# The probe-format graphite calorimeter, Aerrow, for absolute dosimetry in ultrahigh pulse dose rate electron beams

## Abstract

### Purpose

The purpose of this investigation is to evaluate the use of a probe-format graphite calorimeter, Aerrow, as an absolute and relative dosimeter of high-energy pulse dose rate (UHPDR) electron beams for in-water reference and depth–dose-type measurements, respectively.

### Methods

In this paper, the calorimeter system is used to investigate the potential influence of dose per pulses delivered up to 5.6 Gy, the number of pulses delivered per measurement, and its potential for relative measurement (depth–dose curve measurement). The calorimeter system is directly compared against an Advanced Markus ion chamber. The finite element method was used to calculate heat transfer corrections along the percentage depth dose of a 20-MeV electron beam. Monte Carlo–calculated dose conversion factors necessary to calculate absorbed dose-to-water at a point from the measured dose-to-graphite are also presented.

### Results

The comparison of Aerrow against a fully calibrated Advanced Markus chamber, corrected for the saturation effect, has shown consistent results in terms of dose-to-water determination. The measured reference depth is within 0.5 mm from the expected value from Monte Carlo simulation. The relative standard uncertainty estimated for Aerrow was 1.06%, which is larger compared to alanine dosimetry (McEwen et al. https://doi.org/10.1088/0026-1394/52/2/272) but has the advantage of being a real-time detector.

### Conclusion

In this investigation, it was demonstrated that the Aerrow probe–type graphite calorimeter can be used for relative and absolute dosimetries in water in an UHPDR electron beam. To the author's knowledge, this is the first reported use of an absorbed dose calorimeter for an in-water percentage depth–dose curve measurement. The use of the Aerrow in quasi-adiabatic mode has greatly simplified the signal readout, compared to isothermal mode, as the resistance was directly measured with a high-stability digital multimeter.

## 1 INTRODUCTION AND PURPOSE

FLASH radiotherapy (FLASH-RT) is an emerging preclinical treatment paradigm that involves the delivery of therapeutic-level doses with microsecond pulses of ultrahigh dose rate (UHDR) (generally >1 Gy per pulse) radiation, most commonly high-energy electrons.^{1, 2} The typical dose per pulse (DPP) for FLASH-RT is over three orders of magnitude greater than that of conventional radiotherapy, permitting, in principle, the near-instantaneous delivery of a patient's dose prescription. The elevated DPP of FLASH-RT poses a new metrological challenge, as many of the active dosimeters (e.g., air-filled ionization chambers [ICs], solid-state detectors, and scintillators) that are routinely used in conventional therapy beams for clinical reference and relative measurements exhibit less-than-ideal characteristics (e.g., severely reduced ion collection efficiency, quenching, and radiation-induced material degradation) as the DPP is increased. The secondary dose standard of choice due to their precision and stability, reference class ICs suffer from prohibitively large ion recombination effects in the UHDR regime and are, thus, currently unsuitable for accurate UHDR beam dosimetry.^{3-7} At present, no primary absorbed dose standard or validated dosimetry formalism (i.e., codes of practice) for FLASH-RT exists, though these items are amongst the charges of the ongoing project “Metrology for advanced radiotherapy using particle beams with ultra-high pulse dose rates” (“UHDpulse”).^{8, 9}

As a more direct alternative to realize absorbed dose in UHDR beams, new graphite and water calorimeters designed to determine absorbed dose based on first principles have been developed.^{10, 11} Even among primary absorbed dose standards, calorimetry is considered the most direct and absolute method of determining dose-to-water because detector response can be calibrated entirely in terms of quantities that are traceable to temperature and/or electrical standards, independent of ionizing radiation, and is thus considered dose rate independent.^{12, 13} Historically, the principal disadvantages of absorbed dose calorimetry have been a relatively low-dose sensitivity and the need to account for heat transfer in the sensitive volume. The short timescales involved in UHDR deliveries drastically mitigate the latter effect, and arguably, the dose rate magnitudes may mitigate the former depending on the mode of operation. Calorimetry is thus a natural choice to form the basis of a direct absorbed dose calibration of a secondary standard dosimeter, or alternatively, a derivation of a standard dosimeter's correction factors (e.g., ion recombination), in the ultrahigh dose rate field of interest.

Aerrow is a probe-format graphite calorimeter, which has been developed to accurately measure absorbed dose-to-water in the user's beam with a minimum disruption to the existing clinical workflow.^{14} Aerrow's small cylindrical form and intended use in water is more like that of a reference-class IC than most primary standard graphite calorimeters. Notably, it was also the first absorbed dose calorimeter to incorporate an aerogel-based thermal insulation within its nested graphite structure. To date, the use of Aerrow to determine absolute dose in conventional, flattening filter free, small field, and MR-guided MV photon beams has been shown to be comparably accurate to calibrated reference-class ICs.^{15, 16}

The purpose of this study is to evaluate the use of Aerrow as an absolute dosimeter of high-energy UHDR electron beams. In this paper, the calorimeter system is directly compared against an Advanced Markus IC with a dose calibration traceable to Physikalisch-Technische Bundesanstalt (PTB), Germany's national metrology institute, to investigate the potential influence of DPP up to 5.6 Gy. The finite element analysis of heat transfer corrections at various depths along the PDD of a 20-MeV electron beam and the Monte Carlo calculation of dose conversion factors necessary to calculate absorbed dose-to-water at a point from the measured dose-to-graphite are also presented.

## 2 METHODS AND MATERIALS

### 2.1 Calorimetry and reading system

*T*is the radiation-induced temperature rise,

*c*

_{gr}is the specific heat capacity of the graphite-sensitive volume, and $${K}_{{\rm{n}} - {\rm{gr}}}$$ is the impurity correction factor to account for non-graphite materials in the calorimeter core.

*K*

_{sh}is a correction factor to account for the change in the self-heating of the thermistor during the measurement. This correction factor is specific to the thermistor type and the readout system.

^{17}

*K*

_{hc}is the heat loss correction factor due to conductivity, which is obtained by thermal simulation. The ratio in brackets is the absorbed dose ratio of water to graphite calculated by Monte Carlo. The dose ratio includes the correction for beam perturbation due to the heterogeneous composition of the calorimeter, the beam nonuniformity, and the positioning of the detector depth, that is, the effective point of measurement.

The radiation-induced temperature increase, $$\Delta T$$, is measured through the change in resistance of the negative temperature coefficient thermistors (A96N4-GC11KA143L/37C, Amphenol Advanced Sensors), which have a nominal resistance of 22 kΩ at 25°C, embedded in Aerrow's graphite core.^{14} The thermistor resistance was directly measured using a high-stability digital multimeter (DMM), the Agilent 3458A (stability option 002, 8.5 digit). The DMM was operated in the 100-kΩ range, in the two-wire mode, and the measurement frequency was 1 Hz (integration time of 200 ms). The DMM was stored in the control room during measurement for radioprotection purpose. The additional resistance from the cable to connect the Aerrow core thermistor to the DMM was not accounted for as it would have no impact on the measured resistance change. In addition, the resistance of the wire was considered negligible compared to the total resistance of the thermistors at the room temperature, around 25 kΩ.

*R*, the linear extrapolation to the mid-radiation point technique was used.

^{18}Once the change in resistance is obtained from the measured temperature–time trace, the radiation-induced temperature rise, $$\Delta T$$, is obtained by using the sensitivity of the thermistor, which is calculated using the following equation

^{19}:

*R*the total resistance measured,

*T*the temperature of the core, and $${{\Delta}}T$$ is the radiation-induced temperature rise. The resistance–temperature calibration factors, β, of the thermistors were obtained by following the procedure described by Seuntjens et al.

^{19}for a range of temperatures between 15 and 30°C.

The specific heat capacity of graphite, *c*_{gr}, used for this investigation was an average of the values found using the published equation of Williams et al.^{20} and Picard et al.^{21} for the temperature of the core during the measurement, nominally 17.6°C. The relative standard uncertainty of the specific heat capacity was taken to be half of the difference between the two values found, 0.16%. As the specific heat capacity was calculated for each measurement, no additional uncertainty was considered for *c*_{gr}. The average value for the specific heat capacity was calculated to be 695.5(1.1) J (kg K)^{−1}. Throughout the text, the standard uncertainty (*k* = 1) on the last digit will be represented in the parentheses following the value. The standard uncertainties were evaluated following the recommendations of the guide to the expression of uncertainty in measurements (GUM).^{22}

The impurity correction factor, $${K}_{{\rm{n}} - {\rm{gr}}}$$, accounts for the fact that that the specific heat capacity of Aerrow's graphite core is not equal to the specific heat capacity of pure graphite due to the presence of non-graphite material, or impurities, namely, the embedded thermistors. As the exact compositions of the thermistor are not known, the impurity correction factor was taken to be unity with an upper limit on the associated uncertainty of 0.05%.

The thermistor's self-heating is caused by electrical power dissipation that elevates the temperature of the thermistor compared to the surrounding graphite temperature. It depends on the thermistor type, reading system, and their surrounding environment. For this investigation, the self-heating was assumed to be the average of the published values^{17, 23, 24} for comparably sized thermistors, and the uncertainty was considered to be the spread in published values, 1.5(4) mK μW^{−1}, assuming a rectangular probability distribution. The DMM dissipates a DC current of 50 μA in the thermistor, so for a temperature increase of 60 mK during the irradiation period, the radiation-induced thermistor resistance change results in a change in the self-heating of 0.26 mK. The change in thermistor's self-heating represent ∼0.45% of the radiation-induced temperature rise on average. The uncertainty due to the use of published self-heating values is evaluated to be 0.15%.

#### 2.1.1 The Aerrow graphite probe

The calorimeter used in this investigation is the graphite probe-type Aerrow developed at McGill University by Renaud et al.^{14} The graphite components of the calorimeter (Isotropic Grade G458, Tokai Carbon Co., *ρ* = 1.86 g cm^{3}) are arranged in a nested cylindrical geometry. The 6.1-mm diameter, 10.0-mm long graphite core (i.e., the sensitive volume) is separated from a 0.7-mm thick jacket by a 0.7-mm isotropic layer of rigid silica aerogel insulation (Airloy X103H, Aerogel Technologies LLC). Likewise, a 1.0-mm layer of aerogel thermally isolates the jacket from a 1.0-mm thick graphite shield. The mechanical support provided by the solid insulation maintains the constant relative positioning of the graphite components and allows for normal handling by the user.

Aerrow was developed to be used primarily in isothermal mode; however, based on the characteristics of the radiation beam in this investigation, the calorimeter was operated in quasi-adiabatic mode. This decision was made as isothermal was not needed and would have required additional equipment to maintain the temperature of the calorimeter's core stable. Isothermal mode is used to minimize the heat loss during the measurement. As ultrahigh dose rates were used, the radiation exposure time lengths were about 10 times less than conventional dose rates. Therefore, in principle, the heat loss during irradiation could be assumed to be negligible. Furthermore, the active thermal control system used in isothermal has not been tuned for the rapid and drastic changes in temperature from the ultrahigh pulse dose rate (UHPDR).

#### 2.1.2 Thermal simulation

The simulated temperature–time traces, used to obtain *K*_{hc}, were calculated by the finite element method using the software COMSOL Multiphysics v.5.6. (COMSOL AB, Stockholm, Sweden). To this end, heat transfer by conduction was modeled using the differential form of Fourier's law containing a spatially varying heat source defined by the absorbed dose distribution. The graphite calorimeter was modeled in full 3D geometry in a water cube of 30 × 30 × 30 cm^{3} to mimic the measurement. The implemented heat source in the geometry was calculated using the egs_chamber application from EGSnrc,^{25} the details of which are provided in Section 2.4. To obtain the heat loss correction factor, two simulations are compared to each other; one that aims to be as close as possible to the reality, and one in which the heat conductivity around the core is set to zero, leading to no heat loss occurring. From the two simulated temperature–time traces, the temperature rises are extracted using the same linear extrapolation method as for the measurements. The heat loss correction factor is the ratio of the two temperature rises, $$\Delta T$$, extracted from the simulated temperature–time traces without heat loss to the realistic simulation.^{18}

#### 2.1.3 Calorimeter measurements

The calorimetric measurement procedure consisted of a series of five successive radiation exposures with a delay of 90 s between each exposure. The delay between the successive measurements were used as a pre-drift, (30 s), and post-drift, (60 s), for each separated irradiation run. The time delay between irradiation was also useful to avoid a large temperature gradient between the calorimeter core and the surrounding water, which would have resulted in a larger heat loss. The average of the five measurement runs were taken to calculate the total dose deposited, and the standard deviation, σ, was used as the type-A uncertainty of the calorimeter using: $$\sigma /\sqrt {( N )} $$.

Three types of measurements were performed with Aerrow during this investigation: (i) absolute absorbed dose-to-water measurement at the reference depth for a range of DPP between 0.5 and 5.6 Gy (10 pulses delivered), (ii) absolute dose to water measurement at the reference depth for a range of pulses delivered (between 2 and 20 pulses) at a constant pulse rate frequency (delivery time between 0.4 and 4 s) for four fixed DPP settings within the previously mentioned range, and (iii) a depth–dose curve with a constant DPP of 5.6 Gy at the reference depth *z*_{ref} = 0.6*R*_{50} − 0.1 cm (10 pulses delivered).

### 2.2 Radiation and phantom setup

The calorimetric measurements have been performed using an UHPDR 20-MeV electron beams provided by the research linear accelerator (linac) at the Metrological Electron Accelerator Facility (MELAF)^{26} of the German national metrology institute, PTB Braunschweig. The measurements were performed using a pulse repetition of 5 Hz and a pulse length of 2.5 μs. The electron beam energy spectrum was evaluated using a magnetic spectrometer^{27} prior to the experiments. The beam spectrum has a Gaussian shape centered at 20 MeV with a full width at half maximum (FWHM) of 0.05 MeV. To monitor the beam, a nondestructive Integrating Current Transformer (ICT) is integrated in the beamline.^{28} The accelerated electrons exit the beamline through a 0.1-mm thick copper vacuum window. No scattering or flattening filter was used to maximize the DPP.

The measurements were performed in a 30 × 30 × 30 cm^{3} water tank. The water tank wall was made entirely of PMMA, minus the entrance window, which was made of a 0.776-cm thick polycarbonate wall. The entrance window differs from the other wall material because polycarbonate has a greater resistance to radiation than PMMA. A scaling factor of 1.20, equivalent to the density of polycarbonate, was used to convert the thickness of the entrance window to equivalent water depth, as recommended in international dosimetry codes of practice.^{29} Aerrow was positioned using a motorized precision XYZ translation system.

In this investigation, the distance between the water tank and the linac exit window was 70 cm, as measured with a laser range finder (±3.0 mm, Bosch, Gerlingen, Germany). The beam has an approximate Gaussian lateral dose distribution shape with an FWHM of 82(1) mm and a flatness within 1.5 cm of the center of 90.6(4)%. The reference depth in water of the beam is at 46.5 mm.^{30}

### 2.3 Advanced Markus measurement

In the aim to validate the absolute dose-to-water measurements obtained with Aerrow, measurements with a calibrated Advanced Markus were performed in the same radiation setup with the same monitoring system and linac setting. A total of seven dose measurements were performed between 0.95 and 5.3 Gy per pulse. A depth–dose curve was also performed with the ion chamber to compare with the measurement using Aerrow and the Monte Carlo simulation. The absorbed dose-to-water calibration factor,^{31}
$${N}_{{D}_w}$$, was measured in a calibrated ^{60}Co beam traceable to the PTB water calorimeter primary standard.^{24} The beam quality conversion factor,^{31}
$${k}_{{R}_{50}}$$, was calculated using a Monte Carlo simulation of the UHPDR electron beam used in this investigation, which inherently includes the volume averaging correction due to the beam radial nonuniformity. The value of 0.892(6) was obtained, which is comparable with the value obtained from Muir and Rogers^{32} fitting equation for a beam quality specifier (*R*_{50}) of 7.9 cm.

As ion chambers are known to have large recombination effect in UHPDR,^{5} the ion recombination correction factor of this specific ion chamber was obtained, prior to this investigation, by comparing the measured charge per pulse against the estimated dose to water from an alanine calibration of the monitoring system (ICT)^{10} for a range of DPP between 0.1 and 6.3 Gy. To obtain the recombination correction factor, *k*_{ion}, the ion chamber signal measurement, *M*, is corrected as stated in the AAPM protocol (TG-51^{31}), except for the recombination correction factor. The obtained partially corrected measurement, *M*′, is converted to absorbed dose-to-water using the simulated beam quality conversion factor, $${k}_{{R}_{50}}$$, and the measured absorbed dose-to-water calibration factor, $${N}_{{D}_w}$$. The value obtained is the ratio between the absorbed dose of water to the ion recombination factor, $${D}_{{\rm{IC}},{\rm{w}}} \cdot k_{{\rm{ion}}}^{ - 1}$$. As the absorbed dose of water can be estimated from alanine calibration,^{10} it is possible to obtain the ion recombination factor. The relationship between the ion recombination correction factor and the DPP, or charge collected per pulse, can be determined and used in further investigation. The resulting charge collection efficiency dependence as a function of dose rate was consistent with the Petersson et al. model.^{5}

### 2.4 Monte Carlo

Monte Carlo simulations have been performed during this investigation to (i) obtain the dose ratio, as shown in Equation (1), (ii) calculate the dose mapping to be implemented as the heat source in the thermal simulation model, (iii) calculate the depth–dose curve for comparison against measurements, and (iv) calculate the beam quality conversion factor, $${k}_{{R}_{50}}$$, for the Advanced Markus. Monte Carlo simulations have been performed using the egs_chamber application from EGSnrc software^{25} (released version 2020). The validated beam source model was the same as in Bourgouin et al.^{30}.

The detector geometry was modeled following the technical drawing of Renaud et al.^{14} The water tank with the polycarbonate front window was simulated as a simple water cube with the same dimensions as the water tank. The depth–dose curve was simulated in a single voxelized geometry. The electrons and photons were tracked down to a kinetic energy of 5 keV. No variance reduction techniques were used. A custom density correction file was generated for the silica-based aerogel components in the calorimeter assembly with a density of 0.02 g cm^{−3} and an *I*-value of 141 eV, using ESTAR data.^{33} The density correction for the graphite was calculated using the density effect parameter of the crystalline (grain) graphite form (density of 2.265 g cm^{−3}, *I*-value of 81 eV) as recommended by the ICRU report 90.^{34} The density correction file water_icru90, available with the EGSnrc distribution, was used to simulate water material in the geometry.

The dose deposited in the Aerrow core was compared to the dose deposited in a water cylinder of radius 0.25 cm and a thickness of 0.1 cm along the beam axis to calculate the dose ratio. To obtain an uncertainty of 0.05%, typically 10^{9} particles were simulated for the dose to water simulations and 10^{8} particles for the dose to the Aerrow graphite core. The water tank volume was voxelized into 1-cm^{3} cubes in the Monte Carlo simulation used to obtain the dose mapping implemented in COMSOL for the thermal simulation.

## 3 RESULTS AND DISCUSSION

### 3.1 Calorimeter measurements and thermal simulation

#### 3.1.1 Calorimeter measurements

The normalized average temperature–time traces of the five repeated measurement for the different radiation conditions are presented in Figure 1. The curves are normalized, so the extrapolation of the linear fit of the post-drift to the midpoint of the irradiation is 1.00. As shown in Figure 1, the most significant change in the post-drift curvature is for the right panels, that is, the temperature–time traces for the different depths in water. This observation was expected, as the surrounding thermal environment of the detector is changing more drastically as compared to the two other tests, variation of DPP and variation of number of pulses delivered. In general, the heat is expected to travel through the water along the beam axis in the direction of upstream (regions of relatively higher dose) to downstream, where there is almost no energy deposited. Hence, it was expected that similar post-drifts would be observed for all measurements in the plateau region of the depth–dose curve, as well as a decrease in heat loss with depth, which is shown in Figure 1f.

In Figure 1b,d, the heat loss is shown to increase with the DPP and the number of pulses delivered. This observation is expected because as the total dose delivered increases, the thermal gradients between the different materials increase due to the range of specific heat capacities. The noise level appears larger in Figure 1 for the lowest DPP setting; however, this is due to the normalization of the curve rather than a change in the absolute noise level, which is on average 0.102(6) Ω. Another observation that can be made from the lower panels of Figure 1 is the presence of a short-lived transient noise directly proceeding the end of the irradiation. This transient noise was also observed at the start of the irradiation but was not observed when Wheatstone bridges were used for readout (as opposed to directly measuring the thermistor resistance with the DMM). This phenomenon is thought to be a short-lived electrical noise induced by the starting and stopping of the linac. This section of the temperature–time trace was excluded from the fit of the pre- and post-drifts, which are used to calculate the radiation-induced temperature rise.

The radiation-induced temperature rise is measured via the change in resistance of the thermistor. The direct DMM resistance measurement method is not common in absorbed dose calorimetry, because the SNR of the DMM is generally poorer than achievable with Wheatstone bridges^{35} or a DMM used in voltage divider circuit.^{36} De Prez et al.^{17} estimated that the type-A uncertainty from the direct DMM approach would be of the order of 0.20% (*k* = 1) for at least 15 repeated measurements. In this investigation, for five repeated measurements, the average type-A uncertainty from sample standard deviation was found to be 4(1) mGy per pulse, which represents a relative uncertainty smaller than 0.2% for DPP settings greater than 2.3 Gy per pulse.

#### 3.1.2 Heat loss correction factor

The results of the simulated temperature–time traces are shown in Figure 2, which have been normalized in the same way as the measurement data in Figure 1. The five repeated irradiations with 90-s delay between irradiations took about 3 h to simulate on six CPU cores of a desktop computer. As shown in Figure 2a,b, different values for the conductivity of the aerogel in the calorimeter were simulated. The stated value from the vendor^{37} is 43 mW (m K)^{−1}. As observed in another investigation,^{38} the thermal simulations obtained by using the vendor-specified heat conductivity value showed an excess of heat loss from the core when compared to measurements. Therefore, different values of heat conductivity were initially simulated (10, 20, 30, and 43 mW (m K)^{−1}), and based on the level of agreement of these results, a value of 20 mW (m K)^{−1} was selected to be used for the proceeding simulations (Figure 2c–f).

The obtained values for the heat loss correction factors are shown in Table 1. As listed in the table, the heat conductivity selected for the aerogel has a considerable impact on the calculated heat loss correction factor, *K*_{hc}. In Figure 2b, the comparison between the average measured temperature–time trace and the simulation shows that the use of 43 mW (m K)^{−1} would lead to an overestimation of the correction factor. As the spread observed between the curves for the 10 and 30 mW (m K)^{−1} simulations was similar to the spread observed experimentally (see Figure 1b), the difference of the *K*_{hc} values obtained with an aerogel conductivity of 10 and 30 mW (m K)^{−1}, 0.5%, was used in the evaluation of the type-B standard uncertainty of *K*_{hc}. In addition, thermal simulation has been performed for different specific heat capacity values of aerogel within the estimated uncertainty from literature,^{38} 1.05(15) J (g K)^{−1}. It was found that the specific heat capacity used in thermal simulation had a significant impact on the *K*_{hc} evaluated. The difference in *K*_{hc} evaluated for a value of specific heat capacity of 0.90 and 1.05 J (g K)^{−1} was 0.83%. The total relative standard uncertainty of *K*_{hc} was therefore evaluated to be 0.97%.

*K*

_{hc}for the different tests performed with Aerrow in this study

Thermal condition simulated | K_{hc} range |
||
---|---|---|---|

Aerogel heat conductivity: | (10–43) mW (m K)^{−1} |
1.0031 | 0.9757 |

Number of pulses delivered: | (2–20) pulses | 0.9905 | 0.9928 |

Depth in water: | (2.93–8.93) cm | 0.9899 | 0.9919 |

As seen in Figure 2e,f, the heat loss decreases with increasing depth, which is consistent with the results of the measurements. However, the simulated heat loss for the greatest depth in water, 7.93 and 8.93 cm, appears to be overestimated. As these depth-dependent simulations are for relative-type measurements, this overestimation of the heat loss has a nonsignificant impact on the calculated depth–dose curve shape. The simulations involving different numbers of delivered pulses shown in Figure 2c,d have a consistent post-drift, except for the two-pulse delivery, where a smaller heat loss is observed. For validation purposes, the numerical temperature rises simulated without heat conductivity for 2 pulses to 20 pulses delivered were compared. It was found that for the simulation in which 2 pulses were delivered, the calculated temperature rise was 12% greater than expected. This could be due to the smoothing function that is applied to the spatial heat source distribution, the purpose of which is to avoid simulating unrealistic instantaneous heat sources. This 12% difference in the expected temperature rise has, however, a 0.10% impact or less on the calculated *K*_{hc} because it is present in both simulations (i.e., realistic and heat transfer turned off) used to calculate the correction factor and thus largely cancels out.

### 3.2 Dose measurement

#### 3.2.1 Dose per pulse

The Monte Carlo dose ratio from Equation (1) was calculated to be 1.149(4) and was compared to the water-to-graphite stopping power ratio (SPR = 1.137(9)), a difference of 1.2(1)%. The difference in the two values is due to the volume averaging due to the non-homogeneity of the beam and the radiation field perturbation caused by the detector.^{16} The volume averaging can be roughly estimated to be 2.8% for the beam setup, which has a Gaussian shape with an FWHM of about 80 mm. Therefore, it can be estimated that the net perturbation from the presence of the detector would be on the order of 4%. A type-B uncertainty of 0.20%^{39} was assign to the Monte Carlo calculation to account for the uncertainty related to the transport parameters of both graphite and water, and to account for the geometric difference between simulation and measurements.

The absolute dose measurements in the DPP range of 0.6–5.6 Gy is presented in Figure 3. As shown in the figure, the absolute dose measurements of Aerrow are in good agreement with the doses to water measured using the Advanced Markus, considering combine uncertainty. The uncertainty table for the Aerrow absolute dosimetry measurements is presented in Table 2. The heat loss correction factor is by far the greatest contribution to the total uncertainty. This is mainly due to the uncertainty surrounding the value of the thermal conductivity of the aerogel, which significantly impacts the value of the simulated heat loss correction. However, the combined uncertainty on the dose determination remains relatively low compared to the typical level of uncertainty associated with UHPDR electron beam dosimetry. Ion chambers have large uncertainties due to the saturation effect; for instance, in this investigation, the saturation effect was estimated to have a relative standard uncertainty of about 1.3%. The dose determination relative standard uncertainty of alanine is smaller compared to Aerrow, 0.6%^{40}; however, calorimeters have the advantage of being real-time detectors and a primary standard dose method.

*k*= 1) for the determination of dose to water in ultrahigh pulse dose rate (UHPDR) 20-MeV electron beam using the Aerrow graphite calorimeter probe

Component of Equation (1) | Type-A (%) | Type-B (%) | |
---|---|---|---|

Specific heat capacity | c_{gr} |
0.16 | |

Impurity correction factor | $${K}_{{\rm{n}} - {\rm{gr}}}$$ | 0.05 | |

Statistical standard deviation | $$\Delta T$$ | 0.10 | |

Extrapolation method | 0.25^{a} |
||

Thermistor's self-heating | K_{sh} |
0.15 | |

Heat loss correction factor | K_{ht} |
0.97 | |

Monte Carlo dose ratio | $${( {{D}_{\rm{w}}/{D}_{{\rm{gr}}}} )}_{{\rm{MC}}}$$ | 0.10 | 0.20^{b} |

Combined standard uncertainty (k = 1) |
1.06 |

^{a}From Bourgouin et al. estimation.^{18}^{b}From Sander et al.^{39}

The relative standard uncertainty of the absolute dose to water measurements could be improved. As the heat loss correction factor is the greatest contribution, further investigation would be required, including the simulation of convection in water. Moreover, the heat conductivity of the aerogel should be measured experimentally to validate the value used in this investigation. Another approach would be to reduce the heat loss occurring during measurement. The largest factor in the heat loss is the difference in temperature between graphite and water, from the difference in the specific heat capacity. Therefore, a solid phantom with a specific heat capacity nearer to graphite could further reduce heat loss from the detector and improve positioning-related uncertainties; however, this comes with a constraint in the depth(s) of measurement. Alternatively, as the core is the presence of a large water volume that receives relatively little of the deposited energy, reducing the size of the water tank would reduce its influence. In terms of radiation transport, the water volume of 30 × 30 × 30 cm^{3} could be reduced to a 10-cm side length without affecting the fluence spectrum at the reference point.

#### 3.2.2 Independence on the number of pulses delivere

The relative DPP measurements as a function of irradiation time, equivalent to the total number of pulses delivered, is presented in Figure 4. As shown in the figure, the DPP measured is independent of the number of pulses delivered within 0.6%. No trend was observed, indicating that Aerrow has no dependency on the number of pulses delivered. The greatest type-A statistical uncertainty is observed for the lowest DPP (0.6 Gy) with the fewest number of pulses delivered (two). The type-A uncertainty is less than 0.4% for a total dose delivered greater than 10 Gy and converges to a value of about 0.1%.

#### 3.2.3 Depth–dose curve

The depth–dose curve obtained from the Monte Carlo simulation and the measurements using Aerrow and an Advanced Markus are presented in Figure 5. The Aerrow data has been corrected for the variation in the heat loss correction factor, 0.7(1)% and the variation in the water to graphite SPR, 0.7%. The depth–dose curve measured using the Advanced Markus was corrected for the variation in the water-to-air SPR, about 10%, and the saturation effect, which varies between 1.00 and 1.898(13). In the aim of validating both the thermal simulation and the dose mapping calculated with Monte Carlo, a depth–dose curve labeled COMSOL is also presented in Figure 5. This curve was obtained from the simulated temperature increase in the core when no heat loss occurs. The value obtained is therefore proportional to the absorbed dose.

As shown in Figure 5, the obtained depth–dose curves are in good agreement with each other. The calculated reference depth, *z*_{ref}, obtained for the simulations and measurements are all within 0.5 mm. A depth–dose curve has also been calculated for Aerrow without applying the corrections for SPR or heat transfer. The difference in the calculated *z*_{ref} was 0.01 mm, well within the uncertainty. Therefore, the relative depth measurements can be directly used to determine the percentage depth–dose curve without further correction. In the case of the Advanced Markus, if only the variation in SPR is considered and the saturation effect correction is not applied, the resulting *z*_{ref} is 3 mm too large.

## 4 CONCLUSIONS

In this investigation, it was demonstrated that the Aerrow probe-type graphite calorimeter can be used for relative and absolute dosimetries in water in an UHPDR electron beam. The comparison of Aerrow against a fully calibrated Advanced Markus chamber, corrected for the saturation effect, has shown consistent results in terms of dose-to-water determination. The relative standard uncertainty of absorbed dose-to-water estimated for Aerrow was 1.06%, which is larger compared to alanine dosimetry, 0.6%,^{40} but has the advantage of being a real-time detector. The largest component of the measurement uncertainty is the heat loss correction factor due to the uncertainty of the aerogel thermal properties, the specific heat and heat conductivity. Therefore, future work will include jacket measurements to accordingly optimize heat flow control and corrections.

The use of the Aerrow in quasi-adiabatic mode has greatly simplified the signal readout, as the resistance was directly measured with a high-stability DMM. A given radiation-induced signal can be delivered ∼1000 times faster by UHPDR as compared to conventional RT, and hence a higher SNR is achievable within the time scale of the heat transfer processes. This is why the type-A statistical standard uncertainty in this work was estimated to be 0.10(3)% and the SNR was comparable to measurements of total doses delivered greater than 10 Gy with a conventional Wheatstone bridge readout.

## CONFLICT OF INTEREST

Andreas A. Schönfeld, Jakub Kozelka, Jeff Hildreth, and William Simon are employees of Sun Nuclear Corp., Melbourne, FL. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## ACKNOWLEDGMENTS

We thank Christoph Makowski for the operation of the electron accelerator. The assistance of Andreas Schlesner in the preparation of the experiments is gratefully acknowledged. The author would like to thank Leon De Prez for the help with the determination of the self-heating.

## FUNDING INFORMATION

This project 18HLT04 UHDpulse has received funding from the EMPIR program co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation program.

## Open Research

# DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author, Dr. A. Bourgouin, upon request.