Optimization Modeling and Computational Issues in...

Optimization Modeling andComputational Issues in

Radiation Therapy

(lecture developed in collaboration with Peng Sun)

February 5, 2002

Outline

1. Radiation Therapy

2. Linear Optimization Models

3. Computation

4. Nonlinear and Mixed-Integer Models

5. Looking Ahead to the Course

RadiationTherapy

Overview�This year, 1,200,000 Americans will be diagnosed

with cancer

� 600,000+ patients will receive radiation therapy

– beam(s) of radiation delivered to the body inorder to kill cancer cells

�Sadly, only 67% of “curable” patients will be cured

RadiationTherapy

Overview�High doses of radiation (energy/unit mass) can kill

cells and/or prevent them from growing and dividing

– true for cancer cells and normal cells

�Radiation is attractive because the repairmechanisms for cancer cells is less efficient than fornormal cells

RadiationTherapy

Overview�Recent advances in radiation therapy now make it

possible to:

– map the cancerous region in greater detail– aim a larger number of different “beamlets” with

greater specificity

�Spawned the new field of tomotherapy

� “Optimizing the Delivery of Radiation Therapy toCancer Patients,” by Shepard, Ferris, Olivera, andMackie, SIAM Review, Vol. 41, pp. 721–744, 1999.

RadiationTherapy

Overview

Conventional Radiotherapy...

Relative Intensity of Dose Delivered

RadiationTherapy

Overview

...Conventional Radiotherapy...

tumor5

Relative Intensity of Dose Delivered

RadiationTherapy

Overview

...Conventional Radiotherapy...

In conventional radiotherapy

– 3 to 7 beams of radiation

– radiation oncologist and physicistwork together to determine a set ofbeam angles and beam intensities

– determined by manual “trial-and-error” process

RadiationTherapy

Overview

...Conventional Radiotherapy

Complex Shaped Tumor Area

Critical Area Present

With only a small number of beams, it is difficult/impossible to

deliver required dose to tumor without impacting the critical area.

RadiationTherapy

Overview

Recent Advances...�More accurate map of tumor area

– CT — Computed Tomography– MRI — Magnetic Resonance Imaging

�More accurate delivery of radiation

– IMRT: Intensity Modulated Radiation Therapy– Tomotherapy

RadiationTherapy

Overview

...Recent Advances

RadiationTherapy

Overview

Formal Problem Statement...

�For a given tumor and given critical areas

�For a given set of possible beamlet origins andangles

�Determine the weight on each beamlet such that:

– dosage over the tumor area will be at least a targetlevel ��

– dosage over the critical area will be at most atarget level ��

RadiationTherapy

Overview

...Formal Problem Statement

20 40 60 80 100 120 140 160 180 200

LinearOptimization Models

Discretize the Space

Divide up region into a 2-dimensional (or3-dimensional) grid of pixels

pixel (i,j)

Create Beamlet Data

Create the beamlet data for each of � � �� possiblebeamlets.

�� is the matrix of unit doses delivered by beam � .

��

� � � unit dose delivered to pixel �� by beamlet �.

Dosage Equations

Decision variables � � ��

�� intensity weight assigned to beamlet �,

� � �� .��

��

� � ��

(“��” denotes “by definition”)

� ��

��

Definitions of Regions

� is the target area

� is the critical area

� is normal tissue

� ��

Ideal Linear Model��

��

� � ��

� �

��

Ideal Linear Model

��

� � ��

� �

��

Unfortunately, this model is typically infeasible.

Cannot deliver dose to tumor without some harm to criticalarea(s).

Engineered Approaches��

��

� � ��

� �

��

� ��

��

�� )

Engineered Approaches

Some other possible objective functions:

Let �� be the target prescribed dose to bedelivered to pixel ��

��

� � ��

� �

This is the same as:

��

� � ��

� �

Here is another model:

��

� � ��

� �

This is the same as:

��

� � ��

� �

��

ComputationBase Case Model

Consider the “base case” example problem:

��

� � ��

� �

��

ComputationSize of the Model

Dimensional Analysis...��

��

� � ��

� � �

��

Dimensional Analysis:number of pixels � ��

number of beamlets � ��

�� ; �� ; � � � !��

��

...Dimensional Analysis...��

��

� � ��

� � �

��

Decision Variables Number� � ��

� ��

�� Total � � ��

...Dimensional Analysis

��

� � ��

� �

��

Number of Constraints

Simple Variables Upper/Lower Bounds Number

� � � ��

Total ��

Other Constraints* Number��

��

Total ��

*We usually exclude simple variable upper/lower bounds when countingconstraints.

Summary

Variables Constraints*��

*Excludes variable upper/lower bounds.

Optimal Solution

20 40 60 80 100 120 140 160 180 200

Base Case Model Solution

ComputationAnother Model Solution

20 40 60 80 100 120 140 160 180 200

Solution of a nonlinear model.

ComputationDose Histogram

of Solution

0 5 10 15 20 250

Dose Volume HistogramQP model

Fraction ofTotal Dose

tumor normal

critical

Dose Volume

ComputationAnother Model Solution

20 40 60 80 100 120 140 160 180 200

Solution of a nonlinear model, where � � � � � � �.

ComputationComputational Issues

Software/Algorithms

�Software codes:

– CPLEX simplex (pivoting method)– CPLEX barrier– LOQO

�Algorithms:

– Simplex method (“pivoting” method)– Interior-point method (IPM) (“barrier” method)

Counting Iterations

� Iteration Counts:

– Number of pivots for simplex method– Number of Newton steps for IPM

Issues in Running Times

�Running time will be affected by:– number of constraints– number of variables– software code– type of algorithm (simplex or IPM)– properties of linear algebra systems involved

� density/patterns of nonzeroes of matrix systems to besolved

– other problem characteristics specific to problem– idiosyncratic influences– pre-processing heuristics

ComputationBase Case

No Pre-Processing

�Base Case Model

�No Pre-Processing

Running TimeCPU WallCode Algorithm Iterations(sec) (minutes)

CPLEX Simplex 183,530 440 250CPLEX Barrier 49 13 37

ComputationSome Generic Rules

1. The simplex algorithm is designed to handle variables withlower bounds and upper bounds:

��

� � �

� � � � �

where �� and/or �� is allowed.

2. We say �� has no bounds if �� and �� .Otherwise �� is a bounded variable.

��

� � � � �

3. For the simplex method, the work per pivot generally dependson the number of nonzeros in .

4. If is very sparse (its density of nonzero elements is low), thenthe work per pivot will be low.

5. The number of simplex pivots in a “good” model is roughlybetween � and � �.

��

� � �

� � � � �

5. The work per iteration of an interior-point method generallydepends on the structure of the matrix

� ��

6. The structure of � is often (but not always) related to thestructure of the matrix because the following two matricesare “similar”:

� ��

� �

7. The number of interior-point method iterations is typically

��–� (independent of � and/or �).

ComputationPre-Processing

Heuristics...

Pre-Processing Heuristics inCommercial-Grade Software

�Designed to Eliminate Constraints and/or Variables

�Example:

��

� � � � � � � ! � � � � �

...Heuristics...

Example:��

� � � � � � � � � � � � �

� � ��

Therefore we can eliminate the bounds on �

Therefore we can treat � as a free variable

Therefore we can eliminate � from our model altogether.

...Heuristics

�Base Case Model

�With Pre-Processing

CPLEX Simplex 18,428 4.3 4CPLEX Barrier 16 130 133

ComputationEquivalent Formulation

“Small” Model...

Equivalent Formulation: (eliminate �� )

“Small” Model:

��

� � ��

� �

...“Small” Model...

Base Case Model Small ModelVariables ��

Constraints* �� !��

*always excludes simple variable upper/lower bounds

...“Small” Model

�Small Model

CPLEX Simplex 171,656 390 216CPLEX Barrier 57 80 31

ComputationComparisons

Running TimeWallCode Algorithm Model

(minutes)Base Case 250

CPLEX Simplex Pre-Processed 4Small Model 216

Base Case 37CPLEX Barrier Pre-Processed 133

Small Model 31

NonlinearOptimization

Quadratic Model��

��

� � �

��

� �

��

� � ��

� �

Quadratic Model

Quadratic Model Output

20 40 60 80 100 120 140 160 180 200

Quadratic Model

Computational Results

Running TimeCPUModel Code Algorithm Iterations(sec)

Base Case QP Model LOQO Barrier 31 82.7Small QP Model LOQO Barrier 32 149.0

Mixed IntegerOptimization

Limiting the Number of Beamlets

��

� � ��

� � �

��

Mixed IntegerOptimization

Computation

CPLEX MIP Solver

Running TimeMIP Gap Simplex CPU Wall

(%) Pivots (seconds) (minutes)20 11,646 7 415 11,646 7 412 11,646 5 410 14,538 9 67 14,538 7 65 14,538 10 64 14,538 7 63 14,538 5 62 3,655,445 1,700 25.3 hours

Modifications ofthe Model

Partial Volume Constraints

Partial Volume Constraints:

“No more than ! " of the critical region can exceed adose of � ��.”

“No more than �" of the critical region can exceed adose of � ��.”

Approach #1 (Integer Programming Model)

Let � be a very large number,

��

�� !

��

Approach #2 (Error Function Approach)

The error function, or sigmoid function, is of the form:

��

Instead of integer variables, we use

��

�� !

��

Looking AheadModeling Languages

Used in the Course

�Modeling languages and software used in the course

– OPL Studio

linear and mixed-integer programming

solver is CPLEX simplex and/or CPLEX barrier

first half of course– AMPL

linear and nonlinear programming

solver is LOQO

second half of course

Looking AheadModeling Tools

and Issues

� “Column Generation” (week 3)

– generates new decision variables “on the fly”

�Exact optimization and exact feasibility

– in models– in algorithms

�Computational Issues in LP (next lecture)

– simplex method with upper/lower bounds– methods for updating the basis inverse

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