Efficiency improvement in proton dose calculations with an - PowerPoint PPT Presentation

Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism Daniel Maneval 1 , 2 Hugo Bouchard 3 ıt Ozell 4 es 1 , 2 Benoˆ Philippe Despr´ October 16, 2017 1 Universit´ e Laval, D´ epartement de physique, de g´ enie physique et d’optique, Qu´ ebec, Canada 2 Universit´ e Laval, D´ epartement de radiooncologie et centre de recherche du CHU de Qu´ ebec, Qu´ ebec, Canada 3 Universit´ e de Montr´ eal, D´ epartement de physique, Montr´ eal, Canada 4 ´ Ecole polytechnique de Montr´ eal, D´ epartement de g´ enie informatique et g´ enie logiciel, Montr´ eal, Canada

The proton energy loss along a step length Electromagnetic inelastic interactions with atomic electrons • Split in soft and hard collisions. Soft collisions Hard collisions Ionization ranges short (sub-cutoff) significant Transport Deterministic Monte Carlo Physical quantities Stopping power Production cross section Simulation scheme Condensed Analog The proton Continuous Slowing Down Approximation (CSDA) • An approximate variance reduction technique to compute the mean proton energy loss (∆ E ) • The true energy loss (∆ E true ) is obtained with the energy straggling ( δ ∆ E ): ∆ E true = ∆ E ± δ ∆ E 1

The mean energy loss ( ∆ E ) calculations Two schemes EGSnrc for electrons/positons (Kawrakow, 2000; Kawrakow et al., 2016) and Geant4 for all charged particles (Collaboration, 2015). A new one proposed named the equivalent restricted stopping power formalism: the L eq formalism 2

The ∆ E calculations EGSnrc Geant4 formalism Fetch the range Compute the step restriction Fetch the fractional with a T aylor expansion energy loss Compute the stepping and a maximal fractional function energy loss Fetch the equivalent Determine the step length stopping power with 2 step restrictions Determine the step length with the step restriction Fetch the stopping power Fetch the stopping power Compute the mean energy loss Compute with a midpoint rule < 1% of E yes No Accept Compute with an inverse range table 3

L eq formalism benefit Step length hard proton trajectory Interface restriction collision High dose accuracy simulations: Balanced simulations: accuracy - computation time: High accuracy with the formalism: Geant4 or EGSnrc The L eq formalism s = min ( d vox , d hard , d max ) s = min ( d vox , d hard ) 4

The L eq formalism L is the restricted/unrestricted stopping power. E is the proton kinetic energy The distance ( s ) space The energy ( ∆ E ) space d ∆ E d ∆ E = L − 1 ( E − ∆ E ) ds = L ( s ) ds � s � E − ∆ E L ( E ′ ) d s ′ ⇔ ∆ E = L − 1 ( E ′ ) d E ′ ⇔ s = − 0 E Stopping power The midpoint rule of the Newton-Cotes formula: ⇒ ∆ E = s · L eq ( E , ǫ ) L ( x ) L eq ( E , x ) = � 1+ 2 L ′ ( x )2 − L ( x ) L ′′ ( x ) E 2+ x ( x − 2 E ) � L ( x )2 6 � � and ǫ ( s , E ) = ∆ E x = E 1 − ǫ 2 E ǫ links the ∆ E space with the s space. Step 5

L eq Look-up tables L ( x ) � with x = E � � L eq ( E , x ) = 1 − ǫ 1+ 2 L ′ ( x )2 − L ( x ) L ′′ ( x ) E 2+ x ( x − 2 E ) � 2 L ( x )2 6 Material Fixed ǫ Fixed Energies ǫ = 0 % 0.5 MeV ǫ = 10 % 1.5 MeV L eq (MeV · cm − 1 ) ǫ = 25 % 800 5 MeV ǫ = 50 % 15 MeV 10 2 ǫ = 70 % 50 MeV 600 ǫ = 100 % · 350 MeV Water 400 10 1 200 0 0 20 40 60 80 100 10 0 10 1 10 2 ǫ (%) Energy (MeV) ǫ = 0 % 0.5 MeV ǫ = 10 % 1.5 MeV L eq (MeV · cm − 1 ) 3000 ǫ = 25 % 10 3 5 MeV ǫ = 50 % 15 MeV 2500 ǫ = 70 % 50 MeV · ǫ = 100 % 350 MeV 2000 Gold 1500 1000 10 2 500 0 0 20 40 60 80 100 10 0 10 1 10 2 6 ǫ (%) Energy (MeV)

✂ ✂ ǫ look-up tables � E (1 − ǫ ) ǫ ( s , E ) = ∆ E dE ′ determined when solving s + L ( E ′ ) = 0 E E Material Fixed Step lengths Fixed Energies 10 2 10 2 10 1 10 1 10 1 10 1 (%) (%) 10 3 3 10 Water 5 5 10 10 7 10 7 10 10 5 4 3 2 1 0 10 0 10 1 10 2 Energy (M eV) 10 2 10 2 10 1 10 1 10 0 10 0 10 ✁ 1 ✁ 1 10 (%) (%) 10 ✁ 2 10 ✁ 2 Gold 1.0 mm MeV 10 ✁ 3 10 ✁ 3 0.5 M eV 0.1 mm MeV 1.5 M eV ✁ 4 ✁ 2 mm ✁ 4 10 10 10 MeV 5 M eV ✁ 3 mm 10 MeV ✁ 5 ✁ 5 15 M eV 10 10 ✁ 4 mm (c) Gold (d) Gold MeV 10 50 M eV ✁ 6 10 ✁ 6 10 ✁ 5 mm MeV 350 M eV 10 ✁ 7 10 ✁ 7 10 ✁ 5 ✁ 4 ✁ 3 ✁ 2 ✁ 1 10 0 10 1 10 2 10 10 10 10 10 10 0 Energy (MeV) Step (mm) Energy (M eV) Step (mm) 7

Geant4 setups Simulation setups Geant4 CSDA parameters Time (h) Errors (%) d max Linear loss Step function Maximum Falloff Reference 1 µ m 1% (20%, 50 µ m) 140 - - High Accuracy (HA) 10 µ m 1% (20%, 50 µ m) 14 0.2 0.9 Balanced (Bal) 1 mm 0.1% (0.1%, 1 µ m) 2.5 0.8 4.7 Default 1 mm 1% (20%, 50 µ m) 0.8 4.8 16.5 20 Reference Bragg peak Default Dose (Gy) Balanced Maximum step (mm) High Accuracy 10 0 10 Reference 10 − 1 0 16 Error (%) 12 Default 10 − 2 8 Balanced High Accuracy 4 Reference 1 10 − 3 305 310 315 320 325 330 0 50 100 150 200 250 300 Depth (mm) Depth (mm) 8

GPU-based Implementations Graphic processor Unit (GPU) • pGPUMCD : a new GPUMCD branch dedicated to proton MC transport • GPUMCD : a validated GPU-based MC dose calculation code for photons and electrons (Hissoiny et al., 2011c) Efficiency L eq formalism intrinsic efficiency: • pGPUMCD - L : the Geant4 CSDA scheme • pGPUMCD - L eq Material ρ n e ( × 10 23 ) I T min e g . cm − 3 cm − 3 eV MeV Lung ∗ 0.26 0.86189 69.69 0.148 Water 1.0 3.3428 78 0.352 Bone ∗∗ 1.85 5.9056 91.9 0.512 Copper 8.96 24.625 322 1.4 Gold 19.32 46.665 790 2.3 ∗ ICRU inflated lung (ICRU, 1992) ∗∗ Bone, Compact (ICRU) (Berger et al., 2005) 9

L eq formalism validation 0.75 0.75 0.75 Geant4 100 Geant4 Geant4 (b) Water 100 MeV 60 (c) Water 230 MeV 150 pGPUMCD 0.50 pGPUM CD pGPUM CD 0.50 0.50 80 50 Error Error (%) Error (%) Dose (Gy) Error Error Dose (Gy) Error (%) rror (%) Dose (Gy) 0.25 0.25 0.25 40 100 60 0.00 0.00 0.00 30 40 (a) Lung 70 MeV ✁ 0.25 ✁ 0.25 0 25 20 50 ✁ 0.50 20 ✁ 0.50 0 50 10 ✁ 0.75 ✁ 0.75 0 75 0 0 0 0 50 100 150 0 20 40 60 80 0 100 200 300 Depth (mm) Depth (mm) Depth (mm) 0.75 0.75 0.75 (e) Copper 230 MeV 20 (d) Bone 230 MeV (d) Gold 230MeV 60 25 0.50 0.50 0.50 50 Error (%) Error (%) Dose (Gy) Dose (Gy) Dose (Gy) Error (%) Dose (Gy) Error (%) 20 0.25 15 0.25 0.25 40 0.00 15 0.00 0.00 10 30 ✁ 0.25 10 ✁ 0.25 ✁ 0.25 20 5 ✁ 0.50 ✁ 0.50 ✁ 0.50 5 10 ✁ 0.75 ✁ 0.75 ✁ 0.75 0 0 0 0 50 100 150 200 0 20 40 60 0 10 20 30 Depth (mm) Depth (mm) Depth (mm) Depth (mm) Depth (mm) Depth (mm) The ranges ( R 80 ) matched within 1 µ m 10

L eq formalism efficiency Geant4 pGPUMCD - L pGPUMCD - L eq Intrinsic speed up factors Material Energy T HA T Bal T HA T Bal T L eq GPU ( T G4 / T L ) L eq ( T L / T L eq ) G4 G4 L L (MeV) ( hour ) ( second ) ( millisecond ) HA Bal HA Bal Lung 70 6.5 2 12.25 2.85 70 1,900 2,500 175 41 Water 100 3.5 1.9 4.73 2.12 46 2,600 3,200 103 46 Water 230 14 2.5 91 4.34 145 550 2,100 630 30 Bone 230 8 2.3 17.62 3.05 84 1,600 2,700 210 36 Copper 230 2.5 1.6 4.11 1.64 40 2100 3,500 103 41 Gold 230 1.7 1.4 3.23 1.30 31 1900 3,800 103 42 • Geant4/EGSnrc: linear algorithmic time complexity, i.e. O ( n ) where n represents the number of subdivision in a voxel to maintain the mean energy loss accuracy. n is fixed by d max . • the L eq formalism: constant algorithmic time complexity, i.e. O (1) 11

L eq formalism with the energy straggling Water 100 MeV Water 230 MeV Copper 10 2 Gold Lung Dose (Gy) Bone 10 1 Geant4 10 0 pGPUMCD 2 Error (%) 1 0 − 1 − 2 0 50 100 150 200 250 300 Depth (mm) The ranges ( R 80 ) matched within 100 µ m 12

Computation times Geant4: 1.4 to 20 hours per million transported protons pGPUMCD: 31 to 173 milliseconds per million transported protons 13

Conclusion The L eq formalism led to an intrinsic efficiency gain factor ranging between 30-630, increasing with the prescribed accuracy of simulations. It allows larger steps leading to a constant algorithmic time complexity . It significantly accelerates Monte Carlo proton transport while preserving accuracy . The L eq formalism constitutes a promising variance reduction technique for computing proton dose distributions in a clinical context. The L eq formalism could be used for other charged particles. The multiple scattering was validated, not presented here. Under investigations: nuclear interactions More details concerning the L eq formalism: (Maneval et al., 2017) 14

Acknowledgements 15

References I References Martin J. Berger, JS Coursey, M.A. Zucker, and J Chang. Estar, pstar, and astar: Computer programs for calculating stopping-power and range tables for electrons, protons, and helium ions (version 1.2.3). Technical report, National Institute of Standards and Technology, Gaithersburg MD, 2005. URL http://physics.nist.gov/Star . Geant4 Collaboration. Physics reference manual. obtainable from the GEANT4 website: http://geant4. cern , 2015. Sami Hissoiny, Benoˆ ıt Ozell, Hugo Bouchard, and Philippe Despr´ es. Gpumcd: A new gpu-oriented monte carlo dose calculation platform. Medical Physics , 38(2):754–764, 2011c. URL http://link.aip.org/link/?MPH/38/754/1 . arXiv:1101.1245v1. 16