Computational Optimization Techniques Applied to

Brachytherapy, External Beam and IMRT

Eva K. Lee, Ph.D.Industrial & Systems Engineering, Georgia Institute of TechnologIndustrial & Systems Engineering, Georgia Institute of Technology y

Computational Research & Informatics, Radiation Oncology, Emory Computational Research & Informatics, Radiation Oncology, Emory University University Prepared for the NCIPrepared for the NCI--NSF workshop on Operations Research in Radiation NSF workshop on Operations Research in Radiation

Therapy, February 7Therapy, February 7--9 2002, Washington DC.9 2002, Washington DC.

Overview of TalkPart I: Optimization models for treatment planning in

IMRTMIP modelsDose-volume restrictions, dose homogeneity, multiple objectives, and conflicting constraints Clinical results

Part II: Image-guided dose escalation and biological treatment in brachytherapy

Part III: OR issues

Computerized Optimization ApproachProvide an automated planning mechanism which allows clinically desirable constraints to be input into treatment planning modelsImprove patient care:

Improve tumor control, reduce normal tissue complications Produce plans with different clinical properties and allow clinicians to objectively select the “best” one based on RTOG standardsImprove efficiency and reduce planning time

Provide a research tool to push frontier of understandingSuperior plans and negligible planning time open up opportunities for more complex clinical investigations

Part I: Simultaneous beam angles and intensity map optimization in IMRT viamixed integer programming

Lee, Fox, Crocker (Georgia Tech and Emory Collaboration, 1998—present)

Some Goals in Planning Models

Ensure at least 95% dose coverageControl dose within tumor region

Provide good local tumor controlControl dose on tumor contour

Ensure prescription dose conformity to tumorReduce radiation to external tissue

Minimize dose to critical structuresLower normal tissue complication

More Constraints & ConsiderationsStrict dose-volume criteria for different anatomical structuresDose homogeneity Number of beams/angles/collimators usedPrioritize importance of dose delivered to various anatomical structuresUnderdose to tumor (lower bound on dose) Maximum dose to tumor

Dose Calculation for External BeamNon-coplanar Arc Dose Calculation

Circular collimator sizesIMRT Dose Calculation

3D dose computation using convolution-superposition algorithms (which have been implemented by various researchers in the field)Obtain fluence information directly from commercial treatment delivery system

Fluence and dose are not the same. For this talk, will only focus on dose.

Model ParametersPlanning target volume (PTV), critical structures and nearby normal tissue in discretized voxels.A collection of beams, Bi , i in I, from different directions/gantriesDP, ia — dose per monitor unit contribution of a beamlet a (from beam i) to a voxel P PrDose — Prescription dose

LP and UP — Target lower and upper bounds for radiation dose to voxel P

Beam configuration constants — maximum no. of beams, isocenters, collimators, gantry angles, couch angles allowed in plan

Model Variables

Beam GeometriesArcs/beamsCouch anglesGantry anglesIsocentersCollimater sizes

Beamlet intensity

Mixed integer programming-based models: Variables

wia >= 0 — intensity corresponding to beamlet ia. Total radiation dose delivered to a voxel P: ∑i, a DP,ia wiayp — 0/1 variable to record if voxel P received the prescription dose, PrDose.

bi, φi, γi, ti , ci — 0/1 variables to indicate, respectively, the use of beam i, couch anglei, gantry angle i, target i, collimator i.

MIP Model –PTV Objective and Constraints

||95.0)1(0)1(

01.0,

,

PTVyyf

yf

fwDtoSubject

fMaximize

PTVPP

PP

PP

Piia

iP

PTVPP

aa

≥−≥−−

≤−

=+

∑

∑

∑

∈

∈

PrDose PrDose

PrDose Constraints for

.Coverage

.Homogeneity

.Underdose

wia nonnegative, fP free, yP ∈ {0, 1}

Plus DVH constraints, and restrictions on the number of beams,

targets, collimators, couch angles, and gantry angles

DVHs Constraint for Critical Structures

1,1,1

||85.0

||65.0

||25.0

,

,,

,

,,

,

,

,,

,,

=+=+=+≥≥

≥+≤

≥+≤

≥+≤

≤≤

∑∑

∑∑

∑∑

∑

∈

∈

∈

ppppppppp

CriticalPPppi

iaiP

CriticalPPppi

iaiP

CriticalPPppi

iaiP

piia

iPp

zyzvzuuvy

CriticalyzywD

CriticalvzvwD

CriticaluzuwD

UwDL

aa

aa

aa

aa

max100%

max50%

max20%

DD

DD

DD

Variations in the Objectives

Minimize total dose to critical structuresMinimize excess dose on the boundary between tumor and external normal tissue (maximize dose gradient)Minimize maximum dose received by critical structuresMaximize minimum dose received by tumor volumeWeighted sum of different objectives (prioritize importance of achieving target dose levels for different anatomical structures)

Evaluation Criteria

Coverage: Percentage of tumor volume covered by prescription isodose curve.Conformity: Ratio of prescription dose volume to target volume.Homogeneity: Ratio of maximum dose received by tumor to prescription dose Toxicity: Ratio of maximum dose received by normal tissue to prescription dose Dose distribution: DVHs and isodose curves

Transverse CorSag

IMRTMRT

3DCRTDC

PlanPlan CoverageCoverage HomogeneityHomogeneity ConformityConformity ToxicityToxicity

IMRT 0.99 1.4 1.3 0.83DCRT 0.98 1.9 1.6 0.9

PlanPlan CoverageCoverage HomogeneityHomogeneity ConformityConformity ToxicityToxicity

IMRT 0.99 1.2 1.1 0.83DCRT 0.99 1.6 1.5 1.1

Transverse CorSag

IMRTTMRT

3DCRT

Transverse CorSag

16 Beams

6 Beams

PlaPla CoverageCoverage HomogeHomoge ConformConform ToxicityToxicity

16 0.99 1.4 1.1 0.8

10 0.98 1.3 1.5 1.1

8 0.99 1.3 1.6 1.2

6 1.00 1.5 1.4 1.04 0.99 1.6 1.5 0.8

Minimize dose to C-structure and maximize dose gradient

Maximize dose gradient

ReferencesLee, Fox, Crocker, Optimization of Radiosurgery Treatment Planning via Mixed Integer Programming Medical Physics Vol. 27(5), 995-1004, 2000.Lee, Fox, Crocker, Integer Programming Applied to Intensity-Modulated Radiation Therapy Treatment Planning 2001, Annals of Operations Research, accepted for publication*Lee, Fox, Crocker, Beam Geometry and Intensity Map Optimization in IMRT via Mixed Integer Programming, 2002, submitted.*Lee, Fox, Crocker, Effects of beam configuration and tumor representation on dosimetry and plan quality, submitted.*Lee, Analyzing handling of multiple objectives in intensity modulated radiation therapy treatment planning, 2002, submitted.** preprints available upon request: eva.lee@isye.gatech.edu

Part II: Brachytherapy Study

MRS-guided TCP-NTCP Dose Escalation Study in

Prostate Implants

Lee, Zaider etal. (Georgia Tech & Memorial Sloan Collaboration, 1996 — present)

MRS-Guided Dose Escalation

Obj: weighted sum (max coverage & min conformity)Constraints: dose-volume constraints for prostate, escalated dose to tumor pockets, strict dose bounds to urethra and rectum

Minimum and maximum doses in urethra (Gy); prescription dose:144 Gy

computerized plan (Plan A) MRS-guided plan (Plan B) Tumor volume

(cm3) Minimum dose Max dose Minimum dose Max dose

1.36 146.0 172.7 144.3 172.5

2.35 146.0 172.7 144.3 172.5

3.71 146.0 172.7 143.3 174.0

Dose-volume distributions for urethra (percentage of volume in each dose interval)

for three escalated plans (each with different tumor pocket size).

MRS-guided plan Dose interval

(Gy) 1.36 cm3 2.35 cm3 3.71 cm3

Computerized

plan

144-158.4 25% 25% 22.5% 18.75%

158.4-172.8 75% 75% 77.5% 81.25%

Estimated TCP values for 3 different tumor volumes

Estimated TCP values (n=1.36 109 cells, PV=38.1 cm3)

Tumor volume

(cm3)

computerized

plan (Plan A)

MRS-guided plan

(Plan B)

Ratio of Plan B to

Plan A

1.36 0.649 0.943 1.45

2.35 0.650 0.965 1.48

3.71 0.761 0.948 1.25

MRS-guided plan appears consistently superior to the standard plan.

ReferencesEK Lee, M Zaider, Mixed Integer Programming Approaches to Treatment Planning for Brachytherapy -- Application to Permanent Prostate Implants, 2001. Annals of Operations Research (Optimization in Medicine), accepted for publication, in press. M Zaider, M Zelefsky, EK Lee, K Zakian, HA Amols, J Dyke, J Koutcher, Treatment Planning for Prostate Implants Using MR Spectroscopy Imaging. Int J Radiat Oncol BiolPhys., 47(4): 1085-96, 2000.EK Lee, RJ Gallagher, D Silvern, CS Wuu and M ZaiderTreatment Planning for Brachytherapy: an Integer Programming Model, Two Computational Approaches and Experiments with Permanent Prostate Implant Planning. Phys Med Biol., 44, No. 1, 145-165, 1999.

Part III: OR Issues –Computational Effort

MIP models for IMRT can generate problems involving tens to hundreds of thousands of variables. The example shown in this talk involved about 50,000 rows and 50,000 columnsInitial LP relaxation could be difficult to solve. Employing in house parallel linear solver helps.In-house parallel integer programming solver is used for solving the IP. Specialized heuristic routine appears to work well, quickly returns (within a few minutes) treatment plans which satisfy all constraints and of good clinical quality.

OR Issues – Properties of the Constraint Matrix

Objectives: some of the OR techniques seem to work well for handling multiple objectivesConstraints:

Not all dose constraints can be satisfied simultaneously. Need to manage them as in the case for multiobjectives.Do not need to include all constraints initially, can generate them as we proceed with finding feasible solutions (constraint generation).Due to the nature of radiation, dose constraint matrix tends to be denser than instances from industrial applications.Possible nonlinear structure when TCP and NTCP are incorporated.

OR Issues – Properties of the beam variables

Beams: No need to include all beams in initial formulation. Beams can be introduced successively during solution process (column generation).

QuestionsHow much time should we allow for finding the best treatment plan?What type of uncertainty do we expect in dose calculation (sensitivity analysis)?What type of uncertainty in the dose received do we expect in executing the plan? How much improvement (TCP) do we want and how hard are we willing to try to find a better solution?

References

Zaider, Lee, Treatment planning for low dose rate and high dose rate brachytherapy 2002: In ``Basic and Advanced Techniques in Prostate Brachytherapy,” Ed. Dicker, Merrick, Gomella, Valicenti, Waterman.Lee, M. Zaider, Treatment planning optimization in brachytherapy, Handbook of Operations Research/Management Science Applications in Health Care, Kluwer Academic Publishers 2003.Lee, Fox, Crocker, Intensity-modulated treatment planning via computational optimization – models and issues, Handbook of Operations Research/Management Science Applications in Health Care, Kluwer Academic Publishers 2003.