Direct determination of k_Q for Farmer-type ionization chambers in a clinical scanned carbon ion beam using water calorimetry

All calorimetric and IC measurements were performed at HIT, which relies on the intensity-controlled raster scanning method (Haberer et al 1993). Due to the synchrotron-based beam delivery, the irradiation has a pulsed structure with beam-on and beam-off times, where new particles are accelerated to the requested energy, of both about 5 s.

The beam delivery is controlled by a beam application monitoring system (BAMS) by Siemens, which is based on the original design from the Helmholzzentrum für Schwerionenforschung GSI, Darmstadt, Germany (Haberer et al 1993, Kraft and Weber 2011). It features a redundant system of three identical large area ICs regulating the number of particles delivered per irradiation spot, which are framed by two multi-wire proportional chambers (MWPCs) controlling the beam position. The time-resolved measurements of the MWPCs as well as the ICs containing all the irradiation-relevant information such as e.g. beam position, beam width, irradiation duration, and number of delivered particles per spot are recorded for each irradiation within the irradiation records.

For the k_Q measurements, an irradiation plan has been used which nominally should ensure that the irradiation field dependent correction factors for the water calorimeter (e.g. heat conduction corrections) are as small as possible. In summary, a field size of about 5.8 cm  ×  5.8 cm (realized by 26  ×  26 spots with 2.3 mm spacing on a rectangular grid) was chosen and optimized for homogeneity as well as reproducibility by performing a re-painting (figure 1). A pencil beam of about 5.5 mm full width at half maximum (FWHM) was selected, as the corresponding intensity distribution shows the best symmetry (figure 2) which is also beneficial in terms of heat conduction calculations (section 2.4.1). To enable a preferably short irradiation time, the highest clinically used particle flux of 8  ×  10⁷ ions per second was chosen, resulting in an irradiation time per spot of about 32 ms and of about 95 s for the complete scan. In total, an absorbed dose to water of about 1.5 Gy was applied. In order to have a large distance between the calorimetric measurement position (at a nominal water depth of 50 mm) and the Bragg peak, an energy of 429 MeV/u was selected corresponding to a residual range R_res of 24.1 cm in water (R_res  =  R_p  −  d, with d being the measurement depth and R_p the practical range defined as the depth at which the absorbed dose beyond the Bragg peak decreases to 50% of its maximum value (Lühr et al 2011). The residual range is a measure of the radiation quality Q according to DIN 6801-1 (DIN 2016), while TRS 398 assumes Q to be energy-independent. As shown in figure 3, the corresponding depth dose distribution (ddd) is very flat around the calorimetric measurement position exhibiting a small relative dose gradient of  −0.023% mm⁻¹. As the precise knowledge of the irradiation parameters and the resulting dose distribution is essential for the evaluation of correction factors required for the D_w and subsequent k_Q determination, corresponding measurements were directly performed at the measurement position of the water calorimeter and frequently repeated over the course of all k_Q measurements. Hence, an experimental set-up was designed to mimic the real measurement conditions of the water calorimeter (including the phantom and the calorimetric detector) by means of a water-equivalent slab phantom (figure 4). The 2D IC array STARCHECK by PTW (Freiburg, Germany) consisting of 527 air-filled ICs (dimensions: 8 mm  ×  3 mm  ×  2.2 mm, max. spatial resolution: 3 mm) was used for the measurement of relative lateral dose profiles. To increase the spatial resolution, the STARCHECK array has been repositioned multiple times. Prior to its use at HIT, extensive measurements in the well-characterized ⁶⁰Co irradiation field at PTB allowed a reduction of the relative absorbed dose to water calibration uncertainty for all of the array's chambers from 1% (as stated by the manufacturer) to 0.3%. Therefore, it is assumed that the relative response of the different detectors of the STARCHECK array is known within 0.3% also in the carbon beam. The long-term reproducibility of the beam delivery system has been monitored through frequent measurements with a thimble IC. The data showed that the beam delivery system at HIT in combination with the irradiation plan used enables very reproducible measurement conditions with a relative standard deviation of 0.3% found for the delivered dose at the central axis of the field over the course of all k_Q measurements (7 months).

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The PTB transportable water calorimeter is operated at a water temperature of 4 °C. Its design, temperature stabilization system and the detector measuring the radiation-induced temperature rise have been previously described in detail (e.g. Krauss et al 2012, Krauss 2006). Briefly, the radiation-induced temperature rise is measured by two calibrated thermistors each fused in the conically shaped tip of a glass pipette. The glass pipettes themselves are centrally arranged inside a water-filled (high-purity water saturated with hydrogen gas) thin-walled plane-parallel glass cylinder perpendicular to the cylinder axis, with the two thermistors facing each other having a distance of about 7 mm. The glass cylinder is positioned inside the water phantom with the cylinder axis oriented parallel to the beam direction. The measurement depth of the thermistors with respect to the beam entrance window of the water phantom is nominally set to 50 mm as schematically shown in figure 5. The spacing was frequently checked directly before and after each calorimetric beam time (BT). Small distance changes occurring due to a time-dependent bowing of the entrance window of the water phantom were found to be negligible.

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Within this investigation, two calorimetric detectors (using the same type of glass cylinder and the same type of preparation) with slightly different spacings between the thermistors were used. Both detectors were employed in the primary standard water calorimeter at PTB in ⁶⁰Co radiation in order to prove the response of the detectors just before and after their usage at HIT.

The resistance of each thermistor (about 10 kΩ at 4 °C) is independently determined within a separate 1.5 V DC-powered voltage divider circuit with the thermistor being one part of the voltage divider and a calibrated high-precision resistor with a well-known resistance value (nominally 20 kΩ) the second. This allows measuring the resistance of the thermistor with a resolution of better than 1 mΩ (Krauss and Kapsch 2014).

In total, three separate BTs were performed within a time period of 7 months, each comprising between 60 and 80 calorimetric measurements. Figure 6 shows a typical thermistor signal for a series of ten irradiations. For each irradiation, the separate measurement signals of both thermistors were analyzed by performing linear fits over the pre- and post-irradiation drift curves, extrapolating the fits to the mid-run position and taking the corresponding difference as the radiation-induced resistance change of the thermistor. Time intervals of 110 s for the pre- and post-irradiation drift curves, with the fit interval for the post-irradiation drift curve starting 10 s after the end of an irradiation, were applied (Krauss and Kapsch 2014). It was found that the standard uncertainty of the mean value for the relative resistance change during irradiation amounts to 0.15% for each calorimetric experiment.

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The basic definition of the k_Q factor is given as the ratio of the chamber's absorbed dose to water calibration coefficients N_D,w,Q at the radiation quality Q and N_D,w,Co at the reference beam quality (here: ⁶⁰Co):

$$\begin{eqnarray}&&{{k}_{\text{Q}}}=~{{N}_{\text{D},\text{w},\text{Q}}}/{{N}_{\text{D},\text{w},\text{Co}}}.\end{eqnarray} \tag{ 2 }$$

The calibration coefficients N_D,w,Co of the ICs used are traceable to PTB's primary standard water calorimeter operated in a ⁶⁰Co beam under reference conditions (Krauss 2006). The calibration coefficients N_D,w,Q with respect to the raster scanned carbon ion beam at HIT are determined by means of the transportable water calorimeter in a two-step procedure. First, the calorimeter is used to measure D_w,Q in a certain measurement depth z on the central axis of the irradiation field. Second, after heating the water to about 18 °C and replacing the calorimetric detector by an IC positioned with its reference point, P_ref, at z, the reading M_Q of the IC is directly measured in the water phantom of the calorimeter. Then, N_D,w,Q is determined by the following equation:

$$\begin{eqnarray}&&{{N}_{\text{D},\text{w},\text{Q}}}={{D}_{\text{w},\text{Q}}}/\left({{M}_{\text{Q}}}\cdot {{k}_{\text{v}}}\right).\end{eqnarray} \tag{ 3 }$$

Here, M_Q is corrected for the influence quantities temperature and pressure, electrometer calibration, polarity effect and ion recombination. As the measurement of D_w,Q by means of water calorimetry is rather point-like, the volume correction factor k_v is required to account for the volume-averaging effect of the IC to ensure comparable measurement conditions. k_v depends on the lateral dose distribution and the volume of the IC.

It is important to note that the values of the experimentally determined k_Q factors are not directly comparable with the theoretical k_Q factors stated in TRS-398 (Andreo et al 2006) or DIN 6801-1 (DIN 2016), as the procedures described in the protocols regarding chamber positioning and consideration of the displacement effect differ from the above-mentioned experimental procedure. Following the protocols, P_ref of the IC has to be positioned 0.75 · r_cyl (r_cyl: inner radius of the chamber in mm) deeper than z, while the common calibration procedure for the determination of N_D,w,Co implies positioning P_ref at z. Additionally, there is a difference between both protocols in the consideration of the displacement correction (as part of the perturbation factor ${{p}_{{{Q}_{0}}}}$ in equation (1)) for ⁶⁰Co as reference beam quality. While in TRS-398 the displacement correction for ⁶⁰Co is considered within the overall perturbation factor, DIN 6801-1 separately addresses this effect by introducing a further chamber-dependent correction factor k_r. Thus, the experimental k_Q values, referred to as $k_{\text{Q}}^{\text{cal}}$ in the following equations, have to be transformed into $k_{\text{Q}}^{\text{DIN}}$ and $k_{\text{Q}}^{\text{TRS}}$, respectively, in order to be comparable and applicable for reference dosimetry of ion beams according to DIN 6801-1 and TRS-398:

$$\begin{eqnarray}&&k_{\text{Q}}^{\text{DIN}}=\frac{k_{\text{Q}}^{\text{cal}}}{{{k}_{\text{r}}}}\centerdot {{\left(1+0.75\centerdot {{r}_{\text{cyl}}}\centerdot {{\delta}_{12\text{C}}}\right)}^{-1}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&k_{\text{Q}}^{\text{TRS}}=k_{\text{Q}}^{\text{cal}}\centerdot {{\left(1+0.75\centerdot {{r}_{\text{cyl}}}\centerdot {{\delta}_{12\text{C}}}\right)}^{-1}}\end{eqnarray} \tag{ 5 }$$

Here, δ_12C is the relative depth dose gradient at z in the carbon ion field (figure 3) and k_r is given by k_r  =  (1  −  0.003 · r_cyl)⁻¹ and therefore amounts to k_r  =  1.0092 for both thimble ICs used (DIN 2016).

2.4.1. Calorimetric determination of D_w,Q in the carbon ion beam.

The principles of the calorimetric determination of D_w,Q have been described in full detail in e.g. Ross and Klassen (1996), Krauss (2006) and Seuntjens and Duane (2009). Briefly, the measured radiation-induced relative resistance change at the position of the thermistor of the calorimetric detector leads to a corresponding temperature rise ΔT which can be extracted by applying the thermistor's temperature calibration coefficient. D_w,Q at the central axis of the radiation field is then given by the following equation with c_p being the specific heat capacity of water at a temperature of 4 °C and the k's being correction factors for several influence quantities:

$$\begin{eqnarray}&&{{D}_{\text{w},\text{Q}}}=T\cdot \,{{c}_{\text{p}}}\cdot \,{{k}_{\text{h}}}\cdot \,{{k}_{\text{c}}}\,\cdot \,{{k}_{\text{P}}}\,\cdot \,{{k}_{\text{l}}}\,\cdot \,{{k}_{\text{T}}}\,\cdot \,{{k}_{\text{e}}}.\end{eqnarray} \tag{ 6 }$$

As the calorimetric detector comprises two thermistors, D_w,Q has been obtained separately for each, taking into account position depend correction factors for each thermistor. The mean of both D_w,Q values has been taken as the final result of a calorimetric BT.

In the following, detailed information is given on the principal methods for the experimental and/or theoretical determination of the correction factors used in equation (6). Corresponding results for the main correction factors k_l, k_c and k_P are presented in section 3.3.

The factor k_h considers the correction for the so-called heat defect which is a possible deviation between the absorbed radiation energy and the energy which appears as heat. The heat defect is caused by chemical reactions triggered by the radiolysis of water together with potential additives or impurities in the water. For some aqueous systems, the heat defect can be calculated on the basis of a radiolysis model (e.g. Klassen and Ross 2002) allowing to compare the corresponding results on a relative basis with the results from water calorimeter experiments. For hydrogen-saturated water these calculations predict a stationary state for the products of the radiolysis after a small pre-irradiation dose, i.e. the heat defect is zero by definition. It was shown experimentally that this assumption of a zero heat defect is reasonable within a relative standard measurement uncertainty of 0.14% (Krauss 2006). Sassowsky and Pedroni (2005) performed model calculations of the radiolysis of water for radiation with higher linear energy transfer (LET). For the H₂ system, they showed for proton radiation up to an LET of 25 keV µm⁻¹ that the same stationary state with a zero heat defect occurs confirming the results obtained by Palmans et al (1996). As the radiolysis model for the H₂ system predicts this stationary state for all irradiation conditions independent of the LET or type of the radiation, this result can also be taken to be valid for heavier ions within the investigated LET region. The carbon beam at HIT has a maximum LET of about 11.3 keV µm⁻¹ at the calorimetric measurement position (section 3.2) and consequently, for the determination of D_w,Q by means of water calorimetry in the scanned carbon ion beam, the correction k_h is taken to be 1.000 within a relative standard uncertainty of 0.14%. Before and after a detector was used in the carbon ion beam at HIT its response was proven in ⁶⁰Co γ-radiation at PTB to be stable and to coincide with the expected response of a zero heat defect within 0.1%. This possible variation of the detector response has been considered also for the measurements in the carbon beam.

The factor k_c corrects for the influence of heat transport effects on the determination of D_w and is typically investigated by heat transport calculations on the basis of the finite element method. In order to account for the heat conduction effects occurring during and after a calorimetric measurement, the real calorimetric measurement conditions need to be reproduced as precisely as possible within finite element calculations (here: COMSOL Multiphysics version 4.3a). For static irradiation fields this method is well established (e.g. Krauss 2006, Seuntjens and Duane 2009). However, for the raster scan pattern used in this investigation (figure 1) it was found to be not feasible to model the heat conduction effects of the entire irradiation field by computing the time-dependent temperature evolution of each of the 1352 raster spots with 32 ms irradiation time. Therefore, a computational model had to be developed which is based on the assumption that the total time-dependent temperature drift at a given measurement point can be calculated by the undisturbed superposition of the corresponding temperature drifts caused by each raster spot. Thus, the temperature evolution over a total time of 200 s of only a single pencil beam applied for 32 ms to the center of the water calorimeter was calculated within a rotational symmetric 2D model. The corresponding time and space dependent temperatures T(t, r) at the position z corresponding to the water depth of the calorimetric detector were recorded with 1 ms resolution in time and with 0.1 mm resolution in space for r  =  0 to r  =  60 mm. In principle, the temperatures T(t,r) are the same for each spot of the raster scan but a translation in space r and in time t according to the real spatial irradiation pattern and the real time structure used in the calorimetric measurement has to be considered. Then, the total temperature rise with respect to the measurement position of the thermistor probe can be simulated by superimposing the temperature drifts of each spot.

For the finite element calculations of the single pencil beam, the water phantom including the flat glass walls of the detector cylinder was approximated in a rotational symmetric 2D geometry model with the lateral dose distribution of the spot considered by a symmetric 2D Gaussian distribution (figure 2). The dose distribution in z-direction was taken from the measured ddd (figure 3).

Prior to its application, this kind of convolution model was validated in detail for different raster scan patterns comprising only a few pencil beams with different widths. By comparing the corresponding results with the results of full 3D heat conduction calculations, agreement within 0.2% was found. A very similar approach for the calculation of heat conduction effects in scanned ion beams as used here has been independently developed at the Dutch metrology institute VSL (Zavgorodnyaya 2015). It has been shown that this convolution model is suitable for both homogeneous and inhomogeneous irradiation fields realizing a fast and flexible method easily applicable to different scanning patterns.

The factor k_l in equation (6) corrects the non-uniformity of the lateral dose distribution, which causes a difference between the value of D_w,Q measured off-axis with each thermistor of the calorimetric detector and the value of D_w,Q at the central axis of the radiation field. The position-dependent k_l values can be evaluated either from the measured lateral dose profiles by interpolating the corresponding data to the individual thermistor position or from the calculated dose profiles using the modified raster spot positions and the measured pencil beam width (section 3.1).

The perturbation correction k_P accounts for the change of the radiation field due to the presence of the calorimetric detector and has been determined experimentally by using a 'dummy detector' in combination with the thimble IC TM30013 (PTW). By turns, measurements have been performed with the IC placed inside the water phantom of the calorimeter with and without the surrounding 'dummy' glass cylinder. In total, 16 (11) measurements were performed without (with) the 'dummy detector'. Moreover, the radiation field perturbation factor has been verified via a Monte Carlo simulation (section 2.5) by comparing the dose deposition with and without the presence of the glass cylinder at the measurement position of the thermistor probes.

The correction k_T accounts for the effect of the difference in the water temperature between the calorimetric measurements (4 °C) and the IC calibrations (about 18 °C). This correction must be applied to the calorimetrically determined value of D_w,Q in order to obtain its value at 18 °C. Depending on the difference in density between water of 4 °C and 18 °C and considering the very small depth dose gradient of  −0.023% mm⁻¹ at the measurement position of the water calorimeter, it was found that the k_T values were generally very small. A value of k_T  =  0.9990 with a relative standard uncertainty of 0.01% was considered for the calorimetric D_w,Q determination.

A further correction k_e was considered for the change in the thermistor's electrical power during an irradiation, as a change in the thermistor's electrical power also changes the difference between the thermistor temperature and the temperature of the surrounding water. Based on the set-up of the resistance measuring circuit as well as the thermal coupling between thermistor and water, k_e was calculated to 1.0004.

2.4.2. IC measurements.

Within this investigation, N_D,w,Q and thus the k_Q factors for carbon ions have been determined for the two Farmer-type ICs FC65-G by IBA (Schwarzenbruck, Germany), and TM30013 by PTW (Freiburg, Germany) having a sensitive volume of 0.65 cm³ (height: 23 mm, diameter: 6.2 mm) and 0.60 cm³ (height: 23 mm, diameter: 6.1 mm), respectively. Prior to their use at HIT, N_D,w,Co for both chambers has been determined in the ⁶⁰Co reference field at PTB with a relative standard uncertainty of 0.25%.

The measurements with the thimble ICs at HIT were performed directly after the calorimetric measurements. After the water temperature in the phantom of the calorimeter had been increased to about 18 °C, the reference point of the IC was positioned at the same depth of water in the phantom of the calorimeter as the calorimetric detector during the calorimetric measurements. By using an ionometric measuring system developed by PTB, the IC charge, the water temperature inside the calorimeter phantom, and the ambient air pressure were recorded during the irradiation with a sample rate of 1 Hz, allowing continuous correction of the IC reading M_Q for the influence of air temperature and pressure. Analogue to the analysis of the calorimetric measurement data, the integral radiation-induced charge has been determined by extrapolating the linear fits of the pre- and post-irradiation drift curves to the mid-run position. The chambers were operated at voltages of  +300 V (FC65-G) and  +400 V (TM30013), respectively. In total, 100 (90) measurements were performed with the FC65-G (TM30013) chamber over the course of all three BTs. The relative standard deviation of the measurements was found to be 0.3%, which is consistent with the observed long-term reproducibility of the irradiation conditions discussed in section 2.2. The required corrections for the saturation effect and for the polarity effect, k_s and k_p, were determined experimentally following the procedures described in DIN 6801-1.

The exact computation of the volume correction k_v (equation (3)) would require both the knowledge of the dose response function of the IC, which is a measure of the chamber's ability for spatial resolution (Looe et al 2013, 2015), and the dose distribution itself. Extensive work has been recently carried out to measure dose response functions for commonly used ICs with respect to photon beams (e.g. Butler et al 2015, Ketelhut and Kapsch 2015 and references therein). However, this concept has not yet been transferred from the dosimetry of photon beams to the dosimetry of ion beams. On the other hand, it can be expected that in smoothly-varying dose distributions without steep dose gradients the method of simple spatial averaging over the cross-sectional area of the IC already provides a sound approximation for k_v. Therefore, the lateral dose distribution has been numerically integrated over the cross-sectional area of the IC perpendicular to the beam axis without considering their real cylindrical form. The volume correction factor is then given by the ratio of the relative dose value at the position of the reference point of the chamber (located at the central axis) and the result of the integration. Thus, the method is based on the mean relative lateral dose distribution measured with the STARCHECK array with the corresponding vertical and horizontal profiles shown in figure 7. A second approach is based on the calculated 2D dose distribution using the modified raster spot positions derived from the original irradiation records (see section 3.1).

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Monte Carlo (MC) simulations have been performed using the FLUKA code version 2011.2c.0 (Ferrari et al 2005, Böhlen et al 2014) to provide more information about the 'quality' of the radiation by simulating the LET distribution of the particle spectrum at the actual measurement position as well as the contribution of the different particles to the total deposited dose. Furthermore, FLUKA has been used to calculate the perturbation correction k_P of the calorimetric detector. The real measurement condition of the water calorimeter, i.e. all materials including Styrofoam insulation and its inhomogeneous composition, water phantom, and glass cylinder of the calorimetric detector, has been implemented in FLUKA and verified by comparing the simulated ddd with the corresponding measured ddd. The deviation between experimental and simulated data has shown to be in the order of 0.2pp (percentage points) in the entrance channel, maximal 2.0pp in the raising shoulder of the Bragg peak and about 0.4pp in the tail region. Since the calorimetric measurements are performed in the entrance channel of the ion beam with a very small gradient of the ddd, the impact of the differences seen in the raising area of the Bragg peak can be assumed to be minor. The distance between the water calorimeter surface and the synchrotron comprising the vacuum window and the BAMS of the beam nozzle has been taken into account by using the appropriate phase-space file provided by HIT (Tessonnier et al 2016). The recommended default settings for hadron therapy, hadrothe, were used. Moreover, full transport of light and heavy ions was activated, evaporation of heavy fragments considered, δ-ray production by muons and charged hadrons deactivated, charged hadron transport step size decreased to a corresponding 0.5% loss of kinetic energy, and the transport cutoff in terms of kinetic energy reduced to 10 keV for all charged hadrons. The absorbed dose deposited at the measurement position of the thermistor probes was estimated using usrbin in a water region of 1 mm thickness (approximately the sensitive area of the thermistor probes) and a cross-section of 30 cm  ×  30 cm. In addition to the scored dose deposited by all particles, D_all, the dose deposited by particles with atomic number Z  =  1 to Z  =  6 regardless of their mass number M, was determined via FLUKA's auxscore card. The particle spectrum was estimated using FLUKA's usryield detector by scoring the particle yield d²N/(dLET  ×  dE) with respect to LET and energy E in a cross-section of 30 cm  ×  30 cm water at the measuring depth of the thermistor probes, with auxscore filtering the particle yield by atomic number Z.

Figure 7 shows the relative lateral dose profiles as measured by means of the STARCHECK array within the inner 40 mm  ×  40 mm area of the irradiated carbon ion field. Between the different BTs, the reproducibility, calculated as the mean value over the standard deviation for each data set per IC, amounts to 0.28%, demonstrating stable lateral dose distributions. Thus, for the determination of the field-dependent correction factors k_l and k_v, mean relative lateral dose distributions were used.

The data in figure 7 shows pronounced dose inhomogeneities having a maximal difference of about 3% between the central beam and the marginal regions of the radiation field. These large deviations were not expected from the initial irradiation plan. Furthermore, by taking the data of the irradiation records regarding the spatial irradiation pattern, the number of particles delivered to each raster, and the measured width of the ion pencil beam, the 2D dose distribution can be calculated by superimposing the intensity distributions of each spot. As also shown in figure 7, the derived theoretical dose profiles in both horizontal and vertical direction indicate an almost homogenous irradiation field for all three BTs and thus show no agreement with the experimental data. A possible explanation for the disagreement could be deviations of the raster spot positions as regulated by the MWPC located in the beam nozzle. If single wires of the MWPC are not located at their nominal position but within the production tolerance of about  ±0.10 mm, ion beams delivered to this specific position will have a systematic shift due to the misplaced MWPC wires, whereas the irradiation records of the MWPC would record the coordinates of the nominal raster spot position. It could be shown by repetitive simulations that by varying the position of single raster spots as recorded by the MWPC within  ±0.06 mm, assuming a systematic shift of the corresponding MWPC wires, the measured relative lateral dose distributions in both horizontal and vertical direction can be well approximated by the calculated profiles (figure 8). Although this hypothesis could not be verified experimentally yet, these slightly modified raster spot positions have been used for the calculation of the heat conduction correction k_c.

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In order to specify the radiation quality of the carbon ion beam beyond the determination of R_res (section 2.2), figure 9 summarizes the results of the MC simulation regarding the LET distribution of the particle spectrum as well as the contribution of primary particles and fragments to the total deposited dose. Although this additional information is not necessary for the comparison of the experimentally determined k_Q factors with the theoretical k_Q factors stated in current dosimetry protocols, it has been added in foresight as it might be of importance for continuing work. As expected, carbon ions (Z  =  6) show the most narrow peak at the highest median LET of 11.3 keV µm⁻¹ corresponding to a kinetic energy of about 368 MeV/u. This value is in agreement with the expected energy loss of the primary carbon ions (E  =  429 MeV/u) passing the corresponding water-equivalent thickness from the synchrotron to the measurement position of the water calorimeter. The lightest particles with Z  =  1 show the broadest peak at the lowest median LET of 0.4 keV µm⁻¹, while all other particles with Z  =  2–5 are located in between. Protons, deuterons and tritons (Z  =  1) dominate the particle spectrum with a fraction of 48%, whereas carbon ions only contribute with 39% to the total number of particles scored at the measurement position of the water calorimeter. However, due to the difference in LET, the total dose is mainly deposited by carbon ions (85%), while the dose contribution of particles with Z  =  1 is only 8%. Helium ions (Z  =  2) make 10% of the total number of particles, while their contribution to the deposited dose is only 3%. Lithium (Z  =  3), beryllium (Z  =  4), and boron (Z  =  5) are rare in the spectrum (less than 2% each) and deposit about 4% of the total dose all together. Particles with higher Z have not been explicitly considered in the simulation, since the sum over the doses from Z  =  1 to 6 agrees within 99.8% with the total deposited dose scored independent of particle type. Thus, target fragments like oxygen and other heavier fragments only contribute with 0.2% to the total deposited dose and are therefore neglected in the particle spectrum shown.

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The k_l factors are directly given by the reciprocal of the interpolated relative dose at the individual thermistor position, as the dose profiles in figures 7 and 8 have been normalized to the value at the central axis of the irradiation field. Differences well below 0.2% occur if either the mean relative lateral dose distribution measured with the STARCHECK array or the calculated relative dose distribution using the modified raster spot positions of the MWPC is used for the k_l determination. The mean of both approaches was taken as the true thermistor-specific k_l values. Depending on the specific thermistor position, values for k_l lie between 1.0071 and 1.014 with a mean relative uncertainty of 0.36% dominated by the uncertainty of the measured mean lateral dose distribution. In addition, positioning uncertainties of the calorimeter, which are conservatively assumed to  ±1 mm, cause a possible variation of the k_l values within a relative standard measurement uncertainty of 0.14%, which is separately addressed in the overall uncertainty budget for the k_Q determination (table 2).

The perturbation correction k_P was determined as the mean ratio of IC measurements without and with the glass cylinder of the calorimetric detector present to k_P  =  1.0021 with a relative standard error of the mean of only 0.07%. This uncertainty contribution already includes the effects from small positioning variations of the IC during the course of measurements. Nevertheless, a possible systematic difference between the real calorimetric and the 'dummy detector' geometry (e.g. absence of thermistor probes) must be considered and is accounted for by an assumed uncertainty contribution of 0.2%. Thus, k_P was taken to 1.0021 with an overall relative standard uncertainty of 0.21%. As a result from the MC calculations, the value for k_P was found to be 1.0014 and thus confirms the experimental result.

By means of the convolution method (section 2.4.1), the heat conduction correction k_c was determined to 1.0177 within a relative standard uncertainty of 0.50% using the slightly modified raster spot positions according to the knowledge gained from the measured lateral dose distribution (section 3.1), the measured size of the pencil beam (figure 2), and a mean time structure with a total irradiation time of 95 s. The value for k_c is given here as a position-independent heat conduction correction, as the variations of k_c for the different thermistor positions were found to be less than 0.2%, which is therefore incorporated in the given standard uncertainty. Additionally, the uncertainty for k_c comprises the following components: (I) Variations of the time structure occurring from irradiation to irradiation due to different numbers of 'spills' delivered from the synchrotron as well as variations of the lateral dose distribution between different BTs are accounted for by performing calculus of variations using the measured fluctuation range. This component contributes to the total uncertainty with about 0.3%. (II) Uncertainties of the applied convolution model especially with respect to the complex raster pattern and the influence of the detector cylinder wall on raster spots positioned close by are also estimated to be approx. 0.3%. (III) Usually, the exact time evolution of the series of consecutive irradiations is considered in the heat transport calculations (Krauss 2006). This method, however, would lead to almost impractical data handling for the convolution method used here. Therefore, to further validate the convolution method, full heat conduction calculations were performed for a static irradiation field. It was assumed that the lateral dose distribution discussed in section 3.1 was permanently applied during an irradiation time of 95 s. Further, in order to estimate the influence of the consecutive irradiations, the heat conduction calculations have been performed for a series of 10 irradiations interrupted by breaks of 3 min. Maximal differences between the irradiation specific k_c's were found to be 0.2% with a mean k_c value of 1.0118. Thus, this variation is considered as an additional contribution to the overall uncertainty of the irradiation independent heat conduction correction. (IV) The uncertainty of the geometrical water calorimeter model as well as the thermal parameters used within the finite element calculations is assumed to contribute approx. 0.1% to k_c.

According to the procedure described in DIN 6801-1, k_s has been determined to 1.0022 (1.0023) for the FC65-G (TM30013) chamber operated at  +300 V (+400 V). Via calculus of variations, where the number of data points in the Jaffé diagram has been slightly varied and its impact on the resulting k_s value studied, the total measurement uncertainty for k_s was estimated at 0.22% for both chambers including a small uncertainty contribution from the fit parameters used with respect to the analysis of the Jaffé diagram.

The polarity corrections k_p have been determined to 1.0012 (0.9993) for the FC65-G (TM30013) chamber. The relative standard uncertainty was found to be 0.07% in both cases dominated by the standard error of the mean of the repeated measurements.

However, for the determination of k_Q the ratio of k_s as well as k_p between carbon ions and ⁶⁰Co as reference beam quality is required with k_s,Co  =  1.001 (for both chambers) and k_p,Co  =  1.001 (k_p,Co  =  0.999) for the FC65-G (TM30013) chamber as taken from the calibration certificates.

The volume correction factor k_v of the ICs (section 2.4.2) was found to be 1.0129 on average. The relative standard uncertainties are taken to be 0.26% comprising an assumed uncertainty contribution of 0.20% for the simplified method for the determination of k_v itself and 0.17% from the mean lateral dose distribution measured with the STARCHECK array (figure 7). Analogue to the lateral positioning uncertainty of the thermistor probes, an additional uncertainty contribution of 0.10% to k_v results from possible positioning uncertainties (±1 mm) of the IC.

In summary, the main correction factors required for the calorimetric D_w,Q determination and the ionometric measurements are given in table 1.

The combined standard measurement uncertainties of the N_D,w,Q and the k_Q factors were evaluated in accordance with the recommendations of the GUM, Guide to the Expression of Uncertainty in Measurement (JCGM 2008). They are composed of the uncertainty contributions from the calorimetrically determined D_w,Q (as the mean of the absorbed dose values from both thermistors), the ionometric measurement of M_Q, as well as the uncertainty of the calibration factor N_D,w,Co. Because both N_D,w,Q and N_D,w,Co are determined by use of water calorimetry, the uncertainties for the specific heat capacity of water, for the heat defect, and for the uncertainty contribution caused by the Pt-25 standard thermometer used for the calibration of the thermistor probes are common in both cases and thus will be omitted in the calculation of the overall standard measurement uncertainty of k_Q (Krauss and Kapsch 2007a, Krauss and Kapsch 2007b)). Table 2 summarizes the complete uncertainty budget for the experimentally determined N_D,w,Q and for the k_Q factors. As the uncertainties for the ionometric measurements performed with both ICs are very similar, the uncertainty budget is valid for the k_Q factors determined with both ICs. In addition to the uncertainties of the calorimetric and ionometric correction factors discussed in section 3.4 and 3.5, additional contributions need to be considered for the measured relative resistance change ΔR/R (0.15%) with the calorimeter, the charge measurement (0.09%) with the ICs as well as a 0.30% contribution for possible variations in the dose deposition occurring between the calorimetric and ionometric measurements (section 2.2). The first two mentioned contributions comprise statistical measurement uncertainties, uncertainties of the calorimetric and ionometric measurement systems and uncertainties introduced by the data analysis methods.

Thus, an overall standard measurement uncertainty of 0.82% for the k_Q factors determined for each IC per calorimetric/ionometric BT has been achieved. The corresponding k_Q factors are shown in figure 10 agreeing well within the given uncertainties. The final k_Q factor per IC is taken to be the mean value of the three experimentally determined factors. Please note that the mean k_Q values given in the figure need to be transformed into $k_{\text{Q}}^{\text{DIN}}$ (equation (4)) and $k_{\text{Q}}^{\text{TRS}}$ (equation (5)), respectively, in order to be used for the reference dosimetry of ion beams according to DIN 6801-1 (DIN 2016) and TRS-398 (Andreo et al 2006). The corresponding $k_{\text{Q}}^{\text{DIN}}$ and $k_{\text{Q}}^{\text{TRS}}$ values are summarized in table 3 and are compared to the data from literature.

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