Medical considerations to guide mixed integer formulations of

IMRT planning problemsfrom MD to NP to CR

Mark Langer, MD

Radiation Oncology

Indiana University Medical Center

Formulations and fabulations

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Bahr, GK Radiol, 1968

Sonderman, Abrahamson, Op Res 1985; Lee

Langer, M, Int J Rad Onc 1990

Problem Scale• Beams – 9 to 36

– may need to choose “best” 9 of 36

• Beamlets per beam– 100 – 400

• Structures– 3-6

• Constraint tolerances– 3% at any point, or within 3mm if dose

gradient ≥ 30%/cm

• Optimization bound tolerance– 1 fraction size (about 2.5%)

Voxeldose calculated on a grid

.25mm X 0.25mm on a slice, slices 0.5 cm apart

10 cm wide

.25 cm

Dose distribution – matrix stored

Dose distribution - isodose plot

Dose distribution – dose volume histogram

10 cm

long1600 grid pts/slice X 20 = 32 000

4 -6 critical structures

Treatment volume

What kind of accuracy is expected?What is the Point to Take home?

• Dose at individual points– recommended agreement calculations vs measured:

– Low dose gradient (<30%/cm) region: 4% dose– High dose gradient region: 4mm

• Dyk, J. Van et al, for the Comm. Of Medical Physicists of the Ontario Cancer Institute and Foundation

• Int. J. Radiat. Oncol. Biol. Phys. 26:261-73(1993)

What kind of accuracy is expected?

• Dose at individual points• Measured vs. calculated doses

– “The phantoms will be designed to assess the accuracy of a delivered dose (+5%) near the center of the target”

Cumberlin, R. and Kaplan, R; NCI guidelines for IMRT as posted on ITC website (http://itc.wustl.edu; 05/17/04)

What kind of accuracy is expected?Radiation needs to be Fullfilling

Dose distribution across a volumeDose distribution against a standard– any discrepancy between the submitting institution's

DVHs and those computed by the ITC in excess of +5% (or 3 cc for small structures) in total volume or +5% (relative to the absolute structure volume) of the volume calculated to be at or above the appropriate TD 5/5 dose for the particular structure will need to be resolved prior to successfully completing the Dry Run Test

http://itc.wustl.edu/ 11/27/02 Image Guided Therapy Center

What kind of accuracy is required?Optimization must be better than good

• “In order to satisfy [the 1976 ICRU] recommendation [of 5%], each step in the radiotherapeutic process has to be performed at an accuracy better than 5%”

• Declich, F. et al, “dosimetric evaluation of a commercial 3-D treatment planning system using Report 55 by AAPM Task Group 23”Rad Onc 52: 69-77

What kind of accuracy is expected?When are outliers outliars?

• Target dose homogeneity

– “Dose Variation within the Planning Target volume…should be kept within +7% and -5% of the prescribed dose…otherwise, it is the responsibility of the radiation oncologist to decide whether this can be accepted or not ” ICRU report 50, section 2.4.1.

Homogeneity and the Maximum doseBrush off the Biggest

• For 2D, max target dose only counted for an area >2cm2 (ICRU Report 29)

• For 3D, max target dose only counted for a volume whose min. diameter is >15mm (ICRU 50)

• Outside the target, similarly sized volume called “hot spot” (ICRU 50)

Homogeneity and the minimum doseSmall is dutiful

• “In contrast to the situation with the maximum absorbed dose, no volume is limit is recommended when reporting minimum dose.” (ICRU 50, section 2.4.4.)

Normal tissue maximums

• Can apply minimum volume rule if maximum dose is not critical

• Otherwise, a smaller volume is applied• ICRU 50• Cord,e.g.

Structure Fraction held to Dose Limit

Limit on Total Dose

Limit on Dose per Fraction

Spinal Cord 100% 45 Gy+3% 2 Gy

Lung 66%-5% 20 Gy+3% 2 Gy

microscopic tumor

100% 45 Gy-3% 2 Gy

Normal Tissue LimitsNormal Tissue and Tumor Dose Limits

Homogeneity Limit

ObjectiveStructure Maximize

Tumor min dose

Bound on Improvement

2.5%

Structure Volume Homogeneity limit

(max – min dose)

Target 100% -5% ≤10%

Gaining Traction

• Sparsing of [aij ] matrix

• Sampling of point set {i}

• Sampling of beamlet set {j}– Geometric (e.g, by target\normal tissue

projections)– Random, iterative (fill in the holes)– Numerical – column generation

Sources of Discrepancy

c

0.25 cm

Report discrepancies

- isodose plot grid ≠ dose calculation grid - pixel grid finer than dose calculation grid - dvh sampling is

inferior - within voxel

variations, if >50%/cm →>5% in 1mm

2

2

2

10

Optimization Failures

+ +

+

+

++

Types of Discrepancies

• Violations– In volume– in Dose– In homogeneity

• Shortfalls

Dose (in Gy)

% VolumeHeart Volume

100%

50%

60 Gy35 Gy(1) Absolute dose limit

4% vol error

4% e

rror

in d

ose

NEEDED: An Accounting for error in estimates of dose and volume when the dvh is constructed

Dose (in Gy)

% VolumeHeart Volume

100%

50%

60 Gy35 Gy(1) Absolute dose limit

Type I reported violation: A dvh through this point would be clearly be in violation; it lies within the hatched region, that is placed outside the range of uncertainty in volume (4%) and dose (4%)

4% vol error

4% e

rror

in d

ose

Dose (in Gy)

% Volume Heart Volume

100%

50%

60 Gy35 Gy(1) Absolute dose limit

4% vol error

4% e

rror

in d

ose

Type II violation : A point lies outside the 4% dose calc error, but it could lie within the 4% uncertainty in estimating the volume holding that point

Dose (in Gy)

% Volume Heart Volume

100%

50%

60 Gy35 Gy(1) Absolute dose limit

Type III violation :The dose volume histogram lies within the region of ±4% uncertainty in dose and volume. But >4% volume exceeds the dose limit by some amount, and there is an error of >4% in dose within some part of the volume. The maximum recorded dose is plotted, along with the range of error in volume holding that dose (0%-4%)

Maximum violations in lung dose-volume limit in no shortfall 3D trials

# sample points

trials #trials with lung violations

#trials with shortfalls

Volume violation

Dose violation

450 35 11 5 3% 4.9 Gy

600 35 11 6 4% 7.4 Gy

800 34* 8 8 2% 4.1 Gy

* one trial could not be solved

Langer, M.; et al, “The reliability of optimization under dose-volume limits”, Int. J. Radiat Oncol. Biol. Phys. 26:529-538; 1993

Sampling and time

Sampling points

Avg time 95th percentile

Maximum time

450 3.78 min 12.3 min. 36 min.

600 4.03 min 17.9 min 31 min

800 56 min 101 min 1315 min

Langer, M.; et al, “The reliability of optimization under dose-volume limits”, Int. J. Radiat Oncol. Biol. Phys. 26:529-538; 1993

Formulations and fabulations

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Bahr, GK Radiol, 1968

Sonderman, Abrahamson, Op Res 1985

Langer, M, Int J Rad Onc 1990

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Rardin, R.; Langer, M.P.; Preiciado-Walters, F.; Thai, V. Column Generation for IMRT Cancer Therapy Optimization with Implementable Segments. Euro/Informs Joint International Meeting, #2704, July 6-10, 2003, Istanbul. (meeting presentation)

Langer, M.; Thai, V. Delivery of Static Field IMRT is Improved by Avoiding Discretization of Intensity Levels. Int. J. Radiat. Oncol. Biol. Phys. 54(2):156; October 6-10, 2002, New Orleans, LA (meeting presentation).

Langer, M.; Thai, V.; Papiez, L. Improved Leaf Sequencing Reduces Segments or Monitor Units Needed to Deliver IMRT Using Multileaf Collimators. Med. Phys. 28(12): 2450-2458; 2001.

Preciado-Walters, F.; Langer, M.P.; Rardin, R.; Thai, V. Column generation for IMRT Cancer Therapy - Optimization with Implementable Segments.

Cord 45 Gy Target dose homogeneity .85Maximize minimum target dose

Target DVH (optimization sample points[182])

0

10

20

30

40

50

60

70

80

90

100

4500 5000 5500 6000 6500 7000 7500 8000 8500 9000

Dose (cGy)

Continuous LevelsDiscrete Levels

Intensity map

Continuous Levels relaxed homogeneity

Structure Clinical plan Criteria

IMRT constraint template (starting)

PTV PTV1=54 Gy

PTV2=70 Gy

Max dose 120% prescribed

PTV1

prescribed=54Gy

PTV2

prescribed=70Gy

PTV1 =51.3-56.7 Gy

PTV2 =66.5-73.5 Gy

Penalty =25

spinal cord

Max dose 40 Gy Max dose = 35 Gy

Penalty = 100

Brain stem

Max dose 45 Gy Max dose = 35 Gy

Penalty = 100

1.2 x (54)=64.8

1.2 x (70)=84

Hunt, MA et al, IJROBP 49:623-632; 2001

What is the effect of uncertainty in geometry?

Minimum Dose

79.279.6 79.8

McCormick, T.C.; Dink, D.; Orcun, S.; Pekny, J.; Rardin, R.; Baxter, L.; Langer, M. Projecting the Effect of Target Boundary Uncertainty on the Tumor Dose Prescription to Guide Contouring for Treatment Planning. Int. J. Radiat. Oncol.Biol. Phys. 57(2 suppl):S235 (2003) Amer. Soc. Therap. Radiol. Oncol. 45th Annual Meeting (meeting presentation).

Conclusions

• MDs can formulate the planning problems

• They are combinatorially complex

• We can identify levels of required accuracy

• Is there an advantage or disadvantage to incorporate delivery constraints?