dose constraints in imrt
TRANSCRIPT
The Use of Equivalent Uniform The Use of Equivalent Uniform Dose constraints in IMRTDose constraints in IMRT
Thomas BortfeldThomas Bortfeld11, Christian Thieke, Christian Thieke1,21,2, Yair Censor, Yair Censor33
11Department of Radiation Oncology, Department of Radiation Oncology, Massachusetts General Hospital, Boston, USAMassachusetts General Hospital, Boston, USA
22Department of Medical Physics, Department of Medical Physics, Deutsches Krebsforschungszentrum, Heidelberg, Germany,Deutsches Krebsforschungszentrum, Heidelberg, Germany,
33Department of Mathematics, Department of Mathematics, University of Haifa, IsraelUniversity of Haifa, Israel
OutlineOutline
1.1. Projectors in the dose spaceProjectors in the dose space– Max/min dose constraintsMax/min dose constraints– DVH constraintsDVH constraints– EUD constraintsEUD constraints
2.2. Optimizing intensitiesOptimizing intensities– Scaled gradient methodScaled gradient method– POCSPOCS
Maximum dose constraintMaximum dose constraint
Vol
ume
DoseDmax
C aa a
{ }for
otherwise
0
0"Positivity projector":
penalty(weight)
dose atvoxel i
in OAR k
tolerancedose
Penalty function:Penalty function:
Constraints: Maximum (tolerance) doses
"The spinal cord should get less than 40 Gy."
kN
ikikkk dbdCwbF
1
2max, })({)(
Small penalty (w)
Vol
ume
Dosedmax
Large penalty (w)DVH
Vol
ume
dmax
Dose
DVH
2max,
1 1
2
,min
})({
)}({)(
kikk
K
k
N
iikkk
dbdCw
bddCvbFk
Constraints: Minimum and tolerance doses
Vol
ume
Dose
DVH
dmax
dmin
Non-intensity-modulated(4 beams, non-coplanar, MLC)
Intensity-modulated(9 beams, coplanar)
BrainstemBrainstem
Target Target
Example: Clivus chordoma
NTCP = 7% NTCP = 0.7%
Example: Clivus chordoma
Non-intensity-modulated(4 beams, non-coplanar, MLC)
Intensity-modulated(9 beams, coplanar)
Brainstem
Target volume
Lungs
Spinal cord Transversal view
Target
Spinal cord
Lung Lung
Technique: 9 beams, coplanar, intensity-modulated
Example: Thyroid
Example: Thyroid
Vol
ume
Dose
DVH
dmax
Vmax
maxmax )(DVH kkk Vd
Constraints: Dose-volume constraints
"No more than 1/3 of the lung should get more than 15 Gy."
Why are DVH constraints non-convex?Why are DVH constraints non-convex?
Critical structureconsisting of 2 voxels d1 d2
Not more than 50% of the volume (1 voxel) should get more than 30 Gy
30 Gy
30 Gy
feasible region
d1
d2
Non-convexity of DVH constraintsNon-convexity of DVH constraints
• Even though DVH constraints are not Even though DVH constraints are not convex, we can easily determine a convex, we can easily determine a projection of a dose distribution that projection of a dose distribution that violates a DVH constraint, onto the violates a DVH constraint, onto the nearest one that fulfills the constraint: nearest one that fulfills the constraint:
Vol
ume
Dose
DVH
dmax
Vmax
d
Violation of DVH constraintViolation of DVH constraint
k
k
N
ikikdkk dbdCwbF
1
2max,],0[ })({)(
~
otherwise0
for}{],[
vauaaC vu
Modified penalty function:
Interval constraint projector:
Constraints: Dose-volume constraints
Bortfeld et al., ICCR 1997
““Proof” that CProof” that C[0,[0,dk]dk]{ } actually projects onto the nearest { } actually projects onto the nearest
dose distribution that fulfills the DVH constraintdose distribution that fulfills the DVH constraint
• Assume that NAssume that Nvv voxels receive a dose that voxels receive a dose that
is too high is too high
• We need to reduce (down to dWe need to reduce (down to dmaxmax, but not , but not further) the dose in Nfurther) the dose in Nvv voxels voxels
• Which voxels to choose?Which voxels to choose?
• The smallest correction is required for the The smallest correction is required for the ones with the smallest excess dose above ones with the smallest excess dose above ddmaxmax, i.e., with dose values between d, i.e., with dose values between dmaxmax and dand dmaxmax + + dd
Example: Thyroid
• Even though DVH constraints are not Even though DVH constraints are not convex, repeated projections onto the convex, repeated projections onto the feasible space with Cfeasible space with C[0,[0,dk]dk]{ } converge well { } converge well
and there are no problems with local and there are no problems with local minima.minima.– Q. Wu, R. Mohan et al., Med. Phys. 2002Q. Wu, R. Mohan et al., Med. Phys. 2002– J. Llacer et al., PMB 2003J. Llacer et al., PMB 2003
• WHY??WHY??
Volume effectVolume effect Whole lung: 18 Gy50% of lung: 35 Gy
Volume effectVolume effect
nv
TDvTD
)1()(
Power-law relationship for tolerance dose (TD):
n small: small “volume effect”n large: large “volume effect”
0
25
50
75
100
0 20 40 60
Vo
lum
e [%
]
Dose [Gy]80 100
EUD = The homogeneous dose that gives the same clinical effectLung:
EUD = 25 Gy
Spinal Cord:EUD = 52 Gy
Arbitrary (not 0/1) dose distributions Arbitrary (not 0/1) dose distributions
The EUD Concept for OptimizationThe EUD Concept for Optimization
• EUD = equivalent uniform doseEUD = equivalent uniform dose
• Single parameter for each organSingle parameter for each organ
• Example objectives and constraints:Example objectives and constraints:– Maximize EUD(target)Maximize EUD(target)– Minimize EUD(OAR)Minimize EUD(OAR)– EUD(OAR) < ToleranceEUD(OAR) < Tolerance
• EUD has not yet been fully validated EUD has not yet been fully validated
• Use hard physical constraints to limit Use hard physical constraints to limit search spacesearch space
• Use EUD to find Pareto solutions within Use EUD to find Pareto solutions within the limited search space.the limited search space.
Volume effect -> EUD, Power-Law (a-norm) ModelVolume effect -> EUD, Power-Law (a-norm) Model
a
i
aii dv
/1
EUD
“a-norm”
(a=1/n)
Mohan et al., Med. Phys. 19(4), 933-944, 1992Kwa et al., Radiother. Oncol. 48(1), 61-69, 1998Niemierko, Med. Phys. 26(6), 1100, 1999
Examples:
:
:1
a
a
maxEUD
EUD
D
D
• EUD is a convex function of the dose EUD is a convex function of the dose distribution (for a>1 and negative a) distribution (for a>1 and negative a)
Projection onto convex sets (POCS) Projection onto convex sets (POCS) methods will converge to given methods will converge to given EUD-constrained solutionsEUD-constrained solutions
POCS – Projection onto convex setPOCS – Projection onto convex set
x
x
d2
d1
D
Convex set
D‘
})(|{ maxEUDDEUDD
maxEUDEUD
maxEUDEUD
EUD constraint IIEUD constraint II
Vol
ume
Dose
max)( EUDDEUD
EUD constraint IIEUD constraint II
Vol
ume
Dose
max)( EUDDEUD
max)'( EUDDEUD
EUD projectorEUD projector
max)'(. EUDDEUDII
min)'(.1
2
N
iii ddI
Extrema on a bounded surface
Use Lagrange Multipliers
EUD projector: Lagrange multiplierEUD projector: Lagrange multiplier
)'()'(),'( max DEUDEUDDfDL
0'1
'1
)'(2'
11
1
)1(
aN
m
am
k
ai
kii
i
k
dN
dN
ddd
L
EUD projectorEUD projector
• Right-hand side is independent of Right-hand side is independent of ii
• Exact solution for Exact solution for aa=1 and =1 and aa=2: =2:
k
aN
m
am
kkai
ii NiDNNd
dd k
,...,1'1
2'
'1
1
1)1(
)1(max
max)1('
'
aa
i
ii
EUD
EUDEUD
d
dd
EUD projectorEUD projector
• It turns out that this a good approximation It turns out that this a good approximation for all values of for all values of aa
• Easy solution of the implicit equationEasy solution of the implicit equation
• Is there an exact solution??Is there an exact solution??
C. Thieke, T. Bortfeld, A. Niemierko and S. Nill, From physical dose constraints to equivalent uniformdose constraints in inverse radiotherapy planning, Medical Physics, 30 (2003), 2332--2339.
Initialization
OrganConstraint
dpres({organ}) = ... dpres({organ}) = ... dpres({organ}) = ...
all organs processed?
Adjoint dose calculationDose calculationCalc. objective function
converged ?
End
Max/min
DVH
EUD
no
yes
no
yes
POCS – Example Serial OrganPOCS – Example Serial Organ
0 20 40 600
20
40
60
80
100
Current DoseProjected to EUD=33 Gy:
a = 7.4
Re
lativ
e V
olu
me
(%
)
Dose (Gy)
POCS – Example Serial/Parallel OrganPOCS – Example Serial/Parallel Organ
0 20 40 600
20
40
60
80
100
Current DoseProjected to EUD=33 Gy:
a = 7.4 a = 1.0
Re
lativ
e V
olu
me
(%
)
Dose (Gy)
POCS – Example TargetPOCS – Example Target
0 20 40 600
20
40
60
80
100
Current Dose Projection to EUD=66Gy,
a = -10
Rel
ativ
e V
olum
e (%
)
Dose (Gy)
Example: Head and neck caseExample: Head and neck case
Brainstem
Spinal Cord
Parotis
ResultsResults
OrganOrgan EUD-ConstraintEUD-Constraint(Gy)(Gy)
EUD EUD (Gy)(Gy)
BrainstemBrainstem Max=23Max=23 2323
Spinal Spinal CordCord Max=25.5Max=25.5 25.525.5
ParotisParotis Max=13.0Max=13.0 13.113.1
TargetTargetMin=60.0Min=60.0
Max=61Max=61
58.858.8
61.061.0
ResultsResults
0 20 40 600
20
40
60
80
100
Boost
phys EUD/phys
Target
Spinal Cord
Parotid gland
V
olu
me
(%
)
Dose (Gy)
OutlineOutline
1.1. Projectors in the dose spaceProjectors in the dose space– Max/min dose constraintsMax/min dose constraints– DVH constraintsDVH constraints– EUD constraintsEUD constraints
2.2. Optimizing intensitiesOptimizing intensities– Scaled gradient methodScaled gradient method– POCSPOCS
Pre-calculated Pre-calculated DDijij matrix matrix
Voxel i
Bixel j
Source
Patient
j
jiji bDd
Scaled gradient projection techniqueScaled gradient projection technique
• Newton-like iteration (simultaneous update):Newton-like iteration (simultaneous update):
: damping factor: damping factor
2
2
1
j
j
j
dbFd
dbdF
tj
t bb
violated,
2
p
2
)(2
iiji
iijiii
tj Dw
Dddwb
“TBNN” – Thieke, Bortfeld, Niemierko, Nill
i i
ijiiitj
tj s
Dddwbb
)( p1
sconstraint EUDfor 1
sconstraint physicalfor 2
jij
i
Ds
POCS, Censor & Elfving:
Scaled gradient projection (TBNN):
j
iijiii
tj
tj s
Dddwbb
)( p
1
violated,
2
iijj Ds
Physical dose only, Patient 1Physical dose only, Patient 1
Physical dose only, Patient 2Physical dose only, Patient 2
EUD only, Patient 1EUD only, Patient 1
EUD only, Patient 2EUD only, Patient 2
• Why doesn’t the TBNN method work for Why doesn’t the TBNN method work for EUD-only constraintsEUD-only constraints
• Possible answer: EUD violations affect a Possible answer: EUD violations affect a large number of voxels at the same time, large number of voxels at the same time, which may lead to oscillationswhich may lead to oscillations
ConclusionsConclusions
OptimizationIteration 1
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OptimizationIteration 3
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