Time‐resolved diode dosimetry calibration through Monte Carlo modeling for in vivo passive scattered proton therapy range verification

Abstract Purpose Our group previously introduced an in vivo proton range verification methodology in which a silicon diode array system is used to correlate the dose rate profile per range modulation wheel cycle of the detector signal to the water‐equivalent path length (WEPL) for passively scattered proton beam delivery. The implementation of this system requires a set of calibration data to establish a beam‐specific response to WEPL fit for the selected ‘scout’ beam (a 1 cm overshoot of the predicted detector depth with a dose of 4 cGy) in water‐equivalent plastic. This necessitates a separate set of measurements for every ‘scout’ beam that may be appropriate to the clinical case. The current study demonstrates the use of Monte Carlo simulations for calibration of the time‐resolved diode dosimetry technique. Methods Measurements for three ‘scout’ beams were compared against simulated detector response with Monte Carlo methods using the Tool for Particle Simulation (TOPAS). The ‘scout’ beams were then applied in the simulation environment to simulated water‐equivalent plastic, a CT of water‐equivalent plastic, and a patient CT data set to assess uncertainty. Results Simulated detector response in water‐equivalent plastic was validated against measurements for ‘scout’ spread out Bragg peaks of range 10 cm, 15 cm, and 21 cm (168 MeV, 177 MeV, and 210 MeV) to within 3.4 mm for all beams, and to within 1 mm in the region where the detector is expected to lie. Conclusion Feasibility has been shown for performing the calibration of the detector response for three ‘scout’ beams through simulation for the time‐resolved diode dosimetry technique in passive scattered proton delivery.


| INTRODUCTION
Proton therapy may reduce adverse effects of radiation due to the potential for increased normal tissue sparing as compared to other modern external beam radiotherapy techniques. 1 Variation in beam range is the result of variation in stochastic range straggling in the presence of anatomical tissue heterogeneity. Patient setup error, misalignment, and internal organ motion result in different heterogeneities in the path of the beam at time of treatment as compared to CT simulation. Range uncertainties are currently accounted for using range margins, but these margins limit the potential advantages of distal edge normal tissue sparing of proton therapy. Calculation of the position of the Bragg peak for a given heterogeneous tissue distribution is determined using a conversion of CT Hounsfield units (HU) to relative proton stopping power combined with measurements in water. This calibration is subject to the uncertainty of converting HU to water-equivalent density, approximately 1%-2% of the proton beam range, as well as uncertainty in CT HU measurements themselves, mainly due to beam hardening and also approximately 1%-2% of the proton beam range. 2 A range uncertainty estimate of 3.5% AE 1 mm of the proton beam range as a distal margin is commonly implemented in proton therapy treatment planning for spread out Bragg peak (SOBP) fields. 3 Sparing healthy tissue by reducing the applied margins provides motivation to develop millimeter accuracy in range verification techniques. The use of a small volume detector array for range measurements has been proposed, 4,5 which could provide the opportunity for in vivo range verification and be used on-line to adapt the treatment plan and minimize these margins for passively scattered proton delivery.
Because the detector response is beam-specific, experimental measurements in homogeneous media have been employed to establish a calibration curve of the response of the detector to WEPL for every SOBP that may be delivered for each given clinical case. [4][5][6][7][8][9][10] This process is both tedious, as it necessitates a separate set of measurements for every new 'scout' beam (a 1 cm overshoot of the predicted detector depth with a dose of 4 cGy), as well as inconvenient due to the time constraints for access to the clinical beamline. The aim of this work is to investigate whether the calibration response can be simulated with sufficient accuracy to eliminate the necessity of performing these physical measurements.
The characteristic time dependence of the dose rate at a point within the SOBP can be used to determine the water-equivalent path length (WEPL) when a deliberate overshoot of the targeted range is implemented such that the detector lies in the plateau of the SOBP. 4 The relationship between detector signal and WEPL results from the periodic motion of the range modulation wheel (RMW) in the passively scattered delivery mechanism. A measurement of the detector signal at depth in a medium over one period of the rotation of the wheel can determine the residual range of the beam according to a set of calibration data because the dose rate profile per RMW cycle is unique to each depth (Fig. 1).
The detector signal per rotation of the RMW at a given depth (WEPL) in a homogeneous medium is specific to a given set of beam parameters: range, modulation width, and beam current modulation sequence (which varies proton fluence to achieve AE 2% flatness in the plateau region of the SOBP). These selections are made to suit the clinical application (expected depth of detector for a given treatment site) such that the 'scout' SOBP will overshoot the expected WEPL of the detector by 1 cm.
While in previous implementations, a set of calibration measurements in homogeneous media were required to establish a calibration, we now apply Monte Carlo simulations to achieve this calibration curve. As shown previously, a statistical approach for analysis can be applied to correlate the dose rate profile per RMW cycle to WEPL. 6 The root-mean-square (rms) width of each dose rate profile was computed for each detector positional depth, and a relationship between the rms width of the time the diode reads signal and WEPL was established. A fourth-order polynomial was fitted to provide a continuous calibration curve for dose rate per RMW cycle and WEPL as per Gottschalk 2011. 6  Three 'scout' beams were selected to provide sufficient overshoot of targets for different clinical scenarios. A 10 cm range beam with 9.9 cm modulation width, a 15 cm range beam with 14 cm modulation width, and a 21 cm beam with 18 cm modulation width were selected as representative for typical treatment fields. The The diodes were connected to an in-house amplification and digitizing system as detailed in Bentefour 2015 9 (Fig 2), in essence comprising a preamplifier, a digitizer, and custom software for data logging and acquisition. The digitizer was triggered using data from the dose counting electronics unit (DCEU) which monitors the beam current modulation.

2.B | Calibration measurements
For a single diode, the voltage v i was sampled at time t i at a rate of 100,000 samples/sec. The r rms width of the resulting distribution was then computed, resulting in a r rms width value for each of 19 cycles, for each of 12 diodes, for each of 30 WEPLs, for each of three beams. Thus, over 20,000 data points were collected. The mean r rms width was taken over the 12 diodes and 19 cycles to determine a mean r rms width per WEPL, and a fourth-order polynomial was used to fit r rms width as a function of WEPL.

2.C | Simulation methods
The Simulations were repeated for ten different randomization seeds, since each simulation represents a single full rotation of the RMW.
A fourth-order polynomial was fitted to each set of simulated data, and the fit was used to predict the r rms width of the dose rate profile per RMW cycle at each of the measured depths. The standard error over the results from each seed was computed over the mean of these values for each depth.
Dose rate profiles were scored for three different absorption media. Water-equivalent plastic was modeled using material chemical compositions and densities provided by the manufacturer, and these simulations are referred to as 'simulated plastic water.' A CT of water-equivalent plastic was also used as an absorber, and finally a patient CT where the beam penetrated only soft tissues in the abdomen, with no air cavities, was used to evaluate WEPL based on dose rate profiles with depth.

| RESULTS
The dose at the position of the diode was simulated for the 'scout' SOBPs of range 10 cm, 15 cm, and 21 cm (168 MeV, 177 MeV, and 210 MeV), and the fourth-order polynomial fits were compared against the measured data resulting in an adjusted R 2 of 0.999 for all three beams in water-equivalent plastic and an adjusted R 2 of 0.998 in the patient CT (Fig 3-5). The maximum WEPL deviation of fitted values from measurements and adjusted R 2 between the polynomial fitted to simulated data and the polynomial fitted to measured data are shown in Table 1.
Within 1 cm of the range of the "scout" SOBP, where the detector is expected to lie, the deviation between measured and simulationderived WEPL is within 1 mm for all beams for all absorption media.

| DISCUSSION
In this work, a method for the calibration of a diode detector system response was established through simulation for three "scout" SOBP beams of anticipated clinical relevance. In lieu of acquiring measurements to calibrate detector response in a novel "scout" beam, this work has validated the accuracy of using a TOPAS simulation to establish the detector response if a new "scout" beam for which no calibration fit has been measured or simulated is required. Range uncertainty of 0.2% is introduced which results from CT HU to proton stopping power conversion in Monte Carlo simulations. 3 As the maximum WEPL deviation from measurements was of the order of 1-3 mm for all three media, we conclude that there is no significant difference in the accuracy of determining the WEPL from simulated dose rate among each of the three media. Depth (cm) Root-mean-square width of dose rate profile (ms) measured in water-equivalent plastic fit to measured fit to sim water-equivalent plastic fit to CT patient fit to CT of water-equivalent plastic F I G . 3. Calibration function of time-resolved diode dosimetry system to WEPL as determined by fourth-order polynomial fitted to measurements and TOPAS simulation of dose rate profile per range modulation wheel cycle for the SOBP of 10 cm range, 9.9 cm modulation width. Shown are the measured data, a polynomial fitted to measured data, and polynomials fitted to the simulated dose rate profiles in simulated water-equivalent plastic, a water-equivalent plastic CT, and a patient CT. Depth (cm) measured in water-equivalent plastic fit to measured fit to sim water-equivalent plastic fit to CT patient fit to CT water-equivalent plastic F I G . 4. Calibration function of time-resolved diode dosimetry system to WEPL as determined by fourth-order polynomial fitted to measurements and TOPAS simulation of dose rate profile per range modulation wheel cycle for the SOBP of 15 cm range, 14 cm modulation width. Shown are the measured data, a polynomial fitted to measured data, and polynomials fitted to the simulated dose rate profiles in simulated water-equivalent plastic, a water-equivalent plastic CT, and a patient CT.
It was noted that the mean and maximum WEPL deviation of simulation-derived fitted values from measurements were largest for the 15 cm range beam in simulated water-equivalent plastic and in the patient CT. However, the difference among the three beams for the mean WEPL deviation with 1r uncertainty is nonsignificant for all three simulation cases. This is in agreement with results from Testa et al. 11 where agreement between TOPAS-simulated range and measured range of the SOBPs for the RMW configuration options for our three beams was shown to range from 1. The accuracy of determining WEPL from dose rate profiles per RMW cycle is expected to decrease in the presence of range mixing as was confirmed experimentally. 7 Nearly all clinical scenarios will involve some level of range mixing as the "scout" beam penetrates the inhomogeneous patient. In order to use the time-resolved diode dosimetry system to accurately evaluate the WEPL to the dosimeter, diodes whose signal is contaminated by range mixing need to be disregarded.
It was hypothesized that analysis of the skewness and kurtosis of the dose rate profile per RMW cycle could be used to determine range mixing. 8 In this work, measurements and simulations were conducted in homogeneous media in order to establish accurate determination of WEPL through simulated dose rate profiles. Future studies will establish the accuracy of the simulation of detector response in the presence of range mixing by taking advantage of these properties.
The present work focused on implementation of a simulationderived calibration fit for detector response for the purposes of range verification in a passively scattered proton beam delivery. This is realized due to the presence of the range modulator wheel which rotates at a fixed rotational speed, and thus dose rate profiles on the scale of the RMW cycle can be correlated to depth in medium. However, the same principles have been investigated for proton radiography in an active scanned beam delivery based on an energyresolved dose measurement methodology. 17,18

| CONCLUSION
A proof of principle was demonstrated for using TOPAS to establish the dose rate profile for a given WEPL to within 3.4 mm in a patient CT and to within 2.4 mm in water-equivalent plastic. In the region where the detector is expected to lie, within 1 cm of the range of the 'scout' beam, this accuracy was shown to be within 1 mm. This enables performing the calibration procedure of the time-resolved diode dosimetry system without physical measurements. TOPAS simulations of dose rate profiles per RMW cycle established a fit to WEPL with an adjusted R 2 ≥ 0.999 in waterequivalent plastic as compared to measurements for three scout SOBPs.

CONF LICTS OF INTEREST
The authors have no relevant conflicts of interest to disclose.