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

SLIDE 1

Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism

Daniel Maneval 1,2 Hugo Bouchard 3 Benoˆ ıt Ozell 4 Philippe Despr´ es 1,2 October 16, 2017

1Universit´

e Laval, D´ epartement de physique, de g´ enie physique et d’optique, Qu´ ebec, Canada

2Universit´

e Laval, D´ epartement de radiooncologie et centre de recherche du CHU de Qu´ ebec, Qu´ ebec, Canada

3Universit´

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

SLIDE 2

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 (∆Etrue) is obtained with the energy

straggling (δ∆E): ∆Etrue = ∆E ± δ∆E

1

SLIDE 3

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 Leq formalism

2

SLIDE 4

The ∆E calculations

Fetch the range Compute the stepping function Determine the step length with 2 step restrictions Fetch the stopping power Compute the mean energy loss

Geant4

< 1% of E Accept Compute with an inverse range table yes No

EGSnrc formalism

Fetch the fractional energy loss Fetch the equivalent stopping power Compute the step restriction with a T aylor expansion and a maximal fractional energy loss Determine the step length with the step restriction Fetch the stopping power Compute with a midpoint rule

3

SLIDE 5

Leq formalism benefit

Step length restriction Interface proton trajectory

High dose accuracy simulations:

hard collision

Balanced simulations: accuracy - computation time: High accuracy with the formalism:

Geant4 or EGSnrc The Leq formalism s = min(dvox, dhard, dmax) s = min(dvox, dhard)

4

SLIDE 6

The Leq formalism

L is the restricted/unrestricted stopping power. E is the proton kinetic energy The distance (s) space

d∆E ds

= L(s) ⇔ ∆E = s L (E ′) ds′

Stopping power Step

The energy (∆E) space

ds d∆E = L−1(E − ∆E)

⇔ s = − E−∆E

E

L−1(E ′)dE ′ The midpoint rule

• f the Newton-Cotes formula:

⇒ ∆E = s · Leq (E, ǫ) Leq (E, x) =

L(x)

• 1+ 2L′(x)2−L(x)L′′(x)

L(x)2 E2+x(x−2E) 6

• x = E
• 1 − ǫ

2

• and ǫ(s, E) = ∆E

E

ǫ links the ∆E space with the s space.

5

SLIDE 7

Leq Look-up tables

Leq (E, x) =

L(x)

• 1+ 2L′(x)2−L(x)L′′(x)

L(x)2 E2+x(x−2E) 6

with x = E

• 1 − ǫ

2

• Material

Fixed ǫ Fixed Energies Water

100 101 102

Energy (MeV)

101 102

Leq (MeV · cm−1)

ǫ = 0 % ǫ = 10 % ǫ = 25 % ǫ = 50 % ǫ = 70 % ǫ = 100 %

20 40 60 80 100

ǫ (%)

200 400 600 800

·

0.5 MeV 1.5 MeV 5 MeV 15 MeV 50 MeV 350 MeV

Gold

100 101 102

Energy (MeV)

102 103

Leq (MeV · cm−1)

ǫ = 0 % ǫ = 10 % ǫ = 25 % ǫ = 50 % ǫ = 70 % ǫ = 100 %

20 40 60 80 100

ǫ (%)

500 1000 1500 2000 2500 3000

·

0.5 MeV 1.5 MeV 5 MeV 15 MeV 50 MeV 350 MeV

6

SLIDE 8

ǫ look-up tables

ǫ(s, E) = ∆E

E

determined when solving s + E(1−ǫ)

E dE′ L(E ′) = 0

Material Fixed Step lengths Fixed Energies Water

100 101 102

Energy (M eV)

10

7

10

5

10

3

10

1

101 102

(%)

(%)

10

5 4 3 2 1

10

7

10

5

10

3

10

1

101 102

Gold

100 101 102

Energy (M eV)

10

✁ 7

10

✁ 6

10

✁ 5

10

✁ 4

10

✁ 3

10

✁ 2

10

✁ 1

100 101 102

✂

(%)

1.0 mm 0.1 mm 10

✁ 2 mm

10

✁ 3 mm

10

✁ 4 mm

10

✁ 5 mm

(c) Gold

Energy (MeV)

(%)

10

✁ 5

10

✁ 4

10

✁ 3

10

✁ 2

10

✁ 1

100

Step (mm)

10

✁ 7

10

✁ 6

10

✁ 5

10

✁ 4

10

✁ 3

10

✁ 2

10

✁ 1

100 101 102

✂

0.5 M eV 1.5 M eV 5 M eV 15 M eV 50 M eV 350 M eV

MeV MeV MeV MeV MeV MeV Step (mm)

(d) Gold 7

SLIDE 9

Geant4 setups

Simulation setups Geant4 CSDA parameters Time (h) Errors (%) dmax Linear loss Step function Maximum Falloff Reference 1 µm 1% (20%, 50 µm) 140

• High Accuracy (HA)

10 µm 1% (20%, 50 µm) 14 0.2 0.9 Balanced (Bal) 1 mm 0.1% (0.1%, 1 µm) 2.5 0.8 4.7 Default 1 mm 1% (20%, 50 µm) 0.8 4.8 16.5

10 20

Dose (Gy)

Default Balanced High Accuracy Reference 305 310 315 320 325 330

Depth (mm)

1 4 8 12 16

Error (%)

50 100 150 200 250 300

Depth (mm)

10−3 10−2 10−1 100

Maximum step (mm)

Reference Bragg peak

Default Balanced High Accuracy Reference

8

SLIDE 10

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 Leq formalism intrinsic efficiency:

• pGPUMCD-L: the Geant4 CSDA scheme
• pGPUMCD-Leq

Material ρ ne (×1023) 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

SLIDE 11

Leq formalism validation

50 100 150

Depth (mm)

50 100 150

Dose (Gy)

Geant4 pGPUM CD Error

✁ 0.75 ✁ 0.50 ✁ 0.25

0.00 0.25 0.50 0.75

Error (%)

Geant4 pGPUMCD Error

Dose (Gy) (a) Lung 70 MeV

20 40 60 80

Depth (mm)

20 40 60 80 100 Geant4 pGPUM CD Error

✁ 0.75 ✁ 0.50 ✁ 0.25

0.00 0.25 0.50 0.75

Error (%) (b) Water 100 MeV

100 200 300

Depth (mm)

10 20 30 40 50 60

Dose (Gy)

0 75 0 50 0 25 0.00 0.25 0.50 0.75

rror (%) Error (%) (c) Water 230 MeV

50 100 150 200

Depth (mm)

10 20 30 40 50 60

Dose (Gy)

✁ 0.75 ✁ 0.50 ✁ 0.25

0.00 0.25 0.50 0.75

Error (%) Depth (mm) Dose (Gy) (d) Bone 230 MeV

20 40 60

Depth (mm)

5 10 15 20 25

Dose (Gy)

✁ 0.75 ✁ 0.50 ✁ 0.25

0.00 0.25 0.50 0.75

Error (%) (e) Copper 230 MeV Depth (mm)

10 20 30

Depth (mm)

5 10 15 20

Dose (Gy)

✁ 0.75 ✁ 0.50 ✁ 0.25

0.00 0.25 0.50 0.75

Error (%) Depth (mm) (d) Gold 230MeV Error (%)

The ranges (R80) matched within 1 µm

10

SLIDE 12

Leq formalism efficiency

Geant4 pGPUMCD-L pGPUMCD-Leq Intrinsic speed up factors Material Energy THA

G4

TBal

G4

THA

L

TBal

L

TLeq GPU ( TG4/TL) Leq ( TL/TLeq) (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 dmax.

• the Leq formalism: constant algorithmic time complexity, i.e. O(1)

11

SLIDE 13

Leq formalism with the energy straggling

100 101 102 Dose (Gy)

Water 230 MeV Water 100 MeV Lung Bone Copper Gold Geant4 pGPUMCD

50 100 150 200 250 300 Depth (mm) −2 −1 1 2 Error (%)

The ranges (R80) matched within 100 µm

12

SLIDE 14

Computation times

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

13

SLIDE 15

Conclusion

The Leq 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 Leq formalism constitutes a promising variance reduction technique for computing proton dose distributions in a clinical context. The Leq formalism could be used for other charged particles. The multiple scattering was validated, not presented here. Under investigations: nuclear interactions More details concerning the Leq formalism: (Maneval et al., 2017)

14

SLIDE 16

Acknowledgements

15

SLIDE 17

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

SLIDE 18

References II

• ICRU. Photon, Electron, Proton and Neutron Interaction Data for Body
• Tissues. ICRU Report. International Commission on Radiation Units

and Measurements, Bethesda, Md., U.S.A, 1992. URL http://books.google.ca/books?id=E9UqAAAAMAAJ. Iwan Kawrakow. Accurate condensed history Monte Carlo simulation of electron transport. I. EGSnrc, the new EGS4 version. Medical Physics, 27(3):485–498, 2000. URL http://link.aip.org/link/?MPH/27/485/1. Iwan Kawrakow, David Rogers, F. Tessier, and B. Walters. The egsnrc code system: Monte carlo simulation of electron and photon transport (pirs-701). National Research Council (NRC), Report, 2016.

17

SLIDE 19

References III

Daniel Maneval, Hugo Bouchard, Benoˆ ıt Ozell, and Philippe Despr´ es. Efficiency improvement in proton dose calculations with an equivalent restricted stopping power formalism. Physics in Medicine and Biology,

• 2017. URL

http://iopscience.iop.org/10.1088/1361-6560/aa9166.

18