the convolution/superposition method: a model-based dose
TRANSCRIPT
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External Beam Photon Dose Calculations Thomas Rockwell Mackie
ProfessorUniversity of Wisconsin
Co-Founder and Chairman of the Board
TomoTherapy Inc.
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Financial Disclosure
I am a founder and Chairman of TomoTherapy Inc. (Madison, WI) which is participating in the commercial development of helical tomotherapy.
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Acknowledgements
Michael Sharpe – University of TorontoColin Field – University of AlbertaMilton Woo – University of TorontoAnders Ahnesjo – University of UppsalaJeff Siebers – Medical College of VirginiaRobert Jeraj – University of WisconsinJatinder Palta – University of FloridaBruce Curran – Tufts UniversityIsaac Rosen – Lousiana State UniversityThomas Bortfeld – Harvard UniversityGary Ezzel – Mayo Clinic, Scotsdale AZCedric Yu – University of Maryland
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Correction-Based Dose Computation
• Based on measured dose distributions in a water phantom.
• Independent corrections for:– Beam modifiers– Surface contours– Tissue heterogeneities
• Based on effective spatial measures such as depth, field boundaries and path lengths.
• Heterogeneity corrections that produce less accuracy than having no corrections at all.
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Model-Based Dose Computation
• The convolution/superposition, finite pencil beam and Monte Carlo methods are model-based dose calculation systems.
• Unlike correction-based methods, model-based methods compute the dose directly in a patient representation.
• Model-based dose computations use monitor unit calculations based on beam intensity (e.g., energy fluence) rather than dose in a phantom.
• The beam has to be modeled explicitly.
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What Has to Be Accounted ForIn Model Based Methods
• Finite source size.• Angular distribution of photons.• Primary transmission.• Extrafocal radiation mainly from the primary
collimation and the field flattening filter.• Differential hardening of the beam by the field
flattening filter.• Curved leave ends.• Leaf configuration.• Tongue and groove effect.• Leaf transmission.• Electron contamination.• Tissue heterogeneities.
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Modeling the Head
Percent of Energy Fluenceat Central Axis
Direct ~85-95% Primary Collimator ~1-4 %
Field Flattening Filter ~5-10 %
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source
mlc
Issues with the Leaf Shape
Potential errors due to leaf shaping
Rounded Edge
Tongue & Groove
From Bruce Curran
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Model-Based Methods Are Need for IMRT
• IMRT = Summation of small fields• Dose = function(penumbra+leakage+head scatter)• Need accurate treatment head model to get this right
XRT beam profileIMRT beam profile
XRTIMRT
From Michael Sharpe, U. of Toronto
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IMRT Is About Using Small FieldsIntensity pattern
possibly unconstrained intensity levels
Intensity limited to a few discrete intensity levels
What is delivered
Adapted from Michael Sharpe, U. of Toronto
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IMRT Is About Using Small Fields
• Accuracy of dose model at small field sizes is a consideration
• Convolution-superposition or Monte Carlo desirable
Measurement
Conventional RTP
IMRT
From Michael Sharpe, U. of Toronto
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Extend the Beam Model
• Minimum Equivalent Square – Conventional XRT: ~4x4cm2
– IMRT: ~0.5x1cm2 - 2x2cm2
• How to get small field data?– Direct measurement – difficult, errors– Extrapolation of existing data – assumes
an underlying physical model– Verify a “golden model”
• Any case will have uncertainty; evaluate its impact on a clinical IMRT distribution generated by the inverse planning system
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Reasons for Drop in Output with Small Field Size
• Backscatter into monitor unit from beam defining jaws
• Reduced scatter (phantom and head)• Electronic disequilibrium• Obscuration of the source
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Dose Computation as a Linear System
W1W2
D1 D2
In particular:
D1 = M11 W1 + M12 W2
D2 = M21 W1 + M22 W2
In general:
Where M is a matrix giving the contribution of dose from each pencil beam.
M11
M21M12
M22 D M W or D MWi ijj
j= =∑
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Pencil Beam Convolution
( ) ( ) ( ), , ', ', ', 'zD x y z x y z K x x y y dxdy= Ψ − −∫ ∫ ∫
From Colin Field, University of Alberta
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Pencil Beams Generated for Homogeneous Water are Fast
• Fast especially for iterative optimization calculations.
• The importance of speed grows for more beam directions, finer resolution, and more iterations.
• Good for early phases of iteration.• Should do a more accurate calculation for
the last iteration and final dose calculation.
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The Problem with Pre-Computed Pencil Beams
The scattering of rays near the boundary are computed with an environment of water around it when it, in fact, may not always exist.
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Convolution Equation• The dose distribution can be computed using
a convolution equation:
D(r) = r ) A(r - r ) dVvol
µρΨz ′ ′(
′rr r - r′
Source ofRadiation
PrimaryEnergyFluence
Kernel DerivedFrom Monte CarloSimulation
DoseDistribution
Dose Deposition Site
PrimaryInteractionSite
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Graphical Representation of Dose Computation using
Convolution
Primary Energy FluenceDistribution
Convolution Kernel (from MC Simulation)
Dose Distribution
x =
Adapted From Colin Field
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Comparison of Two Kernel Sets
Mackie et al 1988
Chui and Mohan 1986
5 MeV 20 MeV
Mackie et al 1988 PMB 33:1-20
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Spherical Shell Summed Kernels Mackie et al 1988 PMB 33:1-20
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Kernels Describe Photon Scattering
and Electron Transport
10 cm x 10 cmParallel BeamIn Water
0.5 MeV
50 MeV
Ahnesjo and Asparadakis, 1999 Phys Med Biol 44:R99-R155
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Adjusting the Spectra to GetAgreement with Measurements
0 1 2 3 4 5 6 7 80.00
0.02
0.04
0.06
0.08
0.10
Spec. 1 Spec. 2
dφ/d
E (a
.u.)
Energy (MeV)
0.0 2.5 5.0 7.5 10.0 12.5 15.020
40
60
80
100
120
140
160
180 Field Size 1.12cm
Exp. Conv.Sup. Spec. 1 Conv.Sup. Spec. 2
PDD
depth (cm)
The depth-dose curve is very forgiving with respect to spectral shape.
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Source of Photons for a Conventional Linac
85 %Percent of Total 4 %
10 %
Liu HH et al. Med. Phys. 24, 1960-1974 (1997).
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Scatter Fraction from the Flattening Filter
Ahnesjo 1994 Med. Phys. 21;1227-1235
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MC Simulation of the Relative Fluence as a Function of Radius in the Beam
10 MV Varian
Liu HH et al. Med. Phys. 24, 1960-1974 (1997).
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MC Simulation of the Radial Dependence of Mean Energy
10 MV Varian
Liu HH et al. Med. Phys. 24, 1960-1974 (1997).
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Fippel et al 2003 Med. Phys. 30:301-311
MC Simulation of the Radial Dependence of Mean Energy
6 MV Elekta
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Tomotherapy Off-axis Spectral Dependence
0 1 2 3 4 5 6 70.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0-5 cm 5-10 cm 10-15 cm 15-20 cm
Phot
on sp
ectru
m
Energy [MeV] 0 1 2 3 4 5 6 7
1E-3
0.01
0-5 cm 5-10 cm 10-15 cm 15-20 cm
Phot
on sp
ectru
m
Energy [MeV]
Tomotherapy has little off-axis spectral dependence because it has no flattening filter; favorable for TPS
From Jeraj et al Med. Phys. 396-404, (2004).
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Modeling the Energy Fluence from Measurements in Air
Fippel et al 2003 Med. Phys. 30:301-311
Comparison of an energy fluence model and the dose in air. Used to model energy fluence for MC simulation but also could be used for the convolution calculations.
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Effect of Kernel Hardening
No Kernel HardeningKernel Hardening
Monte Carlo
Liu HH et al Med. Phys. 24, 1729-1741 (1997).
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Depth (cm)
Depth-Dose at 85 cm SSD for 1 cm Beam Width
Dose
Measurement
Pinnacle Model
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Percent Depth-Dose (Normalized to Dose at 10 cm Depth)
Tomotherapy Beam
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Transverse Unmodulated Dose Profiles
Calculations using the convolution/superposition method on theTomoTherapy HI-ART II treatment planning system.
Tomotherapy Beam
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Longitudinal (Z-axis) Dose ProfilesTomotherapy Beam
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Superposition Equation
D(r) = T r ) A( r - r ) dVvol
r r-r
z ′ ′⋅ ′ ⋅ ′(ρ ρ
The Primary Terma Depends on theRadiological Pathlength ofthe PrimaryRay
The Kernel is Scaledw.r.t. the RadiologicalPathlength Betweenthe Primary Photon Interaction Site and the Dose Deposition Site
DoseDistribution
′r r - r′
Source ofRadiation
Dose Deposition Site
PrimaryInteractionSite
r
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Superposition Assumptions
• Ray tracing from the primary photon interaction site to the dose deposition site assumes that that is the relevant path.
• This correctly accounts for perturbation in first scatter photons which is the dominant scatter.
• It is not a bad approximation for electrons.
• The approximation is worse for 2nd and higher order scatter.
PrimaryPhotonInteraction Site
DoseDeposition Site
PrimaryPhoton
FirstScatteredPhoton
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No Kernel Stretching
muscle ρ=1 gr/cm3
lung ρ=0.25 gr/cm3
Image from Nikos Papanikolaou, Ph.D.
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Heterogeneities: Kernel Stretching
muscle ρ=1 gr/cm3
lung ρ=0.25 gr/cm3
Image from Nikos Papanikolaou, Ph.D.
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Heterogeneous TermaBut Homogeneous Electron
Transport and Scatter
Homogeneous Vs. Heterogeneous
Homogeneous Terma,Electron Transport
and Scatter
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Non-Invariance of Convolution Kernels in Inhomogeneous Phantoms
From Woo andCunningham
Density Scaled Kernel
Monte Carlo Computation forHetereogeneous Situation
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SuperpositionResult
In low density materials like lung (or cork) the electrons and scattered photons travel much farther.
To first order, radiation penetration is inversely proportional to electron density.
From Anders Ahnesjo
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Small Field Sizesand ElectronicDisequilibrium
6 MV Direct Component Only
Each Beam Normalized Separately to Dmax at the Central Axis
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Dose per Incident Energy FluenceAs a Function of Field Diameter
A=Adipose, M=Muscle, B=Bone, L=Lung 4 MV, Parallel Beam
Ahnesjo and Asparadakis, 1999 Phys Med Biol 44:R99-R155
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Sweeping Contamination
Electrons With an Electromagnet
Electrons Swept Out of the Page and Positrons Swept Into the Page
B
From Jursinic and Mackie PMB 41:1499-1509
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Buildup CurvesWith and Without
Magnetic Field
From Jursinic and Mackie PMB 41:1499-1509
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Surface Doseas a Function
of Magnetic Field Strength
Each Beam Normalized Separately to the Dmax at the Central Axis with No Magnetic Field
Photon Backscatter
From Jursinic and Mackie PMB 41:1499-1509
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Ionization Attributableto Electron
Contamination
Electron Contamination Depth-Dose Curve Has aSmall Buildup at Higher Energies
From Jursinic and Mackie PMB 41:1499-1509
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Electron Contamination
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Modeling Electron Contamination
Done as part of the modeling process – e contamination dose is modeled as a modified exponential curve.
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Convolution Superposition as Done in Pinnacle
1. Model the incident fluence as it exits linac head.
2. Project this fluence through the density representation of the patient (CT data) –compute TERMA.
3. 3D convolution/superposition of TERMA and point kernel (ray tracing)
4. Electron contamination added to dose distribution at the end.
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Monte Carlo Calculation
• Explicitly transport individual photons through the head and patient.
• Geometry and radiation physics is the model.
• EGS4 and BEAM, MCNP, Geant, and Penelope are available for research.
• Models are clinically available (e.g. Monaco from CMS)
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Head and Neck Case
Monte CarloSuperposition
MonteCarlo
Histogram ofMonte Carlo - Superposition
From Jeff Siebers
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Lung Case MonteCarlo
Monte Carlo - Superposition
Histogram ofMonte Carlo - Superposition
From Jeff Siebers
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Comparisons for an IMRT Case
Monte Carlo
Convolution/superposition Pencil Beam
From Robert Jeraj, University of Wisconsin
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Errors in Optimized Planning
Convolution/superposition dose calculation
Pencil beam dose calculation
Error [% Dmax] Tumour Lung
Systematic − 0.1 ± 2 − 1 ± 1
Convergence 2-5 1-4
Error [% Dmax] Tumour Lung
Systematic + 8 ± 3 + 6 ± 5
Convergence 3-5 6-7
Convolution/superposition dose calculation is acceptable,pencil beam dose calculation may not always be.
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MU Calculations For Model Based Systems
MU = D P)
D / MU (P)presc (
DMU
P) = D P)MU
SPD, r)( ( (Ψ
Ψ
ComputedDirectly
Beam Output
P
SPD
ΨMU
SPD, r)(
r
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Beam Output for Model-Based Systems
ΨΨMU
SPD,r) = DMU
(Ref) D (Ref)(
MeasuredBeam OutputWhich is PhantomIndependent
Computed for SameCondition as Measurement
Ref
SPD
ΨMU
SPD, r)(
• Establish Energy Fluence/MU By Measurement in a Phantom.
• But the Result is Phantom Independent.
r
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Extended Phantom Concept and Dose Reconstruction
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Will Monte Carlo Take Over?
• For few-field IMRT Monte Carlo is slower than the convolution/superposition method.
• The computation time for Monte Carlo is nearly independent of the number of beam directions.
• The Monte Carlo method is only slightly more accurate for photon beams.
• It will be easier to take into account IMRT delivery geometries (e.g. collimators).
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Summary• The convolution method is a deterministic
calculation using results of Monte Carlo simulation.
• Beam characterization can be performed using Monte Carlo simulation.
• The superposition method, a variant of the convolution method, and predicts dose within a few percent of the Monte Carlo method and about an order of magnitude faster for few calculation fields.
• Energy fluence calibration is more appropriate for model-based dose calculation systems.