

PET scanner
To image the beam-induced activity on the graphite target, a modified dual-head Siemens Biograph mCT clinical scanner (Jakoby et al 2011) was used.The same scanner was earlier used to investigate 12 N imaging with proton and helium beams (Ozoemelam et al 2020a(Ozoemelam et al , 2020b)).The scanner consists of two opposite heads of block detectors (a full ring consists of 12 identical heads).Each head is 229 × 218 mm 2 in size and is composed of a 4 × 4 block detector array, each of which is composed of a 13 × 13 array of 4 × 4 × 20 mm 3 LSO scintillator crystals.A 2 × 2 array of photomultiplier tubes (PMTs) reads out the signals of these crystals.Anger logic is used to identify the LSO crystal in which an interaction took place.The distance in-between the heads was set at 210 mm.The center of the FoV is at beamline height, 400 mm above the beamline table.The heads were installed such that the curved side was oriented along the vertical direction (see figure 1 right).
The energy signals are transmitted by 7.5 m long Cat 6A twisted pair cables with RJ-45 jacks to two detector electronics assemblies (DEAs), which encode the position, energy, and time of the photon arrival on the detector.A coincidence unit receives the signals from the DEAs and identifies valid coincidence events based on a time coincidence window of 5 ns and an energy window of 435-650 keV.Only coincidences between the 2 scanner heads are recorded.Finally, the data are transmitted to a workstation by an optical fiber cable for their analysis.

Photomultiplier tube gain recovery
In order to avoid potential damage to the PMTs due to the high radiation flux during beam-on, the PMTs were operated in a pulsed mode as follows, remaining on for 90 (or 25) ms and were turned off 1 ms before the beamon and remained in that way up to 4 ms after the beam-off, obtaining a 100 (or 35) ms period.Figure 2 shows the time structure described above and the first five cycles of a graphite irradiation to illustrate the periodic switching on/off on the PMTs and the proton beam.
A relatively slow gain recovery of the PMTs after they are turned on was observed, and due to this, a nonnegligible number of coincidences are missed during the first milliseconds of data acquisition during each pulsing period.A correction of the slow gain recovery for each individual irradiation separately is possible by investigating the data taken during the 40 s data acquisition after the 20 s irradiation time.This data does not contain 12 N decays but only the decay of isotopes with a half-life much longer than the 35 or 100 ms pulsing periods.The decay rate of these isotopes is thus constant on this time scale.The observed coincidence count time spectra over the pulsing period are however not constant, with the deviation from a constant being a measurement of the gain recovery.The time spectra exhibit a saturation behavior that is described by the following function: where t is the time (in ms) since the beam was switched off, S 0 is the initial number of counts (i.e. at t = 4 ms, right after the PMT is switched on), S max is the number of counts when the PMT gain recovery has been completed, and t is the gain recovery time constant (in ms).S max was calculated as the average counts within the interval from 50 to 94 ms for the 100 ms period acquisitions and from 20 to 29 ms for the 35 ms-period acquisitions.A least squares fit then determined the values of S 0 and t.The measured time spectra are then divided point-by-point by equation (1) to obtain the corrected time spectra.This procedure is detailed in our earlier work (Ozoemelam et al 2020a).

Sensitivity measurements
To validate the Monte Carlo model (described in the following section 2.4), a sensitivity measurement was carried out using a 68 Ge point source of 3.24 MBq (±3%, on the day of the measurement).The positron branching ratio for this radionuclide is 0.889.The active area of the source has a diameter of 5 mm and is embedded into an acrylic disk with a diameter of 25.4 mm and a thickness of 6.35 mm.The measurement consisted of a 300 s PET acquisition in list-mode with the source placed at the center of the FoV.Additionally, a 300 s acquisition was done without the source to be able to correct for the intrinsic radiation background counts.

Monte Carlo simulations
To determine the target efficiency factors, the ratio between the coincidences detected by the PET scanner and the number of 12 N nuclides formed in the graphite target was evaluated.The value of this factor depends on three parameters: the intrinsic sensitivity of the PET scanner, the positron annihilation fraction, and the gamma self-attenuation due to the target configuration.The tungsten and graphite thicknesses play a crucial role in determining the last two parameters.Therefore, to calculate the target efficiency factor accurately, Monte Carlo (MC) simulations were used.These simulations enabled to consider the complex interactions between positrons and gamma rays with the target materials, providing reliable target efficiency factors.In addition to these simulations, a simulation of the scanner sensitivity was performed according to the specifications described in section 2.3.2 in order to validate the MC scanner model.In total 6 × 10 5 primary particles ( 68 Ge ions) were simulated.
The MC simulations were carried out using the Geant4 Application for Tomographic Emission (GATE) version 9.1, an MC simulation toolkit widely used for medical physics applications (Jan et al 2004, Sarrut et al 2021).Figure 3 shows the simulation model of the experimental setup.A dual-head PET system model according to the description in section 2.3 was used.
The geometry settings consisted of the target configuration according to the specifications provided in table 2 and the PMMA target holder (shown in figure 1).The breadboard and the steel plates on which the PMMA holder is placed were not simulated because they are outside the scanner FOV.The source geometry was defined as a cylindrical and isotropic 12 N positron source, with a height equal to the graphite thickness (table 2) and a circular gaussian intensity profile (with a FWHM according to the beam properties described in section 2.1).The positron energy spectrum was obtained using an analytical model described by Mougeot (2015).
In total, 1.41 × 10 6 primary positrons were simulated.The target efficiency factors (e) were calculated as the ratio of coincidence events and the number of positrons simulated: For estimating the statistical uncertainty in the efficiency factor, the history-by-history method employing equation (3), described by Sempau et al (2000), was used Basically, the method computes the standard deviation of the quantity set X , i { } where i represents each independent event, i.e. each positron particle, and = X 1 i if that particle produces a coincidence event and = X 0 the target efficiency, the above expression can be rewritten as: Similarly, MC simulations were performed to compute the efficiency factors for the PMMA targets, computing for the main positron emitters produced on PMMA, 11 C (half-life 1221.8 s), 10 C (half-life 19.29 s), and 15 O (half-life 122.24 s).
The results of the efficiency calculations are shown in tables 3 and 5.

12 N production
The number of 12 N nuclei present at the end of the first beam pulse (N N12 0 ) is given by: where N p0 is the number of incident protons per pulse, = t 5 ms 0 (pulse width), r = -1.718g cm 3 (graphite density), = Ń 6.022 10 mol A 23 1 (Avogadro's constant), = M 12.01 (carbon atomic mass), l = -0.063ms 1 (decay constant of 12 N), d is the effective target thickness (in cm), and s ̅ is the average cross section (in mb).As the targets are at an angle of 45 degrees, the effective thickness is 1/cos(45) = 1.41 times larger than the target  plate thickness given in table 2. Note that equation (5) assumes that there are no beam intensity fluctuations within a pulse, which is the case here due to the short duration of the pulses (5 ms).For determining the 12 N production, and thus the reaction cross section, all beam cycles of a 20 s irradiation are added to provide 1-time spectrum of the coincident counts over the 35 or 100 ms cycle period.Because of this addition, contributions from previous cycles need to be considered.With each cycle added, the number of nuclei increases with the contribution from that cycle, equal to N , N12 0 and the contribution of all previous cycles after decay for a time equal for each period, i.e. a factor l e .
t p Thus, the number of nuclei present after the ith pulse is given by: where t p is the cycle period (35 or 100 ms).The number of nuclei after adding all cycles is then: with K the number of cycles in the 20 s irradiation.Because K is large (571 and 200 for the 35 and 100 ms cycle periods respectively), equation (7) simplifies to: Combining equation (8) with equation (5): the total number of protons in the irradiation.The difference with equation (5), i.e. the contribution from previous cycles, is given by the factor -l e 1 t p in the denominator.For the 35 and 100 ms cycle periods, this amounts to 12% and 0.18%, respectively.Given the accuracy of the cross section measurements that is aimed for, this factor is essential for the 35 ms cycle period.
2.6.Data analysis 2.6.1.Data acquisition List-mode data files were collected, containing information on the crystal IDs, numbered from 0 to 51 in both the horizontal and vertical directions for both PET heads, of the coincidences detected by the PET scanner.The files also included data on the detection time difference between the two blocks detectors involved in the coincidence detection, as well as the time event tag relative to the beginning of the data acquisition, recorded at intervals of 1 ms.Due to the inability to synchronize the beam pulsing with the start of data acquisition, a manual adjustment was required to shift the time event tag to ensure that t = 0 ms corresponded to the moment when the beam was turned off after the first beam-on pulse.This moment in time can be accurately identified from the list mode data.This time shift was different for each acquisition.

Image reconstruction
Image reconstruction was performed to ensure that only positron annihilations in the target assembly were used for determining the 12 N production.The image reconstruction method employed in this work was adapted from previous studies (Buitenhuis et al 2017, Ozoemelam et al 2020a).The method involves performing a lowcomputational cost 2D reconstruction of the beam-induced activity in the plane between the scanner heads that contains the proton beam.
The reconstruction method consists of the following steps.Firstly, the coordinates of the center of the front crystal faces are calculated based on the crystal IDs involved in the coincidence event.Next, a uniform randomization over the crystal area (x and y direction) is calculated, followed by an exponential probability randomization in z (up to the crystal thickness of 20 mm) according to the attenuation coefficient of 511 keV photons.This results in the coordinates of the two points that define the line-of-response (LoR).The intersection of the LoR with the image plane, which was located in the center between and parallel to the two scanner heads, is calculated next.Finally, the intersection points for all coincidence events are accumulated into a 2D histogram, which stands for the 2D projection of the positron annihilation distribution.The image size was 222 × 222 mm 2 , and it was segmented into pixels of 2 × 2 mm 2 .
2.6.3. 12N identification and cross section calculation The list-mode data file has the time event tag of each coincidence (see section 2.6.1).By applying a modulus operation on the time tag, the time spectrum over the cycle period (35 or 100 ms) summed over the full irradiation was obtained.Next, the PMT gain recovery correction was applied (see section 2.3.1),resulting in an activity histogram.The starting point for the activity histogram, set to t = 0 ms, corresponds to the time when the beam was turned off. Figure 4 outlines the process described above.
Figure 5 left shows an exemplary 2D image showing that most of the coincidences are originating from the target assembly activation.However, there is a small but non-negligible off-target counts contribution which is being detected.This contribution is hypothetically attributed to thermal neutrons.
A nonlinear least-squares fit, performed using Matlab (The MathWorks Inc. 2022), was used to fit the activity histograms with the following function where A is the coincident count rate (in ms −1 ), A 0 is the initial 12 N count rate (in ms −1 ) at the moment when the beam goes off, A 1 is the initial count rate (in ms −1 ) of the off target counts contribution, which decays at a rate l, and b (in ms −1 ) is the constant activity due to long-lived positron emitters.A 1 and l were found by analyzing the activity histograms computed from off-target counts, these counts come from the outside of a mask that marks off the target position.This mask is bounded by   z 0 50 mm and   y 20 20 mm.The above procedure is outlined in figure 5.For all measurements, the parameter l did not present significant variations or any relation with the beam energy, obtaining a value between 0.147 and 0.171 ms −1 with a 1σ uncertainty between 0.013 and 0.015 ms −1 .This reinforces the hypothesis about the origin of these off-target counts.
The number of 12 N nuclei at the start of the histogram, i.e. the end of the beam pulse, is given as: where e is the detection efficiency calculated by means of MC simulations (section 2.4).Using equations ( 9) and (11) the cross section is finally calculated as:

Uncertainty analysis
As can be seen in equation ( 12), the majority of the elements (or the inverse of them) are incorporated linearly, resulting in the uncertainty propagation following the same way, described by: where f i is any element (or its inverse) in equation (12) and Df i its respective 1σ uncertainty value.An exception to this is the decay constant λ, which is introduced exponentially in the cross section calculation.The contribution of the uncertainty on λ to the systematic uncertainty is given by: where a = 0.578 and 0.839 for 35 and 100 ms period, respectively.Thus, the total statistical and systematic uncertainties on the cross section were calculated as: Cross section measurement validation From the PMMA irradiations, the nuclear cross section of the typically-used positron-emitting isotopes were calculated following the same procedure previously detailed.These measurements were compared with the cross section data obtained from the literature in order to validate the method for the cross section measurements presented in this study.For the comparison, the cross section values were interpolated from previous works, namely Akagi et al (2013)  Additionally, the 12 N cross section measurements were compared with data calculated using nuclear data libraries such as the TALYS-based Evaluated Nuclear Data Library: TENDL-2021 (Koning et al 2019) and the Japanese Evaluated Nuclear Data Library version 5: JENDL-5 (Iwamoto et al 2023).Both of them calculate nuclear reaction cross sections using nuclear models complemented by available measured data.The data were obtained from the Evaluated Nuclear Data File (ENDF) (database version 2023-08-25, software version of 2023-10-31) (2023).

Typically-used positron emitters production
Figure 6 shows the beam-induced activity histogram for PMMA targets and their corresponding multiexponential fitting curve for three different beam energies.The fit parameters required to calculate the cross section of the mentioned reaction channels are presented in table 3. The calculation of the cross section values with their corresponding uncertainties can be found in table 4. Because in the fitted region the irradiation has already ended, the influence of off-target counts is minimal (3% of the total counts), unlike the measurements carried out for the calculation of the 12 N cross section, where the fraction of off-target counts was between 20% and 25% of the total counts.
The cross section values obtained in this study show a discrepancy (depending on the reference and the energy) within 19.7%-22.5% for 11 C and 21.0%-22.9%for 15 O when comparing to values reported in the literature.In both cases, our measurements underestimate the literature values (see figure 7).The sensitivity measurement carried out at the center of the FOV using the 3.24 MBq (±3%) 68 Ge point source results in 69 559 780 coincidences (after background correction), which indicates a sensitivity of 0.085 (±3%), while for the MC simulation, the coincidences were 60 605 indicating a sensitivity of 0.101 (±0.04%).We conclude that a systematic scaling factor of 1.25 (±3%) needs to be taken into account for the correction of the target efficiency.It is relevant to now that a compatible difference is found for the validation measurement presented above.The underlying causes of such disagreement can potentially be attributed to factors such as the disregard of light transport within the LSO crystals or the non-modeled photomultiplier tube efficiency, as discussed in previous works (Lamare et al 2006).

Target efficiency
The results obtained from the MC simulations for the efficiency factors for the different target configurations are shown in table 5.For each simulation, 1412 083 12 N positrons were simulated.The next to last column was used for the cross section value calculations (section 2.4).

PMT recovery correction
Figure 8 presents the coincidence count histograms over the cycle period summed over the post-irradiation period, from 20 to 60 s (i.e.without beam).The PMT gain recovery is clearly seen.The fits of each histogram with equation (1) are also shown.The fitted parameters are given in table 6. S S , 0 max / the initial count rate relative to the count rate after PMT recovery, ranges from 0.88 to 0.92, a small variation with respect to the irradiation carried out and the target configuration.The recovery time constant τ presents a higher variation, which is however compatible with the uncertainties.The small dependence of the fit parameters on the irradiation and target setup is notable.However, the correction factors were calculated and applied to the time spectra during the irradiation time (0-20 s) for each target configuration individually.

Time activity histograms of carbon target irradiations
Figure 9 shows a series of time histograms summed over the irradiation period.The presence of a short-lived decay, which is associated with the annihilation of 12 N positrons, is very clear.Table 7 summarizes the fitting results using equation (10).The amplitude parameter A 0 is used to calculate the cross section of the 12 C(p,n) 12 N reaction, making use of equation (12).
3.5. 12C(p,n) 12 N reaction cross section calculation Table 8 presents the 12 C(p,n) 12 N reaction cross sections calculated using equation ( 12) and the parameter values given in tables 2, 5 and 6, followed by a correction by the systematic factor of 1.21 due to the efficiency difference found on the PMMA measurements (see section 3.1), along with their statistical and systematic uncertainties.
The statistical and systematic uncertainties of these cross section values are affected by the following sources: • Statistical uncertainties: • Uncertainty of fit parameter A 0 (table 7), which for 1σ has a value between 1.9% and 7.6%.
• Time event tag: the time resolution of the data acquisition system is 1 ms, which means that coincidences tagged at time t (ms) could have actually been detected within the interval - + t t 0.5, 0.5 , [ ] following a uniform distribution.The standard deviation of this distribution is given by s = » 1 12 0.289 ms.i.e. 1.8%.
• Uncertainty of the efficiency calculated by means of MC simulations, which resulted in a 1σ value between 0.61% and 0.76% (last column in table 5).• Energy straggling of the primary beam: FWHM = 0.2%, which is negligible in comparison with the energy loss in the target which is between 1.5% and 10% (table 2).
• Systematic uncertainties: • Uncertainty in the determination of t 0 , the beam pulse width, is less than 1 μs, thus, it is negligible.
• Graphite thickness d, which was measured for the targets with a nominal thickness of 5 mm and 0.5 mm.A value of 5.194 ± 0.031 mm and 0.497 ± 0.010 mm, respectively, was measured.
• BIM calibration (section 2.1), whose standard deviation depends on the beam energy: 1.8% (for 66.5 MeV), 3.6% (120 MeV), and 4.1% (150 MeV) (table 1).Table 7. Fitting parameter A 0 obtained by means of double exponential fit (equation ( 10)) using a fixed half-life of 11.0 ms (of 12 N) and a free half-life to fit the off-target counts which was determined to be between 4.0 and 4.7 ms.c n 2 is the reduced chi-square value.• Scanner efficiency calculation (section 3.1), which results in a factor of 1.25 with an uncertainty of e = 3.0%, f sys over all measurements.
The set of 11 measurements within the energy range of 23.5-147 MeV obtained in this work (table 8) were extended with the low energy part, from 20 to 23 MeV, of the measurements done by Rimmer and Fisher (1968) to make a weighted cubic smoothing spline interpolation using Matlab (The MathWorks Inc. 2022) in order to generate a smooth cross section curve for the energy range from 20 to 150 MeV. Figure 10 shows the data measured in this study, the interpolated curve, the data available in the literature (Rimmer andFisher 1968, Knudson et al 1980), and the values calculated by the TENDL-2021 (Koning et al 2019) and JENDL-5 (Iwamoto et al 2023) nuclear data libraries.

Discussion
An accurate and precise measurement of the 12 C(p,n) 12 N cross section was successfully carried out within an energy range spanning from 23.5 to 147 MeV.The statistical and systematic (excluding the scanner efficiency) uncertainties are each below 6% in all, but one of the 11 energies measured (table 8).This represents an improvement and extension of the values reported by Rimmer and Fisher 1968, whose energy range is limited to below 48 MeV and whose values have a systematic uncertainty of 15%.However, these data are significant as they exhibit a fine energy spacing at low energy, enabling the identification of details that were not observed within the scope of this study, such as the cross section resonances (Rimmer and Fisher 1968).
The data calculated by the evaluated nuclear data libraries show values quite a bit lower than those measured in both this and previous studies.TENDL-2021 and JENDL-5 cross section data show a maximum value of 1.62 mb in the range of 23.0-27.0MeV and 0.69 mb in the range of 22.0-24.0MeV, respectively.The interpolated data show a maximum value of 3.96 mb at 21.4 MeV, which is consistent in terms of the energy, but not in terms of the cross section.The cross section ratio (this study-to-library) in the maximum is 2.4 and 5.7 for TENDL-2021 and JENDL-5, respectively.On the other hand, the cross section at 150 MeV is 0.23, 0.067, and 0.95 mb for TENDL-2021, JENDL-5, and this study data, respectively, resulting in a cross section ratio of 4.1 for TENDL-2021 and 14.1 for JENDL-5.This disagreement could be attributed to the limitations of the nuclear data libraries to calculate the cross section accurately.These limitations include the non-availability of experimental data for the high-energy range and the facts that the nuclear models used in TENDL-2021 are only validated for isotopes with a nucleon number A 19 (Koning et al 2019) and that JENDL-5 is recommended for long-lived (half-life> 1 d) reaction products (Iwamoto et al 2023).Our experimental data may help to improve the cross section calculations.
The statistical uncertainty primarily stems from the uncertainty associated with the parameter A , 0 which was obtained from a fit of the activity histograms.This uncertainty is largely determined by the number of 12 N counts measured.The reasons for the limited counts are the low cross section value (σ < 4 mb), the scanner efficiency of the target configuration (1%-2%), and the short irradiation time of 20 s, which was necessary to avoid a large contribution from the relatively long-lived isotopes.This is more noticeable in the lowest-energy beam experiment due to the use of the thinnest graphite target (0.5 mm) where the statistical uncertainty Table 8. 12 C(p,n) 12 N reaction cross-section values versus proton energy.Given are the statistical and systematic uncertainty calculated by equations (15) and (16).In the last column, the value in parenthesis shows the systematic uncertainty excluding the scanner efficiency uncertainty (3.0%).becomes the largest, as long as we do not consider the scanner efficiency uncertainty.Another contribution to statistical uncertainties arises from the MC simulations which keep a 1σ-value of less than 1%.

Beam
In terms of systematic uncertainties (excluding the scanner efficiency uncertainty), it was observed that the BIM calibration uncertainty exhibits a dependence on the beam energy, being smaller the lower the beam energy.This energy dependence arises because the low-energy protons deposit more energy on the BIM, resulting in a higher signal and consequently a more precise calibration.To mitigate this uncertainty in future experiments, the integration time or the number of measurements during the BIM calibration should be increased for high-energy beams (120 and 150 MeV).Another source of systematic uncertainty is associated with the physical target properties, such as the density and thickness, contributing to the uncertainty with a 1σvalue between 1% and 2%.This implies that for the low-energy beam (66.5 MeV), the BIM calibration uncertainty is comparable to the uncertainty due to the target properties.However, for high-energy beams, the primary contribution to the systematic uncertainty originates from the BIM calibration process.
An important finding highlighted in table 8 is the dominant contribution of systematic uncertainty when employing the high-energy beam, while the opposite is observed when utilizing low-energy beams, where the statistical uncertainty becomes the primary contributor.In contrast, for high-energy beams, thicker targets (2 and 5 mm) were used as the proton stopping power is smaller and larger energy losses in the target are acceptable, this results in an increased number of 12 N counts.Moreover, longer irradiation periods (100 ms) were employed, thereby reducing the uncertainty associated with the fitted parameter A .
0 However, when we consider the uncertainty of the scanner efficiency (3%) this becomes more relevant.
The methodology used in this study shows the advantages of positron emission imaging for precisely measuring cross sections of short half-life radionuclides.In-beam measurements are needed for such short halflives, typically involving a more challenging radiation environment compared to off-line measurements in which an irradiated target is transported to a detector setup outside of the irradiation room.This latter method, using a clinical or small animal PET/CT scanner, is e.g.used by Rodríguez-González et al (2022, 2023) to measure the production of 11 C, 13 N and 15 O, and by Fraile et al (2016) to measure the production of 68 Ga and 66 Ga.An advantage in this case is that multiple targets or target areas, e.g.irradiated with different beam energies, can be measured simultaneously because the different targets or target areas can be delineated in the resulting images.For half-lives much shorter than the transportation time (perhaps 1 min in favorable cases), no radioactivity survives transportation to an off-line detector setup, and one is forced to do an in-beam measurement.A representative example is the work by Horst et al (2019), who measure coincidences between 511 keV annihilation photons with two relatively small detectors (3.5 × 3.5 cm cross section), not providing imaging capabilities.Our use of an imaging system with a larger FoV has some advantages.The sensitivity of a larger system varies less over the size of the irradiation spot, making the measurement results less dependent on the exact knowledge of the beam spot size.In the case of 12 N, the die-away time of the neutrons created by the beam (several ms) is not a lot shorter than the half-life.Maximizing the statistics of the measurement requires a good understanding of this neutron-induced component.As we show in figure 5, the imaging capability enables to characterize the time structure of this component in a very clean way.
The measurements were conducted using the same system (dual-head Siemens Biograph mCT PET system) that will be used in future experiments aimed at imaging the 12 N beam-induced activity on anthropomorphic phantoms.By using this system for both purposes, it is guaranteed that the systematic uncertainties associated with the PET scanner are accurately accounted for in subsequent 12 N imaging experiments.This compatibility ensures a reliable assessment of the beam-induced activity on phantoms.
The long range of the 12 N positrons (the root mean square (rms) of the 1D projected range is 2.0 g cm −2 ) produces a blurring in the 12 N PET image, as we have seen in previous studies, such as Ozoemelam et al (2020a).This blurring causes a broadening of the distal activity edge (in the beam direction) and a spreading in the lateral profiles.It is relevant to consider what this blurring means for determining the proton range.For this, it is most important to consider the distal edge broadening.Qualitatively, the accuracy of 12 N imaging will be less than imaging of the longer-lived isotopes (assuming for these a measurement time of at least one half-life in order to observe a substantial amount of the decays).However, the main rationale for 12 N imaging is much faster feedback, not necessarily more precise feedback, on the proton range.Fast feedback promises to enable better treatment plans by enabling smaller safety margins without compromising treatment safety.
In the following, we develop a back-of-the-envelope estimate of the quality of range determination very early in an irradiation (say in the first second) by both 12 N imaging and imaging of long-lived isotopes ( 15 O and 11 C).Three factors are essential: production cross section, number of decays (and thus number of counts) and positron range blurring.We consider that the uncertainty of range determination is largely determined by counting statistics and thus scales with the square root of the number of counts, as we have shown in previous studies using proton (Ozoemelam et al 2020a), helium (Ozoemelam et al 2020b) and radioactive ion beams (Kostyleva et al 2023, Purushothaman et al 2023).The production cross section of 12 N is about 20 times smaller, but it decays about 10 4 times faster than 15 O and 10 5 times faster than 11 C. Compared to 15 O, which is about 10 times more dominant very early on than 11 C, the statistical uncertainty of 12 N imaging is thus about SQRT(10 4 /20) ≈ 20 times better.Assuming a state-of-the-art PET intrinsic image resolution of 4 mm and a positron range blurring of 20 mm for 12 N (the positron range blurring of 15 O and 11 C is negligeable in comparison with the intrinsic image resolution), we might expect that the slope of the PET activity distal edge will be 5 times less steep for 12 N than for the long-lived isotopes, resulting in a 5 times larger uncertainty in determining its position (which is the surrogate for the proton range).Combining this with the statistical factor above, we can expect that the 12 N range accuracy very early in an irradiation is 4 times better than that for 15 O.In a realistic clinical scenario, the most distal energy layer(s) are delivered in less than a second, a time scale at which 12 N is thus the positron emitter giving the most precise information on the proton range.

Conclusion
In this study, the 12 C(p,n) 12 N cross section was measured within an energy range from 23.5 to 147 MeV.To achieve this, we measured the 12 N production on a graphite target during the irradiation with pulsed pencil beams (5 ms on, 30 and 95 ms off) delivering between 0.7 and 2 × 10 8 protons per pulse.The 12 N production was determined by measuring coincidences between 511 keV annihilation photons using a dual-head Siemens Biograph mCT PET scanner.
The measurements obtained in this study not only extend the energy range covered in the previous work by Rimmer and Fisher (1968), but also exhibit enhanced precision.We achieved a reduction in systematic uncertainties, with a 1σ-value below 6% (and below 5% excluding the scanner efficiency), in contrast to the previously reported data with 15% uncertainty.Also, a statistical uncertainty ranging from 2.7% to 5.8% (1σ) was achieved.It is important to note that both datasets are complementary since this study was unable to provide detailed cross section information at low energies due to the limitations imposed by the low 12 N production for thin graphite targets, impeding precise measurements with fine energy spacing.
Incorporating the aforementioned datasets, a weighted spline data interpolation was employed to generate a continuous and smooth curve for representing the 12 C(p,n) 12 N cross section data.

Figure 2 .
Figure2.The time structure consists of a periodic off/on switching of the PMTs synchronized with the on/off switching of the proton beam.The time histogram shows the number of PET coincidences during the first five cycles of the irradiation on a graphite target.It is visible a fast decay component, the analysis of this component is detailed in section 2.6.3.

Figure 3 .
Figure 3. MC model implemented in GATE to compute the target efficiency factors necessary to calculate the 12 N cross section values.The dotted red circle highlights the target configuration comprised of an arrangement of tungsten + graphite + tungsten.1a and 1b: opposing PET scanner heads, 2: individual block detector, 3: graphite target, 4: tungsten plates, and 5: PMMA target holder.

Figure 4 .
Figure 4. Workflow implemented to obtain the activity histogram for one target irradiation.The time event tag is extracted from the list mode file (upper left) and used to build the coincidence time spectrum (bottom).Finally, through a modulus operation over the time tag, the pulsing period time spectrum, called the activity histogram, is created (upper right).

Figure 5 .
Figure 5. Left: 2D image for an activated target assembly which shows a non-negligible contribution of off-target counts.The white box shows the limits of the in-target counts.Note that the color scale is logarithmic.Right: time activity histogram for the off-target counts, which shows a decay with a half-life between 4.0 and 4.7 ms, faster than the 12 N decay.

Figure 6 .
Figure 6.Beam-induced activity on PMMA for 60 s of pulsed beam irradiation, followed by 600 s decay.The curves were fitted by a multi-exponential function to find the initial activities and the decay constants for the 3 main radioisotopes ( 11 C, 10 C and 15 O).(a) beam energy of 66.5 MeV and 2 mm thickness target, (b) beam energy of 120 MeV and 2 mm thickness target, and (c) beam energy of 150 MeV and 5 mm thickness target.
t / Therefore, the uncertainty in determining the parameter A 0 due the time resolution is given by s

Figure 7 .
Figure 7. Cross section values reported in literature and measured in this work for the main reaction channels activated in a PMMA irradiation.The data are presented for the three energies used in this work.An interpolation of the literature values was performed to calculate the cross section for the proton energies used in this work.The black error bars represent statistical uncertainty and the red error bars represent combined uncertainty.It is important to remark that the contribution of the 16 O(p,x) 11 C reaction channel was considered for the comparison with the 11 C cross section.

Figure 8 .
Figure 8. PMT recovery correction.The time activity histograms of the post-irradiation period, normalized to the value after gain recovery, are shown for all irradiations, labelled by the proton energy in the center of the target.t = 0 ms represents the moment when the beam goes off.(a) Proton beam of 66.5 MeV and period of 35 ms (detectors on from 4 to 28 ms).(b) Proton beam of 120 MeV and period of 100 ms (detectors on from 4 to 93 ms).(c) Proton beam of 150 MeV and period of 100 ms (detectors on from 4 to 93 ms).The lines show the results of fitting with equation (1).

Figure 9 .
Figure 9.A few time histograms summed over the irradiation period (for 20 s).The energy at the center of the target is (a) 28.5, (b) 49.5, (c) 109.3, and (d) 146.6 MeV.The blue squares represent the counts per ms detected by the PET system, the red points are the corrected counts taking into account the PMT gain recovery factor calculated with equation (1).The continuous blue line represents the fit with equation (10) of the corrected activity histograms.The fit region was 7-28 ms in (a) and 7-93 ms in (b), (c) and (d).

Figure 10 .
Figure10.Experimental and calculated 12 C(p,n)12 N cross section data.The blue squares show the results obtained in this work, the horizontal error bars represent the energy loss in the target and the vertical error bars represent the statistical uncertainty, calculated using equation (15) and reported in the fourth column of table 8. Green circles represent theRimmer and Fisher (1968) data which have a systematic uncertainty of 15%.Knudson et al (1980) report a cross section of 1.03 mb (±11%) at 99.1 (±0.16)MeV (brown diamond).The black line shows the cross section curve obtained from the smoothing spline interpolation of the experimental data.The red arrows show the TENDL-2021 calculation and the pink arrows show the JENDL-5 calculations (see text for more information).

Table 1 .
BIM calibration (protons per monitor unit) for the different beam energies.

Table 3 .
Fitting parameters obtained by means of the exponential decay fit using a multi-exponential function for the PMMA irradiation at three different energies.The literature decay constants (in s −1 ) for each radionuclide are: 5.68 × 10 −4 for 11 C, 5.68 × 10 −3 for 15 O, and 3.59 × 10 −2 for 10 C. -As

Table 4 .
Cross-section values and their statistical and systematic uncertainties measured at three different energies for the main positron-emitter reaction channels.

Table 5 .
Efficiency factors calculated by MC simulations for the different target configurations.The first column shows the nominal beam energy used in the irradiation.The second column gives the thickness of the aluminum degrader (used for only one measurement).The third to fifth columns give the thicknesses of tungsten (W) and graphite plates in the target configuration.The efficiency e was calculated as the ratio between the number of coincidence events and the number of positrons used in the MC simulation and its uncertainty was calculated following equation (4) (last column).

Table 6 .
Fitting parameters of the PMT recovery curves, equation (1).S S