Optimization Modeling and Computational Issues in...
Transcript of Optimization Modeling and Computational Issues in...
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Optimization Modeling andComputational Issues in
Radiation Therapy
(lecture developed in collaboration with Peng Sun)
February 5, 2002
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Outline
1. Radiation Therapy
2. Linear Optimization Models
3. Computation
4. Nonlinear and Mixed-Integer Models
5. Looking Ahead to the Course
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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
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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
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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.
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RadiationTherapy
Overview
Conventional Radiotherapy...
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9
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tumor
Relative Intensity of Dose Delivered
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RadiationTherapy
Overview
...Conventional Radiotherapy...
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tumor5
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Relative Intensity of Dose Delivered
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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
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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.
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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
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RadiationTherapy
Overview
...Recent Advances
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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 ��
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RadiationTherapy
Overview
...Formal Problem Statement
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LinearOptimization Models
Discretize the Space
Divide up region into a 2-dimensional (or3-dimensional) grid of pixels
pixel (i,j)
i
j
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LinearOptimization Models
Create Beamlet Data
Create the beamlet data for each of � � �� � � � � � possiblebeamlets.
�� is the matrix of unit doses delivered by beam � .
0
0
0
0
0
0
0.9
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1.0
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0.8
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0.9
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0.8
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0
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0
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1.0
1.0
0
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0
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1.0
��
� � � unit dose delivered to pixel ��� �� by beamlet �.
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LinearOptimization Models
Dosage Equations
Decision variables � � ���� � � � ����
�� � intensity weight assigned to beamlet �,
� � �� � � � � �.�� � ��
�������
� � ��
(“��” denotes “by definition”)
� ��
���������
is the matrix of the integral dose (total delivered dose)c�2002 Massachusetts Institute of Technology. All rights reserved. 15.094 15
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LinearOptimization Models
Definitions of Regions
151
151
� is the target area
� is the critical area
� is normal tissue
� �� � � � � �
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LinearOptimization Models
Ideal Linear Model������
���������� �
��
���� �� � �
�������
� � �� ��� � � �
� �
�� � � �� ��� � � �
�� � � �� ��� � � �
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LinearOptimization Models
Ideal Linear Model
������
���������� �
���
���� �� � �
�������
� � �� ��� �� � �
� �
�� � � � ��� �� � �
�� � � ��� �� � �
Unfortunately, this model is typically infeasible.
Cannot deliver dose to tumor without some harm to criticalarea(s).
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LinearOptimization Models
Engineered Approaches������ ��
��������� � � ��
��������� � � �
��������
�� �
������ �� � �
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��� � ��� � � ��� � ��� � � �
� ��
�
������� � � �� � � � � �
������������ � � � )
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LinearOptimization Models
Engineered Approaches
Some other possible objective functions:
Let �������� � be the target prescribed dose to bedelivered to pixel ��� �
������ ���
���������� � �������� ��
��
���� �� � �
�������
� � �� ��� � � �
� �
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LinearOptimization Models
Engineered Approaches
This is the same as:
������
����
���� ��� � �������� � � ��� � � �
�� � �
�������
� � �� ��� � � �
� �
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LinearOptimization Models
Engineered Approaches
Here is another model:
������
����������� � �������� ��
��
���� �� � �
�������
� � �� ��� � � �
� �
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LinearOptimization Models
Engineered Approaches
This is the same as:
������
��������
�� �
����
���� �� � �
�������
� � �� ��� � � �
� �
�� � ��� � �������� � ��� � ��� � � �
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ComputationBase Case Model
Consider the “base case” example problem:
151
151
�������� � � ��� ��� � � �
�������� � � � ��� � � �
�������� � � � ��� � �
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ComputationBase Case Model
������ � ��
��������� � � � �
��������� � � � �
��������� �
����
���� �� � �
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� � �� ��� � � �
� �
�� � ��� � �������� � ��� � ��� � � �
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ComputationSize of the Model
Dimensional Analysis...������ � ��
��������� � � ����
��������� � � �
��������
�� �
�����
���� � � �
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� � �� ��� �� � �
� � �
��� � � � � �������� � �� � ��� �� � �
Dimensional Analysis:number of pixels � �������� � � ��
number of beamlets � ��� ���
�� � � �� ��; ��� � �� ; � � � !���
��� � ������
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ComputationSize of the Model
...Dimensional Analysis...������ � ��
��������� � � ����
��������� � � �
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�� �
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� � �� ��� �� � �
� � �
��� � � � � �������� � �� � ��� �� � �
Decision Variables Number� � �� ��
� ���
�� � �� ��Total � � ��
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ComputationSize of the Model
...Dimensional Analysis
������ � �
��������� � � � �
��������� � � �
��������
�� �
�����
���� �� � �
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� � �� ��� �� � �
� �
��� � ��� � � �������� � ��� � ��� �� � �
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ComputationSize of the Model
Number of Constraints
Simple Variables Upper/Lower Bounds Number
� � � ���
Total ���
Other Constraints* Number�� � �� ��
�� � ���������� ������
Total ������
*We usually exclude simple variable upper/lower bounds when countingconstraints.
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ComputationSize of the Model
Summary
Variables Constraints*����� ������
*Excludes variable upper/lower bounds.
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ComputationBase Case Model
Optimal Solution
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Base Case Model Solution
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ComputationAnother Model Solution
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Solution of a nonlinear model.
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ComputationDose Histogram
of Solution
0 5 10 15 20 250
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Dose Volume HistogramQP model
Fraction ofTotal Dose
tumor normal
critical
Dose Volume
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ComputationAnother Model Solution
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Solution of a nonlinear model, where � � � � � � �.
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ComputationComputational Issues
Software/Algorithms
�Software codes:
– CPLEX simplex (pivoting method)– CPLEX barrier– LOQO
�Algorithms:
– Simplex method (“pivoting” method)– Interior-point method (IPM) (“barrier” method)
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ComputationComputational Issues
Counting Iterations
� Iteration Counts:
– Number of pivots for simplex method– Number of Newton steps for IPM
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ComputationComputational Issues
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
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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
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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.
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ComputationSome Generic Rules
��� � �
�
�� � �
� � � � �
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 � �.
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ComputationSome Generic Rules
��� ��
�
� � �
� � � � �
5. The work per iteration of an interior-point method generallydepends on the structure of the matrix
� ��
�
�
�
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ComputationSome Generic Rules
� ��
�
�
�
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 �).
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ComputationPre-Processing
Heuristics...
Pre-Processing Heuristics inCommercial-Grade Software
�Designed to Eliminate Constraints and/or Variables
�Example:
�� ��� �� � ��
� � � � � � � ! � � � � �
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ComputationPre-Processing
...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.
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ComputationPre-Processing
...Heuristics
�Base Case Model
�With Pre-Processing
Running TimeCPU WallCode Algorithm Iterations(sec) (minutes)
CPLEX Simplex 18,428 4.3 4CPLEX Barrier 16 130 133
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ComputationEquivalent Formulation
“Small” Model...
Equivalent Formulation: (eliminate �� �)
“Small” Model:
������ � �
��������� � � � �
��������� � ��
��������
�� �
������� ��� � �
�������
� � �� � �������� � ��� � ��� �� � �
� �
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ComputationEquivalent Formulation
...“Small” Model...
Base Case Model Small ModelVariables ����� ������
Constraints* ������ �!����
*always excludes simple variable upper/lower bounds
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ComputationEquivalent Formulation
...“Small” Model
�Small Model
Running TimeCPU WallCode Algorithm Iterations(sec) (minutes)
CPLEX Simplex 171,656 390 216CPLEX Barrier 57 80 31
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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
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NonlinearOptimization
Quadratic Model�� � ������ � �
���������� � ������� ���
���
� � �
���������� � ������� ���
� �
��������
��� � ������� ���
���� �� � �
�������
� � �� ��� �� � �
� �
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NonlinearOptimization
Quadratic Model
Quadratic Model Output
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
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NonlinearOptimization
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
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Mixed IntegerOptimization
Limiting the Number of Beamlets
������ � ��
��������� � � ����
��������� � � �
��������
�� �
�����
���� � � �
������
� � �� ��� �� � �
� � �
��� � � � � �������� � �� � ��� �� � �
�� ����� � � �� � � � � �
�� � ���� � � �� � � � � �
������� ���
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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
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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 � ��.”
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Modifications ofthe Model
Partial Volume Constraints
Approach #1 (Integer Programming Model)
Let � be a very large number,
�� � � � �� � �� �� �� � � � ���� �� � � �
�� � � � �� � �� �� �� � � � ���� �� � � �
��� ������ � � ��� � �!
��� ������ � � ��� � � �
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Modifications ofthe Model
Partial Volume Constraints
Approach #2 (Error Function Approach)
The error function, or sigmoid function, is of the form:
������ �
�
�� ����
������ � ��
�� � �
������ � � �� ���
������ � �� �� �
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Modifications ofthe Model
Partial Volume Constraints
Instead of integer variables, we use
��� ����
������ � � � � ��� � �!
��� ����
������ � � � � ��� � � �
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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
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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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