Mixed Integer Approaches to External Beam Radiotherapy

with IMRT

by Ronald L. Rardin, Ph.D.,NSF and Purdue University

Mark Langer, M.D., Indiana UniversityFelicia Preciado-Walters, Purdue University

(supported in part by NSF-0120145)

IMRT External Beam Radiation Therapy

• Applies multiple beams/beamlets of controllable intensity at different angles TargetTarget

• Seeks to maximize tumor dose without unsustainable damage to surrounding healthy tissues

• IMRT allows us to profile/contour beams

• Fix the set of beam angle choices (e.g. one every 10 degrees)

• Discretize tissues as point/voxel sets• Assume approximate linearity of dose as a

function of beam pattern intensities

Simplifications

pppatternsip xai ∑≈pointatdose

• Here the aip are pre-computed doses per unit intensity on IMRT patterns (more later) and the xp are decision variables for intensities

IMRT Beam Patterns

1 3 5 7 91

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0.20

0.40

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FractionTimeOpen

Beamlet Column

BeamletRow

• Decision variables xpchose background intensities for patterns p

• We will come later to the question of how they are generated

Objective Function

• Objective function maximizes minimum tumor dose over all tumor points

∑ ≥p

pip itxa

t

tumor allfor

:subject to maximize

min

min

Tumor Dose Homogeneity

• One set of constraints enforces limit on homogeneity of the tumor dose

• Min tumor dose at least a fraction α of max tumor dose

∑ ≤p

pip it

xa tumor allfor min

α

Simple Dose Limits on Healthy Tissues

• Simple dose limitconstraints keep the dose at any point of healthy tissue k no more than given limit bk

kibxa

k

pkpip in allfor

, ssuehealthy tieach For

∑ ≤

So Far an LP

• Up to now this model is a Linear Program (LP)• All variables xp and tmin are continuous • Objective function is linear in xp and tmin• All constraints are linear in the xp and tmin

• Implies we can solve to an exact optimum over thousands of points in minutes or less

Discrete Modeling of Beam Angle Count

• One discrete element is to limit the number of used beam angles j to at most n

• Auxiliary decision variables zj = 1 if beam is used and =0 otherwise

( )

∑

∑≤

≤

jj

jjforp

p

nz

znobigx

j

.

, angles allFor

Dose-Volume Limits

• Also model dose-volumelimit on healthy tissue k

• Pick points i to satisfy with variables yi = 1 if included, = 0 otherwise

simplelimit bk

dose

tighter limit dkon fraction fk

% ti

ssue

vol

ume

( )

( )∑

∑

≥

−+≤

kiniki

ip

kkkpip

ptsknofy

ki

ybdbxa

k

..

in allfor

,volume-dose subject to ssuehealthy tieach For

Full Example(Prostate)

Tissue

No. ofpoints

b(cGy)

%under

DVrest.

d(cGy)

Tumor 2438 - - -Bladder 1216 10000 80 8000Rectum 1578 10000 80 7500Femoral Head 1 870 7200 60 5000Femoral Head 2 859 7200 60 5000Skin 2654 15000 0 -

Homogeneity limit α = 85%, Beam angle count limit n = 9 of 36

Model Characteristics

• Model is a Mixed Integer Program (MIP)with continuous decision variables xp andtmin, plus discrete ones zj and yi• Generally much harder to find an optimal solution

than with LP’s• Still, every feasible solution satisfies

• Dose homogeneity limits • Simple dose limits on health tissues• Dose volume constraints on healthy tissues• Beam angle count limits

Provable Bound onMin Tumor Dose

• Critical advantage is easy availability of a provable (upper) boundon min tumor dose

• Obtained via relaxingthe MIP to an LP by treating the zj and yi as continuous (i.e. allowing fractions)

x

y

0

1

MIP feasiblepoints

extra solutionsfeasible in LP

(can only improveobjective function

value)

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Beamlet Column

BeamletRow

Pattern Column Generation

• Decision variablesxp chose background intensities for patterns p

• There are an enormous number of possibilities for such patterns

• Each a column of the constraints

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Beamlet Column

BeamletRow

Pattern Column Generation

• We start with conformal therapy patterns (100% on)

• New patterns are generated as the search evolves

• Tremendous flexibility in how these patterns are constructed

Column Generation Approach

• Tissue geometry can guide patterns (block the image of selected healthy tissues)

• We have used LP-based potentials estimating gain from increasing intensity of specific beamlets

LP-BASED-0.0003 -0.0003 -0.0003 -0.0006 -0.0026 -0.0476 -0.0542 POTENTIALS-0.0003 -0.0003 -0.0003 -0.0004 -0.0017 -0.0391 -0.0722-0.0003 -0.0003 -0.0003 -0.0003 -0.0006 -0.0037 -0.0451 -0.0523

-0.0002 0.0014 0.0067 0.0052 -0.0234 -0.0703 -0.0734Image of Tumor 0.0001 0.0035 0.2274 0.0998 0.2385 0.1050 -0.0568

0.0017 0.0090 0.0084 -0.0190 -0.0598 -0.0653

Next Pattern 0.0013 0.0572 0.0441 -0.0594 -0.0581

Some Results

• Case taken from clinical practice• Computation on a Sun SPARC 1104

using CPLEX commercial software to solve the LP relaxations

• Columns generated from LP-based potentials

• MIP feasible solutions constructed by rounding fractional values to 1 (or 0)

Our Best Result

MT.1.2.D (Prostate)No. normal tissues: 5Under d-v restrictions: 4N (max number of angles) N/AAlpha (homogeneity) 0.85

Tissue Points b (cGy) d (cGy) % dv

Skin 2654 15000 X XBladder 1216 10000 8000 80Rectum 1578 10000 7500 80Femoral Head 1 870 7200 5000 60Femoral Head 2 859 7200 5000 60Target 2438 X X X

Model MTD (cGy) CPU (mins) Columns

Comformal Bound 9885 1.2 36Rounded Conformal 9116IMRT Bound 10017 50.5 247Rounded IMRT 10015

• CF to IMRT bound +1.3%

•CF to IMRT rounded +9.9%

•CF max error 7.8%

•IMRT max error 0.02%

Conclusions

• Results so far depend on the case (IMRT error limits range to 10%)

• Still, believe we can make this approach solve IMRT’s to within provable 2-3% in under one hour

• Future focus on• Refining column generation methods• Improving the bounds with stronger relaxations• More sophisticated rounding methods• Time-phasing of delivery