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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