ROBUST OPTIMIZATION

TECHNIQUES THAT ADDRESS

UNCERTAINTY IN INTENSITY-

MODULATED RADIATION THERAPY

Phil Diette

SE 724

April 29, 2015

Outline

• Introduction

• IMRT

• Robust Management of Motion Uncertainty in Intensity-

Modulated Radiation Therapy

• Uncertainty Set

• Formulation

• Results

• Future Work

• Nonconvex Robust Optimization for Problems with Constraints

• Formulation

• Uncertainty Set

• Results

• Future Work

Introduction

• High incidence of cancer

• 1,658,370 new cancer cases are expected to be diagnosed in the

US in 2015

• 589,430 Americans expected to die of cancer in 2015

• 2/3 of cancer patients receive radiation therapy

• Intensity-modulated radiation therapy (IMRT) shown to be

beneficial

IMRT

• Precision radiation

delivery

• Conforms beam to

tumor shape and size

• Beams of radiation

delivered from chosen

angles

• Beams are made up of

beamlets whose

intensities can be

adjusted

Problem

• Radiation kills tumors!

• But… radiation also kills good, healthy tissue and organs

• Optimization problem• Minimize radiation dose to healthy tissue

• Deliver adequate radiation to tumorous tissue

• IMRT Planning• Caregivers create a radiation treatment plan for a patient‘s specific

needs

• Includes planning intensities of radiation beamlets and angles of beams

• Not accounting for uncertainty leads to• Cold spots: underdosed tumorous tissue

• Hot spots: overdosed healthy tissue

Papers

• Case #1

• T. Bortfeld, T. Chan, A. Trofimov, J. Tsitsiklis (2008) Robust

Management of Motion Uncertainty in Intensity-Modulated

Radiation Therapy. Operations Research 56(6)

• Improvements over nominal planning

• Simple robust counterpart

• Case #2

• D. Bertsimas, O. Nohadani and K. M. Teo, (2010) Nonconvex

Robust Optimization for Problems with Constraints. INFORMS

Journal on Computing 22(1):44-58

• More general approach

• Lots of computation

Case 1

• Focuses specifically on breathing motion that effects lung

tumors during radiation therapy

• A common method of addressing breathing motion used a

margin around the tumor

• Guarantee minimum dose delivered to tumor, but also delivered

more dose to healthy tissue

PDF

• Another method proposed using a motion probability

density function

• Poor performance if the exact motion pdf isn’t realized during

treatment (irregular breathing, differences in health)

Uncertainty Set

• Solution: account for uncertainty in motion pdf with upper and lower bounds

• Uncertainty set 𝑃𝑈

𝑃𝑈 = 𝑝 ∈ 𝑝 𝑥 − 𝑝 𝑥 , 𝑝 𝑥 + 𝑝 𝑥 ∀𝑥 ∈ 𝑈; 𝑝 𝑥 = 𝑝 𝑥 ∀𝑥 ∈ 𝑋\𝑈;

𝑥∈𝑋

𝑝 𝑥 = 1

Nominal pdf, p(x)

Lower bound 𝒑 𝒙

Upper bound 𝒑 𝒙

Definitions

• V: set of voxels that tissue is divided into

• A voxel is a small volume of tissue with corresponding location

• T: subset of V that contains tumorous tissue

• N: subset of V that contains non-tumor, healthy tissue

• B: set of beamlets that make up a beam

• 𝐷𝑣,𝑏: dose that voxel v receives from beamlet b

• 𝑙𝑣: prescribed dose for voxel v

• 𝑤𝑏: intensity (or weight) of beamlet b

Problem – motion unaccounted for

min𝑤

𝑣∈𝑉

𝑏∈𝐵

𝐷𝑣,𝑏 𝑤𝑏

s.t.

𝑏∈𝐵

𝐷𝑣,𝑏 𝑤𝑏 ≥ 𝜃𝑣 ∀𝑣 ∈ 𝑇

𝑤𝑏 ≥ 0 ∀𝑏 ∈ 𝐵

Minimize total dose delivered to patient

Each voxel must receive prescribed dose

Nominal Formulation

min𝑤

𝑣∈𝑉

𝑏∈𝐵

𝑥∈𝑋

∆𝑣,𝑥,𝑏𝑝 𝑥 𝑤𝑏

s.t.

𝑏∈𝐵

𝑥∈𝑋

∆𝑣,𝑥,𝑏𝑝 𝑥 𝑤𝑏 ≥ 𝜃𝑣 ∀𝑣 ∈ 𝑇

𝑤𝑏 ≥ 0 ∀𝑏 ∈ 𝐵

• Account for motion with the motion pdf p(x)

• ∆𝑣,𝑥,𝑏: radiation dose delivered to voxel v, when the

anatomy is in breathing phase x, from beamlet b

Motion pdf

Robust Formulation

• Account for imperfect knowledge of motion pdf

• Robust solution: solution remains feasible for any realization of motion pdf (shallow breathing, irregular breathing, etc.)

min𝑤

𝑣∈𝑉

𝑏∈𝐵

𝑥∈𝑋

∆𝑣,𝑥,𝑏𝑝 𝑥 𝑤𝑏

s.t.

𝑏∈𝐵

𝑥∈𝑋

∆𝑣,𝑥,𝑏 𝑝 𝑥 𝑤𝑏 ≥ 𝜃𝑣 ∀𝑣 ∈ 𝑇, ∀ 𝑝 𝑥 ∈ 𝑃𝑈

𝑤𝑏 ≥ 0 ∀𝑏 ∈ 𝐵

𝑝 𝑥 − 𝑝 𝑥 ≤ 𝑝 ≤ 𝑝 𝑥 + 𝑝 𝑥 ∀𝑥 ∈ 𝑈

Robust Formulation – Finite Constraints

• Original robust formulation had infinite constraints

• Rewrite:

Final Robust Form

• Strong duality theory:

Results

• Applied to a clinical case

• Robust solution vs nominal solution

• Nominal solution: assumes no uncertainty in motion pdf

• Nominal solution led to average underdoses to tumor of 6-11%

• Robust solution worst-case realization was a 1% underdose

• Similar doses to healthy tissue

• Robust solution vs margin solution

• Margin solution assumes 100% uncertainty in motion pdf

• Robust solution delivered up 11% less radiation to left lung

Case #2

• Nonconvex Robust Optimization for Problems with

Constraints

• Example application to IMRT planning

• Robust solution: minimizes worst case costs due to

perturbations

• Iterative descent method that moves away from worst-

case errors while maintaining feasibility

Robust Nonconvex Optimization

• Definitions

• 𝒙: design vector

• Δx: implementation error

• U: uncertainty set

• 𝑈 ≔ ∆𝑥 ∈ 𝑹𝑛 ∆𝑥 2 ≤ Γ

• N: neighborhood of 𝒙

• 𝑁 ≔ {𝑥 𝑥 − 𝑥 2 ≤ Γ}

Problem

• min𝑥

𝑓(𝑥)

• Robust problem: minimize

worst-case errors

• min𝑥

max∆𝑥∈𝑈

𝑓(𝑥 + ∆𝑥)

Descent Direction

• Find descent direction away from worst case directions

• Solve following Second-Order Cone Program (SOCP)

Algorithm

• Step 0: Intitialization

• x1 arbitrary initial decision vector,

Set k:=1

• Step 1: Neighborhood

search:

• Search for worst cost neighbors

of 𝒙. Record all function

evaluations

• Step 2: Robust Local Move

• Solve SOCP

• Terminate if infeasible

• Set 𝑥𝑘+1 ≔ 𝑥𝑘 + 𝑡𝑘𝒅∗

• Set k:=k+1. Got to step 1.

Extend to Problem with Constraints

• Robust Formulation:

min𝑥

max∆𝑥∈𝑈

)𝑓(𝒙 + ∆𝒙

s.t. max∆𝑥∈𝑈

ℎ𝑗 𝒙 + ∆𝒙 ≤ 0 ∀𝑗

• Problem w/ constraints:

min𝑥

)𝑓(𝑥

𝑠. 𝑡. ℎ𝑗 𝒙 ≤ 0 ∀𝑗

Robust Optimization w/ Constraints

• Neighborhood search:

• Identify violated

constraints

• max∆𝑥∈𝑈

ℎ𝑗 𝒙 + ∆𝒙

Robust Optimization w/ Constraints

• Robust Local Move

• 𝒙 infeasible under

perturbations:

• Step along descent

direction, 𝒅𝑓𝑒𝑎𝑠∗ , that

maximizes the angle to

𝒚𝑖 − 𝒙

• 𝒅𝑓𝑒𝑎𝑠∗ is found by solving

the following SOCP

Robust Optimization w/ Constraints

• Robust Local Move• 𝒙 feasible under

perturbations: • Search for constraint

violations just outside of neighborhood

• Step along descent direction,𝒅𝑐𝑜𝑠𝑡

∗ , found by solving the following SOCP

Algorithm Termination Criteria

• 𝒙∗ is a robust local minimum for the problem with

constraints if the following conditions apply:

• 𝒙∗ is feasible under all pertubations in the uncertainty set

• No descent direction,𝒅𝑐𝑜𝑠𝑡∗ , exists at 𝒙∗

Application to IMRT Planning

• Simultaneous optimization of beamlet intensity and beam

angle

• First paper to explore this through robust optimization

• Working with same hospital – Massachusetts General

Hospital

Nominal Problem

• Similar to previous case

• Dose, 𝐷𝑣𝑏 𝜃𝑖 , depends on beam angle

• 𝑐𝑣 penalizes important organs more than normal tissue

• Adds a constraint to limit dose to a voxel

minw,𝜃

𝑣∈𝑉

𝑖∈𝐼

𝑏∈𝐵

𝑐𝑣𝐷𝑣𝑏 𝜃𝑖 𝑤𝑖

𝑏

s.t.

𝑖∈𝐼

𝑏∈𝐵

𝐷𝑣𝑏 𝜃𝑖 𝑤𝑖

𝑏 ≥ 𝑙𝑣 ∀𝑣 ∈ 𝑇

𝑖∈𝐼

𝑏∈𝐵

𝐷𝑣𝑏 𝜃𝑖 𝑤𝑖

𝑏 ≥ 𝑢𝑣 ∀𝑣 ∈ 𝑉

𝑤𝑏𝑖 ≥ 0 ∀𝑏 ∈ 𝐵𝑖 , ∀𝑖 ∈ 𝐼

Minimize total dose delivered to patient

Ensure adequate dose

delivered to each voxel

Limit dose delivered to each

voxel

Robust Problem

• Implementation Errors: 𝜃 + ∆𝜃, 𝑤 ∗ 1 ± 𝛿

• 𝛿𝑖𝑏~𝑁 0,0.01 , ∆𝜃𝑖~ 0,

1

3

°

• 𝑈 =𝛿

0.03∆𝜃

𝛿

0.03∆𝜃

2

≤ Γ

• Robust Formulation:

Results

• Several robust solutions

were calculated

• Pareto Frontier

• Give clinicians ability to

trade-off between mean-

cost and probability of

constraint violation

Results

• Robust results compared

to a convex optimization

solution

• Fix θ

• Prob. of violation can be

high

• Convex better

• Prob of violation needs to

be low (near vital organ)

• Robust local search better

• But robust local search is

more general

Robust

local search

Convex opt

Discussion – PDF Approach

• Improvements over

existing planning

methods

• Tractable robust

counterpart

• Future Work:

• Generalize

• Explore other uncertainty

sets

• Does not account

position uncertainties

between treatment

sessions

• Multi-stage approach

• Simultaneous

optimization of beamlet

intensity and beam angle

Discussion – Local Search Approach

• Generalized approach

• Can handle non-

convexities

• Provide clinicians with

trade-offs between

robustness and mean-

cost

• Future Work:

• Robust solution took 20

hours to solve

• Simplify constraints

• Improve neighborhood

search portion of algorithm

• Incorporate ideas from

PDF approach

• Shrink uncertainty set

• Cones of uncertainty?

Questions?