Experimental consolidation and absolute measurement of the nat C(p,x) 11 C nuclear activation cross section at 100 MeV for particle therapy physics

The nat C(p,x) 11 C reaction has been discussed in detail in the past [EXFOR database, Otuka et al. (Nuclear


Introduction
The activation of natural carbon and the production of the radionuclide 11 C by protons ( nat C(p,x) 11 C) is a well studied nuclear reaction and measured cross sections are available from the threshold (18.72 MeV) up to several GeV [31]. However, the absolute cross section values vary by about 15% among the literature references and accumulate around two distinct trends. Interestingly, this difference still persists in three recent publications [5,6,17] where the nat C(p,x) 11 C reaction has been studied independently with two different experimental setups.
In the past, the investigation of this reaction was motivated by the application as a monitor reaction for nuclear physics experiments, e.g. for the determination of the number of protons (e.g. Ref. [33]). Today, the interest on the nat C(p,x) 11 C reaction is motivated by its potential application in monitoring and verification of proton therapy. On the one hand, carbon is a main component of biological molecules. Therefore, 11 C is produced in the irradiated tissue as a by-product of the radiotherapy with protons or heavy ions and its spatial distribution can be determined with a positron emission tomography (PET) camera during or after the treatment [32]. On the other hand, dosimetry of proton fields with activated graphite foils has been proposed as an alternative approach to ionization chamber based dosimetry [28]. Such applications in medical physics require an accurate knowledge of the nat C(p,x) 11 C cross section in the therapeutic energy range (proton energies up to 250 MeV) and suitable uncertainties for the accuracy required for radiotherapy. Monte Carlo transport codes are typically verified and optimized by comparison with experimental cross section data [4].
The goal of the presented study is to investigate the cause of the systematic deviations of measured nat C(p,x) 11 C cross sections and to provide a consensus value at 100 MeV. For this purpose a comparative measurement using two different experimental methods, that from Dortmund / Essen [5,6], which is further called Dortmund Low Background Facility (DLB) method, and that from Giessen / Darmstadt [17], which is named in-beam method, is performed at the West German Proton Therapy Centre Essen (WPE) (Essen). As a third method, a measurement using a commercial PET scanner is performed. Furthermore, different experimental parameters that are suspected to potentially introduce systematic uncertainties (irradiation field geometry, target thickness, target material, target size, field size of the broad proton field) are varied for the DLB method. Figure 1 shows an overview of the nat C(p,x) 11 C cross section data sets currently found in literature [1,2,5,6,10,13,16-18, 21,25,26]. Note, that the dataset by Kettern et al. [20] is not shown because the data points scatter strongly. The data points accumulate around two different curves. In many of the quoted publications only the relative error is given and systematic uncertainties have not been estimated properly. Therefore, due to this lack of standardization, the error-bars are not shown in Fig. 1. A detailed study of the original publications reveals that most measurements are only relative measurements normalizing on prior publications. Only five publications, Aamodt et al. [1], Kavanagh et al. [18], Kettern et al. [20], Bäcker et al. [5] (together with Bäumer et al. [6]), and Horst et al. [17] provide absolute measurements at 100 MeV beam energy. While Aamodt et al. [1] and Bäcker et al. [5] accumulate around the upper curve, Kavanagh et al. [18] and Horst et al. [17] accumulate around the lower one. The cross sections proposed by Kettern et al. [20] are in agreement with both curves because of the larger uncertainty of the experimental setup.

Current status in the literature
In the right panel of Fig. 1 the same data sets are normalized to 100 MeV (normalization done by extrapolation to 100 MeV). In the peak area between 50 and 60 MeV beam energy, the data points of some of the literature references scatter. This leads to possible uncertainties in the estimation of the exact trend of the excitation function. However, the shape of a sharp peak is highly likely based on the different literature sources. The normalization demonstrates that all past experiments have in fact measured the same curve of the nat C(p,x) 11 C excitation function while the discrepancies between the measurements are found in the absolute crosssection values. Therefore, we report a detailed experimental study of the absolute cross section for the nat C(p,x) 11 C reaction at 100 MeV. This value can then in turn be used to obtain a more reliable absolute excitation function from the relative values that are shown in the right panel of Fig. 1.

Materials and methods
In recent measurements of absolute nat C(p,x) 11 C cross sections, two types of experimental methods have been developed, the DLB method [5,6] and the in-beam method [17]. Both methods are compared in a joint experiment performed at WPE. While the DLB experiment features a low-level gamma-ray spectrometry laboratory at the TU Dortmund University which requires the transport of the target from the proton therapy center to Dortmund (about 40 km), the inbeam method bases on an experimental setup to measure the activity of the targets directly at the beam line after the end of irradiation. Only important aspects for the comparison of the two methods are summarized in the following while details can be found in the original publications.

Proton therapy center
All irradiations for this study are performed at the WPE. The WPE is a cyclotron-based proton therapy center with four treatment rooms featuring a variety of delivery modes for clinical patient treatment. The pencil beam scanning technique is chosen for the experiments as it allows to use quasi mono-energetic proton beams with less secondary particles compared to passive beam applications techniques. The proton energy used for the present experiments is 100 MeV since most data from the past overlap at about 100 MeV and the activation cross section obtained by Kavanagh et al. [18], which is considered the gold standard, as the cross section is published with low uncertainties, has been measured at 98 MeV.  In a first step, it is necessary to exclude any systematic errors in the initial 11 C activity determination with the two experimental setups. Therefore, a combined measurement is performed at the WPE. The in-beam setup known from Ref. [17] is installed at the WPE. The setup is based on three BaF 2 crystal scintillation detectors and a coincidence unit and is placed directly at the beam line downstream of the treatment head. Thus, it is possible to start the data acquisition directly after the irradiation is finished. The irradiation of the target is performed with a co-axial spot and the target is tilted by 45 • compared to the beam axis, to achieve a better detection efficiency. The target has a lateral size of (8.00±0.05)×(8.00±0.05) cm 2 and a thickness of (7.1±0.1) mm in beam direction. The density is (1.8±0.1) g/cm 3 . The target material is manufactured from the same supplier of the target material as in Refs. [6,17] (SGL Carbon, Wiesbaden, Germany). Thus, impurities are considered to be negligible. After the activation of the graphite target by a short proton pulse (shorter than 1 s), the in-beam setup immediately monitors the decay of the induced β + -activity. It consists of three BaF 2 scintillators coupled by a coincidence trigger unit. Two scintillators are arranged in 180 • to monitor the 511 keV photon coincidences from the β + -decays of the produced 10 C and 11 C. A third scintillator is arranged at 90 • to monitor the random coincidence rate which has to be subtracted from the 180 • coincidences to obtain the true coincidence rate. For absolute calibration of the system, a 22 Na reference source as used in Ref. [17] is measured right before the experiment. To cross check the calibration, the 22 Na source from Ref. [6] is used in addition.
After about 20 min of data acquisition, the target is brought to the Dortmund Low Background Facility (DLB) to perform a gamma-ray spectrometry measurement. The distance between WPE and DLB is about 40 km resulting in about 40 min for the target transport. The experimental procedure is analogous to Ref. [5,6] and the initial 11 C activity is determined according to DIN ISO 11929:2011 [11]. The DLB is a low-level gamma-ray spectrometry laboratory at TU Dortmund University [12]. The laboratory is located above ground and has a multi-layer shielding with an active muon-veto to reduce the background down to approximately 1.85 counts/(min kg detector ) in the energy range from 40 to 2700 keV [29]. The detector itself is an about 1.2 kg highpurity germanium detector. Because of the sophisticated shielding concept, it is still possible to accurately determine activity even after the measurement at the beam line and the transport from Essen to Dortmund.
Previous activation experiments at the WPE have been performed with a scanned field larger than the target [5,6], while the in-beam setup [17] can only be used with a pencil beam impinging at the center of the target. For the comparative measurement of the in-beam and DLB method, it is necessary to change the irradiation setup from the previous irradiations at the WPE and the subsequent activation measurements at the DLB to a pencil beam.

Variation of experimental parameters
After the direct comparison of both experimental setups, additional experimental parameters are investigated. While the number of incident protons, the target material (different densities, different suppliers) and field and target size have been investigated in a previous study and an influence was demonstrated to be insignificant [6], now, the target thickness, lateral target size and the irradiation technique (central axis pencil beam and scanned field) are compared. Therefore, targets of various thicknesses are irradiated with a co-axial pencil beam spot and a scanned proton field, which is larger by a factor of four compared to the lateral target size, to make sure that the target is activated homogeneously in lateral directions. Therefore, the lateral beam profile has been obtained by the so-called pair-magnification method introduced by Lin et al. [22,23] and optimized at the WPE [34]. The measurement is performed with an IBA Lynx PT (IBA Dosimetry, Schwarzenbruck, Germany). The Lynx PT detector features a scintillation screen of 30×30 cm 2 and a CCD chip to take an image of the scintillation light. For a target of 2.5×2.5 cm 2 and a beam energy of 100 MeV 28% of the protons miss the target which must be taken into account in the analysis. For the 8×8 cm 2 target from Sect. 3.2, the correction for lateral loss of protons is only 1.5%. Additionally, a 7.5×7.5 cm 2 large target with a significant larger thickness (15 mm) is irradiated with a single pencil beam. For this target, the correction is 0.9%, as the target is placed perpendicular to the beam direction. In contrast to the target used in Sect. 3.2, the targets used for the experiments described in this section are perpendicular to the beam axis during the irradiation. The obtained fluence profiles are shown in Fig.  2. It is visible, that a part of the protons, depending on the lateral size of the target does not hit the target for the single pencil beam irradiation. This has to be taken into account for the calculation of the activation cross section. The activation of the target holder for the scanned field does not affect the results of the activity measurement, as the target is extracted from the target holder before the activity is measured.
The uncertainties of the cross section measurements are estimated according to the description in Ref. [6]. Briefly, the systematic uncertainties originate from measurements of the number of protons using the Faraday-cup, gamma-ray spectrometry method, and the determination of the target mass. All uncertainties sum up to about 2.4% [6] (1 standard uncertainty). As increasing target thicknesses lead to decreasing proton energies, the activation is not homogenous in beam direction according to the increasing cross section shown in Fig. 1. For the correction of the beam profile for the sin-gle spot activation and for the inhomogeneous activation for targets thicker than 1 mm an additional 1% uncertainty has been estimated for the efficiency calculation of the gammaray spectrometer. In consequence, for these targets the uncertainty is 2.6% if only one correction is applied or 2.8% if both corrections are necessary.
For the activation with the scanned proton field, the activation cross section is calculated with the equation from Ref. [5]: A 0 is the decay-corrected activity for the time of irradiation, N A is the Avogadro constant, M T and m T are the target's molar mass and mass, proton is the proton fluence (number of protons per field size) and λ = 567.3 · 10 −6 1/s [27] is the decay constant, calculated from the half-life of 11 C. For the single spot activation the cross section is calculated according to Ref. [17]: where z is the target thickness, N is the corrected number of protons across the target area and n V is the particle density of the target. The correction for the number of protons is calculated from the fluence distributions obtained with the Lynx PT described at the beginning of this section. Using conversions, it can be shown that both equations are equivalent. Both equations contain the accessible variables for the number of protons, number of reaction products and number of target nuclei for each experiment.

Validation measurements with a PET/CT scanner
To check the consistency of all performed measurements, two additional measurements are performed using a Siemens Biograph Vision ToF-PET/CT System (Siemens Healthineers, Erlangen, Germany). For the first measurement a 10 cm large stack of 2.5×2.5 cm 2 1 mm thick graphite foils is activated with a scanned field of 130 MeV protons. At 3.7 cm depth, the graphite is activated by protons with an average residual energy of 100 MeV. However, the energy spread of the degraded beam is larger than for the irradiations presented in Sects. 3.2 and 3.3. The activity of the complete stack is measured with the PET scanner and the cross section is calculated with this result.
The second measurement is the activation of a larger target (5×5 cm 2 ) with a single pencil beam directed to the center of the target. The target is divided into four sections. Each section has a different thickness consisting of one, three, five and seven 1 mm thick graphite sheets. In consequence, different activities are obtained for the sections. Furthermore, Before starting with the experiment, the calibration of the PET scanner is checked with the 22 Na-source used for the cross check of the in-beam setup and in Ref. [6]. The measurement results in a precision of 5% of the estimation of the activity. The 5% deviation from the calibration certificate of the source is the assumed systematic uncertainty of the PET system, which is commonly taken in clinical practices [19].

Results and discussion
4.1 Comparison of the activity determined with the two experimental setups The β + activities induced in a graphite target irradiated with 5 × 10 9 protons are measured with both experimental setups.
To compare the determined initial activities, the exponential decay of the 11 C has to be taken into account, since the target has to be brought from Essen to Dortmund for the measurement at the DLB. The decay curve obtained with the coincidence setup at the beam line and at the DLB in the annihilation peak of the gamma-ray spectrum is shown in Fig. 3. It is visible, that both activities match within their uncertainties. The activity obtained at the beam line is A in-beam 0 = (11.1 ± 0.7) kBq. The activity obtained at the DLB is A DLB 0 = (11.7 ± 0.4) kBq. The uncertainty of the DLB is lower compared to the in-beam setup which is given by the special design and characteristics of the DLB. Moreover, within the first seconds of the measurement, a fast decay of produced 10 C is visible. It is not measured at the DLB due to the short halflife. Once, the cross sections for the productions of 11 C has been validated, previous measurements of the cross section for the 10 C production can be reviewed as well.
Data acquisition time at the DLB is about 2 weeks, to identify possible impurities in the graphite target, as they have been found in a previous study [6]. A gamma-ray spectrum of the activated graphite sample is shown in Fig. 4. Compared to Be is produced from carbon as well, but due to the smaller number of impinging protons and short time of measurement compared to Ref. [5] the statistical uncertainty is larger and the radionuclide is not analyzed here.
Within the first hour an additional coincidence peak of the full energy peak and the back-scatter peak of the 511 keV photons is visible at approximately 700 keV. The influence of the coincidence of a primary 511 keV photon and a backscattered photon on the estimated activity is insignificant as the peak is smaller by three orders of magnitude. Figure 5 shows the energy spectra of the events detected by the 180 • and by the 90 • detector pairs obtained with the in-beam setup. In an offline analysis, the 511 keV peaks are separated by applying a cut on the energy spectra. Subtracting the 90 • coincidences from those measured in 180 • gives the true 180 • coincidence rate required for determination of the initial activity.

Variation of experimental parameters and measurements at the DLB
Several targets with different densities and thicknesses are irradiated with a single pencil beam and a scanned proton field. From the measured activity of 11 C, the activation cross section is calculated. Table 1 provides an overview on the experimental parameters and the results of the activation cross section determination. All results are shown in Fig.  6 for comparison. Since the targets have different densities and thicknesses, the areal density zρ, where ρ is the density of the target material, is considered to be a representative for the effective thickness of the targets. For zρ less than 1 g/cm 2 the calculated activation cross sections vary between 67 and 70 mb and agree within one standard deviations. For zρ > 1 g/cm 2 , the calculated cross sections are slightly smaller, but an increasing cross section is expected because of the degradation of the beam energy of 7 MeV in 1.5 g/cm 2 graphite [7]. However, the correction of the efficiency of the DLB is less accurate for increasing zρ because the activation increases with the thickness of the targets as the cross section increases at lower energies (see Fig. 1). This results in an increasing activity to the downstream end of the target. The targets are placed with the upstream end of the target on the detector end-cap. In consequence, the slight decrease of the measured cross section might be induced by systematic uncertainties of the estimated activity profile for the efficiency calculation. In contrast, Kavanagh et al. [18] observed an increasing cross section for thicker target, but the thickness of the target ranged between 0.15 and 0.65 cm and no information on the density is given. Furthermore, the increase is about 1 mb which is well below the differences measured in the current study. However, a plastic scintillator has been used as target material. Today, high-purity graphite sheets are supposed to be superior compared to plastic targets as used in the past. Additionally, the agreement of all measurements can be tested in an alternative way. The activation cross section is representative of the probability of the nuclear reaction. Independent from the target size, zρ is representative of the number of target nuclei for a homogenous distribution of the target particles across the target area. The number of activated residuals per number of incident protons gives the probability of a reaction in each target. For a constant activation cross section for all zρ, all measurements of the study should accumulate around a linear fit, which slope represents the activation cross section. In Fig. 7, it is visible, that indeed all measurements accumulate around a linear fit. The activation cross section calculated from the slope is (67.0 ± 1.8) mb. This value is smaller than the mean value of all values. As it is visible in the Figs. 6 and 7, the cross sections for zρ > 1.0 g/cm 2 are Table 1 Measurement results for the calculation of the activation cross sections for the nat C(p,x) 11 C-reaction at 100 MeV for the different targets. "hd" indicates high-density target material used in the experiments described in Ref. [17], "ld" low-density for the target material used in the experiments described in Refs. [5,6]. The value for the scanned-field activation of the ld-target is taken from Ref. [6]. The correction is the correction for the lateral beam profile. 1 This result is obtained with the in-beam setup, same target and activation as line before smaller than the other activation cross sections obtained at the DLB. Thus, these individual results have a large influence on the result of the fit. The deviation is explained by the uncertainty of the efficiency of the DLB. For the targets with zρ < 1.0 g/cm 2 , (68 ± 3) mb is concluded as the reference cross section as this is the mean of all obtained activation cross sections. As the beam is degraded along the target, (97±3) MeV is the reference energy of the calculated cross section.

Consistency measurements with a PET/CT
In the PET image, a region of interest (ROI) is placed 4 mm around the point where the proton beam is degraded to 100 MeV. The activity of the ROI is measured and subsequently the activation cross section is calculated as (69.8±3.6) mb. This experiment is a test, if thick targets may result in a bias determining the activation cross sections. The calculated activation cross section is in the same magnitude as the ones presented in Sect. 4.3 and added to Fig. 6. The uncertainty is given by the one of the PET scanner (5%). Obviously, this simple study does not result in a high accuracy measurement for cross section determination compared to the gamma-ray spectrometry with a high-purity germanium detector. Furthermore, more uncertainties have to be taken into account, like the energy loss straggling of the protons or the scattered protons, which leave the target. All further uncertainties result in less accuracy of this measurement method. In consequence, the uncertainty of the determined cross section is larger than the 5% contribution of the activity and the 1.1% of the number of protons [6]. In a second step, a target with varying thicknesses is irradiated with a single pencil beam. For the four sections of the

Renormalization of the excitation function
All experimental parameters included in the study can be ruled out as the cause of the systematic difference among past publications as all results from both experiments described in this paper converge at one value. Thus, another experimental parameter has to be the origin of the systematic deviation of the measured cross sections. The experimental setups used in the latest publications have been compared with a joint experiment, analyzing the same target with the in-beam and the DLB method. After a systematic error of one of the measurement setups has been excluded, the experimental parameters have been varied. The activity of the targets has been measured with the well-known gamma-ray spectrometry setup DLB. In summary, it is possible to reduce the systematic uncertainties down to 3%. From the measured cross sections presented in Sects. 4.2 and 4.2, (68±3) mb is derived as the new reference cross section for (97±3) MeV protons as the weighted mean value of all measured cross section in this work. The different relative uncertainties are used as the weights, to consider the different accuracy of the individual experimental setups. The energy of (97±3) MeV is given by the degradation of the beam in the thick targets used in the experiments. The new reference value is shown in Fig. 8  From the results in Sect. 4.2, it can be derived that thin targets should be used for cross sections measurements as the results for the thin targets scatter less than the one for thicker targets. Thin targets manipulate the beam less than thick targets and the properties of the beam can be assumed to be constant. With a well defined proton beam at a clinical therapy center, the targets can be irradiated with different irradiation techniques. In addition, the consistency check with the PET scanner results in a comparable cross section. This third independent method excludes any error for both setups, the in-beam setup and the DLB. A thicker target and a build-up is used in the PET scanner. This results in higher uncertainties for the measured cross section. The linear increase of the activity by increasing the thickness of a target is demonstrated as well, as shown in Fig. 7. However, at some point thick targets will lead to an inaccuracy of the cross sections measurement due to difficult accessible beam properties. With the new reference cross section, the normalized excitation function presented in the right panel of Fig. 1 can be re-normalized to this value. With the normalized data given in the supplementary material, a new excitation function can be calculated. After the re-normalization of the nat C(p,x) 11 C excitation function, more cross sections for other nuclear reactions can be reviewed whether a re-normalization is necessary, since many literature values have been measured relative to this cross section.
At last, there is a potential difference to the cross section published in Ref. [17]. To explain this difference, it has to be mentioned that two different methods for determining the primary proton number are commonly applied: by means of a dose measurement using an ionization chamber or by means of a charge measurement using a Faraday cup, e.g. Refs. [8,9,14]. A systematic difference of up to 3% has been observed along the publications over time, where Faraday cup measurements result in a higher number of incident protons. Furthermore, systematic uncertainties of these translate directly into the cross section value and it is yet unclear which method is more accurate. Both methods are still under investigation and possible systematic errors are discussed in the literature aiming for the correct results, see e.g. Refs. [24,30]. For all cross section calculations presented in this study, a Faraday cup based monitor calibration is used, as it is common for nuclear physics experiments. As presented in Ref. [6], the Faraday cup measurement has been performed according to [15].
In conclusion, we want to note that the observed 3% difference between both measurement methods to determine the number of protons is covered in our uncertainty. While the Faraday Cup measurement is associated with an uncertainty of about 1.2% [6], the reference dosimetry in proton therapy using ionization chambers is associated with an uncertainty of 2.3% according to IAEA TRS 398 [3]. However, further studies on the difference of both methods might reduce the uncertainty of the cross sections further.

Conclusion
The presented study investigates the nat C(p,x) 11 C reaction at a proton energy of 100 MeV. The previously published activation cross sections scatter by about 15%. The experimental parameters and the two recent measurements of activation cross sections are investigated regarding their influence on the obtained results. In result, all investigated parameters do not have any influence on the measured cross section on a significant level. With a well-defined gamma-ray spectrometer system and varying the several experimental parameters and an additional validation measurement with a PET scanner, the cross section is measured with a high accuracy. A new reference value of (68 ± 3) mb is provided at (97±3) MeV proton energy, which gives together with the re-normalized literature data accurate values of the absolute nat C(p,x) 11 C excitation function in the energy range relevant for proton therapy. This allows an accurate modeling of the nat C(p,x) 11 C nuclear reaction and calculation of PET activity distributions in proton therapy patients. Other cross section data from the literature can be reviewed if they have been measured relative to the carbon activation in the past. Moreover, Monte Carlo simulations of activity distributions will benefit from more accurate cross-section data. However, a remaining issue for determination of an accurate absolute cross section is the relatively high uncertainty of the beam monitor calibration. Systematic differences between the two standard methods in proton therapy, based on either Faraday cup or ionization chamber measurements, of up to 3% are an open issue. For this study, the Faraday cup is supposed to be the more accurate estimation of the number of protons as it is common in nuclear physics experiments.