The Challenge of Steering a Radiation Therapy

Planning Optimizationby Ronald L. Rardin

Professor of Industrial EngineeringPurdue University

West Lafayette, Indiana, USA

Caesarea Rothschild Institute, University of Haifa, June 2004

Acknowledgments• Our work at Purdue involves an inter-

disciplinary team (of 10-15) spanning– Indiana University School of Medicine– Purdue University College of Engineering– Advanced Process Combinatorics (an

optimization software firm)

• Dr. Mark Langer = our inspiration and medical mentor

• Sponsored in part by National Science Foundation 0120145, National Cancer Institute 1R41CA91688-01, and Indiana 21st Century Fund 830010403

External Beam Radiation Therapy

• Delivered by an accelerator that can rotate 360 degrees around the patient to treat a target at the isocenter from multiple angles

• Implemented with a Multi-Leaf Collimator varying opening during delivery time

Choices for Beamlet Intensities

Accel

erat

or

Acc

eler

ato

r

Intensity Map

(Profile)

Planning Dose Conflict

Accel

erat

or

Acc

eler

ato

r

• Planning seeks beamlet intensities– Sufficient dose on

tumor to control it– Doses to nearby

tissues within tolerances

• Inherent conflict between higher target dose and safety of critical healthy tissues

Optimization Limitations

• Optimization has made critical contributions to radiation therapy planning, but it is not a perfect fit to the dose tradeoff issues

• Optimization means finding a solution that minimizes (or maximizes) one function of the decision variables– Usually subject to constraints on decision

choices• Multiobjective optimization methods do

exist, but use a sequence of single objective opts

• Any optimization is only practical if mathematical form of the objectives and constraints permits tractability

Interactive Sequence of Solutions

• Keep

0

25

50

75

100

3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Dose (cGy)

Volu

me

(%)

Max. Dose

Threshold Dose

1-Protected Fraction

• Limitations usually result in an interactive sequence of optimizations to find a plan suitable to physicians and dosiometrists

• Clinicians use graphic methods (DVH and isodose) to modify opt until they find one they like

Steerability

• Define steerability as the degree to which the optimization model and solution procedure are convenient for this sort of guided meta-search

• Purpose of this talk is to pose guidelines

• This interactive search is often long and frustrating– Inherently indirect as clinician

changes input to an underlying optimization in order to guide it towards an acceptable plan

Typical Ingredients in Opt Models• Assume the beam angles are fixed• Decision variable = intensity of angle j, beamlet g

• Dose at point i is where are pre-computed unit dose coefficients

• Constraints:

– Min tumor dose

– Tumor homogeneity

– Min 2nd target dose

– Max healthy tissue dose

– Dose-volume limits

on healthy dose

jgxi ijg jgjgD a x ijga

min{ : } max{ : }

vol{ : }

Ti

i i

si s

hi h

h hh i

D D i T

D i T D i T

D D s i S

D D h i H

i H D L F h

Penalties & Importance Factors• The system of constraints for given

cases is almost always infeasible, i.e. there is no solution x

• Leads to a penalized violation format reducing all to one score to be minimized

constraint forms relevant

min ( )

where

( [ ]) measures the violation of for

( ) is an increasing nonneg function of

and is a nonneg importance factor on form

k k k ik i

k i

k

k

Pen Viol D

Viol D k i

Pen p p

k

x

x

Penalties & Importance Factors

• Penalty forms

– Squared violation

– Absolute violation

– Piecewise linear

violation

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

2( )

( ) | |

( ) max { }q q q

Pen v v

Pen v v

Pen v v

Optimization with a Single Score

• Single score fairly tractable for optimization– Unconstrained except for nonnegativity of x– Differentiable if squared penalty is used– Local minima (best only among those near) arise

with dose-volume constraints but manageable• Gradient methods solve quickly with squared• Simulated annealing follows randomized

search that adopts generated x-changes if improving & even if not with probability > 0

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

Steerability with a Single Score

• Claim the single score model is relatively poor on steerability– May input maxdose of 35 Gy in hopes of

getting 45 Gy– Manipulating importance factors by hand

has no guarantee of convergence– Difficult to predict what will change with

requirement relaxation or tightening

• Will use single score to illustrate some issues

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

Issue 1: Meaningful Start

• Steered searches must start somewhere, even when constraints are inconsistent

• Single score model is satisfactory in this regard because constraints are enforced only through penalization in the objective function

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

G1 Meaningful Start. Underlying optimization in a steered interactive search should guarantee a meaningful solution even if constraints are violated

Issue 2: Parameter Relevance

• Steering in the single score model is primarily via changing values of importance factors– This steering is indirect– Arbitrary numerical quantities without

clinical import or predictable impact– Difficult and frustrating to manipulate

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

k

G2 Parameter Relevance. Parameters manipulated in a steered interactive search should be meaningful to the application user

Issue 3: Objective Relevance

• In any optimization, relaxing (resp tightening) a requirement can only help (resp hurt) the optimal objective function value– That is, sign of impact is predictable– Local optima can confuse, but not usually too much

• True for single score, but impact is on the total score not on clinically relevant outcomes

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

G3 Objective Relevance. Underlying optimization in a steered interactive search should have an objective function value meaningful to the application user

Issue 4: Hard Constraints

• In the single score model, all constraints are soft, i.e. weighted but not required– Hard constraints are ones explicitly enforced– Required with e.g. dose to cord <= 45 Gy

• Increasing may fail with squared penalty

constraint forms relevant

min ( )k k k ik i

Pen Viol D x

G4 Hard Constraints. The underlying optimization in a steered interactive search should be able to enforce hard constraints (or prove their inconsistency)

k2e.g. min (viol of ) solves at * .5 /y y b y b

Issue 5: Efficient Frontier

• If we think of all soft constraints as objectives, we should seek a solution on the efficient frontier– No objective can be

improved without hurting at least 1 other

tum

or

do

seprotection of healthy tissue

efficient frontier

dominated solution

G5 Efficient Frontier. The underlying optimization in a steered interactive search should produce a solution on the efficient frontier of soft constraints at every round

Issue 5: Efficient Frontier• If 2 constraints can

be simultaneously satisfied, minimizing violation may not give efficient frontier

tum

or

do

se

protection of healthy tissue

• If 2 constraints are inconsistent an efficient solution can be obtained by minimizing violation

tum

or

do

se

protection of healthy tissue

Better Paradigm: Multiobj Opt• Good start: Hamacher and Kufer, “Inverse

radiation therapy planning-a multiple objective optimization approach”, Discrete Applied Mathematices 118, 145-161, 2002.

• Considers lower bounds on target(s), upper limit on healthy tissues (but no dose-volume)– Implies opts are Linear Programs (highly

tractable)

• Each round optimizes some weighted sum of objectives holding each to a hard limit– Optimizing instead of minimizing violation keeps

solutions on efficient frontier

Better Paradigm: Multiobj Opt• Paper actually proposes an automated search

of the efficient frontier– Returns a collection of candidate plans

• Could be adapted to provide a good basis for interactive steering1. Add some form of dose-volume constraints without

conceding too much tractability2. Use any desired starting solution procedure. Perhaps

single score, or maximizing target doses for fixed healthy tissue limits (which has to be feasible)

3. At each round, select (a) criteria to focus upon (by significantly escalating their weights) and/or (b) hard bound values to tighten or relax

Better Paradigm: Multiobj Opt• G1: Meaningful Start: Your choice• G2: Parameter Relevance: Primarily

changes in hard limits• G3: Objective Relevance: Escalated

weights link hard limit modification to expected changes in other criteria

• G4: Hard Constraints: Inherent with fixed limits

• G5: Efficient Frontier: Automatic with weighted sum optimization