Transcript Document

Image Reconstruction and
Inverse Treatment Planning
– Sharpening the Edge –
Thomas Bortfeld
I.
II.
CT Image Reconstruction
Inverse Treatment Planning
Syllabus
2/13 Advances in imaging for therapy (Chen)
2/20 Treatment with protons and heavier particles;
tour of proton facility (Kooy)
2/27 Treatment delivery techniques with photons,
electrons; tour of photon clinic (Biggs, Folkert)
3/6 Intensity-modulated radiation therapy
(IMRT, IMPT) (Seco, Trofimov)
3/13 Dose calculation (Monte Carlo + otherwise)
(Paganetti, Kooy)
3/20 Treatment planning (photons, IMRT, protons)
(Doppke + NN)
2
Syllabus
3/27 Inverse treatment planning and optimization
(Bortfeld)
4/3 Optimization with motion and uncertainties
(Trofimov, Unkelbach)
4/10 Mathematics of multi-objective optimization
and robust optimization (Craft, Chan)
4/17 Dose painting (Grosu)
4/24 Image-guided radiation therapy (Sharp)
5/1 Special treatment techniques for
moving targets (Engelsman)
3
Planigraphy, Tomosynthesis
“Verwischungstomographie”
x-ray source
x-ray tube
focus slice
object
film
film
Ziedses des Plantes (Netherlands), 1932
4
Computerized Tomography
Greek:
• tomos = section, slice
• graphia = to write, to draw
tomograph = slice drawer
5
Nobel prize 1979
Drs. Hounsfield and Cormack
Sir Godfrey N. Hounsfield
(Electrical Engineer)
EMI
Allan M. Cormack
(Physicist)
South Africa, Boston
6
First CT scanner prototype
Hounsfield apparatus
7
CT “generations”
1st generation
2nd generation
translation-rotation scanners
8
CT “generations”
3rd generation
(fan detector)
4th generation
(ring detector)
Rotation-only scanners
9
Imatron: Electron beam CT (EBT)
10
Imatron: Electron beam CT (EBT)
11
Helical (“spiral”) CT
Trajectory of the
continuously rotating
X-ray tube
Start of spiral scan
Table motion
12
From transmission to projection
A.M. Cormack
13
Projection
14
Sinogram, “Radon” transform, 
(Johann Radon, Vienna, 1917)
p
Object
f
?
f
Sinogram
p
15
Backprojection
16
Backprojection
Shepp&Logan phantom
“Reconstruction” by backprojection
17
Backprojection
18
Backprojection
Projections of point object
from three directions
Back-projection onto
reconstruction plane
19
Backprojection
Object
Image
f(r) 1/r
f(r)
Profile through object
Profile through image
Terry Peters (Robarts Institute)
20
Backprojection
This is a convolution (f * h) of f(r) with
the point spread function (PSF) h=1/|r|
21
What is Convolution?
22
Convolution theorem
• Periodic sine-like functions are the
Eigenfunctions of the convolution operation.
• This means, convolution changes the
amplitude of a sine wave but nothing else.
• Hence, convolution is completely described
by the transfer function or frequency
response (Fourier transform of the PSF),
which determines how much amplitude is
transmitted for different frequencies
23
Convolution theorem
24
r-filtered layergram reconstruction
1. Backproject measured projections, and
integrate over f
2. Fourier transform in 2D
3. Multiply with distance from the origin, |r|,
in the frequency space
4. Inverse Fourier transform
25
Central slice theorem
Filtered backprojection
27
Deconvolution (filtering) in the
spatial domain
Ramp filter in the frequency domain:
sampling interval
Transform back to get filter in spatial domain:
28
Deconvolution (filtering) in the
spatial domain
Sample at discrete points p = nDp:
sampling interval
Filter of Ramachandran and Lakshminarayanan (Ram-Lak)
29
Ram-Lak filter
negative components
30
Shepp&Logan filter
31
Backprojecting filtered projections
Terry Peters (Robarts Institute)
32
Filtered backprojection
Shepp&Logan phantom
“Reconstruction”
Filtered backprojection
by backprojection
33
Filtered Backprojection
Simple backprojection
Filtered backprojection
34
Reconstruction from fan projections
k+l = 7:
parallel
k: counter of source positions
l: counter of projection lines
35
Summary CT image reconstruction
1. In CT we measure projections, i.e., line
integrals
2. The set of projection lines from all
directions is called Radon transform or
sinogram
3. Backprojection leads back into image
space but introduces severe 1/r blurring
36
Summary CT image reconstruction, cont’d
4. Image can be de-blurred with
deconvolution techniques
5. Deconvolution can be done in projection
space using the central slice theorem
6. A common filter function is the Ram-Lak
filter
7. Filters have negative components
(“eraser”) to remove blurring
37
Image Reconstruction and
Inverse Treatment Planning
– Sharpening the Edge –
Thomas Bortfeld
I.
II.
CT Image Reconstruction
Inverse Treatment Planning
Brahme, Roos, Lax 1982
Source
Phantom
Target
OAR
Dose
40
The idea of IMRT
"Classical" Conformation
Intensity Modulation
Treated
Volume
Treated
Volume
Target Volume
Tumor
Tumor
OAR
OAR
Target
Volume
Collimator
41
“Inverse” treatment planning
"Conventional" Planning
Treated
Volume
Target Volume
OAR
Inverse Planning
Target
Volume
Treated
Volume
OAR
Collimator
42
Computer Tomography
Conformal Radiotherapy
Projection
Intensity
Modulation
Radiation
Source
Detectors
Radiation
Source
Target
Image Reconstruction
(Filtered Backprojection)
Conformal Radiotherapy
(Filtered Projection)
Density Distribution
of the Tissue
Set of Prescribed
2D Dose Distributions
x-ray Projection
(CT-Scanner)
Projection
(Computer)
1D Filtering
of the Projections
1D Filtering
of the Projections
Backprojection
IMRT with
Filtered Projections
Set of 2D Slice Images
Dose Distribution
44
G. Birkhoff:
On drawings composed of
uniform straight lines
Journ. de Math.,
tome XIX, - Fasc. 3, 1940.
Negative Intensities
After filtering:
Intensity
-
x
46
The inverse problem has no solution!
Consider the inverse problem
as an optimization problem.
Define the objectives of the treatment
and let the computer determine the
parameters giving optimal results.
Optimization basics
Mathematical optimization:
Minimization of objective functions
actual dose prescribed dose
Objective Function
F(x) = Si (di - pi)2, di = f(x1,.., xn)
F(x) = NTCP  (1-TCP)
Constraints
di < dtol
xi > 0
DVH constraints
NTCP < 5%
Parameters
x = (x1, . . ., xn) (e.g., intensity values)
“Decision variables”
•
•
•
•
•
Intensity profiles
Beam weights, segment weights
Beam angles (gantry angle, table angle)
Number of beams
Energy (especially in charged particle
therapy)
• Type of radiation (photons, electrons, ...)
The “standard model” of inverse planning
minimize

N


F (b )   wi d i (b )  Pi
i 1

d i (b )   Dijb j
dose values

2
beam intensities
j
all the physics is here
Dij : dose contribution of pencil beam j to voxel i
bj  0
50
Dose-volume histogram for OAR
small weight (w)
DVH
Volume
Volume
DVH
large weight (w)
max
D
Dose
max
D
Dose
Critical structure (organ at risk) costlet
 (
N R ,k


FR ,k (b )  wk  d i ,k b  Dmax, k
i 1
weight
importance
“penalty”
dose at
voxel i
in OAR k

2

tolerance
dose
 x for x  0
x  
0 otherwise
52
Dose-volume histogram for the target
Volume
DVH
min
D
max
D
Dose
The “standard model” of inverse planning
Minimize
F  wTarget  FTarget  wRisk1  FRisk1  wRisk2  FRisk2  
objectives, costlets, indicators
weights, penalties, importance factors
54
The “standard model” of inverse planning
•
•
•
•
High-dimensional problem:
Ray intensities bj: 10,000
Dose voxels di: 500,000
Dij matrix: 10,000 x 500,000 entries
20 GByte
55
Optimization with gradient descent
F(x)
local min.
x3
x2 x1 x0
x
global min.



xt 1  xt    grad F ( x ) 
xt
dF
1D: xt 1  xt   
dx
56
Optimization algorithms
• Projecting back and forth between dose
distribution and intensity maps
57
Volume effect
Whole lung: 18 Gy
50% of lung: 35 Gy
78
Volume effect
Power-law relationship for tolerance dose (TD):
TD(1)
TD(v)  n
v
n small: small “volume effect”
n large: large “volume effect”
79
Volume effect -> EUD,
Power-Law (a-norm) Model
Mohan et al., Med. Phys. 19(4), 933-944, 1992
Kwa et al., Radiother. Oncol. 48(1), 61-69, 1998
Niemierko, Med. Phys. 26(6), 1100, 1999
1/ a


a

EUD   vi  Di


 i

Examples:
“a-norm”
(a=1/n)
a  1 : EUD  D
a   : EUD  Dmax
80
Equivalent Uniform Dose (EUD)
Volume [%]
100
75
Lung:
EUD =
25 Gy
50
Spinal Cord:
EUD = 52 Gy
25
0
0
EUD =
The homogeneous
dose that gives the
same clinical effect
20
40
60
Dose [Gy]
80
100
81
Volume
DVH constraint
max
V
max
D
Dose
82
Penalties/
Weights/
…
Dose,
Dose-Volume
Constraints
83
Summary inverse treatment planning
1. Intensity-modulated radiation therapy (IMRT)
uses non-uniform beam intensities from various
(5-9) beam directions
2. “Inverse planning” is the calculation of
intensities that will give the desired spatial dose
distribution
3. CT reconstruction techniques cannot be
(directly) applied here because we cannot
deliver negative intensities
4. Today “inverse planning” is usually defined as
an optimization problem, which is “solved” with
gradient techniques
84