The root cause analysis on failed patient-specific measurements of pencil beam scanning protons using a 2D detection array with finite size ionization chambers

The aim of this report is to present the root cause analysis on failed patient‐specific quality assurance (QA) measurements of pencil beam scanning (PBS) protons; referred to as PBS‐QA measurement. A criterion to fail a PBS‐QA measurement is having a <95% passing rate in a 3.0%‐3.0 mm gamma index analysis. Clinically, we use a two‐dimensional (2D) gamma index analysis to obtain the passing rate. The IBA MatriXX PT 2D detection array with finite size ionization chamber was utilized. A total of 2488 measurements performed in our PBS beamline were cataloged. The percentage of measurements for the sites of head/neck, breast, prostate, and other are 53.3%, 22.7%, 10.5%, and 13.5%, respectively. The measurements with a passing rate of 100 to >94%, 94 to >88%, and <88% were 93.6%, 5.6%, and 0.8%, respectively. The percentage of failed measurements with a <95% passing rate was 10.9%. After removed the user errors of either re‐measurement or re‐analysis, 8.1% became acceptable. We observed a feature of >3% per mm dose gradient with respect to depth on the failed measurements. We utilized a 2D/three‐dimensional (3D) gamma index analysis toolkit to investigate the effect of depth dose gradient. By utilizing this 3D toolkit, 43.1% of the failed measurements were improved. A feature among measurements that remained sub‐optimal after re‐analysis was a sharp >3% per mm lateral dose gradient that may not be well handled using the detector size of 5.0 mm in‐diameter. An analysis of the sampling of finite size detectors using one‐dimensional (1D) error function showed a large dose deviation at locations of low‐dose areas between two high‐dose plateaus. User error, large depth dose gradient, and the effect of detector size are identified as root causes. With the mitigation of the root causes, the goals of patient‐specific QA, specifically detecting actual deviation of beam delivery or identifying limitations of the dose calculation algorithm of the treatment planning system, can be directly related to failure of the PBS‐QA measurements.

Implementation of a double scattering nozzle for Monte Carlo recalculation of proton plans with variable relative biological effectiveness

A constant relative biological effectiveness (RBE) of 1.1 is currently used in clinical proton therapy. However, theRBEvaries with factors such as dose level, linear energy transfer (LET) and tissue type. MultipleRBEmodels have been developed to account for this biological variation. To enable recalculation of patients treated with double scattering (DS) proton therapy, includingLETand variableRBE, we implemented and commissioned a Monte Carlo (MC) model of a DS treatment nozzle. The main components from the IBA nozzle were implemented in the FLUKA MC code. We calibrated and verified the following entities to experimental measurements: range of pristine Bragg peaks (PBPs) and spread-out Bragg peaks (SOBPs), energy spread, lateral profiles, compensator range degradation, and absolute dose. We recalculated two patients with different field setups, comparing FLUKA vs. treatment planning system (TPS) dose, also obtainingLETand variableRBEdoses. We achieved good agreement between FLUKA and measurements. The range differences between FLUKA and measurements were for the PBPs within ±0.9 mm (83% ⩽ 0.5 mm), and for SOBPs ±1.6 mm (82% ⩽ 0.5 mm). The differences in modulation widths were below 5 mm (79% ⩽ 2 mm). The differences in the distal dose fall off (D80%-D20%) were below 0.5 mm for all PBPs and the lateral penumbras diverged from measurements by less than 1 mm. The mean dose difference (RBE= 1.1) in the target between the TPS and FLUKA were below 0.4% in a three-field plan and below 1.4% in a four-field plan. A dose increase of 9.9% and 7.2% occurred when using variableRBEfor the two patients, respectively. We presented a method to recalculate DS proton plans in the FLUKA MC code. The implementation was used to obtainLETand variableRBEdose and can be used for investigating variableRBEfor previously treated patients.

Analysis of measurement deviations for the patient-specific quality assurance using intensity-modulated spot-scanning particle beams

To analyze measurement deviations of patient-specific quality assurance (QA) using intensity-modulated spot-scanning particle beams, a commercial radiation dosimeter using 24 pinpoint ionization chambers was utilized. Before the clinical trial, validations of the radiation dosimeter and treatment planning system were conducted. During the clinical trial 165 measurements were performed on 36 enrolled patients. Two or three fields of particle beam were used for each patient. Measurements were typically performed with the dosimeter placed at special regions of dose distribution along depth and lateral profiles. In order to investigate the dosimeter accuracy, repeated measurements with uniform dose irradiations were also carried out. A two-step approach was proposed to analyze 24 sampling points over a 3D treatment volume. The mean value and the standard deviation of each measurement did not exceed 5% for all measurements performed on patients with various diseases. According to the defined intervention thresholds of mean deviation and the distance-to-agreement concept with a Gamma index analysis using criteria of 3.0% and 2 mm, a decision could be made regarding whether the dose distribution was acceptable for the patient. Based measurement results, deviation analysis was carried out. In this study, the dosimeter was used for dose verification and provided a safety guard to assure precise dose delivery of highly modulated particle therapy. Patient-specific QA will be investigated in future clinical operations.

Utilization of optical tracking to assess efficacy of intracranial immobilization techniques in proton therapy

We present a quantitative methodology to measure head interfraction movements within intracranial masks of commercial immobilization devices used for proton radiotherapy. A three‐points tracking (3PtTrack) method was developed to measure the mask location for each treatment field over an average of 10 fractions for seven patients. Five patients were treated in supine with the Qfix Base‐of‐Skull (BoS) headframe, and two patients were treated in prone with the CIVCO Uni‐frame baseplate. Patients were first localized by an in‐room, image‐guidance (IG) system, and then the mask location was measured using the 3PtTrack method. Measured mask displacements from initial location at the first fraction are considered equivalent to the head interfraction movement within the mask. The trends of head movements and couch displacements and rotation were analyzed in three major directions. The accuracy of 3PtTrack method was shown to be within 1.0 mm based on daily measurements of a QA device after localization by the IG system for a period of three months. For seven patients, mean values of standard deviation (SD) in anterior–posterior, lateral, and superior–inferior directions were 1.1 mm, 1.4 mm, and 1.6 mm for head movements, and were 1.4 mm, 1.8 mm, and 3.4 mm for couch displacements. The mean SD values of couch rotations were 1.1°, 0.9°, and 1.1° for yaw, pitch, and roll, respectively. The overall patterns of head movements and couch displacements were similar for patients treated in either supine or prone, with larger deviations in the superior–inferior (SI) direction. A suboptimal mask fixation to the frame of the mask to the H&N frame is likely the cause for the observed larger head movements and couch displacements in the SI direction compared to other directions. The optical‐tracking methodology provided a quantitative assessment of the magnitude of head motion.

PACS number: 87.55.km

Utilization of optical tracking to validate a software-driven isocentric approach to robotic couch movements for proton radiotherapy

PURPOSE

An optical tracking and positioning system (OTPS) was developed to validate the software-driven isocentric (SDI) approach to control the six-degrees-of-freedom movement of a robotic couch.

METHODS

The SDI approach to movements rotating around a predefined isocenter, referred to as a GeoIso, instead of a mechanical pivot point was developed by the robot automation industry. With robotic couch-sag corrections for weight load in a traditional SDI approach, movements could be accurately executed for a GeoIso located within a 500 mm cubic volume on the couch for treatments. The accuracy of SDI movement was investigated using the OTPS. The GeoIso was assumed to align with the proton beam isocenter (RadIso) for gantry at the reference angle. However, the misalignment between GeoIso and RadIso was quantitatively investigated by measuring the displacements at various couch angles for a target placed at the RadIso at an initial couch angle. When circular target displacements occur on a plane, a relative isocenter shift (RIS) correction could be applied in the SDI movement to minimize target displacements. Target displacements at a fixed gantry angle without and with RIS correction were measured for 12 robotic couches. Target displacements for various gantry angles were performed on three couches in gantry rooms to study the gantry-induced RadIso shift. The RIS correction can also be applied for the RadIso shift. A new SDI approach incorporating the RIS correction with the couch sag is described in this study. In parallel, the accuracy of SDI translation movements for various weight loads of patients on the couch was investigated during positioning of patients for proton prostate treatments.

RESULTS

For a fixed gantry angle, measured target displacements without RIS correction for couch rotations in the horizontal plane varied from 4 to 20 mm. However, measured displacements perpendicular to couch rotation plane were about 2 mm for all couches. Extracted misalignments of GeoIso and RadIso in the horizontal plane were about 10 mm for one couch and within 3 mm for the rest of couches. After applying the RIS correction, the residual target displacements for couch rotations were within 0.5 mm to RadIso for all couches. For various gantry angles, measured target location for each angle was within 0.5 mm to its excepted location by the preset RadIso shift. Measured target displacements for ± 30° of couch rotations were within 0.5 mm for gantry angles at 0° and 180°. Overall, nearly 85% of couch movements were within 0.5 mm in the horizontal plane and 0.7 mm vector distance from required displacements.

CONCLUSIONS

The authors present an optical tracking methodology to quantify for software-driven isocentric movements of robotic couches. By applying proper RIS correction for misaligned GeoIso and RadIso for each couch, and the RadIso shifts for a moving gantry, residual target displacements for isocentric couch movements around the actual RadIso can be reduced to submillimeter tolerance.

Dosimetric evaluation of a novel polymer gel dosimeter for proton therapy

PURPOSE

The aim of this study is to evaluate the dosimetric performance of a newly developed proton-sensitive polymer gel formulation for proton therapy dosimetry.

METHODS

Using passive scattered modulated and nonmodulated proton beams, the dose response of the gel was assessed. A next-generation optical CT scanner is used as the readout mechanism of the radiation-induced absorbance in the gel medium. Comparison of relative dose profiles in the gel to ion chamber profiles in water is performed. A simple and easily reproducible calibration protocol is established for routine gel batch calibrations. Relative stopping power ratio measurement of the gel medium was performed to ensure accurate water-equivalent depth dose scaling. Measured dose distributions in the gel were compared to treatment planning system for benchmark irradiations and quality of agreement is assessed using clinically relevant gamma index criteria.

RESULTS

The dosimetric response of the gel was mapped up to 600 cGy using an electron-based calibration technique. Excellent dosimetric agreement is observed between ion chamber data and gel. The most notable result of this work is the fact that this gel has no observed dose quenching in the Bragg peak region. Quantitative dose distribution comparisons to treatment planning system calculations show that most (> 97%) of the gel dose maps pass the 3%/3 mm gamma criterion.

CONCLUSIONS

This study shows that the new proton-sensitive gel dosimeter is capable of reproducing ion chamber dose data for modulated and nonmodulated Bragg peak beams with different clinical beam energies. The findings suggest that the gel dosimeter can be used as QA tool for millimeter range verification of proton beam deliveries in the dosimeter medium.

Energy spectrum control for modulated proton beams

In proton therapy delivered with range modulated beams, the energy spectrum of protons entering the delivery nozzle can affect the dose uniformity within the target region and the dose gradient around its periphery. For a cyclotron with a fixed extraction energy, a rangeshifter is used to change the energy but this produces increasing energy spreads for decreasing energies. This study investigated the magnitude of the effects of different energy spreads on dose uniformity and distal edge dose gradient and determined the limits for controlling the incident spectrum. A multilayer Faraday cup (MLFC) was calibrated against depth dose curves measured in water for nonmodulated beams with various incident spectra. Depth dose curves were measured in a water phantom and in a multilayer ionization chamber detector for modulated beams using different incident energy spreads. Some nozzle entrance energy spectra can produce unacceptable dose nonuniformities of up to +/-21% over the modulated region. For modulated beams and small beam ranges, the width of the distal penumbra can vary by a factor of 2.5. When the energy spread was controlled within the defined limits, the dose nonuniformity was less than +/-3%. To facilitate understanding of the results, the data were compared to the measured and Monte Carlo calculated data from a variable extraction energy synchrotron which has a narrow spectrum for all energies. Dose uniformity is only maintained within prescription limits when the energy spread is controlled. At low energies, a large spread can be beneficial for extending the energy range at which a single range modulator device can be used. An MLFC can be used as part of a feedback to provide specified energy spreads for different energies.

Range and modulation dependencies for proton beam dose per monitor unit calculations

Calculations of dose per monitor unit (D/MU) are required in addition to measurements to increase patient safety in the clinical practice of proton radiotherapy. As in conventional photon and electron therapy, the D/MU depends on several factors. This study focused on obtaining range and modulation dependence factors used in D/MU calculations for the double scattered proton beam line at the Midwest Proton Radiotherapy Institute. Three dependencies on range and one dependency on modulation were found. A carefully selected set of measurements was performed to discern these individual dependencies. Dependencies on range were due to: (1) the stopping power of the protons passing through the monitor chamber; (2) the reduction of proton fluence due to nuclear interactions within the patient; and (3) the variation of proton fluence passing through the monitor chamber due to different source-to-axis distances (SADs) for different beam ranges. Different SADs are produced by reconfigurations of beamline elements to provide different field sizes and ranges. The SAD effect on the D/MU varies smoothly as the beam range is varied, except at the beam range for which the first scatterers are exchanged and relocated to accommodate low and high beam ranges. A geometry factor was devised to model the SAD variation effect on the D/MU. The measured D/MU variation as a function of range can be predicted within 1% using the three modeled dependencies on range. Investigation of modulated beams showed that an analytical formula can predict the D/MU dependency as a function of modulation to within 1.5%. Special attention must be applied when measuring the D/MU dependence on modulation to avoid interplay between range and SAD effects.

Clinical characterization of a proton beam continuous uniform scanning system with dose layer stacking

A proton beam delivery system on a gantry with continuous uniform scanning and dose layer stacking at the Midwest Proton Radiotherapy Institute has been commissioned and accepted for clinical use. This paper was motivated by a lack of guidance on the testing and characterization for clinical uniform scanning systems. As such, it describes how these tasks were performed with a uniform scanning beam delivery system. This paper reports the methods used and important dosimetric characteristics of radiation fields produced by the system. The commissioning data include the transverse and longitudinal dose distributions, penumbra, and absolute dose values. Using a 208 MeV cyclotron's proton beam, the system provides field sizes up to 20 and 30 cm in diameter for proton ranges in water up to 27 and 20 cm, respectively. The dose layer stacking method allows for the flexible construction of spread-out Bragg peaks with uniform modulation of up to 15 cm in water, at typical dose rates of 1-3 Gy/min. For measuring relative dose distributions, multielement ion chamber arrays, small-volume ion chambers, and radiographic films were employed. Measurements during the clinical commissioning of the system have shown that the lateral and longitudinal dose uniformity of 2.5% or better can be achieved for all clinically important field sizes and ranges. The measured transverse penumbra widths offer a slight improvement in comparison to those achieved with a double scattering beam spreading technique at the facility. Absolute dose measurements were done using calibrated ion chambers, thermoluminescent and alanine detectors. Dose intercomparisons conducted using various types of detectors traceable to a national standards laboratory indicate that the measured dosimetry data agree with each other within 5%.

Neutron scattered dose equivalent to a fetus from proton radiotherapy of the mother

Scattered neutron dose equivalent to a representative point for a fetus is evaluated in an anthropomorphic phantom of the mother undergoing proton radiotherapy. The effect on scattered neutron dose equivalent to the fetus of changing the incident proton beam energy, aperture size, beam location, and air gap between the beam delivery snout and skin was studied for both a small field snout and a large field snout. Measurements of the fetus scattered neutron dose equivalent were made by placing a neutron bubble detector 10 cm below the umbilicus of an anthropomorphic Rando phantom enhanced by a wax bolus to simulate a second trimester pregnancy. The neutron dose equivalent in milliSieverts (mSv) per proton treatment Gray increased with incident proton energy and decreased with aperture size, distance of the fetus representative point from the field edge, and increasing air gap. Neutron dose equivalent to the fetus varied from 0.025 to 0.450 mSv per proton Gray for the small field snout and from 0.097 to 0.871 mSv per proton Gray for the large field snout. There is likely to be no excess risk to the fetus of severe mental retardation for a typical proton treatment of 80 Gray to the mother since the scattered neutron dose to the fetus of 69.7 mSv is well below the lower confidence limit for the threshold of 300 mGy observed for the occurrence of severe mental retardation in prenatally exposed Japanese atomic bomb survivors. However, based on the linear no threshold hypothesis, and this same typical treatment for the mother, the excess risk to the fetus of radiation induced cancer death in the first 10 years of life is 17.4 per 10,000 children.

Limited accuracy of dose calculation for large fields at deep depths using the BrainSCAN v5.21 treatment planning system

The Varian 120 multileaf collimator (MLC) has a leaf thickness of 5 mm projected at the isocenter plane and can deliver a radiation beam of large field size (up to 30 cm) to be used in intensity‐modulated radiotherapy (IMRT). Often the dose must be delivered to depths greater than 20 cm. Therefore, during the commissioning of the BrainSCAN v5.21 or any radiation treatment‐planning (RTP) systems, extensive testing of dose and monitor unit calculations must encompass the field sizes (1 cm to 30 cm) and the prescription depths (1 cm to 20 cm). Accordingly, the central‐axis percent depth doses (PDDs) and off‐axis percentage profiles must be measured at several depths for various field sizes. The data for this study were acquired with a 6‐MV X‐ray beam from a Varian 2100EX LINAC with a water phantom at a source‐to‐surface distance (SSD) of 100 cm. These measurements were also used to generate a photon beam module, based on a photon pencil beam dose‐calculation algorithm with a fast‐Fourier transform method. To commission the photon beam module used in our BrainSCAN RTP system, we performed a quantitative comparison of measured and calculated central‐axis depth doses and off‐axis profiles. Utilizing the principles of dose difference and distance‐to‐agreement introduced by Van Dyk et al. [Commissioning and quality assurance of treatment planning computers. Int J Radiat Oncol Biol Phys. 1993; 26:261—273], agreements between calculated and measured doses are <2% and <2 mm for the regions of low‐ and high‐dose gradients, respectively. However, large errors (up to ~5% and ~7% for 20‐cm and 30‐cm fields, respectively, at the depth 20 cm) were observed for monitor unit calculations. For a given field size, the disagreement increased with the depth. Similarly, for a given depth the disagreement also increased with the field size. These large systematic errors were caused by using the tissue maximum ratio (TMR) in BrainSCAN v5.21 without considering increased field size as depth increased. These errors have been reported to BrainLAB.

PACS number: 87.53.‐j

A feasibility study of Dynamic Phantom scanner for quality assurance of photon beam profiles at various gantry angles

The effect of gantry rotation on beam profiles of photon and electron beams is an important issue in quality assurance for radiotherapy. To address variations in the profiles of photon and electron beams at different gantry angles, a Dynamic Phantom scanner composed of a 20×12×6 cm3 scanning Lucite block was designed as a cross‐beam‐profile scanner. To our knowledge, differences between scanned profiles acquired at different gantry angles with a small size Lucite block and those acquired a full‐size (60×60×50 cm3) water phantom have not been previously investigated. We therefore performed a feasibility study for a first prototype Dynamic Phantom scanner without a gantry attachment mount. Radiation beams from a Varian LINAC 21EX and 2100C were used. Photon beams (6 MV and 18 MV) were shaped by either collimator jaws or a Varian 120 Multileaf (MLC) collimator, and electron beams (6 MeV, 12 MeV, and 20 MeV) were shaped by a treatment cone. To investigate the effect on profiles by using a Lucite block, a quantitative comparison of scanned profiles with the Dynamic Phantom and a full‐size water phantom was first performed at a 0° gantry angle for both photon and electron beams. For photon beam profiles defined by jaws at 1.0 cm and 5.0 cm depths of Lucite (i.e., at 1.1 cm and 5.7 cm depth of water), a good agreement (less than 1% variation) inside the field edge was observed between profiles scanned with the Dynamic Phantom and with a water phantom. The use of Lucite in the Dynamic Phantom resulted in reduced penumbra width (about 0.5 mm out of 5 mm to 8 mm) and reduced (1% to 2%) scatter dose beyond the field edges for both 6 MV and 18 MV beams, compared with the water phantom scanner. For profiles of the MLC‐shaped 6 MV photon beam, a similar agreement was observed. For profiles of electron beams scanned at 2.9 cm depth of Lucite (i.e., at 3.3 cm depth of water), larger disagreements in profiles (3% to 4%) and penumbra width (3 mm to 4 mm out of 12 mm) were observed. Additional profiles with the gantry at 90° and 270° were performed for both MLC‐ and jaw‐shaped photon beams and electron beams to evaluate the effect of gantry rotation. General good agreement is seen (less than 1 % variation) at all field sizes for collimator‐shaped 6 MV and 18 MV photon beams. Similar variations observed for MLC‐shaped photon beams indicate that the uncertainty in MLC position is similar to that for the collimator jaws. We conclude that the Dynamic Phantom scanner is a useful device for the routine quality assurance on beam profiles of photon beams and for constancy check on electron beams at various gantry angles. Caution should be taken when using this device to acquire basic electron dosimetry data.

PACS number: 87.53.‐j

Dosimetric characteristics of the Leipzig surface applicators used in the high dose rate brachy radiotherapy

The nucletron Leipzig applicator is designed for (HDR) 192Ir brachy radiotherapy of surface lesions. The dosimetric characteristics of this applicator were investigated using simulation method based on Monte Carlo N-particle (MCNP) code and phantom measurements. The simulation method was validated by comparing calculated dose rate distributions of nucletron microSelectron HDR 192Ir source against published data. Radiochromic films and metal-oxide-semiconductor field-effect transistor (MOSFET) detectors were used for phantom measurements. The double exposure technique, correcting the nonuniform film sensitivity, was applied in the film dosimetry. The linear fit of multiple readings with different irradiation times performed for each MOSFET detector measurement was used to obtain the dose rate of each measurement and to correct the source transit-time error. The film and MOSFET measurements have uncertainties of 3%-7% and 3%-5%, respectively. The dose rate distributions of the Leipzig applicator with 30 mm opening calculated by the validated MC method were verified by measurements of film and MOSFET detectors. Calculated two-dimensional planar dose rate distributions show similar patterns as the film measurement. MC calculated dose rate at a reference point defined at depth 5 mm on the applicator's central axis is 7% lower than the film and 3% higher than the MOSFET measurements. The dose rate of a Leipzig applicator with 30 mm opening at reference point is 0.241+/-3% cGy h(-1) U(-1). The MC calculated depth dose rates and profiles were tabulated for clinic use.

Dose perturbation induced by radiographic contrast inside brachytherapy balloon applicators

Phantom measurements and Monte Carlo calculations have been performed for the purpose of characterizing the dose perturbation caused by radiographic contrast inside the MammoSite breast brachytherapy applicator. Specifically, the dose perturbation is quantified as a heterogeneity correction factor (HCF) for various balloon radii and contrast concentration levels. The dose perturbation is larger for larger balloon radii and higher contrast concentrations. Based on a validated Monte Carlo simulation, the calculated HCF values are 0.99 for a 2 cm radius balloon and 0.98 for a 3 cm radius balloon at 6% contrast concentration levels, and 0.89 and 0.87 for 2 and 3 cm radius balloons, respectively, at 100% contrast concentrations. For a typical implanted balloon radius of 2.4 cm, the HCF values decrease from 0.99 at 6% contrast concentration to 0.90 at 100% contrast concentration. For balloons implanted in patients at our institution, the mean HCF is 0.99, corresponding to a dose reduction of approximately 1%. The contrast effect results in a systematic reduction in the delivered dose, therefore the minimal amount of radiographic contrast necessary should be used.

Modulation of radiotherapy photon beam intensity using magnetic field

The purpose of this study was to explore the potential advantages of using strong magnetic fields to increase tumor dose and to decrease normal tissue dose in radiation therapy. Strong magnetic fields are capable of altering the trajectories of charged particles. A magnetic field applied perpendicularly to the X-ray beam forces the secondary electrons and positrons to spiral and produces a dose peak. The same magnetic field also prevents the electrons and positrons from traveling downstream and produces a lower dose region distal to the dose peak. The locations of these high- and low-dose regions are potentially adjustable to enhance the dose to the target volume and decrease the dose to normal tissues. We studied this effect using the Monte Carlo simulation technique. The EGS4 code was used to simulate the effect produced by a coil magnet currently under construction. The coil magnet is designed to support up to 350 A operating current and 15 T peak field on windings. Dose calculations in a water phantom show that the transverse magnetic field produces significant dose effects along the beam direction of radiation therapy X-rays. Depending on the beam orientation, the radiation dose at different depths along the beam can be increased or reduced. This dose effect varies with photon energy, field size, magnetic field strength, and relative magnet/beam geometry. The off-axis beam profiles also show considerable skewness under the influence of the magnetic field. The magnetic field-induced dose shift may result in high dose regions outside the geometrical boundary of the initial radiation beam. We have demonstrated that current or near-term magnet technology is capable of producing significant dose enhancement and reduction in radiation therapy photon beams. This technology should be further developed to improve our ability to deliver higher doses to the tumor and lower doses to normal tissues in radiation therapy.

Effects of overlapping parametric resonances on the particle diffusion process

The evolution of the beam distribution in a double-rf system with a phase modulation on either the primary or secondary rf cavity was measured. We find that the particle diffusion process obeys the Einstein relation if the phase space becomes globally chaotic. When dominant parametric resonances still exist in the phase space, particles stream along the separatrices of the dominant resonance, and the beam width exhibits characteristic oscillatory structure. The particle-tracking simulations for the double-rf system are employed to reveal the essential diffusion mechanism. Coherent octupolar motion has been observed in the bunch beam excitation. The evolution of the longitudinal phase space in the octupole mode is displayed.