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GPUMCD: a new GPU-oriented Monte Carlo dose calculation platform

Sami Hissoiny, Hugo Bouchard, Benoît Ozell, Philippe Després

arXiv:1101.1245v1physics.med-ph

TL;DR

Monte Carlo dose calculation offers high dosimetric accuracy but remains hindered in clinical radiotherapy by execution time. GPUMCD is a fully coupled GPU-oriented photon-electron Monte Carlo platform, and it achieved close agreement with EGSnrc alongside subsecond simulations and large speedups over EGSnrc and DPM.

  • Problem

    Monte Carlo methods are considered the gold standard for radiotherapy dosimetry, but their execution time limits routine clinical use.

  • Method

    GPUMCD implements a fully coupled photon-electron Monte Carlo dose calculation platform designed for GPU execution.

  • Results

    At least 98% of significant voxels passed the 2%-2mm gamma criteria in all but one case, while GPUMCD completed tested simulations in less than 0.3 s.

  • Takeaways & Limitations

    The reported accuracy and execution speed make GPUMCD an interesting solution for full Monte Carlo dose calculation in radiation oncology.

Abstract

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Purpose: Monte Carlo methods are considered the gold standard for dosimetric computations in radiotherapy. Their execution time is however still an obstacle to the routine use of Monte Carlo packages in a clinical setting. To address this problem, a completely new, and designed from the ground up for the GPU, Monte Carlo dose calculation package for voxelized geometries is proposed: GPUMCD. Method : GPUMCD implements a coupled photon-electron Monte Carlo simulation for energies in the range 0.01 MeV to 20 MeV. An analogue simulation of photon interactions is used and a Class II condensed history method has been implemented for the simulation of electrons. A new GPU random number generator, some divergence reduction methods as well as other optimization strategies are also described. GPUMCD was run on a NVIDIA GTX480 while single threaded implementations of EGSnrc and DPM were run on an Intel Core i7 860. Results : Dosimetric results obtained with GPUMCD were compared to EGSnrc. In all but one test case, 98% or more of all significant voxels passed a gamma criteria of 2%-2mm. In terms of execution speed and efficiency, GPUMCD is more than 900 times faster than EGSnrc and more than 200 times faster than DPM, a Monte Carlo package aiming fast executions. Absolute execution times of less than 0.3 s are found for the simulation of 1M electrons and 4M photons in water for monoenergetic beams of 15 MeV, including GPU-CPU memory transfers. Conclusion : GPUMCD, a new GPU-oriented Monte Carlo dose calculation platform, has been compared to EGSnrc and DPM in terms of dosimetric results and execution speed. Its accuracy and speed make it an interesting solution for full Monte Carlo dose calculation in radiation oncology.

I. INTRODUCTION 14

Monte Carlo methods provide high dosimetric accuracy but are limited in routine radiotherapy by computation time. GPUMCD addresses this constraint with a coupled photon-electron Monte Carlo code rewritten specifically for GPU execution.

  • Motivation: Long computation times required for sufficient statistical significance have limited routine Monte Carlo use in radiation therapy.
  • Motivation: Monte Carlo methods are considered the gold standard for benchmarking analytical dose-calculation approaches.
  • Motivation: Semi-empirical methods trade accuracy for computation time and can differ from Monte Carlo results by up to 10% in complex heterogeneous geometries.
  • Prior fast approaches: Fast Monte Carlo packages improve efficiency by sacrificing some generality and absolute accuracy, including through restricted incoming-particle energy ranges.
  • GPUMCD: GPUMCD is a complete rewrite of a coupled photon-electron Monte Carlo code designed specifically for GPU execution.
  • GPUMCD: The implementation introduces particle-memory management techniques and methods to reduce Monte Carlo divergence during GPU execution.

II. MATERIAL AND METHODS 65

GPUMCD models photon transport with analogue interactions across 10 keV–20 MeV and uses Woodcock raytracing to handle heterogeneous voxelized geometries without a separate geometry engine.

  • Photon transport: GPUMCD simulates Compton scattering, the photoelectric effect, and pair production; Rayleigh scattering is ignored from 10 keV to 20 MeV.Photon interactions are modeled with analogue transport.
  • Heterogeneous geometry: Woodcock raytracing replaces heterogeneous-material tracking with a fictitious attenuation interaction that leaves particle direction and energy unchanged.The volume is treated as homogeneously attenuating using the maximum voxel attenuation coefficient.
  • Heterogeneous geometry: The next interaction distance is sampled using µmax, while interaction type is selected at the interaction position including fictitious events.Sampling continues until the particle exits the geometry or falls below the transport energy threshold.
  • Heterogeneous geometry: Woodcock raytracing produces the correct result even in arbitrarily heterogeneous geometries.The passage explicitly states that this method is not an approximation.
  • Interaction models: Compton scattering uses a free-electron Klein–Nishina model, while binding effects and atomic relaxation are omitted.The approximation is considered reasonable when particle energy is large compared with electron binding energy.
  • Interaction models: The photoelectric model omits atomic relaxation and shell sampling, ejecting electrons from the K-shell with Eelec = Ephot.Accuracy becomes material-dependent near characteristic X-ray energies because characteristic X-rays are not produced.

B. Electron simulation

GPUMCD transports electrons with a Class II condensed history method: soft interactions are grouped into steps, while hard events are simulated explicitly.

  • Electron transport: Analogue electron simulation is impractical because charged particles undergo many interactions before depositing all their energy.Soft collisions are regrouped between hard collision or bremsstrahlung events.
  • Electron transport: The Class II condensed history method explicitly simulates interactions that cause orientation changes greater than θc or energy changes larger than Ec.Interactions below these thresholds are modeled collectively.
  • Hard interactions: Above thresholds, inelastic collisions and bremsstrahlung are simulated analogously to photon interactions.The method therefore combines condensed treatment of soft processes with analogue treatment of hard events.
  • Step selection: Electron step length is the minimum of the next voxel boundary, a user-supplied maximum, and the next hard interaction.The step selection is es = min{dvox, es−max, ehard}.
  • Energy loss: Energy loss during multiple-scattering steps is modeled with the continuous slowing down approximation and restricted stopping powers.Only sub-threshold interactions are included in the CSDA treatment.
  • Multiple scattering: The random hinge method models electron angular deflection and lateral displacement within a step by rotating the electron at a sampled intermediate point.Bielajew’s alternate random hinge refinement samples the deflection before or after energy deposition.

C. GPU implementation

GPUMCD is implemented around GPU-specific random-number generation, particle-array management, voxel-data caching, and divergence-reduction strategies.

  • Random number generation: A lightweight multiply-with-carry PRNG was implemented because the NVIDIA SDK generator uses substantial resources.The generator uses few registers and requires ten integer operations per number.
  • Random number generation: The PRNG passed all but one TestU01 test and achieved 96.7% of peak integer bandwidth.These results characterize its statistical testing and hardware throughput.
  • Memory management: GPUMCD stores electrons and photons in separate arrays, alongside voxel composition identifiers and density values.Particle arrays are allocated once with a graphics-card-dependent MAXSIZE.
  • Memory management: Secondary particles are added through atomic counters and are discarded only after the allocated MAXSIZE is reached.If generated particles remain below MAXSIZE, none are discarded.
  • Divergence reduction: Photons and electrons are simulated in separate phases, eliminating divergence caused by photon–electron coupling.Particles of the other type wait in their corresponding array until the current type is exhausted.
  • GPU memory: Voxel composition and density are stored as 3D textures, while cross sections and stopping powers use a preinterpolated 1-keV grid.Texture storage supports caching for three-dimensional data locality.
  • Divergence reduction: GPU SIMD divergence arises because stochastic particle paths differ, splitting warps into subwarps and reducing performance.The smallest parallel unit is a 32-thread warp.

D. Performance evaluation

Performance was evaluated through dosimetric comparison with EGSnrc and efficiency comparison with EGSnrc and DPM under matched simulation conditions.

  • Evaluation design: The evaluation compares GPUMCD dose distributions with EGSnrc and execution efficiency with EGSnrc and DPM.DPM was included because it was designed for fast Monte Carlo dose calculations.
  • Evaluation design: All platforms used a 643 grid with 0.5 cm^3 voxels, although this resolution was considered insufficient for patient-specific clinical calculations.The setup was judged adequate for benchmarking.
  • Results: Preliminary results indicated no efficiency loss for GPUMCD as the number of voxels increased.This observation was reported for the tested scaling condition.
  • Evaluation design: Dose distributions were normalized to Dmax, and 4 million photon or 1 million electron particles were used for execution-time comparisons.Sources were monoenergetic parallel beams.
  • Metrics: Efficiency was defined using statistical uncertainty and computation time, considering voxels with dose above 0.2·Dmax.The measure uses s and T and applies the stated dose threshold.
  • Metrics: The gamma criterion considered a calculation acceptable when the gamma index was at most 1.0.The gamma comparison uses dose and distance tolerance values.
  • Hardware: Benchmarking used an Intel Core i7 860 and NVIDIA GeForce GTX480, with GPUMCD restricted to one processor core unless using multiple GPUs.DPM and DOSXYZnrc were not modified for multicore architectures.

III. RESULTS 358

The results evaluate GPUMCD using several slab-geometry phantoms, report execution times and efficiency gains, and test a multi-GPU implementation.

  • III. RESULTS: Three slab-geometry phantoms are used to evaluate GPUMCD.The study also reports execution times, efficiency gains, and multi-GPU acceleration.

A. Dosimetric results

Dosimetric evaluation compares GPUMCD with EGSnrc across homogeneous and heterogeneous slab phantoms using PDD, profiles, isodose curves, and gamma analysis. The gamma criterion is 2% and 2 mm, with most comparisons showing close agreement and larger electron-beam discrepancies in the high-Z geometry.

  • Evaluation procedure: PDD curves and dose profiles are evaluated for slab-geometry phantoms, while gamma indices summarize dose differences.Gamma statistics include maximum and average values and proportions of significant voxels exceeding selected thresholds.
  • Evaluation procedure: The gamma criterion is set to 2% and 2 mm, described as generally accepted for clinical dose calculation.The summaries report gamma values for voxels with dose above a specified fraction of the maximum dose.
  • Water phantom: Homogeneous-phantom results show no efficiency gain because GPUMCD transports particles as though the medium were heterogeneous.Electron- and photon-beam results for the water phantom are presented with gamma comparisons.
  • Water-lung phantom: The water-lung geometry tests GPUMCD under lateral and longitudinal disequilibrium conditions using distinct lung-box dimensions for electron and photon beams.PDD and isodose results are presented for both beam types.
  • Tissue-lung-bone phantom: The tissue-lung-bone geometry uses different slab arrangements for electron and photon beams to test transport through multiple materials.Its PDD and profile results are accompanied by gamma results.

B. Execution time and efficiency gain

Execution-time and efficiency experiments compare GPUMCD with EGSnrc and DPM across three phantom geometries, quantify optimization effects, and test dual-GPU execution. The reported timings include memory transfers and reach fractions of a second for the stated 15 MeV workloads.

  • Comparison setup: Execution times and efficiency gains are evaluated for GPUMCD against EGSnrc and DPM in three geometries.Geom1 is water, Geom2 is water-lung, and Geom3 is tissue-lung-bone.
  • Workloads: GPUMCD timing and acceleration measurements simulate 1M 15 MeV electrons and 4M 15 MeV photons.The absolute execution times include the stated GPU workload comparisons.
  • Optimization strategies: Stream compaction and pool divergence-reduction methods are evaluated as acceleration strategies relative to the base configuration.The corresponding acceleration factors are defined with respect to base execution time.
  • Absolute execution times: 0.269 s, 0.275 s, and 0.366 s are reported for photon beams in Geom1, Geom2, and Geom3, respectively.Electron-beam times are 0.118 s, 0.147 s, and 0.162 s for the same geometries.
  • Multi-GPU execution: A dual-GPU configuration is tested using execution time per complete history, with reported times including transfers to and from the GPU.Single- and dual-GPU execution times are compared using Tesla C1060 cards.

A. Dosimetric evaluation

GPUMCD shows close dosimetric agreement with EGSnrc across homogeneous and heterogeneous phantoms, while discrepancies remain localized mainly to buildup regions and the high-Z electron-beam case. Nearly all significant voxels satisfy the 2%-2 mm gamma criterion.

  • Localized differences: Differences of less than 2% occur near the end of the buildup region, without affecting the remainder of the curves or particle ranges.This pattern is reported for both photon and electron beams.
  • Low-density heterogeneity: In low-density heterogeneous simulations, electron-beam buildup differences reach 2% but do not affect particle range, while photon dose agrees well near the water-lung interface.The photon result is reported for the water-lung phantom.
  • High-Z geometry: In the high-Z heterogeneous simulation, electron-beam differences reach 2%, while photon PDD differences remain below 1% and profile differences reach 1.5%.These values describe the presented PDD and profile comparisons.
  • Gamma agreement: At most 2% of significant voxels fail the gamma test for electron beams in the first two geometries.The agreement is described as excellent for homogeneous phantoms and the low-density heterogeneous geometry.
  • Gamma agreement: 98% or more of significant voxels meet the gamma criteria for photon beams in all tested geometries.The reported criterion is 2% dose difference and 2 mm distance-to-agreement.

B. Execution times

GPUMCD achieves sub-second execution times and large acceleration over CPU-based Monte Carlo codes, although performance depends on beam type, geometry, and transport complexity.

  • Limitations: Monoenergetic test beams may favor GPU implementations, while polyenergetic beams could introduce additional stream divergence.
  • Absolute execution times: Less than 0.2 s is required for all tested electron-beam cases.
  • Absolute execution times: Less than 0.3 s is required for the first two photon-beam simulations, whereas the bone-slab simulation exceeds 0.36 s.
  • Acceleration: 203x is the minimum acceleration over DPM, reaching 210x for electron beams.
  • Acceleration: More than 1200x acceleration is observed over EGSnrc for electron beams and more than 940x for photon beams.
  • Optimization effects: Electron beams gain up to 26% from divergence-reduction methods and optimizations, while photon-beam acceleration factors can fall below 1.
  • Multi-GPU scaling: GPUMCD scales linearly with the number of GPUs, but its acceleration factor decreases as the number of simulated histories is reduced.
  • Limitations: The comparisons exclude variance-reduction techniques, whose uncertain GPU applicability could impose a performance penalty relative to VRT-enabled codes.

V. CONCLUSION AND FUTURE WORK

GPUMCD is a fully coupled GPU-oriented Monte Carlo dose calculation platform evaluated against established codes for accuracy and speed. It generally agrees with EGSnrc while delivering very short calculation times, though its current scope excludes several planned capabilities.

  • GPUMCD is a new fully coupled GPU-oriented Monte Carlo dose calculation platform.
  • The platform was evaluated against EGSnrc for accuracy and compared with DPM for execution speed.
  • 98% or more of significant voxels passed the 2%-2mm gamma criteria in all but one case; the exception achieved 92%.
  • The authors suggest that GPUMCD is suitable for clinical use and an interesting solution for full Monte Carlo dose calculation in radiation oncology.
  • Less than 0.12 s is required for a 15 MeV electron-beam calculation in water with 1M histories, and less than 0.27 s for 4M photon histories.
  • Future work: Future work includes phase-space and spectrum files, CT-based phantoms, and GPU-compatible variance-reduction techniques.
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