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![Page 1: Radiotherapy Planning Stephen C. Billups University of Colorado at Denver billups sbillups@carbon.ucdenver.edu.](https://reader038.fdocuments.in/reader038/viewer/2022110321/56649cf75503460f949c79f2/html5/thumbnails/1.jpg)
Radiotherapy Planning
Stephen C. BillupsUniversity of Colorado at Denverhttp://www-math.ucdenver.edu/[email protected]
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Goals
• Deliver enough radiation to a tumor to destroy the tumor.
• Minimize damage to the patient.
Bad News: Radiation must travel through healthy tissue to get to the tumor.
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Radiation Delivery
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Guiding Principles
• Healthy tissue can recover from small doses.– So, hit the tumor from different directions.
• Avoid hitting critical organs.
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A Treatment Plan
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Radiotherapy Planning
• Determine which gantry angles to use
• For each angle used, determine – How much radiation to deliver – How to “shape” the radiation beam
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Shaping the Radiation Beam
Multileaf Collimator
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Outline• Physics of radiation oncology• The geometry of Radiotherapy• Dose deposition operator• A “Simple” linear programming model for radiotherapy
planning• Other issues:
– Dose-volume constraints – Minimum-support plans– Dynamic planning– Uncertainty issues.
• Summary/Conclusions
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Some physics
• High energy photons, through collisions, set fast electrons in motion, which
• Kick atomic electrons off molecules, which
• Lead to chemical reactions, which
• Lead to impaired biological function of DNA, which
• Leads to cell death
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Fluence
number of crossing photons
Fluence = ---------------------------------
surface area crossed
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Fluence vs. dose
Fluence is exponential in depthDose is nearly exponential
Dose
Fluence
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Geometry
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Terminology
• Beam – A cone emanating from the accelerator and enclosing the entire target area. (Corresponds to a single gantry position).
• Pencil – Part of a beam, along which a nearly constant dose is delivered.
• Pixel/Voxel – Smallest subdivision of the target area. Pixel=square, Voxel=cube
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Dose Deposition Operator
• As a pencil of radiation passes through the body, it deposits a certain fraction of its energy in each pixel it passes through.
• The dose deposition operator specifies what fraction of each pencil is deposited in each pixel.
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Dose Deposition Operator
• For pixel i, beam b, pencil p
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Dose Deposition Operator
b)(p,
b) x(p,b)p,D(i, i pixel toDose
where x(p,b) is the intensity of pencil p in beam b.
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Dose Deposition Operator
• The dose deposition operator allows for accurate modeling of the physics.– nonlinearities due to depth of penetration
– scattering
– etc.
• But the resulting optimization model is still linear! (Tractable) – so long as dose is proportional to beam intensity
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Linear Programming Model
0doses all
boundupper pixeleach todose
dose prescribedpixel each tumor todose
structures critical todosemax
subject to
minimize
not linear
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Linear Programming Model
0
,),(),,(
,),(),,(
,),(),,(
subject to
min
BbP,p
BbP,p
BbP,p
,
x
bodyoObpxbpoD
tumortTbpxbptDT
criticalcbpxbpcD
u
ul
x
Standard Trick
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More Simply
Sx
x
),(subject to
min ,
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Other Goals
• Dose Volume Constraint: – No more than x % of a structure can exceed “y”
dose.
• Minimum support plans. – Keep the number of gantry angles small
• Dynamic plans.
• Uncertainty
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Dose Volume Histogram
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Dose Volume Constraint
}1,0{
,
,),(),,(,
c
organc c
BbPp c
y
Ny
organcMyUbpxbpcD
M is a really big numberN is the maximum number of pixels in the organ that can get “fried”.
Integer Constraint=Hard
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Dose Volume Constraint
• The Integer Programming formulation is too hard for general purpose solvers to solve– Requires specialized code. – Don’t try this at home!
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Minimum Support Plans
See: S.C. Billups and J. M. Kennedy, Minimum-Support Support Solutions for Radiotherapy Planning, Annals of Operations Research (to appear).
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•Many beams used•Expensive to administer
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• Few beams.• Clinically, the plan is nearly as good.• Practical to administer.
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Finding Minimum-support Plans
used) beams(# min S x),(
otherwise1
0 if0and
),()(,),(:),,(where
)(min
*
*),,(
zz
bpxbzSxzxT
bz
p
BbTzx
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Integer Programming Formulation
beams1,0)(
beams)(),(
Sx),(subject to
)(min ),,(
bby
bbybpx
by
p
byx
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Exponential Approximation
• Approximate *-norm by exponential function.
b
bze )1(min )(
T z)x,,(
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Exponential Approximation
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Successive Linearization Algorithm
1. Solve LP model
2. Linearize exponential problem around latest solution (generating new LP)
3. Solve new model.
4. Repeat steps 2 and 3 until solution stops changing.
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Penalized Subproblems
If a beam is “barely” turned on in one solution, it will be penalized heavily in the next subproblem.
min ))()(()(min )(
),,(bzbze i
b
bzi
Tzx
i
osolution t theas ),,( Choose 111 iii zx
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Choosing Parameters
• β balances the therapeutic goals against the number of beams used.
• α controls the size of beams that are penalized significantly.– Large α – only weakest beams are penalized
– Small α – all beams penalized to varying degrees.
– If α is large enough, exponential problem has same solution as integer programming problem.
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Quality of Solutions
• Successive linearization algorithm generates a local solution to the exponential problem.
• Compared to integer programming solution, the SLA solution may use slightly more beams,
• but is just as good clinically.
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Uncertainty
• The dose actually delivered differs from the plan:– Modeling approximations– Patients move during treatment!– etc.
• How sensitive are solutions to this uncertainty?
• Can we devise more robust models?
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Dynamic Planning
• Radiation is delivered over 20 days. (Same drill every day).
• Is it possible to measure the effects of the plan and adjust the plan each day?
• See Ferris and Voelker. Neuro-dynamic programming for radiation treatment planning, Numerical Analysis Research Report NA-02/06, Oxford University Computing Laboratory, Oxford University, 2002.
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Summary/Conclusions
• Good IP algorithms exist for doing 3-dimensional planning with difficult constraints.
• Handling uncertainty was the biggest concern at the workshop last February
• Also, growing interest in dynamic planning.
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More Information
• http://www.trinity.edu/aholder/HealthApp/oncology