Measurement of PET isotope production cross sections for protons and carbon ions on carbon and oxygen targets for applications in particle therapy range verification

The method for the measurement of PET isotope production presented in this work is conducted in the several steps, described in the following: the experimental setup, consisting of three scintillators and a coincidence unit is positioned at the beamline and aligned according to the laser positioning system. The next step is a calibration measurement using a $\mathrm{^{22}Na}$ point source with known activity positioned at the center of the detection system guided by the positioning lasers. To be able to calculate the detection efficiency properly, the beamspot has to be characterized before the actual activation measurement can be performed. For this purpose, a Gafchromic EBT3 film is positioned in the target holder and the laser markings are transfered to the film before it is irradiated by a short pulse of protons or ions with the same beam settings used for the activation measurement later. Finally, the irradiated film is exchanged with the target, the data acquisition system is turned on and the target is irradiated. The induced $\beta^+$ activity can then be monitored as long as necessary (typically 15–30 $\mathrm{min}$ depending on the isotopes of interest) and afterwards the next measurement can be performed.

In this section each of the above mentioned steps and components are described in detail.

A schematic of the experimental setup is shown in figure 1.

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Three $\mathrm{BaF_2}$ scintillators (crystal dimensions: $3.5 \times 3.5 \times 7~\mathrm{cm^3}$) with a thin wrapping and a Hamamatsu R1668 photomultiplier are arranged around a thin graphite or beryllium oxide target tilted by $45^\circ$. The $\mathrm{BaF_2}$ scintillators are positioned at a distance of $5~\mathrm{cm}$ from the target center. Two of them ($\# 1$ and $\# 2$) are arranged at $180^\circ$ to measure the coincidence rate of the $511~\mathrm{keV}$ annihilation photons following the $\beta^+$ decays and a third one ($\# 3$) is arranged at $90^\circ$ to measure the random coincidence rate. The targets and films are positioned in a 3D-printed holder with a modular setup and can easily be exchanged without affecting the detector setup.

The irradiations were performed as treatment plans with a single beam spot. The raster scanning control system monitored the irradiation and the number of incident particles measured by the monitor IC within the nozzle was documented in the machine records. All irradiations were done using the beam line settings for the smallest focus (FWHM at the isocenter: 8.1–30.5 $\mathrm{mm}$ for protons and 3.4–9.3 $\mathrm{mm}$ for carbon ions) and at the highest intensity that can be extracted from the synchrotron ($1.5 \cdot 10^9~\mathrm{protons~s^{-1}}$ and $6.5 \cdot 10^7~\mathrm{carbon~ions~s^{-1}}$). The beam pulses had a duration of  ∼1.3 $\mathrm{s}$ for protons and  ∼1.0–2.5 $\mathrm{s}$ for carbon ions. These short pulse durations were chosen to ensure that the time of isotope production was well defined and small against the decay times. The number of particles per measurement were $1.9 \cdot 10^{9}$ for protons and between $6.5 \cdot 10^{7}$ and $1.9 \cdot 10^{8}$ for carbon ions.

The trigger unit was built from a set of NIM modules (discriminator, gate generator, coincidence module, dual timer) and generated a trigger signal for coincident signals either from detector $\# 1$ and $\# 2$ ($180^\circ$) or from detector $\# 1$ and $\# 3$ ($90^\circ$). The coincidence window was adjusted to $30~\mathrm{ns}$, which provided a good noise suppression for the $180^\circ$ scintillator pair. However, there was still the possibility to measure random coincidences caused by two independent $\beta^+$ decays that occur both within the coincidence window. This random coincidence rate was monitored by the $90^\circ$ scintillator pair. Subtracting the coincidence rate measured by detector $\# 1$ and $\# 3$ from the rate measured by $\# 1$ and $\# 2$ gives the true $180^\circ$ coincidence rate as shown in figure 2.

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This example shows that the random coincidence rate strongly depends on the present activity. In the time shortly after the irradiation, when the activity is at its maximum, random coincidences contribute more than 30% to the total coincidences measured at $180^\circ$. However, as the activity decreases, after  ∼2 $\mathrm{min}$ the contribution of random coincidences to the total coincidence rate subsides significantly.

A Tektronix DSA 72004C oscilloscope was used for data acquisition. The signals of the three scintillators were recorded as waveforms by means of fast sampling (sample rate $3.1~\mathrm{GS~s^{-1}}$) triggered by the coincidence unit. The oscilloscope can store data in its RAM with low deadtime between consecutive events, but every few 1000 events (depending on the exact acquisition settings like sample rate or digital resolution) it needs to save the stored data on a hard drive. During these few seconds, the oscilloscope does not accept any triggers which causes a gap in the measured decay curves every few minutes. Examples of such storage deadtimes are marked by arrows in figure 2.

In an offline analysis the $511~\mathrm{keV}$ peaks were separated by applying a cut on the energy spectra. In order to suppress the measurement of prompt gamma photons which are produced during the irradiation, the end-of-plan signal from the accelerator control system was used to start the data acquisition immediately after the end of the spill (the trigger pulse is created $120~\mathrm{ms}$ after the end of beam extraction). Furthermore, the end-of-plan signal was used to precisely determine the time when the beam pulse ended.

Two different target materials were irradiated in the present experiments. To obtain the production cross sections for the isotopes $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ on carbon, graphite targets (SGL Carbon R 6550, density: $1.83~\mathrm{g~cm^{-3}}$) with thicknesses of $5~\mathrm{mm}$ and $10~\mathrm{mm}$ (depending on the beam energy) and lateral dimensions of $80 \times 80~\mathrm{mm^2}$ were irradiated with protons and carbon ions. To obtain the production cross sections for $\mathrm{^{15}O}$ on oxygen, beryllium oxide targets (Materion Thermalox 995, density: $2.85~\mathrm{g~cm^{-3}}$) with a thickness of $3.9~\mathrm{mm}$ and lateral dimensions of $114 \times 114~\mathrm{mm^2}$ were used. For the measurements presented in this work (production of positron emitters) beryllium oxide acts as a pure oxygen target because the beryllium does not fragment into $\beta^+$-isotopes (as discussed by Tobias et al (1971)) and therefore does not contribute to the measured activity. To enhance the efficiency of the detection system, the targets were tilted by $45^\circ$ (see figure 1). The local roughness at the center of the targets (uncertainty of the thickness) was  <1%.

The calculation of the detection efficiency (as described in the next section) relies on the knowledge of the spatial distribution of the induced $\beta^+$ radioactivity within the target relative to the detectors. To estimate the activity distribution within the target for each individual species-energy-target combination, fluence measurements using EBT3 films located exactly at the target position (also with the $45^\circ$ tilt) were performed in advance before all target irradiations. From these films measured vertical and horizontal fluence profiles were obtained and fitted with single gaussian functions to be used for the calculation of the efficiency (see section 2.7). Also shifts of the beamspots relative to the scintillator setup could be detected and taken into account for the efficiency calculation. The parameters characterizing a beamspot are called FWHM_horizontal, FWHM$_{vertical}$, shift_horizontal and $shift_{vertical}$.

To convert the grey values of the EBT3 films into fluence values, a calibration function was determined for both protons and carbon ions. Carbon ion measurements were restricted to the isocenter (distance from isocenter to nozzle:  ∼108 $\mathrm{cm}$) where the beamspot sizes are daily checked and documented in the QA protocols. For the proton irradiations the setup was moved to a distance of $50~\mathrm{cm}$ from the nozzle. This closer distance was of advantage especially for the low proton energies because proton beams scatter much stronger within the nozzle (∼2 $\mathrm{mm}$ water equivalent thickness) than carbon ion beams, which leads to relatively large proton beamspots at the isocenter (see above).

Figure 3 shows examples of films irradiated with protons and carbon ions. As expected the carbon ion beamspots are sharper compared to the proton beamspots. The beam spot sizes get larger for lower energies due to the increased lateral scattering within the nozzle. Slight shifts of the beamspots relative to the positioning lasers can be observed depending on the energy. The film response to the carbon ions is stronger than to protons due to their higher LET. The beamspots appear stretched in horizontal direction because the films were irradiated with the $45^{\circ}$ tilt like the targets.

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For the calculation of the production cross sections for the different isotopes, their absolute activities produced by the proton or ion pulses within the targets need to be determined. For such an absolute measurement the efficiency of the detection system (count rate per activity) must be known. Because the beamspot sizes and therefore the spatial distribution of the induced radioactivity varied considerably among the different beams used (see figure 3), the efficiency had to be determined for each measurement separately.

To calculate the detection efficiency for each individual measurement, a numerical algorithm was developed which takes all relevant effects into account: the detection efficiency depends (a) on the spatial distribution of the $\beta^+$ activity relative to the detector setup (for activities located in the center the efficiency is maximum and drops at the sides), (b) on the amount of material between the activity and the detectors (causing attenuation of the $511~\mathrm{keV}$ annihilation photons) and (c) on the distance between the activity and the target edge (positrons may escape from the target and annihilate outside the detection zone).

The efficiency algorithm requires the following input parameters: the first parameter is the maximum efficiency of the detection system determined with a $\mathrm{^{22}Na}$ point source with known activity positioned at the center of the detection zone. This calibration was repeated each time the experiment was re-built to take account of e.g. small variations in the electronic thresholds of the coincidence unit or a slight geometrical misalignment of the detectors. Secondly, the beamspot parameters (FWHM_horizontal, FWHM$_{vertical}$, shift_horizontal, $shift_{vertical}$) obtained from the film measurement have to be taken into consideration to model the activity distribution in the target. Lastly, the target thickness and material have to be specified to enable an accurate estimation of the photon (self-) absorption and the fraction of positrons that escape the target without annihilation.

Based on the beamspot parameters, the algorithm models the spatial distribution of the $\beta^+$ radioactivity within the target divided into voxels ($80 \times 80$ voxels lateral and 100 voxels in depth) considering that the induced activity is proportional to the fluence. In the next step, the activity in each voxel is weighted by the efficiency of the corresponding voxel position divided by the total activity. To be able to calculate the efficiency for every position, a high resolution efficiency map was recorded in advance by moving a $\mathrm{^{22}Na}$ point source in $1~\mathrm{mm}$ steps through the detection zone (shown in figure 4) using a mechanical positioning device. This efficiency map is normalized to unity at the detection zone center and can be converted into absolute values by applying the calibration factor measured with the $\mathrm{^{22}Na}$ source right before the measurements (see above). The reduction of the efficiency due to the absorption of one of the $511~\mathrm{keV}$ photons within the target or the detector wrapping and due to positrons escaping from the target is taken into account for each individual voxel by using the photon attenuation coefficients from the NIST XCOM database (Berger et al 2010) and by applying a positron loss model based on FLUKA simulations (Ferrari et al 2005, Böhlen et al 2014, Battistoni et al 2015, 2016) considering published $\mathrm{^{11}C}$ and $\mathrm{^{15}O}$ positron spectra given by Eckerman et al (1994) and a $\mathrm{^{10}C}$ positron spectrum calculated according to a model given by Levin and Hoffman (1999). Finally, the resulting detection efficiency, which relates the measured coincidence rate with the activity for the particular measurement (true $180^\circ$ coincidences per second per Becquerel) is obtained by averaging the activity-weighted efficiency over all voxels.

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Due to the target thicknesses of $3.8~\mathrm{mm}$ for beryllium oxide or 5 and $10~\mathrm{mm}$ for graphite, the efficiency reduction due to positron loss was only in the order of 4%–8% since they could only escape from the last 1–2 mm of the graphite targets (depending on the isotope) and from the last mm of the beryllium oxide targets. The calculated efficiencies varied between 0.30% and 0.65% for the proton measurements and between 0.41% and 0.73% for the carbon ion measurements. The generally very low efficiencies are due to the small solid angle covered by the scintillators and the differences between the measurements at different energies are mainly due to the varying beamspot sizes, which are also the reason for the higher efficiencies for the carbon ion measurements compared to the proton measurements (see figure 3).

The measured $180^\circ$ coincidence count rate as a function of time $A(t)$ can be fitted by a composite exponential decay function (Stöckmann 1978) (one exponential function for each produced isotope) according to equation (1)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A(t) = {A_0^{X_1}} \cdot 0.5^{\frac{t}{T_{1/2}^{X_1}}} + {A_0^{X_2}} \cdot 0.5^{\frac{t}{T_{1/2}^{X_2}}} + ... \label{Eq1} \nonumber \end{align} \tag{ 1 }$$

where $A_0^{X_i}$ are the initial count rates and ${T_{1/2}^{X_i}}$ are the half lives of the isotopes X_i.

Using the initial count rates obtained from fitting the measured decay curve, the production cross sections $\sigma_{X_i}$ can be calculated according to equation (2)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma^{X_i} = \frac{\frac{A_0^{X_i}}{\epsilon}}{z \cdot \frac{n}{V} \cdot N \cdot \lambda^{X_i}} \label{Eq2} \nonumber \end{align} \tag{ 2 }$$

where $\newcommand{\e}{{\rm e}} \epsilon$ is the detection efficiency, z is the target thickness in beam direction, $n/V$ is the number of target nuclei per volume, N is the number of primary particles in the beam pulse and $\lambda^{X_i}$ is the decay constant of isotope X_i.

A fitting function according to equation (1) assumes that the irradiation time $\Delta t$ is much shorter than the half lives T_1/2 of the isotopes produced because the competition between build-up and radioactive decay during the irradiation is not taken into account. For isotopes where the duration of the irradiation is non-negligible compared with the half life—in this work this was only the case for $\mathrm{^{10}C}$—the term for a single isotope can be split into multiple terms having different zero time points. With this approach, the temporal course of the activity production can be taken into account in good approximation. For the fitting model in this work the time point of the $\mathrm{^{10}C}$ production was split into three according to equation (3).

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A^{^{10}C}(t)=\frac{A^{^{10}C}_0}{3} \cdot 0.5^{\frac{t-\Delta t /3}{T^{^{10}C}_{1/2}}} + \frac{A^{^{10}C}_0}{3} \cdot 0.5^{\frac{t}{T^{^{10}C}_{1/2}}} + \frac{A^{^{10}C}_0}{3} \cdot 0.5^{\frac{t+\Delta t /3}{T^{^{10}C}_{1/2}}}. \label{Eq3} \nonumber \end{align} \tag{ 3 }$$

The approximation described by equation (3) is illustrated in figure 5.

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The energy loss within the targets smeared the energy where the observed reactions took place. This energy interval was kept small by using thin targets but is non-negligible, especially for the low energy measurement points. It is accounted for by giving each cross section value for the mean energy at the target center with an uncertainty interval covering the energies before and after. These energy uncertainty intervals were calculated by transport calculations through the different targets using the FLUKA code. The initial beam energy spread from the accelerator (∼0.1%–1% depending on the energy) was not taken into account because it is negligible compared with the energy loss effects described before.

Besides this energy smearing, there are also different uncertainties to consider that propagate directly into the uncertainty of the cross section value: the activity of the $\mathrm{^{22}Na}$ source used for calibration of the detection system has a manufacturing uncertainty of 3%. The number of primary ions impinging on the target was determined by the monitor IC in the nozzle. Its calibration by means of an absorbed dose to water measurement under dosimetric reference conditions has an uncertainty which was assumed to be 4% (where 2% results from the uncertainty of the k_Q value used for absorbed dose to water determination and another 2% from the beam model which was used to convert the measured absorbed dose to water into fluence). The algorithm used to calculate the detection efficiency considers all relevant effects but uses some simplified models (e.g. the single Gaussian beam profile), therefore the calculated efficiency is not free of uncertainty either. This was estimated to be 3% based on variations of the input parameters within reasonable limits. Following the rules of error propagation, these individual uncertainties add up to an estimated total systematic cross section uncertainty of 10%. Lastly, the produced initial activities are estimated by fitting the measured decay curve with a composite exponential decay function, whose accuracy is mainly affected by the amount of produced activity and the resulting counting statistics. This uncertainty was estimated by the fitter individually for each measurement and added to the generalized systematic uncertainty of 10% given above.

Uncertainties associated with the targets (homogeneity or misplacement) are small against the above mentioned sources of uncertainty and are therefore neglected.

The measured production cross sections for the main PT-PET isotopes $\mathrm{^{11}C}$ and $\mathrm{^{15}O}$ by protons were validated against published activity profiles measured in a tissue equivalent gel phantom with a clinical PET scanner after proton irradiation (Espana et al 2011). For this purpose, look-up tables with the $\mathrm{^{12}C(\,p,pn){}^{11}C}$ and $\mathrm{^{16}O(\,p,pn){}^{15}O}$ excitation functions representing the measured cross sections presented here were created and convoluted with the proton spectrum to obtain the produced $\beta^+$-activity (the method and also how to model the temporal progress of the activity has been described in detail by Parodi et al (2002) and Bauer et al (2013)).

In this work, the proton spectrum as a function of depth was obtained from simulations with the Monte Carlo toolkit TOPAS (Perl et al 2012). The proton source spectrum for the simulation was optimized to reproduce the depth dose profile published in the article of Espana et al (2011). The main energy peak was found to lie around $116~\mathrm{MeV}$ as also stated by Espana et al (2011) but notably, also a low-energy proton component was required in the input spectrum to reproduce the entrance region of the reference dose profile. This can be explained by the fact that Espana et al (2011) performed their experiment at a passive scattering beamline where secondary protons are produced in the scatter foils. The elemental composition of the phantom material was given as 9.6% $\mathrm{H}$, 14.6% $\mathrm{C}$, 1.46% $\mathrm{N}$ and 73.8% $\mathrm{O}$ and the density was given as $1.13 ~\mathrm{g~cm^{-3}}$. Due to the high carbon and oxygen content in the tissue equivalent gel, it is well suited to validate the production cross sections on both target materials. To take account of the PET scanner resolution the calculated activity profiles were convoluted with a Gaussian kernel with $7~\mathrm{mm}$ FWHM, also given by Espana et al (2011). The image acquisition protocols used in the experiment should mimic two different PT-PET methods: the $5~\mathrm{min}$ protocol ($5~\mathrm{min}$ image acquisition started directly after the irradiation) corresponds to an in-room PET measurement, while the $30~\mathrm{min}$ protocol ($30~\mathrm{min}$ image acquisition after a $15~\mathrm{min}$ break) is more similar to an offline PET measurement.

The reaction channels that had to be considered in the transport calculation were $\mathrm{^{12}C(\,p,pn){}^{11}C}$ and $\mathrm{^{16}O(\,p,pn){}^{15}O}$ which were characterized in the present work, but also $\mathrm{^{16}O(\,p,X){}^{11}C}$ and $\mathrm{^{16}O(\,p,X){}^{13}N}$ which were not measured in this work. The cross section tables for the latter two channels were taken from Bauer et al (2013). Contributions of $\mathrm{^{10}C}$ can be neglected for the acquisition protocols used by Espana et al (2011) because also for the $5~\mathrm{min}$ measurement they could not avoid a break of  ∼1 $\mathrm{min}$ between irradiation stop and start of the PET imaging which is long enough for the majority of the $\mathrm{^{10}C}$ to decay.

Figure 6 shows the measured decay curves for graphite and beryllium oxide targets irradiated with both protons and carbon ions.

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It can be observed that for the graphite targets the activity decreases fast in the first two minutes after irradiation because the short-lived $\mathrm{^{10}C}$ (half life: $19.29~\mathrm{s}$) dominates the activity while later only the produced $\mathrm{^{11}C}$ (half life: $20.334~\mathrm{min}$) remains. In contrast to these distinct two decay components, the decay curves of the activated beryllium oxide are dominated by the produced $\mathrm{^{15}O}$ (half life: $122.24~\mathrm{s}$) and the other produced isotopes ($\mathrm{^{10}C}$, $\mathrm{^{11}C}$, $\mathrm{^{14}O}$, $\mathrm{^{13}N}$) only contribute a few percent to the total activity. It can also be seen, that the produced activity per irradiation pulse was considerably lower for carbon ions than for protons which results in a lower signal to noise ratio (and thus less accurate cross section measurements as seen below). This can be explained by looking at the requirements for the radiotherapy accelerator operated at MIT: in the facility design phase the maximum intensities (ions per second) that need to be extracted from the synchrotron were defined in terms of dose rate. Due to the much higher LET of carbon ions compared to protons, the maximum carbon ion intensity can be a factor of  ∼30 lower than the proton intensity and still generate the same dose rate. Also the PET counting statistics that can be collected for range verification during patient treatments with carbon ions suffers from this relation compared to proton therapy (Parodi et al 2002). However, in our experiment the lower particle numbers in the carbon ion beam pulses were partially compensated by their larger nuclear reaction cross sections and sharper beamspots (better detection efficiency) compared to protons. Additionally, the carbon ion irradiation times were increased up to twice the length of the proton pulses to further increase the number of primary particles and thereby the amount of produced activity.

The initial activities obtained from the fit functions shown in figure 6 could be converted into the production cross sections by applying equation (2). Figure 7 shows the $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ production cross sections as a function of energy (excitation functions) for protons impinging on carbon targets. For comparison also the Q-values (the energy that is absorbed in a nuclear reaction) for both channels calculated by comparing the masses of the nuclei in the initial and in the final state are shown.

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For the $\mathrm{^{12}C(\,p,pn){}^{11}C}$ reaction channel, there are a lot of reference data available and the cross section data obtained in the present experiments fit rather well into the general systematics. There is a good agreement between the present data and the reference data from the literature except those from the very early publications by Hintz and Ramsey (1952) and Aamodt et al (1952) and the very recent work by Bäumer et al (2019) and Bäcker et al (2019). While deviations of our recent measurements from values published in the 1950s are not particularly surprising, the deviation from the cross section at $100~\mathrm{MeV}$ reported by Bäumer et al (2019) (∼5% from error bar to error bar and  ∼17% from value to value) needs more attention and is therefore discussed in the following. The approach to determine the $\mathrm{^{11}C}$ production cross section via the measurement of the amount of induced activity is comparable to our approach. However, the experimental method of Bäumer et al (2019) is quite different from the one presented in this work. While our experiment was set up in-beam and the induced $\beta^+$ activity was measured with scintillation detectors coupled by a coincidence unit, Bäumer et al (2019) transported their irradiated targets from the proton therapy center in Essen to a low background gamma spectrometry facility in Dortmund  ∼35 $\mathrm{km}$ away to analyze the samples there using a well-characterized high purity germanium detector. They used the same graphite target material type with high purity as used in this work, therefore no differences can be expected from the material, but the irradiation fields used are quite different: while in the present work the targets were irradiated with a single pencil beam impinging on the target center, Bäumer et al (2019) irradiated their targets with a scanned beam producing a homogeneous fluence on the target. Only slight differences (up to  ∼3% according to a dosimetric study by Gomà et al (2014)) may originate from the different way of determining the number of primary particles impinging on the target. In this work the clinical monitor calibration determined by an absorbed dose to water measurement under dosimetric reference conditions was used while Bäumer et al (2019) did separate measurements with a Faraday cup. The possible origin of the discrepancies between primary particle fluences determined by means of IC measurements and Faraday cup measurements has been widely discussed (Palmans and Vanitsky 2016, Gomà et al 2016) and it is still not finally resolved which method gives the more accurate result. Therefore, a conservative estimate on the accuracy of the IC based monitor calibration is included in our uncertainty calculation (see above). The origin of the remaining discrepancy between the $\mathrm{^{11}C}$ production cross sections determined in the present experiment and that of Bäumer et al (2019) are not clear yet and could be investigated in future work.

The $\mathrm{^{12}C(\,p,p2n){}^{10}C}$ reaction channel has been less investigated in previous studies than the channel for $\mathrm{^{11}C}$ production, however, there is also reasonable agreement between the $\mathrm{^{10}C}$ production cross sections presented here and the few data available in the literature. The decrease to higher energies measured by Matsushita et al (2016) might be an edge artifact due to their experimental method (PET imaging of thick targets after irradiation). Also the fact that they measured $\mathrm{^{10}C}$ production cross section values greater than zero below the Q-value of the reaction is probably an artifact because they obtained their energy information by correlation of the induced activity with the depth in their target. However, this relation is strongly smeared at high depths by the energy loss straggling.

Figure 8 shows the $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ production cross sections as a function of energy for carbon ions impinging on carbon targets. In the case of carbon projectiles the simple calculation of a single Q-value is not sufficient to describe the reaction threshold appropriately due to the several different projectile fragmentation channels that are possible. Therefore, the reaction thresholds shown in figure 8 were calculated using an appropriate built-in FLUKA routine that returns the corresponding energy threshold for each fragmentation channel. The minimum energy threshold among all possible fragmentation channels is used here as reaction threshold. However, it is important to note that unless complete fusion occurs, usually not all nucleons take part in a nucleus-nucleus reaction, and therefore the required energy per nucleon is actually greater for most of the collision processes than that given by the reaction threshold. Moreover, due to its not clear and straightforward determination, the Coulomb barrier has not been considered in the calculation of the reaction thresholds. Therefore, the values shown in figure 8 should only be used as a rough indicator of the actual minimum energy per nucleon needed for the reaction to occur.

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The available data for target fragmentation induced by carbon projectiles is much more sparse than for protons. The few datapoints from experiments at the Bevalac (Smith et al 1983) and at the HIMAC accelerator (Yashima et al 2003, 2004) compare reasonably well with the present cross sections while the newer datapoints from Salvador et al (2017) lie significantly higher. In contrast to the proton data (see figure 7), no rise of the $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ production cross sections on $\mathrm{^{12}C}$ targets towards lower energies can be observed for incident carbon ions in the energy range investigated in this work.

Figure 9 shows the same data as figures 7 and 8, but as the ratio of the $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ production cross sections both for protons (left panel) and carbon ions (right panel) impinging on carbon targets as a function of energy.

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Most of the systematic uncertainties cancel out when calculating the ratio, therefore the error bars are considerably smaller than in figures 7 and 8. However, the remaining uncertainties due to the counting statistics are larger for the carbon ion measurements than for the proton measurements due to the lower beam intensities (see above). A comparison with literature data is only possible for experiments where $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ production were measured simultaneously. This is only the case for the datasets from Matsushita et al (2016) and Salvador et al (2017). The present proton data compare reasonably well with those by Matsushita et al (2016) (their values below the Q-value of $31.83~\mathrm{MeV}$ are not shown, see discussion above) while for carbon ions the ratios presented here are again lower than those reported by Salvador et al (2017). For protons, the two-neutron-removal reaction cross section ($\mathrm{^{10}C}$ production) relative to the cross section for removal of only one neutron ($\mathrm{^{11}C}$ production) decreases with decreasing energy while for carbon projectiles, no energy-dependency could be observed at all in the investigated energy range of this study (down to  ∼75 $\mathrm{MeV~u^{-1}}$).

Figure 10 shows the $\mathrm{^{15}O}$ production cross section as a function of energy for protons and carbon ions impinging on oxygen targets.

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For the $\mathrm{^{16}O(\,p,pn){}^{15}O}$ reaction channel, the measured dataset fits well into the literature data and extends them towards higher energies. For the corresponding carbon ion reaction, there is fair agreement with the higher energy datapoint by Salvador et al (2017) but again (see also figure 8) the rise of the cross section towards lower energies which they report could not be reproduced in the present experiment. Their experimental method is comparable to our approach (monitoring of the $511~\mathrm{keV}$ photon emission with a pair of scintillators and a coincidence logic), but they do not report about any random coincidence correction. However, as shown in figure 2, without this correction the produced activity of the generated isotopes and consequently their production cross sections may be overestimated (depending on the produced activity). Therefore, one could speculate that the discrepancy between the cross sections reported by Salvador et al (2017) and the values presented here could be due to a missing correction for random coincidences in the method by Salvador et al (2017). Another point where their experiment and the measurements presented in this work differ considerably is the method how the beam energy was varied: while at MIT the energy could be actively changed by the synchrotron, Salvador et al (2017) had to use degrader plates which introduces the issue that the beam gets already contaminated by fragments before hitting the target. Another point where their method differs from the one presented here is that their target had to be moved to the measurement position after the irradiation while our setup could measure in-beam. Further research or comparisons could help to clarify where exactly these differences come from.

Generally, the threshold energies (expressed in $\mathrm{MeV~u^{-1}}$) are considerably lower for the carbon ion reactions than for the proton-induced reactions. For protons this leads to a gap of  ∼2–5 $\mathrm{mm}$ between the end of the activity profiles and the Bragg peak (Parodi et al 2002), while carbon ions produce target fragments almost until the end of their range. Thus for carbon ions there is a clear range correlation not only in the $\beta^+$-activity profiles of the $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ projectile fragments (due to kinematics they stop shortly before the primary $\mathrm{^{12}C}$ range (Enghardt et al 1999)) but also in those of the $\mathrm{^{10}C}$, $\mathrm{^{11}C}$ and $\mathrm{^{15}O}$ target fragments created along the beam path.

The cross section data presented in figures 7, 8 and 10 are also compiled in tables 1 and 2.

Table 1. Measured cross sections for the production of $\mathrm{^{10}C}$ and $\mathrm{^{11}C}$ target fragments by protons and carbon ions on carbon targets.

Projectile	Target	Thickness / mm	Kinetic energy / $\mathrm{MeV~u^{-1}}$	$\mathrm{^{10}C}$ production cross section $\sigma^{^{10}C}$ / $\mathrm{mb}$	$\mathrm{^{11}C}$ production cross section $\sigma^{^{11}C}$ / $\mathrm{mb}$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.91	$215.1^{+4.8}_{-4.9}$	$2.53 \pm 0.33$	$40.4 \pm 4.8$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.90	$184.3^{+5.3}_{-5.4}$	$2.40 \pm 0.33$	$41.9 \pm 5.1$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.91	$152.8^{+6.0}_{-6.2}$	$2.53 \pm 0.34$	$45.6 \pm 5.6$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.85	$121.7^{+7.0}_{-7.3}$	$2.63 \pm 0.38$	$50.0 \pm 6.3$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.83	$96.5^{+8.2}_{-8.8}$	$2.91 \pm 0.38$	$58.0 \pm 6.8$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.91	$75.4^{+9.6}_{-9.8}$	$3.21 \pm 0.43$	$65.7 \pm 7.9$
$\mathrm{p}$	$\mathrm{^{12}C}$	9.83	$49.3^{+12.9}_{-16.5}$	$2.69 \pm 0.40$	$75.8 \pm 9.6$
$\mathrm{p}$	$\mathrm{^{12}C}$	4.99	$40.7^{+7.8}_{-9.2}$	$2.15 \pm 0.41$	$83.4 \pm 11.7$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.91	$417.5^{+10.2}_{-10.3}$	$4.18 \pm 0.99$	$72.3 \pm 14.5$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.90	$367.3^{+10.8}_{-11.0}$	$3.75 \pm 0.67$	$70.9 \pm 9.7$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.91	$316.4^{+11.7}_{-11.9}$	$4.33 \pm 0.66$	$71.9 \pm 9.2$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.91	$264.0^{+12.9}_{-13.3}$	$4.21 \pm 0.68$	$72.2 \pm 9.3$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.91	$158.8^{+17.4}_{-18.9}$	$4.15 \pm 0.65$	$75.2 \pm 9.5$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	9.90	$101.4^{+23.0}_{-27.9}$	$4.42 \pm 0.69$	$76.9 \pm 9.8$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	5.01	$92.0^{+12.8}_{-14.3}$	$3.99 \pm 0.99$	$75.9 \pm 11.7$
$\mathrm{^{12}C}$	$\mathrm{^{12}C}$	4.99	$65.9^{+15.9}_{-19.7}$	$4.37 \pm 1.17$	$80.2 \pm 13.7$

Table 2. Measured cross sections for the production of $\mathrm{^{15}O}$ target fragments by protons and carbon ions on oxygen targets.

Projectile	Target	Thickness / $\mathrm{mm}$	Kinetic energy / $\mathrm{MeV\,u^{-1}}$	$\mathrm{^{15}O}$ production cross section $\sigma^{^{15}O}$ / $\mathrm{mb}$
$\mathrm{p}$	$\mathrm{^{16}O}$	3.93	$217.0^{+2.8}_{-2.9}$	$39.7 \pm 4.8$
$\mathrm{p}$	$\mathrm{^{16}O}$	3.92	$145.0^{+3.7}_{-3.8}$	$48.2 \pm 6.3$
$\mathrm{p}$	$\mathrm{^{16}O}$	3.91	$104.2^{+4.6}_{-4.8}$	$53.2 \pm 6.8$
$\mathrm{p}$	$\mathrm{^{16}O}$	3.88	$72.4^{+6.0}_{-6.4}$	$68.2 \pm 9.4$
$\mathrm{p}$	$\mathrm{^{16}O}$	3.91	$39.4^{+9.1}_{-11.4}$	$71.7 \pm 9.6$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.92	$422.4^{+5.9}_{-6.0}$	$65.1 \pm 15.4$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.91	$322.0^{+6.8}_{-6.9}$	$69.8 \pm 12.2$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.88	$240.3^{+8.0}_{-8.2}$	$76.4 \pm 11.8$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.82	$113.0^{+12.9}_{-14.0}$	$79.7 \pm 12.7$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.91	$84.5^{+15.6}_{-18.1}$	$78.0 \pm 17.2$
$\mathrm{^{12}C}$	$\mathrm{^{16}O}$	3.90	$65.5^{+18.3}_{-23.6}$	$72.7 \pm 15.5$

Figure 11 shows the cross section models for the $\mathrm{^{12}C(\,p,pn){}^{11}C}$ and $\mathrm{^{16}O(\,p,pn){}^{15}O}$ reactions used for the radiation transport calculation. The calculated Q-values (see figures 7 and 10) were assumed as reaction thresholds. For sake of clarity the reference data from the literature which were also used as guidance for the models are not shown in figure 11.

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The results of the proton transport calculation using the $\mathrm{^{12}C(\,p,pn){}^{11}C}$ and $\mathrm{^{16}O(\,p,pn){}^{15}O}$ cross sections presented in this work and the $\mathrm{^{16}O(\,p,X){}^{11}C}$ and $\mathrm{^{16}O(\,p,X){}^{13}N}$ cross sections published by Bauer et al (2013) are shown in figure 12. The calculated profiles are compared with profiles measured with a PET scanner (Espana et al 2011). The depth in the tissue equivalent gel phantom was converted into water equivalent depth with a conversion factor determined by TOPAS simulations. Considering the noise in the measured activity profiles their shapes are well reproduced for both irradiation protocols (the $5~\mathrm{min}$ protocol being dominated by $\mathrm{^{15}O}$, and the $30~\mathrm{min}$ protocol being dominated by $\mathrm{^{11}C}$) by the transport calculation using the cross section tables shown in figure 11. This is in contrast to the original work by Espana et al (2011) where standard ICRU and EXFOR tables were applied and none of them could reproduce both the $\mathrm{^{11}C}$ and $\mathrm{^{15}O}$ profiles accurately. As already pointed out by Espana et al (2011) the tails behind the distal edges of the measured activity profiles are artifacts due to background noise and PET image reconstruction and can therefore be neglected. As seen in comparison with the dose profiles the distal edges of the activity profiles lie some mm before the distal edge of the Bragg curve. This shift is well-known and is due to the threshold energies of the reactions producing the $\beta^+$-emitters (see also figures 7 and 10).

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This example shows that the calculation of activity depth profiles for protons does not require a full Monte Carlo simulation but only the proton spectrum as a function of depth (which is e.g. available in typical treatment planning systems as basic data) because the created target fragments do not need to be transported. For this kind of calculations cross section tables can be generated using the cross section data presented in this work and the respective Q-values. However, in the case of carbon ions, the projectile fragments are produced with velocities similar to the projectiles and therefore also need to be transported by the code used for calculation of the reference pattern. This is more complex and a typical problem where at least a partial (Pönisch et al 2004) or even a full Monte Carlo simulation (Sommerer et al 2009) is required.