Proton Dose Calculations
Semi-empirical models for scanning beams and IMPT
Uwe Oelfke
DKFZ Heidelberg
Department of Medical Physics (E040)
Im Neuenheimer Feld 280
69120 Heidelberg, Germany
u.oelfke@dkfz.de
Outline
Phase Space, Physics and Dose
Dose Kernels: Water
Tissue Inhomogeneities
Consequences
Seite 1
1
Phase space and dose of a beam spot
Dose = Phase Space ** Physics Pencil Beam (H20)
Depth dose curve
A simple model
Stopping power, CSDA Range
Straggling, energy-spektrum
Inelastic nuclear interactions
Basic physics
Dose factorizes in depth dose and lateral spread
Laterally integrated depth dose curve
Energy loss due to collisons with e-
Straggling
Energy spread of dose delivery system
Primary and nuclear interactions
Seite 2
2
Fluence and depth dose curve
Protons
D(z,r) = ???
H2O
E(0)
dN p dN p Φ=
Ψ=
Ee
Energy -Fluence:
Fluence:
dA ⊥
dA ⊥
Dose and vectorial energy fluence
D(x) =
dE ab
1 (x) = ∇ Ψ (x)
ρ
dm
Vectorial energy fluence
Depth dose curve: a simple model
Protons
D(z,r) = ???
E(0)
D(z) = -
H2O
1 ∂
ρ ∂z
(Φ(z) E(z)) = -
1
ρ
E(z)
Inelastic nuclear inter.
∂Φ( z) 1 ∂E( z)
Φ( z )
∂z
ρ ∂z
Mass Stopping Power (collision)
Seite 3
3
Coulomb interaction with electrons
Stopping power - Range
Stopping Power
Microscopic def. :
S(E) =
E tr
( E)
L
L = mean free pathlength
E
R CSDA = ∫ dE'
CSDA Range:
0
1
S(E' )
Bethe Bloch Formula
2
Z eff
e4
 2m c 2 β 2

dE
= −4πN
Z T ln e
− β 2
dx el .
me c 2 β 2  I (1 − β 2 )

[
(
Zeff = Z p 1 − exp − 125 ⋅ β ⋅ Z −p2 / 3
)]
Barkas-Formel
Mass stopping power
Seite 4
4
CSDA Range
Coulomb interaction with electrons
Straggling and Bragg Peak
Statistical fluctuations: broadening of energy spectrum
Straggling width:
Microscopic def. :
∝ R
Bragg Peak:
Σ(E) =
E tr E tr
( E)
L
L = mean free pathlength
∗∗ )
Depth dose: Straggling – Bragg Peak
Seite 5
5
Proton Bragg Peaks – Straggling
Energy spectrum of accelerator
Inelastic Nuclear Interactions
Strong interaction with nuclei in the body
creation of slower secondary protons,
recoil heavier fragments
creation of new background ‚Bragg peaks‘
Lateral scattering at lower energies - halo
Seite 6
6
Inelastic nuclear interactions
Lateral Scattering
Coulomb interaction with nuclei shielded by
electron cloud
Moliere Theory: Gaussian width + Broader
Background
Drifting + Scattering = Transport Eq.
2 Gaussians for Modelling – Primary + Secondary
(Field Size Effect)
Distribution of scattering angles
Seite 7
7
Lateral spread of beam (Highland)
FWHM = σ * 2.35
160 MeV
200 MeV
Multiple Coulomb scattering
The Source(s): phase space of the accelator
Energy spectra
Lateral distributions
Angular distributions
Source size, position
Collimator scattering
Extrafocal radiation, ...
Calculation ?
f i (E, r, z,θ x , θ y )
i = particle type
Calibration via a set of measurments
or
Monte Carlo Techniques !
Seite 8
8
Characterization of Phase Space
‚Physics‘ dose in water is defined by either semiempirical source models or MC calculations
Use a set of measurements to calibrate the
parameters of a phase space model
Aim: To determine a few parameters of your
‚machine‘ such that the dose for all beam spots in
water is correct
Assumptions:
Parameters:
Seite 9
9
Beyond Gaussians….
Seite 10
10
Seite 11
11
• Pencil beam in water:
• Standard 1D scaling: pathlength scaling
Independent of
inhomogeneity
position and
scattering properties
• New 2D scaling: pathlength AND radial scaling
Depends on
inhomogeneity
position and
scattering properties
Pencil beam scaling methods
m
 r

S
1
P (r , z , E 0 ) =  
P w 
, zeq , E 0 
2
 ρ  w (Fr )
 Fr

• 1D scaling:
Fr=1
• 2D scaling: Fr =
σ (z , E 0 )
σ w (zeq , E 0 )
Use of Highland’s formula for
Multiple Coulomb Scattering
Seite 12
12
Implementation of the 2D scaling
We need:
Depth dose curves in water
CT calibration curve relating Hounsfield numbers to
relative stopping powers
Simple analytical formula for calculating the standard
deviation of lateral spread (function of depth and
energy)
CT calibration curve relating Hounsfield numbers to
material specific lateral scaling factors
CT Calibration curves
Lateral Scaling Factors
Relative Stopping Powers
1,5
1,0
0,5
0,0
-1000
Lateral Scaling Factors
1000
2,0
100
10
1
-200
-100
0
100
200
Hounsfield Units
1
0,1
-500
0
500
1000
1500
2000
-1000
-500
Hounsfield Units
0
500
1000
1500
2000
Hounsfield Units
Example: Homogeneous Media
Hanitra Szymanowski and Uwe Oelfke: Two-dimensional pencil beam scaling: an improved proton dose algorithm for heterogeneous media 2002 PMB 47.
Seite 13
13
Example: Homogeneous Media
Water
Bone
0,4
MC
1D
2D
0,3
0,2
Sigma (cm)
Sigma (cm)
0,4
0,1
0,0
0
5
10
MC
1D
2D
0,3
0,2
0,1
0,0
0
15
Depth (cm)
5
10
Depth (cm)
Homogeneous medium: bone
Lateral dose distribution
Bone at z=4cm
Bone at z=8cm
20
MC
2D
1D
15
10
10
5
5
0
-0,6 -0,4 -0,2 0,0 0,2
Tiefe (cm)
0,4
0
-0,6 -0,4 -0,2 0,0 0,2
Tiefe (cm)
0,6
0 ,3
MC
1D
2D
0 ,1
0
2
4
6
Sigma (cm)
S ig m a (c m )
0,4
bone
0 ,2
0,2
2
4
0 ,1
0 ,0
0
2
4
6
8 10 12 14 16 18
D e p th (c m )
S ig m a (c m )
S ig m a (c m )
0 ,2
6
8 10 12 14 16 18
A ir s la b
MC
1D
2D
0 ,3
bone
Depth (cm)
A ir s la b
a ir
0,6
0,1
0,0
0
8 10 12 14 16 18
MC
1D
2D
0,3
D e p th (c m )
0 ,4
0,4
Bone slab
B o n e s la b
0 ,4
0 ,0
MC
2D
1D
15
Dosis (%)
Dosis (%)
20
0 ,4
MC
1D
2D
0 ,3
a ir
0 ,2
0 ,1
0 ,0
0
2
4
6
8 10 12 14 16 18
D e p th (c m )
Seite 14
14
Comparison to GEANT4
Lateral inhomogeneities
Influence of dose algorithm on IMPT
Clinical example:
• Lung tumor
• 5 equally spaced proton beams, 3mm wide proton spots
• 3D spot scanning technique
• Distal-Edge-Tracking technique
Treatment plans:
• Optimisation with 1D-scaling - Recalculation with 1D-scaling
• Optimisation with 1D-scaling - Recalculation with 2D-scaling
• Optimisation with 2D-scaling - Recalculation with 2D-scaling
Seite 15
15
Lung: Systematic error
Optimisation: 1D scaling
Lung DET
Volume (%)
Recalc:1D scaling
Recalc:2D scaling
110
100
90
80
70
60
50
40
30
20
10
0
PTV opt:1D calc:1D
Lung opt:1D calc:1D
PTV opt:1D calc:2D
Lung opt:1D calc:2D
0
10
20
30
40
50
60
70
80
Dose (Gy)
%
D m in
P TV
Lun g
D m ax D m e an D m in D m a x D m ea n
∆ s (D E T) 7.5
3.9
5.4
0.0
3.9
0.2
3.8
9.0
4.6
0.0
9.0
0.2
∆ s (3D )
Lung: Convergence error
Recalculation: 2D scaling
Optim:2D
Volume (%)
Optim:1D
Lung DET
110
100
90
80
70
60
50
40
30
20
10
0
PTV opt:1D calc:2D
Lung opt:1D calc:2D
PTV opt:2D calc:2D
Lung opt:2D calc:2D
0
10
20
30
40
50
60
70
80
Dose (Gy)
%
D m in
P TV
Lung
D m ax D m ean D m in D m ax D m ean
∆ c (D ET) -7.1 -1.9
-5.4
0.0 -1.9 -0.5
-4.9 -8.6
-5.8
0.0
∆ c (3D)
-8.6
-0.6
Conclusions
Calibration of dose for beam spots in water
has to be done with outmost accuracy
Integration of phase space and physics is done
with phenomenological models
There are clear limitations of this approach for
tissue inhomogenieties
Seite 16
16