Development and evaluation of a standard for absorbed dose to water from alpha-particle-emitting radionuclides

Targeted radionuclide therapy and brachytherapy with alpha particles has gained significant clinical relevance recently. Absorbed dose traceability to a standard is currently lacking in the dosimetry chain. The short range of alpha particles in water of <100 μm complicates the absorbed dose measurements in the form of significant attenuation and perturbation effects. The aim of this work was to develop and validate a standard for absorbed dose to water from alpha-particle-emitting radionuclides. A dosimetric formalism to realize surface absorbed dose to water per unit activity using a windowless cylindrical parallel-plate ion chamber (IC) was introduced. IC-based and Monte Carlo (MC)-based correction factors were calculated for a planar circular 210Po alpha-particle emitter. The measured absorbed dose to air was compared to the MC-calculated absorbed dose. A parallel-plate IC with a nominal 4 mm collector diameter composed of polystyrene-equivalent material was utilized as a standard. MC simulations were performed using the TOPAS MC code and finite source size, backscatter, and emission angle divergence correction factors were calculated by modeling the IC and the source. Multiple measurement trials were performed to measure ionization current at air gaps in the 0.3 mm to 0.525 mm range. The proposed dosimetric formalism was employed to calculate the surface absorbed dose to water from a point-like 210Po source. The recombination and polarity correction factors were measured to be <0.50% when a 150 V mm−1 electric field strength was applied. The MC-calculated and measured absorbed dose to air agreed within 2.05%. The finite source size and backscatter corrections were <10% with the emission angle divergence correction being in the 93%–239% range. The surface absorbed dose to water was measured to be 2.8913 × 10−6 Gy s−1 Bq−1 with a combined uncertainty of 4.23% at k = 1. This work demonstrated the ability of a windowless parallel-plate IC as a standard for absorbed dose from alpha-particle-emitting radionuclides.


Introduction
Treatment of cancers with alpha particles has seen significant clinical growth in the past few decades [1][2][3].Alpha particles have been employed in both targeted radionuclide therapy (TRT) and brachytherapy for such treatments [4][5][6][7].Similar to external beam radiotherapy, it is pertinent that accurate and traceable dosimetry methods are used to standardize the unit of absorbed dose and to correlate the treatment outcomes with absorbed dose.Further, the determination of absorbed dose to the organs at risk allows estimation of possible complications from the treatment and risk of secondary cancers.Currently, radioactivity-based traceability is maintained and computational methods, such as Monte Carlo (MC) simulations, are utilized to either calculate the absorbed dose using the measured activity or to calculate dose point kernels (DPKs) used for convolution-based algorithms [8][9][10][11][12].Microdosimetry is often used to assess cell survival after exposure to alpha particles due to the wide distribution of energies deposited inside the nucleus [13][14][15][16][17]. Calculations of DPKs and microdosimetry can be performed using MC methods [10,11,18].
The conversion of radioactivity to absorbed dose is a highly non-standard process since multiple algorithms and software exist to perform such a conversion.These software are dependent on the relevant nuclear decay data and specific transport parameters, and most have not been experimentally validated.To establish absorbed dose-based traceability, a reference standard alpha-emitting radionuclide and an instrument could be used.Measured absorbed dose data acquired with the standard could then be utilized to validate the computational methods currently being employed for dosimetry.Billas et al previously proposed a standard for absorbed dose from betaemitting radionuclides [19].To the authors' knowledge, this is the first study that aims to directly realize absorbed dose from alpha-particle-emitting radionuclides.It is of note that the proposed measurements do not intend to replace absolute activity measurements and will potentially function as a component of a traceability chain that includes both activity and absorbed dose measurements.
Alpha particles used in TRT and brachytherapy have an energy range of 5 MeV to 9 MeV and traverse only <100 µm distance in water.Therefore, for unsealed radionuclides used in TRT, a solution of alpha-particle-emitting radionuclides will self-absorb the majority of the emitted particles before they reach the surface of the volume.Furthermore, the presence of an entrance window on the dosimeter can significantly attenuate the incoming alpha particles.Thus, a windowless cylindrical parallel-plate ionization chamber (IC) was deemed suitable as a standard for absorbed dose from alpha-particleemitting radionuclides.Vented air-filled ICs are ideal dosimeters for alpha particles due to the low density of air that leads to minimal charged particle fluence perturbation and attenuation.
For unsealed TRT alpha sources, self-absorption in volumetric source can be minimized by depositing a thin film of alphaparticle-emitting radionuclide on an electrode and measuring absorbed dose directly from the alpha particles emitted by the thin film source.
This work aimed to construct and evaluate a standard for absorbed dose from alpha-particle-emitting radionuclides.This work proposes a dosimetric formalism for alpha-particleemitting radionuclides used in TRT based on thin film circular planar sources.The constructed standard was evaluated using a thin film 210 Po source and the surface absorbed dose to water was measured using the extrapolation method.The measured absorbed dose data were also compared to the MC-calculated dose to evaluate the accuracy of MC methods.

Dosimetric formalism
Figure 1 illustrates the cylindrical geometrical setup chosen in this work.A cross-section of the experimental geometry is shown.By employing a thin film circular alpha source on a substrate mounted on a source holder, a cylindrical sensitive volume can be created by applying an electric field across the surfaces of the substrate and the detector.The electric field lines are shaped by the collector, collector-guard insulator, and guard.By varying the air gap between the two electrodes, absorbed dose to air can be measured at distinct air gaps.The quantity of interest in this work was surface absorbed dose to water, which can be measured by operating the IC as an extrapolation chamber.Using this method, the ionization current versus air gap curve can be extrapolated to zero air gap and the surface absorbed dose to water can be determined.Since alpha-particle-emitting radionuclides are often approximated as point sources distributed over the tumor volume, a correction factor to account for the finite radius of the thin film source was proposed in order to measure surface absorbed dose to water from point-like alpha-particle-emitting radionuclides.
The surface absorbed dose to water, normalized by the radioactivity, can be given by where A o is the activity of the radionuclide of interest, W e air is the mean energy required to liberate an ion pair in dry air, ρ o is the physical density of air at standard temperature and pressure, r is the radius of the cylindrical air cavity, I is the ionization current, and ℓ is the air gap between the collector and source surfaces.Jesse and Bay et al previously experimentally determined the magnitude of W e air to be 34.96J/C with a total uncertainty of 0.2% for alpha particles [20,21].The water-to-air stopping power ratios, averaged over the charged particle spectra, were calculated using the data tabulated in the International Commission on Radiation Units and Measurements report 90 [22].
The first series of correction factors in equation ( 1) are related to ICs in general.k pol is the polarity correction that corrects for the differences in the signal due to the polarity of the applied voltage: where M + is the positive charge collected by the detector and M − is the negative charge.k ion is the recombination correction that accounts for the signal loss due to the recombination of the liberated electrons with ions in the cavity.The total recombination correction accounts for both initial and general recombination effects.It is worth noting that the initial and general recombination collection efficiencies are proportional to V and V 2 , respectively, where V is the applied voltage.For charged particles with low linear energy transfer (LET), the recombination correction is usually determined by plotting 1/V 2 against 1/Q, where Q is the collected charge.The saturation charge, Q sat , is calculated as the inverse of the y-intercept.The total recombination correction at a given electric field strength is determined by Qsat Q .However, initial recombination must be considered for heavy charged particles due to their dense ionization tracks that lead to significant initial recombination [23].Therefore, the recombination correction can be modeled by: where α, β, and γ are fitting parameters.The exponential term dictates the collected charge in the multiplication region.k TP in equation ( 1) is the temperature and pressure correction factor that accounts for the difference in the mass of the cavity due to non-standard temperature and pressure of the air: where T is the temperature in degrees Celsius and P is the air pressure in kPa.k elec is the electrometer correction factor that converts the displayed current value to a traceable current value.
The second set of correction factors in equation ( 1) is specifically-related to the dosimetry of unsealed radionuclides and can be calculated using MC methods.k point corrects for the finite radius of the thin film source to allow measurement from a point-like source: where D cav,point is the absorbed dose to air cavity calculated using a point source and D cav,circ is the absorbed dose to air cavity calculated using the measured radius of the circular source.k backscatter accounts for the difference in backscatter between the medium surrounding the air-cavity and water and can be calculated by: where D cav,point,water is the absorbed dose to the air cavity using a point source with the IC, substrate, and source holder materials set to water and D cav,point,water is the absorbed dose to air cavity using a point source with the nominal materials of the IC, substrate, and source holder.As the air gap between the source and the detector increases, the side-loss of alpha particle fluence also increases leading to a sharp decrease in absorbed dose as a function of air gap.Therefore, the ionization current at each air gap must be corrected.To determine the surface absorbed dose, the side-loss of fluence can be accounted for using a MC-calculated divergence correction factor given by: where (D cav,point,water ) ℓ→0 is the absorbed dose to the cavity extrapolated to a zero air gap from a point source with the substrate, detector, and source holder composed of water.
(D cav,point,water ) ℓ→0 can be determined by fitting the absorbed dose versus air gap curve and calculating the intercept.In this work, a third-order polynomial fit was utilized to determine the extrapolated surface absorbed dose.All MC-based correction factors are dependent on the air gap.It is of note that the absorbed dose ratios in equations ( 5)-( 7) are for the same activity.

210 Po source
The constructed standard of absorbed dose was evaluated using a 210 Po pure alpha-particle emitter.This radionuclide decays into 206 Pb and emits a 5.305 MeV alpha with an intensity of ∼100% for each nuclear disintegration.A custom 210 Po source was requested from Eckert & Ziegler (E&Z) with a NIST-traceable activity of 46.36 kBq measured to within 1% uncertainty at k = 1.The source was chemically plated on a silver-coated steel substrate of 11.1 mm diameter and 1 mm thickness.In order to measure the surface absorbed dose, the alpha particles emitted from the thin film 210 Po source must have minimal self-absorption before they escape the source.Sources with large thicknesses will have greater self-absorption.Since the thickness of the deposited layer of 210 Po is challenging to measure directly, the full width at half maximum (FWHM) of the alpha peak was used as an indicator of self-absorption.Our preliminary work using MC simulations showed that sources with FWHM <40 keV have a negligible impact on absorbed dose.Therefore, sources with <40 keV can be utilized without requiring a correction factor for self-absorption.
The FWHM of the E&Z 210 Po source was measured to be 25.58 keV using alpha spectroscopy (ORTEC Alpha Aria).The radius of the source was measured using unlaminated radiochromic films (EBT3 Ashland, NJ) and was determined to be 1.60 ± 0.30 mm [24].Following the alpha spectroscopy and unlaminated EBT3 measurements, the source substrate was mounted on the source subassembly for ionization current measurements.

Apparatus assembly
A SolidWorks ® schematic of the constructed apparatus is shown in figure 2. The detector and the source are supported by two breadboards placed orthogonal to each other using 90 • aluminum brackets.The source subassembly was mounted on a miniature hexapod stage (Physik Instrumente H-811.I2) that is capable of both translation and rotational motion in all three axes.The translational motion allows variation of the air gap and the rotational motion allows the establishment of parallelism between the detector and source planes.The hexapod also allows rotation about an arbitrary pivot point and can manipulate the coordinate system based on rotational or translational shifts.Since the travel range along the z-axis of the hexapod is only 6.5 mm, a linear stage (Physik Instrumente L-509) was mounted on a breadboard that is capable of 52 mm travel.A rotational stage (Thor labs KS2RS) with a 2 ′′ diameter was mounted on top of the linear stage to allow rotational freedom about the z-axis.With the use of the linear and rotational stages, the detector subassembly can be translated or rotated about the z-axis.The source subassembly was placed on top of the hexapod stage with an aluminum shield placed around the source to reduce electronic noise.The thin film source was placed on a source holder constructed of aluminum.The electrical bias to the source substrate was provided using a DC voltage generator (Fluke 343A).The applied bias was measured and confirmed using a multimeter (HP 34401A).The D400 cylindrical planar IC was employed in this work for absorbed dose measurements.The collector and the guard of this IC were made of D400 material, which is a conducting plastic equivalent to polystyrene in atomic composition and has a physical density of 1.16 g cm −3 .The nominal collector diameter and guard outer diameter were 4.00 mm and 10.16 mm, respectively.The surface flatness of the D400 IC and the source substrate was evaluated using coherence scanning interferometry (Zygo NewView 9000) to ensure a flatness within 20 µm.The charge generated inside the air cavity was collected by connecting the IC to a MAX4000 electrometer (Standard Imaging, Middleton WI).
With the detector and the source planes facing each other, a parallel alignment between them is needed to ensure an accurate assessment of the cavity volume.Any contact between the source and the detector can contaminate the detector, therefore, a partial contactless method was devised to align the two plates parallel to each other.By rotating the source plate about its center in both positive and negative directions and measuring the angles at which the edges of the source plate make contact with the guard electrode, any tilts can be detected and corrected.This method operates under the assumption that the two plates are perfectly flat.While rotating the source substrate, the edge of the substrate makes contact with the guard ring instead of the collector due to the dimensions of the two planes.This contact prevents contamination and allows measurement of the tilt angles based on the electric conductivity between the two plates.
Once the detector and the source planes were aligned in a parallel configuration, the absolute air gap between the two plates must be known.Capacitance measurements were acquired to determine the absolute air gap as well as the effective diameter of the air cavity.Assuming the detector and the source plates act as a parallel-plate capacitor, any change in the applied potential, ∆V, leads to a flow of electric charge, Q = C o ∆V, across the air gap depending on the capacitance of the air, C o , which is given by where ϵ r is the dielectric constant of air, ϵ o is the permittivity of vacuum, r is the radius of the cavity, and ℓ o is the air gap between the two plates.With the presence of a radionuclide, the measured electric charge is a combination of ionization current and current induced due to the capacitance of the air cavity.In order to deconvolve these two effects, a voltage increase method was utilized as previously proposed by Selbach et al [25].Multiple 30 s charge readings were initially collected at a voltage V 1 and defined to be Q o .The Q ∆ was measured by initializing a 30 s charge reading and increasing the voltage by ∆V.The amount of time between the charge initialization and the voltage increase was defined to be ∆t.After measuring Q ∆ , Q f was measured as a 30 s charge reading at a bias of V 1 + ∆V.When ∆t is kept close to zero, the charge induced by the capacitance of the air cavity can be given by where Q dis is the charge induced by the capacitance of the air cavity and t ref was chosen to be 30 s. Q dis can then be used to calculate C o = Q dis /∆V.Since the initial air gap is unknown, the offset between the actual air gap, ℓ o , and the assumed air gap, ℓ assumed , can be calculated by Equations ( 8) and ( 10) can be combined to be By plotting 1/C o against ℓ assumed , the ℓ offset can be determined by the y-intercept and the slope can be used to calculate the effective radius of the cavity.

Measurement procedure
Multiple trials were run to determine the D400 ionization current from the 210 Po source.Independent measurements were acquired for each trial and the entire measurement process was repeated from the beginning.With the 210 Po source mounted on the source subassembly, the D400 IC was lowered using the linear motion stage to an air gap <1 mm.The source was then biased with an electric potential of −40 V and lateral scans were acquired using the hexapod motion stage with 0.1 mm steps.At each dwell position, a single 45 s charge reading was taken.The position with the maximum signal was selected for measurements.Following the lateral alignment, the rotational alignment is performed for the D400 IC.
Once the lateral and rotational alignment has been performed, the initial air gap between the source and the D400 IC is determined using the capacitance method described in section 2.3.A voltage change from −10 V to −100 V is applied and a 30 s charge reading is collected.Following this measurement, a 30 s charge reading is collected at the −100 V bias to distinguish the charge due to the capacitance and the charge due to ionization.These measurements are acquired at least three times at four separate air gaps with increments of 50 µm.The recombination correction was measured by acquiring three 60 s charge readings at voltages ranging from 1-200 V with the air gap set to 300 µm.At least 10 min of time was provided after switching the voltage to allow an equilibrium to be reached.The recombination correction was calculated using the fitting function reported in equation (3) for the 150 V mm −1 electric field strength.A least square optimization function was used in MATLAB v2020 (Natick, MA) to fit the measured data to equation (3).Both positive and negative bias was applied and three 60 s charge readings were collected to calculate the polarity correction factor.The temperature and pressure were noted to apply an air density correction to the ionization current readings.Three 60 s charge readings were collected at each air gap ranging from 300 µm to 525 µm in 25 µm increments.The k TP , k pol , k ion , and k elec corrections were applied to the charge readings, which were then normalized by the decay-corrected radioactivity of 210 Po.

MC simulations
The TOol for PArticle Simulation (TOPAS) MC code was utilized for correction factor calculations.The TOPAS MC code is a GEANT4 wrapper developed for heavy-charged particle simulations [26].In this work, a modular physics list was used consisting of G4RadioactiveDecay, G4Decay, G4HadronElasticPhysicsHP, G4HadronPhysicsQGSP_BIC_ HP, G4IonElasticPhysics, G4IonQMDPhysics, and G4StoppingPhysics.A modified G4EMStandardOpt4 physics list was used for electromagnetic physics.The electromagnetic parameters for electrons were unchanged since the G4EMStandardOpt4 physics list is considered to be highly accurate for light-charged particle transport [27].For alpha particles, the optimal electromagnetic physics parameters determined in the work of Khan and DeWerd were utilized [28].The internal GEANT4 nuclear decay data were used to simulate the 210 Po source.
The D400 IC was simulated according to the drawings provided by the manufacturer.The capacitance measurements, described in section 2.2, were used to simulate the radius of the air cavity.The substrate was simulated as a steel disk with a silver layer on the surface.The atomic composition of the D400 material is equivalent to polystyrene.The physical density and mean excitation energy of the D400 material were set to 1.16 g cm −3 and 68.7 eV, respectively.The GEANT4 internal materials G4_AIR and G4_WATER were selected to represent air and water.The mean excitation energies of air and water were 85.7 eV and 78 eV, respectively.The atomic deexcitation was turned on and the production thresholds were set to 1 µm for these simulations.The MC-calculated absorbed dose to air cavity was compared with the measured absorbed dose, which was calculated by

Results
The D400 IC was aligned with the source disk so that the long axis of the two cylinders are coincident.However, throughout the measurement trials, a lateral shift of 0.2 mm was applied in both X and Y directions since the center of the deposited thin film source had a positional offset from the center of the source substrate.Rotational offsets of up to 1.2 deg were noted about the Y axis. Figure 3 shows the capacitance results for one of the trials.The initial air gap was determined to be −0.312mm with an effective diameter of 4.00 mm using a linear fit.These measurements were repeated eight times and the average effective diameter was measured to be 4.01 mm with a percent standard deviation of 1.00%.Figure 4 shows an example of the measured and fitted data where 1/Q is plotted as a function of both 1/V and 1/V 2 .The recombination correction is also plotted for the applied electric field strengths.For this example, the fitting parameters Qsat, α, β, and γ were calculated to be 529.94pC, 2.06 × 10 −4 V/pC, 5.68 × 10 −5 V 2 /pC, and 1.59 × 10 −4 V −1 , respectively.Therefore, the recombination correction was determined to be <0.5% for an electric field strength of 150 V mm −1 .As hypothesized, the recombination correction was found to decrease as the strength of the electric field was increased.However, beyond 150 V mm −1 , the multiplication region was encountered requiring a correction of less than unity.
Figure 5 displays the current readings for the three trials conducted in this study.The signal-to-noise ratio (SNR) was ∼1000 for all of the current readings.The ionization current was normalized by the decay-corrected activity of 210 Po, which varied <2% between the trials.The IC correction factors, such as k TP , k ion , k pol , and k elec , were applied to the current readings for each of the trial.The k ion , k pol , and k TP were measured to be 0.999 ± 0.007, 1.004 ± 0.008, and 1.027 ± 0.003, respectively.The k elec correction factor was 0.999 for the current range used on the MAX4000 electrometer.Therefore, the variation in the IC-specific corrections was found to be relatively small.The Type A uncertainty in the charge readings was measured to be <0.3%.The standard deviation of the mean ionization current was observed to decrease with the increasing air gap.A percent standard deviation of the mean in the range of 2.19%-3.30%was found in the ionization current readings when results from all trials were considered.Therefore, the dominant cause of variations between the different trials was the variation in the cavity volume due to positioning errors.

Comparison between MC and experimental data
The accuracy of the MC-predicted absorbed dose to cavity, normalized by radioactivity, was assessed by comparing it with the measured absorbed dose.Figure 6 compares the measured and MC-simulated absorbed dose to air cavity subtended by the D400 IC.The absorbed dose decreases as a function of air gap due to the divergence of the source and the loss of fluence from the sides of the cavity.Good agreement was found between the two curves with average and maximum deviations of 1.43% and 2.05%, respectively.These results demonstrate the ability of MC methods to accurately calculate absorbed dose for alpha-particle-emitting radionuclides.
Table 1 shows the combined uncertainty in the measured absorbed dose to air cavity for the 0.30 mm air gaps.The random fluctuations in the signal led to a 0.2% Type A uncertainty in the signal for a given trial.The current repeatability was calculated by computing the percent standard deviation of the mean current readings considering all measurement trials.The current repeatability uncertainty component was the largest contributor to the total uncertainty and is attributed to the positional uncertainty in the initial air gap calculated using the capacitance method and uncertainty in the rotational alignment of the detector with respect to the source.The ionization signal was observed to be highly sensitive to the positional offsets in the air gap.Based on the MC simulated absorbed dose, deviations of ∼20 µm between the actual and measured air gap can lead to deviations >3% in the absorbed dose.Any tilts in the geometry further add to the uncertainty.Additionally, the uncertainty in the current readings was noted to decrease with increasing air gap since the slope of the dose versus air gap curve decreases with the increasing air gap.Therefore, current readings acquired at smaller air gaps were found to be more sensitive to positional offsets.The combined uncertainty at k = 1 ranged between 3.31%-4.12%for the 0.525 mm to 0.3 mm air gaps.

MC correction factors
The k point correction factor, calculated using equation ( 5), as a function of air gap is shown in figure 7(a).The correction was calculated to be in the 7.0%-9.3%range and observed to be increasing with the air gap.The mean track length of the alpha particles in the air cavity increases with decreasing source diameter leading to an increase in energy deposited in the cavity.This leads to a k point correction greater than unity.Additionally, the k point correction was found to be dependent on the air gap. Figure 7(b) shows the k backscatter correction factor calculated using equation (6).A mean correction of 5.35% was found with a maximum correction of 5.96%.The backscatter correction was observed to have a dependence on the air gap with the magnitude of the correction decreasing with increasing air gaps.The backscatter correction is expected to be largely dependent on the angular distribution of the alpha particles striking a given material rather than the material composition itself.Since the alpha particle source was simulated as an isotropically-emitting point source on the surface of the source substrate, the trajectory angle of the alpha particle inside the substrate has a wide distribution.For alpha particles traveling orthogonally to the substrate's surface, a large number of multiple Coulomb scattering events are required to deflect the trajectory of the particle back toward    the air cavity.However, the particles with oblique trajectories closer to the air cavity can be easily deflected by a small number of scattering events.Thus, the backscatter correction is dominated by alpha particles traversing close to the surface of the substrate.The MC-calculated absorbed dose versus air gap curve with the fitted polynomial function is shown in figure 8.The divergence correction is also plotted as a function of air gap.The divergence correction was found to be the largest correction factor for the extrapolation method.The associated uncertainty in the polynomial fit was calculated to be 1.70% with an R 2 of 0.997.Since the absorbed dose has a non-linear relationship with the air gap, the extrapolated absorbed dose was found to be dependent on the fitting parameters leading to relatively higher uncertainty.A linear relationship, with R 2 of 0.999, was observed between the k div and air gap.The magnitude of the correction ranged from 93% to 239% within the simulated range of air gaps.

Surface absorbed dose to water
The extrapolation method measures the surface absorbed dose to water by extrapolating the ionization curve to a zero air gap.This dosimetric formalism was previously described in section 2.1 using equation (1).The mean ionization current at each air gap corrected by the IC-specific correction factors and MC-calculated correction factors is shown in figure 9.The current data were found to be increasing with the air gap.The mean slope of the curve, ∆I ∆l ℓ→0 , was calculated to be 1.11 × 10 −18 A Bq −1 µm −1 .Based on this calculation, the mean surface absorbed dose to water was measured to be 2.8913 × 10 −6 Gy s −1 Bq −1 .
Table 2 shows the combined uncertainty in the surface absorbed dose to water using the extrapolation dosimetric formalism.The largest contributors to the total uncertainty were the stopping power ratio and the dosimetric uncertainty in the k point correction factor.Using MC simulations, it was concluded that a 0.3 mm change in source diameter leads to a maximum change of 2% in the absorbed dose to air.A large contributor to the total uncertainty was the current slope repeatability, calculated by considering the standard deviation of the mean slope of the corrected ionization current versus air gap curve.This uncertainty was observed to be relatively smaller than the repeatability uncertainty reported in table 1.This can be partially explained by the usage of a highly precise hexapod motion stage capable of relatively moving with high repeatability.Therefore, despite having possible rotational and positional errors, the slope of the ionization versus air gap curve was found to be repeatable over multiple trials.The uncertainty in the divergence correction was the second largest contributor.This uncertainty was calculated by considering both the statistical uncertainty in the MC simulations and the polynomial fitting error when calculating the dose extrapolated to zero air gap.The surface absorbed dose to water uncertainty using the extrapolation method was found to be 4.23% at k = 1.

Discussion
This work reports the construction and evaluation of the first absorbed dose standard for alpha-particle-emitting radionuclides.A planar circular thin film source was utilized along with a planar cylindrical windowless parallel-plate IC to measure surface absorbed dose to water from point-like alpha sources.The overall combined uncertainty in the surface absorbed dose to water measurements was found to be 4.23% at k = 1.
Billas et al previously proposed an absorbed dose primary standard for unsealed beta-emitting radionuclides [19].The quantity of interest in their work was absorbed dose to water per nuclear decay at the center of an infinite solution of radionuclide, which differs from the quantity of interest proposed in this work.Similar to their work, the absorbed dose to cavity at each air gap was accurately predicted by MC simulations.These results demonstrate the ability of general-purpose MC codes to accurately calculate absorbed dose using given radioactivity.In our previous work, an extensive evaluation of the GEANT4 electromagnetic physics parameters was performed for alpha particles and the internal decay library was compared to validated databases [28].Therefore, validation of MC transport parameters should be considered given the good agreement observed in this work between measured and MC-calculated absorbed dose.A major limitation of this work was using a single MC code for all simulations.While the work of Billas et al allowed measurement of absorbed dose with an overall combined uncertainty of 1.56% at k = 1, the uncertainty determined in this work was observed to be much higher.Their work reported a repositioning uncertainty of 0.50%, whereas, the current slope repeatability uncertainty was found to be 2.79% in this work.The higher uncertainty encountered in this work was attributed to the higher LET and mass of the alpha particles as well as the geometry of the source where the absorbed dose gradient from point-like sources is much higher than the gradient encountered for volumetric sources.Additionally, the windowless setup makes positioning the IC more challenging due to possible contamination of the detector when in contact with the source.The MCbased correction factor in their work was reported to be in the 241%-255% range, which is larger than the MC-based correction factors introduced in this work.
The D400 IC used in this work was previously used by Hansen et al to measure surface absorbed dose from beta ophthalmic applicators [29,30].The IC from the work of Hansen et al was modified by reducing the guard's outer diameter.Overall combined uncertainty in the absorbed dose of ∼3.1%-4.8% at k = 1 was reported in both of these previous studies, which is similar to the results found in this study.The variation in the ionization current readings between different trials was the leading cause of the higher uncertainty in both works.Therefore, positional and rotational alignment uncertainty must be reduced to achieve a more repeatable measurement of absorbed dose.While the work of Hansen et al reported an uncertainty of 0.50% in the divergence correction, a much larger uncertainty of 1.79% was noted in this work due to the differences in the beta and alpha particle physics.The higher uncertainty in the divergence correction in this work can be attributed to the steeper and non-linear absorbed dose versus air gap curve calculated for alpha particles with a larger uncertainty in the fitting function when compared to their work.The larger gradient of the absorbed dose versus air gap curve also led to a much larger k div correction for alpha particles.The maximum divergence correction in the works of Hansen et al was reported to be ∼19% for beta particles and this work calculated a maximum divergence correction of up to 239% for alpha particles.The k div correction follows a linear relationship with the air gap and can be minimized  by acquiring ionization current readings at smaller air gaps.However, since the absorbed dose gradient is much steeper at smaller air gaps, the positional and rotational alignment errors can lead to much higher uncertainty in the absorbed dose measurements.Although alpha particles tend to scatter less than beta particles due to their larger mass, the magnitude of the backscatter correction was found to be similar between both works.However, their definition of the k backscatter differs from this work.Their work calculates the backscatter correction by changing the detector materials to water, while this work also changes the materials of the source substrate and holder to water.It was hypothesized that the backscatter from the substrate was the dominant contributing factor.In future work, absorbed dose measurements will be acquired for clinically-relevant alpha-particle-emitting radionuclides such as 223 Ra, 225 Ac, 227 Th, 212 Pb, and 211 At.The mixed particle emission from these sources can further characterize the absorbed dose standard.The dosimetric formalism needs to be revised to incorporate absorbed dose from both the beta and the alpha particle components.Methods to reduce uncertainty in the positional and rotational alignment will also be explored.The measured absorbed dose must be compared to commonly-utilized dose calculation platforms.This work validated the GEANT4 Monte Carlo code by comparing the calculated absorbed dose with the measured absorbed dose.This process must be performed for other platforms as well such as MCNP and TOPAS n-Bio.Such comparisons will foster confidence in the dose calculation techniques for alpha particles.Engagement with the clinical community and methods to disseminate the standardized absorbed dose to water quantity to clinics will also be explored.

Conclusions
A standard for activity-normalized surface absorbed dose to water was constructed, with a combined uncertainty of 4.23% at k = 1, and evaluated using a pure alpha emitter.The dominant contributor to the overall uncertainty was the repeatability of the ionization current readings between different measurement trials.By using a windowless IC setup with a thin film source, the attenuation of alpha particles was minimized and a contactless method was employed to determine the absolute air gap between the IC and the source substrate.The IC-specific correction factors, except k TP , were found to be <0.5% for all measurement trials while the MCcalculated correction factors were calculated to be much larger.The divergence correction was noted to be the highest correction factor due to the sharp fall-off of the absorbed dose as a function of air gap.Using the extrapolation dosimetric formalism, the activity-normalized surface absorbed dose to water was measured to be 2.8913 × 10 −6 Gy s −1 Bq −1 for a pointlike 210 Po source.

Figure 1 .
Figure 1.A schematic of the measurement setup is shown.

Figure 2 .
Figure 2. (a): An XZ view of the entire absorbed dose standard is shown.(b): The D400 IC and the source subassembly are shown.

Figure 3 .
Figure 3. Determination of the absolute air gap and the cavity diameter using the capacitance method.Measurements with an arbitrary unknown initial air gap are shown.

Figure 4 .
Figure 4. Measured and fitted charge readings to the recombination equation as a function of (a) 1/V and (b) 1/V 2 .(c) The total recombination correction as a function of electric field strength.

Figure 5 .
Figure 5.The ionization current, normalized by radioactivity, collected by the D400 IC at each air gap.The error bars represent the uncertainty in the radioactivity, standard deviation in the current readings for each trial, and uncertainties in the air density, polarity, and recombination corrections.

Figure 6 .
Figure 6.The measured and MC simulated absorbed dose to cavity as a function of air gaps in the 300-525 µm range.The error bars correspond to combined uncertainty at k = 1.

Figure 7 .
Figure 7.The (a) k point and (b) k backscatter correction factors as a function of air gap for a 210 Po source.The error bars correspond to Type A 1σ uncertainty in the MC simulations.

Figure 8 .
Figure 8.(a) Absorbed dose to cavity plotted as a function of air gap with a 3rd-order polynomial fit.(b) The divergence correction factor for 0.3 mm to 0.525 mm air gaps.The error bars correspond to 1σ uncertainty in the MC simulations and are smaller than the marker size.

Figure 9 .
Figure 9. Corrected ionization current at each air gap for the 210 Po source.

Table 1 .
Uncertainty budget for an absorbed dose to air measurement, acquired with the D400 IC, at a 0.30 mm air gap.

Table 2 .
Uncertainty budget for the surface absorbed dose to water measured using the D400 IC.