Measurement of characteristic prompt gamma rays emitted from oxygen and carbon in tissue-equivalent samples during proton beam irradiation

The first probe of a FLASH proton beam by PET

The recently observed FLASH effect related to high doses delivered with high rates has the potential to revolutionize radiation cancer therapy if promising results are confirmed and an underlying mechanism understood. Comprehensive measurements are essential to elucidate the phenomenon. We report the first-ever demonstration of measurements of successive in-spill and post-spill emissions of gammas arising from irradiations by a FLASH proton beam. A small positron emission tomography (PET) system was exposed in an ocular beam of the Proton Therapy Center at MD Anderson Cancer Center to view phantoms irradiated by 3.5 × 1010protons with a kinetic energy of 75.8 MeV delivered in 101.5 ms-long spills yielding a dose rate of 164 Gy s-1. Most in-spill events were due to prompt gammas. Reconstructed post-spill tomographic events, recorded for up to 20 min, yielded quantitative imaging and dosimetric information. These findings open a new and novel modality for imaging and monitoring of FLASH proton therapy exploiting in-spill prompt gamma imaging followed by post-spill PET imaging.

Investigating the hard X-ray production via proton spallation on different materials to detect elements

Various atomic and nuclear methods use hard (high-energy) X-rays to detect elements. The current study aims to investigate the hard X-ray production rate via high-energy proton beam irradiation of various materials. For which, appropriate conditions for producing X-rays were established. The MCNPX code, based on the Monte Carlo method, was used for simulation. Protons with energies up to 1650 MeV were irradiated on various materials such as carbon, lithium, lead, nickel, salt, and soil, where the resulting X-ray spectra were extracted. The production of X-rays in lead was observed to increase 16 times, with the gain reaching 0.18 as the proton energy increases from 100 MeV to 1650 MeV. Comparatively, salt is a good candidate among the lightweight elements to produce X-rays at a low proton energy of 30 MeV with a production gain of 0.03. Therefore, it is suggested to irradiate the NaCl target with 30 MeV proton to produce X-rays in the 0-2 MeV range.

Design and performance evaluation of a slit-slat camera for 2D prompt gamma imaging in proton therapy monitoring: A Monte Carlo simulation study

PURPOSE

We investigated the design of a prompt gamma camera for real-time dose delivery verification and the partial mitigation of range uncertainties.

METHODS

A slit slat (SS) camera was optimized using the trade-off between the signal-to-noise ratio and spatial resolution. Then, using the GATE Monte Carlo package, the camera performances were estimated by means of target shifts, beam position quantification, changing the camera distance from the beam, and air cavity inserting. A homogeneous PMMA phantom and the air gaps induced PMMA phantom were used. The air gaps ranged from 5 mm to 30 mm by 5 mm increments were positioned in the middle of the beam range. To reduce the simulation time, phase space scoring was used. The batch method with five realizations was used for stochastic error calculations.

RESULTS

The system's detection efficiency was 1.1 × 10 - 4 PGs Emitted PGs ( 1.8 × 10 - 5 $1.1 \times {10}^{-4}\frac{{\rm PGs}}{{\rm Emitted}\ {\rm PGs}}\ (1.8 \times {10}^{-5}$ PGs/proton) for a 10 × 20 cm² detector (source-to-collimator distance = 15.0 cm). Axial and transaxial resolutions were 23 mm and 18 mm, respectively. The SS camera estimated the range as 69.0 ± 3.4 (relative stochastic error 1-sigma is 5%) and 67.6 ± 1.8 mm (2.6%) for the real range of 67.0 mm for 10⁷ and 10⁸ protons of 100 MeV, respectively. Considering 160 MeV, these values are 155.5 ± 3.1 (2%) and 152.2 ± 2.0 mm (1.3%) for the real range of 152.0 mm for 10⁷ and 10⁸ protons, respectively. Considering phantom shift, for a 100 MeV beam, the precision of the quantification (1-sigma) in the axial and lateral phantom shift estimation is 2.6 mm and 1 mm, respectively. Accordingly, the axial and lateral quantification precisions were 1.3 mm and 1 mm for a 160 MeV beam, respectively. Furthermore, the quantification of an air gap formulated as gap d e t = 0.98 × gap real ${{\rm gap}}_{det}=0.98 \times {{\rm gap}}_{{\rm real}}$ , where gap d e t ${{\rm gap}}_{det}$ and gap_real are the estimated and real air gap, respectively. The precision of the air gap quantification is 1.6 mm (1 sigma). Moreover, 2D PG images show the trajectory of the proton beam through the phantom.

CONCLUSION

The proposed slit-slat imaging systems can potentially provide a real-time, in-vivo, and non-invasive treatment monitoring method for proton therapy.

Secondary Particle Interactions in a Compton Camera Designed for in vivo Range Verification of Proton Therapy

The purpose of this study was to determine the types, proportions, and energies of secondary particle interactions in a Compton camera (CC) during the delivery of clinical proton beams. The delivery of clinical proton pencil beams ranging from 70 to 200 MeV incident on a water phantom was simulated using Geant4 software (version 10.4). The simulation included a CC similar to the configuration of a Polaris J3 CC designed to image prompt gammas (PGs) emitted during proton beam irradiation for the purpose of in vivo range verification. The interaction positions and energies of secondary particles in each CC detector module were scored. For a 150-MeV proton beam, a total of 156,688(575) secondary particles per 10⁸ protons, primarily composed of gamma rays (46.31%), neutrons (41.37%), and electrons (8.88%), were found to reach the camera modules, and 79.37% of these particles interacted with the modules. Strategies for using CCs for proton range verification should include methods of reducing the large neutron backgrounds and low-energy non-PG radiation. The proportions of interaction types by module from this study may provide information useful for background suppression.

A filtering approach for PET and PG predictions in a proton treatment planning system

Positron emission tomography (PET) and prompt gamma (PG) detection are promising proton therapy monitoring modalities. Fast calculation of the expected distributions is desirable for comparison to measurements and to develop/train algorithms for automatic treatment error detection. A filtering formalism was used for positron-emitter predictions and adapted to allow for its use for the beamline of any proton therapy centre. A novel approach based on a filtering formalism was developed for the prediction of energy-resolved PG distributions for arbitrary tissues. The method estimates PG yields and their energy spectra in the entire treatment field. Both approaches were implemented in a research version of the RayStation treatment planning system. The method was validated against PET monitoring data and Monte Carlo simulations for four patients treated with scanned proton beams. Longitudinal shifts between profiles from analytical and Monte Carlo calculations were within -1.7 and 0.9 mm, with maximum standard deviation of 0.9 mm and 1.1 mm, for positron-emitters and PG shifts, respectively. Normalized mean absolute errors were within 1.2 and 5.3%. When comparing measured and predicted PET data, the same more complex case yielded an average shift of 3 mm, while all other cases were below absolute average shifts of 1.1 mm. Normalized mean absolute errors were below 7.2% for all cases. A novel solution to predict positron-emitter and PG distributions in a treatment planning system is proposed, enabling calculation times of only a few seconds to minutes for entire patient cases, which is suitable for integration in daily clinical routine.

PIBS: Proton and ion beam spectroscopy for in vivo measurements of oxygen, carbon, and calcium concentrations in the human body

Proton and ion beam therapy has proven to benefit tumour control with lower side-effects, mostly in paediatrics. Here we demonstrate a feasible technique for proton and ion beam spectroscopy (PIBS) capable of determining the elemental compositions of the irradiated tissues during particle therapy. This follows the developments in prompt gamma imaging for online range verification and the inheritance from prompt gamma neutron activation analysis. Samples of water solutions were prepared to emulate varying oxygen and carbon concentrations. The irradiation of those samples and other tissue surrogate inserts by protons and ion beams under clinical conditions clearly showed a logarithmic relationship between the target elemental composition and the prompt gamma production. This finding is in line with the known logarithmic dependence of the pH with the proton molar concentration. Elemental concentration changes of 1% for calcium and 2% for oxygen in adipose, brain, breast, liver, muscle and bone-related tissue surrogates were clearly identified. Real-time in vivo measurements of oxygen, carbon and calcium concentrations will be evaluated in a pre-clinical and clinical environment. This technique should have an important impact in the assessment of tumour hypoxia over the course of several treatment fractions and the tracking of calcifications in brain metastases.

A New Method to Reconstruct in 3D the Emission Position of the Prompt Gamma Rays following Proton Beam Irradiation

A new technique for range verification in proton beam therapy has been developed. It is based on the detection of the prompt γ rays that are emitted naturally during the delivery of the treatment. A spectrometer comprising 16 LaBr₃(Ce) detectors in a symmetrical configuration is employed to record the prompt γ rays emitted along the proton path. An algorithm has been developed that takes as inputs the LaBr₃(Ce) detector signals and reconstructs the maximum γ-ray intensity peak position, in full 3 dimensions. For a spectrometer radius of 8 cm, which could accommodate a paediatric head and neck case, the prompt γ-ray origin can be determined from the width of the detected peak with a σ of 4.17 mm for a 180 MeV proton beam impinging a water phantom. For spectrometer radii of 15 and 25 cm to accommodate larger volumes this value increases to 5.65 and 6.36 mm. For a 8 cm radius, with a 5 and 10 mm undershoot, the σ is 4.31 and 5.47 mm. These uncertainties are comparable to the range uncertainties incorporated in treatment planning. This work represents the first step towards a new accurate, real-time, 3D range verification device for spot-scanning proton beam therapy.

In vivo range verification in particle therapy

Exploitation of the full potential offered by ion beams in clinical practice is still hampered by several sources of treatment uncertainties, particularly related to the limitations of our ability to locate the position of the Bragg peak in the tumor. To this end, several efforts are ongoing to improve the characterization of patient position, anatomy, and tissue stopping power properties prior to treatment as well as to enable in vivo verification of the actual dose delivery, or at least beam range, during or shortly after treatment. This contribution critically reviews methods under development or clinical testing for verification of ion therapy, based on pretreatment range and tissue probing as well as the detection of secondary emissions or physiological changes during and after treatment, trying to disentangle approaches of general applicability from those more specific to certain anatomical locations. Moreover, it discusses future directions, which could benefit from an integration of multiple modalities or address novel exploitation of the measurable signals for biologically adapted therapy.

Spectroscopic Compton imaging of prompt gamma emission at the MeV energy range

This work explores a novel tomographic approach to PGAA that is both quantitative and spatially resolved, adapted from a clinical "proton beam range finder" in which MeV gamma rays are imaged by coincidence measurements of Compton scattered gamma rays with multi-detector arrays. We performed preliminary measurements using a Compton camera made with CdZnTe detector arrays on a series of test samples with high-energy (> 1 MeV) gamma emission lines. 3D image reconstructions were performed on the 2.2 MeV peak from H. The image reconstruction methods were also evaluated using the emission data generated by Monte Carlo simulations under ideal conditions.

Imaging of monochromatic beams by measuring secondary electron bremsstrahlung for carbon-ion therapy using a pinhole x-ray camera

A feasibility study on the imaging of monochromatic carbon-ion beams for carbon-ion therapy was performed. The evaluation was based on Monte Carlo simulations and beam-irradiation experiments, using a pinhole x-ray camera, which measured secondary electron bremsstrahlung (SEB). The simulation results indicated that the trajectories of the carbon-ion beams with injection energies of 278, 249 and 218 MeV/u in a water phantom, were clearly imaged by measuring the SEB with energies from 30 to 60 keV, using a pinhole camera. The Bragg-peak positions for these three injection energies were located at the positions where the ratios of the counts of SEB acquisitions to the maximum counts were approximately 0.23, 0.26 and 0.29, respectively. Moreover, we experimentally demonstrated that it was possible to identify the Bragg-peak positons, at the positions where the ratios coincided with the simulation results. However, the estimated Bragg-peak positions for the injection energies of 278 and 249 MeV/u were slightly deeper than the expected positions. In conclusion, for both the simulations and experiments, we found that the 25 mm shifts in the Bragg-peak positions can be observed by this method.

3D prompt gamma imaging for proton beam range verification

We tested the ability of a single Compton camera (CC) to produce 3-dimensional (3D) images of prompt gammas (PGs) emitted during the irradiation of a tissue-equivalent plastic phantom with proton pencil beams for clinical doses delivered at clinical dose rates. PG measurements were made with a small prototype CC placed at three different locations along the proton beam path. We evaluated the ability of the CC to produce images at each location for two clinical scenarios: (1) the delivery of a single 2 Gy pencil beam from a hypo-fractionated treatment (~9  ×  10⁸ protons), and (2) a single pencil beam from a standard treatment (~1  ×  10⁸ protons). Additionally, the data measured at each location were combined to simulate measurements with a larger scale, clinical CC and its ability to image shifts in the Bragg peak (BP) range for both clinical scenarios. With our prototype CC, the location of the distal end of the BP could be seen with the CC placed up to 4 cm proximal or distal to the BP distal falloff. Using the data from the simulated full scale clinical CC, 3D images of the PG emission were produced with the delivery of as few as 1  ×  10⁸ protons, and shifts in the proton beam range as small as 2 mm could be detected for delivery of a 2 Gy spot. From these results we conclude that 3D PG imaging for proton range verification under clinical beam delivery conditions is possible with a single CC.

Spectroscopic study of prompt-gamma emission for range verification in proton therapy

We present the results of an investigation of the prompt-gamma emission from an interaction of a proton beam with phantom materials. Measurements were conducted with a novel setup allowing the precise selection of the investigated depth in the phantom, featuring three different materials composed of carbon, oxygen and hydrogen. We studied two beam energies of 70.54 and 130.87MeV and two detection angles: 90° and 120°. The results are presented in form of profiles of the prompt-gamma yield as a function of depth. In the analysis we focused on the transitions with the largest cross sections: ¹²C_4.44→g.s. and ¹⁶O_6.13→g.s.. We compare the profiles obtained under various irradiation conditions, with emphasis on the shape of the distal fall-off. The results are also compared to calculations including different cross-section models. They are in agreement with the model exploiting published cross-section data, but the comparison with the Talys model shows discrepancies.

Monte Carlo simulation of photon emission below a few hundred kiloelectronvolts for beam monitoring in carbon ion therapy

The purpose of this study is to determine whether the main component of the low-energy (63-68 keV) particles emitted perpendicularly to the ¹²C beam from the ¹²C-irradiated region in a water phantom is secondary electron bremsstrahlung (SEB). Monte Carlo simulations of a ¹²C-beam (290 MeV/u) irradiated on a water phantom were performed. A detector was placed beside the water phantom with a lead collimator between the phantom and the detector. To move the Bragg-peak position, a binary filter was placed in an upper stream of the phantom. The energy distributions of the particles incident on the detector and those deposited in the detector were analyzed. The simulation was also performed with suppressed delta-ray and/or bremsstrahlung generation to identify the SEB components. It was found that the particles incident on the detector were predominantly photons and neutrons. The yields of the photons and energy deposition decreased with the suppression of SEB generation. It is concluded that one of the predominant components of the yields in the regions shallower than the Bragg-peak position is due to SEB generation, and these components become significantly smaller in regions deeper than the Bragg-peak position.

Towards clinical application: prompt gamma imaging of passively scattered proton fields with a knife-edge slit camera

Prompt γ-ray imaging with a knife-edge shaped slit camera provides the possibility of verifying proton beam range in tumor therapy. Dedicated experiments regarding the characterization of the camera system have been performed previously. Now, we aim at implementing the prototype into clinical application of monitoring patient treatments. Focused on this goal of translation into clinical operation, we systematically addressed remaining challenges and questions. We developed a robust energy calibration routine and corresponding quality assurance protocols. Furthermore, with dedicated experiments, we determined the positioning precision of the system to 1.1 mm (2σ). For the first time, we demonstrated the application of the slit camera, which was intentionally developed for pencil beam scanning, to double scattered proton beams. Systematic experiments with increasing complexity were performed. It was possible to visualize proton range shifts of 2-5 mm with the camera system in phantom experiments in passive scattered fields. Moreover, prompt γ-ray profiles for single iso-energy layers were acquired by synchronizing time resolved measurements to the rotation of the range modulator wheel of the treatment system. Thus, a mapping of the acquired profiles to different anatomical regions along the beam path is feasible and additional information on the source of potential range shifts can be obtained. With the work presented here, we show that an application of the slit camera in clinical treatments is possible and of potential benefit.

Nuclear physics in particle therapy: a review

Charged particle therapy has been largely driven and influenced by nuclear physics. The increase in energy deposition density along the ion path in the body allows reducing the dose to normal tissues during radiotherapy compared to photons. Clinical results of particle therapy support the physical rationale for this treatment, but the method remains controversial because of the high cost and of the lack of comparative clinical trials proving the benefit compared to x-rays. Research in applied nuclear physics, including nuclear interactions, dosimetry, image guidance, range verification, novel accelerators and beam delivery technologies, can significantly improve the clinical outcome in particle therapy. Measurements of fragmentation cross-sections, including those for the production of positron-emitting fragments, and attenuation curves are needed for tuning Monte Carlo codes, whose use in clinical environments is rapidly increasing thanks to fast calculation methods. Existing cross sections and codes are indeed not very accurate in the energy and target regions of interest for particle therapy. These measurements are especially urgent for new ions to be used in therapy, such as helium. Furthermore, nuclear physics hardware developments are frequently finding applications in ion therapy due to similar requirements concerning sensors and real-time data processing. In this review we will briefly describe the physics bases, and concentrate on the open issues.

Monitoring of Hadrontherapy Treatments by Means of Charged Particle Detection

The interaction of the incoming beam radiation with the patient body in hadrontherapy treatments produces secondary charged and neutral particles, whose detection can be used for monitoring purposes and to perform an on-line check of beam particle range. In the context of ion-therapy with active scanning, charged particles are potentially attractive since they can be easily tracked with a high efficiency, in presence of a relatively low background contamination. In order to verify the possibility of exploiting this approach for in-beam monitoring in ion-therapy, and to guide the design of specific detectors, both simulations and experimental tests are being performed with ion beams impinging on simple homogeneous tissue-like targets (PMMA). From these studies, a resolution of the order of few millimeters on the single track has been proven to be sufficient to exploit charged particle tracking for monitoring purposes, preserving the precision achievable on longitudinal shape. The results obtained so far show that the measurement of charged particles can be successfully implemented in a technology capable of monitoring both the dose profile and the position of the Bragg peak inside the target and finally lead to the design of a novel profile detector. Crucial aspects to be considered are the detector positioning, to be optimized in order to maximize the available statistics, and the capability of accounting for the multiple scattering interactions undergone by the charged fragments along their exit path from the patient body. The experimental results collected up to now are also valuable for the validation of Monte Carlo simulation software tools and their implementation in Treatment Planning Software packages.

Compton Camera and Prompt Gamma Ray Timing: Two Methods for In Vivo Range Assessment in Proton Therapy

Proton beams are promising means for treating tumors. Such charged particles stop at a defined depth, where the ionization density is maximum. As the dose deposit beyond this distal edge is very low, proton therapy minimizes the damage to normal tissue compared to photon therapy. Nevertheless, inherent range uncertainties cast doubts on the irradiation of tumors close to organs at risk and lead to the application of conservative safety margins. This constrains significantly the potential benefits of protons over photons. In this context, several research groups are developing experimental tools for range verification based on the detection of prompt gammas, a nuclear by-product of the proton irradiation. At OncoRay and Helmholtz-Zentrum Dresden-Rossendorf, detector components have been characterized in realistic radiation environments as a step toward a clinical Compton camera. On the one hand, corresponding experimental methods and results obtained during the ENTERVISION training network are reviewed. On the other hand, a novel method based on timing spectroscopy has been proposed as an alternative to collimated imaging systems. The first tests of the timing method at a clinical proton accelerator are summarized, its applicability in a clinical environment for challenging the current safety margins is assessed, and the factors limiting its precision are discussed.

Range Verification Methods in Particle Therapy: Underlying Physics and Monte Carlo Modeling

Hadron therapy allows for highly conformal dose distributions and better sparing of organs-at-risk, thanks to the characteristic dose deposition as function of depth. However, the quality of hadron therapy treatments is closely connected with the ability to predict and achieve a given beam range in the patient. Currently, uncertainties in particle range lead to the employment of safety margins, at the expense of treatment quality. Much research in particle therapy is therefore aimed at developing methods to verify the particle range in patients. Non-invasive in vivo monitoring of the particle range can be performed by detecting secondary radiation, emitted from the patient as a result of nuclear interactions of charged hadrons with tissue, including β (+) emitters, prompt photons, and charged fragments. The correctness of the dose delivery can be verified by comparing measured and pre-calculated distributions of the secondary particles. The reliability of Monte Carlo (MC) predictions is a key issue. Correctly modeling the production of secondaries is a non-trivial task, because it involves nuclear physics interactions at energies, where no rigorous theories exist to describe them. The goal of this review is to provide a comprehensive overview of various aspects in modeling the physics processes for range verification with secondary particles produced in proton, carbon, and heavier ion irradiation. We discuss electromagnetic and nuclear interactions of charged hadrons in matter, which is followed by a summary of some widely used MC codes in hadron therapy. Then, we describe selected examples of how these codes have been validated and used in three range verification techniques: PET, prompt gamma, and charged particle detection. We include research studies and clinically applied methods. For each of the techniques, we point out advantages and disadvantages, as well as clinical challenges still to be addressed, focusing on MC simulation aspects.

Intensity modulated proton therapy

Intensity modulated proton therapy (IMPT) implies the electromagnetic spatial control of well-circumscribed "pencil beams" of protons of variable energy and intensity. Proton pencil beams take advantage of the charged-particle Bragg peak-the characteristic peak of dose at the end of range-combined with the modulation of pencil beam variables to create target-local modulations in dose that achieves the dose objectives. IMPT improves on X-ray intensity modulated beams (intensity modulated radiotherapy or volumetric modulated arc therapy) with dose modulation along the beam axis as well as lateral, in-field, dose modulation. The clinical practice of IMPT further improves the healthy tissue vs target dose differential in comparison with X-rays and thus allows increased target dose with dose reduction elsewhere. In addition, heavy-charged-particle beams allow for the modulation of biological effects, which is of active interest in combination with dose "painting" within a target. The clinical utilization of IMPT is actively pursued but technical, physical and clinical questions remain. Technical questions pertain to control processes for manipulating pencil beams from the creation of the proton beam to delivery within the patient within the accuracy requirement. Physical questions pertain to the interplay between the proton penetration and variations between planned and actual patient anatomical representation and the intrinsic uncertainty in tissue stopping powers (the measure of energy loss per unit distance). Clinical questions remain concerning the impact and management of the technical and physical questions within the context of the daily treatment delivery, the clinical benefit of IMPT and the biological response differential compared with X-rays against which clinical benefit will be judged. It is expected that IMPT will replace other modes of proton field delivery. Proton radiotherapy, since its first practice 50 years ago, always required the highest level of accuracy and pioneered volumetric treatment planning and imaging at a level of quality now standard in X-ray therapy. IMPT requires not only the highest precision tools but also the highest level of system integration of the services required to deliver high-precision radiotherapy.

Detecting prompt gamma emission during proton therapy: the effects of detector size and distance from the patient

Recent studies have suggested that the characteristics of prompt gammas (PGs) emitted from excited nuclei during proton therapy are advantageous for determining beam range during treatment delivery. Since PGs are only emitted while the beam is on, the feasibility of using PGs for online treatment verification depends greatly on the design of highly efficient detectors. The purpose of this work is to characterize how PG detection changes as a function of distance from the patient as a means of guiding the design and usage of clinical PG imaging detectors. Using a Monte Carlo model (GEANT4.9.4) we studied the detection rate (PGs per incident proton) of a high purity germanium detector for both the total PG emission and the characteristic 6.13 MeV PG emission from (16)O emitted during proton irradiation. The PG detection rate was calculated as a function of distance from the isocenter of the proton treatment nozzle for: (1) a water phantom irradiated with a proton pencil beam and (2) a prostate patient irradiated with a scanning beam proton therapy treatment field (lateral field size: ∼6 cm × 6 cm, beam range: 23.5 cm). An analytical expression of the PG detection rate as a function of distance from isocenter, detector size, and proton beam energy was then developed. The detection rates were found to be 1.3 × 10(-6) for oxygen and 3.9 × 10(-4) for the total PG emission, respectively, with the detector placed 11 cm from isocenter for a 40 MeV pencil beam irradiating a water phantom. The total PG detection rate increased by ∼85 ± 3% for beam energies greater than 150 MeV. The detection rate was found to be approximately 2.1 × 10(-6) and 1.7 × 10(-3) for oxygen and total PG emission, respectively, during delivery of a single pencil beam during a scanning beam treatment for prostate cancer. The PG detection rate as a function of distance from isocenter during irradiation of a water phantom with a single proton pencil beam was described well by the model of a point source irradiating a cylindrical detector of a known diameter over the range of beam energies commonly used for proton therapy. For the patient studies, it was necessary to divide the point source equation by an exponential factor in order to correctly predict the falloff of the PG detection rate as a function of distance from isocenter.