Optimization Techniques and their Suitability and Robustness for IMRT Planning
Overview of Talk • Optimization approaches for intensitymodulated radiation therapy – Modeling approaches (MIP versus nonlinear)
Eva K. Lee, Ph.D. Kyungduck Cha, Jin Shi Center for Operations Research in Medicine, Industrial & Systems Systems Engineering, Georgia Institute of Technology Winship Cancer Institute, Emory University School of Medicine Radiation Oncology, Emory University School of Medicine
Size of Treatment Models • Beam Geometry – Number of input candidate beams – 0/1 variables – Couch angles, collimator sizes, energy, isocenters – Beamlet intensity – continuous variables
• Management of solution spaces • Clinical Comparisons • Discussion
Treatment Models and Tractability Polynomial-time Convex •Linear •Nonlinear
• Constraints/Objectives – Planning target volume (PTV), Organs-at-risks (OARs) in discretized representation – Clinical constraints (dose-based, dose-volume, spatial, EUD, biological TCP, NTCP)
• Uncertainties – Estimate in dose calculation – Uncertainty in voxel positions (Time-Dependent Planning) – Delivery parameters (Setup errors)
NP-Hard Non-convex •Nonlinear •Discrete/combinatorial Optimization (linear/nonlinear mixed integer programming)
NP-Complete
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Treatment Models and Tractability Polynomial-time Convex
Polynomial-time Some Algorithms
•Linear •Nonlinear
NP-Hard Non-convex •Nonlinear •Discrete/combinatorial Optimization (linear/nonlinear mixed integer programming)
NP-Complete
Treatment Models and Tractability
•Exact Algorithms •Simplex •Gradient-based /interior-point based/Newton-type methods •Branch-and-bound •Heuristic Algorithms •Simulated annealing •Genetic algorithms •Evolutionary-genetic algorithms
Issues to Expand in this Talk • Selection of an optimization method and the associated “steering” objectives for solution procedure • DVH objectives – discrete vs continous models • Multiple-objective IMRT treatment planning
Convex
Constraint Types •Dose-based •Fluence intensity •DVHs
•Linear •Nonlinear
•Coverage •Beam orientation •Couch angles, energy type
NP-Hard Non-convex •Nonlinear •Discrete/combinatorial Optimization (linear/nonlinear mixed integer programming)
•EUD •Biological TCP/NTCP
NP-Complete
Methods C
• In-house modeling (CAMELA , Lee, Cha, Shi ’01-’03) and optimization modules (MIPSOLC , Lee, ’97-’03) developed based on (linear and nonlinear) mixed integer programming • Treatment planning models incorporate multiple objectives, dose constraints, spatial DVHs, PTV coverage, homogeneity (underdose/overdose), and a large collection of candidate beam orientations. • Compare MIP approach versus nonlinear programming approach with (DVHs) as the steering objective. • Compare two multiobjective strategies for IMRT planning with 4 steering objectives: PTV Homogeneity, min OAR doses, PTV Conformity, OAR-DVHs • Analyze plans based on clinical metrics: coverage, conformity, homogeneity, DVHs and isodose curves
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Treatment Planning Model Variables • Continuous variables to represent intensity for each beamlet • Continuous variables to measure dose discrepancy of each voxel from desired dose bounds, and 0/1 variable to record if voxel meets certain dose bounds (PTV coverage/over/underdose, OARs’ DVHs) • 0/1 variables to indicate, respectively, the use of beam directions, couch angle, energies, and collimator size.
A Head-and-neck Case
Observations
Modeling of DVHs: MIP/discrete approach versus nonlinear programming approach •Imposed Constraints: –Upper/lower/mean dose-based constraints, spatial DVHs, homogeneity constraints (PTV Underdose / Overdose), PTV coverage, maximum number of beams/angles allowed •Steering objective: –Optimize DVHs (maximizing volume versus penalizing total dose deviation)
A Head-and-neck Case
–Little change in PTV coverage, dose homogeneity, and DVH
–Little change in PTV coverage, dose homogeneity, and DVH
–DVHs of OARs change slightly with spatial information
–DVHs of OARs change more significantly with spatial information
–Lower dose to brainstem and spinal cord –Slightly higher dose to left parotid –Max dose remains the same for OARs –Mean dose difference is within 0.5% of each other MIP With spatial information MIP without spatial information
Observations
–Significant lower dose to brainstem and spinal cord, drastic reduction in max/mean/min dose –Slightly lower dose to right parotid –Slighter higher dose to left parotid Nonlinear NLP With spatial information Nonlinear NLP without spatial information
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A Head-and-neck Case (MIP vs NP)
MIP With spatial MIP without spatial
Nonlinear NLP With spatial Nonlinear NLP without spatial
Modeling of Multiple-Objectives MIP/discrete nonlinear approach Comparison – 4-stage preemptive versus goal programming
Prostate: Comparing Spatial Information
MIP with spatial MIP without spatial
Nonlinear with spatial Nonlinear without spatial
4-stage Preemptive Programming vs Goal Programming 4 Objectives: PTV Homogeneity, min OAR doses, PTV Conformity, OAR-DVHs
Observations –Compatible PTV coverage, DVHs and homogeneity –OARs (some tradeoffs in DVHs)
•Imposed Constraints: –Upper/lower/mean dose-based constraints, spatial DVHs, homogeneity constraints (PTV Underdose / Overdose), PTV coverage, maximum number of beams/angles allowed
–Lower dose to brainstem, spinal cord, left parotid and normal tissue in goal programming
•Steering objectives: PTV Homogeneity, min total OAR doses, PTV Conformity, OAR-DVHs
4-stage preemptive programming Goal programming
A Head-and-Neck Case
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Table 1. Dose Statistics from Optimal Plans Obtained by 4-stage preemptive programming vs Goal Programming
4-stage Preemptive Programming vs Goal Programming 4 Objectives: PTV Homogeneity, min OAR doses, PTV Conformity, OAR-DVHs
Coverage
Conformity
Homogeneity max/min
Mean dose(Gy)
Min dose(Gy)
Max dose(Gy)
4-stage preemptive programming
0.962
1.21
1.103
74.0
70.9
78.2
goal programming
0.964
1.19
1.114
74.2
70.4
78.4
PTV
Spinal Cord
Left Parotid
Right Parotid
Brain Stem
Spinal Cord
Left Parotid
Right Parotid
Brain Stem
Vol (cm 3 )
10.7
14.0
11.2
25.38
10.7
14.0
11.2
25.38
23.6%
49.8%
56.1%
66.3%
17.8%
44.5%
56.0%
59.3%
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–More homogeneous PTV dose from 4-state preemptive programming
Goal programming
4-stage preemptive programming Structure
†V
Observations
V30
8.9%
17.4%
46.2%
27.8%
3.4%
19.1%
46.0%
24.1%
V40
0.7%
10.1%
35.3%
7.4%
0
10.2%
35.2%
6.4%
V50
0
2.3%
25.4%
0
0
2.2%
25.0%
0
mean dose
12.0
22.7
29.7
23.4
12.1
21.3
29.7
22.3
max dose
42.0
55.1
75.1
54.0
39.8
54.7
75.2
53.9
min dose
0.2
1.5
0.8
1.5
0.2
1.0
0.3
1.4
–OARs –Lower dose to rectum, and bladder in goal programming –Slightly higher normal tissue max dose for goal programming plan 4-stage MIP-Preemptive Programming Goal Programming
A Head-and-Neck Case
Observations
Table 2. Dose Statistics from Optimal Plans Obtained by 4-stage preemptive programming vs Goal Programming Coverage
Conformity
Homogeneity max/min
Mean dose(Gy)
Min dose(Gy)
Max dose(Gy)
4-stage preemptive
0.953
1.20
1.082
76.9
74.0
80.1
Goal programming
0.938
1.19
1.152
78.0
72.6
83.6
PTV
A Prostate Case
•
Under the same solution space with explicit constraints in the treatment model: Upper/lower/mean dose-based constraints, spatial DVH constraints, Homogeneity constraints (PTV Underdose/Overdose), PTV Coverage constraints, OAR EUDs, maximum number of beams/angles allowed
•
With DVH-driven objective: – 1) maximize volume of OARs satisfying DVHs (MIP) versus
4-stage Preemptive Programming Structure Vol
(cm3
)
Bladder
Rectum
Tissue
– 2) Quadratic dose difference penalization (NLP)
Goal Programming Bladder
Rectum
Tissue
98.3
67.0
8072.1
98.3
67.0
8072.1
V20
45.0%
72.3%
39.4%
43.2%
51.8%
40.0%
V40
22.0%
36.3%
7.9%
19.4%
17.0%
7.1%
V60
12.5%
15.8%
1.4%
10.5%
6.3%
1.1%
mean dose
27.0
34.8
19.5
25.0
24.8
19.0
max dose
80.8
79.9
84.7
80.6
79.1
85.6
min dose
5.2
0.2
0.1
3.1
2.0
0.1
A Prostate Case
•
Findings: – NLP optimal plans are very sensitive to incorporation of spatial information within DVH-driven objective: In both head-and-neck and prostate cases, including spatial information drastically improves the dose-distribution in OARs while PTV maintains homogeneous and good coverage – MIP optimal plans are not as sensitive to spatial information. – MIP plans are superior to NLP plans in both tumor sites – Tradeoffs – MIP is NP-complete, there is no known polynomial-time algorithm, hence proven-optimal solution time can be very significant.
•
The type of models dictate the optimization algorithms used, and vice versa
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Observations • Under the same solution space with explicit constraints in the treatment model: Upper/lower/mean dose-based constraints, spatial DVH constraints, Homogeneity constraints (PTV Underdose/Overdose), PTV Coverage constraints, OAR EUDs, maximum number of beams/angles allowed
• With 4 clinical objectives: PTV Homogeneity, min total OAR doses, PTV Conformity, OAR-DVHs, contrast multi-stage preemptive programming versus goal programming • Findings: – Both approaches arrive at Pareto optimal points on the efficient frontier, and result in plans with improvement over single objective plans – Tradeoffs occur in PTV homogeneity versus OARs dose improvement – More detailed discussion on multiple objective IMRT planning on Tuesday: TU-C-BRA-8 (11:10 am, Ballroom A)
Related Manuscripts Recent Manuscripts (materials of this talk were extracted from these two papers) EK Lee, Handling Multiple Clinical Objective in Intensity Modulated Radiation Therapy Treatment Planning 2004. Selected oral presentation for AAPM 2004. Paper: EK Lee, KD Cha, J Shi, submitted to medical physics. EK Lee, Optimization Techniques and their Suitability and Robustness for IMRT Planning, invited presentation for AAPM 2004. Paper, EK Lee, KD Cha, J shi, to submit to medical physics Previous work EK Lee, T Fox, I Crocker, Integer Programming Applied to Intensity-Modulated Radiation Therapy Treatment Planning. Annals of Operations Research, Optimization in Medicine 119: 165-181, 2003 EK Lee, T Fox, I Crocker, Simultaneously Beam Geometry and Intensity Map Optimization in IMRT, 2003. Int. J of Rad Onc, Biology and Physics, in review. Modeling and Computational Engines E.K. Lee (1997), Computational Experience with a General Purpose Mixed 0/1 Integer Programming Solver (MIPSOL C ),'' Software Report, School of Industrial and Systems Engineering, Georgia Institute of Technology. EK Lee, KD Cha, J Shi, CAMELAC, An Algebraic ModEling LAnguage for mathematical programming in C, User Guide, 2001.
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