Radiotherapy Planning

Stephen C. BillupsUniversity of Colorado at Denverhttp://www-math.ucdenver.edu/~billupssbillups@carbon.ucdenver.edu

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.

Radiation Delivery

Guiding Principles

• Healthy tissue can recover from small doses.– So, hit the tumor from different directions.

• Avoid hitting critical organs.

A Treatment Plan

Radiotherapy Planning

• Determine which gantry angles to use

• For each angle used, determine – How much radiation to deliver – How to “shape” the radiation beam

Shaping the Radiation Beam

Multileaf Collimator

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

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

Fluence

number of crossing photons

Fluence = ---------------------------------

surface area crossed

Fluence vs. dose

Fluence is exponential in depthDose is nearly exponential

Dose

Fluence

Geometry

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

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.

Dose Deposition Operator

• For pixel i, beam b, pencil p

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.

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

Linear Programming Model

0doses all

boundupper pixeleach todose

dose prescribedpixel each tumor todose

structures critical todosemax

subject to

minimize

not linear

Linear Programming Model

0

,),(),,(

,),(),,(

,),(),,(

subject to

min

BbP,p

BbP,p

BbP,p

,

x

bodyoObpxbpoD

tumortTbpxbptDT

criticalcbpxbpcD

u

ul

x

Standard Trick

More Simply

Sx

x

),(subject to

min ,

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

Dose Volume Histogram

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

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!

Minimum Support Plans

See: S.C. Billups and J. M. Kennedy, Minimum-Support Support Solutions for Radiotherapy Planning, Annals of Operations Research (to appear).

•Many beams used•Expensive to administer

• Few beams.• Clinically, the plan is nearly as good.• Practical to administer.

Finding Minimum-support Plans

used) beams(# min S x),(

otherwise1

0 if0and

),()(,),(:),,(where

)(min

*

*),,(

zz

bpxbzSxzxT

bz

p

BbTzx

Integer Programming Formulation

beams1,0)(

beams)(),(

Sx),(subject to

)(min ),,(

bby

bbybpx

by

p

byx

Exponential Approximation

• Approximate *-norm by exponential function.

b

bze )1(min )(

T z)x,,(

Exponential Approximation

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.

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

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.

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.

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?

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.

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.

More Information

• http://www.trinity.edu/aholder/HealthApp/oncology